# SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

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1 SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering and D225 Dynamics of Mechanical Systems. On completion of this short tutorial you should be able to do the following. Describe a mechanism. Define relative and absolute velocity. Define relative and absolute acceleration. Define radial and tangential velocity. Define radial and tangential acceleration. Describe a four bar chain. Solve the velocity and acceleration of points within a mechanism. Use mathematical and graphical methods. Construct velocity and acceleration diagrams. Define the Coriolis Acceleration. Solve problems involving sliding links. It is assumed that the student is already familiar with the following concepts. Vector diagrams. Simple harmonic motion. Angular and linear motion. Inertia force. Appropriate level of mathematics. All these above may be found in the pre-requisite tutorials. D.J.DUNN 1

2 1. INTRODUCTION A mechanism is used to produce mechanical transformations in a machine. This transformation could be any of the following. It may convert one speed to another speed. It may convert one force to another force. It may convert one torque to another torque. It may convert force into torque. It may convert one angular motion to another angular motion. It may convert angular motion into linear motion. It may convert linear motion into angular motion. A good example is a crank, connecting rod and piston mechanism. Figure 1 If the crank is turned, angular motion is converted into linear motion of the piston and input torque is transformed into force on the piston. If the piston is forced to move, the linear motion is converted into rotary motion and the force into torque. The piston is a sliding joint and this is called PRISMATIC in some fields of engineering such as robotics. The pin joints allow rotation of one part relative to another. These are also called REVOLUTE joints in other areas of engineering. Consider the next mechanism used in shaping machines and also known as the Whitworth quickreturn mechanism. Figure 2 D.J.DUNN 2

4 2. DISPLACEMENT, VELOCITY AND ACCELERATION All parts of a mechanism have displacement, velocity and acceleration. In the tutorial on free vibration, a mechanism called the Scotch Yoke was examined in order to explain sinusoidal or harmonic motion. The wheel turns at a constant speed and the yoke moves up and down. Figure 4 It was shown that the displacement x, velocity v and acceleration a of point p was given as follows. Angle θ = ωt Displacement x = R sin(ωt). Velocity v = dx/dt = ωr cos(ωt) Acceleration a = dv/dt = -ω 2 R sin(ωt) The values can be calculated for any angle or moment of time. The acceleration could then be used to calculate the inertia force needed to accelerate and decelerate the link. Clearly it is the maximum values that are needed. Other mechanisms can be analysed mathematically in the same way but it is more difficult. The starting point is to derive the equation for displacement with respect to angle or time and then differentiate twice to get the acceleration. Without the aid of a computer to do this, the mathematics is normally much too difficult and a graphical method should be used as shown later. WORKED EXAMPLE No.1 A crank, con rod and piston mechanism is shown below. Determine the maximum acceleration of the piston when the crank speed is 30 rev/min clockwise. Figure 5 D.J.DUNN 4

5 SOLUTION When θ = 0 the piston will be furthest left at a distance of 170 mm from point O. Take this as the reference point and measure displacement x from there. Remember that θ = ωt and ω = 2π x 30/60 = rad/s. The displacement is then Differentiate to get the velocity Differentiate again to get the acceleration. The diagram shows a plot of displacement, velocity and acceleration against angle. It should be noted that none of them are sinusoidal and not harmonic (in particular, the acceleration). Figure 6 The units are all in mm and seconds. The above was done with a computer package. Plotting the above functions over a complete rotation shows that the maximum acceleration occurs at t = 0 (θ = 0) and evaluating gives an answer of 700 mm/s 2. D.J.DUNN 5

6 If the radius of the crank is small in comparison to the length of the connecting rod, the motion becomes close to sinusoidal. To illustrate this, here is the plot with the crank radius reduced to 10 mm. The acceleration is now almost a cosine curve. Figure 7 Solving these problems mathematically is difficult so we will now look at a graphical method. 3. VELOCITY DIAGRAMS This section involves the construction of diagrams which needs to be done accurately and to a suitable scale. Students should use a drawing board, ruler, compass, protractor and triangles and possess the necessary drawing skills. ABSOLUTE AND RELATIVE VELOCITY An absolute velocity is the velocity of a point measured from a fixed point (normally the ground or anything rigidly attached to the ground and not moving). Relative velocity is the velocity of a point measured relative to another that may itself be moving. TANGENTIAL VELOCITY Consider a link A B pinned at A and revolving about A at angular velocity ω. Point B moves in a circle relative to point A but its velocity is always tangential and hence at 90 o to the link. A convenient method of denoting this tangential velocity is (v B ) A meaning the velocity of B relative to A. This method is not always suitable. Figure 8 D.J.DUNN 6

8 Every point on every link has a velocity through space. First we label the centre of rotation, often this is the letter O. Point A can only move in a tangential direction so the velocity of A relative to O is also its absolute velocity and the vector is normal to the crank and it is designated (v A ) O. (Note the rotation is anticlockwise). Now suppose that you are sat at point A and everything else moves relative to you. Looking towards B, it would appear the B is rotating relative to you (in reality it is you that is rotating) so it has a tangential velocity denoted (V B ) A. The direction is not always obvious except that it is normal to the link. Consider the fixed link OC. Since both points are fixed there is no velocity between them so so (v C ) O = 0 Next consider that you at point C looking at point B. Point B is a sliding link and will move in a straight line in the direction fixed by the slider guides and this is velocity (v B ) C. It follows that the velocity of B seen from O is the same as that seen from C so (v B ) C = (v B ) O The absolute velocity of B is (v B ) C = (v B ) O and this must be the vector sum of (V A ) O and (v B ) A and the three vectors must form a closed triangle as shown. The velocity of the piston must be in the direction in which it slides (conveniently horizontal here). This is a velocity diagram. METHODOLOGY Figure 12 First calculate the tangential velocity (v A ) O from v = ω x radius = ω x OA Draw the vector o - a in the correct direction (note lower case letters). We know that the velocity of B relative to A is to be added so the next vector ab starts at point a. At point a draw a line in the direction normal to the connecting rod but of unknown length. We know that the velocity of B relative and absolute to O is horizontal so the vector ob must start at a. Draw a horizontal line (in this case) through o to intersect with the other line. This is point b. The vectors ab and ob may be measured or calculated. Usually it is the velocity of the slider that is required. In a design problem, this velocity would be evaluated for many different positions of the crank shaft and the velocity of the piston determined for each position. Remember that the slider direction is not always horizontal and the direction of o - b must be the direction of sliding. D.J.DUNN 8

10 SELF ASSESSMENT EXERCISE No.1 Find the velocity of the piston for each case below and the angular velocity of AB about point A. 1. The crank OA rotates anti-clockwise at 3000 rev/min. Answer 34 m/s and rad/s Figure The crank revolves clockwise at 300 rev/min. Note that the vector ob is still horizontal because the piston can only slide horizontally relative to O. Also the rotation of the crank is opposite to the previous cases so the direction of oa is down to the right. Figure 16 Answer 1.11 m/s to the right and 5.55 rad/s 3. The crank O-A rotates at 200 rev/min clockwise. Note the vector ob is at 45 o to the horizontal as the piston must slide in this direction. Answer 0.49 m/s and 6.92 rad/s. Figure 17 D.J.DUNN 10

11 4 BAR CHAIN The input link rotates at a constant angular velocity ω 1. The relative velocity of each point relative to the other end of the link is shown. Each velocity vector is at right angles to the link. The output angular velocity is ω 2 and this will not be constant. The points A and D are fixed so they will appear as the same point on the velocity diagram. The methodology is the same as before and best shown with another example. Figure 18 WORKED EXAMPLE No. 3 Find the angular velocity of the output link when the input rotates at a constant speed of 500 rev/min. The diagram is not to scale. Figure 19 D.J.DUNN 11

12 SOLUTION First calculate ω 1. ω 1 = 2π x 500/60 = rad/s. Next calculate the velocity of point B relative to A. (V B ) A = ω 1 x AB = x 1 = m/s. Draw this as a vector to an appropriate scale. Figure 20a Next draw the direction of velocity C relative to B at right angles to the link BC passing through point b on the velocity diagram. Next draw the direction of the velocity of C relative to D at right angles to link DC passing through point a (which is the same as point d). Point c is where the two lines intersect, Figure 20b Figure 20c Determine velocity cd by measurement or any other method. The velocity of point C relative to D and is 43.5 m/s. Convert this into angular velocity by dividing the length of the link DC into it. ω 2 = 43.5/0.7 = 62 rad/s. D.J.DUNN 12

13 SELF ASSESSMENT EXERCISE No. 2 Determine the angular velocity of the link DC for each case shown and the direction of rotation. The diagrams are not to scale and should be constructed first. You are advised to use the best drawing instruments possible for accuracy. 1. The input rotates at 500 rev/min. Link BC is horizontal. (Ans. 76 rad/s clockwise.) Figure The input link AB rotates at 60 rev/min in a clockwise direction. (Ans. 16 rad/s) Figure 22 D.J.DUNN 13

14 4. ACCELERATION DIAGRAMS It is important to determine the acceleration of links because acceleration produces inertia forces in the link which stress the component parts of the mechanism. Accelerations may be relative or absolute in the same way as described for velocity. We shall consider two forms of acceleration, tangential and radial. Centripetal acceleration is an example of radial. CENTRIPETAL ACCELERATION A point rotating about a centre at radius R has a tangential velocity v and angular velocity ω and it is continually accelerating towards the centre even though it never moves any closer. This is centripetal acceleration and it is caused by the constant change in direction. It follows that the end of any rotating link will have a centripetal acceleration towards the opposite end. The relevant equations are: v = ωr a = ω 2 R or a = v 2 /R. The construction of the vector for radial acceleration causes confusion so the rules must be strictly followed. Consider the link AB. The velocity of B relative to A is tangential (v B ) A. The centripetal acceleration of B relative to A is in a radial direction so a suitable notation might be a R. It is calculated using a R = ω x AB or a R = v 2 /AB. Note the direction is towards the centre of rotation but the vector starts at a and ends at b 1. It is very important to get this the right way round otherwise the complete diagram will be wrong. Figure 23 D.J.DUNN 14

15 TANGENTIAL ACCELERATION Tangential acceleration only occurs if the link has an angular acceleration α rad/s 2. Consider a link AB with an angular acceleration about A. Figure 24 Point B will have both radial and tangential acceleration relative to point A. The true acceleration of point B relative to A is the vector sum of them. This will require an extra point. We will use b 1 and b on the vector diagram as shown. Point B is accelerating around a circular path and its direction is tangential (at right angles to the link). It is designated a T and calculated using a T = α x AB. The vector starts at b 1 and ends at b. The choice of letters and notation are arbitrary but must be logical to aid and relate to the construction of the diagram. Figure 25 D.J.DUNN 15

16 WORKED EXAMPLE No.4 A piston, connecting rod and crank mechanism is shown in the diagram. The crank rotates at a constant velocity of 300 rad/s. Find the acceleration of the piston and the angular acceleration of the link BC. The diagram is not drawn to scale. SOLUTION Figure 26 First calculate the tangential velocity of B relative to A. (v B ) A = ω x radius = 300 x 0.05 = 15 m/s. Next draw the velocity diagram and determine the velocity of C relative to B. From the velocity diagram (v C ) B = 7.8 m/s Figure 27 Next calculate all accelerations possible and construct the acceleration diagram to find the acceleration of the piston. The tangential acceleration of B relative to A is zero in this case since the link has no angular acceleration (α = 0). The centripetal acceleration of B relative to A a R = ω 2 x AB = x 0.05 = 4500 m/s 2. The tangential acceleration of C relative to B is unknown. The centripetal acceleration of C to B a R = v 2 /BC = /0.17 = m/s 2. The stage by stage construction of the acceleration diagram is as follows. D.J.DUNN 16

17 First draw the centripetal acceleration of link AB (Fig.a). There is no tangential acceleration so designate it ab. Note the direction is the same as the direction of the link towards the centre of rotation but is starts at a and ends at b. Figure 28a Figure 28b Figure 28c Next add the centripetal acceleration of link BC (Figure b). Since there are two accelerations for point C designate the point c 1. Note the direction is the same as the direction of the link towards the centre of rotation. Next add the tangential acceleration of point C relative to B (Figure c). Designate it c 1 c. Note the direction is at right angles to the previous vector and the length is unknown. Call the line a c line. Next draw the acceleration of the piston (figure d) which is constrained to be in the horizontal direction. This vector starts at a and must intersect the c line. Designate this point c. Figure 28d The acceleration of the piston is vector ac so (a C ) B = 1505 m/s 2. The tangential acceleration of C relative to B is c 1 c = 4000 m/s 2. At the position shown the connecting rod has an angular velocity and acceleration about its end even though the crank moves at constant speed. The angular acceleration of BC is the tangential acceleration divided by the length BC. α (BC) = 4000 / 0.17 = rad/s 2. D.J.DUNN 17

18 WORKED EXAMPLE No.5 The diagrams shows a rocking lever mechanism in which steady rotation of the wheel produces an oscillating motion of the lever OA. Both the wheel and the lever are mounted in fixed centres. The wheel rotates clockwise at a uniform angular velocity (ω) of 100 rad/s. For the configuration shown, determine the following. (i) The angular velocity of the link AB and the absolute velocity of point A. (ii) The centrifugal accelerations of BC, AB and OA. (iii) The magnitude and direction of the acceleration of point A. The lengths of the links are as follows. BC = 25 mm AB = 100 mm OA = 50 mm OC = 90 mm SOLUTION Figure 29 The solution is best done graphically. First draw a line diagram of the mechanism to scale. It should look like this. Figure 30 Next calculate the velocity of point B relative to C and construct the velocity diagram. D.J.DUNN 18

20 SELF ASSESSMENT EXERCISE No.3 Solve the acceleration of the piston for each case shown. You should draw the space diagram out accurately first. 1. (Ans. 153 m/s) Figure (Ans m/s 2 ) Figure 34 D.J.DUNN 20

21 WORKED EXAMPLE No. 6 Find the angular acceleration of the link CD for the case shown. SOLUTION Figure 35 First calculate or scale the length CB and find it to be 136 mm. Next find the velocities and construct the velocity diagram. Start with link AB as this has a known constant angular velocity. (v B ) A = ω x radius = 480 x 0.08 = 38.4 m/s Figure 36 Next calculate all the accelerations possible. The centripetal acceleration of B to A is /0.08 = m/s 2 The centripetal acceleration of C to D is 15 2 /0.16 = 1406 m/s 2 The centripetal acceleration of C to B is 31 2 /0.136 = 7066 m/s 2. We cannot calculate any tangential accelerations at this stage. The stage by stage construction of the acceleration diagram follows. D.J.DUNN 21

22 First draw the centripetal acceleration of B to A (Figure a). There is no tangential to add on). Figure 37a Figure 37b Figure 37c Next add the centripetal acceleration of C to B (figure b) Next draw the direction of the tangential acceleration of C to B of unknown length at right angles to the previous vector (figure c). Designate it as a c line. We cannot proceed from this point unless we realise that points a and d are the same (there is no velocity or acceleration of D relative to A). Add the centripetal acceleration of C to D (figure d). This is 1406 m/s 2 in the direction of link CD. Designte it d c 2. Figure 37d Figure 37e Finally draw the tangential acceleration of C to D at right angles to the previous vector to intersect the c line (figure e). From the diagram determine c 2 c to be m/s 2. This is the tangential acceleration of C to D. The angular acceleration of the link DC is then: α (CD) = 24000/0.16 = rad/s2 in a clockwise direction. Note that although the link AB rotates at constant speed, the link CD has angular acceleration. D.J.DUNN 22

23 WORKED EXAMPLE No. 7 The same arrangement exists as shown for example 5 except that the link AB is decelerating at 8000 rad/s 2 (i.e. in an anticlockwise direction). Determine the acceleration of the link CD. SOLUTION The problem is essentially the same as example 5 except that a tangential acceleration now exists for point B relative to point A. This is found from a T = α x AB = x 0.08 = 6400 m/s 2 T he direction is for an anticlockwise tangent. This is vector b1 b which is at right angles to a b 1 in the appropriate direction. The new acceleration diagram looks like this. Figure 38 Scaling off the tangential acceleration c 2 c we get m/s 2. Converting this into the angular acceleration we get α = /0.16 = rad/s 2 in a clockwise direction. D.J.DUNN 23

25 5. INERTIA FORCE One of the reasons for finding the acceleration of links is to calculate the inertia force needed to accelerate or decelerate it. This is based on Newton s second law. Force = mass x acceleration F = M a And Torque = moment of inertia x angular acceleration T = I α WORKED EXAMPLE No.8 A horizontal single cylinder reciprocating engine has a crank OC of radius 40 mm and a connecting rod PC 140 mm long as shown. The crank rotates at 3000 rev/min clockwise. For the configuration shown, determine the velocity and acceleration of the piston. The sliding piston has a mass of 0.5 kg and a diameter of 80 mm. The gas pressure acting on it is 1.2 MPa at the moment shown. Calculate the effective turning moment acting on the crank. Assume that the connecting rod and crank has negligible inertia and friction. SOLUTION Figure 41 Draw the space diagram to scale. Figure 42 The moment arm should be scaled and found to be 34 mm (measured at right angles to the connecting rod PC. Calculate the velocity of C relative to O. ω = 2πN/60 = 2π x 3000/60 = rad/s (v C ) O = ω x radius = x 0.04 = m/s D.J.DUNN 25

26 Draw the velocity diagram. Figure 43 From the velocity diagram we find the velocity of the piston is 11 m/s. Next calculate all the accelerations possible. Point C only has a radial acceleration towards O Radial acceleration of C is v 2 /radius = /0.04 = 3950 m/s 2 Point P has radial and tangential acceleration relative to C. Tangential acceleration is unknown. Radial acceleration = (v P ) C 2 /CP = 9 2 /0.14 = m/s 2 Now draw the acceleration diagram and it comes out like this. The acceleration of the piston is 2839 m/s 2. Now we can solve the forces. Figure 44 Pressure force = p x area = 1.2 x 10 6 x π x /4 = 6032 N and this acts left to right. Inertia force acting on the piston = M a = 0.5 x 2839 = N and this must be provided by the pressure force so the difference is the force exerted on the connecting rod. Net Force = = N. D.J.DUNN 26

27 The connecting rod makes an angle of 11 o to the line of the force (angle scaled from space diagram). This must be resolved to find the force acting along the line of the connecting rod. Figure 45 The force in the connecting rod is cos 11 o = 4528 N. This acts at a radius of 34 mm from the centre of the crank so the torque provided by the crank is T = 4528 x = 154 N m. SELF ASSESSMENT EXERCISE No.5 1. The piston in the mechanism shown has a mass of 0.8 kg. Determine its acceleration and the inertia force needed for the position shown. Figure 46 (Ans m/s 2 and 3200 N) D.J.DUNN 27

28 6. CORIOLIS ACCELERATION Consider a link rotating at ω rad/s and accelerating at α rad/s 2. On the link is a sliding element moving away from the centre of rotation at velocity v R = dr/dt (positive if getting larger) The link has a tangential velocity v T = ωr The component of this velocity in the x direction is v T sin θ = v T sin ωr The velocity v R also has a component in the x direction And this is v R cos θ The total velocity in the x direction of the sliding link is v x = -v T sin θ + v R cos θ v x = -(dr/dt) cos ωt + ωr sin ωt The acceleration in the x direction is a x Figure 47 2 dv d R dr 2 dt 2 dt dt a x = x = - cos( ω t) + ω sin(ω t) + ω R cos( ω t) + ω sin( ω t) + R sin( ω t) 2 d R dr 2 2 dt dt dω dt dr dt a x = cos( ω t) + 2 ω sin( ω t) + Rω cos( ω t) + R sin( ω t) 2 d 2 R dr a x = cos θ + 2 ω sin θ + R ω 2 dt dt When θ = 90 o dr a x = a T = 2 ω + α R dt a T = 2 ω vr + α R cosθ + α R sin θ The tangential acceleration is not simply α R as is the case for a constant radius but an extra term of 2ωvR is added and this term is called the Coriolis acceleration and must be taken into consideration when solving problems with changing radius. dω dt D.J.DUNN 28

29 WORKED EXAMPLE No.9 Figure 48 The diagram shows part of a quick return mechanism. The pin A slides in the slot when the disc is rotated. Calculate the angular velocity and acceleration of link BC when θ = 60 o and ω = 100 rad/s. SOLUTION The tangential velocity of A relative to O is ωr = 100 x 0.04 = 4 m/s. The velocity diagram is constructed as shown. Figure 49 The tangential velocity of pin A relative to B is (V A1 ) B = a a 1 = 4 cos (38.95 o ) = 3.11 m/s The radial velocity of A relative to B is (V A ) B = 4 sin(38.95 o ) = m/s The length of BA is easily calculated from the diagram. BA = ( ) = mm Figure 50 The angular velocity link BC = 3.11/BA = 32.2 rad/s D.J.DUNN 29

30 ANALYTICAL METHOD The angle of link BC is 1 40sin θ 1 sin θ α = tan = tan cos θ 7/4 + cos θ The angular velocity is dα/dt and the tools for doing the differentiation are given in the question as follows. sin θ dα 1 1 dx 1+ 7/4cos θ Let x = = = = 7/4 + cos θ 2 2 dx 1+ x sin θ dθ ( 7/4 + cos θ ) /4 + cos θ dα dα dx /4cos θ = = dt dx dθ sin θ 2 ( 7/4 cos θ ) 2 put θ = 60 o and evaluate /4 + cos θ dα/dθ = θ = ω t so dθ = ω dt answer found before. dt = dθ/ω so dα/dt = x ω = 31.6 rad/s which is close to the Next construct the acceleration diagram. Figure 51 Link O A only has centripetal acceleration inwards (a A ) O = ω 2 R = x 0.04 = 400 m/s 2 The pin A has a tangential acceleration and Coriolis acceleration normal to the link. It has centripetal acceleration and radial acceleration towards the centre of rotation. B. The diagram can be constructed without calculating them. The Corioilis acceleration is 2 ω v where ω = 32.2 and v is the radial velocity = m/s The Coriolis term is hence m/s 2 The tangential acceleration of A relative to B is a 1 a = 400 sin = m/s 2 Part of this is the Coriolis so the tangential acceleration is = m/s 2 The angular acceleration of link AC is α = /BA = 89.49/ = 928 rad/s 2 The direction is negative (clockwise) so it is decelerating. D.J.DUNN 30

31 SELF ASSESSMENT EXERCISE No.6 A link OA is 80 mm long and rotates at a constant speed of 50 rad/s. A sliding link attached to it slides on link BC and makes BC rotate about B as shown. Calculate the angular velocity and acceleration of BC when angle θ = 70 o. (22.8 rad/s and rad/s 2 ) Figure 52 D.J.DUNN 31

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