MECHANICAL PRINCIPLES OUTCOME 4 MECHANICAL POWER TRANSMISSION TUTORIAL 1 SIMPLE MACHINES

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1 MECHANICAL PRINCIPLES OUTCOME 4 MECHANICAL POWER TRANSMISSION TUTORIAL 1 SIMPLE MACHINES Simple machines: lifting devices e.g. lever systems, inclined plane, screw jack, pulley blocks, Weston differential pulley block, simple and compound gear trains, wheel and axle, differential wheel and axle; mechanical advantage, velocity ratio, law of a machine, force, torque and power transmitted, overhauling, limiting efficiency Belt drives: flat and v-section; velocity ratio, angles of lap, limiting tight and slack side tensions, maximum power transmitted for given conditions Linkage mechanisms: plane mechanisms eg slider-crank, four-bar linkage and quick return mechanisms; determination of magnitude and nature of output velocity for given input conditions On completion of this tutorial you should be able to do the following. Describe the principles of lifting machines. Define Velocity Ratio. Define Mechanical Advantage. Explain the efficiency of a machine and the factor affecting it. Explain the law of a machine. Explain the limiting efficiency. Describe a range of lifting machine including: Levers Screw Jacks Pulleys Differential Axles Weston Differential Pulley Geared Systems Winches Calculate the force required to slide a block on an inclined plane. Relate the inclined plane to screw threads. Calculate the torque needed to rotate a turnbuckle. Explain Mechanical Power Make calculations involving Mechanical power. It is assumed that the student is already familiar with friction theory. D.J.Dunn freestudy.co.uk 1

2 1. GENERAL INFORMATION FOR LIFTING MACHINES A lifting machine is any machine designed to enable a load (F L ) to be raised by a much smaller effort (F E ). The ratio is called the Mechanical Advantage or Force Ratio. M.A. = F L /F E The distance moved by the load is X L and the distance moved by the effort is X E. The ratio of the two is called the Velocity Ratio or Movement Ratio. V.R. = X E / X L If we plot a graph of Effort against Load for an ideal frictionless machine we get a straight line through the origin as shown. The gradient of the straight line is F E /F L = 1/M.A. The law of energy conservation tells us that if there is no energy lost, the work done by the effort must equal the work done by the load so it follows that : F E X E = F L X L In a real machine, there is friction and energy is lost. The effort has to overcome the dead load of the lifting machine so some effort F S is required before any external load is applied. If F L is the external load and this is plotted against effort, we get a graph as shown labelled actual. EFFICIENCY Symbol η (eta) η = Work done by the load/work done by the Effort η = F L X L /F E X E η = M.A./V.R This is the same formula for ideal and actual machines so long as we use the actual load and effort. When the external load is zero the Effort is F s and the Mechanical Advantage is zero so it follows that the efficiency is zero. At large loads the Mechanical Advantage tends to become constant and the efficiency reaches a limiting value. In between these extremes the efficiency varies and the plot is a curve as shown. Figure 1 The law of the straight line is F E = F S + F L x gradient F E = F S + F L x B/A F E = F S + F L M.A. F E = F S + F L (V.R. η) D.J.Dunn freestudy.co.uk 2

3 WORKED EXAMPLE No.1 The table shows the values of load and corresponding effort for a lifting machine with a velocity ratio of 140. Plot the graph and also plot the efficiency. Load F L (N) Effort F E (N) Calculate the efficiency at each point using η % = (M.A./V.R) x 100% = (M.A./40) x 100% For example when F L = 400 N M.A. = 400/17 = η % = (23.54/40) x 100 = 59% Load F L (N) Effort F E (N) η % Figure 2 SELF ASSESSMENT EXERCISE No.1 1. A lifting machine has a velocity ratio of 5. The Effort required to raise a load of 5 KN is 2 kn. Calculate the mechanical advantage and the efficiency. (Answer 2.5 and 50%) 2. A Lifting machine has a velocity ratio of 8. Given the efficiency is 30% when the load is 580 N, calculate the effort required. (Answer 242 N) Next we will look at particular lifting devices. D.J.Dunn freestudy.co.uk 3

4 2. LEVERS A lever is one of the simplest lifting devices but it is limited to small movements. It is very efficient however and it is found in many mechanisms. A crow bar is an example of a lever used to lift heavy objects a small distance. The lever principle is used in cutting tools such as pliers, wire cutters, bolt cutters and branch loppers. Figure 3 The ratio of the movements is strictly related to the lengths A and B so the velocity ratio is V.R. = X E / X L = A/B WORKED EXAMPLE No.2 The load on a lever is 2 kn and acts 20 mm from the fulcrum. The effort is 200 mm from the fulcrum. Assuming 100% efficiency, calculate the effort. η = M.A./V.R = 1.0 (100%) hence M.A. = V.R V.R = A/B = 200/20 = 10 M.A. = 10 = F L /F E = 2000/ F E F E = 2000/10 = 200 N SELF ASSESSMENT EXERCISE No A lever has a total length of 1.2 m with the pivot 0.2 m from one end. Assuming 100% efficiency, calculate: (i) The velocity Ratio. (6) (ii) The Mechanical Advantage. (6) (iii) The effort required to move a load of 2000 N. (333.3 N) (iv) The distance moved by the end of the handle when the load is raised 50 mm. (8.33 N) D.J.Dunn freestudy.co.uk 4

5 3. SCREW JACKS Screw jacks work on the principle that the nut forms the lifting platform or point and the screw is rotated with a lever. Every complete revolution of the screw raises the load by the pitch of the thread. The Effort is applied at radius R to produce a torque F E R. They are commonly used as car jacks in various forms such as the scissors screw jack. There is a lot of friction in screw threads so they are not very efficient. In order to understand this you need to study friction on inclined planes. Figure 4 The distance moved by the effort when the screw is turned once = 2πR The distance moved by the load = pitch = p The velocity ratio = 2πR / p WORKED EXAMPLE No. 3 A screw jack has a thread with a pitch of 5 mm. It must raise a load of 6000 N by turning the thread with a handle 500 mm long. The efficiency is 20% and the effort required to overcome the dead weight and friction with no external load is 20 N. Calculate the effort required. The velocity ratio =2πR / p = 2π x 500/5 = We need to use the law of the machine in this case. F E = F S + F L ( V.R. η) F E = (628.3 x 0.2) = 67.7 N SELF ASSESSMENT EXERCISE No A screw jack has a thread with a pitch of 6 mm. It must raise a load of 2 kn by turning the thread with a handle 250 mm long. The effort is 25 N Calculate the efficiency. (30.6 %) D.J.Dunn freestudy.co.uk 5

6 4. PULLEYS A pulley is two sets of wheels as shown. The rope starts from the axle of one set and goes around the pulleys wheels before coming off to the point where the effort is applied. Each rope between the two sets of wheels gets shorter by the same amount so if there are N ropes, the distance moved by the effort is N times more than the distance moved by the load so the velocity ratio is N. V.R. = Number of ropes connecting the blocks. If the end of the rope is attached as shown the V.R. will always be an even number. The number of pulley wheels in each block will be half this. For example two wheels in each block will give 4 ropes. If the rope is attached to the bottom block instead, the V.R. will always be on odd number with one extra wheel in the top block. Figure 5 WORKED EXAMPLE No. 4 A pulley as shown is 50% efficient. Calculate the effort required to lift 12 KN if there are 6 lengths of rope between the blocks. If the dead weight of the pulleys and hook is 500 N what would the effort be then? V.R. = 6 M.A. = η x V.R. = 50% x 6 = 3 Effort = 12/3 = 4 kn With the dead weight taken into consideration the total load is 12.5 kn Effort = 12.5/3 = 4.18 kn SELF ASSESSMENT EXERCISE No A pulley system similar to that shown above has 5 ropes. The load is 850 N and the effort is 300 N. Calculate (i) The velocity Ratio. (5) (ii) The Mechanical Advantage. (2.83) (iii) The efficiency. (57%) D.J.Dunn freestudy.co.uk 6

7 5. DIFFERENTIAL AXLE The diagram shows a typical arrangement of the differential axle. The load is suspended on a rope and one end if the rope unwinds from the small diameter D 3 and the other end winds in about the larger diameter D 2. The effort is applied by another rope wrapped around a wheel diameter D 1 on the same axle. When the effort is pulled so that the axle turns one revolution the effort moves one circumference πd 1 X E = πd 1 The rope on the differential gets shorter by πd 2 on the one side and longer by πd 3 on the other. The change in length is the difference and the load is raised by half this amount. πd2 πd3 X L = 2 XE 2π D1 2D1 The velocity ratio is V.R. = = = XL πd2 πd3 D2 D3 The efficiency of this system is normally high. Figure 6 WORKED EXAMPLE No. 5 A differential axle as shown above has two diameters 100 mm and 150 mm. The wheel is 200 mm diameter. The load is 80 N and the effort is 13 N. Calculate (i) The velocity ratio (ii) The Mechanical Advantage. (ii) The efficiency. XE 2D1 2 x 200 V.R. = = = = 8 XL D2 D FL 80 M.A. = = = FE 13 M.A η = = x 100 = 77% V.R. 8 SELF ASSESSMENT EXERCISE No A differential axle as shown above has two diameters 75 mm and 100 mm. It is expected that the efficiency will be 80% when raising a load of 500 N and the effort must be 25 N. Calculate (i) The Mechanical Advantage (20) (ii) The Velocity Ratio (25) (iii) The diameter of the wheel. (312.5 mm) D.J.Dunn freestudy.co.uk 7

8 6. WESTON DIFFERENTIAL PULLEY This is the basis of modern chain hoists and uses the same principle as the differential axle. When the effort is pulled so that the wheel rotates one revolution, the distance moved is one circumference πd 1 X E = πd 1 The rope on the differential gets shorter by πd 2 on the one side and longer by πd 1 on the other. The change in length is the difference and the load is raised by half this amount. πd1 πd2 X L = The velocity ratio is 2 XE 2π D1 2D1 V.R. = = = XL πd1 πd2 D1 D2 Although the diagram shows a rope, in reality a chain is used and the wheel and axle has flats that interlock with the links so that the chain does not slip. The two free ends shown are in fact joined so that a continuous loop is formed. It is not practical to diameters and a better measure is the number of flats on each (similar to using the number of gear teeth instead of diameter). 2 x Flats on (1) V.R. = = Flats on (1) - Flats on (2) Figure 7 WORKED EXAMPLE No. 6 A Weston differential pulley has a wheel with 12 Flats on it and an axle with 10 Flats. The load is 1200 N and the effort is 150 N. Calculate (i) The velocity ratio (ii) The Mechanical Advantage. (ii) The efficiency. 2 x 12 V.R. = = M.A. 8 η = = x 100 = 66.7% V.R. 12 FL 1200 M.A. = = = 8 F 150 E SELF ASSESSMENT EXERCISE No A Weston differential pulley as shown above has a wheel with 15 flats. It is expected that the efficiency will be % when raising a load of 1000 N and the effort must be 125 N. Calculate (i) The Mechanical Advantage (8) (ii) The Velocity Ratio (15) (iii) The flats on the axle. (13) D.J.Dunn freestudy.co.uk 8

9 7. GEARS The simplest gear is the simple gear train shown. The speed ratio of the wheels is exactly the same as the ratio as the diameters and also the ratio of the number of teeth in each. The small wheel always goes around faster than the large wheel. In order to get larger ratios compound gears are used which are really two or more sets of wheels each driving the next. Figure 8 There are many other types of gears such as the worm gear and bevel gear which allows the axis of the wheels to be changed. To use a gear box in a lifting machine requires that a pulley be placed on the shaft that will raise the load. A good example is that of winch in which a motor with a high speed and low torque is geared down to turn the drum at a low speed with a large torque. The diagram shows a typical winch that has a reduction gear box built inside the drum. Figure 9 D.J.Dunn freestudy.co.uk 9

10 WORKED EXAMPLE No.7 A simple winch is shown in the diagram. The small gear is turned by a handle 300 mm long. This rotates the larger gear. A drum on the same shaft as the large gear rotates and winds in a rope and raises the load. The gears have 50 teeth and 300 teeth respectively. The drum is 100 mm diameter. The efficiency is 30% Calculate the (i) Velocity Ratio (ii) Mechanical Advantage (iii) Effort required to raise 800 kg Figure 10 Let the handle rotate one revolution. The distance moved by the effort is the circumference of a circle 300 mm radius. X E =2π x 300 = 600π mm The gear ratio is the ratio of the teeth so the gear ratio is 300/50 = 6. This means that the large wheel and drum rotates 1/6 of a revolution. The distance moved by the load is hence 1/6 of the circumference of the drum. X L =2π x 50/6 = 16.67π mm The velocity ratio is hence X E /X L = 600π /16.67π = 36 The Mechanical Advantage is M.A. = η x V.R. = 0.3 x 36 = 10.8 Load = 800 x 9.81 = 7848 N Effort = 7848/10.8 = N SELF ASSESSMENT EXERCISE No A simple winch as shown previously has a drum 100 mm diameter and a gear ring with 125 teeth is attached to it. The gear meshes with a small gear with 25 teeth. This gear is attached to a handle 300 mm long. The efficiency is expected to be 30% when a load of 1200 N is raised. Calculate (i) The Velocity Ratio (30) (ii) The Mechanical Advantage (9) (iii) The Effort. (133.3 N) (iv) The torque produced by the effort. (40 Nm) D.J.Dunn freestudy.co.uk 10

11 8. TORQUE and POWER TRANSMISSION When a force moves work is done and the rate of doing work is the mechanical power. This power is transmitted through the machine from the effort to the load with some being lost on the way through friction. Mechanical power is defined as work done per second. Work done is defined as force times distance moved. Hence Power = P = Fx/t F is the force x is distance moved. t is the time taken. Since distance moved/time taken is the velocity of the force (x/t = v) we may write P = F v where v is the velocity. When a force rotates at radius R it travels one circumference in the time of one revolution. Hence the distance moved in one revolution is x = 2πR If the speed is N rev/second then the time of one revolution is 1/N seconds. The mechanical power is hence P = F 2πR/(1/N) = 2πNFR Since FR is the torque produced by the force this reduces to P = 2πNT In the context of handles, gears and winches we may define Torque = Force x Radius If the speed is in rev/min then S.P. = 2πNT/60 WORKED EXAMPLE No. 8 The drive shaft which connects a motor to the drum of a winch transmits 45 kw of power at 2000 rev/min. This is geared down to the winch drum 100 mm diameter and the drum revolves at 120 rev/min. The system is 30% efficient. Calculate the: (i) Torque in the motor shaft. (ii) Torque on the drum. (iii) The speed at which the load is moving. (iv) The load T = 60P/2πN = 60 x 45000/(2π x 2000) = 215 Nm on the motor shaft. The power available at the drum is 30% x 45 kw = 13.5 kw T = 60P/2πN = 60 x 13500/(2π x 120) = 1074 Nm on the drum. Mechanical Power = Force x velocity Velocity = πnd/60 = π x 120 x 0.1/60 = m/s Power = load x velocity Load = 13500/0.628 = N Check T = Load x Radius = x 0.05 = 1074 Nm D.J.Dunn freestudy.co.uk 11

12 SELF ASSESSMENT EXERCISE No.8 The diagram shows a simple winch. The motor drives the small gear and the large gear rotates the drum and raises the load. The load is 3 kn and it is raised at 0.5 m/s. The system is 25% efficient. Calculate the following. (i) The torque acting on the drum. (120 Nm) (ii) The power produced by the load. (1500 W) (iii) The speed of the drum. (119.4 rev/min) (iv) The power produced by the motor. (6000 W) (v) The speed of the motor. (596.8 rev/min) Figure 11 D.J.Dunn freestudy.co.uk 12

13 9. FRICTION ON INCLINED THREADS The inclusion of inclined planes in the syllabus makes this outcome rather large. It is doubtful that it is intended that all the following should be included but it is meaningless to study inclined planes without going into friction and this does not appear in the syllabus so it must be assumed that it has been already studied. This theory applies to screw threads as a nut turning on a screw is the same problem. You should already know that the coefficient of friction is defined as µ = F/R where F is the force parallel to the surface and R is the force normal to the surface. Consider a block on an inclined plane at angle φ to the horizontal. The weight acts vertically downwards. This must be resolved into two components parallel and perpendicular to the plane. Resolving R = W cos α and F 1 = W sin α If no other force is involved then the block will slide down the plane if F 1 is greater than the friction force. In this case F 1 > µr or F 1 > µ Wcos α Figure 12 The block will just slide when F 1 = µwcos α so it follows that µ = Wsin α/ Wcos α = Tan α and this is a way of finding µ and is called the friction angle. WORKED EXAMPLE No.9 A block rests on an plane and the angle is increased until it just slides. This angle is 13 o. Determine the coefficient of friction. µ= Tan α = Tan 13 o = Now consider the case of a block sliding under the action of a horizontal force such that the block slides up the plane. We must resolve the weight and the force parallel and perpendicular to the plane as shown. Figure 13 The total force acting parallel to the plane is F 1 F 2 and the total reaction is R = R 1 + R 2 The block will just slide up the plane if F 1 F 2 = µ (R 1 + R 2 ) D.J.Dunn freestudy.co.uk 13

14 WORKED EXAMPLE No. 10 A block rests on a plane at 12 o to the horizontal. The weight is 80 N and the coefficient of friction is 0.4. Calculate the force that will just make it slide up the plane. R 1 = F sin 12 o R 2 = W cos 12 o F 1 = F cos 12 o F 2 = W sin 12 o R = (R 1 + R 2 ) = F sin 12 o + W cos 12 o F = F 1 F 2 = F cos 12 o W sin 12 o µ = F/R = (F cos 12 o W sin 12 o )/( F sin 12 o + W cos 12 o ) 0.4 = (0.978F 16.63)/( F ) = F F = N It can be shown that the following formula also works for this problem. F/W = tan (β + α) where β is the friction angle (found from µ = tan β) and α is the angle of the plane. In the worked example we would have used: Β = tan -1 (0.4) = 21.8 o F = W tan (β + α) = 80 tan( ) = o 10. APPLICATION TO SCREW THREAD The motion of two mating threads is the same as the previous problem. The vertical load is the thrust acting axially on the nut (e.g. the load on a screw jack). The angle of the plane is given by: Tan α = pitch/circumference = p/πd WORKED EXAMPLE No.7 The diagram shows a turnbuckle. The threads have a pitch of 3 mm and a mean diameter of 12 mm. The coefficient of friction is 0.2. Calculate the torque needed to turn it when the force in the ropes is 4 kn. Figure 14 The pitch is 3 mm and the circumference is 12π so the angle of the plane is α = tan -1 (3/12π) = 4.55 o The friction angle is β = tan = o The axial force is the force equivalent to the weight W. The torque T is the product of the force F and radius at which it acts which is the radius of the thread (6 mm). F = W tan (β + α) = 4000 tan(15.86 o ) = 1136 N T = F x radius = 1136 x = 6.8 Nm Since there are two ends on a turnbuckle with opposite threads, the same torque is required at both ends so to tighten the turnbuckle requires 2 x 6.8 = 13.6 Nm D.J.Dunn freestudy.co.uk 14

15 SELF ASSESSMENT EXERCISE No Calculate the horizontal force required to make a block weighing 60 N slide up a ramp inclined at 20 o to the ground given µ = 0.2 (Answer 36.4 N) 2. The pitch of a thread on a screw jack is 6.38 mm and the mean diameter is 30 mm. Calculate the torque needed to raise a load 500 N. (Answer 4.5 Nm) 3. A block weighing 600 N rests on an inclined plane at 12 o to the horizontal. The coefficient of friction is Show that the formula to slide the box down the plane is F = W tan (β - α). Calculate the horizontal force required to slide it DOWN the plane. (Answer 109 N) 4. A turnbuckle is used to adjust the tension in a wire to 600 N. Each end has a single start square thread with a mean diameter of 10 mm and pitch of 2 mm. The coefficient of friction is 0.2. (Answer 1.6 Nm) D.J.Dunn freestudy.co.uk 15

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