Kinematics, Dynamics, And Design of Machinery, K. J. Waldron and G. L. Kinzel SOLUTIONS

SUPPLEMENTAL EXERCISE PROBLEMS for Textbookhttp://www.iust.ac.ir/files/mech/madoliat_bcc09/pdf/SUPPLEMENTAL%20EXERCISE%20 PROBLEMS.pdf Kinematics, Dynamics, and Design of Machinery by K. J. Waldron and G. L. Kinzel ©1996-97 by K. Waldron and G. Kinzel Department of Mechanical Engineering - ii - Supplemental Exercise Problems for Kinematics, Dynamics, and Design of Machinery by K. J. Waldron and G. L. Kinzel Supplemental Exercise Problems for Chapter 1 Problem S1.1 What are the number of members, number of joints, and mobility of each of the planar linkages shown below? (a) (b) (c) Problem S1.2 Determine the mobility and the number of idle degrees of freedom of each of the planar linkages shown below. Show the equations used to determine your answers. (a) (b) (c) -1- L3 = 4 mm. (a) Problem S1. L4 = 3. and L4 = 5 m m and link 1 is fixed. what type of four-bar linkage is it? Also. L2 = 3 mm. Show the equations used to determine your answers. L2 = 8”. L3 = 4”. Problem S1. L4 = 19”. a) L1 = 5”.6 You are given two sets of links. and L5 = 2.4 (b) Determine the mobility and the number of idle degrees of freedom associated with the mechanism. 2 3 4 Problem S1.5” -2- .Problem S1.5 If the link lengths of a four-bar linkage are L1 = 1 mm. Sketch the linkage and identify the type of four-bar mechanism.5”. is the linkage a Grashof type 1 or 2 linkage? Answer the same questions if L1 = 2 mm. Select four links from each set such that the coupler can rotate fully with respect to the others. L2 = 2”.3 Determine the mobility and the number of idle degrees of freedom of the linkages shown below. and L5 = 28” b) L1 = 5”. Show the equations used to determine your answers. L3 = 15”. 5" ω 2 = 3 rad s 2 3 34˚ 4 A 50˚ F 6 2.4" 5 D E 29.0" DF = 21.5" -3- .9" CE = 28.1 Draw the velocity polygon to determine the following: a) Sliding velocity of link 6 b) Angular velocities of links 3 and 5 C B AB = 12.4" DC = 27.0" 10.5" BC = 22.Supplemental Exercise Problems for Chapter 2 Problem S2. 2 In the mechanism shown. Draw the velocity polygon. 1vA2 = 15 m/s.7" BC = 1. and determine the velocity of point D on link 6 and the angular velocity of link 5 (show all calculations).4" 45˚ X -4- .2" 5 D 6 B 15 m/s C 4 2. Y 2 A 3 2.Problem S2.05" 1vA 2= AC = 2.4" BD = 3. Find the velocities of points B.5" AC = 1.3 In the mechanism shown below.Problem S2.55" AB = 2. and D of the double-slider mechanism shown in the figure if Crank 2 rotates at 42 rad/s CCW.05" C E 1ω 4 B 2 2 3 3 60˚ A -5- . C. D 6 0. points E and B have the same vertical coordinate.75" CD = 2.75" 5 EA = 0.0" CB = 1. 4 In the figure below. B4 AC = 1 in BC = 3 in r = 2. Identify the sliding velocity between the block and the slide. Draw and dimension the velocity polygon. and 1ω3 = 30 rad/s.Problem S2. and find the angular velocity of link 2. points A and C have the same horizontal coordinate.8 in -6- . 4 1ω 3 3 A 2 C 45˚ B 3 . 1") Y B 2 A 55˚ X C 3 4 AB = DE = 1.5 rad/ s (CCW).0 in CD = 1. 1α 2 = 40 rad/ s2 (CCW) Draw the velocity polygon. 1.5 The following are given for the mechanism shown in the figure: 1ω 2 = 6. and locate the velocity of Point E using the image technique. D (2.2".5 in E -7- .0 in BC = 2.Problem S2. 3 B 4.Problem S2.0" 2 4 50˚ A 120˚ -8- .6 In the mechanism shown. Draw the velocity and acceleration polygons and solve for the angular velocity and acceleration of link 2. link 4 moves to the left with a velocity of 8 in/s and the acceleration is 80 in/s2 to the left. C 3 G 4 B 2 A 90˚ D ω 2.Problem S2. link 2 is turning CCW at the rate of 10 rad/s (constant).0" BG = 1.0" AD = 3.0" BC = 2. Draw the velocity and acceleration polygons for the mechanism.0" CD = 3. α2 AB = 1.7 In the mechanism shown below.0" -9- . and record values for 1aG3 and 1α4. 1ω2 = 24 rad/s (CW).8 For the data given in the figure below.10 - . 1aA = 400 ft/s2.Problem S2. Assume 1vA = 20 ft/s.0" B . C vA 90˚ A 15o aA ω2 α2 AB = 4.5" BC = 2.05" AC = 2. and 1α2 = 160 rads2 (CCW). find the velocity and acceleration of points B and C. 0" BD = 1.11 - . determine the acceleration of D3 by image.4" AB = 3.05" BC = 1. Y B D vA 2= 10 in/s (constant) 2 A 6 5 3 E C 4 81˚ X CD = 1.5" FE = 1.Problem S2.9 In the mechanism shown.45" ED = 1. Also.0".75") .0" F (-1. -0. find 1ω6 and 1α3. 8m BE = 0. 1α5. 1aE6 for the position given. Also find the point in link 5 that has zero acceleration for the position given.67m 6 E 0.8m CD = 0. 1ω2 = 1 rad/s (CCW) and 1α2 = 0 rad/s2. 1vE6.52 m 2 A 30˚ 5 B C 3 4 D .5m BC = 0.12 - .Problem S2. AD = 1m AB = 0.10 In the mechanism shown. Find 1ω5. Also find the velocity of G7 by image.6" G DE = 1.5" DG = 0. Find 1ω7 and 1α7 for the position given. and the velocity and acceleration of point D7 are given. Links 7 and 8 are drawn to scale.Problem S2.65" v D 7 = 5.7" GE = 1. Y E 7 D 6 1.25" 8 X .13 - .11 Part of an eight-link mechanism is shown in the figure.0 320˚ in/sec a D 7 = 40 260˚ in/s 2 1. and 1aC3. points A. B.14 - . and C are collinear. If 1vA2 = 10 in/s (constant) downward.Problem S2.12 In the figure shown below.35" 0.45" . find 1vC3. Cam Contact 2 A 1" 45˚ Rolling Contact B C 3 0. Problem S2.13 Part of an eight-link mechanism is shown in the figure. There is rolling contact at location B. Links 7 and 8 are drawn to scale, and the velocity and acceleration of points A6 and C5 are as shown. Find 1ω8 and 1α7 for the position given. Also find the velocity of E7 by image. E D 7 AD = 2.25" DB = 1.0" AC = 2.85" A 6 72˚ 5 B 8 vA6 = 10 60˚ in/s 2 a C5 = 20 270˚ in/s 2 aA6 = 20 180˚ in/s vC5 = 10 0˚ in/s C - 15 - Problem S2.14 In the position shown, find the velocity and acceleration of link 3 using (1) equivalent linkages and (2) direct approach. C 2 B 1 ω 2 3 45˚ ΑΒ = 0.5" BC = 1.1" A ω 2 = 100 rad/ s CCW 1 α 2 = 20,000 rad / s 2 CCW 1 - 16 - Problem S2.15 For the mechanism shown, find 1ω3, 1α3, 1aB3, and the location of the center of curvature of the path that point B3 traces on link 2. 3 C B 2 A 117˚ 1 ω ,1α 2 2 12" AC = 9 in AB = 2 in 1ω = 50 2 1α 2 =0 rad/s D - 17 - 16 For the mechanism shown. and the location of the center of curvature of the path that point B3 traces on Link 2.1aB3. Point B2 moves in a curved slot on link 3. find 1ω3. 1α3.1aD3.7 m = 3 rad/ s2 CCW D B 3m E 2 3 60˚ A C . 1ω 2 AB = AC = 5 m = 2 rad/ s CCW 1α 2 CD = 7 m CE = 5. points C.18 - . 1vD3. 1vB3. B and D are collinear. For the position given.Problem S2. 19 - . find 1ω3.3" B 2 60˚ 1ω 2 = 200 rad (constant) sec A . and the location of the center of curvature of the path that point B3 traces on Link 2. 3 1. Assume that Link 2 is driven at constant velocity.25" C AB = 1" AC = 4.Problem S2.17 If the mechanism shown is drawn full scale. 1α3. D 3 B AC = 5.18 if 1ω2 = 2 rad/sec (constant) . Determine the following: v B4 . a B4 .18 In the mechanism below. α 4 .19 Resolve Problem S2. v D4 .0 in 1α 2 2 4 1ω 2 75˚ A C Problem S2. a D4 .0 in CD = 10. ω 4 .Problem S2.2 in AB = 6.20 - . and the center of curvature of the path that B4 traces on Link 2. the angular velocity of Link 2 is 2 rad/s CCW and the angular acceleration is 5 rad/s2 CW. 35" AB = 3. 1α3. and the center of curvature of the path that C3 traces on Link 2.6" AD = 4.22" Center of curvature of slot B Y R Pin in Slot C 2 102˚ X 1ω 3 A D 2 . CD = 0. find 1ω3.0" R = 1.21 - .Problem S2.20 If 1ω2 = 20 rad/s (constant). 6. 1ω2 = 10 rad/s. 2. 1ω6.35 in BC = 3. Complete the velocity polygon.Problem S2.0 in DE = 4. 1vF6.0 in BD = 4.3 in AE = 3. and determine the following: 1vD4.0 in GF = 3.22 - .21 In the mechanism below.65) D C 75˚ 6 AB = 1. 1ω4.4 in B 2 A 100˚ X 3 E 5 F . Y 4 G (-5. find 1α3.5 cm BC = 13.22 If 1ω2 = 10 rad/s (constant).0 cm 3 2 105˚ A 1ω 2 C . B AC = 10.23 - .Problem S2. find a) 1ω3 b) The center of curvature of the path that B2 traces on link 3 (show on drawing).24 - . c) The center of curvature of the path that B3 traces on link 2 (show on drawing). AB = 2" AC = 4" B 2 A 40˚ 3 C .Problem S2.23 If 1ω2 = 10 rad/s CW (constant). Problem S2. 10 in 2 Y 1 ω 1α 4 4 B 3 = 1 rad/s =0 4 A 30˚ X .24 Determine the velocity and acceleration of point B on link 2.25 - . 1" AD = 3.45" BC = 1.Problem S2.5" E D 2 157.0" B 6 0. use the instant-center method to find 1ω2 and then use 1ω2 to find vB6 (direction and magnitude).26 - .3" 5 1.5˚ X .9" CD = 1.25 Given vA4 = 1. Y 4 A 3 C DE = 1.5" AC = 2.0 ft/s to the left. Problem S2. B BC = CD BD = 3.27 - .06" 3 150˚ C A 50˚ D 4 2 .26 Find all of the instant centers of velocity for the mechanism shown below. 75" BE = 1. 1.5" GF = 1.27 If the velocity of point A on link 2 is 10 in/s as shown.25" 5 G 5 4 E X C5 6 .5" AD = 0. use the instant center method to find the velocity of point C on link 5.45" B 0.75" AB = 1.Problem S2.5" D A 2 75˚ 3 0.9") Y vA DE = 2.15". F (3.28 - . 85" GF = 1.75" C B 3 6 5 D 7 1.28 Assume that link 7 rolls on link 3 without slipping. find 1ω7 using instant centers.0" G 4 F (-2.8" BG = 0.9" DE = 3.0" 0.29 - . For the given value for 1ω2.0". and I27 .25" AE = 2.95") 2 A 1 ω2 90˚ E X = 2 rad/s .Problem S2.7" BD = 3. and find the following instant centers: I13 . I15 . 0. Y AB = 1. 95" DF = 2.45" AB = 1.Problem S2.8" BE = 0. find 1vC5 using the instant-center method.85" EG = 2.30 - . DA = 0.45" BF = 1.2" EC = 1.29 If 1vA2 = 10 cm/s as shown.25" C B G 6 1vA 2 A 2 3 E 4 5 1.2" CG = 1.9" D 70˚ F . 6" Α 18˚ 1ω 2 110˚ 3 Β 4 2 45˚ D C .5" CD = 4.31 - . Show the velocity vector 1vB3 on the figure. find the velocity of point B using the instant-center method.5" DE = 2.30 If 1ω2 = 10 rad/s CCW. E CA = 1.0" AB = 1.Problem S2. 0" DE = 0.8" AD = 3. Show the velocity vector 1vE4 on the figure.0" BC = 1.Problem S2.0" .31 If 1ω2 = 100 rad/s CCW. 1ω A 2 D 2 B 3 C 70˚ E 4 AB = 1.32 - . find the velocity of point E using the instant center method.75" CD = 2. 32 In the linkage shown below.9" BC = 0.0" 2 A F B C 5 55˚ 3 E 4 D .9" DE = 0. 3. locate all of the instant centers.Problem S2.33 - .9" CF = 2.35" BD = 3.3" 6 AB = 1. 0" DE = 1. find 1ω6 using instant centers. E 5 6 F AC = 1.5" EF = 1.33 If 1ω2 = 5 rad/s CCW.0" AB = 2.0" BD = 4.1" D 4 A 50˚ 3 2 1ω 2 B C .Problem S2.34 - .5" AF = 4. find 1vB4 using instant centers and the rotating radius method.7" CF = 0.65" G 125˚ B4 3 C F 4 A D 95˚ 2 E .35 - .34 If 1ω2 = 100 rad/s CCW.8" CD = 0. AD = 1.75" AE = 0.45" FG = 1.Problem S2.0" DB = 1.75" CB = 1. find the angular velocity (1ω6) of link 6 using the instant-center method.0" EF = 1.0" AD = 2. 1v A2 E 5 A 6 D 4 AB = 1.Problem S2.36 - .95" CE = 2.85" 2 B 27˚ C 3 F .35 If 1vA2 = 10 in/s as shown.25" BF = 3.0" AC = 0. 20" G .16" BC = 0.30 EG = 2. 3 B 2 A 1ω 2 C 4 32˚ E 5 75˚ D 6 F AB = 1.30" DF = 1.Problem S2.45" DE = 1. Show the velocity vector 1vG5 on the figure.16" AD = 1.37 - .70" CD = 1.36 If 1ω2 = 50 rad/s CCW. find the velocity of point G using the instant center method. 0".37 If 1ω2 = 100 rad/s CCW.38 - .0" = 3.0" = 4.Problem S2.0" Y 3 1ω 2 C 5 A F 4 B D 2 X . 6. E (2.2" = 6. find 1ω6.0" = 3.0") 6 AB BC CD AD BF = 1. φ velocity of point B4 for the position defined by φ = 60˚. find 1vC4 and 1aC4 if link 2 rotates with constant angular velocity.3 In the mechanism given. B 4 3 2 10 inches φ Problem 3. Use the loop equation approach to determine 1vA4 and 1aA4. 1vB2 is 10 in/s to the right (constant). For the position defined by the data below.1 For the mechanism shown.39 - . AB = 1 cm. and acceleration. B 2 θ A 3 C 4 θ = 90°.Supplemental Exercise Problems for Chapter 3 Problem 3. 4 A AB = 10 inches 3 O 30˚ 2 B 1v B2 . write the loop equations for position. BC = 4 cm. Use the loop equation approach to determine the In the mechanism shown. Use point O as the origin of your coordinate system. Show all angles and vectors used on the drawing and identify all variables.2 ˙ is 10 rad/s CCW. ˙ = 1 rad/ s θ Problem 3. velocity. Problem 3. If 1ω2 = 10 rad/s CCW.5 Use loop equations to determine the velocity and acceleration of point B on link 4. y 3 in A 2 1ω = 1 rad 2 sec 1α 2 = 0 3 B 4 30˚ x Problem 3. Link 4 also slides on the frame (link 1). use the loop-equation approach to determine the velocity of link 4 for the position defined by φ = 45˚.6 Use loop equations to determine the angular velocity and acceleration of link 3 if vB4 is a constant 10 in/s to the left and 1θ3 is 30˚.40 - . Link 3 slides on link 2.4 In the mechanism shown. y 3 in 2 A 1 θ3 v B4 3 B 4 x . and link 4 is pinned to link 3. 3 4 B 10 inches 2 φ A Problem 3. Compute the linear velocity of point D2. The mechanism is an inverted slider-crank linkage.88" 1. There are two identical linkages one on each side of the car. the angular velocity of link 2 relative to the frame is 1ω2 is 205 (rad/s). This is done by the use of an air cylinder. both in the clockwise direction. and the connecting linkage are links 2 and 4. and the angular acceleration is 1α2 is 60 (rad/s2). Compute the angular velocity and angular acceleration of link 3 for the position shown 201˚ A B 2 1ω 2 61˚ 1.7 The shock absorber mechanism on a mountain bicycle is a four-bar linkage as shown. Assume that the angular velocity of the driver (link 2) is a constant 0.55" 1 D 1.Problem 3.82 rad/s clockwise and the angular acceleration is 0.41 - . and the angular velocity and angular acceleration of link 3 for the position shown .8 The purpose of this mechanism is to close the hatch easily and also have it remain in an open position while the owner removes items from the back. As the bicycle goes over a bump in the position shown. the fork and tire assembly is link 3. The frame of the bike is link 1.22" Problem 3. which helps to support the hatch while it closes and holds it up in the open position. where the car acts as the base link.64" 3 4 C 2. r r2 in terms of the r ˙ ˙ 1 . Link 3 slides on link 2 and link 1.50" C Problem 3.60" 40.D 42. δ.9 The wedge mechanism has three links and three sliding joints. are known. r 2 . ˙˙ variables given. 2 γ r2 β r1 x δ 3 . Determine the relationship among β.91" A 2 35. Link 2 slides on link 3 and link 1. Derive expressions for r 2 . and ˙˙ 1 . r1.30" B 3 2 4 1ω 12. Assume that link 3 is the input link so that r4.74˚ 145˚ 12. Compute the degrees of freedom of the mechanism. and γ which permits the mechanism to move.42 - . Y B1 A3 (-2. and positions A2B2 and A3B3 are vertical. 0. 3) A1 (6. AB = 4 cm.Supplemental Exercise Problems for Chapter 4 Problem S4.3) C *(0.2 Design a four-bar linkage to move its coupler through the three positions shown below using points A and B as moving pivots. The moving pivot (circle point) of one crank is at A and the fixed pivot (center point) of the other crank is at C*.1 Design a four-bar linkage to move a coupler containing the line AB through the three positions shown. and use Grashof’s equation to identify the type of four-bar linkage designed. AB = 6 cm. Position A1B1 is horizontal. Draw the linkage in position 1. What is the Grashof type of the linkage generated? Y B3 B2 60˚ A3(2.43 - .85) A1 (0. 3) A2 (2. 0) X B3 B2 Problem S4. 2. 0) B1 X .4) 50˚ A 2 (2. 4. 4. AB = 4 cm.5) 45˚ A2 (2.3. 1. .Problem S4. Y D* (3. Y C* (2. Position A1B1 and position A3B3 are horizontal.6) C* 45˚ A2 (2.1. . 2. 4.5) B2 A3 (2.0) A1 B1 X B3 .7.3 Synthesize a four-bar mechanism in position 2 that moves its coupler through the three positions shown below if points C* and D* are center points.4 Synthesize a four-bar mechanism in position 2 that moves its coupler through the three positions shown below. Point A is a circle point.6) B2 A 3 (3.7) A1 (0.44 - .0.0.0.7. AB = 4 cm. 3. and point C* is a center point. 1.8) B1 B3 X Problem S4. Position A1B1 and position A3B3 are horizontal.7. and positions A2B2 and A3B3 are vertical.. 0) B3 X .5) A3 C* X (-0. 0.5) A1 B 1(-1. and the fixed pivot for the other crank is to be at C*. -2. Draw the linkage in position 2 and use Grashof’s equation to identify the type of four-bar linkage designed. double crank).45 - . and position A3B3 is vertical.0. The moving pivot for one crank is to be at A.0) B 3 Problem S4.5. 2) B1 C1 * (0. The moving pivot (circle point) of one crank is at A and the fixed pivot (center point) of the other crank is at C*. Y A2 B2 (0.g. Positions A1B1 and A2B2 are horizontal.0 in. AB = 2. 1.6 Design a four-bar linkage to move a coupler containing the line AB through the three positions shown.Problem S4. crank rocker. Y A2 A3 A1 (0.5 Design a four-bar linkage to move the coupler containing line segment AB through the three positions shown. Draw the linkage in position 1 and determine the classification of the resulting linkage (e. 0) B2 (4. AB = 4 in.0. Position A1B1 is horizontal. 0) (2. 7 Design a four-bar linkage to generate the function y=ex -x for values of x between 0 and 1. Accurately sketch the linkage at the three precision point positions.7* π + 3p cos θ + 2p φ = 4.Problem S4. Determine: ∆θ = 60˚ ∆φ = 40˚ a) The number of precision points required to complete the design of the system. and the base length be 2 in. The base length of the linkage must be 0. (15 points)A mechanical device characterized by the following input-output relationship: 2. Exterior constraints on the design require that the parameter a1 = 1.10 2. Use Chebyshev spacing and determine the values of x1 and x2 which will allow the device to best approximate the function.8 Use Freudenstein's equation to design a four-bar linkage to generate approximately the function y=1/x2. φ 1 = 135˚. Use Chebyshev spacing with three position points.8 + 3a tanθ + a is to A mechanical device characterized by the input-output relationship φ = 2a1 2 3 3 be used to generate (approximately) the function y = 2 x over the range 0 ≤ x ≤ π / 4 where x. y. Problem S4. Use the following angle information: θ0 = 65˚ φ0 = 40˚ Problem S4. and determine the values for the unknown design variables which will allow the device to approximate the function. Use three precision points with Chebyshev spacing.8 cm. ∆ θ = 90˚. b) Use Chebyshev spacing.6p1 2 3 .46 - .11 3. Problem S4. θ is to be proportional to x and φ is to be proportional to y. and θ are all in radians. ∆ φ = -90˚. Assume that the device will be used over the ranges 0 ≤ θ ≤ π / 4 and 0 ≤ φ ≤ π / 3 . c) Compute the error generated by the device for x = π/6.9 A device characterized by the input-output relationship φ = x1 + x2 sinθ is to be used to generate (approximately) the function φ = θ 2 (θ and φ both in radians) over the range 0 ≤ θ ≤ π / 4 . Let θ1 = 45˚. 1 ≤ x ≤ 2. φ . Problem S4. a time ratio of 1. The problem is such that you may pick any value for any one of the 3 3 design variables and solve for the other two. Determine: 1) The number of precision points required to complete the design of the system.e. . The oscillation is to be symmetric about a vertical line through O4.0.47 - . Specify the length of each of the links. φ . 2) Use Chebyshev spacing. y. i. Assume that the device will be used over the ranges 0 ≤ θ ≤ π and 0 ≤ φ ≤ π . p2.13 Design a crank-rocker mechanism which has a base length of 1. Problem S4. pick a value for one of p1. the oscillating lever will have a time ratio of advance to return of 3:2.. and the length of the base link is to be 2 in. and determine the values for the unknown design variables which will allow the device to approximate the function. 3) Compute the error generated by the device for x = π/6. Problem S4.is to be used to generate (approximately) the function y = 2 x over the range 0 ≤ x ≤ π where 4 x.25.12 Design a crank-rocker mechanism such that with the crank turning at constant speed. and θ are all in radians. and a rocker oscillation angle of 40˚. p3 and solve for the other two. The lever is to oscillate through an angle of 80o. and angles CAB and CBA both equal 45˚.48 - . Notice that the cognates will also be drag-link mechanisms. Draw the cognates in the position for θ = 180˚. AM = BQ = 4 m.Problem S4. θ B Q M C 45˚ A . AB =2 m.14 Determine the two 4-bar linkages cognate for the drag-link mechanism shown. The dimensions are MQ = 1 m. The rise is 1. determine the number of terms required and the corresponding values for the coefficients Ci if the cam velocity is constant. The rise is h after β radians of rotation. The displacement equation for the follower during the rise period is s = h∑ Ci  θ   β i=0 n i At A.0 cm Dwell β B θ Problem S6. The displacement equation for the follower during the rise period is s = h∑ Ci  θ   β i=0 n i a) If the position. and acceleration must be continuous there. At B only displacement and velocity are important. S Slope = 1/β 1. and the position and velocity are continuous at θ = β. so displacement. b) If the follower is a radial flat-faced follower and the base circle radius for the cam is 1 cm. and the rise begins and ends at a dwell. determine the n required in the equation. Show all equations used.0 cm after β degrees of rotation.2 Assume that s is the cam-follower displacement and θ is the cam rotation. . and the rise begins at a dwell and ends with a constant-velocity segment. and find the coefficients Ci which will satisfy the requirements. dynamic effects are important. velocity.49 - .Supplemental Exercise Problems for Chapter 5 No supplemental problems were developed for Chapter 5. If h is 1 cm and β is 90˚. To satisfy these conditions.1 Assume that s is the cam-follower displacement and θ is the cam rotation. and acceleration are continuous at θ = 0. determine the minimum base circle radius to avoid cusps if the cam is used with a flat-faced follower. velocity. determine whether the cam will have a cusp corresponding to θ = 0. Supplemental Exercise Problems for Chapter 6 Problem S6. and s is the displacement at any given angle θ . determine the minimum base circle radius to avoid cusps if the cam is used with a flat-faced follower.5 Resolve Problem 6. Note: Use the minimum number of Ci possible.S h Β Dwell Dwell Α β θ . h is the rise.12 if the cam velocity is 2 rad/s constant.. If h is 3 cm and β is 60˚. S h Β Dwell Dwell Α β θ . a5 such that the displacement. The displacement equation for the follower during the rise period is s = h∑ ai  θ   β i=0 5 i Determine the required values for a0 . . β is the angle through which the rise takes place. If h is 1 cm and β = π / 3 .50 - . velocity. Determine the number of terms required and the corresponding values for the constants (Ci ) if the displacement and velocity are continuous at the end points of the rise (points A and B) and if the cam velocity is constant. Problem S6.4 For the cam displacement schedule given.. and acceleration functions are continuous at the end points of the rise portion. determine the minimum base circle radius to avoid cusps if the cam is used with a flat-faced follower. If h is 1 cm and β is 90˚.3 Resolve Problem 6.14 if the cam will be used for a low-speed application so that acceleration effects can be neglected. rad Problem S6. determine the minimum base circle radius to avoid cusps if the cam is used with a flat-faced follower. rad Problem S6. The cam will rotate clockwise.2 in. . Lay out the cam profile using 20˚ plotting intervals. and the base circle radius is to be 2 in. Assume that the follower is to dwell at zero lift for the first 100˚ of the motion cycle and to dwell at 1 in lift for cam angles from 220˚ to 240˚. The cam is to have a translating.Problem S6. flat-faced follower which is offset in the clockwise direction by 0.6 Lay out a cam profile using a harmonic follower displacement profile for both the rise and return.51 - . 52 - . 1aB3/A1. ˙ = 0 in / s . . link 3 is vertical (parallel to Z). Supplemental Exercise Problems for Chapter 9 No supplemental problems were developed for Chapter 9. ˙˙ AB = 1in. link 2 lies in a plane parallel to the XY plane and points along a line parallel to the Y axis. and 1 α 3. θ s X θ B ˙ = 1rad / s. For the position to be analyzed.Supplemental Exercise Problems for Chapter 7 Problem S7. and that at B is a revolute joint. respectively.1 In the manipulator shown. The joint between link 2 and the frame at A is a cylindrical joint. Determine 1vC3/A1. BC = 1in φ m Supplemental Exercise Problems for Chapter 8 No supplemental problems were developed for Chapter 8. Z C A 2 3 θ = 90°. In the position to be analyzed. θ ˙˙ = 0 rad / s2. and link 3 is perpendicular to link 2. φ ˙˙ = 0 rad / s2. the joint axes at A and B are oriented along the z and m axes. ˙ = 1rad / s. φ Y s s = 1in / s2 . 1 An alternative gear train is shown as a candidate for the spindle drive of a gear hobbing machine. select gears 3 and 5 which will satisfy the ratio. If the gear blank is to be driven clockwise. Gears are available which have all of the tooth numbers from 15 to 40. N3 = 30T. determine the hand of the hob. N4 = 20T and N5 = 130.53 - . Make up a table of possible gear ratios where the maximum number of teeth on all gears is 100. The gear blank and the worm gear (gear 9) are mounted on the same shaft and rotate together.3 Resolve Problem 10. . Finally. Next determine the velocity ratio ( ω3 / ω5 ) to cut 75 teeth on the gear blank. Idler Gear 3 2 Right-Handed Single Tooth Worm 4 6 (35T) 5 7 (180T) To motor 105T Hob (1T) 9 Gear blank Problem S10. Identify the gear set that most closely approximates the desired ratio. What is the error? Problem S10.20 when N2 = 80T.Supplemental Exercise Problems for Chapter 10 Problem S10.105399. Note that this can be done most easily with a computer program.2 A simple gear reduction unit is to be used to generate the gear ratio 2. N4 = 50. N2 = 60. Determine the angular velocity of gear 8. Derive an expression for the angular velocity of gear 8. N4.5 In Problem S10. ω8 ω2 3 2 4 5 8 7 6 Problem 10. N5 = 140. N6. N3 = 30. N7. N5. N6 = 120.4 In the mechanism shown.Problem S10. . let the input be gear 2 and assume that all of the gear tooth numbers (N2. assume that ω2 = 50 rpm . N3. and N8 = 80.4. and N8) are known. N7 = 20.54 - . Consider friction at the slider only and neglect the masses of the links.0 and 0. B ω2 T12 A 2 40˚ 3 AB = 2 in BC = 3.2.Supplemental Exercise Problems for Chapter 11 Problem S11.55 - .1 Find the torque T12 for a coefficients of friction µ of 0.4 in 1.0" 0.45" 4 C 20 lb . Hence find its initial angular acceleration and the initial acceleration of point A. The moment of inertia of a uniform rectangular plate with sides 2a and 2b about an axis normal to its plane passing through its centroid is m(2a+2b)/12. You may also assume that the angular displacement from the initial position is small.1 A uniform rectangular plate is suspended from a rail by means of two bogies as shown.Supplemental Exercise Problems for Chapter 12 Problem S12. You may assume that the rollers which support point A are frictionless and that they remain in contact with the rail. allowing the plate to swing downward. The plate is connected to the bogies by means of frictionless hinge joints at A and at B. where m is the mass of the plate. Write the equations of motion of the plate as it starts to move.56 - . At time t=0 the pin of joint B breaks. A b b a B G a . lump the weight of the connecting rod at the crank pin and piston pin and locate the counterbalancing weight at the crank radius. Determine the optimum counter balancing weight which will give 1.1085 lb-s2-in B 3 W3= 3.4 The four-cylinder V engine shown below has identical cranks.85 lb IG3= 0. and the phase angle is φ2 = 180˚ . The smallest horizontal shaking force 2.Supplemental Exercise Problems for Chapter 13 Problem S13.1 For the mechanism and data given.11. Are the primary or secondary shaking forces balanced? What about the primary and . 1ω 2= 210 rad/s CCW 1α = 2 rad/s2 W2 = 3.16 for the following values for the phase angles and offset distances. The smallest vertical shaking force Problem S13. and pistons. Draw the shaking force vector on the figure. determine the shaking force and its location relative to point A. φ1 = 0 a1 = 0 φ2 = 180˚ a2 = a φ3 = 90˚ a3 = 2a φ4 = 270˚ a4 = 3a Problem S13. connecting rods.3 Resolve Problem 13. The rotary masses are perfectly balanced. The V angle is ψ = 60˚ .57 - .40 lb W4= 2.2 For the engine given in Problem 13. Derive an expression for the shaking forces and shaking moments for the angles and offset values indicated.40 lb G3 45˚ A G2 AB = 3 in BC = 12 in BG 3 = 6 in C 4 G4 Problem S13. φ3 = 180˚. a6 = 5a.58 - . a3 = 2a. and φ7 = 240˚ a1 = 0. φ4 = 180˚. The engine is characterized by the following phase angles: φ1 = 0. and a7 = 6a Determine the type of unbalance which exists in the engine. a5 = 4a.6 Assume that a seven-cylinder in-line engine is being considered for an engine power plant.5 Resolve Problem S13. X y B1 B4 Cylinders 2 and 4 Cylinders 1 and 3 B3 L B2 ψ φ 2 θ A1 φ1 = 0 φ 2 = 180˚ a R A2 Problem S13. φ2 = 90˚. φ5 = 0˚. a4 = 3a. Problem S13. a2 = a. Estimate the optimum value for ψ . . φ6 = 120˚.secondary shaking moments? Suggest a means for balancing any shaking forces or couples which result.4 if the V angle is ψ = 90˚ .