TEST BANK FOR Theory Of Machine And Mechanisms 4th Edition By Gordon R. Pennock & Joseph E. Shigley John J. Uicker

Exam (elaborations) TEST BANK FOR Theory Of Machine And Mechanisms 4th Edition By Gordon R. Pennock & Joseph E. Shigley John J. Uicker Solutions Manual to accompany THEORY OF MACHINES AND MECHANISMS Fourth Edition International Version John J. Uicker, Jr. Professor Emeritus of Mechanical Engineering University of Wisconsin – Madison Gordon R. Pennock Associate Professor of Mechanical Engineering Purdue University Joseph E. Shigley Late Professor Emeritus of Mechanical Engineering The University of Michigan Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. Chapter 1 The World of Mechanisms 1.1 Sketch at least six different examples of the use of a planar four-bar linkage in practice. They can be found in the workshop, in domestic appliances, on vehicles, on agricultural machines, and so on. Since the variety is unbounded no standard solutions are shown here. 1.2 The link lengths of a planar four-bar linkage are 0.2, 0.4, 0.6 and 0.6 m. Assemble the links in all possible combinations and sketch the four inversions of each. Do these linkages satisfy Grashof's law? Describe each inversion by name, for example, a crankrocker mechanism or a drag-link mechanism. s  0.2, l  0.6, p  0.4, q  0.6 ; these linkages all satisfy Grashof’s law since 0.20.6  0.40.6. Ans. Drag-link mechanism Drag-link mechanism Ans. Crank-rocker mechanism Crank-rocker mechanism Ans. Double-rocker mechanism Crank-rocker mechanism Ans. 1.3 A crank-rocker linkage has a 250 mm frame, a 62.5 mm crank, a 225 mm coupler, and a 187.5 mm rocker. Draw the linkage and find the maximum and minimum values of the transmission angle. Locate both toggle positions and record the corresponding crank angles and transmission angles. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. Extremum transmission angles:  min  1  53.1 max  3  98.1 Ans. Toggle positions: 2 2 4 4   40.1  59.1  228.6  90.9 Ans. 1.4 In Fig. P1.4, point C is attached to the coupler; plot its complete path. 1.5 Find the mobility of each mechanism illustrated in Fig. P1.5. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. (a) n  6, j1  7, j2  0; m  361  27 10 1 Ans. (b) 1 2 n  8, j 10, j  0; m  381 21010 1 Ans. (c) 1 2 n  7, j  9, j  0; m  37 1  29 10  0 Ans. Note that the Kutzbach criterion fails in this case; the true mobility is m=1. The exception is due to a redundant constraint. The assumption that the rolling contact joint does not allow links 2 and 3 to separate duplicates the constraint of the fixed link length 2 3 O O . (d) 1 2 n  4, j  3, j  2; m  341  23 12 1 Ans. Notice that each coaxial pair of sliding ground joints is counted as only a single prismatic pair. 1.6 Use the Kutzbach criterion to determine the mobility of the mechanism illustrated in Fig. P1.6. 1 2 n  5, j  5, j 1; m  351  25 11 1 Ans. Notice that the double pin is counted as two single j1 pins. 1.7 Find a planar mechanism with a mobility of one that contains a moving quaternary link. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. How many distinct variations of this mechanism can you find? To have at least one quaternary link, a planar mechanism must have at least eight links. The Grübler criterion then indicates that ten single-freedom joints are required for mobility of m = 1. According to H. Alt, “Die Analyse und Synthese der achtgleidrigen Gelenkgetriebe”, VDI-Berichte, vol. 5, 1955, pp. 81-93, there are a total of sixteen distinct eight-link planar linkages having ten revolute joints, seven of which contain a quaternary link. These seven are shown below: Ans. 1.8 Use the Kutzbach criterion to detemine the mobility of the planar mechanism illustrated in Fig. P1.8. Clearly number each link and label the lower pairs (j1) and higher pairs (j2) on Fig. P1.8. n  5, j1  5, j2 1; m  351  25 11 1 Ans. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. 1.9 For the mechanism illustrated in Fig. P1.9, determine the number of links, the number of lower pairs, and the number of higher pairs. Using the Kutzbach criterion determine the mobility. Is the answer correct? Briefly explain. n  4, j1  3, j2  2; m  341  23 12 1 Ans. If it is not evident that the input shown will increment this device in the direction shown, then consider incrementing link 3 downward. Since it seems intuitive that this determines the position of all other links, this verifies that mobility of one is correct. 1.10 Use the Kutzbach criterion to detemine the mobility of the planar mechanism illustrated in Fig. P1.10. Clearly number each link and label the lower pairs (j1) and higher pairs (j2) on Fig. P1.10. Treat rolling contact to mean rolling with no slipping. 1 2 n  5, j  5, j 1; m  351  25 11 1 Ans. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. 1.11 For the mechanism illustrated in Fig. P1.11 treat rolling contact to mean rolling with no slipping. Determine the number of links, the number of lower pairs, and the number of higher pairs. Using the Kutzbach criterion determine the mobility. Is the answer correct? Briefly explain. n  7, j1  8, j2 1; m  37 1  28 11 1 Ans. This result appears to be correct. If all parts remain assembled and within the limits of travel of the joints shown, then it appears that when any one member is locked the total system becomes a structure. 1.12 Does the Kutzbach criterion provide the correct result for the planar mechanism illustrated in Fig. P1.12? Briefly explain why or why not. 1 2 n  4, j  2, j  3; m  341  22 13  2 Ans. This result appears to be correct. If any part except the wheel is moved, other parts are required to follow. However, after all other parts are in a set position, the wheel is still able to rotate because of slipping against the frame at A. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. 1.13 The mobility of the mechanism illustrated in Fig. P1.13 is m = 1. Use the Kutzbach criterion to determine the number of lower pairs and the number of higher pairs. Is the wheel rolling without slipping, or rolling and slipping, at point A on the wall? Suppose that we identify the number of constraints at A by the symbol k. Then if we account for all links and all other joints as follows, the Kutzbach criterion gives n  5; j1  4; j2 1; jk 1; m  351 24 11 k 1  3 k; Therefore, to have mobility of m 1, we must have k  2 constraints at A. The wheel must be rolling without slipping. Ans. 1.14 Devise a practical working model of the drag-link mechanism. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. 1.15 Find the time ratio of the linkage of Problem 1.3. From the values of 2 and 4 we find  188.5 and  171.5 . Then, from Eq. (1.5), Q   1.099. Ans. 1.16 Plot the complete coupler curve of the Roberts' mechanism illustrated in Fig. 1.24b. Use AB = CD = AD = 62.5 mm and BC = 31.25 mm. 1.17 If the crank of Fig. 1.11 is turned 25 revolutions counterclockwise, how far and in what direction does the carriage move? Screw and carriage move by (25 rev)/(6 rev/mm) = 4.17 mm to the right. Carriage moves (7 rev)/(18 rev/mm) = 3.57 mm to the left with respect to the screw. Net motion of carriage = 25/6 – 25/7 = 25/42 = 0.59 mm to the right. Ans. More in-depth study of such devices is covered in Chapter 9. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. 1.18 Show how the mechanism of Fig. 1.15b can be used to generate a sine wave. With the length and angle of crank 2 designated as R and 2, respectively, the horizontal motion of link 4 is x4  Rcos2  Rsin2 90 . 1.19 Devise a crank-and-rocker linkage, as in Fig. 1.14c, having a rocker angle of 60. The rocker length is to be 0.50 m. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. 1.20 A crank-rocker four-bar linkage is required to have a time ratio Q = 1.2. The rocker is to have a length of 62.5 mm and oscillate through a total angle of 60. Determine a suitable set of link lengths for the remaining three links of the four-bar linkage. Following the procedure of Example 1.4, the required time ratio gives 180 1.2 180 Q         and, therefore, we must have  16.36. Then, with the X-line chosen at 30°, the drawing shown below (dimensioned 10 times size) gives measured distances of 4 2 1 108.5 mm O O R  r  , 2 2 3 2 160.5 mm B O R  r  r  , and 1 2 3 2 111 mm BO R  r  r  . From these we get one possible solution, which has link lengths 1 2 3 4 r 111 mm, r  24.8 mm, r 135.8 mm, r  62.5 mm Ans. 111 mm 160.5 mm Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. Chapter 2 Position and Displacement 2.1 Describe and sketch the locus of a point A that moves according to the equations   cos 2 xA R  at t , sin2  y A R at t   , and 0 zA R  . The locus is the spiral shown. Ans. 2.2 Find the position difference to point P from point Q on the curve 2 16 y x x    , where 2 xPR  and 4 xQ R  .  2 2 2 16 10 y P R      ; 2ˆ 10ˆ P R  i  j  2 4 4 16 4 y Q R     ; 4ˆ 4ˆ Q R  i  j 2ˆ 14ˆ 14.142 98.1 PQ P Q R  R R   i  j    Ans. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. 2.3 The path of a moving point is defined by the equation y  2x2  28 . Find the position difference to point P from point Q if 4 xP R  and 3 xQ R   .  2 2 4 28 4 y P R    ; 4ˆ 4ˆ P R  i  j  2 2 3 28 10 y Q R      ; 3ˆ 10ˆ Q R   i  j 7ˆ 14ˆ 15.652 63.4 PQ P Q R  R R  i  j    Ans. 2.4 The path of a moving point P is defined by the equation y  60  x3 / 3. What is the displacement of the point if its motion begins when 0 xP R  and ends when 3 xP R  ?    3 0 60 0 / 3 60 y P R    ; 0 60ˆ P R  j    3 3 60 3 / 3 51 y P R    ; 3 3ˆ 51ˆ P R  i  j (3) (0) 3ˆ 9ˆ 9.487 71.57 P P P R  R R  i  j    Ans. 2.5 If point A moves on the locus of Problem 2.1, find its displacement from t = 1.5 to t = 2. 1.5 1.5 cos3 ˆ 1.5 sin3 ˆ 1.5 ˆ A R  a  i  a  j   ai 2.0 2.0 cos4 ˆ 2.0 sin 4 ˆ 2.0 ˆ A R  a  i  a  j  ai 2.0 1.5 3.5 ˆ A A A ΔR  R R  ai Ans. 2.6 The position of a point is given by the equation R 100e j2t . What is the path of the point? Determine the displacement of the point from t = 0.10 to t = 0.60. The point moves in a circle of radius 100 with center at the origin. Ans.   0.10 100 0.628 80.902ˆ 58.779ˆ j R  e  i  j   0.60 100 3.770 80.902ˆ 58.779ˆ j R  e   i  j ΔR  R0.60R0.10  161.804ˆi 117.557ˆj  200.0216 Ans. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. 2.7 The equation R  t2  4e jt /10 defines the position of a point. In which direction is the position vector rotating? Where is the point located when t = 0? What is the next value t can have if the orientation of the position vector is to be the same as it is when t = 0? What is the displacement from the first position of the point to the second? Since the polar angle for the position vector is    t /10, then d / dt is negative and therefore the position vector is rotating clockwise. Ans. R0  02  4e j0  40 The position vector will next have the same direction when  t /10  2 , that is, when t=20. Ans.     2 2 20 20 4 404 0 j e   R      R  R20R0  4000 Ans. 2.8 The location of a point is defined by the equation   2 / 30 R  4t  2 e jt , where t is time in seconds. Motion of the point is initiated when t = 0. What is the displacement during the first 3 s? Find the change in angular orientation of the position vector during the same time interval. R0  0 2e j0  20  2ˆi R3  12 2e j9/ 30 1454  8.229ˆi 11.326ˆj ΔR  R3 R0  6.229ˆi 11.326ˆj 12.92661.19 Ans.   540  54 ccw Ans. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. 2.9 Link 2 in Fig. P2.9 rotates according to the equation   t / 4 . Block 3 slides outward on link 2 according to the equation r  t2  2 . What is the absolute displacement 3 RP from t = 1 to t = 3? What is the apparent displacement 3/ 2 RP ?   3 2 / 4 2 j j t P re t e   R      3 1 3 45 2.121ˆ 2.121ˆ P R     i  j   3 3 11 135 7.778ˆ 7.778ˆ P R      i  j     3 3 3 3 1 9.899ˆ 5.657ˆ 11.402 150.26 P P P ΔR  R R   i  j    Ans.   3 0 2 / 2 2 2 ˆ j P R  re  t  i   3 / 2 2 1 3ˆ P R  i   3 /2 2 3 11ˆ P R  i     3 /2 3 /2 3 /2 2 3 1 8ˆ P P P ΔR  R R  i Ans. 2.10 A wheel with center at O rolls without slipping so that its center is displaced 250 mm to the right. What is the displacement of point P on the periphery during this interval? Since the wheel rolls without slipping, O PO R   R . / 250 mm/150 mm 1.667 rad 95.51 O PO   R R         For PO R ,      27095.51 174.49 150 mm 174.49 149.3ˆ 14.4ˆ mm PO R      i  j   250ˆ 149.3ˆ 14.4ˆ 150ˆ mm P O PO PO         ΔR ΔR R R i i j j 6 in PO R  100.7ˆ 164.4ˆ mm 192.8 mm 58.51 P R  i  j    Ans. 2.11 A point Q moves from A to B along link 3 whereas link 2 rotates from 2   30 to 2   120 . Find the absolute displacement of Q. 3 0.3 m 30 259.8ˆ 150.0ˆ mm Q R     i  j 3 0.3 m 120 150.0ˆ 259.8ˆ mm QR      i  j 3 3 3 409.8ˆ 109.8ˆ mm Q Q Q ΔR  R R   i  j 5 /3 600.0ˆ mm Q BA ΔR  R  i Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. 2 4 0.3 m AO BO R  R  ; 4 2 0.6 m BA O O R  R  Q5 Q3 Q5 /3 ΔR  ΔR ΔR 5 190.2ˆ 109.8ˆ mm 219.6 mm 30 Q ΔR  i  j    Ans. 2.12 The linkage is driven by moving the sliding block 2. Write the loop-closure equation. Solve analytically for the position of sliding block 4. Check the result graphically for the position where   45 . The loop-closure equation is A B AB R  R R Ans. j /12 j  A B AB j B AB R e R R e R R e          500 mm, 15 AB R     Taking the imaginary components of this, we get sin15 sin A AB R   R  sin sin 45 500 mm 1365 mm sin15 sin15 A AB R R           Ans. 2.13 The offset slider-crank mechanism is driven by the rotating crank 2. Write the loopclosure equation. Solve for the position of the slider 4 as a function of 2  . 20 mm AO R  , 50 mm BA R  , and 140 mm CB R  C A BA CB R  R R R j / 2 j 2 j 3 C A BA CB R R e R e R e       Taking real and imaginary parts, 2 3 cos cos C BA CB R  R   R  and 2 3 0 sin sin A BA CB  R  R   R  and, solving simultaneously, we get 1 2 3 sin sin A BA CB R R R            with 3 90   90 Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. 2  2 2 2 2 2 2 2 cos sin 50cos sin 2 500sin C BA CB A BA R R  R R R             Ans. 2.14 For the mechanism illustrated in Fig. P2.14, define a set of vectors that is suitable for a complete kinematic analysis of the mechanism. Label and show the sense and orientation of each vector on Fig. P2.14. Write the vector loop equation(s) for the mechanism. Identify suitable input(s) for the mechanism. Identify the known quantities, the unknown variables, and any constraints. If you have identified constraints then write the constraint equation(s). One suitable set of two vector loop equations is ? ? ? 2 3 4 5 1      R R R R R  0 and ? 1 2 3 I  C C C R R R R R  0 Ans. The angle 2  is a reasonable input. Three constraint equations are required. 44 4   (C1) 24 4    (C2) 22 2    (C3) Ans. There are four unknowns 3 4 5 24  ,  ,  , and R . Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University Press 2015. All rights reserved. 2.15 Assume rolling with no slip between pinion 5 and rack 4 in the mechanism illustrated in Fig. P2.15. Define a set of vectors that is suitable for a complete kinematic analysis of the mechanism. Label and show the sense and orientation of each vector on Fig. P2.15. Write the vector loop equation(s) for the mechanism. Identify suitable input(s) for the mechanism. Identify the known quantities, the unknown variables, and any constraints. If you have identified constraints then write the constraint equation(s). One suitable set of vectors is as shown. The vector loop equation is ? ? R2R3R6 R4 R15 R1  0 Ö I Ö Ö Ö Ö Ö Ö Ö Ö with 5 5 6    R Ans. The angle 2  is a suitable input. Ans. There are two unknown variables, 3  and R6. Ans. Theory of Machines and Mechanisms, 4e Uicker et al. © Oxford University