Worm gears are used to transmit power between two nonintersecting
shafts, which are right angles to each other. Crossed helical gears are
also used for applications involving nonparallel, non intersecting
shafts; but they are limited in their load transmission capacity. Worm
gear drives are used for large speed reduction ratio of 100:1 or more in
a single stage. This large amount of speed reduction is not possible
with any other gears in a single stage. They are very compact compared
to other gears. Worm gear drives consists of a worm and a worm gear or
wheel which is a helical gear Fig.5.16.1.
The worm is similar to a screw. The threads of the worm have an
involute helicoid profile. The pair of teeth on meshing worm and worm
gear must have the same hand. The teeth on the worm wheel envelop the
threads on the worm giving either a line or an area of contact between
meshing parts.
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Fig. 5.16.1 Worm and worm gear on loom
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One
of the advantages associated with the use of worm gears is that the
tooth engagement occurs without shock prevalent in other gear types. The
meshing of teeth occurs with a sliding action resulting in very quiet
operation. The sliding friction may produce overheating, which must be
dissipated to the surroundings by lubrication. The power transmission
efficiency of worm gears is lower compared to spur gears, parallel
helical gears, and bevel gears; but higher than that of crossed helical
gears. Worm and worm gears produce thrust load on shaft bearings. The
power transmission capacity is low and limited to 100kW.
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Worm
gears are very compact compared to other gears for the same speed
reduction. Provision can be made for self-locking operation, where the
motion is transmitted only from the worm to the worm wheel. This is
advantageous in lifting devices. The worm wheel in general made from
phosphor-bronze alloy, which is costly. The worm is usually made of
hardened alloy steel. The worm is usually cut on a lath, whereas the
gear is hobbed. All the worm gears must be carefully mounted to ensure
proper operation.
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5.17
TERMINOLOGY OF WORM GEARS
A pair of worm gears is designated by four quantities in the order: number of start on worm (z1), numbers of teeth on worm wheel (z2), diametral quotient of the worm (q) and module in mm (m) as, z1/z2/q/m. A simplified diagram of the worm and worm wheel is shown in Fig.5.17.1. The diametral quotient (q) and module (m) are related as,
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q = d1/m .....................................................................(5.15)
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d2 = mz2 ....................................................................(5.16)
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| Where, d1 and d2 are the pitch circle diameter of the worm and worm wheel respectively. |
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Fig. 5.17.1 Terminology of worm gears
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5.18 CLASSIFICATION OF WORM GEARS
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Worm gears can be classified into:
(a) Single envelope/single start worm gear set; and (b) Double
envelope/double start worm gear set. In the former, a single spiral
starts from one end of worm (left) and finishes at other end (right),
forming the threads. In the later, two spirals with phase difference of
180° start at one end and finishes at other end, forming the threads.
Both the set of threads maintain the phase difference all around. When
the worm gear/wheel having z numbers of teeth is rotated through one revolution, the worm will complete z revolution for single start threads. For double start threads, the number revolutions of the worm will be z/2.
This implies that the speed reduction with single start worm gear set
is twice that of double start worm gear set. When the worm gear is
having 100 teeth, the speed reduction ratios (ratio of output speed and
input speed) are 1/100 and 1/50 respectively for the single start and
double start worm gear sets.
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Single envelop worm gear
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In
a single enveloping set, the width of worm gear is cut into concave
surface, thus partially enclosing the worm in meshing as shown in Fig.5.18.1. They are used in applications requiring a high speed reduction and low load transmission.
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Fig. 5.18.1 Single envelope worm gear set on wrap reel
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Double envelop worm gear
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In double envelope worm gear set, both the width of the helical gear and the length of the worm are cut concavely as shown in Fig.5.18.2.
These results in both the worm and gear partially enclose each other.
The double envelop worm set have more teeth in contact; and area contact
rather than line contact, thus permitting greater load transmission.
The double enveloping gears are difficult to mount compared with single
envelope gears. They are used for higher load transmission compared with
single start gears.
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Fig. 5.18.2 Double envelope worm gear set on ring spinning machine
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5.19 APPLICATIONS OF WORM GEARS
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Worm gears find applications in almost all textile machines. Few applications are listed below:
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Drive between cylinder and flat
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Drive to builder mechanism in ring spinning machine
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Drive to pedal roller of scutcher from top cone pulley to feed roller
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Drive to bottom calender roller of scutcher from lap stop lever.
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Drive to cams
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6.1 GEAR TRAIN
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In
machines, rotary motion is transmitted from one shaft to other. A set
of gears are employed to transmit motion from main shaft of machine to
various revolving elements. A combination of gears employed to transmit
motion from one shaft to other(s) is called ‘Gear train’. (Fig 6.1.1
)
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Fig. 6.1.1 Spur gear train on the head stock of roving machine
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6.2 CLASSIFICATION OF GEAR TRAINS
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Gear trains are classified into the following:
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Simple gear trains.
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Compound gear trains.
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Reverted gear trains.
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Epicyclic (or planetary) gear trains
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6.3 SIMPLE GEAR TRAIN
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| Simple gear trains are shown in
Fig. 6.3.1. Each shaft is mounted with one gear. |
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Fig. 6.3.1 Simple train of gears
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6.4 COMPOUND GEAR TRAIN
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In compound gear trains (Fig.6.4.1),
at least one pair of gears is rigidly mounted on a same shaft, thus
that pair has the same numbers of revolution. They are widely used in
textile machines such as drafting and twisting gearing and head stock
gearing.
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Fig. 6.4.1 Compound train of gears
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| The gear transmission ratio of the compound train shown in figure 5.3 is |
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6.5 REVERTED GEAR TRAIN
In a reverted gear train, the first and the last gears have the same axis of rotation (Fig.6.5.1).
If these two gears are mounted on the same shaft, one of them must be
loosely mounted. They find applications in epicyclic gear trains. They
are also used in clocks and machine tools.
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Fig. 6.5.1 Reverted gear trains
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6.6 EPICYCLIC/ PLANETARY GEAR TRAIN
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Epicyclic
gear train is the one in which the axes of some of the gears have
motion. The said gear(s) would be revolving about external axis or axes.
Whereas in other gear trains, the axes of all the gears do not have
motion, only the gears rotate on their axes. Planetary gear trains are
often employed to make more compact gear reducer (large speed reduction
in a small volume) compared to other gear trains. Multiple kinematic
combinations (multiple inputs) are possible with planetary gear trains.
Since few gears are revolving around, the bearings are subjected to high
loads; requiring constant lubrication. Hence, planetary gears are
placed in box with lubricants, sometimes in a sealed box inaccessible to
maintenance crew. Their design and manufacturing is complex and require
a very high degree of balance.
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An epicyclic gear train with one degree of freedom is shown in Fig.6.6.1.
The sun gear A is grounded. In other words, it is held stationary. The
arm/lever is pivoted on the axis of gear A and on its other end it
carries a planetary gear B. The gear B is meshing with the sun gear A.
As the arm rotates, the planetary gear B revolves around the periphery
of the gear A and also rotates on its axis since it is meshing with the
sun gear A. The gear B is the output gear. Since the sun gear is
grounded, the gear B gets its input only from the rotation of arm. This
is called ‘one degree of freedom’.
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Fig. 6.6.1 Epicyclic gear trains: One degree of freedom
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6.7 VELOCITY RATIO OF EPICYCLIC GEAR TRAIN
The velocity ratio of an epicyclic gear train is determined by the following methods: (a) Tabulation method; (b) Formula method; and (c) Instant centre method or tangential velocity method. The tabular and formula methods are discussed below.
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Tabulation method
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This
method determines motion of every element in the gear train. This
procedure is based on a kinematic inversion, where two easily
describable parts of the total motion are analyzed separately, then
added together:
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| (1) |
Motion of all components rigidly fixed to the rotating arm; |
| (2) |
Motion of all the components relative to the arm. |
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| The superposition of the two components is carried out by the following steps: |
a)
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In the
first step, motion with arm is determined. The gears which are grounded
are disconnected from the ground. All the gears are fixed rigidly to
the rotating arm. The arm is rotated with the rigidly attached gears by a
number of revolutions proportional to the angular velocity of the arm.
If the angular speed of arm is not known, then, rotate the arm by ‘+y’
revolutions (+ve rotation corresponds to counterclockwise direction; and
–ve rotation corresponds to clockwise direction). In doing so, all the
gears will get +y revolutions.
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| b) |
In the second
step, motion of every gear relative to the arm is determined when the
arm is held stationary. In this step, the gears are unlocked from the
arm, and the sun gear is rotated +x revolution (i.e. counterclockwise),
holding the arm stationary. Then, the number of revolutions and signs of
rotations of other elements/gears are noted.
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| c) |
In the third and
final step, the total number of revolution of each element is found by
algebraically adding its numbers of rotations. This is the sum of
revolution from step 1 and step 2. The basic equations for speeds of all
the elements are obtained in this step. Then these equations are solved
by putting the boundary conditions.
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With reference to the Fig 6.6.1, the tabulation of speeds and signs of rotation of all the elements are given in Table 6.7.1.
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Table 6.7.1 Tabulation method to determine speeds of elements of gear train
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Formula method
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| This method is useful for preliminary design of gear train as it is rapid. Referring to Fig.6.6.1
, |
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6.8 EPICYCLIC GEAR TRAIN WITH TWO DEGREES OF FREEDOM
Another
important application of the planetary gear train is to make use of two
degrees of freedom of this mechanism when the sun gear is released from
the ground (Fig.6.8.1). This
two-degrees-of-freedom mechanism requires two input conditions to
determine completely the motion of gear train. The
two-degrees-of-freedom mechanism allows two separate input speeds to be
combined to give an output speed proportional to the sum or difference
of the two inputs. Two degrees of freedom is used in roving and combing
machines and auto-levellers. Using two or more degrees of freedom PIV
(Positively Infinitesimally Variable) drives can be constructed. The
input speeds are preciously controlled by speed-control motors to vary
the output speed infinitesimally and if requires continuously also.
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Click on image to run the animation
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Animation 6.8.1 Inputs and outputs of epicyclic gear train
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Fig. 6.8.1 Epicyclic gear trains with sun gear released from ground: -Two degrees of freedom
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| Required boundary conditions from Fig. 6.8.1 are: |
| If the rpm of sun gear is 120, then, N A = 120; and using the tabulation method ( refer the Table 6.7.1), |
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6.9 EPICYCLIC GEAR TRAINS ON ROVING MACHINE
The
epicyclic gear train is used in roving machine to combine a fixed
rotational speed of flyer and a variable rotational speed of winding of
roving around the bobbin. Both these speeds are combined to get the
rotational speed of bobbin. The rotational speed of winding is reduced
from bare/empty bobbin to full bobbin after completion of each and every
layer of winding. The speed reduction is carried out using a pair of
cone pulleys and belt shift mechanism. This variable speed is given to
the epicyclic gear train via a simple gear train from the bottom cone
pulley.
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Bobbin diameter and speeds of bobbin
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| Bobbin lead winding principle is universally employed to control the bobbin speed on cotton roving machines which is shown in Fig. 6.9.1 . |
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Fig. 6.9.1 Bobbin-lead winding principle on roving machine
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Fig. 6.9.2 Change of bobbin diameter on winding
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As the winding of roving proceeds from the empty bobbin to the full bobbin, the winding rpm (NW) decreases, since the delivery rate of roving (v) is constant ( refer the Eq. 6.21).
As a result, the bobbin rpm must be reduced in proportion to the
changes in the bobbin diameter after winding each layers of roving.
Referring to equation (6.20), the bobbin speed is a function of a
constant flyer speed (NF) and a variable winding speed (NW).
The required inputs of fixed and variable speeds are given to the
epicyclic gear train to process and then the output is transmitted to
the bobbin through a simple gear train.
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| Epicyclic gear train on conventional roving machine |
An epicyclic gear train used on an old roving machine is shown in Fig. 6.9.3.
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Fig.6.9.3 Epicyclic gearing in an old roving machine
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Two compounded bevel gears 3 & 4
of the epicyclic gear train are mounted on the main shaft of roving
machine that constitutes the arm of epicyclic gearing. They revolve
around the main shaft with same speed. The main shaft drives the top
cone pulley and flyers each through a separate gear train. The top cone
pulley drives the bottom cone pulley through a shifting belt.
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Each time, one layer of roving is
wound on to the bobbin; the belt is shifted so that the speed of bottom
cone pulley is reduced. Through a gear train, the variable speed from
bottom cone pulley is supplied to the spur gear ‘1’, which is compounded
to the bevel gear ‘2’. Both the gears ‘1’ & ‘2’ are loosely mounted
on the main shaft of machine and have the same rotational speed around
their axes. The gear ‘4’ meshes with the gear ‘5, which is compounded
with the sprocket 6; both are also loosely mounted on the main shaft and
have the rotational speed. From the sprocket ‘6’, drive to the bobbins
is transmitted through gear a train. Gears, ‘1’, ‘2’, ‘5’ and sprocket
‘6’ are coaxial to the main shaft.
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Speeds of various elements
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Speed of various elements can be calculated using the tabulation method, which is given in
Table 6.9.1.
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Operation
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Speed (rpm) of various elements
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| Arm |
Gears/elements
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1
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2
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3
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4
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5
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6
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| Whole system rotates with +y rpm |
y
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y
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y
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y
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y |
y
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y
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| Arm is fixed. Gear ‘1’ rotates with +x rpm |
0
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x
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x
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-x(z2/z3)
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-x(z2/z3)
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x .e2
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x .e2
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(add column wise)
Step 1 + Step 2 |
y
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y + x |
y + x
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y - x(z2/z3)
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y - x(z2/z3)
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y +x.e2
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y + x.e2
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| Putting the boundary conditions: |
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Epicyclic gear train on a new roving machine
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An epicyclic gear train used on a roving machine is shown in Fig. 6.9.4.
Front drafting roller diameter is 27 mm. The diameters of top cone and
bottom cone pulleys while winding on bare bobbin are: 216 mm and 114 mm
respectively. The diameter of bare bobbin is 48mm. The rpm of flyer and
main shaft are 1000 and 827 respectively.
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Fig. 6.9.4 Gear trains on Lakshmi roving machine
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Speeds of elements of epicyclic gear train
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The speeds of various gears are given in
Table 6.9.2
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Gears/
Operations |
Speeds (rpm) of various gears
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Gear, 50T (Arm)
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Sun gear, 59T
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Gear, 28T
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Gear, 25T
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Gear, 62T
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Arm is locked.
Sun gear (59T) is given x rpm |
0
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x
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-x(59/28)
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-x(59/28)
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x (59/28)(25/62)
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| Whole system revolves at y rpm |
y
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y
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y
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y
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y
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| Total |
y
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x + y
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-2.11x + y
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-2.107x + y
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0.85x + y
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The rpm of main shaft or gear 59T is 827. Therefore,
The rpm of gear 50T is y, i.e,
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Therefore, x = 191
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The rpm of 62T gear is, y+0.85x = 798
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Roving stretch or slackness
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Therefore, the roving stretch
is ~zero. The use of larger gear on the top cone (position E), larger
sized front bottom drafting roller (>27 mm) create slackness on the
roving. The use of bobbin of smaller diameter (<48mm 3.8="" a="" accumulates="" adjusted.="" adjustment="" attempted.="" be="" belt="" bobbin="" bottom="" break.="" by="" can="" compensating-rails="" cone="" finer="" for="" gear="" greater="" guide="" if="" increasing="" is="" leading="" met="" might="" more="" moves="" must="" number="" of="" on="" one="" only="" otherwise="" p="" position:="" requires="" roller="" roving="" rpm.="" rpm="" slackness="" teeth="" tension="" than="" the="" then="" to="" which="" winding="">
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Changing the number of teeth
on the twist changing gears (positions at B, C and D) would change both
the delivery- and winding- rates of roving in proportion. These do not
alter the roving slackness or stretch.
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| Relation between the speeds of bobbin and bottom cone pulley |
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6.10 EPICYCLIC GEAR TRAIN ON COMBING MACHINE
Motion of detaching rollers
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In a
combing machine, when the operation of cylinder comb has been completed,
the detaching rollers bring a part of the fibre fringe formed from the
preceding cycle in the reverse direction. The nippers then swing forward
and lay the newly combed fiber fringe onto this fringe, projecting from
the detaching rollers. When the detaching rollers rotate in the web
take-off direction again, they draw the newly combed fibres through the
top comb and out of the lap sheet. Thus, a new fibre web or fringe is
created.
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The
detaching rollers should perform a back-and-forth movement to carry out
the operation of piecing the fibre fringe. Obviously, the forward motion
of detaching rollers must be greater compared with their backward
motion, for an effective delivery of fibre web. The back-and-forth
movement of the detaching rollers is derived from an epicyclic gear
train.
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An intermittent rotation (I) is superimposed on to a constant rotation (C) generated from the cylinder comb shaft in the epicyclic gear train. The intermittent rotation (I) is also derived from the cylinder comb shaft. The intermittent rotation is faster than the constant rotation (C), i.e.,
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The
superimposed motion from the epicyclic gear train is transmitted to the
detaching rollers through a gear train. If the intermittent and constant
rotations are acting in the same sense, the result is that the
detaching rollers rotates in the forward direction rapidly, i.e., the
rotational speed of the detaching rollers O1 is positive and assumes a greater value.
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If the direction of intermittent rotation is opposite in sense to the constant rotation (I is negative; C
is positive), then the intermittent rotation being dominant cancels out
the effect of the constant rotation and the net effect is that a
backward movement to the detaching rollers, i.e., the rotational speed
of detaching rollers is O2, and is in negative sense.
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If the intermittent motion is ceased to exist, then the rotational speed of detaching rollers is O3, and it is positive.
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The net movement of fibre fringe (s) is a function of all these speeds of the detaching rollers, their time duration, t1, t2 and t3, and detaching roller diameter (dd) as
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Operation of epicyclic gear train
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Figure 6.10.1 shows the epicyclic gear train used on the comber and a sketch of the same is given in Fig. 6.10.2.
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Fig 6.10.1 Epicyclic gear trains on comber
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Fig. 6.10.2 Epicyclic gear train to drive detaching roller on comber
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In the epicyclic gear train shown in Fig. 6.10.2,
the gear F rotates in counterclockwise direction with a constant rpm,
as one standing near the headstock looks towards the delivery side of
the comber. This gear gets its drive from cylinder-comb shaft through
simple gearing. On one side of this gear F (near head stock end), three
identical gears B; are mounted at 120 ° interval, away from the axis of
gear F. Similarly, on the other side of the gear F, three identical
gears C are mounted with angular separation at 120° , away from the axis
of gear F. Gears B and C are compounded having the same rotation. The
purpose of having three gears is to dynamically balance the epicyclic
gear train. Otherwise, the mass imbalance created at the epicyclic
gearing would transmit vibrations to the detaching rollers, and hence,
mass variations on the delivered web. From the point of view of
rotational speed of output gear, the three gears at each position B and C
have the same effect as that of having single gear at these positions.
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Gear A
is mounted coaxial to the gear F, and can rotate either in positive or
negative directions. A complex mechanism comprising a crank mounted on
the gear of cylinder comb shaft, cam, swivel plate and swinging arm
drives the gear A. Gear C drives the gear D that is loosely mounted on
the axis of gear F. Gears A, F, and D are mounted on the same axis, but
rotate at different speeds. Gears B and C revolve around the axis of
gear F, and in addition they rotate with respect to their own axes. The
superimposed speed from the gears F and A is transmitted to the output
gear D of epicyclic gear train which in turn transfer motion to the
front and back detaching rollers through a gear train which is not shown
in figure.
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| Speeds of elements of epicyclic gear train |
The equations governing the rotational speeds of all the elements of
epicyclic gear train are determined using the tabulation method (
Table 6.10.1 ).
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Operation
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Speeds of various elements (rpm)
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A
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B
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C
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D
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F
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| The epicyclic gear train as a whole rotate with +y rpm |
+y
|
+y
|
+y
|
+y
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+y
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| Arm F is locked & Gear A is given +x rpm |
+x
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-x(33/21)
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-x(33/21)
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+x(33/21)(29/25)
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Zero
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| Resultant rpm |
x + y
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y -(33/21)x
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y -(33/21)x
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y +(33/21)(29/25)x
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+y
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Rpm of the gear A = x + y ..................................................................................(6.45)
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Rpm of the gears B & C = -1.57x + y ..................................................................(6.46)
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Rpm of the gear D = 1.82x + y ...........................................................................(6.47)
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Rpm of the gear F = y ........................................................................................(6.48)
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From
the gearing plan of comber, the rpm of gear F can be calculated for
various throughput rates (nips/minute) of comber. For 240 nips/minute of
the comber, it is +36 rpm (i.e. counterclockwise direction). The speed
and direction of rotation of the gear F is always constant throughout
the combing cycle.
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Observations from combing cycle index
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The observations made from the combing cycle at 240 nips per min are:
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The index numbers per cycle is 0 to 40
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The observations from the gearing plan are
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The gear train ratio between the output gear D and the bottom detaching roller is -3.1 (negative).
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Counterclockwise
rotation of the detaching rollers corresponds to forward movement of
detaching rollers and vice versa (viewed from headstock)
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The diameter of detaching rollers is 25mm
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Web delivery per cycle of combing
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7.1 CONE PULLEYS ON ROVING MACHINE
It has been discussed in module
5 that the bobbin speed is a function of bobbin diameter. The reduction
in bobbin speed is effected through cone pulleys (Fig.7.1).
After the completion of winding of each layer of roving, the builder
mechanism shifts the belt on the cone pulleys, in such a way that the
speed of bottom cone pulley is reduced. From the bottom cone pulley,
through gear train, drive is transmitted to the epicycle gear train.
This speed is superimposed on the fixed speed of the main shaft that
corresponds to flyer speed. The output from the epicycle gearing is
transmitted to the bobbins through gears.
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Fig. 7.1.1 Cone pulleys used on a roving machine
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7.2 DESIGN ASPECTS OF CONE PULLEYS
Design aspects of cone pulleys
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Certain assumptions must be made in order to
simplify the approach in designing cone pulleys. The boundary conditions
for the design can be set based on the space consideration of whole
machine and dimensions related to various elements including gear
trains. Based on the space consideration and simplicity of design, the
dimensions of the following can be selected.
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- The maximum and minimum diameter for the cone pulleys.
- Sum of top and bottom cone pulley diameters
for any cone belt position is constant as the belt moving over the
pulleys is fixed in length.
- Length of cone pulleys.
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Based on the dimensions of the following elements, a design criterion must be set.
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- Width of cone belt.
- Minimum and maximum bobbin diameters of empty & full bobbin to be used.
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The consideration of the following points could simplify the design approach further.
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(a)
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Winding
principle employed in winding the roving on to the bobbins is a ‘Bobbin
lead’, which is universally adapted on all commercial cotton machines
for various practical considerations.
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(b)
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The length of the roving delivered by the front drafting rollers per unit time is constant (delivery rate).
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(c)
|
Twist density along the length of roving is constant (i.e. flyer speed is constant).
|
| (d) |
Roving diameter is assumed to be infinitesimal for very accurate design of the cone pulleys. |
(e)
|
Roving is
incompressible. There is no flattening effect of roving due to winding
tension. Roving remains circular even after winding several layers on to
the bobbin. Of course, in practice this does not happen. Ideal and
accurate design approach should consider the degree of flattening of
roving as winding of each layer proceeds. This involves the knowledge of
shape, specific gravity and compression resilience of fibres,
dimensional changes of certain fibres to moisture absorption, blend
compositions, roving tension and twist. However, this is not a serious
issue in designing the cone pulleys, once the roving is assumed to be
infinitesimal. But these factors do affect the belt initial position and
its subsequent shifting.
|
|
The delivery rate of roving is constant with a
well maintained drafting system (i.e., no front roller nip-movement)
and the machine is under minimal vibrations (speed variation of front
roller due to vibrations are negligible). With perfect gearing and
balancing of the flyer mass, the flyer speed is constant. Assumption (d)
is useful for minimizing the errors in developing smooth profiles for
the cone surfaces. In practical situations, the roving is highly
compressible and assumes the shape of an ellipse, with the major axis of
ellipse lying along the circumferential direction; and the minor axis
of ellipse along the radial direction of the bobbin. In addition, as the
roving layers are wound, the radial pressure on the inner layers built
up continuously, hence the inner layers deviate further from their
circular cross-section, the major diameter of the ellipse increases and
the minor diameter decreases till they become incompressible. Further,
the rate at which they are compressed with respect to the number of
layers are not linear, initially the rate of compression is maximum and
may reach zero after winding a certain number of layers. It is very
difficult to simulate the conditions of winding of roving, and measure
their diameter, especially the minor diameter. Considering the
difficulty, the roving diameter may be assumed to be circular and
infinitesimal.
7.3 STEPS TO DESIGN CONE PULLEYS
Steps to design cone pulleys
|
Assume a roving of infinitesimal thickness. Calculate the bobbin diameter (dBm) corresponding to winding of various layers of roving using the Eq. (6.22)
|
Using the Eq. (6.21), (Nw = ν/π dBm), calculate the winding rpm (NW) for winding each layer of roving of infinitesimal diameter. Using the Eq. (6.19), (NW = NB - NF), the bobbin rpm (NB) could be determined. The bottom cone pulley rpm (N2m) for winding the mth layer of roving is determined using the Eq. (6.42) as
|
 |
|
The rpm of top cone pulley is 622. The rpm of top and bottom cone pulleys (assuming v as 20.5 m/min. and dB0 < 48 mm) are plotted in Fig. 7.3.1
|
|
|
Fig. 7.3.1 Rotational speed of cone pulleys in relation to number of layers wound
|
The ratio between top and bottom cone
diameters (or radii) for winding various numbers of layers, assuming
there is no slippage at the interface between the surfaces of cone
pulleys and belt is
|
 |
The values of k m are plotted in
Fig.7.3.2. (Note: Plotting this ratio is not essential, and only the values are needed for design).
|
|
|
Fig. 7.3.2 Ratio of top and bottom cone pulley diameters (km) in relation to the number of layers
|
The radii of cone pulleys, r1m and r2m for various bobbin diameters are related by the expression.
|
r1m + r2m = k0 ............................................................................................................(7.4)
|
Where, k0 is constant by design, as the length of belt moving over the pulleys is fixed. The value of k0 is set at 160 cm.
7.4 HYPERBOLIC CONE PULLEYS :
|
| Combining the Eq. (6.3) and (6.4), the radii of bottom cone pulleys with respect to numbers of layers wound can be found as |
 |
| Substituting the value of r2m in Eq. (7.4), the corresponding values for r1m could be obtained. Mark the r1m and r2m
values (i.e., radii of cones) as the radial distances from the axes of
both cones, by shifting along the cone axes, each shift equal to l/m (l, denotes length of cone). The initial radii of the cones r11 and r21 can be taken as rmax and rmin
respectively. These values definitely correspond to a bobbin of
diameter < 48mm. This means that belt is not placed on the edge of
cones during winding of roving on bare bobbin whose diameter is 48 mm;
in other words, the pulleys are designed to accommodate bare bobbins
whose diameters are less than 48 mm to take care of dimensional
variation in moulding the bobbins. Draw smooth curve tracing the points
to obtain the profiles for hyperbolic cone pulleys as shown in Fig. 7.4.1. |
|
|
Fig. 7.4.1 Profile of top and bottom cone pulleys: Hyperbolic cones; and right- Straight cones
|
|
|
|
|
|
|
|
7.5 STRAIGHT CONE PULLEYS
|
In the case of straight cone pulleys, rmax and rmin are placed apart at a distance equal to the cone length (l) and straight lines connect them ( Fig.7.5.1).
Hyperbolic cone pulleys are difficult to design. Further, the belt is
always moved on the surfaces of varying inclination. During initial
shifting of belt (first few layers of winding), the inclination of cone
surfaces is much sharper; and the belt envelops the larger diameters of
pulleys and hence, may result in lower winding rate. As a result, cone
pulleys today are mostly made straight sided.
|
|
|
Fig. 7.5.1 Profile of top and bottom cone pulleys: Straight cones
|
|
Since the cone length (l) is fixed, Eq. (7.4) also applies to straight cone pulleys. The maximum and minimum radii (rmax and rmin)
are fixed and are the same for both top and bottom pulleys. From these,
the profiles of cone pulleys are determined. The taper angle of the
cone is related to maximum and minimum radii and the length of cone as,
|
tan Υ = (rmax - rmin) / l ......................................................................................(7.6)
|
The value of tanΥ is 0.09123; that is the slope/taper angle is about 5.20.
|
Shifting of belt on straight cone pulleys
|
The radii of top and bottom cone pulleys (r1m and r2m) must vary from their initial values after completion of each layer of winding. The total shift of belt (lm) from its initial position along the axes of pulleys while winding the second layer (of arbitrary thickness) onwards (m = 2, 3, 4…) is related:
|
r1m = r11 - lm tan y ............................................................................................(7.7)
|
r2m = lm tan y ...................................................................................................(7.8)
|
Where, r11 and r21 are the radii of the top and bottom pulleys, while winding the first layer of roving. These are decided as per the values of km obtained from the Eq.(7.9)
|
r11 / r22 = N21 / N1 = km ....................................................................................(7.9)
|
Where, N21
is the rpm of bottom cone pulley while winding the first layer of
roving on the bare bobbin, which could be obtained from Eq. (6.42). Note
that the rotational speed of top cone is fixed for the given gearing
plan.
|
Adding the Eq. (7.7) and (7.8), and substituting k0 = 160, we get
|
r2m + 1m = k0 = 160 ........................................................................................(7.10)
|
Subtracting the Eq. (6.7) from (6.8), we get
|
 |
| If the initial values of r11 and r21 are 113.3 mm and 46.7 mm (i.e. rmax for top cone and rmin for bottom cone), then, lm for winding second layer (of arbitrary thickness) onwards is
|
|
|
The cumulative axial shifts of belt for
both the cases of hyperbolic- and straight-cone pulleys from the extreme
left ends of cones are plotted in Fig.7.5.2.
The shifting of belt is uniform for the former. For the later, the
shifts are longer at the beginning of winding, and then it progressively
reduces at later stages of winding.
|
 |
Fig. 7.5.2 Profiles of shifting of belt along the axes of straight and hyperbolic cone pulleys from extreme left end of cones
|
Belt slippage and corrections for belt position.
|
In case of belt slipping over the pulley (usual case), the Eq.(7.2) must be modified taking into account the amount of slippage. If the belt slip by s % over the cone pulleys, then,
|
 |
The belt must be moved towards the starting
side of the cone pulleys by some distance to get the right ratio of top-
and bottom cone pulley diameters.
|
Considering the belt slipping about 3% (quite normal), the radii of bottom and top pulleys could be set at 106 mm (r11) and 54 mm (r21) respectively while winding the first layer of roving on a bare bobbin diameter of 48 mm. Then,
|
 |
The shifting of belt (that slips by 3%) from the initial position (i.e., r11 and r21 are 106 and 54 mm respectively) is plotted in Fig. 7.5.3.
|
 |
Fig.
7.5.3 Profile of shifting of belt (with 3% slip) on straight cone
pulleys from its starting position corresponding to winding on empty
bobbin of 48 mm (Total shift is 570 mm)
|
Belt shifting mechanism for straight cone pulleys
|
The unequal but progressively reducing in the shift of belt is
achieved by rotating a cam by a fixed degree (that depends on the
thickness of roving) is shown in figures 7.5.4 and 7.5.5.
|
|
|
Fig. 7.5.4 Cam to control belt shifting on cone
|
|
|
Fig. 7.5.5 Belt shifting mechanism for straight cone pulleys
|
|
One
end of a first steel cord is fastened to the cam and wrapped over the
cam surface. The other end passes over a fixed guide, roller of belt
guide and then attached to the frame of the belt guide. A second steel
cord attached to the belt guide passes over a pulley of a stationery
frame and weight is hung on to the free end of cord. The roller of belt
guide is positioned over a compensating rail that is usually placed
parallel to the bottom side of the top cone and the top side of bottom
cone. This means that the slopes (taper) of both the cones and
compensating rail are the same.
|
The
hanging weight always tries to shift the belt guide towards the right
side of pulleys (towards the minimum diameter of top cone pulley), but
resisted by the cam when it is not in motion. After completion of each
layer of winding, a ratchet is rotated by half a tooth. From this,
motion is transmitted through gear train to the cam. The angular drift
of cam can be changed by the numbers of teeth selected for the traverse
change gear of gear train that depends on the coarseness of roving. When
the cam turns, the roller of belt guide, rolls over the compensating
rail towards right due to the force exerted by the hanging weight, thus,
shifting the belt.
|
If the belt shifts by a distance smduring winding of mth layer (which is lm –lm-1), the distance moved by the roller of belt guide over the compensating rail is
|
|
|
During
first layer of winding, the diameter of the cam is at its maximum, and
decreases progressively towards the end. If the working length of cone
is 570 mm, the cam can be designed to have 570 radial lengths (R1, R 2, R3 Rm) corresponding to 5700 rotation in order to have fine adjustment of belt shift. The path length of cord between the guide and cam (sm'') progressively increases as the winding proceeds from the first to last layer (Fig. 7.5.6).
|
 |
Fig. 7.5.6 Belt positions, radial position of cam and path length of cords
|
|
|
|
Further, the circumferential length of cam from R1 to R2 and R2 to R3… Rm-1 to Rm denoted as sm'''
progressively decreases and hence, the shifting of belt reduces from
second to last layer of winding, evident from the equation.
|
 |
Finer adjustment on belt shifting
|
In spite of precise design
and manufacture of cam, belt guide and compensating rails and their
controlled operations in moving the belt over the cone pulleys, the
roving might encounter slackness or stretch. Changes in humidity, roving
twist, and fibre properties such as bulkiness, compressibility,
micronair/denier and shape of fibres, affect the effective thickness of
roving. Accordingly, the effective bobbin diameter varies, warrants
correction on the bobbin speed by changing the incremental movement of
belt.
|
The compensating rail comprises of three parts that corresponds to
first, second and third quarters of winding, each of its inclination can
be changed independently by screws and pivot arrangement. If the roving
experiences slackness during first quarter of winding, it implies that
bobbin speed is less, which requires slower rates of belt shifting. Then
the taper angle (Υ) of the first part of compensating rail (nearest to
the cam) must be increased. This reduces the amount of belt shift for
each angular drift of cam. When a very high roving tension is
encountered during winding, the taper angle must be reduced. Similarly,
the remaining parts of winding can be controlled with the help of second
and third parts of compensating rail. This requires fair degree
experience and skill on the part of operator.
|
.
7.6 CONE PULLEYS FOR PIANO-FEED REGULATION :
Piano feed regulation controls the thickness
of lap delivered by the scutcher. Several pedals below the feed roller
of scutcher moves up and down independently, depending on the localized
variation in the thickness of material. Through links and levers, these
movements are mechanically integrated. The integrated output is used to
move the belt on cone pulleys, to vary the rotational speed of top cone
pulley (output cone pulley). The output from the top cone pulley is
transmitted to the feed roller, through gears, thus, adjusting its speed
corresponding to average thickness of material sensed at the interface
of pedals and feed roller. When the average thickness of material is
high, the speed of the feed roller must be reduced and vice versa.
|
The following assumptions are useful in the design of the cone pulleys :
|
- The density of web is constant, i.e. mass flow rate of web (g/min.) is proportional to the thickness of web/lap.
- The sum of the top and bottom cone pulley diameters for any cone belt position shall be constant.
- Specification for cone dimensions, viz., maximum and minimum diameters, and length.
- The rotational speed of driving cone (bottom cone) is constant for a given production rate.
|
The mass of a unit length of a web/lap (m) is
|
m = ραt .....................................................................................(7.18)
|
Where ρα is the areal density of web in g/cm2, which is constant; and t is the web thickness in cm. Therefore, the mass flow rate of web (M) in g/min is
|
M = πdf nf ραt .............................................................................(7.19)
|
Where df is the diameter of feed roller in cm; and nf is the rpm of feed roller.
|
When the gear transmission ratio from top cone to feed roller is ‘ e ’; and the rpm of top cone pulley is n2, then,
|
M = πd fe n2 ραt ..........................................................................(7.20)
|
Since, the parameters M, ρα, π , df, and e are constant, under practical condition, then, n2 is proportional to (1/t). Then,
|
1/t = C1 n2 .................................................................................(7.21)
|
Where, C1 is a constant equal to π df e ρα / M.
|
Also,
|
n2 = (d1n1)/d2 .............................................................................(7.22)
|
| Where |
d1 = Bottom cone pulley diameter in cm |
|
d2 = Top cone pulley diameter in cm |
|
n1 = bottom cone pulley rpm |
|
Combining the Eq. (6.21) & (6.22), we get
|
1/t = C1 n1 (d1/d1) ......................................................................(7.23)
|
Since n1, the rpm of driving cone pulley is kept constant. Therefore,
|
1/t = C(d1/d1) ............................................................................(7.24)
|
Where, C is a constant equal to C1 n1.
|
By assuming various values for the thickness of web passing between feed roller and the pedals, the ratio d1 /d1 could be found. From this ratio, d1 and d1 values can be computed considering that d1 + d1 = X, a constant. This is given in the Table 7.6.1.
|
Table 7.6.1: The diameters of cone pulleys in piano-feed regulation
|
Web thickness, t (cm)
|
(1/t) = (C.d1/d2)
|
d1+d2 |
d1 |
d2 |
0.1
|
1/0.1
|
X
|
calculate
|
calculate
|
0.2
|
1/0.2
|
X
|
|
|
0.3
|
1/0.3
|
X
|
|
|
0.4
|
1.0.4
|
X
|
|
|
0.5
|
1/0.5
|
X
|
|
|
and so on
|
|
X
|
|
|
|
|
Construct
the cones similar to the procedure followed in the design of cone
pulleys for roving machine. In practice, the web thickness variation
should be considered as infinitesimal to get smooth profile for the cone
surfaces. The resulting profiles of the cones are hyperbolic. Straight
cone pulleys can also be designed with belt shifting mechanism using a
cam and other elements as discussed earlier.
|
When processing different fibres (synthetics and cotton), the parameter, ραmust be considered in the equations (must be kept as a variable; not a part of constant C1), as the fibre specific gravity affects this value.
|
| 8.0 : DESIGN OF TRANMISSION SHAFTS AND DRAFTING ROLLERS |
8.1 INTRODUCTION
|
|
Transmission shafts are
rotating elements and are mostly circular in cross-section. Shafts are
classified as straight, cranked, stepped and flexible. They could be
either solid or hollow. Shafts are supported by bearings for smooth
running. Shafts support transmission elements like gears, pulleys and
sprockets to transmit power from one rotating member to another. The
portion of shaft that carries pulley or gear is cut as slot (keyway) on
which a key is placed. The key of rectangular cross section partially
sinks in the slot and projects from the shaft. The projected part of key
lies in the slot cut on the inside hub of the gear and holds the gear
securely.
|
Transmission shafts may
be subjected to tensile, bending or torsional shear stresses or
combinations of these. They are subjected to torque due to power
transmission and bending due to reactions on the members that are
supported by them. While designing a transmission shaft for a correct
diameter, knowledge on the type of stresses involved in its application,
interaction of these stresses and the material properties of shaft must
be known. Further, the material of shafts must have (a) high strength
(b) low notch sensitivity, (c) ability to be heat treated and case
hardened to increase wear resistance of journals and (d) good machine
ability. Shafts are made from ductile materials like mild steel, carbon
steels or alloy steels such as nickel, nickel-chromium or
chromium-vanadium steels or ductile cast iron.
|
Drafting rollers of
different surface contours used to attenuate/draft the fibre assemblies
and also to transmit power to other drafting rollers. Spindle is a short
rotating shaft. Crank shafts are used in loom to carry out beat-up
operation. Shafts used for clutching operations are splined.
|
|
.
8.2 MATERIAL PROPERTIES
|
| Engineering materials are
broadly classified into ductile and brittle materials. The stress-strain
diagrams of ductile and brittle materials are shown in Fig. 8.2.1. |
 |
Fig. 8.2.1 Stress-strain diagrams of engineering materials under tensile load (a) ductile material (b) brittle material
|
From O
to P, the strain is linearly proportional to stress. This region is
called ‘elastic region’. The Hook’s law is applicable for this region.
After point P, the stress-strain relationship deviates from the linear
relationship, and the material exhibit more strain for a given stress.
Point E in the curve is called ‘elastic limit of the material’. When a
ductile material is subjected to a tensile stress corresponding to its
elastic limit, and then the load is removed, the material comes back to
its original length without any permanent deformation left in the
material. Point Y is called ‘yield point’. At yield point, material
yields i.e. it undergoes considerable strain without any increase in
stress. Brittle materials do not exhibit a characteristic yield point.
Point U refers to ultimate tensile strength (UTS) of the material. This
is the maximum stress that a material (both ductile and brittle) can
undergo without fracture.
|
A
ductile material has about 5% or more tensile strain before fracture
takes place. A brittle material has a tensile breaking strain about 5%
or less. Structural steels and aluminum are ductile materials, while
cast iron is a brittle material.
|
A shaft may fail, if it is unable to perform its function satisfactorily. The failure of a shaft may occur due to:
|
- Elastic deflection.
- General yielding
- Fracture.
|
For
transmission shafts (including drafting rollers) supporting gears, the
maximum force acting on the shaft, without affecting its performance, is
limited by the permissible elastic deflection of shaft. Lateral or
torsional rigidity may also be considered as the criterion of design in
such cases. For drafting rollers, this permissible elastic deflection,
especially the lateral defection should be much lower compared to
transmission shafts, as eccentricity due to the deflection of the roller
would result in roller nip movement, and consequently irregular
drafting of fibres. Elastic deflection may result in unstable conditions
like vibrations of bearings. The design of shaft is based on the
permissible lateral or torsional deflection.
|
The
stresses induced on the shaft should not be significant to the extent of
general yielding or fracture. In other words, shafts must be subjected
to stresses below the yield point. Therefore, the ultimate tensile
strength is not important. The modulus of elasticity (E) and modulus of rigidity (G)
are the important properties of ductile materials used for making
shafts. The dimensions of the shaft are determined by the
load-deflection equations.
|
|
| | |
|
| | | | | |
|
|
8.3 FACTOR OF SAFTEY AND ALLOWABLE STRESS
|
| In designing a shaft, it is
essential to guarantee sufficient reserve strength left on it in case of
an accident. It must be assumed that the shaft would be subjected to
extremely high load under unforeseeable situations while it is
performing. Taking a suitable factor of safety (fs) can ensure this. The factor of safety is defined as |
|
|
The allowable
stress is the one, which is used in design calculations to determine
the dimensions of shaft. It is considered as a stress, which the
designer expects will not be exceeded under normal operating conditions.
For ductile materials, the failure stress is limited to the yield
stress or yield strength (Syt); and the allowable stress (σ) is
|
|
|
For brittle materials, the failure stress is limited to the ultimate tensile stress (Sut); and the allowable stress is
|
|
|
|
8.4. STRESS-STRAIN RELATIONSHIPS OF MATERIALS
|
Tensile stresses
|
| When a component of a material of length (l in mm) is subjected to a static tensile load (F in N) over a cross-sectional area (A in mm2) results in extension (δ, in mm), the tensile stress (δt ) and strain are given by |
|
|
Shear stresses
|
Riveted plates are shown in Fig.8.4.1. When they are subjected to two equal but opposing external forces (F)
which are not collinear, the adjacent planes may slide with respect to
each other. The stresses on these planes are called direct shear
stresses.
|
 |
Fig. 8.4.1 Shear stresses in a riveted joint: (a) Riveted joint (b) Shear deformation (c) Shear stresses
|
| The average shear stress on the rivet having cross-sectional area (A) is given by |
|
|
A rectangular element of a component subjected to shear force is shown in Fig. 8.4.2.
|
 |
Fig. 8.4.2 Rectangular element subjected to pure shear force: (a) Pure shear stress; (b) Shear strain
|
|
|
Bending stresses
|
A circular shaft which is subjected to a bending moment (Mb) is shown in Fig. 8.4.3. The shaft is subjected to tensile stresses below and compressive stresses above its neutral axis.
|
 |
Fig.
8.4.3 Shaft subjected to bending moment: (a) distribution of bending
stresses at the plane X-X; (b) Section of the shaft at section X-X
|
The bending stress at a distance y from the neutral axis or shaft axis is given by
|
 |
Torsional stresses
|
A transmission shaft, subjected to an external torque, is shown in Fig.8.4.4.
The torque induces internal stresses in the shaft which resist the
action of twist. The internal stresses are called torsional shear
stresses.
|
 |
Fig. 8.4.4 Stresses on a shaft due to torsional moment: (a) shaft is twisted (b) distribution of torsional shear stresses
|
| The torsional shear stress is give by |
|
|
|
8.5 DESIGN OF TRANSMISSION SHAFT
|
|
Transmission shafts can be designed by various approaches :
- Design against static load
- Design for torsional rigidity
- Design for lateral rigidity
-
8.6 DESIGN OF SHAFT AGAINST STATIC LOAD
|
|
| Most of the transmission
shafts supporting gears, pulleys, sprockets and flywheels are subjected
to a combined bending and torsional moments. The shaft materials are
ductile and the maximum shear-stress theory of failure is used to
determine the shaft diameter |
8.7 MAXIMUM SHEAR STRESS THEORY OF FAILURE IN DESIGN OF SHAFTS
|
The maximum shear stress in the shaft can be found by constructing a Mohr’s circle as given in Fig. 8.7.1. The bending stress (σ) at a point on the shaft is a normal stress represented in X direction. The shear stress (σ) is in XY plane.
|
|
|
Fig. 8.7.1 Mohr’s circle for an element of shaft
|
|
|
|
-
8.8
DESIGN OF SHAFT USING A.S.M.E CODE
|
A.S.M.E. code (American Society of Mechanical Engineers) is
one of the approaches followed in design of transmission shaft.
According to this code, the permissible shear stress for shaft without
keyways is taken as 30% of the yield strength in tension (Syt), or 18% of the ultimate tensile strength of material (Sut), whichever is lower. Therefore, the permissible shear stress (σd) is
|
|
|
If shafts have keyways ( shown in Fig. 8.8.1), these values have to be reduced by 25%.
|
|
|
Fig. 8.8.1 Shaft, key and pulley assembly
|
|
The Eq.(8.22) does not
consider the effect of fatigue and shock loads. To account for these,
A.S.M.E code incorporates multiplication factors kb and kt for bending and torsional moments respectively. So the Eq. (8.22) is modified as
|
|
|
| Where |
kb = combined shock and fatigue factor applied to bending moment |
|
kt = combined shock and fatigue factor applied to torsional moment |
|
| The values of kb and kt for rotating shafts are given in the Table 8.8.1. |
Table 8.8.1 Multiplication factors for bending and torsional moments
|
Load type
|
kb
|
kt
|
Gradually applied
|
1.5
|
1.0
|
Suddenly applied
Minor shock
|
1.5-2.0
|
1.0-1.5
|
Heavy shock
|
2.0-3.0
|
1.5-3.0
|
|
|
A
transmission shaft designed for heavy shock load would have larger
diameter followed by shafts designed for minor shock load and then
gradually applied load. As heavy shocks are not involved in the case of
drafting rollers, the load can be considered as ‘gradually applied’.
|
The following example illustrates the design of shaft: A main
shaft of machine receives power from an electric motor (not shown in
figure) through flat belt ( Fig. 8.8.2).
|
|
|
Fig. 8.8.2 Main shaft carrying a pulley and gear supported by two bearings
|
|
The rpm of the motor is 1435. The diameters of the motor and
machine pulleys are 180 and 430 mm respectively. Motor is placed below
the machine shaft such that the axes of both pulleys are in a vertical
plane. The main-shaft transmits power through spur gear (in the plane D)
to a drafting system through gear trains (not shown in figure). The
driven spur gear is placed above the gear D such that the axes of shaft,
Gear D and driven gear are in the same vertical plane. The power
transmitted by the motor to the main shaft is 15kW. The pitch circle
diameter and pressure angle of the gear are 300 mm and 20 °
respectively. The ratio of the tight- and slack- tensions on the belt is
3. Two bearings A and B support the shaft. The properties of material
of shaft are: Sut = 700 N/mm 2 and Syt = 460 N/mm2 and G =79300 N/mm2. Determine the shaft diameter using A.S.M.E code. The pulley and gear are mounted on shaft using keyways.
|
Solution:
|
|
|
| The net vertical downward force acting on the shaft in the plane of the pulley is |
| (Tt+Ts) = 2220.77N |
| Also, |
Mt = Tangential force acting on the gear * radius of pitch circle of gear
|
238732.41 = Pt * 150
|
Pt =1591.55 N (acts horizonally in the plane of the gear)
|
Radial force acting on the gear is Pr = Pt * tan 200 = 579.28 N (acts vertically downwards in the plane of gear).
|
The reactions at the bearings are: P and Q in the vertical plane and R and S in the horizontal plane containing the shaft ( Fig. 8.8.2). Taking moments in the vertical plane about A,
|
(2200.77 * 900) + (579.28 * 300) = (Q * 600), we get Q = 3620.8 N.
|
Taking moments in the vertical plane about B,
|
(2200.77 * 300) = (P * 300) + (579.28 * 300), we get , P = 820.75 N
|
Similarly, the values and direction of R and S
in the horizontal plane could be found out. The bending moment diagram
is constructed from left to right, at various planes considering the
forces that are acting on the plane and those on the left side of the
plane. Counterclockwise- and clockwise moments are assigned positive and
negative signs respectively.
|
The bending moments in the vertical plane
|
|
|
The bending moment diagrams are shown in Fig 8.8.3.
|
 |
Fig. 8.8.3 Forces and bending moments at different planes of the shaft: Left- vertical plane; Right- horizontal plane
|
| From the bending moment diagrams, the maximum
bending moment is observed at the plane of bearing B. The resultant
bending moment at the plane B is |
BMR at B = (BMH2) + BMY2)1/2 ...............................................................................................(8.30)
|
| Where, BMH = Bending moment at B in the horizontal plane. BMV = Bending moment at B in the vertical plane. |
Therefore, BMR at B = (6662312 + 02)1/2 = 666231 N-mm
|
|
|
|
8.9 DESIGN OF SHAFT FOR TORSIONAL RIGIDITY
|
Machine tool spindles and
some line shafts are designed on the basis of torsional rigidity
considerations. The total angle of twist θ in degrees is given by the Eq. (8.20).
The permissible angle of twist or limiting value of twist for line
shaft and machine tool (or spindle) applications are 3º and 0.25° per m
length of shaft respectively.
|
Let us assume that the modulus of rigidity (G) of the shaft material is 79300 N/mm2, and the permissible angle of twist (θ) is 3° /m length of shaft. The distance between the bearings (l)
is 600mm. This is the span length of shaft on which the maximum bending
takes place and as a result, the maximum deflection also occurs.
|
|
|
8.10 DESIGN OF SHAFTS FOR LATERAL RIGIDITY
|
For some
applications, the shafts have to be designed on the basis of lateral
rigidity or the deflection of shafts. A rigid shaft does not deflect or
bend too much due to bending moments. These shafts should be designed on
the basis of permissible lateral deflection. When a shaft supporting a
gear is deflected, the meshing of gear teeth would not be proper. In
addition, the misalignment between the bearing and journal results in
early wear at the gear and bearing surfaces.
|
For a transmission shaft with span length, L (distance between the two adjacent bearings), the maximum deflection (δ) is in the range
|
δ = (0.001) L to (0.003) L.
|
The maximum permissible radial deflection (δ)
at any gear is limited to 1 mm. In the case of drafting rollers, the
eccentricity of rollers and the maximum permissible lateral deflection
is 0.05 to 0.075 mm. Values above this range would result in
considerable periodic irregularity to the drafted fibre assemblies. The
rigidity of a transmission shaft or drafting roller can be increased by
the following methods.
|
- Reducing the span length of shaft by increasing number of supports provided to it
- Reducing the number of joints on shaft
|
The maximum
span length of drafting roller (in between bearing support) used in a
ring spinning machine is around 60 mm. Drafting roller segments of 60 mm
are joined together to cover the full width of ring spinning machine
having spindles in the range 1000 to 1400 or more.
|
The lateral deflection of a shaft or roller depends on the
dimensions of shaft (span length and diameter), forces acting on the
shaft, and the modulus of rigidity of the material of shaft. The modulus
of rigidity (G) is practically same for all types of steel viz., plain carbon steel and alloy steel.
|
The important methods for determining the lateral deflection are as follows:
|
- Castigliano’s theorem for complex structures using strain-energy principle.
- Graphical integration method
- Area moment method
- Double integration method.
|
| Lateral deflection can also be
calculated by simple formulae from strength of materials. The cases
pertaining to (a) simply supported shaft subjected to central load; (b)
simply supported shaft subjected to intermediate load; and (c) simply
supported shaft subjected to uniform load are given below : |
Simply supported shaft subjected to central load
|
A simply supported shaft of length L and diameter d, supported at its
extreme ends by two bearings, is subjected to a central load P. For
this the bending moment diagram is shown in Fig. 8.10.1.
|
 |
Fig. 8.10.1 Bending moments and deflection of simply supported shaft with central load
|
The bending moments (Mb) and deflections (δ) are as follows:
|
 |
Simply supported shaft subjected to intermediate load
|
The bending moment diagram for a simply supported shaft subjected to an intermediate load is shown in Fig. 8.10.2.
|
 |
Fig. 8.10.2 Bending moments and deflection of simply supported shaft with intermediate load
|
The bending moments (Mb) and deflections (δ) are as follows:
|
Bending moments
|
|
|
Simply supported shaft subjected to uniform load
|
A simply supported shaft supported at the ends by bearings is
subjected to a uniform load (w); and the bending moment diagram is shown
in Fig. 8.10.3.
|
 |
Fig. 8.10.3 Bending moments and deflection of simply supported shaft with uniformly distributed load
|
The bending moments and deflections are as follows :
|
Bending Moments
|
|
|
Where, w = applied load per unit length of shaft.
|
|
|
|
|
8.11 DESIGN OF BOTTOM-DRAFTING ROLLERS
|
Design perspective
|
The lateral deflection of drafting roller introduces eccentricity on
rotation. The lateral deflection is an important criterion in design of
drafting rollers. Design against static load and torsional rigidity
(particularly for long rollers) can also be considered.
|
Processing perspective
|
The process ability of fibres
during drafting depends to a larger extent on the roller- lapping
tendency of fibres being processed. The following discussion will
highlight the importance of using large diameter-drafting roller to
control roller-lapping tendency of fibres, vibration forces on the
bearings supporting the drafting roller, and the roller-nip movement.
|
Fibres forced to
bend over an element of small radius of curvature (smaller-diameter
drafting roller) are subjected to more bending and compressive forces,
and make large frictional contact with that element. As a result, they
develop more frictional contacts with the drafting roller, which
increase the tendency of roller lapping. The roller lapping would be
severe with long, fine and less rigid fibres. Therefore, from the point
of view of minimizing the roller-lapping tendency, the roller diameter
must be kept larger, subject to limitation imposed by the distance
between two consecutive drafting rollers (roller settings in front and
back zones).
|
In the latest
generation drawing machine (single delivery drawing machine), the
throughput speed has gone beyond 600 m/min from the earlier speed of 250
m/min. At this high speed, the generation of negative air pressure
around the surface of drafting roller would be high, that increases the
roller lapping tendency. Obviously, high drafting speed with
smaller-diameter drafting roller would further increase the
roller-lapping tendency of fibres. In addition, the loading on the
rollers has been increased for better drafting. These lead to more
bending stresses on the roller and compressive stresses on the fibres.
This necessitated the use of larger-diameter rollers. In the present day
drawing machine, the diameter of the front drafting roller is about 52
mm compared with earlier ones of 35 mm. The rotational speed of front
drafting roller has also gone up to 3060 rpm or more from 2275 rpm.
|
If the diameter of
the front drafting roller of the latest drawing machine were kept at 35
mm, then the rpm of the roller would be 5460 rpm with a 2.4 fold
increase from the level of 2275 rpm to obtain high production rate. For
smooth running of drafting roller, the vibration forces on the bearings
should be less. Lack of roundness of rollers and the eccentricity due to
fixing of the rollers in the bearings would also result in vibrations
on the bearings.
|
When drafting
rollers are mounted on rolling contact bearings, there would be some
eccentricity (e) between the geometric centre of rotation of roller
(bearing axis) and the roller axis due to clearance. As a result, the
drafting roller would be out of balance to a certain degree. The effect
of this unbalance in terms of vibration forces acting on the bearings
would depend on the diameter and rotational speed of drafting roller;
the effect of rotational speed is predominant than the diameter. The
following illustration explains this.
|
The cross-sectional area of
excess material of roller (A), on one side of roller due to eccentricity
in mounting the roller inside the bearings is
|
 |
The bearings
would be subjected to this amount of force for every rotation of roller.
This force of vibration on bearings can be calculated for a front
roller delivery speed 500 m/minute with the roller diameters 35 mm and
52 mm using the above equation. When these rollers are mounted in
rolling contact bearings with an eccentricity of 0.003 cm, they are
subjected to a vibration force of 25.6 N. But the frequencies of the
vibrations are: 76 Hz and 51 Hz for the rollers of diameters 35 mm and
52 mm respectively. It is clear that for the smooth running of drafting
rollers, it is preferable to have large diameter. This is especially
important for the front drafting roller which runs at higher rotational
speed compared with middle and back rollers. Otherwise, bearing life
would reduce considerably. In addition, the centrifugal force of
unbalance of masses of drafting roller would create certain amount of
irregularity in drafting the fibres due to roller-nip movement.
|
For controlling the roller-nip movement, the ratio of eccentricity to roller diameter (e/d) must be kept low. For a given eccentricity, e/d
is low for a large-diameter roller compared with a small-diameter
roller, as the eccentricity is only influenced by the inaccuracies in
mounting the roller in the bearings. In addition, the radial run out of
the drafting roller is also very important in controlling the drafting
irregularity.
|
The bottom drafting rollers
are made of steel. To improve their ability to carry the fibres along,
they are formed with flutes ( Fig. 8.11.1) of the following types.
|
- Axial flutes
- Spiral flutes
- Knurled flutes
|
 |
Fig. 8.11.1 Spiral flutes on front- and back drafting rollers
|
Spiral fluting gives quieter running and more even clamping of
fibres compared to axial fluting. In addition, rolling of top rollers on
spiral flutes takes place in a more even manner and with fewer jerks.
Knurled fluting is used on middle rollers receiving aprons, to improve
the transfer of drive to aprons. The diameters of bottom drafting
rollers normally lie in the range of 25 to 50 mm. In long machines (e.g.
ring spinning machines) the bottom rollers are made by screwing
together short segments of roller.
8.12 DESIGN OF BOTTOM-DRAFTING ROLLER AGAINST TORSIONAL RIGIDITY
|
The following example illustrates the calculation of diameter of
drafting roller (for a front bottom-drafting roller) on a single
delivery drawing machine with a throughput speed 500 m/min. The example
is hypothetical only with assumption of power, distance between the
bearings that support the roller, since the exact details are not
available. The motor transmits power to the front drafting roller which
transmits to other drafting rollers. The power transmitted from motor to
front drafting roller is 4 kW. The front drafting roller would be
rotating at a speed of 3060 rpm, if its diameter were 52 mm.
|
The torque on the front drafting roller is
|
 |
With the following assumptions :
|
- The modulus of rigidity of material (steel) of bottom roller (G) is 79300 N/mm2
- Span length of drafting roller l, between the two bearings is 500 mm.
- If the allowable angle of twist (θ ) per m length of drafting roller is 10.
|
Then, the allowable angle of twist in degrees = 1 * (500/1000) = 0.50
|
From Eq. (7.20), and d4 = (584 * 12840 * 500) / (79300 * 0.5) and d = 17.41 mm for 10 of twist per m length of roller.
|
Similarly, the diameters of the front bottom roller would be 20.71mm for 0.50 of twist per m length of roller and 24.62 mm for 0.250 of twist per m length of roller.
|
8.13 DESIGN OF BOTTOM-DRAFTING ROLLER AGAINST LATERAL RIGIDITY
|
| Distributed load on bottom drafting roller : |
The load is applied on the top-drafting roller which is distributed
along the nip of bottom-drafting roller over a length that is equal to
the width of fibre spread during drafting the slivers ( Fig. 8.12.1 ). The width of fibre spread is about 20-25 cm. This is for a single delivery drawing machine.
|
 |
Fig. 8.12.1 Schematic diagram of front drafting rollers with supports and fibre web
|
Let us assume that the total load acting on the bottom drafting roller (P) is 60 kgf or 588 N. Referring to Fig. 8.12.2, the distributed load per unit length of bottom roller (assuming the slivers spread is 200-mm) is
|
w = P / 200..................................................................................................... (8.35)
|
 |
Fig. 8.12.2 Distributed load on bottom drafting roller
|
| The maximum deflection of bottom drafting roller in this case is given by : |
δmax = w(5L4 - 24L2a2 + 16a4)/ (384EI)............................................................................. (8.36)
|
The value of w is = 2.94 N/mm. (i.e. 588 N is distributed over 200 mm at middle of the roller. The value of ‘a’
= (500 –200)/2 = 150 mm. From this, the diameters of roller at 60-kgf
load are: 40.9 mm for 0.05 mm deflection; and 48.6 mm for 0.025 mm
deflection. The diameters of roller at 80-kgf load are: 44 mm for 0.05
mm deflection; and 52.3 mm for 0.025 mm deflection.
|
.
| 9.0 :
CLUTCHES |
| 9.1 INTRODUCTION |
Clutches and brakes are used in machines for
effective control and transmission of torque, speed and power. Clutch
transfers torque from an input shaft to an output shaft, whereas a brake
is used to stop and hold a load. A clutch may be used in emergency
situation to disconnect the main shaft from the motor in the event of a
machine jam. In such cases a brake will also be fitted to bring the
machine to a rapid stop. Under normal circumstances, the brake is
disengaged and the clutch is engaged. During emergency situation, power
fails and brakes are engaged. Clutches and brakes can be classified in a
number of ways, by the technique used to engage or stop the load or
torque transfer (mechanical lockup, friction, and electromagnetic) and
by the method used to actuate them (mechanical, pneumatic, hydraulic,
electric and self-activating).
|
Mechanical actuation is the simplest and
cheapest way to engage a clutch/brake. The actuation is by rods, cables,
levers or cams. But the actuation force is limited to about 300N. This
low clamping force also limits response times and cycling rates. Air
actuation (pneumatic) is the most common method used in industrial
machines. Air pressure up to 1.4 N-mm-2 is used. They can
operate at about one Hz; and generate less heat in the actuator.
Hydraulic actuation depends on oil pressure as high as 3.5 N-mm-2
and is faster; provides smooth engagement and is costly. In both
pneumatic and hydraulic actuation, fluid pressure is delivered to a
piston that acts against a rod, lever, or cam to engage or disengage the
clutch or brake.
|
Electric actuated clutches and brakes operate
at extremely fast about 25 Hz. But the actuating force is much less
compared with pneumatic and hydraulic ones. Self-actuating clutches rely
on centrifugal forces to generate actuating force. They are mostly used
on motors, where the motor speed is an adequate clutch control
parameter.
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|
|
|
|
9.2
CLUTCHES
|
Clutch is an important part of automobiles.
Clutch is also used widely in heavy industrial machines. Clutch is a
mechanical device, which is used to connect or disconnect the source of
power or motion either manually or automatically. An automotive clutch
when disconnected (neutral) can permit the engine to run without moving a
car. This is desirable when the engine is to be started or stopped or
when the gears have to be shifted. Similarly when a motor of a heavy
textile machine is to be started, the drive to the machine should be
disconnected till the motor attains the full or safe-speed. This
safeguards the motor; otherwise the motor draws more current to move the
static elements of machine, in doing so the coils of motor may burn
out. Based on the type of contact between the elements of clutch,
clutches are classified as
|
(1) Mechanical lockup or positive contact clutches
|
(2) Friction clutches
|
The most popular type of clutch or brake uses
the friction developed between two matting surfaces to engage or stop a
load. Disk, drum and cone clutches/brakes are of this type.
9.3 MECHANICAL LOCKUP CLUTCHES
Mechanical lockup clutches are classified as
square jaw, spiral jaw, multi-tooth, sprag and wrap spring clutches. The
square jaw clutch consists of square teeth that lock into mating
recesses in facing member. It provides positive lockup, but because it
cannot slip, the operating speed is under 10 rpm. Spiral jaw have
sloping tooth that permits smooth engagement up to 150 rpm. Actuating
forces in these clutches are mostly mechanical. Multi-tooth or saw
toothed clutches can engage up to 300 rpm and have the advantage of
using electric, pneumatic or hydraulic actuation. In all these clutches,
the torque generating force is mechanical. Both the spiral jaw and
toothed clutch are used on looms, warping and spinning machines.
|
The advantage of mechanical clutches is
positive engagement and, once coupled, can transmit large toque with no
slip. They are sometimes combined with a friction type clutch, which
drag the two elements to nearly the same velocity before the jaws or
teeth engage. This is called synchromesh clutch.
|
A spiral jaw clutch is shown in Fig. 9.3.1.This
consists of two disks; both have jaws that can fit with each other. The
disk ‘1” is fastened to the driving shaft ‘A’. The other disk ‘2’ is
fitted on a hub mounted over a splined output shaft (driven shaft) ‘B’
(Refer for splined shaft and hub in Fig. 9.3.2)
and can be shifted along that shaft by a shifting mechanism ‘C’. Both
the shafts are coaxial. When the jaws are engaged, motion is transmitted
by direct interference between the projections on two parts of clutch.
|
Click on Image to run the animation
|
Animation 9.3.1 Operation of jaw clutch
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|
|
Fig. 9.3.1 Jaw clutch
|
|
|
|
Fig 9.3.2 Splined shaft and hub
|
|
Characteristics of jaw clutch
|
The satisfactory operation of jaw clutch
depends on the accurate machining of teeth on two halves to ensure
perfect contact. The correct mounting of two halves on the respective
shafts is essential for accurate engagement. Jaw disks require higher
accuracy in machining compared with toothed disks for trouble free
meshing of disks. For high-speed application, the engagement causes
violent shock and noise due to metal contact. Jaw clutches are
comparatively smaller in size compared to friction clutches for the same
power transmission.
9.4 APPLICATIONS OF MECHANICAL CLUTCHES
Mechanical clutches are used on the main shaft
of conventional loom. The drive to the loom is disengaged till the
motor attains its required torque and speed. In some looms, the drive
from main shaft to tappet shaft is through a jaw clutch. Disengaging
this clutch will allow the tappet shaft to rotate freely. This will be
useful to set the angular position of tappet shaft synchronizing with
other mechanisms of loom. In conventional blow rooms, the drive to pedal
roller of piano-feed regulating mechanism is through a clutch. Once the
lap attains a pre-set length, a lever from the lap stop mechanism
disengages the clutch and hence, the drive to the pedal roller of last
beater of blow room is disconnected to severe the lap. Mechanical clutch
is used in the drive from cylinder to flat on high production card (Fig. 9.4.1).
To set the flat with respect to cylinder, a hand lever is pushed which
disengages the drive to the flat. Then the flat can be moved freely
while setting it. Few application of mechanical clutches in textile
production machines are described in the subsequent sections.
|
|
|
Fig. 9.4.1 Mechanical clutch on drive from cylinder to card
|
|
Feed roller drive on card
|
Both the jaw and toothed clutches are used to transfer motion from doffer to feed roller on low production carding machines (Fig. 9.4.2).
The driving shaft gets its motion from the doffer through straight
bevel gears (1 and 2). On the driving shaft, disk (A) is fastened. On
the output shaft, disk (B) is mounted that can be moved axially by a
manually operated lever. The feed roller is connected to the driven
shaft by means of straight bevel gears (3 and 4). When the feed roller
is to be stopped, the disk (B) is moved away from the disk (A) which
disengages the clutch.
|
|
|
Fig. 9.4.2 Mechanical clutch to control feed roller on carding machine
|
|
Lap roller drive on sliver doubling machine
|
In sliver doubling machine, front bottom
calender roller drives the lap rollers through chain and sprockets via a
pneumatically operated clutch. When the lap attains the pre-set length,
the clutch disengages the positive drive to the lap rollers. A drum
brake mounted on the main shaft applies brake which slows down/or
momentarily stop the calender rollers. The lap rollers continue to roll.
This severs the lap behind the back lap rollers; and then the formed
lap is ejected out.
|
Resetting of belt on cone pulleys on roving machine
|
During starting of roving machine, the belt
moving on the cone pulleys must be brought to its initial position. The
belt is moved by a mechanism comprising ratchet; cam and compensating
rails. An electromagnetic actuated toothed clutch is fitted to the
ratchet shaft. The clutch is always engaged during winding so that the
ratchet transmits motion intermittently to the cam via a driven shaft.
To bring the belt to its initial position, the bottom cone pulley is
angularly shifted up, thus the belt gets slackened. Then the clutch is
disengaged, the driven shaft is rotated in opposite direction by other
mechanism, which reverses the direction of cam, pulling the belt to its
starting position.
|
Yarn under winding in ring spinning
|
Yarn is wound in the form of chases on ring
bobbin. Before the end of a chase, the ring rail must be raised
corresponding to the diameter of yarn. This is done by a controlled
rotation of a ratchet wheel. A belt on the top is connected to ring rail
and one of the rollers (Fig.9.4.3).
The other belt is fastened to the larger roller carrying a projection
(for building a right profile of cop base), passes over a roller mounted
on the extreme end of a pivoted lever which gets translating motion
from cam followers and then joined to a winding roller. The winding
roller is loosely mounted on a shaft. One side of winding roller is
profiled as saw toothed. This can mesh with a movable toothed disk
mounted on the same shaft. The shaft and movable toothed disk rotate
together. The ratchet wheel transmits motion to the shaft through worm
gear and a gear train. When winding the yarn from empty to full cop, the
moveable toothed disk is always in contact with the winding roller.
When the ratchet transfers motion to the shaft at the end of each chase,
the winding roller draws the belt and wound on it, raising the ring
rail a bit.
|
Click on Image to run the animation
|
Animation 9.4.3 Clutch disengagement and yarn under-winding
|
|
|
Fig. 9.4.3 Yarn under-winding in ring spinning
|
|
At the end of full cop, the serrated disks are
disengaged; hence the winding roller could rotate freely on the shaft.
The ring rail moves down due its weight, drawing the belt that is
previously wound on to the winding roller. The yarn is wound on to the
spindle below the bobbin. When the empty bobbin is inserted on to the
spindle, spinning can proceed without piecing.
|
|
9.5 FRICTION CLUTCHES
|
Friction clutches are gradually engaging clutches. Driving shaft may be
rotating at full speed while the driven shaft either stationary or
rotating at much lower speed is brought into connection with the former.
As the engagement of clutch proceeds, the speed of driven shaft attains
the speed of driving shaft. The torque transmitting efficiency of
friction clutches depends on the frictional force between two bodies
which are pressed together. To increase the friction between the two
bodies, a special material, ‘friction material’ is provided on one of
these bodies.
|
| 9.6 SINGLE FRICTION DISK CLUTCH |
| Single friction disk clutch consists of two disks or flanges or plates (shown in Fig. 9.6.1). One of the disks (friction disk) is lined with friction material. It is also called as ‘Single plate friction clutch’. |
Click on Image to run the animation
|
Animation 9.6.1 Operation of single friction disk clutch
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|
|
Fig. 9.6.1 Single friction disk clutch
|
|
One disk ‘driving disk’ is fastened to the
driving shaft. The driven disk is free to move along the driven shaft
due to splined connection. Both the shafts are coaxial. During
disengagement of the clutch, a contact lever keeps the driven disk away
from the driving disk. To engage the clutch, the contact lever is
gradually released. Then a spring provides an actuating force to the
driven disk forcing it to move towards the driving disk and finally
makes contact with it. The driven disk starts rotating at low speed due
to the friction between the disks. When the contact lever is fully
released, the spring provides the required axial force to press the
driven disk against the driver disk, the friction force between them
increases, and the driven disk attains the speed of the driver disk.
Torque is transmitted by means of frictional force between these plates.
|
The friction clutches are classified as
two-plane disks or multiple-lane disks depending upon the number of
friction surfaces. Based on the shape of the friction lining, they are
classified as disk clutches, cone clutches or expanding shoe clutches.
Friction clutches permit smooth engagement at any speed. In the event of
over loads, the friction clutches slip momentarily, safeguarding the
machine or mechanism against breakage.
|
Torque transmission capacity of single friction disk clutch
|
A two-plane disk friction clutch is shown in Fig. 9.6.2.
|
|
|
Fig. 9.6.2 Notations of single disk friction clutch
|
|
F = total actuating force (axial force) (N)
|
Mt = Torque transmitted by friction (N-mm)
|
m = Coefficient of friction between friction disks
|
Consider an elemental ring of radius, r and radial thickness dr. For this ring, the cross-sectional area of the element,
|
|
|
Torque transmission capacity of old and new disk clutches
|
There are two criteria to obtain the torque
transmission capacity of friction clutches, viz., uniform pressure and
uniform wear.
|
Torque transmission under uniform pressure
|
This theory is applicable to new clutches. In
new clutches employing a number of springs, the pressure can be assumed
as uniformly distributed over the entire surface area of the friction
disk. With this assumption, the intensity of pressure between disks, p is regarded as constant. From Eq. (9.1) and (9.2)
|
|
|
The above equation is valid for a single pair of mating disk surfaces.
|
Torque transmission under uniform wear
|
This theory is based on the fact that wear is
uniformly distributed over the entire surface area of friction disk.
This assumption can be used for worn out clutches/old clutches. The
axial wear of the friction disk is proportional to frictional work. The
work done by the friction is proportional to the frictional force (μp) and the rubbing velocity (2πrn ) where n is the speed of the disk in revolution per minute. When the speed n and the coefficient of friction m are constant for a given configuration, then
|
Wear α pr ..............................................................................................(9.6)
|
According to this assumption,
|
pr = Constant .......................................................................................(9.7)
|
When the clutch plate is new and rigid, the wear at
the outer radius will be more, which will reduce pressure at the outer
edge due to rigid pressure plate. This will change pressure
distribution. During running condition, the pressure distribution is
adjusted such that the product (pr) is constant. Therefore,
|
p.r = pa.r ............................... ...............................................................(9.8)
|
Where pa is the pressure at the inner edge of plate, which is also the maximum pressure. From equation (9.1) and (9.2)
|
|
The above equation gives the torque
transmitting capacity for a single pair of contacting surfaces. The
uniform-pressure theory is applicable only when the friction lining is
new. When the friction lining is used over a period of time, wear
occurs. Therefore, the major portion of the life of friction lining
comes under uniform-wear criterion. Hence, in the design of clutches,
the uniform wear theory is used.
|
From Eq. (9.11), it is clear that the torque transmitting capacity can be increased by three methods:
|
| (a) |
Using the friction material with a higher coefficient of friction (m); |
| (b) |
Increasing the intensity of pressure (p) between disks; and |
| (c) |
Increasing the mean radius of friction disk (R + r)/2. |
|
Delayed starting of drafting in ring spinning
|
The drafting rollers are driven through gear
trains. When the ring spinning machine is started, the spindles lag
behind the drafting rollers due to slackness of tape. The tape takes
time to build up sufficient tension ratio around the spindle wharves to
turn the spindle. Meanwhile, the front roller has started delivering the
fibre strand. This creates slackness on the yarn that leads to yarn
breaks. To avoid this, the starting of drafting rollers must be delayed
synchronizing with the start of spindles. A gear (A) mounted on the main
shaft drives a gear (B) that is loosely mounted on an intermediate
shaft ( Fig. 9.6.3). A friction disk
can be moved along the intermediate shaft by means of piston actuated
by compressed air in a cylinder. When the machine is started, this disk
is kept away from the gear, B (clutch is disengaged); the gear B
revolves freely without transmitting motion to the intermediate shaft,
the drive to the front drafting roller is disconnected. After a pre-set
time delay, the clutch is engaged, the gear B and the intermediate shaft
revolve together, driving the drafting rollers.
|
Click on Image to run the animation
|
Animation 9.6.3 Delayed starting of drafting rollers
|
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Fig. 9.6.3 Disk clutch for delayed starting of drafting on ring spinning machine
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9.7 MULTI DISK FRICTION CLUTCH
A multi-disk friction clutch is shown in Fig. 9.7.1.
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Fig. 9.7.1 Multi-disk friction clutch
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It consists of two sets of disks, A and B. The
set of driven disks, ‘A’, are mounted on the output shaft by means of
splined sleeve, so that they are free to move in the axial direction. An
L-shaped plate or drum is fastened to the driving shaft. The drum
rotates along with driving shaft. Holes (three or four) are drilled on
the rim of plate and also on the drum with equal angular separation, and
bolts are passed through each set of holes. The driving set of disks,
‘B’ is also made with holes. The bolts pass through the holes of the
drum, driving disks, ‘B’ and the rim of plate. A clearance fit between
the bolts and the holes in the driving disk allows the disks B to move
in axial direction. The bolts are rigidly fixed to a revolving drum.
Normally, the disks, ‘A’ are placed compressed under spring force, so
that they pressed against the driving disks, B’, and torque is
transmitted to the driven shaft. For disengagement of the clutch,
contact levers move the driven disks away from the driving ones.
Hardened steel and hardened bronze are used to make the driven and
driving disks respectively.
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Torque transmission capacity of multi disk friction clutch
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| For the uniform-pressure criterion, torque is |
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Where z is the number of pairs of contacting surfaces
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The multi disk clutch has higher torque
transmission capacity compared to single plate clutch, due to more
number of contacting surfaces. For a given torque capacity, the size of a
multi plate clutch is smaller compared with single plate clutch. The
work done by frictional force during engagement or disengagement is
converted into heat. Because of the large number of friction surfaces,
heat dissipation is a serious problem in multi plate-clutch. Therefore,
multi plate clutches are made as wet clutches using oil. The oil reduces
the coefficient of friction between the disks, reducing the torque
capacity of multi plate clutches. Further, the holes reduce the area of
contact between the disks by around 5%.
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Multi-disk friction clutch on bale opener
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Electrically actuated multi-disk friction clutches are used on bale opener (Fig. 9.7.2).
When the material height (sensed by photo electric sensor) in the
vicinity of inclined spiked apron is enough, feeding of fibres by the
creel apron must stop. The inclined apron drives (chain) the feed apron.
Feed apron transmits motion to creel apron by chain drive via a clutch.
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The operation of clutch is schematically illustrated for a single disk clutch in Fig.9.7.3.
The inclined apron drives the sprocket (A) of the feed apron; the later
transfer motion to the sprocket (B) mounted on the intermediate shaft.
Sprocket (C) is loosely mounted on the intermediate shaft. The sprocket
(C) usually transfer motion to sprocket (D) mounted on the creel apron
shaft. When the photo electric sensor receives light (material height
below that of sensor), electromagnetic force actuate and moves the disk
mounted on splined shaft towards the sprocket, engaging the clutch and
hence, the sprocket C rotates driving the creel apron. When the height
of material exceeds above the height of photoelectric sensor,
electromagnetic forces cease to exist, disengaging the clutch, the
sprocket (C) stops rotating. Hence, the creel apron stops feeding the
fibres into the bale opener.
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Fig. 9.7.2 An electromagnetic actuated multi-disk friction clutch in bale opener
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Fig. 9.7.3 Schematic representation of disk friction clutch on bale opener (for clarity only one pair of disk is shown)
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Multi-disk friction clutch for fabric roll-up mechanism on loom
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The drive to fabric roller is realized from
the shaft of take-up (or draw-off) roller via chain/other drives and
pneumatic or electric actuated multi-disk friction clutch. As the radius
of fabric roll (Rfi) increases and a constant tension (Ff) must be kept on the fabric (in order to wind the fabric uniformly on the fabric roller), the torque on the fabric roll (Mt) must also increase as
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The actuating force must be varied
continuously by automatically adjusting electromagnetic forces in the
clutch. In order to take out the fabric roller while the loom is
running, the clutch is disengaged, which allows the fabric roller to be
rotated freely.
9.8 CONE CLUTCHES
Cone clutches are friction clutches. They are
simple in construction and are easy to disengage. However, the driving
and driven shafts must be perfectly coaxial for efficient functioning of
the clutch. This requirement is more critical for cone clutch compared
to single plate friction clutch. A cone clutch consists of two working
surfaces, viz., inner and outer cones, as shown in Fig.9.8.1.
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Click on Image to run the animation
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Animation 9.8.1 Operation of cone clutch
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Fig. 9.8.1 Cone clutch
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The outer cone is fastened to the driving
shaft and the inner cone is free to slide axially on the driven (output)
shaft due to splines. A spring provides the necessary axial force to
the inner cone to press against the outer cone, thus engaging the
clutch. A contact lever is used to disengage the clutch. The inner cone
surface is lined with friction material. Due to wedging action between
the conical working surfaces, there is considerable normal pressure and
friction force with a small engaging force. The semi cone angle a
is kept greater than a certain value to avoid self-engagement;
otherwise disengagement of clutch would be difficult. This is kept
around 12.50.
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Torque transmission capacity of cone clutch
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An elemental frustum of the cone bounded by circles of radii r and (r + dr) is shown in Fig. 9.8.2.
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Fig. 9.8.2 Notations of cone clutch
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Fig. 9.8.3 An element of cone clutch
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| For this elemental frustum, (from Fig 9.8.3), area of contacting surfaces between the cones (δA) is expressed as following: |
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Driving of bobbin carriage on roving machine
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The up and down movement of bobbin carriage is controlled by two conical clutches and bevel gears ( Fig. 9.8.4).
Two bevel gears (1 on right side and 2 on left side of vertical shaft)
are loosely mounted over a horizontal driving shaft. These two gears
constitute the outer cones of two clutches. Two friction lined conical
disks (A & B) can be moved axially over the splines of horizontal
shaft using pneumatically operated piston inside a cylinder. These
friction disks constitute the inner cones of clutch. When the piston
retracts (moving towards left) as shown in the figure, the clutch on the
right side (A) engages, whereas the one at the left side (B) disengages
simultaneously. Gear 1 revolves along with the horizontal shaft,
transmits motion to the bevel gear 3 mounted on the vertical shaft; and
the bobbin carriage moves downwards. Since the bevel gear 3 is always
engaged with both the bevel gears 1 and 2, the bevel gear 2 revolves
freely, but opposite to that of horizontal shaft.
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Click on Image to run the animation
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Animation 9.8.4 Double cone clutch transmitting motion to bobbin carriage
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Fig. 9.8.4 Double cone clutch to drive the bobbin carriage on roving machine
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When the bobbing carriage is to be moved
upwards, the piston moves from left to right that simultaneously engages
the left side clutch (B) and disengages the right side clutch (A). Then
the gear 2 rotates along with the horizontal shaft, transmit motion to
the vertical shaft in opposite direction and the bobbin carriage moves
upwards. The bevel gear 3 transmits motion to the loosely held bevel
gear 1 which now rotates opposite to that of horizontal shaft.
9.9 CENTRIFUGAL CLUTCHES
The centrifugal clutch permits the drive motor
to start, warm up and accelerate to the operating speed without load.
Then the clutch is automatically engaged and the driven equipment is
smoothly brought up to the operating speed. These clutches are highly
useful for heavy loads (large machines) where the motor cannot be
started under that load.
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Operating principle of centrifugal clutch
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The centrifugal clutch works on the principle
of centrifugal force, which increases proportional to the square of
rotational speed. A centrifugal clutch is shown in Fig. 9.9.1.
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Fig. 9.9.1 Centrifugal clutch
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Guides or spiders are mounted on the driver or
motor shaft. They are equally spaced such that if there are four
guides, they are separated by 90º. Sliding shoes are placed between the
guides and each is retained by a spring. The outer surface of the
sliding shoe is provided with a lining of friction material. A co-axial
drum, which is mounted on the output or driven shaft, encloses the
assembly of spider, shoes and spring.
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When the motor is started, the rotational
speed of input shaft increases, and hence the centrifugal force acting
on the sliding shoe increases. This causes each shoe to move outward.
The shoe continues to move with increasing speed until they contact the
inner surface of drum, overcoming the spring force. Torque is
transmitted due to frictional force between the shoe lining and the
inner surface of drum ( Refer animation 9.9.1).
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Torque transmission capacity of centrifugal clutch
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The forces acting on the shoe are shown in Fig. 9.9.2, with the following notations:
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Fig. 9.9.2 Forces acting in centrifugal clutch
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The engagement/disengagement and torque
transmission capacity of clutch depend on the centrifugal force
generated in the clutch which rely mainly on the motor speed. So, the
centrifugal clutch is called ‘self-activating clutch’. The engaging
speed depends on the selection of spring constant.
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Applications of centrifugal clutches
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Centrifugal clutch is widely used in textile
machinery such as looms, carding-, knitting-, drawing-, roving- and
spinning machines. The spider is fixed on the main motor shaft. The
outer surface of drum forms a pulley which is either crowned or grooved
depends on whether a flat or v-belt is used. From this pulley, drive is
transmitted to the pulley on the main shaft. For example, a centrifugal
clutch is used in the drive from motor to cylinder and lickerin on a
high speed card. Once the motor attains the required speed, the
centrifugal clutch engages, transmitting the drive to lickerin and
cylinder, thus safe guarding the motor during start-up. Centrifugal
clutch is economical and requires less maintenance to any other motor
safety device such coupling.
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9.10 MATERIALS FOR FRICTION LINING
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Asbestos-based materials and sintered metals
are commonly used for friction lining. There are two types of asbestos
friction disks: woven and moulded. A woven asbestos friction disk
consists of asbestos fibre woven with endless circular weave around
brass, copper or zinc wires and then impregnated with rubber or asphalt.
The endless circular weave increases the bursting strength. Moulded
asbestos friction disks are prepared by moulding the wet mixture of
brass chips and asbestos. The woven materials are flexible, have higher
coefficient of friction, conform more readily to clutch surface, costly
and wear at faster rate compared to moulded materials. Asbestos
materials are less heat resistant even at low temperature.
Sintered-metal friction materials have higher wear resistance, high
temperature-resistant, constant coefficient of friction over a wide
range of temperature and pressure, and are unaffected by environmental
conditions. They also offer lighter, cheaper and compact construction of
friction clutches.
| 10.0 : BRAKES |
| 10.1 INTRODUCTION |
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Brake is a machine element which is used
either to stop the machine or retard the motion of a moving system, such
as a rotating rollers or drums or vehicle where the driving force has
ceased to act or is still acting. In practice most brakes act upon drums
mounted on the driving shafts or driven shafts. In such cases brake
will act either upon the internal surface or external surface of the
drum. The brakes acting on the brake drums do not make contact along the
whole periphery and the part making contact with the drum is called
shoe. The shoe has to expand for internal contact and close in for
external contact. When the braking action takes place, the energy
absorbed by the brake shoe is converted into heat energy and dissipated
to surroundings. Heat dissipation is a serious problem in brake
applications.
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10.2 MECHANICAL OR FRICTION BRAKES
In the design of mechanical brakes the first
step is to determine the braking torque capacity for the given
application. This depends on the amount of energy absorbed by the brake.
When a brake drum of mass ‘m’ moving with a angular velocity of ‘ω2’ is slowed down to a velocity of w 1 during the period of braking, the kinetic energy absorbed by the brake is given by
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| Where |
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F = tangential force on the brake drum |
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r = radius of the brake drum |
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θ = Angle through which the brake drum rotates during braking period (rad.) |
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Mt = Braking torque (N-m) |
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The mechanical brakes are classified into:
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| (a) |
Block brakes (simple block and pivoted block). They are also called as ‘drum brakes’ |
| (b) |
Internal expanding brakes |
| (c) |
Band brakes and |
| (d) |
Disc brakes |
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| 10.3 BLOCK BRAKE WITH SHORT SHOE |
A block brake consists of a rotating drum
(brake drum) against which a brake shoe is pressed by means of a pivoted
lever (shown in Fig.10.3.1). The block brake is also referred as ‘Drum brake’.
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Fig. 10.3.1 Drum or Simple block brake
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The friction force between the shoe and the
brake drum acts against the direction of rotation of the drum at the
contact region. This causes retardation of the drum. When the friction
force is very high, the drum stops rotating. The angle of contact, ‘θ’, between the shoe and the brake drum is usually kept less than 450,
to obtain uniform pressure between them. The main disadvantage of the
drum brake is the tendency of the drum shaft to bend under the action of
normal force, N ( Fig. 10.3.2)
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Analysis of forces acting on the drum
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The free body diagram of forces acting on the drum and the lever is shown in Fig.10.3.2.
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Fig. 10.3.2 Free body diagram of forces acting on the drum and the lever of simple drum brake
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When the shoe is rigidly attached to the lever, then the torque acting on the brake drum is
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Mt = μNr............................................................................................................ (10.3)
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Where μ = Coefficient of friction between the drum and shoe
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N = normal force acting on the drum (N). The normal force acting on the drum;
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N = pl1w............................................................................................................ (10.4)
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Where p = permissible pressure between the shoe and the brake drum (N/mm2)
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l1 = length of the shoe (mm)
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w = width of the shoe (mm)
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The reaction forces on the pivot of the lever in horizontal and vertical directions are denoted as Rx and Ryrespectively. Considering the equilibrium of forces on the lever in horizontal and vertical directions,
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Rx = μN ..............................................................................................................(10.5)
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Ry = (N-F) ..........................................................................................................(10.6)
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Where F is the actuating force on the lever. Taking moment of forces acting on the lever about the pivot point, it can proved that,
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Depending
upon the magnitude of coefficient of friction (μ) and the position of
pivot point, there are three possibilities emerge.
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Case 1
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a > μ b .......................................................................................................(10.8)
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In this case, the friction force (μN ) helps to reduce the magnitude of the actuating force F. From the free body diagram, the moments (Fl), and (μNb )
are counterclockwise. Such a brake is called a partially
self-energizing brake. However, the brake is not self-locking, because a
small magnitude of positive force F is required for the braking action.
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Case 2
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a = μ b .......................................................................................................(10.9)
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In this case, the actuating force F is
zero, as seen from the Eq. (9.7). This indicates that no external force
is required for the braking action. Such a brake is called’
self-energizing’. Since, the actuating force is zero; it is also called
as ‘self-locking’ or ‘self acting’ brake.
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Case 3
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a < μ b .......................................................................................................(10.10)
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Under this condition, the actuating force F becomes negative, as seen from the Eq. (10.7). This results in uncontrolled braking.
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Drum
brakes should be designed so that it is not self-locking and, at the
same time, full advantage of the partial self-energizing effect should
be taken to reduce the magnitude of the operating force F.
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9.3.2 Block brake with short shoe on lap former
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Block brake with short shoe is used
on lap former of conventional blow room. In order to build a compact lap
suitable for feeding the card, a brake applies pressure on the lap when
it is wound on a spindle. A brake drum is mounted on a shaft which
carries two pinions on both sides of the lap former (Fig. 10.3.3). These pinions mesh with racks.
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Fig. 10.3.3 Brake drum, shoe, racks, pinions and gear train on lap former
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When winding of lap progress, its diameter increases, pulling the racks up (Fig. 10.3.4). This rotates the pinions.
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Fig.10.3.4 Racks, pinions and lap spindle
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A brake shoe mounted on a lever with adjustable weight (W) is in contact with the brake drum during lap winding ( Fig.10.3.5).
The upward movement of the lap-rack must overcome the braking action.
The work done in raising the racks or lap must be equal to work done in
turning the brake drum.
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Fig.10.3.5 Brake in lap former
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Balancing the moments about the pivot of the actuating lever, we get the normal reaction force on the lever as,
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The parameters in the bracket are
fixed by design. The hanging weight and its position on the actuating
lever are the variable parameters. The force applied on the lap depends
on the resilience of fibres under compression. The usual forces applied
on the lap while processing synthetic, cotton and viscose rayon are: 40
kN, 25 kN and 18 kN respectively.
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| 10.5 INTERNAL EXPANDING BRAKE |
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An internal expanding brake is shown in Fig.10.5.1. It consists of a shoe, which is pivoted at ‘A’ and on the other end ‘B’ an actuating force F acts.
A friction lining is provided on the shoe. The complete assembly of
shoe with lining and pivot is placed inside the brake drum. Under the
action of the actuating force the shoe contact the inner surface of
drum. Internal shoe brakes, with two symmetrical shoes, are used in all
automobile vehicles. The actuating force is usually provided by a
hydraulic cylinder or a cam mechanism.
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Fig.10.5.1 Internal expanding brake
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10.6 BAND BRAKES
In
band brake, a flexible steel band lined with friction material, presses
against the rotating brake drum. The braking action is performed either
to slow down or halting the drum. The braking action is obtained by
tightening the band around the drum. This kind of brake is used on
sectional warping machine and warp let-off motion on conventional looms.
Band brakes are classified into simple and differential band brakes.
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A simple band brake is shown in Fig. 10.6.1,
where one end of the steel band passes through the fulcrum of the
actuating lever (O). The other end of the band is connected to the lever
at point (A) a distance ‘a’ from the pivot point.
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Fig.10.6.1 Simple band brake
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Actuating
force is applied at point (B) on the lever. Since the band is
stationary, centrifugal force will not lift the band. Therefore, the
working of steel band is similar to that of a stationary flat belt on
rim of a pulley and hence, the ratio of tensions on the steel band is
given by
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Tt / Ts = eμθ ........................................................................................................(10.16)
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The torque absorbed by the brake is given by
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Mt = (Tt - Ts)r ..................................................................................................... (10.17)
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Considering the forces acting on the lever and taking moments about the pivot (O),
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Ts.a = F.l ........................................................................................................... (10.18)
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F = Ts.a / l ..........................................................................................................(10.19)
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A differential band brake is shown in Fig.10.6.2.Thedifferential
band brake is similar to the simple band brake, except that the ends of
the band are joined to the actuating lever other than the pivot of the
lever (O). In this configuration, the tight side tension on the band
helps in reducing the actuating force (the moments due to actuating
force and tight side tension acting on the lever are in the direction,
i.e., clockwise). Differential band brakes may be designed for the
condition of self-energizing or self-locking.
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Fig. 10.6.2 Differential band brake
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Considering the forces acting on the lever and taking moments about the pivot, we can get,
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If the brake is already designed for
the self locking condition for the drum rotating in clockwise direction,
then substituting the values of a and b as per Eq.
(10.24) in the Eq. (10.26), yields a positive actuating force for the
drum rotating in counter clockwise direction. This means that a brake
designed to be self-locking in one direction of rotation, can be free to
rotate in the opposite direction. Therefore, a self locking brake can
be used for those applications where rotation of drum is permitted in
only one direction.
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Negative let-off warp on loom using band brakes
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Brake is used on loom to let off warp
yarns that commensurate with the length of fabric produced. The
required total force or tension on the warp yarns (nT) to let off the warp is generated from a brake mechanism ( Fig.10.6.2). The construction of brake is similar to the simple band brake shown in Fig. 10.6.3 . Instead of a band, two chains, each envelopes the ruffle (radius, r) of warp beam on both sides. The radius of warp beam is R. The tension on each warp yarn is T and the numbers of warp yarns are n. The angle of wrap of chain over the ruffle in radians is θ. The coefficient of friction between the ruffle and chain is μ. Two weights (W) are placed on the both the levers placed on the left and right sides of the warp beam.
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Fig. 10.6.3 Brake on warp beam on a conventional loom
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Usually the angle of wrap is 540 0 which results in, e μ θ <
1, and hence, the brake is not self-energizing one. By varying the
weight and its placement and the number of turns of chain wrapped over
the ruffle, the warp tension and consequently the warp breaks are
controlled.
10.7 DISK BRAKES
Disk brakes are similar to disk clutches in
terms of construction and operating principles. The former is employed
to stop or retard the motion of machine or machine elements, whereas the
later is for engaging or disengaging a drive system. The force analysis
described for the disk clutches (given in module 9) are the same for
the disk brakes. Disk brakes are highly suitable for heavy duty
industrial application. In yarn winding machine, when yarn breaks or
bobbin run-out, braking of the package takes place until the package has
come to a complete stop. For this, a brake lining is pressed against
brake ring located in the adapter of package. Disk brakes are also used
in many textile machines. A disc brake is shown in Fig.10.7.1.
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Fig.10.7.1 Disk brake
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| 10.8 NON-FRICTION TYPE CLUTCHES AND BRAKES |
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Electromagnetic clutches and brakes use
electromagnetic attraction rather than friction to perform their
function. In other words, the torque transmission is through
electromagnetic force rather than the friction force. Three types of
non-friction electric clutches and brakes are available: (a) magnetic
particle, (b) eddy current, and (c) hysteresis. They are mainly used in
applications that require variable slip.
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10.9 DISC CLUTCH AND DICS BRAKE COMBINATION
Disk brakes along with disk clutches can be
fitted on the main shafts of looms, comber, roving and ring spinning
machines and warp beam in warping machines. During emergency situation
the machine must be stopped. The connection between the main shaft
pulley and main shaft is through multi-disk friction clutch. On the main
shaft a multi-disk friction brake can be attached. During emergency
situation, the clutch disengages the main shaft from the motor. Then the
brake applies on the shaft to stop the machine. While stopping the
machine, the disengagement of clutch must precede the braking action by a
short time interval, but not in a reverse sequence.
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Pneumatic actuated multi-disk friction clutch and brake
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Piston operated multi-disk friction clutch and brake is shown in Fig.10.9.1.
A motor transmits motion to the input shaft (main shaft of machine)
usually through V-belt. A drum is rigidly mounted on the input shaft
which carries a set of disks (A) through bolts. The output shaft is
splined and coaxial to the input shaft. Two set of disks (clutch disks B
and brake disks C) are placed on the splined sleeve of output shaft, so
that they are free to move axially. A stationary outer shaft is coaxial
to the output shaft which carries a fourth set of disks (D) through
bolts. The arrangement of disks in the clutch is such that the disks A
and B are alternately placed with clearance. Similar is the case with
disks C and D in the brake. From the output shaft power is transmitted
to the machine.
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Click on Image to run the animation
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Animation 10.9.1 Operation of multi-disk friction clutch and brake
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Fig.10.9.1 Multi-disk friction clutch and brake
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When the motor is stopped, a piston moves
towards right. The disks B disengages from the disks A (clutch is
disengaged); followed by the disks C contacting the disks D (braking),
thus the output shaft and machine will halt rapidly. While starting the
machine, the motor first starts rotating; after the motor acquires
enough torque, the piston moves from towards left. This disengages the
brake (clearance between the disks C and D) first followed by engagement
of clutch (contact between the disks A and B); thus transferring power
to the machine.
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Hysteresis disk clutch and brake
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These types of clutch and brakes provide
constant torque for a given control current. They can be used to provide
any amount of slip, as long as heat dissipation capacity of the unit is
not exceeded. Hysteresis losses transmit torque in this type of clutch
and brake. A hysteresis clutch cum brake used on the main shaft of loom
is shown in Fig.10.9.2.
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Click on Image to run the animation
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Animation 10.9.2 Operation of hysteresis disk clutch and brake
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Fig. 10.9.2 Hysteresis disk clutch and brake on loom
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An electro-magnet (clutch magnet) on the input
rotor generates magnetic field in the rotor and drag disk. The drag
disk is mounted on a splined output shaft and can move along the shaft. A
stationary outer shaft is mounted on the output shaft; both are
co-axial to each other. An electro-magnet (brake magnet) fixed on the
outer shaft can also generate a magnetic field on the drag disk.
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To transmit motion to the machine, the clutch
magnet is actuated which pulls the drag disk away from the de-energized
brake magnet. In doing so, the brake is disengaged first followed by
engagement of clutch. This transfers drive to the output shaft, and
hence the machine. While stopping the machine, motor is turned off.
Simultaneously, the clutch magnet is de-energized followed by energizing
of the brake magnet. The drag disk moves away from the clutch magnet
and towards the brake magnet. Thus, the power transmission is cut off to
the machine followed by rapid halting of the machine. The clearance
between the drag disk and brake magnet must be lesser compared with that
between the drag disk and clutch magnet in order to carry out the
operations properly.
| 11.0 : BEARINGS |
| 11.1 INTRODUCTION |
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Shafts and the parts supported by them are
carried by machine elements called ‘bearings’. In general any rotating
part of the machine must to be supported by a relatively stationary part
which is called ‘bearing’. The main requirement of any bearing is to
offer minimum frictional resistance to the rotating part in order to
reduce loss in power during transmission. Bearings are classified into
(a) sliding contact bearings and (b) rolling contact bearings.
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11.2 SLIDING CONTACT OR BUSH BEARINGS
These bearings are also called as ‘Journal bearings’ or ‘Bush bearings’. A sliding contact bearing is shown in Fig.11.2.1.
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Fig. 11.2.1 Sliding contact bearing
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The shaft called ‘journal’ is mounted
inside a hollow cylinder termed ‘bearing’. When the journal rotates,
there is a relative motion between two surfaces of the bearing namely
the journal and the bearing inner surface. This results in friction.
Placing lubricant between the journal and the inner surface of the
bearing can reduce the friction. The journal bearings are used to
support load in radial direction. In majority of the applications, the
journal rotates while the bearing is stationary. In few applications,
the bearing rotates and the journal is stationary. In some cases, both
the bearing and journal rotate.
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Solid bush and lined bush bearings
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Sliding contact bearing is constructed either with solid bushing or lined bushing, as shown in Fig.11.2.2.
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Fig. 11.2.2 Sliding-contact bearing (a)Solid bush (b)Lined bush
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A
solid bushing is made either by casting or by machining from a roller.
Bronze bearings are of this type. A lined bushing consists of a steel
outer body with a thin inner lining of bearing materials like Babbitt
(alloy of tin-copper-lead-antimony). These bearings are usually split
into two halves, provided with a locking arrangement that prevents the
two halves from being displaced in radial and axial directions from the
housing.
|
| The
footstep bearings of ring spindles are of bush type. The bearing is
supported by springs, so that it can swing laterally to a limited
degree. Thus, offering self-centering or self-aligning to the ring
spindles. Nipper frame on comber is supported by bush bearings at its
extreme ends. |
Full and partial bush bearings
|
Journal bearings are made with either ‘Full bearing’ or ‘Partial bearing’ shown in Fig.11.2.3.
In the former, the whole circumference of the journal is covered by the
bearing; and in the later, a portion of the circumference of the
journal is supported at its bottom and an oil cap is placed around the
remaining.
|
 |
Fig. 11.2.3 Journal bearings: (a) Full bearing (b) Partial bearing
|
The radii of bearing and journal are depicted as rb and ri respectively.
The clearance ‘c’ is the gap between the journal and the bearing inner
surface measured along the radial distance (rb - rj). The distance ‘L’ is the bearing length.
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11.3 LUBRICATION IN BUSH BEARINGS
In
a sliding contact bearing, the journal is directly inserted into the
bearing. This results in direct metal to metal contact between them. As a
consequence the friction is higher between the inner surface of the
bearing and the outer surface of journal, if there is no lubricating
film present in between them. Bearings can be lubricated with three
kinds of lubricants, viz. liquids like mineral oil or vegetable oils,
semi-solids like grease, and solids like graphite or molybdenum
disulfide. These lubricants are used to reduce friction and wear,
dissipate the frictional heat and to protect against corrosion. There
are two basic modes of lubrication: (a) thick film and (b) thin film
lubrication.
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Thick film lubrication
|
In thick film lubrication, two
surfaces of bearing in relative motion, (viz., the journal and the
bearing inner surface) are completely separated by a fluid film. The
resistance to relative motion arises from the viscous resistance of the
fluid. This does not depend on the structure of journal surface and
bearing inner surface as they are not in contact with each other. Thick
film lubrication is classified into: hydrodynamic and hydrostatic
lubrication.
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Hydrodynamic lubrication
|
Hydrodynamic lubrication is defined
as a system of lubrication in which the load supporting fluid film is
created by the shape and relative motion of the sliding elements. The
principle of hydrodynamic lubrication in journal bearing is shown in Fig.11.3.1.
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 |
Fig. 11.3.1 Hydrodynamic Lubrication (a) Journal at rest (b) Journal starts to rotate (c) Journal at full speed
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When
the shaft (centered at O’) is at rest, it goes to the bottom of bearing
(centered at O) under the action of load W. This load is due to the
weights of shaft and various elements (gears, pulleys) supported by the
shaft. The outer surface of journal and inner surface of bearing touch
each other during rest, with no clearance at the bottom. The letter ‘e’ denotes the eccentricity, the offset between the axes of the journal and the bearing.
|
As
the journal starts to rotate, it will climb the bearing surface. When
the speed is increased further, it forces the fluid into the
wedge-shaped region between the journal and bearing. As more and more
fluid is forced into the wedge shaped region, pressure is generated
within the fluid as shown in Fig.11.3.2. This fluid pressure generated in the clearance space supports the external load (W).
It can be seen that the pressure distribution around journal varies
greatly. Hydrodynamic lubrication does not need a supply of lubricants
at high pressure from external source (pumps), as enough fluid pressure
is generated within the system. Bearings that use ‘hydrodynamic
lubrication’ are called, ‘Hydrodynamic bearings’.
|
 |
Fig. 11.3.2 Pressure distribution in hydrodynamic bearing
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| Hydrostatic lubrication |
In
the case of hydrostatic lubrication, the load supporting fluid film is
created by an external source, like pump, supplying fluid at sufficient
pressure. Bearing using such system is called ‘externally pressurized
bearing’ or ‘Hydrostatic bearing’. The principle of this lubrication is
demonstrated in Fig. 11.3.3.
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Fig. 11.3.3 Hydrostatic lubrication
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|
When the shaft is at rest, there is not enough oil pressure as the pump supplying the oil is not yet started (
( Fig 11.3.3b). When the shaft
starts rotating, the pump supply oils at high pressure into the
clearance zone. This lifts the journal, reduces the starting friction
and also the eccentricity (e).
|
Hydrodynamic and hydrostatic bearings
|
Hydrodynamic
bearings are simple in construction, have lower initial and running
cost and are easy to maintain compared to the hydrostatic bearings.
However, hydrostatic bearings offer the following advantages.
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- High load carrying capacity even at low speeds
- No starting friction
- No contact between sliding surfaces at any range of speeds and loads.
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Thin film lubrication
|
Thin film lubrication is defined as a
condition of lubrication where the lubricant film is relatively thin
which may rupture that leads to more metal to metal contact; resulting
in higher friction. This type of lubrication is also called ‘Boundary
Lubrication.’ The boundary lubrication is neither planned by the
designer and nor desirable. In sliding contact bearing, heavy load on
the bearing, insufficient oil supply, low speed of the journal, chemical
reactions and misalignment between the journal and the bearing create
conditions for boundary lubrication.
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Viscosity of lubricants
|
The friction in a lubricated sliding
contact bearing is due to the viscous resistance of the fluid in
shearing. If a fluid film of thickness ‘t’ and of cross-section ‘A’, is subjected to a shearing force ‘F’ with a relative velocity between the top and bottom surface of the fluid as ‘v’, (refer Fig. 11.3.4) then,
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Fig. 11.3.4 Fluid film between stationary and moving surfaces
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11.4 POWER LOSS IN BUSH BEARING
Petroff’s relation can be used to calculate
the power loss in sliding contact bearings. It is assumed that shaft and
bearing are concentric, and a constant thickness layer of lubricant
film exists between them. Since the oil film thickness is small, the
journal and bearing may be considered as two parallel plates. If one of
the plate (journal) is moving at a velocity ‘v’, with a N rpm, and the clearance between the journal and bearing is ‘c’, then
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 |
 |
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11.5 COEFFICIENT OF FRICTION IN BUSH BEARINGS
| If the radial force acting on the journal is Fr, (force acting perpendicular to thickness plane of the film) is creating a pressure of ‘p’ in the oil film, then the frictional torque |
 |
The relationship between μ and N η/ p, is plotted in Fig. 11.5.1.
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 |
Fig. 11.5.1 Coefficient of friction (μ) of journal bearing as a function of viscosity, speed and fluid pressure
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If
the bearing is operating at the hydrodynamic region (thick film), and
some change in operating condition causes the temperature to rise, this
will result in lowering of viscosity (η), lowering the parameter (Nη/p). This shifts the point of operation to the left. This shift in effect will reduce the coefficient of friction (μ), and thus produce less heat and lower the temperature. The lowering of temperature will cause an increase in viscosity (η) and original value of the parameter (N.η/p)
will be restored. Hence, the operation in this zone of thick film
lubrication is stable, and it is known as stable lubrication.
|
If
the operation is performed at the boundary lubrication (thin film
lubrication), that is close to y-axis, a slight increase in temperature
would decrease the viscosity, shift the point of operation zone towards
left and increase the coefficient of friction. This will produce more
heat, an increase in the temperature, resulting in further lowering of
viscosity of lubricant. The operating region further shifts towards
left. This process of shifting the operating region of bearing towards
left will be continuous. This operating region (boundary lubrication) is
called ‘unstable lubrication.’ At boundary lubrication, the film
thickness is very small (either due to low speed or low viscosity of
film or due to very high pressure across the film); as a result metal to
metal contact is high. So the operation would be highly unstable,
resulting in stick-slip (unsteady) movement of the journal.
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11.6 OIL-GROOVES IN BUSH BEARINGS
Oil
grooves are constructed on journal bearings either by circumferential
or cylindrical patterns. A circumferential oil-groove bearing is shown
in Fig. 11.6.1.
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 |
Fig. 11.6.1 Circumferential oil groove in sliding contact bearing
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The
oil-groove divides the bearings into two short bearings in the axial
direction, each of length (L/2). The presence of groove reduces the
pressure developed in the fluid in the plane of groove considerably, as
well as the overall pressure. This reduces the load carrying capacity of
bearing. Further, the centrifugal force acting on the oil in the
circumferential groove may build pressure higher than the supply
pressure, restricting the flow of the lubricant into the bearing. These
bearings find application in automobiles.
|
The cylindrical oil-groove bearing is shown in Fig.11.6.2.
The bearing has an axial groove along the full length of bearing. It
has higher load carrying capacity compared to circumferential oil-groove
bearing. It is more susceptible to vibrations. It is used for gearboxes
and high-speed applications. Different patterns of oil-grooves are also
made by combination of cylindrical and circumferential grooves.
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 |
Fig. 11.6.2 Cylindrical oil-groove in sliding contact bearing
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11.7 ROLLING CONTACT BEARINGS
|
A rolling contact bearing is called as
‘anti-friction bearing. It is an assembly of rolling elements (balls or
rollers) placed between the shaft and housing, maintaining radial space
between them. The bearing has usually two rings with hardened raceways
(outer and inner races), in between hardened steel balls or rollers
roll. These balls or rollers are called ‘rolling elements’ and are held
in angularly spaced relationship by a cage or separator. The rolling
contact bearing can be classified into ball bearings and roller bearings
based on the geometry of the sliding elements. Rolling contact bearings
are used to carry radial or thrust loads or the combination of both.
Rolling contact bearings are lubricated with grease. The friction
coefficients of rolling contact bearings are about 0.0013 to 0.0050.
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11.8 BALL BEARINGS
Single row radial deep groove ball bearings are most commonly used. The nomenclature of such bearing is shown in Fig.11.8.1.
The bearing consists of four parts: the outer ring, inner ring, the
balls and the separators. The separators prevent the balls from
colliding with each other.
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Fig. 11.8.1 Nomenclature of a deep-groove-ball bearing
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Conformity of radii of balls and raceways
|
For successful design of bearing, the conformity of the ball radius to the raceways radii is very important. Figure 11.8.2 shows an example of low conformity of ball to raceway.
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Fig. 11.8.2 Conformity of ball radius to raceway radius
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Increasing
the conformity (i.e., the radius of ball is increased so that it is
closer to the radii of the curvatures of the race ways) increases the
area of contact between the balls and raceway. This increases the
friction. However, the unit surface stress on the ball is reduced which
in turn supports a greater load. Thus, the selection of curvatures for
the raceways is a matter of design compromise between friction and load.
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Bearing
manufacturers establish their own conformity values based on research
and experience. Most commercial bearings have inner and outer raceways
curved to radii between 51.5 and 53% of ball diameter. When bearing is
loaded, elastic and plastic deformations of the balls and raceways
increases the conformity; then the balls do not have pure rolling
motion. As a result, a small amount of sliding always occurs between the
balls and raceways which affect both the frictional loss and life of
bearing.
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The
balls are inserted between the inner and outer rings by moving the
inner ring to an eccentric position. After placing the balls, the inner
ring is brought into the position of concentricity with the outer ring
and then the separator is placed on the balls. Bearings are available
with shields or seals to prevent dirt from entering and also to retain
grease.
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11.9 TYPES OF BALL BEARINGS
Various types of ball bearings are shown in Fig.11.9.1.
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Fig. 11.9.1 Ball bearings: (a) Deep-groove; (b) Angular-contact; (c) Self-aligning; and (d) Thrust
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Ball bearings can be broadly classified into the following:
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- Deep-Groove ball bearing
- Angular-contact ball bearing
- Self-aligning ball bearing
- Thrust ball bearing
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Deep-groove ball bearings
|
The widely used ball bearing to support radial load is ‘Deep-Groove ball bearing’ or ‘Conrad-bearing’ as shown in Fig.11.9.1(a).
They are primarily designed to support high radial load and moderate
thrust load. They have deep raceways that are continuous (i.e. there are
no openings or recesses) over all of the ring circumferences. This type
of construction permits the bearings to support relatively high thrust
load in either direction. In fact the thrust load capacity is about 70%
of the radial load capacity. A ball bearing primarily designed to
support radial load can also support high thrust load; because only few
balls carry the radial load, whereas all the balls can withstand the
thrust load.
|
The double-row deep-groove ball bearings have
two rows of balls rolling in two pairs of races. They have more radial
load capacity than that of single row bearings. In other words they are
smaller in diameter compared to single row ball bearings for comparable
radial load capacity. However, the proper load sharing between the balls
mainly depends on the accuracy of manufacturing.
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|
The angular contact bearings (Fig.11.9.1b)
are designed such that the centerline of contact between balls and
raceways is at an angle to a plane perpendicular to the axis of
rotation. This angle is called “contact angle”. The angular contact ball
bearing may be of single or two rows of balls. They are meant to carry
radial and axial load together or only axial load depending on the
magnitude of the angle of contact. The bearings having large contact
angle support heavy thrust. The groove curvature radii are generally 52
to 53% of ball diameter. Angular contact single row ball bearings have
high radial load and high unidirectional thrust load capacity than the
deep groove ball bearings.
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The contact angle is usually less than 40°.In the case of angular
contact ball bearings, one side of the outer race is cut to insert
balls. This permits the bearing to take the thrust load in only one
direction. Therefore, single row angular contact ball bearings are
generally used in pairs. In the case of double row angular contact ball
bearings (duplex), the balls can be arranged ‘back to back’ and face to
face’ or ‘tandem’ configurations (Fig.11.9.2).
The back to back and face to face duplex bearings can accommodate
radial load and axial loads in both directions. The tandem bearings can
accommodate radial load and heavy axial load in only one direction.
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 |
Fig.11.9.2 Duplex angular contact ball bearings
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Self aligning ball bearings
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For assembly of shaft and housing which cannot
be made perfectly coaxial, the self- aligning ball bearings are best
used. They consist of two rows of balls on a common spherical outer race
(Fig. 11.9.1c). In such bearings
the assembly of inner ring and balls can tilt in the outer ring. The
loss of load-carrying capacity is inherent in this construction, due to
non-conformity of outer raceway with the balls. This is compensated by
having large number of balls in the bearings. Self-aligning ball
bearings are used in top drafting rollers and main shaft of ring
spinning machine.
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Thrust ball bearings
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If the contact angle of angular contact
bearings exceeds 45°, it is classified as ‘thrust bearing’. The maximum
value this angle can assume is 90° . In such case, races are on the
sideways as shown in Fig.11.9.1(d).
Such a bearing cannot take any radial load, and is used only for thrust
loads. The shafts carrying bevel or worm or helical gears should be
mounted with thrust bearings, except the shafts carrying honeycomb
(Herringbone) gears or crossed helical gears of left- and right hands
placed alternatively along the shafts.
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11.10 BALL BEARINGS IN TEXTILE MACHINES
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Shafts of beater, cages, condensers, and fans
in blow room, coiler wheels (both top and bottom) of carding machine,
creel rollers in roving machine, gear-shafts, tension pulleys for
spindle tapes in ring spinning machine are mounted in ball bearings. In
some ring spinning machines, the neck-bearings of ring spindles are
fitted with ball bearings. In rotor- spinning machine, shaft of rotor is
mounted with ball bearings for rotor speed up to 60,000 rpm. Beyond
this speed, the life of ball bearings reduces drastically, and hence,
‘in-direct bearings drives’ should be used. In this system, the shaft of
rotor is driven by tangential belts and rests in a cradle formed by
four supporting discs. The discs act as the bearings for the rotor shaft
and are themselves fitted with ball bearings. To reduce the vibrations
of rotor, the outer circumference of each disc is fitted with a
synthetic-fibre ring.
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11.11 ROLLER BEARINGS
Roller bearings have an ideal line contact between the rollers and races
against the point contact exhibited by ball bearings. Because of the
greater contact area between the rollers and races, the load carrying
capacity of straight roller bearings is higher compared with ball
bearings of similar size. They are stiffer and have longer fatigue life
than comparable ball bearings, and costlier. Roller bearings require
almost perfect geometry for the raceways and rollers. A slight
misalignment will cause the rollers to skew and get out of line.
Straight roller bearings do not take thrust loads. For higher radial
load capacity, two or more rows of rollers may be provided. For mounting
the ring-spindles (neck bearing), roller bearings are used. The
different types of roller bearings are shown in Fig.11.11.1.
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Fig.11.11.1 Roller bearings: (a) Plain; (b) Tapered; and (c) Spherical
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Cylindrical or plain roller bearings
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They are the simplest types of roller bearings (Fig.11.11.1a).
The length to diameter ratio of rollers is from 1:1 to 3:1.The outside
diameter of roller is often crowned to increase the load carrying
capacity by eliminating any edge loading.
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Needle bearings
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For
limited radial space, needle bearings are used. In needle bearing, the
ratio between the roller length and roller diameter is very large
compared with plain roller bearing. There are two basic forms of needle
roller bearings. In one form, the needles are not separated, and in the
other form, a roller cage separates the needles. The bearing that does
not have the needle separator has a full complement of rollers and
therefore, can hold higher load compared with the bearings having roller
separators. However, the bearing with needle separator is capable of
operating at much higher speeds because the separator keeps the needles
from one another, preventing collision.
|
They are often used to support oscillating
shafts. The needles, in many cases are directly placed on the shaft
journal eliminating the necessity of inner ring. Needle bearings are
mainly lubricated by grease. For high load or high-speed application,
oil lubrication is required. Needle bearings are used for mounting
bottom-drafting rollers of drawing, combing, roving and spinning
machines and detaching rollers with inner ring placed on the journal.
They are also used on table bottom rollers, lap tension rollers,
circular comb shaft and nipper shaft of comber. Needle bearing used on
bottom drafting of ring spinning machine is shown in Fig.11.11.2.
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 |
Fig.11.11.2 Needle bearings on ring spinning machine
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In tapered roller bearings, the rollers are frustums of a cone shown in Fig.11.11.1(b).
They are arranged in such a way that tangents of raceways intersect in a
common apex point on the axis of the bearing as shown in Fig.11.11.3.
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Fig.11.11.3 Forces acting on a tapered-roller bearing
|
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Tapered roller bearings are capable of
carrying both radial and axial loads; but largely used for applications
where axial load component predominates. They are often used in pairs to
take the thrust load in both directions. Since the inner and outer race
contact angles are different, there is a force component, which drives
the tapered rollers against the guide flange resulting in heating due to
friction. Therefore, these bearings are not suitable for high speeds.
Tapered roller bearings are ideally suited to withstand repeated shock
loads. Multiple-row tapered roller bearings have high radial-load
carrying capacity.
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Spherical roller bearings
|
Spherical roller bearings are called as ‘Self-aligning roller bearings’. Spherical roller bearings shown in Fig.11.11.1 (c)
consist of two rows of spherical rollers, which run on a common
spherical outer race. The inner race can freely adjust itself to the
angular misalignment of shaft in the bearings due to mounting errors or
shaft deflection under heavy load. They are especially good against
heavy loads. Shafts of cylinder, lickerin, doffer, stripper roller and
calender roller are mounted in self-aligning roller bearings, which are
grease lubricated.
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| |
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11.12 MATERIALS OF BEARING
| The bearing material should have following characteristics from the service point of view. |
- High strength to sustain bearing load, high compressive and fatigue strength.
- High thermal conductivity to dissipate the heat quickly.
- Low coefficient of friction.
- Less wear and tear.
- Low cost.
- Bearing materials should not readily weld itself to the shaft material.
- Good corrosion resistance in case the lubricant has the tendency to oxidize the bearing.
- Good conformability. The bearing should
adjust to misalignment or geometric errors. Materials with low modulus
of elasticity usually have good conformability.
|
Cast
iron, brass and alloy materials viz., bronzes (copper-tin), Babbitt
(alloys of tin-copper-lead-antimony), copper-lead alloys and
aluminum-tin alloys are used for making sliding contact bearings. Rubber
and synthetic composite materials are also used for certain
applications (synthetic bearings).
|
The
materials for rolling contact bearings should have the capability of
being hardened to required level. They require high resistance against
wear and fatigue and stability up to 125°C. The inner and outer rings
and rolling elements are made from alloy steel based on Cr-Ni, Mn-Cr,
and Cr-Mo.
|
11.13 STATIC LOAD CAPACITY OF BALL BEARING
| Fig. 11.13.1 shows the forces acting on the inner race through the rolling elements, support the static load C0. |
|
|
Fig.11.13.1 Forces acting on inner race through rolling elements
|
|
| Considering the equilibrium of forces in vertical direction, the static load C 0 is balanced by the reaction forces from the inner race as, |
C0 = F1 + 2F2 cos y + 2F3 cos (2y) + 2F4cos (3y) + ............................................................ (11.14)
|
The basic static load rating or capacity (Cs)
is defined as a load that will cause a permanent deformation of 0.01%
of diameter of rolling element at maximum stressed contact region of any
element. The basic static load rating or capacity for rolling bearing
is related to types of material, hardness, numbers and diameters of
rolling elements and their contact angles.
|
|
|
11.14 DYNAMIC LOAD CARRYING CAPACITY
|
The surfaces of rolling elements and races
undergo fatigue failure while the machine is running which imposes limit
on the life of bearings. Therefore, dynamic load carrying capacity ( C)
of the bearing must be considered in designing the bearings. The
dynamic load carrying capacity is defined as the radial load in radial
bearings (or thrust load in thrust bearings) that can be carried for a
minimum life of one million revolutions by 90% of the bearings before
fatigue crack appears.
11.16 BEARING LOAD AND LIFE
|
|
11.17 COMPARISON OF BEARINGS
Load carrying capacity/types of load
|
| The load carrying capacity of various bearings is shown in Fig. 11.17.1. |
 |
Fig. 11.17.1 Load Characteristics of bearings
|
For hydrostatic bearing, the load
capacity is independent of speed, as constant thickness of fluid film is
presented by external pump throughout the operation. In the case of
hydrodynamic bearing, the load capacity increases linearly with speed.
Any point below this curve, such as point ‘M’, indicates that the life
corresponding to this load-speed combination is infinity. When the load
exceeds, (such as point, ‘N’), the fluid film breaks resulting in metal
to metal contact, lowering of the life of the bearing. Hydrodynamic
bearings are highly suitable for high load-high speed conditions from
the point of view of longer life. They can be used for journal speed up
to 5000 m/min.
|
For a
finite life of rolling-contact bearings, the load carrying capacity
should decrease with an increase in the speed of operation. At higher
speeds (above 3000 m/min. at the centre of the rolling elements), the
centrifugal forces acting on the rolling elements are considerable,
lowering the life of the bearings. Rolling contact bearings with the
exception of straight roller bearings are capable of supporting both
radial and thrust loads. Journal bearings can support only radial load.
|
Shock-Loads
|
Rolling-contact bearings are
vulnerable to shock loads due to poor damping capacity. The rolling
elements and raceways are subjected to plastic deformation under shock
loads or fluctuating loads leading to noise, heat and fatigue failure.
Hydrodynamic bearings are better suited for shock loads, which occur in
connecting rod or crank- shaft applications.
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Starting torque
|
Rolling contact bearings require a
lower starting torque compared to hydrodynamic bearings due to low
coefficient of static or starting friction. Hydrodynamic bearings
exhibit high starting friction due to metal to metal contact during
starting-up. Ball bearings are therefore suitable for applications where
the machines are started frequently. If there is comparatively light
load at the start, and if the load gradually increases with speed,
hydrodynamic bearings are better choice.
|
Power loss
|
Power loss is high during starting
with hydrodynamic bearings. While running with hydrodynamic bearing, a
full lubricant film is developed leading to lower dynamic friction and
less power loss compared to rolling-contact bearing.
|
Space requirement
|
Sliding-contact bearings require
more axial space, while rolling-contact bearings require more radial
space. Sliding-contact bearings require additional space for lubrication
system like pump (for hydrostatic bearings), filter, sump and pipelines
etc. The overall space requirement for rolling-contact bearings is much
less.
|
Precision of mounting of bearings
|
For precise location of the shaft
axis, rolling-contact bearings are preferred. The axes of shaft and
bearing are co-linear for rolling-contact bearings. In hydrodynamic
bearings, the journal moves eccentrically with respect to the bearing,
and the eccentricity varies with load and speed. Because of the
standardization and employment of close tolerances with rolling contact
bearings, they are preferred for cams and gears. Rolling contact
bearings can be used for mounting a shaft placed in any position, and
they offer wide versatility with respect to mounting because they are
supplied in special housings.
|
Noise level
|
The noise level for rolling contact bearings are higher compared to sliding-contact bearings due to metal to metal contact.
|
Cost
|
The cost of sliding-contact bearings
is much higher compared with rolling-contact bearing due to additional
accessories. The cost of maintenance is also high for sliding-contact
bearings due to the maintenance of lubrication system. From the economic
point of view, rolling-contact bearings are better choice.
|
Life
|
The life of rolling-contact bearings
is finite and less compared with journal bearings. A properly
maintained journal bearing has indefinite life. Dirt, metal chips, and
so on, entering the rolling contact bearings can limit their life
causing early failure. The journal bearings do not suffer from this
problem because foreign matter is either washed away by the lubricant or
becomes embedded in the softer bearing material.
|
Ease of inspection and maintenance
|
With rolling contact bearings, the
inspection and maintenance of bearings are easier than sliding contact
bearings. Lubrication with rolling contact bearings is easier with
prepackaged grease or with relatively simple oil systems. In addition,
rolling contact bearings give early warning of impeding failure signaled
by increasing noise. Journal bearings can suddenly fail without any
indication. In general rolling contact bearings are readily replaceable.
|
Sensitiveness to changes in temperature
|
Rolling-contact bearings are less
sensitive to temperature changes. The frictional characteristic of
hydrodynamic bearing is highly sensitive to temperature changes.
|
|
|
| 12.0 : CAM DEVICES IN TEXTILE MACHINES |
| 12.1 INTRODUCTION |
|
A cam device consists of two moving elements,
the cam and the follower, mounted on machine frame. The cam is a
curved-outlined or grooved machine element transfers a predetermined
specified motion to the follower while it oscillates or rotates. Cam
device is simple, versatile and compact, and almost transmit a great
variety of motion to a machine element. Cam devices are widely used in
textile machines such as movement of healds in looms, movement of
needles in knitting machines, belt shifting over cone pulleys in roving
machines and movement of ring rail, balloon control rings and lappets in
ring spinning machines.
12.2 CLASSIFICATIONS OF CAM MECHANISMS
Cam mechanisms can be classified based on
|
- Modes of input/output motion
- Configuration and arrangement of the follower
- Shape of the cam.
|
The combinations of motions of cam and follower are:
|
- Rotating cam and translating follower (cams used in looms, ring spinning machines)
- Rotating cam and follower as an arm swings with respect to its pivot point (cam in comber).
- Translating cam-translating follower
- Stationary cam-rotating followers that also oscillate (cam in knitting machine)
|
The
configuration of follower may be in the forms of knife-edge, roller,
flat, oblique flat and spherical. Classification of follower can also be
done based on the arrangement of follower with respect to the cam as
‘In-line follower’ and ‘Offset follower’. Based on the shape of cam,
plate or disk cam, grooved cam, cylindrical and end cam can be
classified. In disk cam device, the follower moves in a plane normal to
the axis of the rotation of cam shaft. In grooved cam device, the
follower rides in the groove on the face of the cam. In a cylindrical
cam, the follower operates in the groove cut on the periphery of a
cylinder. The cam device used in roving machine to shift belt on cone
pulleys is of this type. A cord attached to the groove of the cam
unwinds as the com rotates intermittently.
|
Depending
on the requirement of motion of the element in a machine, the cam
profile of cam is designed so that the follower is made to move with
constant velocity or constant acceleration or harmonic motion or desired
motion.
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|
12.3 CAM DEVICE FOR COP BUILDING IN RING SPINNING MACHINE
The
schematic diagram of cam device and other related mechanisms for
controlled winding of yarns on plastic tube mounted on ring spindle can
be found in the NPTEL course in the following URL:
|
| http://iitmweb.iitm.ac.in/phase2/courses/116102038/ (R. Alagirusamy’s course) |
| The desired profile of cop is given in Fig 12.3.1 . |
|
|
Fig. 12.3.1 Profile of a ring cop
|
|
The
base building (belly built-up) of cop is mainly controlled by a
mechanism of projector mounted on an oscillating disk which superimposes
its motion on the basic motion derived from the profile of cam. For
details of this mechanism, the above mentioned course can be referred.
|
The plastic tube on which the yarn is wound is conical. Figure 12.3.2
represents schematic representation of laying winding coils on to the
cop. The base of cop is shown in grey and white colour. Each of the
winding coils laid during formation of cylindrical portion of the cop is
shown in separate colour. For simplicity, the binding coils are not
shown and the plastic tube is shown as cylindrical. These
simplifications do not alter significantly the following discussion.
|
 |
Fig.12.3.2 Schematic representation of laying winding coils during built up of cylindrical portion of ring cop
|
|
It shows that the pitch of yarn coils (p)
must be same in order to achieve proper shape to the cop, uniform
density of packing the yarn and to obtain maximum yarn content in cop
(not a too thin package). The conicity of cop greatly increases during
building the base of cop compared with the conicity of plastic tube.
Hence, during the upward motion of ring rail, the yarn coils (winding
coils) are wound on decreasing diameter of cop from bottom to top; the
lowest diameter of yarn coil (ds) being the bare diameter of plastic tube. The maximum diameter of yarn package or the yarn coil (dl) can be set about 0.9 to 0.95 times the ring diameter.
|
Time required to wind one coil of yarn (ti) is related to package diameter (di) and yarn delivery rate (k), the later is constant. This can be expressed as
|
|
|
| The pitch of yarn coils is related to velocity of the rail (vi) during its upward motion |
|
|
| Combining these two equations, we get |
|
|
| Where, K is a constant, equal to Π/k. |
| When the velocity of rail while rising varies from v1, v2, v3 while winding on to the package with decreasing diameters d1, d2, d3 respectively, the above equation can be written as |
|
|
Since, d3< d2< d1; then, v3> v2> v1.
Hence the velocity of rail increases while rising and the displacement
of cam follower must increase for every angular rotation of cam. The
profile of cam transferring motion to ring rail is given in Fig. 12.3.3 .
|
 |
Fig. 12.3.3 Profile of cam for building yarn package in ring spinning
|
|
The measured profiles of displacement, velocity and acceleration of cam follower for one rotation of cam are given in Fig. 12.3.4 .
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|
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|
|
Fig 12.3.4 Profiles of displacement, velocity and acceleration of cam
follower for one rotation of cam for cop building in ring spinning
|
|
Since the cam follower transfer
motion to ring rail through lever and rollers, the above profiles must
be slightly enlarged by some magnitudes. It is observed that the winding
coils are laid corresponding to cam’s angular rotation, 0º to 240º
while the ring rail rises with increasing velocity. The binding coils
are laid corresponding to cam’s angular rotation, 240º to 360º while the
ring rail receding with decreasing velocity. The pitch of yarn binding
coils must be twice that of winding coils. The pitch between binding
coils must be same. This is because, the time required winding the
binding coils must increase from top to bottom of chase (with increasing
diameter of package) and the ring rail moves with decreasing velocity
in accordance with increasing package diameter.
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12.4 CAM DEVICE FOR SHIFTING OF BELT ON STRAIGHT CONE PULLEYS IN ROVING MACHINE
The belt shifting mechanism and the arrangement of cam are discussed in section 6.3.3
of Module 6 (Design of cone pulleys). The cam is of grooved type with
grooves cut on its periphery. The cord acts as follower rests in the
groove of cam and transfer motion to the belt. The cam and cord
transferring motion to belt are shown in Fig. 12.4.1 .
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 |
Fig 12.4.1 Cam for shifting of belt on cone pulleys in roving machine
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The cam rotates in clockwise direction by some degree (say θ=
1º) at the end of winding of each layer of roving, thus releasing the
cord and shifting the belt so that required diameters of cones are
selected. This reduces the rotational speed of winding. The degree of
rotation cam can be controlled by ratchet wheel depending on roving
thickness.
|
When the rotation of cam in radians (θ) for each layer of roving is small, the peripheral length of cam or the corresponding length of cord released (S2’’’) for winding second layer of roving is related to the initial radius (R1) and radius of cam (R2) after rotation by θ can be written as
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|
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The geometry of the system comprising the cam, cord and supporting roller are given in Fig. 12.4.2 .
|
 |
Fig. 12.4.2 Geometry of cam device and supporting roller
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The length of cord from the tip of cam and apex point of roller while winding the first layer of roving is S1’’ and that during winding the second layer is S2’’.
While winding the second layer of roving, the vertical distance between
the apex point of roller and cam tip increases from the initial value, h1 to h2 as
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|
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The actual length of cord released (S2’), at the beginning of winding second layer of roving and the corresponding belt shifting (S2) are related as
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The belt shifting length (S2) for winding second layer and so on the m th layer can be found out knowing the roving thickness; the protocol for the same and notations are given in Module 6. In the above equation, R1, h, l, S2, cosγ and S1’’ are known. When θ is very small (one degree), the unknown parameter R2,
can be solved. Similarly, the shifting of cord and belt required at the
beginning of winding 3rd layer of roving can be written as
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|
|
The radius of cam while winding the third layer R3 can be found as the value S3
is known. The radii of cam corresponding to angle of rotation can be
calculated to generate complete profile of cam. If the cam profile is to
be generated up to 720º rotation, then the roving thickness t can be found (based on equation 5.22 Module 5) from the maximum and minimum diameters of roving bobbin (d max and d min) as
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Accordingly, the number of shifting
of belt or layers of roving to be wound is 720 with 1º angular rotation
of cam for each layer of roving. If the minimum and maximum diameters of
bobbin are 48 mm and 170 mm respectively, the thickness of roving would
be 0.085 mm (85 microns). This indicates that precise profiling of cam
and cones could be generated for precise belt shifting.
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|
| 13.0 : BALANCING OF MACHINES |
13.1 UNBALANCE
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|
Unbalance is the
unequal distribution of weight of a rotor about its rotating axis. When a
rotor is unbalanced, it imparts vibratory force or motion to its
bearings due to centrifugal forces of the rotor. Unbalance causes
vibrations on machine. The consequences are:
- Reduced bearing life
- Inconsistent product quality
- More noise
- Slight increase in energy consumption
- Increased maintenance costs
- Fatigue failures of support structures
- Increased structural degradation.
- Operator fatigue
A properly balanced machine has less vibration. Thus, the machine must be balanced.
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13.2 CAUSES OF UNBALANCE
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Machine manufacturer must
minimize the cost of manufacturing. Perfect manufacturing results in
escalated cost and many times it is not possible to do so. Machine
manufacturer usually resorts to ‘less than perfect manufacturing’ of
machine components. The followings must have been compromised during
manufacturing of machine which creates unbalance on the machine :
|
| (1) |
Porosity in the element/components, especially in castings.
|
| (2) |
Eccentricity – the shaft journal is not concentric with the rotor.
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| (3) |
Shifting of mass center of rotor on tightening support elements.
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| (4) |
Presence of keys and keyways in rotor shaft creates a built-in unbalance.
|
| (5) |
Loose parts like dirt,
water, or welding slag moving around in a hollow place (typical examples
are: tin roller of a ring spinning machine and drum type beaters).
|
| (6) |
Asymmetry of a rotating part (motor windings).
|
| (7) |
Cracks developed in rotor or shaft.
|
| (8) |
Deformation of shaft (shaft bow) due to the relaxation of residual stresses
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|
Further, a machine properly balanced after installation can go out of balance during its service life due to:
|
| (1) |
Changes in
the mass distribution of some elements during maintenance activities
such as cleaning, drilling, grinding, changing fasteners may results in
change or mix of bolts, nuts and screws etc.
|
| (2) |
Bending of shaft due to thermally induced distortion or gravity.
|
| (3) |
Corrosion and erosion of machine elements
|
| (4) |
Deposit buildup on various elements of machine
|
| (5) |
Changes in geometric axis of rotation due to wear at the journal or bearing
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|
. | |
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13.3 PRODUCTION AND FIELD BALANCING
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There are several ways of
correcting excessive vibrations in machines. Machine balancing is one of
the approaches to reduce machine vibrations. In sophisticated balancing
methods, vibrations on the machine are measured and weights are added
or removed to adjust the mass distribution around the rotor, thereby
reducing vibration.
|
Production/shop balancing is
done at the machine manufacturer’s factory after assembling various
machine parts. Unbalance of machine elements or machine might occur due
inaccurate manufacturing of components resulting from faulty design,
selection of inappropriate materials and variation in form and fit of
components and the whole assembly.
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Even if the above factors
are taken care, while building-up a machine, an imbalance is expected.
For example, crankshafts due to their non-symmetrical shape and motor
windings due to the difficulty in winding the coils symmetrically
introduce mass unbalance. To assemble various elements of a machine
without excessive force, there must be some clearance between them, and
this clearance varies within a tolerance range. The centre of gravity of
each rotating assembly varies with shape and size of mating parts and
thus, the balance condition of machine that is to be built.
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But the manufacturing
practice followed in industry is to manufacture machine elements as
quickly as possible and economically, assemble the moving parts, and
then apply a correction for smooth operation in the later stages of
manufacture. This is the main reason for production balancing.
|
A machine that is properly
balanced during manufacturing can also develop an unbalance while the
machine is in use as discussed in section 13.2. For these reasons, in situ balancing is performed on installed machines. This is called ‘field balancing’
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. | | | |
13.4 MEASURES OF UNBALANCE
|
Unbalance can be
visualized in terms of two quantities. Imaginary heavy spot on the
rotor; and vibratory forces caused due to centrifugal forces of heavy
spot. These are discussed briefly.
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Imaginary heavy spot
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Fig. 13.4.1 shows a rotor having two voids present at radii r1 and r2. Let, the deficiency of mass at each voids are represented as -m1 and -m2. These correspond to presence of additional masses opposite to the voids on rotor, m1 and m2.
Two vectors could be drawn to represent the voids. Each vector
originates at the center of the rotor and points away from the void that
its represents. A heavy spot present at a larger radius would create
more unbalance than the one at smaller radius for a given mass of heavy
spot. The magnitude of each vector (length) corresponds to the product
of additional mass on that side and the radius of cg of additional mass (m1r1 and m2r2).
The resultant vector defines the imaginary heavy spot. The magnitude of
imaginary heavy spot corresponds to the length of resulting vector.
Note that the unit of heavy spot is g-cm.
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|
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Figure 13.4.1 Rotor with two voids and the vector sum representing an imaginary
heavy spot
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This process of defining a
single heavy spot can be extended to any number of voids or variations
in mass density of a rotor. In general, unbalance is distributed
throughout a rotor, but, theoretically, all of the unequal weight
distribution in a thin, rigid body can be combined into a single
imaginary heavy spot. This heavy spot pulls the rotor around with it,
causing bending on the shaft, and also exert cyclic forces or vibrations
on the bearings. These vibrations are measured for balancing. The first
step in balancing is to find out amount of heavy spot and its location.
Then an equal counterbalancing weight is placed 1800
opposite the heavy spot. Alternatively, a weight is removed at the
location of the heavy spot. After properly balancing, the rotor runs
smoothly without vibration.
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Centrifugal force of heavy spot
|
Centrifugal force is the
operative force that causes vibration due to unbalance. Unbalance exists
in a rotor whether it is rotating or stationary. The centrifugal force
due to a heavy spot becomes active only when rotor starts rotating. The
centrifugal force due to heavy spot is
|
Fc = mrω2 ...............................................................................(13.1)
|
Where
|
m = mass of the heavy spot |
|
r = radius at which the heavy spot is located from the axis of the rotor |
|
ω = Angular velocity of the rotor. |
|
When the speed is doubled, the centrifugal force quadruples.
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Relationship between unbalance and vibrations on bearings
|
The centrifugal force from an
unbalanced rotor while spinning transmits vibration on the stationary
supports (bearings). The vibration transmitted to the bearing is
oscillatory. This vibration is dependent on speed, the mass of the rotor
and the stiffness of bearing supports. Vibration transfer both force
and motion on the stationary supports. The unbalance is measured by
measuring the vibrations on the stationary supports. Vibration amplitude
and phase angle are the two physical quantities associated with
vibration. Accelerometers, velocity pickups and proximity probes and
other sensors are used to measure the former; while strobe and trigger
sensor methods are used to measure the phase angle. The unbalance ‘U’
can be calculated directly from the acceleration measurements of
vibration. When the rotor is rigid and supported in rigid bearings, the
unbalance at each plane can be determined from the following formula
|
U = Ma/ω2 ............................................................................(13.2)
|
Where
|
M = Total mass in motion |
|
a = Acceleration |
|
ω = Angular velocity of the rotor. | | |
|
. |
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13.5 STATIC BALANCING
|
The static balancing is the earliest method of mass balancing. This type of balancing is shown in Fig. 13.5.1.
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Click on Image to run the animation
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Animation 13.5.1 Statically unbalanced rotor favouring the heavy spot to the bottom
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 |
Figure 13.5.1 Static Balancing of rotor placed on rigid rollers
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To balance a rotor, the rotor
mounted with its shaft is allowed to roll on two hard and smooth rollers
or knife-edges. The rotor rolls such that heavy spot roll to the bottom
due to gravity. A weight is then added to the top (opposite to the
heavy spot) or, some amount of material is removed at the heavy spot
either by drilling or grinding. The amount of unbalance of mass is
unknown, but location of unbalance is known. By trial and error,
material is added or removed until no spot on the rotor favors the
bottom. The balanced rotor could be rolled into any position and
released, it would remain stationary there.
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Static balancing is still widely practiced today. It is very
effective for thin disks and slow speed rotors. Static balancing of
rotor can be performed either when the rotor is removed from the machine
or in its place. In the first approach, the rotor along with its shaft
is rolled over horizontal, smooth, and hard supporting rollers or tandem
rollers as described above. The gravity moves the heavy spot to the
bottom. In the second approach, the rotor is disconnected from belts or
gears and rolled. If any heavy spot is there, it rolls to the bottom.
13.6 STATIC UNBALANCE
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Static unbalance is the result of displacement of principal
mass axis of rotor with respect to the shaft axis. The principal axis
passes through the center of gravity (cg.). Due to clearance between the
journal and bearings, the shaft axis is not the same as rotational
axis. As a result, some machine parts mounted on poor quality bearings
can’t be balanced below a certain level. Fig.13.6.1 shows
the presence of static unbalance on a disk (top), and drum(bottom). In
static unbalance, the heavy spot and the center of gravity of the rotor
are in the same plane
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 |
Figure 13.6.1 Static unbalance on rotor: Top- disk/pulley; Bottom- drum/cylinder/ beater/tin roller
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A driven pulley mounted on a beater shaft has a serious static unbalance due to defect in casting ( Fig.13.6.2). By removing materials from the pulley, the pulley is statically balanced.
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 |
Fig. 13.6.2 Statically balanced V-pulley
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In a pure static unbalance
condition, the vibration amplitude and phase measurements will be
identical at both bearings. This can be corrected with a single mass
placed 1800 opposite to the heavy spot. Using
knife-edge method, pure static unbalance can be detected and corrected
if the unbalance is moderate to large. However, the knife edge method of
static balancing is ineffective for heavy rotors, high-speed rotors, or
where finer balance grades are required. This is because a small
residual unbalance does not create a large enough turning moment to
overcome friction. This method is also not suitable for detecting couple
unbalance present in multi-axial rotors. But this method of balancing
is still widely used, as this method is very effective under certain
situations. In addition, it does not require any instrument. The
location of heavy spot is evident from the part itself.
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Pure static unbalance is rarely found. But, every unbalance
condition has some component of static unbalance. Static unbalance is
most common form of unbalance, and if is compounded with couple
unbalance in long rotors leads to dynamic unbalance.
13.7 QUASI-STATIC UNBALANCE
|
A quasi-static unbalance condition is shown in Fig.13.7.1.
Quasi-static unbalance exists when the principal axis intersects the
shaft axis at a point other than the cg of the rotor. (i.e., the
principal axis is not parallel to the shaft axis).
|
 |
Figure 13.7.1 Quasi-Static unbalance
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Quasi-static unbalance is
very common on motor pulley combinations where the motor is balanced,
but the pulley is unbalanced. It can be corrected with a single
correction weight on the plane of pulley (Single-plane balancing).
Quasi-static unbalance could be considered as an impure static
unbalance. Even though it is a single-plane unbalance, it causes a
dynamic effect with both static and a couple components
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13.8 COUPLE UNBALANCE
|
Pure couple unbalance is shown in Fig.13.8.1. Pure couple unbalance rarely occurs, but leads to dynamic unbalance. In couple unbalance, two equal heavy spots exist at 1800
apart on opposite ends of the rotor. The two heavy spots do not
actually have to be on ends of the rotor. They can be anywhere along
axial direction of the rotor, but the greater their separation, the more
unbalance effect they produce.
|
 |
Figure 13.8.1 Pure Couple Unbalance
|
Rotor having pure couple unbalance (as shown in Fig. 13.8.1 )
will not turn on knife-edge supports, as they are statically balanced.
However, when the rotor rotates, the centrifugal force from the two
heavy spots causes the rotor to oscillate in a conical manner. The
vibration measured at the bearings will be equal in magnitude but 180 °
out of phase for the bearing supports of equal mass and stiffness. This
implies that when one bearing is moving up, the other is simultaneously
moving down. When a bearing on the left side of rotor moves forward, the
one on the right side moves backward. Couple unbalance can only be
detected, while the rotor is rotating
|
| 13.9 COUPLE UNBALANCE IN CARD CYLINDER |
A single mass, can never correct couple unbalance presents on a
rotor. A minimum of two masses is required in two correction planes. On
rigid rotors, the locations of the correction weights do not have to be
in the same plane and radial position as the heavy spots. But they must
create couple effects identical to that of unbalance. This is depicted
in Fig. 13.9.1 for a card cylinder.
|
 |
Figure 13.9.1 Correction weights and heavy spots in different planes for a rigid rotor
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Two heavy spots, each 60 g located at radius 25
cm are separated by one m along the axis of cylinder. Each would have
an unbalance of 1500 g-cm (60 x 25). The couple effect due to this is
(1500 g-cm x 100 cm) = 150 kg-cm2. If the
correction weights were to be located on the surface of cylinder 50-cm
apart, then they require to have a magnitude of 150/50=3kg-cm (i.e.,
couple divided by the distance between the correction weights along the
axis).
|
The correction weights can be
placed anywhere on the cylinder, as long as they create the same couple
effect of the heavy spots and placement of them in such places would not
hinder the function of the cylinder. Placing correction weights on the
base of the carding cylinder can never be considered as they hinder the
carding process. They could be placed at smaller radii. For example two
correction weights, each of 150 g at 20-cm radius or 200 g at 15-cm
radius, separated by 50-cm would balance the cylinder.
|
It is also possible to place
one correction weight on the rotor between the bearings and other could
be outside the bearings on a separate disk mounted on the same shaft,
like a pulley. Both correction weights could be outside the bearings (on
pulleys mounted on the rotor shaft) as long as they produce the same
couple effect to balance the turning tendency of heavy spots.
13.10 DYNAMIC UNBALANCE
|
Any rotating body may be considered as composed of large
number of thin disks mounted on the shaft with their centre of gravity
(cg) not perfectly coincident with the rotational axis. Fig. 13.10.1
illustrates this as condition of dynamic unbalance. This is something
like that disks are drilled with holes not at their centers but each is
offset to a different extent from the center and all are mounted on
shaft. Such a rotor has a combination of static and couple unbalance
present in varying degrees. This is called dynamic unbalance, where the
principal mass axis and the rotating shaft axis do not coincide.
|
 |
Figure 13.10.1 Dynamic unbalance by improperly mounted disks on the shaft
|
The axial distribution of
disks will cause bending forces in the shaft, in addition to a static
unbalance on each disk. If the rotor is rigid enough, it will not bend,
however, the cumulative effect of couple unbalance will cause a turning
moment trying to topple the rotor end over end. This turning force
causes vibration at the bearings. To properly detect the couple
component of unbalance, the rotor must be rotating. The vibration
measured on the left and right side bearings are generally unequal in
amplitude, and the phase difference is neither 0° nor 180° exactly, but
somewhere in-between them.
|
Dynamic unbalance can be
corrected on rigid rotors by two weights in two separate planes. The two
weights are not exactly 180° apart because they must also compensate
for static unbalance. After performing two-plane balancing, the amount
of static and couple unbalances that were originally present in the
rotor could be determined by the angular separation of two correction
weights. If they end up on the same side, mostly static unbalance was
present. If the two-correction weights are nearly 180 ° opposite, mostly
the couple unbalance was originally present. Usually, the two weights
are unequal in weight and their angular separation lies between 00 and 1800 to compensate for both the static and couple components
|
|
| 13.11 DYNAMIC UNBALANCE IN AN OPENING ROLLER |
|
An opening roller used in one of the machines of a blow room had a
dynamic unbalance. Placing two correction weights (unequal) on the
periphery of the roller with angular separation about 500 could balance the roller.
|
|
|
In most cases, the static component of unbalance is more important
than the couple unbalance because a static unbalance mass will usually
cause a greater disturbance in vibration than two equal unbalances in
opposite directions.
|
|
|
. |
| 13.12 DYNAMIC UNBALANCE IN A GROOVED WINDING DRUM |
|
The grooved winding drums used in yarn clearing operations have
dynamic unbalance due to variation of mass distribution in the axial and
radial planes exhibiting different amount of static and couple
unbalance ( Fig.13.12.1). The rotor
must be dynamically balanced by doing corrections only at the two
extreme ends of drum, since adding correction weights or removal of
material is not possible at other places as the yarn would be
traversing. Materials are removed by drilling at few spots both at the
left and right edges of drum in order to dynamically balance the grooved
drum.
|
 |
Fig. 13.12.1 A dynamically balanced grooved winding drum
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|
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13.13 PLANE TRANSPOSITION
|
|
Plane transposition is a process of moving the two
correction weights on rigid rotors along the axis. This method is useful
when correction weights could not be placed on certain locations as
observed in the case of grooved drums. Plane transposition could be
carried on microprocessors controlled balancing machines. In the first
step, two correction weights are calculated and the planes on which they
have to be placed to balance the rotor are determined. The next step is
to select two separate planes (convenient to place for the correction
weights that do not affect the regular operation of machine) along the
rotor axis. Third step involves calculation of the two new correction
weights required for the newly selected planes.
|
.
13.14 BALANCING OF A CYLINDER
|
For example, it is possible to correct an unbalance condition in
cylinder of carding machine by placing correction weights on the
exterior pulleys, after assembling the machine. This will be easier.
Consider a cylinder of width 0.9 m, supported by four supporting
elements equally spaced along on the shaft. Due to inaccuracy of the
casting of one of the support, it shows 10-g unbalance in its plane on
the base of the cylinder as shown in Fig. 13.14.1.
Ideally a correction weight of 10-g should be placed on the same plane
180 ° opposite to the heavy spot on that support element itself. But
this cannot be done, as it is not accessible without dismantling the
cylinder base. One can place the correction weights on the pulleys, i.e.
pulley 1 (left side) and 2 (right side) and can balance the cylinder.
Let us assume that both radii of the cylinder and pulleys are 30 cm.
Other dimensions are given in the figure. One 10-g weight (A) is placed
on the cylinder to correct the static balance of the cylinder.
|
 |
Figure 13.14.1 Plane transposition in balancing a cylinder
|
Obviously, this is not
possible, and this correction weight must be moved to one of the
pulleys. Two 10-g weights can be placed on the pulley 2,1800
opposite each other. (B, placed at the bottom and C, on the top of the
pulley). This does not affect the static balance condition on the
pulley. A and B form a couple of 13500 gcm2
|
= [(10 * 30 * 22.5) + (10 * 30 * 22.5)]
|
Note : Couple = Mass x radius x axial separation about central plane. This couple can be replaced by placing weights ‘y’, one, on top of the pulley 1 and the other on the bottom of the pulley 2. Therefore,
|
13500 = [(y * 30)(15 + 30 +30 +22.5) + (y * 30 * 22.5)] = 3600 yg - cm2
|
y = 3.75 g
|
Now weights on the pulley 2
are 10-g on top (original static correction weight) and 3.75- g on
bottom. This is equivalent to a weight of 6.25g placed on top of the
pulley 2 (i.e. 10 – 3.75).
|
Finally, a balance weight of
3.75g on top of pulley 1 and 6.25g on the top of pulley 2 would balance
the cylinder statically (10 = 3.75 + 6.25).
|
The cylinder is also
dynamically balanced, since both the static and couple unbalances are
corrected. Then the moment about any normal plane of rotor must be zero.
This can be verified with respect to the plane XX. Taking moment of
inertia forces about the plane XX :
|
- (3.75 * 30 * 15) - (10 * 30 * 60) + (6.25 * 30 * 105)
|
| = - 1687.5 - 18000 + 19687.5 = 0 |
|
|
13.15 TRIAL WEIGHTS
|
|
A trial weight is temporary weight placed on the rotor for
the purpose of calibrating the vibration measuring instruments. This
must not be confused with correction weight. Trial weights are placed on
the rotor during balancing process, not in a trial-and-error fashion,
but to add to the exiting unbalance and modify the exiting forces. Rotor
is spun at operating speed or balancing speed (on a balancing stand)
and vibrations are measured. With trial weights placed on the rotor,
again, vibrations are measured while the rotor is spinning at previous
speed(s). From these measurements, calibration of vibration measuring
instrument is carried out. Further, this can be used to calculate the
correction weights. The trial weights should be removed from the rotor,
once the final correction weights are put in place.
13.16 RUN OUT
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Run out is the total linear displacement measured on the outside
diameter of rotor when it is turned using dial indicator. Both the run
out and mass unbalance are separate and independent quantities. A
noncircular part (cam), as shown in Fig. 11.16.1 (left),
can be well balanced (by adding balance weight) to run smoothly but it
would measure a very large run out. On the other hand, a perfectly round
disk (having zero run out) but having a heavy spot (due to bolt, nut
and washer) might have a serious unbalance ( Fig 11.16.1 right).
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Figure 11.16.1 A balanced cam with large run out (left); Disk with zero run out but with serious unbalance (right)
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Rotors such as,
lickerin, cylinder and doffer of carding machine, and drafting rollers
must have negligible run out for smooth operation. A non-circular
drafting roller creates periodic fault. For example, if it is
elliptical-shaped, then the periodicity (wavelength) of drafted fibre
stand would be one-half of the circumferences of that roller. All these
elements should be mass balanced preciously in addition to having very
low run out. For rotors that do not transmit power over their
circumference, such as fans and pump, only mass balancing is sufficient
to ensure smooth operation. The permissible radial run out for plastic
bobbin (tube) used in ring spinning is 0.2 mm.
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Axial run out on a disk, creates a couple leading to a couple unbalance condition (refer Fig. 13.16.2). The centrifugal forces Fc around the disk are not in same plane and are separated by a distance (d).
This couple unbalance is felt at the bearings and that requires
two-plane balancing. This unbalance effect becomes significant on large
diameter rotors or high-speed rotors.
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Figure 13.16.2 Couple due to axial run out of disk
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13.17 UNBALANCE DUE TO ECCENTRICITY IN MOUNTING THE SHAFT
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If the cg of a perfectly
round and perfectly balanced rotor is displaced from its rotational axis
and forced to rotate in that condition it will go out of balance ( Fig 13.17.1).
This happens if there is a clearance between the bearing inner surface
and journal surface. This shifts the journal axis away from the bearing
axis by an amount e called eccentricity which is half that of clearance. The unbalance of rotor of mass m, due to eccentricity (e) in mounting the rotor shaft in the bearing is
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U = em .............................................................................(13.3)
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| When e/r is very small |
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Figure 13.17.1 Eccentricity due to displacement of the rotating axis of shaft in sliding contact bearing
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Assume that a perfectly
round and balanced thin disk is mounted on shaft whose axis is displaced
from that of bearing by 0.005 mm. If the disk and shaft together have a
mass of 150 kg, then the unbalance (U) is
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U = em = 0.0005 x 150000 = 75 g-cm.
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It is obvious that the most
fundamental principle of balancing is the direct relationship between
the unbalance of a rotor and the displacement of its center of gravity.
13.18 UNBALANCE DUE TO NONUNIFORM MASS DISTRIBUTION
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The centre of gravity of a perfectly round disk could also be
shifted from the center of rotation due to ‘less than perfect’
manufacturing processes. This might be either the disk is inhomogeneous
(non-uniform mass distribution), or the hole that is drilled on the disk
for the shaft to pass through is not made exactly at the center of the
dick. A perfectly round and thin disc of radius r, having
inhomogeneous mass distribution, exhibits a heavy spot on the right
side. The rotor is balanced by placing a correction weight (Wc) on the left side ( Fig. 13.18.1).
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Figure 13.18.1 Eccentricity due to displacement mass cg of a disk
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Before balancing the rotor, the cg of rotor moved towards right by a distance e (eccentricity) from the bearing/rotational axis. The centrifugal force due to unbalance (F U) of spinning rotor is
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FU = Weω2 / g ..............................................................................(13.4)
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Where
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W = Weight of rotor
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ω = Rotational speed of rotor
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g = Acceleration due to gravity.
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After balancing the rotor, the centrifugal force due to a correction weight Wc placed at a radius rc from the center is
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FWC = Wc rc ω2 / g ........................................................................(13.5)
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Where FW c is the centrifugal force due to correction weight.
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If the rotor is perfectly balanced by the correction weight, then
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FU = FW c ..................................................................................(13.6)
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Hence, M e = Mc rc ...................................................................(13.7)
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Where M and Mc are the masses of rotor and correction weight respectively.
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From the above equation, it is
clear that the required correction weight is a function of the rotor
weight multiplied by the displacement of its cg. Static unbalance of a
rotor is the result of displacement of its cg from its rotating axis.
Anything that causes a variation in the displacement of the cg of rotor
from its rotational axis will introduce unbalance effects. Therefore for
rotating parts, manufacturing tolerances and balancing tolerances are
inseparable.
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Rearranging the Eq. (11.3) and (11.7), and if rc = r, we get
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e = (U/M) = ( Mcrc/M) = ( Mcr/M) ..................................................(13.8)
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This is called the “specific unbalance per unit mass of rotor” or “eccentricity”.
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13.19 DYNAMIC BALANCING OF SINGLE PLANE ROTOR
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Fig. 13.19.1 shows a rigid single plane rotor consisting of three masses (M1, M2 and M3), rotating all in the same normal plane, about the axis.
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Figure 13.19.1 Dynamically-balanced single-plane rigid-rotor
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A fourth mass, MC(from
the correction weight to balance the rotor) is added to the system so
that the sum of the inertia forces (oscillatory) is zero and thus, the
rotor is balanced. For constant rotational velocity of the rotor, ‘ω ’,
the inertia force (Fc) for any given mass (M), placed at a radius of r from the axis, is (FC) = Mrω2
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For dynamic balancing of the rotor, the vector sum of the inertia forces (ΣFc ) of the system is zero.
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| Where WiMiand r i.are the weight, mass and radii of the ith element respectively. |
If the system is balanced by a correction weight (Wc) placed at a radius rc, then
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Where θi and θc are the phase angles of ith element and correction weight, counter-clockwise measured from x-axis.
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Any number of masses rotating in a common radial plane may be
balanced with a single mass (a single correction weight). If the mass
and radii of centre of gravity of all elements with their phase angles
are known, then one can calculate the product of correcting mass and its
radial position and phase angle using the above two Eq. (13.13) and (13.14). The unbalance can also be determined analytically by summing x- and y- components. This is illustrated in the following Table 13.19.1.
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Table 13.19.1 Balance of inertia forces
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From this, the weight times the radius of correcting element (i.e. the resultant force R) can be calculated as,
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13.20 DYNAMIC BALANCING OF MULTI-PLANE ROTOR
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Consider a case in which the rotating masses of a rigid rotor lie in a common axial plane shown in Fig. 13.20.1. In this case, the inertia forces are parallel vectors.
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Figure 13.20.1 Dynamically-balanced multi-plane rigid-rotor
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Balance of inertia forces is achieved by satisfying the equation ( 13.11).
However, balance of the moments of inertia is also required to
dynamically balance this rotor. In the case of single plane rotor ( Fig.13.19.1), moment equilibrium is inherent since the inertia force vectors are concurrent. However, in Fig.13.20.1,
it is oblivious that the inertia forces are not concurrent when viewed
in the axial plane. In order to balance the moments, the moment of
inertia forces about an arbitrarily chosen axis normal to the axial
plane (selected at O) must be zero, i.e.
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| Where, ac is the moment arm of correction weight about the normal plane at O. |
Tabulation method may be used to calculate the resultant unbalance, (ΣWr) and aC. This is shown in the following Table 13.20.1.
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Table 13.20.1 Balance of moments
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In Table 13.20.1, upward Wr values are taken as positive. The counterclockwise values of Wra are positive. The distance aC from the moment centre O locates the line of action of correction weight (R). To satisfy the Eq. (13.11) and (13.17), the equilibrant (ΣWr ) is equal, opposite, and collinear with R. Then,
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13.21 DYNAMIC AND STATIC BALANCING
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Fig. 13.21.1 shows a rigid rotor with shaft laid on horizontal parallel ways.
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Figure 13.21.1 Dynamically unbalanced single-plane rigid-rotor
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If the rotor is statically balanced, it will not roll under the
action of gravity, regardless of the angular position of the rotor. The
requirement for static balance is that the center of gravity of the
system of masses be at the axis of rotation. For the center of gravity
to be at the axis of rotation, the moment of inertia of masses about the
x-axis and y- axis, respectively, must be zero.
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So, this rotor does not meet the conditions for static balance (equations 13.20 & 13.21). So the rotor is not statically balanced and hence, it is not dynamically balanced.
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For dynamic balancing of rotors, Eq. (13.11) and (13.17)
must be satisfied for single plane rotor and multi-plane rotors
respectively. It can be said that if the rotor is dynamically balanced
then it is also statically balanced (as demonstrated in Fig. 13.19.1& 13.20.1).
The converse is not true for all the rotors. A statically balanced
rotor is not always dynamically balanced, the exception being the single
plane rotors. This is illustrated by the following example.
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Consider the multi-plane rotor shown in Fig. 13.21.2.
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Figure 13.21.2 Multi-plane
rotor statically balanced but dynamically unbalanced without correction
weights and dynamically balanced after placing correction weights
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The above rotor is statically balanced without the correction weight (W C), as
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(27 X 10.5) – (27 X 10.5) = 275 - 275 = 0
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But the rotor is not dynamically balanced (condition for Eq. 13.17 is not met), as,
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(27 X 10.5 X 5.5) – (27 X 10.5 X 15) ≠ 0
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By placing two correction weights (Wc), each 64.3 N at the locations (radial and axial planes shown in the figure) the rotor is dynamically balanced, as
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(27 X 10.5 X 5.5) – (27 X 10.5 X 15) + (64.3 X 7 X 14) – (64.3 x 7 X 8) = 0
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Thus, static balance fails to
indicate moment balance required for the dynamic case. A static balance
is a reliable test of dynamic balance only in the case of single plane
rotors (example: Fig. 13.19.1 ), where all the masses lie in a common transverse plane and dynamic unbalance of moment is unlikely.
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Practical examples of balanced multi-plane rotors can be found in chapter 6. The epicycle gear train shown in Fig 6.10.2
has two sets of planetary gears (B and C) of different size/mass
mounted on the same arm (i.e. on sides of the sun gear F). The planes
containing the gears B and C is statically balanced by having three
gears in each plane with 1200 angular separations (i.e., Σ(Wr) = 0). Similar is the case with plane containing the gear set B.
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The wholes system (rotor)
comprising of sun and two sets of planetary gears is dynamically
balanced. This is by placing the bigger gear set 'C’ at a short distance
(aC) from the mid plane of gear F; while the smaller gear set B is place at longer distance (aB) from the mid plane of gear F. If the weight of the gear sets are WB and W C respectively, then,
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WBr aB = WC r aC
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13.22 PRACTICAL ASPECTS OF BALANCING
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Balancing of machines is
sometimes risky. During balancing, the equipment to be balanced is in
unsafe position. Before balancing, many practical aspects have to be
considered. A machine or machine part may go out of balance after
maintenance. The following actions carried out during maintenance of a
machine might affect its balance :
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- Changing of bearings, pulleys and couplings.
- Rearranging the fasteners (bolts, washers, and nuts) that are not at all identical on weight.
- Thermal distortion due to welding.
- Changing the orientation of assembly clearances. (eccentricity due to fit) .
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Maintenance activity related to shaft fit
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Consider a pulley with shaft mounted on rolling contact bearing.
Some clearance is required between the inner race of bearing and outer
surface of shaft, so that the shaft is inserted into the inner race of
bearing without interference. With this position, the shaft is held with
bolt or screws. On spinning the rotor, it will be out of balance. The
unbalance must be corrected by placing correction weight(s). This is
shown in Fig. 13.22.1(a). Note that
the shaft is moved towards right in the bearing and the rotor has excess
material on it right side. Hence, the rotor is balanced by placing
correction weight at left side. If the clearance between the shaft outer
surface and bearing inner surface is 0.005 cm, then the eccentricity is
0.0025 cm ( refer the figure).
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Figure 13.22.1 Eccentricity and unbalance in rolling contact bearing before and after maintenance operation
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Let us calculate the magnitude
of unbalance originally present on the rotor and correction weight
added to balance the rotor as depicted in Fig 13.22.1(a). From the following data for the rotor, these can be calculated.
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Rotor speed = 900 rpm, corresponds to angular speed, ω = 94.25 radians/s.
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Radius of pulley, r = 20 cm
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Thickness of pulley, t = 4 cm
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The area of excess material on the right side of the pulley based on Eq. ( 7.31) is equal to : π[(20)2 – (20 – 0.0025) 2] ÷ 4 = 0.0785 cm2. From Eq. ( 7.32)
the mass of excess material of the pulley on one side is 0.0785 x 4 x
7.8 = 2.4504 g. Since there is an absence of material at the left side,
the heavy spot on the right side of pulley from Eq. ( 7.33) is 2 x 2.4504 = 4.894 g at a radius 20 cm. The centrifugal force due to the heavy spot of wheel from Eq. (7.34) is 4.894x 20 x (94.25)2 = 85.2 N. The correction weight of 4.894 g is placed on the left side of pulley to balance the rotor.
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After maintenance operation, when the shaft is refit into the bearing with inner race rotated by 1800, the heavy spot and the correction weight will be on the left side of pulley as shown in Fig. 13.22.1(b).
Then the excess material on the right side of pulley is: 2 x 4.894 =
9.79 g at a radius of 20 cm. Thus, the unbalance and the amplitude of
vibratory force on the bearing is doubled; and the effect is much
serious than the one before the maintenance operation.
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Removing materials and adding weights
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For removing excess weights, drilling, milling,
grinding and laser vaporizing are used. Care should be taken so that the
structural integrity of rotating body is not compromised while removing
the excess material. The outside diameter is the most favoured location
for weight removal, because to attain the same effect, least amount of
material is removed at outer surfaces compared to removing materials at
the inner side. In addition, the stresses are lowest on the outer
circumference.
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Correction weights may be
in the form of clips, weld, adhesive backing, soldering, epoxy resins,
screws, washers, bolts, nuts and rivets. In the case of unbalance due to
the coils of armature of an electric motor, the unbalance should be
corrected at planes other than the armature such as pulley on the motor
or cooling fan blade etc
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