Upcoming SlideShare
×

# Ch.08

6,233 views

Published on

Published in: Technology
7 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

Views
Total views
6,233
On SlideShare
0
From Embeds
0
Number of Embeds
9
Actions
Shares
0
539
0
Likes
7
Embeds 0
No embeds

No notes for slide

### Ch.08

1. 1. CONTENTSCONTENTS174 l Theory of MachinesFeatures eatur tures Warping Machine1. Introduction. Acceleration 82. Acceleration Diagram for a Link.3. Acceleration of a Point on a in Mechanisms Link.4. Acceleration in the Slider 8.1. Introduction Introduction Crank Mechanism. We have discussed in the previous chapter the5. Coriolis Component of Acceleration. velocities of various points in the mechanisms. Now we shall discuss the acceleration of points in the mechanisms. The acceleration analysis plays a very important role in the development of machines and mechanisms. 8.2. Acceleration Diagram for a Link Consider two points A and B on a rigid link as shown in Fig. 8.1 (a). Let the point B moves with respect to A, with an angular velocity of ω rad/s and let α rad/s2 be the angular acceleration of the link AB. (a) Link. (b) Acceleration diagram. Fig. 8.1. Acceleration for a link. 174CONTENTSCONTENTS
4. 4. Chapter 8 : Acceleration in Mechanisms l 177 We know that centripetal or radial acceleration of B with respect to O or acceleration of B(because O is a fixed point), v2 aBO = aB = ωBO × OB = BO r 2 OBNote : A point at the end of a link which moves with constant angular velocity has no tangential componentof acceleration. (a) Slider crank mechanism. (b) Acceleration diagram. Fig. 8.3. Acceleration in the slider crank mechanism. The acceleration diagram, as shown in Fig. 8.3 (b), may now be drawn as discussed below: 1. Draw vector o b parallel to BO and set off equal in magnitude of aBO = aB , to some rsuitable scale. 2. From point b, draw vector bx parallel to B A. The vector bx represents the radial componentof the acceleration of A with respect to B whose magnitude is given by : aAB = vAB / BA r 2 Since the point B moves with constant angular velocity, therefore there will be no tangentialcomponent of the acceleration. 3. From point x, draw vector xa perpendicular to bx (or A B). The vector xa represents the ttangential component of the acceleration of A with respect to B i.e. aAB .Note: When a point moves along a straight line, it has no centripetal or radial component of the acceleration. 4. Since the point A reciprocates along A O, therefore the acceleration must be parallel tovelocity. Therefore from o, draw o a parallel to A O, intersecting the vector xa at a. t Now the acceleration of the piston or the slider A (aA) and aAB may be measured to the scale. 5. The vector b a, which is the sum of the vectors b x and x a, represents the total accelerationof A with respect to B i.e. aAB. The vector b a represents the acceleration of the connecting rod A B. 6. The acceleration of any other point on A B such as E may be obtained by dividing the vectorb a at e in the same ratio as E divides A B in Fig. 8.3 (a). In other words a e / a b = AE / AB 7. The angular acceleration of the connecting rod A B may be obtained by dividing thetangential component of the acceleration of A with respect to B ( aAB ) to the length of A B. In other twords, angular acceleration of A B, α AB = aAB / AB (Clockwise about B) t Example 8.1. The crank of a slider crank mechanism rotates clockwise at a constant speedof 300 r.p.m. The crank is 150 mm and the connecting rod is 600 mm long. Determine : 1. linearvelocity and acceleration of the midpoint of the connecting rod, and 2. angular velocity and angularacceleration of the connecting rod, at a crank angle of 45° from inner dead centre position.
5. 5. 178 l Theory of Machines Solution. Given : N BO = 300 r.p.m. or ωBO = 2 π × 300/60 = 31.42 rad/s; OB = 150 mm =0.15 m ; B A = 600 mm = 0.6 m We know that linear velocity of B with respect to O or velocity of B, vBO = v B = ωBO × OB = 31.42 × 0.15 = 4.713 m/s ...(Perpendicular to BO) (a) Space diagram. (b) Velocity diagram. (c) Acceleration diagram. Fig. 8.4 Ram moves Ram moves outwards Load moves inwards outwards Oil pressure on lower side of piston Oil pressure on upper side of piston Load moves inwards Pushing with fluids Note : This picture is given as additional information and is not a direct example of the current chapter.1. Linear velocity of the midpoint of the connecting rod First of all draw the space diagram, to some suitable scale; as shown in Fig. 8.4 (a). Now thevelocity diagram, as shown in Fig. 8.4 (b), is drawn as discussed below: 1. Draw vector ob perpendicular to BO, to some suitable scale, to represent the velocity ofB with respect to O or simply velocity of B i.e. vBO or vB, such that vector ob = v BO = v B = 4.713 m/s 2. From point b, draw vector ba perpendicular to BA to represent the velocity of A withrespect to B i.e. vAB , and from point o draw vector oa parallel to the motion of A (which is along A O)to represent the velocity of A i.e. vA. The vectors ba and oa intersect at a.
6. 6. Chapter 8 : Acceleration in Mechanisms l 179 By measurement, we find that velocity of A with respect to B, vAB = vector ba = 3.4 m / sand Velocity of A, vA = vector oa = 4 m / s 3. In order to find the velocity of the midpoint D of the connecting rod A B, divide the vectorba at d in the same ratio as D divides A B, in the space diagram. In other words, bd / ba = BD/BANote: Since D is the midpoint of A B, therefore d is also midpoint of vector ba. 4. Join od. Now the vector od represents the velocity of the midpoint D of the connectingrod i.e. vD. By measurement, we find that vD = vector od = 4.1 m/s Ans.Acceleration of the midpoint of the connecting rod We know that the radial component of the acceleration of B with respect to O or theacceleration of B, 2 vBO (4.713)2 aBO = aB = r = = 148.1 m/s 2 OB 0.15and the radial component of the acceleraiton of A with respect to B, 2 vAB (3.4)2 aAB = r = = 19.3 m/s 2 BA 0.6 Now the acceleration diagram, as shown in Fig. 8.4 (c) is drawn as discussed below: 1. Draw vector o b parallel to BO, to some suitable scale, to represent the radial component rof the acceleration of B with respect to O or simply acceleration of B i.e. aBO or aB , such that vector o′ b′ = aBO = aB = 148.1 m/s 2 rNote: Since the crank OB rotates at a constant speed, therefore there will be no tangential component of theacceleration of B with respect to O. 2. The acceleration of A with respect to B has the following two components: r (a) The radial component of the acceleration of A with respect to B i.e. aAB , and t (b) The tangential component of the acceleration of A with respect to B i.e. aAB . These two components are mutually perpendicular. Therefore from point b, draw vector b x parallel to A B to represent aAB = 19.3 m/s 2 and rfrom point x draw vector xa perpendicular to vector b x whose magnitude is yet unknown. 3. Now from o, draw vector o a parallel to the path of motion of A (which is along A O) torepresent the acceleration of A i.e. aA . The vectors xa and o a intersect at a. Join a b. 4. In order to find the acceleration of the midpoint D of the connecting rod A B, divide thevector a b at d in the same ratio as D divides A B. In other words b′d ′ / b′a ′ = BD / BANote: Since D is the midpoint of A B, therefore d is also midpoint of vector b a. 5. Join o d. The vector o d represents the acceleration of midpoint D of the connecting rodi.e. aD. By measurement, we find that aD = vector o d = 117 m/s2 Ans.
7. 7. 180 l Theory of Machines2. Angular velocity of the connecting rod We know that angular velocity of the connecting rod A B, vAB 3.4 ωAB = = = 5.67 rad/s2 (Anticlockwise about B ) Ans. BA 0.6Angular acceleration of the connecting rod From the acceleration diagram, we find that aAB = 103 m/s 2 t ...(By measurement) We know that angular acceleration of the connecting rod A B, t aAB 103 α AB = = = 171.67 rad/s 2 (Clockwise about B ) Ans. BA 0.6 Example 8.2. An engine mechanism is shown in Fig. 8.5. The crank CB = 100 mm and theconnecting rod BA = 300 mm with centre of gravity G, 100 mm from B. In the position shown, thecrankshaft has a speed of 75 rad/s and an angular acceleration of 1200 rad/s2. Find:1. velocity ofG and angular velocity of AB, and 2. acceleration of G and angular acceleration of AB. Fig. 8.5 Solution. Given : ωBC = 75 rad/s ; αBC = 1200 rad/s2, CB = 100 mm = 0.1 m; B A = 300 mm= 0.3 m We know that velocity of B with respect to C or velocity of B, vBC = vB = ωBC × CB = 75 × 0.1 = 7.5 m/s ...(Perpendicular to BC) Since the angular acceleration of the crankshaft, αBC = 1200 rad/s2, therefore tangentialcomponent of the acceleration of B with respect to C, aBC = αBC × CB = 1200 × 0.1 = 120 m/s2 tNote: When the angular acceleration is not given, then there will be no tangential component of the acceleration.1. Velocity of G and angular velocity of AB First of all, draw the space diagram, to some suitable scale, as shown in Fig. 8.6 (a). Now thevelocity diagram, as shown in Fig. 8.6 (b), is drawn as discussed below: 1. Draw vector cb perpendicular to CB, to some suitable scale, to represent the velocity ofB with respect to C or velocity of B (i.e. vBC or vB), such that vector cb = vBC = vB = 7.5 m/s 2. From point b, draw vector ba perpendicular to B A to represent the velocity of A withrespect to B i.e. vAB , and from point c, draw vector ca parallel to the path of motion of A (which isalong A C) to represent the velocity of A i.e. vA.The vectors ba and ca intersect at a. 3. Since the point G lies on A B, therefore divide vector ab at g in the same ratio as G dividesA B in the space diagram. In other words, ag / ab = AG / AB The vector cg represents the velocity of G. By measurement, we find that velocity of G, v G = vector cg = 6.8 m/s Ans.
8. 8. Chapter 8 : Acceleration in Mechanisms l 181 From velocity diagram, we find that velocity of A with respect to B, vAB = vector ba = 4 m/s We know that angular velocity of A B, vAB 4 ωAB = = = 13.3 rad/s (Clockwise) Ans. BA 0.3 (a) Space diagram. (b) Velocity diagram. Fig. 8.62. Acceleration of G and angular acceleration of AB We know that radial component of the acceleration of B withrespect to C, 2 2 * a r = vBC = (7.5) = 562.5 m/s2 BC CB 0.1and radial component of the acceleration of A with respect to B, 2 vAB 42 aAB = r = = 53.3 m/s 2 BA 0.3 Now the acceleration diagram, as shown in Fig. 8.6 (c), is drawnas discussed below: 1. Draw vector c b parallel to CB, to some suitable scale, to (c) Acceleration diagram.represent the radial component of the acceleration of B with respect to C, Fig. 8.6 ri.e. aBC , such that vector c ′ b′′ = aBC = 562.5 m/s 2 r 2. From point b, draw vector b b perpendicular to vector c b or CB to represent the ttangential component of the acceleration of B with respect to C i.e. aBC , such that vector b′′ b′ = aBC = 120 m/s 2 t ... (Given) 3. Join c b. The vector c b represents the total acceleration of B with respect to C i.e. aBC. 4. From point b, draw vector b x parallel to B A to represent radial component of the racceleration of A with respect to B i.e. aAB such that vector b′x = aAB = 53.3 m/s 2 r 5. From point x, draw vector xa perpendicular to vector bx or B A to represent tangential tcomponent of the acceleration of A with respect to B i.e. aAB , whose magnitude is not yet known. 6. Now draw vector c a parallel to the path of motion of A (which is along A C) to representthe acceleration of A i.e. aA.The vectors xa and ca intersect at a. Join b a. The vector b arepresents the acceleration of A with respect to B i.e. aAB. t r* When angular acceleration of the crank is not given, then there is no aBC . In that case, aBC = aBC = aB , as discussed in the previous example.
9. 9. 182 l Theory of Machines 7. In order to find the acceleratio of G, divide vector a b in g in the same ratio as G dividesB A in Fig. 8.6 (a). Join c g. The vector c g represents the acceleration of G. By measurement, we find that acceleration of G, aG = vector c g = 414 m/s2 Ans. From acceleration diagram, we find that tangential component of the acceleration of A withrespect to B, aAB = vector xa′ = 546 m/s 2 t ...(By measurement) ∴ Angular acceleration of A B, t aAB 546 α AB = = = 1820 rad/s 2 (Clockwise) Ans. BA 0.3 Example 8.3. In the mechanism shown in Fig. 8.7, the slider C ismoving to the right with a velocity of 1 m/s and an acceleration of 2.5 m/s2. The dimensions of various links are AB = 3 m inclined at 45° with thevertical and BC = 1.5 m inclined at 45° with the horizontal. Determine: 1. themagnitude of vertical and horizontal component of the acceleration of thepoint B, and 2. the angular acceleration of the links AB and BC. Solution. Given : v C = 1 m/s ; aC = 2.5 m/s2; A B = 3 m ; BC = 1.5 m First of all, draw the space diagram, as shown in Fig. 8.8 (a), to some Fig. 8.7suitable scale. Now the velocity diagram, as shown in Fig. 8.8 (b), is drawn asdiscussed below: 1. Since the points A and D are fixed points, therefore they lie at one place in the velocitydiagram. Draw vector dc parallel to DC, to some suitable scale, which represents the velocity ofslider C with respect to D or simply velocity of C, such that vector dc = v CD = v C = 1 m/s 2. Since point B has two motions, one with respect to A and the other with respect to C,therefore from point a, draw vector ab perpendicular to A B to represent the velocity of B withrespect to A , i.e. vBA and from point c draw vector cb perpendicular to CB to represent the velocityof B with respect to C i.e. vBC .The vectors ab and cb intersect at b. (a) Space diagram. (b) Velocity diagram. (c) Acceleration diagram. Fig. 8.8 By measurement, we find that velocity of B with respect to A , vBA = vector ab = 0.72 m/sand velocity of B with respect to C, vBC = vector cb = 0.72 m/s
10. 10. Chapter 8 : Acceleration in Mechanisms l 183 We know that radial component of acceleration of B with respect to C, 2 vBC (0.72) 2 aBC = r = = 0.346 m/s2 CB 1.5and radial component of acceleration of B with respect to A , 2 vBA (0.72)2 aBA = r = = 0.173 m/s 2 AB 3 Now the acceleration diagram, as shown in Fig. 8.8 (c), is drawn as discussed below: 1. *Since the points A and D are fixed points, therefore they lie at one place in the accelerationdiagram. Draw vector d c parallel to DC, to some suitable scale, to represent the acceleration of Cwith respect to D or simply acceleration of C i.e. aCD or aC such that vector d ′ c′ = aCD = aC = 2.5 m/s 2 2. The acceleration of B with respect to C will have two components, i.e. one radial componentof B with respect to C (aBC ) and r the other tangential component of B with respect toC ( aBC ). Therefore from point c, draw vector c x parallel to CB to represent aBC such that t r vector c ′x = aBC = 0.346 m/s 2 r 3. Now from point x, draw vector xb perpendicular to vector c x or CB to represent atBCwhose magnitude is yet unknown. 4. The acceleration of B with respect to A will also have two components, i.e. one radialcomponent of B with respect to A (arBA) and other tangential component of B with respect to A (at BA).Therefore from point a draw vector a y parallel to A B to represent arBA, such that vector a y = arBA = 0.173 m/s2 t 5. From point y, draw vector yb perpendicular to vector ay or AB to represent aBA . Thevector yb intersect the vector xb at b. Join a b and c b. The vector a b represents the accelerationof point B (aB) and the vector c b represents the acceleration of B with respect to C.1. Magnitude of vertical and horizontal component of the acceleration of the point B Draw b b perpendicular to a c. The vector b b is the vertical component of the accelerationof the point B and a b is the horizontal component of the acceleration of the point B. By measurement, vector b b = 1.13 m/s2 and vector a b = 0.9 m/s2 Ans.2. Angular acceleration of AB and BC By measurement from acceleration diagram, we find that tangential component of accelerationof the point B with respect to A , aBA = vector yb′ = 1.41 m/s 2 tand tangential component of acceleration of the point B with respect to C, aBC = vector xb′ = 1.94 m/s 2 t* If the mechanism consists of more than one fixed point, then all these points lie at the same place in the velocity and acceleration diagrams.
11. 11. 184 l Theory of Machines We know that angular acceleration of A B, t aBA 1.41 α AB = = = 0.47 rad/s 2 Ans. AB 3and angular acceleration of BC, t aBA 1.94 α BC = = = 1.3 rad/s 2 Ans. CB 1.5 Example 8.4. PQRS is a four bar chain with link PS fixed. The lengths of the links are PQ= 62.5 mm ; QR = 175 mm ; RS = 112.5 mm ; and PS = 200 mm. The crank PQ rotates at 10 rad/sclockwise. Draw the velocity and acceleration diagram when angle QPS = 60° and Q and R lie onthe same side of PS. Find the angular velocity and angular acceleration of links QR and RS. Solution. Given : ωQP = 10 rad/s; PQ = 62.5 mm = 0.0625 m ; QR = 175 mm = 0.175 m ;R S = 112.5 mm = 0.1125 m ; PS = 200 mm = 0.2 m We know that velocity of Q with respect to P or velocity of Q, v QP = v Q = ωQP × PQ = 10 × 0.0625 = 0.625 m/s ...(Perpendicular to PQ)Angular velocity of links QR and RS First of all, draw the space diagram of a four bar chain, to some suitable scale, as shown inFig. 8.9 (a). Now the velocity diagram as shown in Fig. 8.9 (b), is drawn as discussed below: (a) Space diagram. (b) Velocity diagram. (c) Acceleration diagram. Fig. 8.9 1. Since P and S are fixed points, therefore these points lie at one place in velocity diagram.Draw vector pq perpendicular to PQ, to some suitable scale, to represent the velocity of Q withrespect to P or velocity of Q i.e. v QP or v Q such that vector pq = v QP = v Q = 0.625 m/s 2. From point q, draw vector qr perpendicular to QR to represent the velocity of R withrespect to Q (i.e. vRQ) and from point s, draw vector sr perpendicular to S R to represent the velocityof R with respect to S or velocity of R (i.e. vRS or v R). The vectors qr and sr intersect at r. Bymeasurement, we find that vRQ = vector qr = 0.333 m/s, and v RS = v R = vector sr = 0.426 m/s We know that angular velocity of link QR, vRQ 0.333 ωQR = = = 1.9 rad/s (Anticlockwise) Ans. RQ 0.175
13. 13. 186 l Theory of Machines Example 8.5. The dimensions andconfiguration of the four bar mechanism, shown inFig. 8.10, are as follows : P1A = 300 mm; P2B = 360 mm; AB = 360mm, and P1P2 = 600 mm. The angle AP1P2 = 60°. The crank P1A hasan angular velocity of 10 rad/s and an angularacceleration of 30 rad/s 2 , both clockwise.Determine the angular velocities and angularaccelerations of P2B, and AB and the velocity and Fig. 8.10acceleration of the joint B. Solution. Given : ωAP1 = 10 rad/s ; αAP1 = 30 rad/s2; P1A = 300 mm = 0.3 m ; P2B = A B =360 mm = 0.36 m We know that the velocity of A with respect to P1 or velocity of A , vAP1 = vA = ωAP1 × P1A = 10 × 0.3 = 3 m/sVelocity of B and angular velocitites of P2B and AB First of all, draw the space diagram, to some suitable scale, as shown in Fig. 8.11 (a). Nowthe velocity diagram, as shown in Fig. 8.11 (b), is drawn as discussed below: 1. Since P1 and P2 are fixed points, therefore these points lie at one place in velocity diagram.Draw vector p1 a perpendicular to P1A , to some suitable scale, to represent the velocity of A withrespect to P1 or velocity of A i.e. vAP1 or vA, such that vector p1a = v A P1 = v A = 3 m/s 2. From point a, draw vector ab perpendicular to AB to represent velocity of B with respectto A (i.e. vBA) and from point p2 draw vector p2b perpendicular to P2B to represent the velocity of Bwith respect to P2 or velocity of B i.e. vBP2 or v B. The vectors ab and p2b intersect at b. By measurement, we find that vBP2 = v B = vector p2b = 2.2 m/s Ans.and vBA = vector ab = 2.05 m/s We know that angular velocity of P2B, vBP 2 2.2 ωP2B = = = 6.1 rad/s (Clockwise) Ans. P2 B 0.36and angular velocity of A B, vBA 2.05 ωAB = = = 5.7 rad/s (Anticlockwise) Ans. AB 0.36Acceleration of B and angular acceleration of P2B and AB We know that tangential component of the acceleration of A with respect to P1, atA P1 = α A P1 × P A = 30 × 0.3 = 9 m/s 2 1 Radial component of the acceleration of A with respect to P1, 2 vA P1 aAP1 = r = ωAP1 × P A = 102 × 0.3 = 30 m/s2 2 1 PA 1
14. 14. Chapter 8 : Acceleration in Mechanisms l 187 Radial component of the acceleration of B with respect to A . 2 vBA (2.05) 2 aBA = r = = 11.67 m/s 2 AB 0.36and radial component of the acceleration of B with respect to P2, 2 vBP 2 (2.2)2 aBP 2 = r = = 13.44 m/s2 P2 B 0.36 (a) Space diagram. (b) Velocity diagram. Fig. 8.11 The acceleration diagram, as shown in Fig. 8.11 (c), isdrawn as follows: 1. Since P1 and P2 are fixed points, therefore these pointswill lie at one place, in the acceleration diagram. Draw vectorp1 x parallel to P1A , to some suitable scale, to represent theradial component of the acceleration of A with respect to P1,such that vector p1′ x = aA P = 30 m/s 2 r 1 2. From point x, draw vector xa perpendicular to P1A torepresent the tangential component of the acceleration of A withrespect to P1, such that vector xa ′ = aA P1 = 9 m/s 2 t 3. Join p1 a. The vector p1 a represents the acceleration (c) Acceleration diagramof A . By measurement, we find that the acceleration of A , Fig. 8.11 aA = aAP1 = 31.6 m/s2 4. From point a, draw vector a y parallel to A B to represent the radial component of theacceleration of B with respect to A , such that vector a′y = aBA = 11.67 m/s 2 r 5. From point y, draw vector yb perpendicular to A B to represent the tangential component tof the acceleration of B with respect to A (i.e. aBA ) whose magnitude is yet unknown. 6. Now from point p ′ , draw vector p ′ z parallel to P B to represent the radial component 2 2 2of the acceleration B with respect to P2, such that vector p2′ z = aBP 2 = 13.44 m/s 2 r
15. 15. 188 l Theory of Machines 7. From point z, draw vector zb perpendicular to P2B to represent the tangential component tof the acceleration of B with respect to P2 i.e. aBP 2 . 8. The vectors yb and zb intersect at b. Now the vector p2 b represents the acceleration ofB with respect to P2 or the acceleration of B i.e. aBP2 or aB. By measurement, we find that aBP2 = aB = vector p2 b = 29.6 m/s2 Ans. Also vector yb′ = aBA = 13.6 m/s 2 , and vector zb′ = aBP 2 = 26.6 m/s 2 t t We know that angular acceleration of P2B, t aBP 2 26.6 α = = = 73.8 rad/s 2 (Anticlockwise) Ans. P2B P2 B 0.36 t aBA 13.6and angular acceleration of A B, α AB = = = 37.8 rad/s 2 (Anticlockwise) Ans. AB 0.36 Bicycle is a common example where simple mechanisms are used. Note : This picture is given as additional information and is not a direct example of the current chapter. Example 8.6. In the mechanism, as shown in Fig. 8.12, the crank OA rotates at 20 r.p.m.anticlockwise and gives motion to the sliding blocks B and D. The dimensions of the various linksare OA = 300 mm; AB = 1200 mm; BC = 450 mm and CD = 450 mm. Fig. 8.12 For the given configuration, determine : 1. velocities of sliding at B and D, 2. angularvelocity of CD, 3. linear acceleration of D, and 4. angular acceleration of CD. Solution. Given : NAO = 20 r.p.m. or ωAO = 2 π × 20/60 = 2.1 rad/s ; OA = 300 mm = 0.3 m ;A B = 1200 mm = 1.2 m ; BC = CD = 450 mm = 0.45 m
16. 16. Chapter 8 : Acceleration in Mechanisms l 189 We know that linear velocity of A with respect to O or velocity of A , vAO = v A = ωAO × O A = 2.1 × 0.3 = 0.63 m/s ...(Perpendicular to O A)1. Velocities of sliding at B and D First of all, draw the space diagram, to some suitable scale, as shown in Fig. 8.13 (a). Nowthe velocity diagram, as shown in Fig. 8.13 (b), is drawn as discussed below: (a) Space diagram. (b) Velocity diagram. (c) Acceleration diagram. Fig. 8.13 1. Draw vector oa perpendicular to O A, to some suitable scale, to represent the velocity ofA with respect to O (or simply velocity of A ), such that vector oa = v AO = v A = 0.63 m/s 2. From point a, draw vector ab perpendicular to A B to represent the velocity of B withrespect to A (i.e. vBA) and from point o draw vector ob parallel to path of motion B (which is alongBO) to represent the velocity of B with respect to O (or simply velocity of B). The vectors ab and obintersect at b. 3. Divide vector ab at c in the same ratio as C divides A B in the space diagram. In otherwords, BC/CA = bc/ca 4. Now from point c, draw vector cd perpendicular to CD to represent the velocity of D withrespect to C (i.e. vDC) and from point o draw vector od parallel to the path of motion of D (whichalong the vertical direction) to represent the velocity of D. By measurement, we find that velocity of sliding at B, vB = vector ob = 0.4 m/s Ans.and velocity of sliding at D, v D = vector od = 0.24 m/s Ans.2. Angular velocity of CD By measurement from velocity diagram, we find that velocity of D with respect to C, vDC = vector cd = 0.37 m/s
18. 18. Chapter 8 : Acceleration in Mechanisms l 191 ∴ Angular acceleration of CD, t aDC 1.28 α CD = = = 2.84 rad/s2 (Clockwise) Ans. CD 0.45 Example 8.7. Find out the acceleration of the slider D andthe angular acceleration of link CD for the engine mechanism shownin Fig. 8.14. The crank OA rotates uniformly at 180 r.p.m. in clockwisedirection. The various lengths are: OA = 150 mm ; AB = 450 mm;PB = 240 mm ; BC = 210 mm ; CD = 660 mm. Solution. Given: N AO = 180 r.p.m., or ωAO = 2π × 180/60 =18.85 rad/s ; O A = 150 mm = 0.15 m ; A B = 450 mm = 0.45 m ;PB = 240 mm = 0.24 m ; CD = 660 mm = 0.66 m We know that velocity of A with respect to O or velocityof A , v AO = v A = ωAO × O A = 18.85 × 0.15 = 2.83 m/s All dimensions in mm. ...(Perpendicular to O A) Fig. 8.14 First of all draw the space diagram, to some suitable scale,as shown in Fig. 8.15 (a). Now the velocity diagram, as shown in Fig. 8.15 (b), is drawn as discussedbelow: (a) Space diagram. (b) Velocity diagram. (c) Acceleration diagram. Fig. 8.15 1. Since O and P are fixed points, therefore these points lie at one place in the velocitydiagram. Draw vector oa perpendicular to O A, to some suitable scale, to represent the velocity of Awith respect to O or velocity of A (i.e. vAO or vA), such that vector oa = v AO = v A = 2.83 m/s 2. Since the point B moves with respect to A and also with respect to P, therefore drawvector ab perpendicular to A B to represent the velocity of B with respect to A i.e. vBA ,and from pointp draw vector pb perpendicular to PB to represent the velocity of B with respect to P or velocity ofB (i.e. vBP or vB). The vectors ab and pb intersect at b. 3. Since the point C lies on PB produced, therefore divide vector pb at c in the same ratio asC divides PB in the space diagram. In other words, pb/pc = PB/PC.
19. 19. 192 l Theory of Machines 4. From point c, draw vector cd perpendicular to CD to represent the velocity of D withrespect to C and from point o draw vector od parallel to the path of motion of the slider D (which isvertical), to represent the velocity of D, i.e. v D. By measurement, we find that velocity of the slider D, v D = vector od = 2.36 m/s Velocity of D with respect to C, vDC = vector cd = 1.2 m/s Velocity of B with respect to A , vBA = vector ab = 1.8 m/sand velocity of B with respect to P, v BP = vector pb = 1.5 m/sAcceleration of the slider D We know that radial component of the acceleration of A with respect to O or accelerationof A , aAO = aA = ωAO × AO = (18.85)2 × 0.15 = 53.3 m/s2 r 2 Radial component of the acceleration of B with respect to A , 2 vBA (1.8)2 aBA = r = = 7.2 m/s 2 AB 0.45 Radial component of the acceleration of B with respect to P, 2 vBP (1.5) 2 aBP = r = = 9.4 m/s 2 PB 0.24 Radial component of the acceleration of D with respect to C, 2 vDC (1.2)2 aDC = r = = 2.2 m/s 2 CD 0.66 Now the acceleration diagram, as shown in Fig. 8.15 (c), is drawn as discussed below: 1. Since O and P are fixed points, therefore these points lie at one place in the accelerationdiagram. Draw vector o a parallel to O A, to some suitable scale, to represent the radial component rof the acceleration of A with respect to O or the acceleration of A (i.e. aAO or aA), such that vector o′a′ = aAO = aA = 53.3 m/s 2 r 2. From point a, draw vector a x parallel to A B to represent the radial component of the racceleration of B with respect to A (i.e. aBA ), such that vector a′x = aBA = 7.2 m/s 2 r 3. From point x, draw vector xb perpendicular to the vector ax or AB to represent the ttangential component of the acceleration of B with respect to A i.e. aBA whose magnitude is yetunknown. 4. Now from point p, draw vector p y parallel to PB to represent the radial component of rthe acceleration of B with respect to P (i.e. aBP ), such that vector p′y = aBP = 9.4 m/s 2 r 5. From point y, draw vector yb perpendicular to vector by or PB to represent the tangential tcomponent of the acceleration of B, i.e. aBP . The vectors xb and yb intersect at b. Join p b. Thevector p b represents the acceleration of B, i.e. aB.