Simple Lifting Machines
7
R E T P A H C
Key Concepts
De�nition of simple and compound machines Discussion on effort, load, mechanical advantage, velocity ratio, and ef�ciency De�nition of ideal machine, ideal effort, and ideal load Discussion on reversibility, irreversibility, irreversibility, and law of machine Description of inclined plane, simple screwjack, and differential screwjack Discussion on pulley systems, differential pulley block, and gear pulley block Discourse on simple wheel and axle, wheel and differential axle, and worm and worm wheel Description of single and double purchase crab and winch, worm geared screwjack, and worm geared pulley block
7.1 INTRODUCTION A machine can be de�ned as a device or a contrivance which receives energy in some form or the other and utilizes this energy for for doing some useful work. The term useful work is is very signi�cant in this context. If a heavy heavy box is pulled from one point to another on a rough horizontal �oor, the net work done by the load is zero, although the pulling force does non-zero work. Thus the useful work at output is not useful work. A machine, whatever its type, either simple or compound, must always perform some useful work.
7.2 SIMPLE AND COMPOUND MACHINES A simple machine is a mechanical device which can change the direction and magnitude of a force or effort and makes work easier. The following are the basic characteristics of a simple machine Has a few few or no no moving moving parts Uses energy to perform work Works with one movement
Simple Lifting Machines
421
Makes work easier by using less mechanical effort for for moving an object Uses the concept concept of spreading force force over over distance Offers the scope for attaining attaining advantage by changing the magnitude, speed, or direction of force Allows to use a smaller force to overcome larger resistance Offers a trade-off trade-off of energy The Re Renaissance naissance scientists de�ned six classical classical simple machines as: (1) inclined plane, plan e, (2) wedge, (3) lever, lever, (4) screw, (5) pulley, and (6) wheel and axle. a xle. All these simple machines machin es and their different types are generally used to lift loads. Hence, these are also called simple lifting machines or simple hoisting machines. mac hines. When two or more simple machines work work together, they are called compound or complex machines. These simple machines are connected in series such that the output force force of one machine serves as the input force force of another machine. For For example, a bench vice is a compound machine ma chine which comprises compris es two simple machines, machi nes, namely a lever and a screw. screw. Similarly a simple si mple gear train tr ain consists of a number of wheels and axles in series.
7.3 SOME BASIC BASIC DEFINITIONS DEFINITIONS Before starting the study of simple machines or simple lifting machines, we need to get acquainted with some basic terms which are needed for analysis and problem-solving purposes.
7.3.1 Effort and Load A simple machine is driven by an applied force, and by the application of this force the resistance of force is overcome. This applied force is de�ned as effort and is conventionally symbolized by P. The resistance of force which is overcome, or in other sense, is hoisted up to a certain distance, is called the load and is generally expressed expressed as W .
7.3.2 Mechanical Advantage In the absence of any sort of friction, for maintaining equilibrium, the work done on the load must be equal to the work done by the applied effort. This allows an increase in the output force at the cost of decrease in the distance traversed by the load. The ratio of output force to input force is de�ned as mechanical advantage ( M A) of the machine. In other words, mechanical advantage advantage is de�ned as the ratio of load lifted to the applied effort. effort. Load Lo ad lif lifte ted d = W . In cases where the machine neither absorbs Thus M A = Effort Effo rt appl applied ied P nor dissipates energy, the mechanical advantage is computed from the geometry of the machine. For example, in case of levers, mechanical advantage is equal to the ratio of its effort arm to load arm. For a compound machine consisting of n number of simple machines, machines, the resultant mechanical advantage can be expressed as ( M A )compound
= ( M A )1 ¥ ( M A )2 ¥ ( MA )3 ...( M A )n
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Engineering Mechanics
7.3.3 Velocity Ratio In an ideal simple machine, the rate rate of input power must be equal to the rate rate of output power. We We know that power is the product of the force and its velocity velocit y at the point of applica application. tion. Assuming that the velocity of applied effort effort at the point of application is vin and that of load is vout , we can write
P ¥ vin or,
W P
So,
M A
=
= W ¥ vout
vin vout
= W = P
vin vout
Thus, the ratio of input velocity to output velocity is de�ned as velocity ratio ( V r). Hence, for an ideal machine, M A = V r . Simplifying a bit, it can be written as
V r
=
Distance Distan ce tra travel velled led by eff effort ort Dist Di stan ance ce tr trav avel elle led d by lo load ad
7.3.4 Efficiency and Loss of Energy Energy The ratio of output power to input power is de�ned as the mechanical ef�ciency of the simple machine and is generally symbolized by h. Ef�ciency is also a measure of loss of energy which mainly occurs due to friction and also due to deformation and wear. v Output Out put pow power er W ¥ vout W vout Now, h= = = ¥ = M A ¥ out P ¥ vin P vin vin Input Inp ut pow power er or,
MA
=h¥
vin vout
= h ¥ V r
As velocity is determined by the dimensions of the machine, mechanical ef�ciency ef�ciency is reduced by the losses. In an ideal machine, h = 100% or 1. For a nonideal machine h < 1, so Vr > M A . In a compound machine consisting of n number of simple machines, the resultant ef�ciency can be expressed as, (h )compound
= (h )1 ¥ (h )2 ¥ (h )3 ...(h )n
7.3.5 Ideal Machine An ideal machine is a hypothetical system, where energy and power are neither lost nor dissipated through through friction, deformation, or wear. It possesses theoretical maximum performance, thereby maximum mechanical ef�ciency of 100 per cent. In an ideal machine, M A = V r . Hence, an ideal machine is considered as a baseline for evaluating evaluating performance performance of real machine systems. systems.
7.3.6 Ideal Effort, Ideal Load, and and Loss Let us consider an ideal machine, where Pideal
Vr
= M A = W . Thus ideal effort, P
= W . But in a non-ideal machine, actual effort required to raise a given V r
load W is is more than the ideal effort.
Simple Lifting Machines
Again we know that, h ¥ Vr
So, actual effort Pactual
423
= M A = W . P
= W ¥ 1 .
V r h Hence, loss in effort due to friction, wear, etc.
Ê ˆ = Pactual - Pideal = W ¥ 1 - W = W Á 1 - 1˜ Vr h Vr V r Ë h ¯ Similarly, for a given effort P, Wideal = P ¥ V r . In a non-ideal machine, machine, it is Ploss
noticed that the amount of load that can be lifted by a given effort P is less than the ideal load. So, actual load that can be lifted Wactual = hP ¥ V r Hence the decrease in the amount of load,
Wdecrease
= Wideal - Wactual = P ¥ Vr - hP ¥ Vr = P ¥Vr (1 - h )
7.3.7 Reversibility and Irreversibility Irreversibility In a simple machine machine,, let us consider an effort P moving a distance S E is required to hoist a load W to to a distance S L. If the effort is removed entirely, two situations may occur. Either the machine moves in the reverse direction and the load comes down to its initial position, or the load does not move down and remains static at the position wherefrom the effort is removed. Occurrence of the �rst situation is due to reversibility of the machine, whereas the second case happens due to the irreversibility property of the machine. machine. Irreversibility condition is also sometimes termed as self-locking condition condition as the machine gets locked at at the point of remo removal val of eff effort. ort. Hoisting any load by means of a simple pulley is an example of reversibility because as we release the pull, the load falls down. In case of a screwjack, self-locking situation can be achieved if the inclination of thread becomes less than the limiting angle of static friction. Now we will investigate the condition of irreversibility. In an irreversible simple machine, Loss of work done done due to friction Work done by the effort – Work done by the load P × S E – W × × S L When effort is withdrawn withdrawn and the loss of work done due to friction is higher than the work done by the load, then only self-locking can occur.
So,
( P ¥ S E
or
P
or or
¥ S E
W P
¥
- W ¥ S ) > W ¥ S > 2W ¥ S L
S L S E
L
L
<1
2
h < 1
2
Thus, if the ef�ciency Thus, ef�ciency of the machine becomes less than than 50 per cent, cent, irrever irreversisibility or self-locking condition can be achieved.
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Engineering Mechanics
7.3.7 Law of Machine Machine If an effort P is applied on a machine to hoist a load W , the relationship between load and effort can be idealized as P = aW + b , where a and and b are constants. This is the equation of a straight line. If the equation is plotted considering W along along x-axis and P along y-axis, then y-intercept is de�ned as can be considered as initial effort effort at no load condition. The slope of b . So b can
D P = dP . The governing equation dW DW Æ0 DW
the line (a ) can be de�ned as a = lim
of actual machine is shown as line AB in Fig. 7.1. For an ideal machine, the y-intercept b 0 and thus the governing equation is P = a W , passing through the origin of the coordinate system. This is shown as line OC in Fig. 7.1. For both cases, the slopes of the lines are same and hence will be parallel to each other.
For a particula particularr load denoted by W OM, if we want to �nd the amount of effort required for both ideal and actual machines, we need to draw a normal at point M which intersects line OC at N and AB at R. So the ideal effort Pideal MN and actual effort Pactual MR. Hence, loss of effort, Ploss MR – MN NR. Now the mechanical advantage can be computed as,
M A
=W = P
W = 1 aW + b a +
b W
Ê b ˆ decreases, and thus M A ÁË W ˜ ¯ b increases. At the extremity, when W Æ •, = 0 and thus M A = 1 . The The equation suggests that when W increases,
a variation of mechanical advantage M A with respect to load W has has been plotted in Fig. 7.2, which is basically a hyperbolic plot, and the horizontal line W
passing through
M A
max
=
max
1 is an asymptote of the curve.
a
Effort (P)
B
a W
P
a
A
P
a W
1 a
b C N
(M A)
a
b O
Mechanical advantage
Load (W (W )
M Load (W )
Fig. 7.1 Law of machine in ideal and actual conditions
Fig. 7.2 Variation of mechanical advantage advanta ge with load
Mechanical ef�ciency can thus be computed as, h = 1 ¥ W = 1 ¥ W = 1 ¥ 1 Vr P Vr aW + b Vr b a + W
Simple Lifting Machines
It is observed that at the extremity when b = 0 and thus h = 1 . The W Æ •,
1 Vr a Vr
max
a V r variation of mechanical ef�ciency h with respect to load W has has been plotted in Fig. 7.3, W
which is a hyperbolic plot, and the horizontal 1 , line passing through hmax = is an
425
Mechanical ef�ciency (h)
a V r
asymptote of the plot.
Load (W (W )
Fig. 7.3 Variation of mechanical ef�ciency with load
7.4 INCLINED PLANE An inclined plane is one of of the simplest types of simple machines. machines. It consists of a sloping surface which makes an acute angle above or below the horizontal. Inclined planes are a re used to raise very heavy objects which are dif�cult to hoist vertically. Gangways, ramps, chutes are some common examples of inclined planes. A schematic diagram of an inclined plane is shown in Fig. 7.4. The inclination with the horizontal is a , vertical elevation of the tip from the H L (P) horizontal is H , and the inclined length Effort (P Load (W (W ) from the horizontal to tip is L. Load a (W ) is to be lifted from the bottom to the top at height H and and thus effort ( P) Fig. 7.4 Inclined plane is applied. Now, the velocity ratio can be computed as
V r
=
Distan Dist ance ce tr trav aver erse sed d by th thee ef effo fort rt Distance tr traversed by by th the lo load
=
L H
=
L L s in a
=
1 sin a
Assuming this as an ideal machine, we can say that mechanical advantage M A = V r = 1 sin a
7.5
SIMPLE SCREWJACK
A screwjack is a simple lifting device which is used to lift heavy loads such as large vehicles. It mainly consists of three parts—a nut attached to a pedestal or stand, a large screw �tted within the nut, and a lever attached to the head of the screw, as shown in Fig. 7.5. The weight which is to be lifted is placed either on the head of the screw or on a platform attached to the screw. screw. A screw thread t hread is cut just jus t like an inclined plane. The distance which the screw advances in one turn is called
Fig. 7.5
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Engineering Mechanics
lead distance (L), and the distance measured between two consecutive threads is called pitch distance ( p). Except for single-threaded screws, lead and pitch distances are different. The slope of screw thread measured with respect to lead is called lead angle ( q ). When a load ( W ) is to be raised by screwjack, differentt situations arise that are explained in Figs 7.6, 7.7 and 7.8. differen Let us assume P as the force generated at at the circumference of the spindle by virtue of the application of an external force Q at the end of the lever. Now we will discuss different cases which arise during the working working of a simple screwjack. shown in Fig. 7.6. Considering Motion impending upwards This situation is shown equilibrium of forces along the incline and along the normal to the incline, we obtain P
P cos q N q
Fig. 7.6
P (cos q
or
P
= P sin q + W cos q
Eliminating N from above the equations, we obtain
W
or
= W sin q + m N
P cos q
= W sin q + m( P sin q + W cos q )
- m sin q ) = W (sin q + m cos q )
=W
sin q + tan fs cos q sin q + m cos q = W cos q - m sin q cosq - tanf s sinq
= W
sin q cos fs + sin f s cos q cos q cos fs - sin f s sin q
[multiplying numerator and denominator by cos f s ] sin(q + f s ) =W = W tan(q + f s ) cos(q + f s ) From moment equilibrium about the axis of the screw, we obtain P d = QR 2 or
Ê tan q + tan f s ˆ = Wd tan(q + f s ) = Wd Á 2R 2R Ë 1 - tan q tan f s ˜ ¯ Ê p Ld - m ˆ Wd Ê L + mp d ˆ W d = Á ˜ = [substituting 2 R Ë 1 - L m ¯ 2 R ÁË p d - m L ˜ ¯ p d
Q=P d 2R
tan q = L ] p d
Ê p + mp d ˆ For single-threaded screws, L = p and thus Q = Wd Á 2 R Ë p d - m p ˜ ¯ Ê np + mp d ˆ For n-threaded screws, L = np and thus Q = Wd Á 2R Ë p d - m np ˜ ¯ Motion impending downwards can write P = W tan(fs - q ).
Proceeding in the same manner as abov above, e, we
Taking moment equilibrium eq uilibrium about the axis of the screw, screw, we obtain P d = QR. 2
Simple Lifting Machines
or
QP d 2R
427
tan tan Wd tan(s ) Wd s 2R 2 R 1 tan s tan
Ld Wd 2 R 1 L
Wd d L 2R d L d Wd d p For single-threaded screws, L p and thus Q 2 R d p . Wd d np . Q L np For n-threaded screws, and thus 2 R d np The movement of W in in upwards direction by means of short successive strokes strokes of the lever is shown in Fig. 7.7. After raising up to the desired level, if we do not keep holding the lever, either of two situations occurs. The screw may unwind automatically or may remain �xed at the previous level to which it was raised. The second situation occurs due to self-locking arrangement. arrangement. If s , the screw is said to be self-locking. Otherwise it will unwind if the lever is not hold.
Fig. 7.7
Efficiency of machine while hoisting load The actual effort effort available at the circumference of the screw or spindle can be written as, Pactual W tan( s ). The expression for ideal effort for no-friction condition is Pideal W tan .
Thus the ef�ciency of the machine P W tan ideal tan Pactual W tan( s ) tan( s ) Condition for attaining maximum efficiency sin cos( s ) tan tan( s ) cos sin( s )
[sin( 2
or
[sin( 2
1 1
s ) sin s ] s ) sin s ]
2
2 sin( 2 s ) sin s sin( 2 s ) sin s
sin( 2 sin( 2
We know that
s ) sin s s ) sin s
2 sin s sin( 2 s ) sin s
For attaining attainin g max , (1 ) should be of minimum value value.. As in s has a constant value, max is possible for the maximum value of sin( 2 s ). in( 2 s ) 1 sin and so( 2 s ) . Thus 2 2 Substituting in the expression, we �nd 1 sin s max 1 sin s Expression for torque In some specific cases, the rotation of load W is is not desirable when the loading loadi ng lever is being rotated. To To accomplish accompli sh this, the head of the screw is made up of two parts, as shown in Fig. 7.8, where upper part
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Engineering Mechanics
W through ball bearings. Thus A rests on lower part B through when the lever is rotated, upper part A moves up A vertically without getting rotated. Let us assume that the external and internal radii at the contact surface of A and B are r1 and r2, respectively. Thus the r +r r2 mean radius is rmean = 1 2 at which the frictional 2 B r1 force of magnitude m W acts between A and B . Hence, the applied force Q at the end of the lever Fig. 7.8 will have to overcome the effect of this frictional resistance in addition to raising the load. Considering moment equilibrium equation about the axis of the screw, we obtain
Q ¥ R = mW
Hence, Q
=
Ê r + r ˆ ¥ rmean + P ¥ d = mW Á 1 2 ˜ + W tan (q + fs ) d 2 2 Ë 2 ¯
m W ( r1 + r2 ) Wd + tan(q 2R 2R
= Required Torque
+ f s )
7.6 7. 6 DI DIFF FFER EREN ENTI TIAL AL SCREW SCREWJA JACK CK A differential screwjack is used in precision precisi on equipment where slow movement or �ner adjustment is necessary. It consists of two threads having different pitches wound on the same cylinder or different cylinders. When the threads are wound on the same cylinder, two nuts are necessary. But if the threads are wound on different cylinders, then one nut will suf�ce. These two types of arrangements are shown in Fig. 7.9. During a single single rota rotation tion of the lever lever,, the the load W gets gets raised by a distance ( p2 - p1 ). By then the effort Q tra traverses verses a distance distan ce 2p R. Thus the velocity ratio (V r ) for a differential screwjack can be computed as V r = 2p R , mechanical p2 - p1 advantage as M A
= W and ef�ciency as p
Fig. 7.9
h=
M A Vr
=
W ( p2 - p1 ) . 2p RP
Simple Lifting Machines
7.7
429
SYSTEM OF PULLEYS
Hero (or Heron) of Alexandria (c. AD 10–70), a Greek mathematician and engineer,, identi�ed pulley as one of the six simple machines used for hoisting engineer loads. Basically pulley is a wheel mounted on an axle designed des igned for supporting movement of belt or cable around its circumference. A pulley is sometimes called drum or o r sheave and may also have have grooves between �anges around the circumference. The driving element of a pulley system can be a rope, chain, cable, or belt. Other than hoisting loads, pulleys are also used for application of forc forces es and transmission of power power.. Pulley systems consisting of one or more number of pulleys can be classi�ed in three ways depending on the arrangement 1 of pulleys. First-order pulley system In this system, out of n numb number er of 2 pulleys comprising the system, there exists one fixed pulley P and (n – 1) number of mova movable ble pulleys. The fixed pulley 3 is designated as 1 and at the end of n-th pulley the load is suspended, while effort is applied through the rope wrapped over pulley 1. A first-order pulley system consisting of one 4 fixed pulley and three movable pulleys is shown in Fig. 7.10. Effort P is applied through the rope wrapped over pulley 1 W and load W is is suspended from pulley 4. If we consider the Fig. 7.10 Firstvertical force force equilibrium of pulley 4, we can write order pulley Load (W ) system 8P W , or effort ( P) = 8 Thus the mechanical advantage movable ble pulleys M A = Load = W 8 23 (2)number of mova Effort W 8 Considering it as an ideal machine, velocity ratio
P
P
P (c)
all components P
W
W
( a)
(b)
W (d)
Fig. 7.11 Second-order pulley system
Vr = M A = 23 For n-number of movable pulleys, Vr
= M A = 2n
Second-order pulley system In this case, we need two blocks of pulleys. The top block is attached to a fixed support and the bottom block is movable. One end of the rope is attached to the hook at the bottom-most pulley of the top block and then the rope is wound around each and every pulley successively, as shown in Fig. 7.11(a). At the
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Engineering Mechanics
end of bottom block, the load W is suspended and the effort P is applied through the top-most pulley of the top block. If the system is separated through a section plane X–X , as shown in Fig. 7.11(b), and the freebodies as shown in Figs 7.11 (c) and (d) are analysed, we can easily write Load(W ) 6 P = W or effort (P) 6 Thus the mechanical advantage of the system, M A = Load = W 6 2 × 3 Effort W 6 2 × number of pulleys in the bottom block.
Considering it as an ideal machine, velocity ratio V r M A 6 2 × number of pulleys in the bottom block. block. If the number number of pulleys in the bottom block block is n, then V r M A 2n.
Third-order pulley system In this system, one pulley is attached to the fixed support and is called fixed pulley, 1 as shown in Fig. 7.12(a) as 1. The rope passes over this fixed pulley and one end is attached to a rigid base. The P 2P 4P 2 other end of the rope holds pulley 2 hanging. One end of a separate rope passing over over pulley 2 is attached to the rigid base and the other end of the rope holds pulley 3 W 3 hanging. Another rope is wound over pulley 3, having its (b) one end attached to rigid base and the other end serves P as the point of application of effort P. The load W is hooked hooke d at the bottom of the rigid base. Depending on the W (a) requirement, we can easily change the number of mova movable ble Fig. 7.12 Thirdpulleys.. Considering the free-body diagram of the rigid order pulley system pulleys base, as shown in Fig. 7.12(b), we can easily infer that Load(W ) 7P W or or effort (P) . 7 Thus the mechanical advantage advantage of the system M A = Load = W = 7 = 23 – 1 = [(2)total number of pulleys in the system –1]. Effort W 7 Considering it as an ideal machine, velocity ratio R B Vr = M A = 7 = 23 – 1 = [(2) total number of pulleys in the A system –1]. r
Effort (P)
If the total number of pulleys in the system is n, then Vr = M A = ( 2) n - 1
7.8
WESTON DIFFERENTIAL PULLEY BLOCK
Weston differential differential pulley block is a special type of pulley system, which is normally used to hoist very large masses to a small distance, dist ance, for example, the pulley system syst em used for manually lifting car engines. This differential pulley was invented by Thomas Aldridge Weston from King’ K ing’ss Norton, England, in 1854. Hence, this simple machine m achine is also called Weston differential pulley block. The schematic view of this machine is shown in Fig. 7.13(a).
Endless chain Load (W )
(a) Fig. 7.13 Weston differential pulley block.
Simple Lifting Machines
431
2R This system consists of two �xed pulleys of unequal 2r radii, which are coaxially attached to each other and can rotate together and are �xed to the support, a single pulley hanging at the bottom and holding load W , and P an endless rope wrapped around the pulleys. pul leys. In order to avoid slipping, generally rope is substituted by a chain and connected to pulleys by sprockets. W 2 W 2 To determ determine ine velocity ratio ( V r) of the system, we W 2 W 2 (b) need to consider the pulley block as an ideal machine. For an ideal machine, we know that V r M A. Consider Fig. 7.13 FBD of the radii of the smaller and larger pulleys as r and R, �xed pulley respectively, and a weight W is is hoisted with an effort P. The free-body diagram of the upper part of the system is shown in Fig. 7.13(b). Considering moment equilibrium of forc forces es about the axis of the pulleys, pulleys, we can write
W 2
¥ R = W ¥ r + P ¥ R 2
or
W ¥ ( R - r ) = P ¥ R 2
or
W = 2 R P R-r
Thus mechanical advantage M A
=
2R R-r
As we assumed the system as an ideal machine, velocity velocity ratio V r
=
2R . R-r
When r ≈ R or (R – r) ≈ 0, the mechanical advantage advantage of the system M A = •. This implies that no force (besides friction) is required to move the chain, but movement movement of chain fails to hoist the load. If r 0, then M A = 2 and this system becomes a simple gun tackle.
7.9
GEAR PULLEY BLOCK
In a gear pulley machine, an axle is coaxially attached to an effort wheel having T 1 number of teeth. A pinion having having teeth T 2 and a ratchet and clutch are attached coaxially on the axle. A pawl presses against this ratchet and clutch with the help of a spring. The pinion is geared with a spur wheel having having teeth T 3. On the same axle as spur wheel a load drum ha having ving teeth T 4 is keyed on its circumference. An endless rope or chain is wound over the effort wheel with which the effort ( P ) is applied. A schematic diagram of the gear pulley block is shown in Fig. 7.14. The motion is transmitted from effort wheel to load drum through pinion and spur wheel. A separate rope is wound around half the perimeter of load drum. One end of it is �xed to the frame and and other end holds the load ( W ). ). When the load is hoisted, the ratchet passes under the pawl. On the removal of effort, the pawl prevents the load from falling down. Hence, it is a self-locking arrangement.
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Engineering Mechanics Effort wheel T 1 Load drum T 4
Spring Spur wheel T 3
Spur wheel T 3
Pinion T 2 Pawl
Load drum T 4
Effort wheel T 1
W
Pini Pi nion on T 2
Endless chain
Ratchet and clutch
Fig. 7.15 Fig. 7.14 Schematic diagram of gear pulley block
Working illustration of gear pulley block
In a single single rota rotation tion of effort wheel, (Fig. 7.15) eff effort ort ( P) moves through a distance proportional to T 1. At the same time, the spur wheel and the load drum Ê T ˆ rotate by Á 2 ˜ of a rotation. In a single rotation rotation of load drum, the load ( W ) is Ë T 3 ¯ lifted through a distance proportional to T 4. So for a single rotation rotation of effort wheel, Ê T ˆ the load is lifted by a distance Á 2 ˜ T 4 . Hence, the velocity ratio of the machine, ma chine, Ë T 3 ¯
V r
7.10
T 1
T 2 T T 4 3
T1 .T 3 T2 .T 4
SIMPLE WHEEL AND AXLE
This simple machine is basically a wheel or pulley R which is rigidly attached to an axle or a drum of smaller diameter. Since the wheel and the axle are �tted coaxially, both of them can rotate together. r We see the common use of this simple machine in steering wheels of automobiles, doorknobs, etc. The effort (P) is applied with a rope wound over the wheel. Another rope, keeping it �xed with some P W suitable point on the axle, is wrapped over the axle, Fig. 7.16 from which a load ( W ) is suspended. A schematic diagram of a simple wheel and axle is shown in Fig. 7.16. Let us assume the radii of wheel and axle are R and r, respectively. respectively. As the th e axle and wheel rotate together, the distance traversed by the wheel during a single rotation is 2p R, while that by the axle, and thereby the load, is 2 p r. Thus the velocity ratio of the machine, machine,
Simple Lifting Machines
V r
= 2p R = 2p r
433
R. r
Considering no-friction condition, that is, for 100 per cent ef�ciency, mechanical advantage will be M A = V r = R . r Let us analyse the effect effect of friction on on the machine machine.. In an actual machine machine,, if Pactual is the effort required to lift load W , Work done by the effort Work done by the load
Pactual
¥ 2p R
W × × 2p r
Hence,, the actual ef�cien Hence ef�ciency cy of the machine,
hactual
=
¥ 2p r = Ê W ˆ r = ( M ) r A actual ¥ Á ˜ Pactual ¥ 2p R Ë Pactual ¯ R R W
Thus, the actual mechanical advantage, ( M A )actual
7.11
= hactual ¥ ÊË R ˆ ¯ r
WHEEL AND DIFFERENTIA DIFFERENTIALL AXLE
A wheel and differential axle set-up different from a simple wheel machine because of its axle con�guration. Instead of prismatic single axle in simple wheel and axle, step-down axles are used in wheel and differential axle. This machine has a better mechanical advantage advantage as compared to single wheel and axle. Referring to Fig. 7.17(a) two axles of different diameters are coaxially �tted with a spindle, with which a wheel is also coaxially attached. The effort (P ) is applied through a wrapped string or rope wound around this wheel. Another string or rope is wound over two axles and carries a load ( W ) with the help of a movable pulley. The rope on the wheel and smaller axle are wound in the same direction, whereas that on the larger axle is in opposite direction. When an effort is applied through the wheel, the rope on the wheel and smaller axle gets unwound but gets wound on the larger axle, thus lifting the load. For a single rotation rotation of wheel and axles Distance moved by P through wheel p D Length of rope unwound unwound on smaller axle p d d2 Length of rope wound wound on larger axle p d d1 So, net length of rope wound wound on larger axle p ( d1 - d 2 )
Thus the load W gets gets lifted by a distance
p ( d1 - d 2 )
2 p D = 2D Hence, velocity ratio, V r = p ( d1 - d 2 ) / 2 ( d1 - d 2 ) Considering the free-body diagram of wheels and axle and taking moment equilibrium at point O, we obtain P ¥ D 2 or
+W ¥
P ¥ D =
2
d2 2
= W ¥
W ( d1 - d 2 ) 2
2
d 1 2
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Engineering Mechanics D d 1
Wheel Larger axle Spindle
D
d 2 O
Smaller axle
d 2
d 1
P
Ball bearing
W 2
Ball bearing
W 2
P ovable ova ble pulley pulle y W
W
Fig. 7.17(a) Schematicc diagram of wheel and differential axle Schemati
Fig. 7.17 (b) Free body diagram of wheel and differential axle
= W =
2D P ( d1 - d 2 ) As the velocity ratio and the mechanical advantage bear the same expresexpression, we can infer that ef�ciency is 100 per cent; thus an ideal machine with no friction. Let us evalua evaluate te the actual case case by by considering considering friction, friction, when when the actual effort is denoted by Pactual. Now,, for Now for a single rotation of wheel and axles, work done by the effort effort Pactual ¥ p D So, mechanical advantage, M A
Work done by the load
W
¥ p ( d1 - d 2 ) 2
So, the ef�ciency of the machine,
( d1 - d 2 ) ¥ p (d - d ) 2 h= = W ¥ 1 2 = M A 2D Pactual ¥ p D Pactual W
Thus actual mechanical advantage, advantage, M A
actual
7.12
=
actual
¥
( d1 - d 2 ) 2D
2D h ( d1 - d 2 )
WORM AND WORM WHEEL
A worm is a square-threaded screw and a worm wheel is a toothed wheel. In this machine, a worm and worm wheel are geared together maintaining their axes at right angle to each other. An effort wheel or pulley is attached to the worm coaxially so that effort ( P) can be applied through a rope wound over the pulley. pulley. A load drum is securely mounted coaxially coaxial ly on worm wheel and load is connected with a separate rope wound around the load drum. The pictorial and schematic views of this machine are shown in Fig. 7.18(a) and (b). For a single rotation of effort wheel, effort traverses a distance p D. D.
Simple Lifting Machines
435
Effort wheel Worm
Bearing
D Worm wheel
r
Load drum
P W
Fig. 7.18 (a) Worm and worm wheel
Fig. 7.18 (b) Schematic diagram
For an n-threaded worm, worm pushes the worm wheel through one tooth during a single rotation rotation of eff effort ort wheel. If the total number of teeth in a worm wheel is T , push of one
Ê n ˆ rotations. Thus, when the ÁË T ˜ ¯ radius of load drum is r, distance moved by the load 2p r ¥ n . tooth means the load drum traverses through
T p D DT Therefore, the velocity ratio, V r = . = 2p rn / T 2 nr In ideal condition, neglecting friction loss, mechanical advantage, M A = V r = DT 2nr For a single-threaded worm, in ideal condition, n 1 and thus M A = V r = DT . 2 r
7.13
CRAB AND WINCH
Crab and winch are machines used for hoisting heavy loads applying smaller amount of effort. These machines use gear systems in order to augment velocity ratio. Depending on the number of gear assemblies, crab and winch systems can be classi�ed into two types.
7.13.1
Single Purchase Crab and Winch
In this system, one set of gears gears,, one pinion of teeth T 1, and one spur wheel of teeth T 2 are deployed. The pinion is �tted coaxially with the effort axle and effort pulley, as shown in Fig. 7.19. Generally a rope is wound around the effort pulley or wheel of diameter d iameter D through which effort ( P) is applied. Effort then moves the pinion and thereby the spur wheel gets rotated. As the spur wheel is mounted coaxially with a load drum of diameter d , the load drum will get rotated. A strong rope is attached with load drum, at the end of which load (W ) is connected. Thus the load is lifted by the rotation of effort wheel.
436
Engineering Mechanics
Pinion T 1
Effort axle
D
Load drum
Effort pulley
d P
Spur wheel T 2 W
Fig. 7.19 Single purchase crab and winch
For a single rotation rotation of effort effort wheel, distance traversed traversed by the effort p D. For a single rotation of pinion, spur wheel and thereby the load drum rotate T 1 times T 2 T So, displacement of load p d ¥ 1 T 2
Hence, velocity ratio, V r
=
p D T p d ¥ 1 T 2
=D¥ d
T 2 T 1
In the absence of friction, the work done by the effort equals the work done by the load. So,
P ¥ pD = W
Ê T ˆ ¥ Á p d ¥ 1 ˜ T 2 ¯ Ë
W D T 2 Thus, mechanical advantage, M A = P = d ◊ T 1
As it is no-friction no-friction condition, velocity ratio and mechanical advanta advantage ge are are same and thereby ef�ciency is 100 per cent. When friction is present for actual machine, let us assume the actual effort is Pactual. Now, for a single rotation of effort wheel, work done by the effort Pactual ¥ p D
Work done by the load
W
Ê T ˆ ¥ Á p d ¥ 1 ˜ T 2 ¯ Ë
So, the ef�ciency ef�ciency of the machine,
Ê T ˆ ¥ Á p d ¥ T 1 ˜ Ë T 2 ¯ h= = W ¥ d ¥ 1 = M A Pactual ¥ p D Pactual D T2 W
actual
Thus actual mechanical advantage, M Aactual
=hD ¥ d
T 2 T 1
Ê T ˆ ¥ Á d ¥ 1 ˜ Ë D T2 ¯
437
Simple Lifting Machines
7.13.2
Double Purchase Crab and Winch
In this machine, two two sets of gear assemblies are used, as shown in Fig. 7.20. One additional axle, called an intermediate axle, is deployed. The pinion of teeth T 1 mounted on effort effort axle meshes with spur wheel of teeth T 2 mounted on the intermediate axle. Similarly, the pinion of teeth T 3 on intermediate axle meshes with spur wheel of teeth T 4 mounted on load drum. The effort (P) is applied with a rope wrapped around the effort wheel, and load ( W ) is attached to another rope wound around the load drum. Effort wheel Pinion T 1
Effort axle
D Pinion T 3
Load drum
Spur wheel T 2
P
Intermediate axle
d
Spur Wheel T 4 W
Fig. 7.20 Double purchase crab and winch
For a single rotation rotation of the effort wheel, distance traversed traversed by by the effort p D. For a single rotation rotation of pinion on effort axle, spur wheel on interm intermediate ediate T 1 axle rotates times. T 2 T 1 Now the pinion on the intermediate inter mediate axle also rotates times T 2
Ê T 1 T 3 ˆ times ÁË T2 ¥ T 4 ˜ ¯ Ê T T ˆ p d ¥ Á 1 ¥ 3 ˜ Ë T2 T 4 ¯
So, the spur wheel of the load drum rotates Thus the distance traversed by the load
T T p D =D¥ 2 ¥ 4 d T1 T3 T T p d ¥ 1 ¥ 3 T2 T 4 In an ideal condition, when there is no friction, mechanical advantage, Hence the velocity ratio, V r
MA
= V r = D ¥ d
T2 T1
¥
=
T 4 T 3
and ef�ciency is 100 per cent. In actual condition, considering the effect friction, actual mechanical advantage, M Aactual
= h D ¥ d
T2 T1
¥
T 4 T 3
438
Engineering Mechanics
7.14
WORM GEARED SCREWJACK
In order to attain large velocity ratio in a simple screwjack, a worm and worm wheel can be attached to a screwjack without changing its basic principle. principl e. The sectioned pictorial view of a worm geared screwjack is shown in Fig. 7.21. The effort may be applied with the help of either a handle of length L or a pulley of radius R, attached to the worm (not shown in �gure). Assume the number of start of threads in the the worm is n. For single start thread n 1, and for double start thread n 2. If an effort effort P is applied through one single rotation of handle or pulley, the worm moves the worm wheel through n teeth. Assume the pitch of the thread of the screwjack is p and the total number of teeth in the worm worm wheel is T . Now as the effort (P ) traverses traverses a distance of pulley, the load ( W ) vertically 2p R or 2p L, for a single rotation of handle or pulley,
moves a distance
np . T
Load Pad End Condition Jack must be attached to load and rotation must be restrained. Keyed machine screwjacks are available (wj 1000 and larger). Lifting Screw Standard end conditions, plain (t1), load pad (t2), (t2), threaded (t3) and male clevis (t4). Thrust Bearing Upper (shown) and lower (not shown) permit jack to bear load in both directions.
Sleeve Cap Threaded onto sleeve and secured with set screws.
Sleeve (Housing) Material varies based on size of jack.
Wormgear made from aluminum bronze material.
Grease fitting
Input Shaft (worm) Standard input shaft extends to the right and the left. Shaft modifications are available. Input Shaft Bearing One bearing supports at each end of the input shaft.
Input Shaft Seal Standard on 2-ton and large jacks.
Mounting Bolt Holes
Sleeve/Sleeve Cap Material Sleeve/Sleeve 250-lb–1-ton Aluminum 2-ton–35-ton Ductile iron 50-ton–250-ton Steel Options 2-ton 5-ton–25-ton 5-ton–35-ton
Bearing Cap 2-ton and larger jacks – smaller jacks have retaining retaining rings.
Stainless steel Stainless steel Steel
Protection Tube
(Courtesy : ©2011 Joyce/Dayton Corp.) screwjack k Fig. 7.21 Worm geared screwjac
Simple Lifting Machines
439
Thus the velocity ratio V r = 2p R np / T
= 2p RT [for pulley attachment]
or
V r = 2p L np / T
= 2p LT [for handle attachment]
7.15
WORM GEARED PULLEY BLOCK
np
np
Load drum In a worm geared pulley machine, an endless chain is wound around the t he effort pulley. pulley. A worm Effort pulley is �tted coaxially with the effort pulley, a spur Spur r wheel is geared with the worm, and a load drum is wheel mounted coaxially over the spur wheel. When the Worm effort (P ) is applied by pulling the endless rope or D chain, motion is transmitted transmitt ed from effort effort pulley to load drum through worm and spur wheel. A separate rope �xed to the rigid base passes over the Rigid base load drum and holds a movable pulley or snatch block and then is �xed at a separate point on the Movable rigid base again. The load ( W ) is attached to this pulley Endless chain movable pulley. A schematic diagram of worm W geared pulley block is shown in Fig. 7.22. In a single rotation rotation of effort pulley, effort ( P) Fig. 7.22 Worm geared moves through a distance p D. If the worm is pulley block single threaded, the worm pushes forward the spur wheel by by one tooth. If T is is the number of teeth of spur wheel, the spur wheel and thereby
the load drum rotates through
Ê 1 ˆ of a rotation. rotation. Hence the Ë T ¯
length of the rope between load drum and rigid base decreases by
Ê ˆ 2p r ¥ 1 . Ë T ¯
So, the distance the load moves
p r .
T Hence,, the velocity ratio of the machine, V = p D Hence r p r / T
= TD . r
RECAPITULATION
A simple machine is a mechanical device which can change change the direction and magnitude of a force or effort and makes work work easier. When two or more simple machines work together, they are called compound or complex machines. Six classical simple machines are: inclined plane, wedge, lever, screw, pulley, and wheel and axle. Load Loa d lif lifted ted Mechanical advanta advantage ge M A = = W Effort appl applied ied P distance travelled by by effort Velocity ratio V r = distance travelled by by load
440
Engineering Mechanics
Output power M A = V r Input power For an ideal machine, h 1, otherwise h < 1.
In an actual machine, for a given load W , actual effort Pactual
Mechanical ef�ciency h =
corresponding loss in effort due to friction, etc., is Ploss effort P, actual load to be hoisted Wactual decrease in load is Wdecrease
= hP ¥ V r and
the corresponding
=
b = 0 and thus W
1 and h max a
=
1 a V r
1 = M . A sin a In a simple screwjack having single-threaded screw screw of inclination q and and limiting angle of static friction f s , if P is the force generated at the circumference of the spindle and Q is the t he force applied at the lever end then For an inclined plane, Vr
=
and
+ f s )
Ê P + mp d ˆ Q = Wd Á 2 R Ë p d - m p ˜ ¯
(ii) for downward motion, P = W tan(f s
Ê ˆ For or a given give n = W Á 1 - 1˜ . F V r Ë h ¯
If eff effort ort P is required to hoist a load l oad W for for a machine, the relationship between = dP , and b is the load and effort P = aW + b , where a is is the slope of the line, a = dW initial effort at no-load condition. Thus mechanical advanta advantage, ge, 1 M A = W = W = P aW + b b a + W
(i) for upward motion, P = W tan(q
h
In a simple machine, if the effort is removed entirely entirely and the machine moves in the reverse direction and the load comes down to its initial position, it is called reversibility of the machine. But if the load does not move down and remains static at the position wherefrom the effort is removed removed,, it is called irreversibility or selflocking property property of the machine. machine. In self-locking condition, ef�ciency is less than 0.5.
M Amax
V r
= P ¥ V r (1 - h ).
At the extremi extremity, ty, when W Æ •,
= W ¥ 1 , and
and
- q )
Ê mp d - p ˆ . + m p ˜ ¯
Q = Wd Á 2 R Ë p d
Ef�ciency of the screwjack during hoisting load, h =
tan q . tan(q + f s )
The condition for attaining maximum ef�ciency, ef�ciency, (2q
+ f s ) = p and
thus hmax
1 - sin f s = . 1 + sin f s
The expression for torque,
Ê r1 + r 2 ˆ + W tan(q + f s ) d ˜ 2 Ë 2 ¯
Q × R = m W W Á
2
441
Simple Lifting Machines
In a differential screwjack, velocity ratio, W ( p2 - p1 ) V r = 2p R and ef�ciency, h = p2 - p1 2p RP In a �rst-order pulley system, for n number of movable pulleys, Vr
= M A = 2 n.
In a second-order pulley system, for n number of pulleys in the bottom block,
= M A = 2 n. In a third-order pulley system, for total n number of pulleys in the system, Vr = M A = ( 2 )n - 1. Vr
2R R-r T ¥ T = 1 3. T2 ¥ T 4
In a Weston differential pulley block, M A
In a gear pulley block, V r
In a simple wheel and axle M A
=
T 1
Ê T 2 ˆ T ÁË T 3 ˜ ¯ 4
= V r = R r
mechanicall advantage, ( M A )ac mechanica acttua uall
=
= V r .
in no-friction condition. In reality, actual
Ê R ˆ . = hac actu tual al ¥ Á Ë ˜ ¯ r
In wheel and differential axle, for an idealized condition, Vr In actual condition, M Aactual
=
2 Dh . ( d1 - d 2 )
=
2 D ( d1 - d 2 )
= V r = DT .
In a worm and worm wheel, for idealized condition, M A
For single purchase crab and winch, in no-friction condition, M A In the presence of friction, M Aactual
d
T 2 . T 1
2 r T = D ¥ 2 d T 1
= V r .
For double purchase crab and winch, in no-friction case, mechanical advantage a dvantage,, T T T T M A = V r = D ¥ 2 ¥ 4 . In actual condition, M Aactual = h D ¥ 2 ¥ 4 . d T1 T 3 d T1 T 3 In worm geared screwjack, V r V r
= h D ¥
= M A .
=
2p L np / T
=
2p R np / T
=
2p RT [for pulley attachment] or np
= 2p LT [for handle attachment] np
In worm geared pulley block, V r
=
p D p r / T
= TD r
NUMERICAL EXAMPLES A Example 7.1 A handle drives pinion D, which in turn drives drum M, through gear wheels A, B and C, as shown in Fig. M E7.1. The length of the handle and the diameter of drum M are 170 mm and 85 mm, respectively. Wheel D has 25 teeth gearing with B of 75 teeth and wheel C W has 25 teeth gearing with A of 100 teeth. Considering the ef�ciency of the system as
B
Handle
C D
Fig. E7.1
442
Engineering Mechanics
64.5%, compute the weight W that that can be raised by the drum M, if an effort of 250 N is applied through t hrough the handle. Let the revolutions made by the wheels at a �xed time interval be denoted by RA, RB, RC, and RD and the number of teeth in the corresponding wheels are T A, T B, T C, and T D. Solution
Now,
R B R D
=
TD TB
RA RC
and
=
Multiplying the above relations,
T C T A R B R D
¥ RA T D ¥ T C = . ¥ RC TB ¥ T A
(1)
As wheels B and C are mounted on same shaft, RB RC. Substituting in Eq. (1) and with T D 25, T B 75, T C 25, T A 100
R A R D
=
25 ¥ 25 75 ¥ 100
=
1 12
While the handle makes one revolution, wheel D revolves once, by virtue of which wheel A revolves 1 times. Thus drum M revolves 1 times. In one revolution of 12 12 handle, effort thus moves through (2 π × × 170) mm, and the load attached on drum
Ê 1 ¥ p ¥ 85ˆ mm. ÁË 20 ˜ ¯ p (V r ) = 2 ¥ 170 = 80 1 ¥ p ¥ 85
gets lifted by an amount So, the velocity ratio
20 Here ef�ciency h 0.645 and eff ort ort P 250 N.
Now
h = W or 0.645
or
W 12900
PV r
W 250 ¥ 80
So, the magnitude of load to be lifted is 12.9 kN. machine, an effort of 310 N raised a load of 10,000 N. Wha Whatt Example 7.2 In a lifting machine, is the mechanical advantage? If the ef�ciency is 0.75, what is the velocity ratio? If on this machine, an effort of 610 N raised a load of 20,000 N, what what is the new ef�ciency? What will be the effort required to raise a load of 5000 N? What is the maximum mechanical advantage and what is the maximum ef�ciency? For P 310 N and W 10,000 N 10,000 Mechanical advanta advantage ge M A = W = = 32.258 P 310 Solution
Now,
10,000 h = W or 0.75 =
or
V r 43.01
PV r
310 V r
So, the velocity ratio is 43.01. Considering the same velocity ratio, for P 610 N and W 20,000 N, 20,000 = 0.7623 or 76.23% ef�ciency h = 610 61 0 ¥ 43 43.0 .01 1 The linear relation between load and effort can be idealized as, P = a W + b Substituting the two sets of values of P and W , we obtain 310 10,000 a + + b
610 20,000 a + + b
(1) (2)
Simple Lifting Machines
443
Solving Eqs (1) and (2), we obtain a 0.03 and b 10. So, the governing equation: P 0.03 W + + 10 Hence, the effort required to raise a load of 5000 N will be P 0.03 × 5000 + 10 160 N
Thus mechanical advanta advantage ge M A
For M A to be maximum, Hence
M A
max
=
1
a
=
=W = P
W
a W + b
1
=
a +
b W
Ê b ˆ is to be minimum; and this is possible for W . ÁË W ˜ ¯
1 0.03
= 33.33
Thus, maximum ef�ciency hmax
=
M A
max
V r
= 33.33 = 0.7749 or 77.49% 43.01
screwjack has a mean diameter 80 mm and pitch 15 mm. The Example 7.3 A screwjack coef�cient of friction between its screw and nut is 0.075. Find the effort required to be normally applied at the end of its operating lever 800 mm long to (i) raise a load of 2 kN, and (ii) lower the same load. Find the ef�ciency under this load. Solution
Here, m tan f 0.075, d 80 mm and p 15 mm
So,
tan a =
Now,
tan (a
+ f ) =
tan a + tan f 0.0596 5968 8 + 0.0 0.075 75 = 0.0 1 - tan a tan f 1 - 0.05968 ¥ 0.075
= 0.13528
and
tan (f - a ) =
tan f - tan a 0.075 75 - 0.0 0.0596 5968 8 = 0.0 1 + tan f tan a 1 + 0 .0 .05968 ¥ 0.075
= 0.01525
p = 15 = 0.05968 p d 80p
Effort required at the circumference of the screw to raise 2000 N load
= W tan(a + f ) = 2000 ¥ 0.13528 = 270.56 N So, the effort required at the end of the lever 270. 0.56 56 ¥ 40 = 13.528 N = 27 800
Corresponding ef�ciency
W tan a W tan (a + f )
= 0.05968 = 0.44 0.4412 12 or 44.12% 44.12% 0.13528
Effort required at the circumference of the screw to lower 2000 N load
= W tan (f - a ) = 2000 ¥ 0.01525 = 30.5 N So, the effort required at the end of the lever
= 30.5 ¥ 40 = 1.525 N 800
screwjack is used horizontally horizontally in sliding a bed-plate into position on Example 7.4 A screwjack its foundation. The bed-plate weighs 4 kN and coef�cient of friction between it and the foundation is 0.25. The screw of the jack has a mean diameter of 50 mm and a pitch of 12.5 mm, and the coef�cient coef�cient of friction is 0.1. The axial thrust is carried on a collar of mean diameter 80 mm for which the coef�cient of friction is 0.15. Find the torque required on the jack and the ef�ciency of the operation. Solution
We know torque required for a screwjack
444
Engineering Mechanics
Ê r + r ˆ = Pl = mW Á 1 2 ˜ + W tan (f + a ) d 2 Ë 2 ¯ Here W is is equal to the friction force between the bed-plate and its foundation so that its value is 4000 × 0.25 1000 N. Now, m coef�c coef�cient ient of friction of collar 0.15 r1 + r 2 = 40 mm = mean radius of collar = 80 2 2
tan a =
p = 12.5 = 0.07957 p d 50p
tan f coef�cient of friction of the screw 0.1
So,
tan (a
+ f ) =
tan a + tan f 0.07 0795 957 7 + 0. 0.1 1 = 0.181 = 0. 1 - tan a tan f 1 - 0. 0 .07957 ¥ 0.1
Hence the torque
= 0.15 ¥ 1000 ¥ 40 + 1000 ¥ 0.181 ¥ 50 2
= 10525 Nmm thread 7.5 mm mean diameter and 15 mm Example 7.5 A screwjack has a square thread pitch. The load on the jack revolves with the screw. The coef�cient of friction at the screw thread is 0.05. (a) Find the tangential force required at 360 mm radius to lift a load of 6 kN, and (b) state whether the jack is self-locking. If it is, �nd the torque necessary to lower the load. If it is not, �nd the torque which must be applied to keep the load from descending. Here p 15 mm, d 75 mm, W 6000 N, m tan f 0.05
Solution
(a)
So,
tan a =
and
tan (a
p = 15 = 0.06366 p d 75p
+ f ) =
tan a + tan f 0.06 0636 366 6 + 0. 0.05 05 = 0. 1 - tan a tan f 1 - 0.06366 ¥ 0.05
= 0.114
Hence the force required at the circumference to lift 6000 N load W tan tan ( a + f ) 6000 × 0.114 684 N If P is the tangential force required at a radius of 360 mm, then
P ¥ 360 = 684 ¥ 75 or P 71.25 N 2 (b) As we �nd, tan a > > tan f , thereby a > > f . Thus, we can conclude that the jack is not self-locking. Now,, the force required at the circumference to Now t o prevent the load from descending
W tan tan (a – – f )
6000 ¥
0.06366 6 - 0.0 0.05 5 = 81.69 N 6000 ¥ 0.0636 1 + 0. 0.06 0636 366 6 ¥ 0. 0.05 05
So, the torque required 81.69
tan a - tan f 1 + tan a tan f
¥
75 2
3063.375 N mm.
Example 7.6 An inextensible string wound around all eight pulleys, four in each block, is shown in Fig. E7.6. The upper block is �xed with the ceiling and load to be lifted is attached with the lower block. One end of the string is tied to the bottom end of the upper block and the string is passed successively through the pulleys and the force P is applied at the end of the string. If the tension of the string as it passes
Simple Lifting Machines
445
over each pulley increases by 15%, compute the force that will over have ha ve to be applied at the end of the string to lift li ft 5 kN through this pulley system. Let P be the required effort to raise 5 kN load. Let us assume the tension in the string wrapped around largest pulley in the upper block is S . When the load W is is about to be lifted, overcoming the frictional force force,, we can write P 1.15 S Solution
P
S = P 1.15 Similarly, the tension in the successive portions of the string will be P , P , P 2 3 (1.15 ) (1.15 ) (1.15 )8 From the condition of equilibrium, we can say or
or
P (1.15 )1
+
P (1.15 )2
+
P (1.15 )3
+
È PÍ 1 Í 1.15
+
1 (1.15 )2
+
1 (1.15 )3
+
Î
È
or
P
8
1 Í1 - Ê 1 ˆ ˜ 1.15 Í ÁË 1.15 ¯
Î
1- 1 1.15
+
P (1.15 )8
+
1 ˙ = 5000 (1.15 )8 ˚˙
= 5000 ˘
W
Fig. E7.6
˘ ˙ ˙˚ = 5000
8
or or
P
Ê ˆ 1 - Á 1 ˜ Ë 1.15 ¯
0.15 P 1114.25
= 5000
Hence the required force is 1114.25 N. Example 7.7 In a Weston differential pulley block, the larger pulley has 12 recesses while the smaller has 10. Compute Compute the ef�ciency of of the machine if an effort effort of 1235 N is required to lift a load of 13.5 kN. kN. Solution
Here P 1235 N, W 13500 N , N 1 12, N 2 10
So, the velocity ratio V r
=
2 N 1 N1 - N 2
W Hence the ef�ciency h = PV r
=
=
2 ¥ 12 12 - 10
13500 1235 ¥ 12
= 12
0.91 9109 09 or 91 91.0 .09% 9% = 0.
and wheel wheel are Example 7.8 In a simple wheel–axle system, the diameters of axle and 120 mm and 500 mm, respectively. respectively. The thickness of the rope is 10 mm. Find the velocity ratio. While hoisting a load of 3000 N at 15 m per minute, if the ef�ciency becomes 75%, determine the power necessary for supply to the machine m achine.. Solution
Mean radius of wheel and rope
255 5 mm. = 500 + 10 = 25
Mean radius of wheel and rope
mm.. = 120 + 10 = 65 mm
2
2
446
Engineering Mechanics
So, distance travelled travelled by effort per revolution of wheel 2 p × 255 510 p mm. mm. and distance travelled by load per revolution of axle 2p × 65 130 p mm. mm.
Veloci elocity ty ratio V r
= 255 = 3.923
65 Work done by the machine per minute 3000 × 15 Nm
So, power obtained from the machine
= 3000 ¥ 15 = 750 watt
Considering ef�ciency, supplied power
60 750 = 1000 watt 0.75
=
kW = 1 kW
Two rigidly connected shafts A and B of a differential winch are driven by handle C of length R 600 mm. A load D of weight A Q 720 N is �xed to the moving moving pulley E driven by by a rope. When When the handle C starts rotating, the left side of the rope uncoils from from the shaft A of radius r1 100 mm, while the right side coils on the shaft B of radius r2 120 mm. Calculate for this mechanism (a) the E magnitude of force P required at C in order to lift the weight, (b) the mechanical advantage, and (c) the D velocity ratio.
Example 7.9
P
R C
In one revolution of C, the effort moves through a distance 2p R. The length of rope that uncoils from A 2p r1 and coils on B 2p r2 Hence, decrease in length of rope passing around D 2p (r2 – r1). So, during lifting, load passes through a distance p ( (r2 – r1). Solution
Fig. E7.9
Therefore,, velocity ratio V r = Therefore
2p R p ( r2 - r 1 )
2p ¥ 600 p (120 - 100)
=
= 60
As ef�ciency has not been provided, provided, let us assume an ideal machine ha having ving 100% ef�ciency. So, mechanical advantage M A 60
Now,
h = W
or
1 = 720 P ¥ 60
PV r
or
P 12 N.
crab and winch, following is the relevant data: data: Example 7.10 For a single purchase crab Effort 250 N; Number of teeth on spur wheel wheel 150, Number of teeth on pinion 30, Pitch of teeth 18 mm, Diameter of drum 200 mm, Length of the handle 420 mm, Diameter of lifting rope 15 mm.
Compute velocity ratio. Determ Determine ine the pressure between the teeth for (a) no-frictio no-friction n condition and (b) ef�ciency 75%. Solution
Here, length of the handle ( R) 420 mm. Mean radius of drum and rope (r) 200 + 15 2 Number of teeth on wheel (T w) 150
= 107.5 mm
Number of teeth on pinion (T P) 30
Now, velocity ratio ( V r)
R r
¥
T w T P
=
420 ¥ 150 107.5 30
Circumference of pitch circle of pinion
= 19.5349
T P × pitch 30 × 18 540 mm.
Simple Lifting Machines If r p is the radius of pitch circle, 2 p r p
540
or
447
r p 85.9437 mm.
P¥R r p
250 0 ¥ 420 = 1221.73 N = 25
(a)
In no-friction condition, pressure between teeth
(b)
Considering 75% ef�ciency ef�ciency,, pressure between teeth 1221.73 × 0.75 916.29 N
85.9437
purchase crab crab and winch, as shown in Fig. E7.11, the Example 7.11 In the double purchase crank handle H and pinion A are �xed to the shaft MN. Wheel D and pinion E are �xed together but loose on the shaft MN. Wheel F and drum are als o �xed together but loose on the shaft QR. Wheel B and pinion C are �xed to the shaft QR. The number of teeth in i n wheels and pinion are T A 18, T B 72, T C 20, T D 60, T E 18, T F 72. The effective diameter of drum is 300 mm and radius of crank is 500 mm. Determine the effort required at the crank to lift a weight of 68 kN.
D
A
E
M
N
H Drum Q
R C
F
B
W
Fig. E7.11
For each rotation of the crank handle, the effort moves through a distance of 2p × × 500 1000 p mm. mm. During rotation of crank handle, pinion A rotates and thus drives wheel B. As wheel B and and pinion C are are rigidly connected to shaft QR, due to rotation of B, pinion C also also rotates and thus it drives wheel D. As wheel D and pinion E are rigidly connected, due to rotation of D, pinion E rotates and thus it drives wheel F. F. As F is rigidly connected to drum, due to the rotation of F, drum will rotate.. Hence we can say that pinions A, C, and E are drivers, while wheels B, D, and rotate F are followers. So, we can write, Solution
Rotationall spee Rotationa speed d of drum Rotationa Rota tionall spee speed d of cra crank nk hand handle le
=
Rotationall spee Rotationa speed d of fol followe lowerr F Rotationa Rota tionall spee speed d of drum A
T A T B
¥
T C TD
=
¥
T E T F
= 18 ¥ 20 ¥ 18 = 72
60
72
1 48
Thus, when crank handle rotates once, drum revolves 1 rotations and thus the 48 distance travelled by weight W during during hoisting p ¥ ¥ 300 ¥ 1 mm 6.25 p mm. mm. 48
So, velocity ratio V r
1000 =
p
6.25 p
=
160
448
Engineering Mechanics
Hence the effort required to hoist 68 kN weight
68,000 160
= 425 N.
geared screwjack, screwjack, number number of teeth of worm wheel is 110 Example 7.12 In a worm geared and pitch of the screw is 11 mm. Determine the amount of load that can be lifted by an effort of 250 N applied through a 300 mm long handle, considering the worm is single threaded and of ef�ciency 20%. In one revolution of handle, effort moves through a distance of 2p × × 300 600 p mm mm Thus when worm wheel is pushed forward by one tooth, the worm wheel and als o the screw of the jack rotates through 1 of a revolution. Hence load is lifted through 110 11 0.1 mm 110 So, velocity ratio V r = 600p = 18849.56 0.1 Solution
Using the relationship h = W , we obtain W hPV r PV r 0.20 × 250 × 18849.56
942478 N.
So, the amount of load to be lifted is 942478 N.
EXERCISES Multiple Choice Questions 7.1
7.2
7.3
Which of the following is/are not simple machine(s)? (a) inclined plane (b) bench vice (c) screw (d) lever Which of the following combination of simple machines machines comprises a bench vice? (a) Inclined plane and screw (b) wedge and pulley (c) lever and pulley (d) screw and lever Which of the following is the correct relationship amongst mechanical advantage (M A), velocity ratio (V r), and mechanical ef�ciency ( h)? (a) M A hV r (b) V r hM A (c) M AV r h (d) M AV r h 1 In an ideal simple machine, machine, which of the following following relation holds good? (a) M AV r 1 (b) M A < V r (c) M A V r (d) M A > V r In an actual simple machine, machine, which of the following following relation holds good? good? 2 (a) h 1 (b) h 1 (c) h > 1 (d) h < 1 Which of the following following is the correct expression for mechanical advanta advantage ge of a compound machine comprising three simple machines?
7.4 7.5 7.6
= M A1 ¥ M A2 ¥ M A3 M + M A 2 + M A3 M A = A1
(a) M A
(c)
3
(b) M A
= M A1 + M A2 + M A3
(d) M A
=
1
M A1
+ M A 2 + M A3
Simple Lifting Machines
449
7.7
Which of the following following is the correct expression for mechanical ef�ciency of a compound machine comprising three simple machines?
(a) h = h1 ¥ h 2
(b) h = h1 + h 2
(c) h =
(d) h =
7.8
In an actual machine, machine, which of the following following is the valid valid expression expression for for loss of effort, for a given load?
(a) Ploss
(b) Ploss
(c)
(d)
7.9
In an actual machine, machine, which of the following following is the valid expression for decrease of load, for a given effort? (a) Wdecrease = P ¥ V r (h - 1) (b) Wdecrease = P ¥ V r (1 - h )
(c) W decrease
¥ h3 h1 + h2 + h3 3
Ê 1 - h ˆ = W Á V r Ë h ˜ ¯ V Ê 1 - h ˆ Ploss = r Á W Ë h ˜ ¯
=
P (1 - h ) V r
1
+ h3
h1 + h2
+ h3
Ê h - 1 ˆ = W Á V r Ë h ˜ ¯ V Ê h - 1 ˆ Ploss = r Á W Ë h ˜ ¯
(d) W decrease
=
V r (1 - h ) P
7.10 To attain attain the condition of irreversibility of a machine, the value value of mechanical ef�ciency h is (a) more than 0.50 (b) more than 0.75 (c) exactly 1.0 (d) less than 0.50 7.11 Velocity ratio ratio for for an ideal inclined plane (a) sin a (b) sin -1 a 1 1 (c) (d) sin a sin -1 a 7.12 In a simple screwjack, if q is is the inclination of the thread and f s is the limiting angle of static friction, the necessary condition for attaining self-locking is (a) f = 1 (b) q > f s s
q (c) q < f s
1 (d) q = f s
7.13 In a simple screwjack, if q is the inclination of the thread and f s is the limiting angle of static s tatic friction, the necessary condition conditi on for attaining maximum ef�ciency during hoisting of load is p (a) (q + f s ) = p (b) (q + 2f s ) = 2 2
(c)
(2q
+ f s ) = p
(d) (2q
(c)
22 n
-1
(d) 2n +1
+ f s ) = p
2 4 7.14 In a simple screwjack, if q is the inclination of the thread and f s is the limiting angle of static friction, the expression for maximum ef�ciency during hoisting load is 1 - sin f s 1 (a) (b) 1 - sin f s 1 + sin f s 1 + sin f s 1 (c) (d) 1 + sin f s 1 - sin f s 7.15 The velocity velocity ratio ratio of a �rst-order pulley pulley system comprising n number of moving pulleys is (a) 22 n (b) 2n
450
Engineering Mechanics
7.16 The mechanical mechanical advantage advantage of a third-order pulley system comprising n number of pulley pulleyss is (a) 2n - 1 (b) 2n +1 (c) 2n -1 (d) 2n + 1 7.17 In a simple gun gun tackle, tackle, how much is mechanical mechanical advantage? advantage? (a) 1 (b) 2 (c) 3 (d) 4 7.18 Which of the following is the correct relationship between mechanical advantage and mechanical ef�ciency of a simple wheel and axle having diameter D and d, respectively,, in actual condition? respectively
(a) h = 1
M A
¥
d D
(b) h = M A
¥ D
d
d (d) h = M A ¥ 2 d D D 7.19 In a differential wheel wheel and axle, axle, the actual mechanical advanta advantage ge can be expressed as 2 Rh Rh (a) (b) ( r + r 2 ) 2( r1 - r 2 )
(c) h = M A
(c)
¥
Rh ( r + r 2 )
(d)
2 Rh ( r1 - r 2 )
7.20 In an n-threaded worm and worm wheel, velocity ratio can be expressed as (a) rT (b) RT nR 2 nr (c) 2 RT (d) RT nr nr 7.21 Mechanical advantage advantage for a single purchase crab crab and winch in actual condition condition can be expressed as T T (a) D ¥ 2 (b) h D ¥ 2 d T 1 d T 1 T T (c) h D ¥ 1 (d) d ¥ 1 d T 2 D T 2 7.22 Mechanical advantage advantage for for a double double purchase crab crab and winch in ideal condition can be expressed as T T T T (a) h D ¥ 2 ¥ 4 (b) D ¥ 2 ¥ 4 d T1 T 3 d T1 T 3
T T T T (c) D ¥ 1 ¥ 4 (d) h D ¥ 2 ¥ 3 d T2 T 3 d T1 T 4 7.23 Velocity ratio of a single-t single-threaded hreaded worm worm geared pulley block can be expressed expressed as
(a)
Tr 2 D
(b) TD 2 r
TD (d) Tr r D 7.24 In a pulley attached attached worm geared geared screwjack with double start thread, velocity ratio is expressed as 2p RT RT (a) p (b) p p p RT 2p RT (c) (d) 2 p np
(c)
7.25 In a handle handle attached worm geared screwjack with single start thread, velocity velocity ratio is expressed as
Simple Lifting Machines 2p LT np
(a)
2p LT p
(b)
(c)
p LT
LT (d) p
p
451
2 p
Review Questions 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9
7.10 7.11
7.12 7.13
7.14 7.15 7.16 7.17 7.18 7.19 7.20
Write down the basic characteristics of a simple machine. machine. What What do you mean by a complex machine? Write short notes on (i) mechanical advanta advantage ge (ii) velocity velocity ratio ratio (iii) mechanical ef�ciency. Whatt are the differences Wha differences between an actual machine machine and an ideal ideal machine? Whatt do Wha do you mean by ideal load and ideal effort? Derive for a real machine, the expressions expressions for for actual load for for a given effo effort rt and the decrease in load amount. Derive for a real machine, the expressions expressions for for actual effort for a given given load and corresponding loss in effort due to friction, wear, etc. Distinguish between the properties of reversibility and irreversibility of a machine machine.. Provee that the limiting value of ef�ciency to attain Prov attain self-locking property of a machine is 0.5. Write down the law law of machine and explain it with suitable suitable sketches sketches for for every every possible case. Derive the expression for maximum mechanical advantage and maximum ef�ciency. With the help of suitable schematic schematic diagrams, diagrams, explain the working working principle of a simple screwjack. For a simple screwjack, derive derive the expressions for for effort effort applied at the circumference of spindle and force applied at the end of the lever, when the load is hoisted upwards upwards.. Whatt is self-locking condition for Wha for a simple screwjack? screwjack? Is it advantageous advantageous and how can it be achieved? While hoisting a load using a simple screwjack, deduce the expression for for mechanical ef�ciency. ef�ciency. What is the condition for attaining maximum ef�ciency and derive its expression? Explain with suitable suitable sketches, sketches, how a differential screwjack is different from a simple screwjack. Whatt are the various systems of pulleys? In each case, Wha case, derive the expression for for velocity ratio. Describe the Weston differential differential pulley block. How is a wheel wheel and differential axle different different from a simple wheel and axle? Compare the actual mechanical advantage of both. Amongst single purchase and double purchase crab crab and winch, which one is advantageous for use and why? Describe the working working principles of a double purchase crab crab and winch with a neat sketch. Explain the working principle of worm geared geared screwjack. screwjack.
Numerical Problems 7.1
A double square-threaded screwjack screwjack of lead 12 mm has coef�cient of friction 0.15. Outer diameter diameter of the screw is 55 mm and length of the lever rod rod is 650 mm. Compute the force required to (a) hoist and (b) lower a load of 4500 N.
452 7.2
Engineering Mechanics The mean diameter of of a single square-threaded screw of 6 mm pitch is 60 mm. The head consists of two parts so that the load does not rotate with the screw. The upper part of the head rests on the lower part through ball bearings having outer diameter 90 mm and inner diameter 70 mm and coef�cient of friction µ 0.05. The coef�cient of friction between the nut and the screw µ2 0.1. Evaluate the amount of torque required to hoist a load of 5000 N. A screwjack screwjack having having mean diameter 90 mm, pitch of the screw 15 mm, and coef�cient of friction 0.1 is used to raise a load of 4500 N through 165 mm. Determine the torque required, work done, and ef�ciency of the machine. In a simple wheel wheel and axle, as shown in Fig. P7.4,W 6500 N, R 325 mm, r 175 mm, coef�cient of friction in spindle bearing 0.12. It weighs 2.5 kN and is mounted on a spindle of diameter 60 mm. Compute (a) velocity ratio, (b) effort required, (c) mechanical advanta advantage, ge, and (d) ef�ciency ef�ciency.. A differential pulley lifting tackle in which the upper block consists of two pulleys of diameters D and d, rotating on a �xed axis and a movable pulley below, to which is attached the load, is shown in Fig. P7.5. The tackle is operated by an endless chain and effort P is applied to length coming off the larger pulley. pulley. If D 350 mm, d 250 mm, ef�ciency 80%, determine the t he velocity ratio. ratio. Compute the effort to lift a load of 850 N.
7.3
7.4
7.5
O
r R
P
W
Fig P7.4 d O P
r
r
s
l W
s
A
P
A
Fig P7.5 W
7.6
7.7
1200 N
In a double purchase crab and winch the Fig. P7.7 number of teeth in spur wheels are 50 and 45, while those in pinions are 20 and 25. The length of the handle with which effort is applied is 450 mm long. The effective effective diameter of the drum is 160 mm. Assuming the ef�ciency of the winch as 42%, determine the amount of load that can be lifted by applying an effort of 200 N. Compute the pressure between teeth of each pair of wheels when the pitch of the teeth is 25 mm and friction is neglected. A load W of 1.2 kN is to be lifted by the pulley arrangement, as shown in Fig. P7.7. Determine the magnitude of effort P for hoisting the load if r 300 mm, d 75 mm, and the coef�cient of friction in the journals supporting large
Simple Lifting Machines
453
pulley m 0.3. Neglect friction of the small pulley. Compute also the ef�ciency of the system. 7.8 The table of a planing machine is driven driven by by a screw of pitch 60 mm. The driving pulley on the screw has a diameter of 800 mm and the difference in tensions of the belt on it is 950 N. If the coef�cient of friction between the table and its guide is 0.08 and the ef�ciency of the screw is 55%, determine the weight of the table. 7.9 In an unloaded pulley block, initial frictional resistance is 22 N and and it increases at the rate 17 N per kN load lifted by the block. If the velocity ratio of the pulley-block is 17.5, calculate the amount of effort required to raise 13 kN load and also �nd the ef�ciency of the machine at that time. 7.10 A double purchase purchase crab crab and winch a has number of teeth on spur wheels wheels as 80 and 95, on pinions 25 and 35, effective diameter of loading drum 175 mm, and length of the handle providing effort 380 mm. In test condition it was found that 100 N and 135 N effort can hoist loads of 1900 N and 3200 N, respectively. respectively. Determine Determ ine the law of machine, mach ine, required effort for for hoisting hoisti ng 4500 N load, and corresponding ef�ciency of the machine. Also determine the maximum m aximum ef�ciency. ef�ciency.
