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Tuesday, August 22, 2017

MECHANICS OF TEXTILE MACHINERY - II


5.0 HELICAL, BEVEL AND WORM GEARS
5.16 WORM GEARS
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.
Fig. 5.16.1 Worm and worm gear on loom
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.
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.

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,
q = d1/m .....................................................................(5.15)
d2 = mz2 ....................................................................(5.16)
Where, d1 and d2 are the pitch circle diameter of the worm and worm wheel respectively.
Fig. 5.17.1 Terminology of worm gears

5.18 CLASSIFICATION OF WORM GEARS
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.
Single envelop worm gear
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.
Fig. 5.18.1 Single envelope worm gear set on wrap reel
Double envelop worm gear
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.
Fig. 5.18.2 Double envelope worm gear set on ring spinning machine
5.19 APPLICATIONS OF WORM GEARS
Worm gears find applications in almost all textile machines. Few applications are listed below:
Drive between cylinder and flat
Drive to builder mechanism in ring spinning machine
Drive to pedal roller of scutcher from top cone pulley to feed roller
Drive to bottom calender roller of scutcher from lap stop lever.
Drive to cams
6.1 GEAR TRAIN
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 )

Fig. 6.1.1 Spur gear train on the head stock of roving machine


6.2 CLASSIFICATION OF GEAR TRAINS
Gear trains are classified into the following:
Simple gear trains.
Compound gear trains.
Reverted gear trains.
Epicyclic (or planetary) gear trains
6.3 SIMPLE GEAR TRAIN
Simple gear trains are shown in Fig. 6.3.1. Each shaft is mounted with one gear.
Fig. 6.3.1 Simple train of gears





6.4 COMPOUND GEAR TRAIN
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.
Fig. 6.4.1 Compound train of gears
The gear transmission ratio of the compound train shown in figure 5.3 is





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.
Fig. 6.5.1 Reverted gear trains



     
6.6 EPICYCLIC/ PLANETARY GEAR TRAIN
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.
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’.
Fig. 6.6.1 Epicyclic gear trains: One degree of freedom

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.
Tabulation method
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:
(1) Motion of all components rigidly fixed to the rotating arm;
(2) Motion of all the components relative to the arm.
The superposition of the two components is carried out by the following steps:
a)

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.
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.
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.
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.
Table 6.7.1 Tabulation method to determine speeds of elements of gear train
Formula method
This method is useful for preliminary design of gear train as it is rapid. Referring to Fig.6.6.1 ,

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.
Click on image to run the animation
Animation 6.8.1 Inputs and outputs of epicyclic gear train
Fig. 6.8.1 Epicyclic gear trains with sun gear released from ground: -Two degrees of freedom
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),
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.
Bobbin diameter and speeds of bobbin
Bobbin lead winding principle is universally employed to control the bobbin speed on cotton roving machines which is shown in Fig. 6.9.1 .
Fig. 6.9.1 Bobbin-lead winding principle on roving machine
Fig. 6.9.2 Change of bobbin diameter on winding
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.
Epicyclic gear train on conventional roving machine
An epicyclic gear train used on an old roving machine is shown in Fig. 6.9.3.
Fig.6.9.3 Epicyclic gearing in an old roving machine
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.
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.
Speeds of various elements
Speed of various elements can be calculated using the tabulation method, which is given in Table 6.9.1.
Operation
Speed (rpm) of various elements
Arm
Gears/elements
1
2
3
4
5
6
Whole system rotates with +y rpm
y
y
y
y
y
y
y
Arm is fixed. Gear ‘1’ rotates with +x rpm
0
x
x
-x(z2/z3)
-x(z2/z3)
x .e2
x .e2
(add column wise)
Step 1 + Step 2
y
y + x
y + x
y - x(z2/z3)
y - x(z2/z3)
y +x.e2
y + x.e2
Putting the boundary conditions:
Epicyclic gear train on a new roving machine
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.
Fig. 6.9.4 Gear trains on Lakshmi roving machine
Speeds of elements of epicyclic gear train
The speeds of various gears are given in Table 6.9.2 .
Gears/
Operations
Speeds (rpm) of various gears
Gear, 50T (Arm)
Sun gear, 59T
Gear, 28T
Gear, 25T
Gear, 62T
Arm is locked.
Sun gear (59T) is given x rpm
0
x
-x(59/28)
-x(59/28)
x (59/28)(25/62)
Whole system revolves at y rpm
y
y
y
y
y
Total
y
x + y
-2.11x + y
-2.107x + y
0.85x + y
The rpm of main shaft or gear 59T is 827. Therefore,
The rpm of gear 50T is y, i.e,
Therefore, x = 191
The rpm of 62T gear is, y+0.85x = 798
Roving stretch or slackness
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="">
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.
Relation between the speeds of bobbin and bottom cone pulley



6.10 EPICYCLIC GEAR TRAIN ON COMBING MACHINE
Motion of detaching rollers
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.
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.
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.,
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.
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.
If the intermittent motion is ceased to exist, then the rotational speed of detaching rollers is O3, and it is positive.
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
Operation of epicyclic gear train
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.
Fig 6.10.1 Epicyclic gear trains on comber
Fig. 6.10.2 Epicyclic gear train to drive detaching roller on comber
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.
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.
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 ).
Operation
Speeds of various elements (rpm)
A
B
C
D
F
The epicyclic gear train as a whole rotate with +y rpm
+y
+y
+y
+y
+y
Arm F is locked & Gear A is given +x rpm
+x
-x(33/21)
-x(33/21)
+x(33/21)(29/25)
Zero
Resultant rpm
x + y
y -(33/21)x
y -(33/21)x
y +(33/21)(29/25)x
+y
Rpm of the gear A = x + y ..................................................................................(6.45)
Rpm of the gears B & C = -1.57x + y ..................................................................(6.46)
Rpm of the gear D = 1.82x + y ...........................................................................(6.47)
Rpm of the gear F = y ........................................................................................(6.48)
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.
Observations from combing cycle index
The observations made from the combing cycle at 240 nips per min are:
The index numbers per cycle is 0 to 40
The observations from the gearing plan are
The gear train ratio between the output gear D and the bottom detaching roller is -3.1 (negative).
Counterclockwise rotation of the detaching rollers corresponds to forward movement of detaching rollers and vice versa (viewed from headstock)
The diameter of detaching rollers is 25mm
Web delivery per cycle of combing




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.
Fig. 7.1.1 Cone pulleys used on a roving machine

7.2 DESIGN ASPECTS OF CONE PULLEYS
Design aspects of cone pulleys
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.
  • 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.
Based on the dimensions of the following elements, a design criterion must be set.
  • Width of cone belt.
  • Minimum and maximum bobbin diameters of empty & full bobbin to be used.
The consideration of the following points could simplify the design approach further.
(a)
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.
(b)
The length of the roving delivered by the front drafting rollers per unit time is constant (delivery rate).
(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 lmlm-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.
    Deflections
     
    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.


    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
    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
    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
    Fig. 9.6.3 Disk clutch for delayed starting of drafting on ring spinning machine

    9.7 MULTI DISK FRICTION CLUTCH
    A multi-disk friction clutch is shown in Fig. 9.7.1.
    Fig. 9.7.1 Multi-disk friction clutch
    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.
    Torque transmission capacity of multi disk friction clutch
    For the uniform-pressure criterion, torque is
    Where z is the number of pairs of contacting surfaces
    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%.
    Multi-disk friction clutch on bale opener
    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.
    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.
    Fig. 9.7.2 An electromagnetic actuated multi-disk friction clutch in bale opener
    Fig. 9.7.3 Schematic representation of disk friction clutch on bale opener (for clarity only one pair of disk is shown)
    Multi-disk friction clutch for fabric roll-up mechanism on loom
    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
    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.
    Click on Image to run the animation
    Animation 9.8.1 Operation of cone clutch
    Fig. 9.8.1 Cone clutch
    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.
    Torque transmission capacity of cone clutch
    An elemental frustum of the cone bounded by circles of radii r and (r + dr) is shown in Fig. 9.8.2.
    Fig. 9.8.2 Notations of cone clutch
    Fig. 9.8.3 An element of cone clutch
    For this elemental frustum, (from Fig 9.8.3), area of contacting surfaces between the cones (δA) is expressed as following:
    Driving of bobbin carriage on roving machine
    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.
    Click on Image to run the animation
    Animation 9.8.4 Double cone clutch transmitting motion to bobbin carriage
    Fig. 9.8.4 Double cone clutch to drive the bobbin carriage on roving machine
    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.
    Operating principle of centrifugal clutch
    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.
    Fig. 9.9.1 Centrifugal clutch
    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.
    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).
    Torque transmission capacity of centrifugal clutch
    The forces acting on the shoe are shown in Fig. 9.9.2, with the following notations:
    Fig. 9.9.2 Forces acting in centrifugal clutch

    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.
    Applications of centrifugal clutches
    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.

    9.10 MATERIALS FOR FRICTION LINING

    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

    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.

    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
    Where
    F = tangential force on the brake drum
    r = radius of the brake drum
    θ = Angle through which the brake drum rotates during braking period (rad.)
    Mt = Braking torque (N-m)
    The mechanical brakes are classified into:
    (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

    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’.
    Fig. 10.3.1 Drum or Simple block brake
    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)
    Analysis of forces acting on the drum
    The free body diagram of forces acting on the drum and the lever is shown in Fig.10.3.2.
    Fig. 10.3.2 Free body diagram of forces acting on the drum and the lever of simple drum brake
    When the shoe is rigidly attached to the lever, then the torque acting on the brake drum is
    Mt = μNr............................................................................................................ (10.3)
    Where μ = Coefficient of friction between the drum and shoe
    N = normal force acting on the drum (N). The normal force acting on the drum;
    N = pl1w............................................................................................................ (10.4)
    Where p = permissible pressure between the shoe and the brake drum (N/mm2)
    l1 = length of the shoe (mm)
    w = width of the shoe (mm)
    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,
    Rx = μN ..............................................................................................................(10.5)
    Ry = (N-F) ..........................................................................................................(10.6)
    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,
    Depending upon the magnitude of coefficient of friction (μ) and the position of pivot point, there are three possibilities emerge.
    Case 1
    a > μ b .......................................................................................................(10.8)
    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.
    Case 2
    a = μ b .......................................................................................................(10.9)
    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.
    Case 3
    a < μ b .......................................................................................................(10.10)
    Under this condition, the actuating force F becomes negative, as seen from the Eq. (10.7). This results in uncontrolled braking.
    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.
    9.3.2 Block brake with short shoe on lap former
    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.
    Fig. 10.3.3 Brake drum, shoe, racks, pinions and gear train on lap former
    When winding of lap progress, its diameter increases, pulling the racks up (Fig. 10.3.4). This rotates the pinions.
    Fig.10.3.4 Racks, pinions and lap spindle
    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.
    Fig.10.3.5 Brake in lap former
    Balancing the moments about the pivot of the actuating lever, we get the normal reaction force on the lever as,
    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.

    10.5 INTERNAL EXPANDING BRAKE
    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.
    Fig.10.5.1 Internal expanding brake




    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.
    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.
    Fig.10.6.1 Simple band brake
    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
    Tt / Ts = eμθ ........................................................................................................(10.16)
    The torque absorbed by the brake is given by
    Mt = (Tt - Ts)r ..................................................................................................... (10.17)
    Considering the forces acting on the lever and taking moments about the pivot (O),
    Ts.a = F.l ........................................................................................................... (10.18)
    F = Ts.a / l ..........................................................................................................(10.19)
    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.
    Fig. 10.6.2 Differential band brake
    Considering the forces acting on the lever and taking moments about the pivot, we can get,
    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.
    Negative let-off warp on loom using band brakes
    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.
    Fig. 10.6.3 Brake on warp beam on a conventional loom
    Usually the angle of wrap is 5400 which results in, eμ 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.
    Fig.10.7.1 Disk brake

    10.8 NON-FRICTION TYPE CLUTCHES AND BRAKES

    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.

    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.
    Pneumatic actuated multi-disk friction clutch and brake
    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.
    Click on Image to run the animation
    Animation 10.9.1 Operation of multi-disk friction clutch and brake
    Fig.10.9.1 Multi-disk friction clutch and brake
    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.
    Hysteresis disk clutch and brake
    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.
    Click on Image to run the animation
    Animation 10.9.2 Operation of hysteresis disk clutch and brake
    Fig. 10.9.2 Hysteresis disk clutch and brake on loom
    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.
    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

    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.
    .
    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.
    Fig. 11.2.1 Sliding contact bearing
    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.
    Solid bush and lined bush bearings
    Sliding contact bearing is constructed either with solid bushing or lined bushing, as shown in Fig.11.2.2.
    Fig. 11.2.2 Sliding-contact bearing (a)Solid bush (b)Lined bush
    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.
    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.
    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.
    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.
    Fig. 11.3.1 Hydrodynamic Lubrication (a) Journal at rest (b) Journal starts to rotate (c) Journal at full speed
    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
    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.
    Fig. 11.3.3 Hydrostatic lubrication
    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.
    • High load carrying capacity even at low speeds
    • No starting friction
    • No contact between sliding surfaces at any range of speeds and loads.
    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.
    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,
    Fig. 11.3.4 Fluid film between stationary and moving surfaces
    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
    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.
    Fig. 11.5.1 Coefficient of friction (μ) of journal bearing as a function of viscosity, speed and fluid pressure
    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.
    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.
    Fig. 11.6.1 Circumferential oil groove in sliding contact bearing
    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.
    Fig. 11.6.2 Cylindrical oil-groove in sliding contact bearing
    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.

    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.
    Fig. 11.8.1 Nomenclature of a deep-groove-ball bearing
    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.
    Fig. 11.8.2 Conformity of ball radius to raceway radius
    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.
    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.
    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.
    11.9 TYPES OF BALL BEARINGS
    Various types of ball bearings are shown in Fig.11.9.1.
    Fig. 11.9.1 Ball bearings: (a) Deep-groove; (b) Angular-contact; (c) Self-aligning; and (d) Thrust
    Ball bearings can be broadly classified into the following:
    • Deep-Groove ball bearing
    • Angular-contact ball bearing
    • Self-aligning ball bearing
    • Thrust ball bearing
    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.
    Angular contact bearings
    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.
    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.
    Fig.11.9.2 Duplex angular contact ball bearings
    Self aligning ball bearings
    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.
    Thrust ball bearings
    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.

    11.10 BALL BEARINGS IN TEXTILE MACHINES
    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.

    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.
    Fig.11.11.1 Roller bearings: (a) Plain; (b) Tapered; and (c) Spherical
    Cylindrical or plain roller bearings
    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.
    Needle bearings
    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.
    Fig.11.11.2 Needle bearings on ring spinning machine
    Tapered roller bearings
    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.
    Fig.11.11.3 Forces acting on a tapered-roller bearing
    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.
    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.
    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.
    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:
    1. Rotating cam and translating follower (cams used in looms, ring spinning machines)
    2. Rotating cam and follower as an arm swings with respect to its pivot point (cam in comber).
    3. Translating cam-translating follower
    4. 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.
    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 .
    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.

    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 .
    Fig 12.4.1 Cam for shifting of belt on cone pulleys in roving machine
    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
    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
    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
    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
    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
    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
    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.

    13.0 : BALANCING OF MACHINES
    13.1 UNBALANCE
    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.


    13.2 CAUSES OF UNBALANCE
    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.
    (3)
    Shifting of mass center of rotor on tightening support elements.
    (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
    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
    .

    13.3 PRODUCTION AND FIELD BALANCING
    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.
    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.
    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’
    .
    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.
    Imaginary heavy spot
    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.
    Figure 13.4.1 Rotor with two voids and the vector sum representing an imaginary heavy spot
    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.
    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.
    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.
    .

    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.
    Click on Image to run the animation
    Animation 13.5.1 Statically unbalanced rotor favouring the heavy spot to the bottom
    Figure 13.5.1 Static Balancing of rotor placed on rigid rollers
    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.
    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
    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
    Figure 13.6.1 Static unbalance on rotor: Top- disk/pulley; Bottom- drum/cylinder/ beater/tin roller
    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.
    Fig. 13.6.2 Statically balanced V-pulley
    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.
    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
    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


    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
    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


    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
    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).
    Figure 11.16.1 A balanced cam with large run out (left); Disk with zero run out but with serious unbalance (right)
    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.
    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.
    Figure 13.16.2 Couple due to axial run out of disk 

    13.17 UNBALANCE DUE TO ECCENTRICITY IN MOUNTING THE SHAFT
    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
    U = em .............................................................................(13.3)
    When e/r is very small
    Figure 13.17.1 Eccentricity due to displacement of the rotating axis of shaft in sliding contact bearing
    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
    U = em = 0.0005 x 150000 = 75 g-cm.
    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
    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).
    Figure 13.18.1 Eccentricity due to displacement mass cg of a disk
    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
    FU = Weω2 / g ..............................................................................(13.4)
    Where
    W = Weight of rotor

    ω = Rotational speed of rotor

    g = Acceleration due to gravity.
    After balancing the rotor, the centrifugal force due to a correction weight Wc placed at a radius rc from the center is
    FWC = Wc rc ω2 / g ........................................................................(13.5)
    Where FW c is the centrifugal force due to correction weight.
    If the rotor is perfectly balanced by the correction weight, then
    FU = FW c ..................................................................................(13.6)
    Hence, M e = Mc rc ...................................................................(13.7)
    Where M and Mc are the masses of rotor and correction weight respectively.
    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.
    Rearranging the Eq. (11.3) and (11.7), and if rc = r, we get
    e = (U/M) = ( Mcrc/M) = ( Mcr/M) ..................................................(13.8)
    This is called the “specific unbalance per unit mass of rotor” or “eccentricity”.
    13.19 DYNAMIC BALANCING OF SINGLE PLANE ROTOR
    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.
    Figure 13.19.1 Dynamically-balanced single-plane rigid-rotor
    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
    For dynamic balancing of the rotor, the vector sum of the inertia forces (ΣFc ) of the system is zero.
    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
    Where θi and θc are the phase angles of ith element and correction weight, counter-clockwise measured from x-axis.
    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.
    Table 13.19.1 Balance of inertia forces
    From this, the weight times the radius of correcting element (i.e. the resultant force R) can be calculated as,
    .
    13.20 DYNAMIC BALANCING OF MULTI-PLANE ROTOR
    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.
    Figure 13.20.1 Dynamically-balanced multi-plane rigid-rotor
    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.
    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, Wr) and aC. This is shown in the following Table 13.20.1.
    Table 13.20.1 Balance of moments
    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,



    13.21 DYNAMIC AND STATIC BALANCING
    Fig. 13.21.1 shows a rigid rotor with shaft laid on horizontal parallel ways.
    Figure 13.21.1 Dynamically unbalanced single-plane rigid-rotor
    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.
    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.
    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.
    Consider the multi-plane rotor shown in Fig. 13.21.2.
    Figure 13.21.2 Multi-plane rotor statically balanced but dynamically unbalanced without correction weights and dynamically balanced after placing correction weights
    The above rotor is statically balanced without the correction weight (W C), as
    (27 X 10.5) – (27 X 10.5) = 275 - 275 = 0
    But the rotor is not dynamically balanced (condition for Eq. 13.17 is not met), as,
    (27 X 10.5 X 5.5) – (27 X 10.5 X 15) ≠ 0
    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
    (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
    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.
    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.
    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,
    WBr aB = WC r aC

    13.22 PRACTICAL ASPECTS OF BALANCING
    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 :
    • 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) .
    Maintenance activity related to shaft fit
    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).
    Figure 13.22.1 Eccentricity and unbalance in rolling contact bearing before and after maintenance operation
    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.
    Rotor speed = 900 rpm, corresponds to angular speed, ω = 94.25 radians/s.
    Radius of pulley, r = 20 cm
    Thickness of pulley, t = 4 cm
    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.
    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.
    Removing materials and adding weights
    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.
    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