SUPPLEMENTL EXERISE PROLEMS for Textbook 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
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Supplemental Exercise Problems for Kinematics, Dynamics, and Design of Machinery by K. J. Waldron and G. L. Kinzel Supplemental Exercise Problems for hapter 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. 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 -
Problem S1. Determine the mobility and the number of idle degrees of freedom of the linkages shown below. Show the equations used to determine your answers. (a) (b) Problem S1. Determine the mobility and the number of idle degrees of freedom associated with the mechanism. Show the equations used to determine your answers. Problem S1.5 If the link lengths of a four-bar linkage are L 1 = 1 mm, L = mm, L = mm, and L = 5 mm and link 1 is fixed, what type of four-bar linkage is it? lso, is the linkage a Grashof type 1 or linkage? nswer the same questions if L 1 = mm. Problem S1.6 You are given two sets of links. Select four links from each set such that the coupler can rotate fully with respect to the others. Sketch the linkage and identify the type of four-bar mechanism. a) L 1 = 5, L = 8, L = 15, L = 19, and L 5 = 8 b) L 1 = 5, L =, L =, L =.5, and L 5 =.5 - -
Supplemental Exercise Problems for hapter Problem S.1 Draw the velocity polygon to determine the following: a) Sliding velocity of link 6 b) ngular velocities of links and 5 = 1.5" =." D = 7.9" E = 8.0" DF = 1.5" = ω rad s 50.0" 6 F 10." 5 D E 9.5" - -
Problem S. In the mechanism shown, 1 v = 15 m/s. Draw the velocity polygon, and determine the velocity of point D on link 6 and the angular velocity of link 5 (show all calculations). Y =." D =.7" = 1." D 6.05" 5 1v = 15 m/s 5 X." - -
Problem S. In the mechanism shown below, points E and have the same vertical coordinate. Find the velocities of points,, and D of the double-slider mechanism shown in the figure if rank rotates at rad/s W. D 6 0.75" 5 E = 0.55" =.5" = 1.0" = 1.75" D =.05" E 1ω 60-5 -
Problem S. In the figure below, points and have the same horizontal coordinate, and 1 ω = 0 rad/s. Draw and dimension the velocity polygon. Identify the sliding velocity between the block and the slide, and find the angular velocity of link. 1ω, 5 = 1 in = in r =.8 in - 6 -
Problem S.5 The following are given for the mechanism shown in the figure: ω = 65. rad/ s (W); α = 0rad/ s (W) 1 1 Draw the velocity polygon, and locate the velocity of Point E using the image technique. Y D (.", 1.1") E 55 X = DE = 1.0 in =.0 in D = 1.5 in - 7 -
Problem S.6 In the mechanism shown, link moves to the left with a velocity of 8 in/s and the acceleration is 80 in/s to the left. Draw the velocity and acceleration polygons and solve for the angular velocity and acceleration of link..0" 10 50-8 -
Problem S.7 In the mechanism shown below, link is turning W at the rate of 10 rad/s (constant). Draw the velocity and acceleration polygons for the mechanism, and record values for 1 a G and 1 α. G = 1.0" =.0" G = 1.0" D =.0" D =.0" 90 ω, α D - 9 -
Problem S.8 For the data given in the figure below, find the velocity and acceleration of points and. ssume 1 v = 0 ft/s, 1 a = 00 ft/s, 1 ω = rad/s (W), and 1 α = 160 rads (W). v α =.05" =.5" =.0" 90 15 o ω a - 10 -
Problem S.9 In the mechanism shown, find 1 ω 6 and 1 α. lso, determine the acceleration of D by image. v = 10 in/s (constant) E 5 6 Y D 81 X D = 1.0" D = 1.05" = 1.5" ED = 1.5" FE = 1." =.0" F (-1.0", -0.75") - 11 -
Problem S.10 In the mechanism shown, 1 ω = 1 rad/s (W) and 1 α = 0 rad/s. Find 1 ω 5, 1 α 5, 1 v E 6, 1 a E 6 for the position given. lso find the point in link 5 that has zero acceleration for the position given. 6 D = 1m = 0.5m = 0.8m D = 0.8m E = 0.67m E 5 0.5 m 0 D - 1 -
Problem S.11 Part of an eight-link mechanism is shown in the figure. Links 7 and 8 are drawn to scale, and the velocity and acceleration of point D 7 are given. Find 1 ω 7 and 1 α 7 for the position given. lso find the velocity of G 7 by image. Y 7 E 6 D 1.5" G 8 X 1.6" DE = 1.5" DG = 0.7" GE = 1.65" v D7 a D7 = 5.0 0 in/sec = 0 60 in/s - 1 -
Problem S.1 In the figure shown below, points,, and are collinear. If 1 v downward, find 1 v, and 1 a. = 10 in/s (constant) am ontact 1" 5 Rolling ontact 0.5" 0.5" - 1 -
Problem S.1 Part of an eight-link mechanism is shown in the figure. There is rolling contact at location. Links 7 and 8 are drawn to scale, and the velocity and acceleration of points 6 and 5 are as shown. Find 1 ω 8 and 1 α 7 for the position given. lso find the velocity of E 7 by image. E D =.5" D = 1.0" =.85" 7 D 7 8 v5 = 10 0 in/s v6 = 10 60 in/s a 5 = 0 70 in/s a 6 = 0 180 in/s 6 5-15 -
Problem S.1 In the position shown, find the velocity and acceleration of link using (1) equivalent linkages and () direct approach. ΑΒ = 0.5" = 1.1" 1ω 5 1 ω = 100 rad/ s W 1 α = 0,000 rad /s W - 16 -
Problem S.15 For the mechanism shown, find 1 ω, 1 α, 1 a, and the location of the center of curvature of the path that point traces on link. 117 1 ω, 1 α 1" = 9 in 1ω = 50 rad/s = in 1 α = 0 D - 17 -
Problem S.16 For the mechanism shown, points, and D are collinear. Point moves in a curved slot on link. For the position given, find 1 ω, 1 α, 1 v,1 a, 1 v D,1 a D, and the location of the center of curvature of the path that point traces on Link. = = 5m D= 7m E= 5. 7m ω = rad/ s W α = rad/ s W 1 1 D m E 60-18 -
Problem S.17 If the mechanism shown is drawn full scale, find 1 ω, 1 α, and the location of the center of curvature of the path that point traces on Link. ssume that Link is driven at constant velocity. 1.5" = 1" =." 60 1ω = 00 rad (constant) sec - 19 -
Problem S.18 In the mechanism below, the angular velocity of Link is rad/s W and the angular acceleration is 5 rad/s W. Determine the following: v, vd, ω, a, ad, α, and the center of curvature of the path that traces on Link. D = 5. in = 6.0 in D = 10.0 in 1α 1ω 75 Problem S.19 Resolve Problem S.18 if 1 ω = rad/sec (constant) - 0 -
Problem S.0 If 1 ω = 0 rad/s (constant), find 1 ω, 1 α, and the center of curvature of the path that traces on Link. D = 0.6" D =.0" R = 1.5" =." enter of curvature of slot Y R Pin in Slot D 10 X 1 ω - 1 -
Problem S.1 In the mechanism below, 1 ω = 10 rad/s. omplete the velocity polygon, and determine the following: 1 v D, 1 ω, 1 v F 6, 1 ω 6. Y G (-5.6,.65) D E 75 6 5 F = 1.0 in D =. in E =.0 in DE =.0 in GF =.5 in =. in 100 X - -
Problem S. If 1 ω = 10 rad/s (constant), find 1 α. = 10.5 cm = 1.0 cm 105 1ω - -
Problem S. If 1 ω = 10 rad/s W (constant), find a) 1 ω b) The center of curvature of the path that traces on link (show on drawing). c) The center of curvature of the path that traces on link (show on drawing). = " = " 0 - -
Problem S. Determine the velocity and acceleration of point on link. 10 in Y 1ω = 1 rad/s 1 α = 0 0 X - 5 -
Problem S.5 Given v = 1.0 ft/s to the left, use the instant-center method to find 1 ω and then use 1 ω to find (direction and magnitude). v 6 Y D 0.5" 157.5 X DE = 1.9" D = 1.5" = 1.1" D =.5" =." 5 6 1.0" E - 6 -
Problem S.6 Find all of the instant centers of velocity for the mechanism shown below. = D D =.06" 50 150 D - 7 -
Problem S.7 If the velocity of point on link is 10 in/s as shown, use the instant center method to find the velocity of point on link 5. DE =.5" D = 0.75" = 1.75" E = 1.5" GF = 1.5" v D Y 75 0.5" G5 0.5" 5 5 E 6 F (.15", 1.9") X - 8 -
Problem S.8 ssume that link 7 rolls on link without slipping, and find the following instant centers: I 1, I 15, and I 7. For the given value for 1 ω, find 1 ω 7 using instant centers. = 1.8" G = 0.85" GF = 1.7" D =.9" DE =.5" E =.0" G Y 1.0" 7 0.75" 5 D 6 F (-.0", 0.95") 90 E X 1ω = rad/s - 9 -
Problem S.9 If 1 v = 10 cm/s as shown, find 1 v using the instant-center method. 5 D = 0.95" DF =.5" = 1.5" F = 1.8" E = 0.85" EG =." E = 1." G = 1.5" G 6 1v D 70 5 E F 1.9" - 0 -
Problem S.0 If 1 ω = 10 rad/s W, find the velocity of point using the instant-center method. Show the velocity vector 1 v on the figure. E 110 = 1.5" DE =.5" D =.0" = 1.6" Α 18 Β 1ω 5 D - 1 -
Problem S.1 If 1 ω = 100 rad/s W, find the velocity of point E using the instant center method. Show the velocity vector 1 v E on the figure. D 1ω 70 E = 1.0" = 1.75" D =.0" DE = 0.8" D =.0" - -
Problem S. In the linkage shown below, locate all of the instant centers.." = 1.5" D =.9" DE = 0.9" = 0.9" F =.0" 5 F 6 55 E D - -
Problem S. If 1 ω = 5 rad/s W, find 1 ω 6 using instant centers. E 5 D F 6 = 1.0" =.0" D =.0" DE = 1.5" EF = 1.5" F =.1" 50 1ω - -
Problem S. If 1 ω = 100 rad/s W, find 1 v using instant centers and the rotating radius method. D = 1.8" D = 0.75" E = 0.7" F = 0.5" FG = 1.75" = 1.0" D = 1.65" G 15 E F D 95-5 -
Problem S.5 If 1 v = 10 in/s as shown, find the angular velocity (1 ω 6 ) of link 6 using the instant-center method. = 1.0" D =.0" = 0.95" E =.0" EF = 1.5" F =.85" 1v 7 5 D E F 6-6 -
Problem S.6 If 1 ω = 50 rad/s W, find the velocity of point G using the instant center method. Show the velocity vector 1 v G on the figure. 5 75 E 5 = 1.16" = 0.70" D = 1.5" DE = 1.16" D = 1.0" DF = 1.0 EG =.0" 1 ω D 6 F G - 7 -
Problem S.7 If 1 ω = 100 rad/s W, find 1 ω 6. E (.0", 6.0") 6 = 1." = 6.0" D =.0" D =.0" F =.0" 1ω Y 5 F D X - 8 -
Supplemental Exercise Problems for hapter Problem.1 For the mechanism shown, write the loop equations for position, velocity, and acceleration. Show all angles and vectors used on the drawing and identify all variables. For the position defined by the data below, find 1 v and 1 a if link rotates with constant angular velocity. = 1cm, = cm, θ = 90, θ = 1 rad/ s θ Problem. In the mechanism shown, φ is 10 rad/s W. Use the loop equation approach to determine the velocity of point for the position defined by φ = 60. 10 inches φ Problem. In the mechanism given, 1 v is 10 in/s to the right (constant). Use the loop equation approach to determine 1 v and 1 a. Use point O as the origin of your coordinate system. O = 10 inches 0 1v - 9 -
Problem. In the mechanism shown, Link slides on link, and link is pinned to link. Link also slides on the frame (link 1). If 1 ω = 10 rad/s W, use the loop-equation approach to determine the velocity of link for the position defined by φ = 5. 10 inches φ Problem.5 Use loop equations to determine the velocity and acceleration of point on link. y in 1ω =1 rad sec 1α = 0 0 x Problem.6 Use loop equations to determine the angular velocity and acceleration of link if v is a constant 10 in/s to the left and 1 θ is 0. y in v 1θ x - 0 -
Problem.7 The shock absorber mechanism on a mountain bicycle is a four-bar linkage as shown. The frame of the bike is link 1, the fork and tire assembly is link, and the connecting linkage are links and. s the bicycle goes over a bump in the position shown, the angular velocity of link relative to the frame is 1 ω is 05 (rad/s), and the angular acceleration is 1 α is 60 (rad/s ), both in the clockwise direction. ompute the angular velocity and angular acceleration of link for the position shown 01 1ω 61 1.55" 1.6" 1.88"." 1 D Problem.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. This is done by the use of an air cylinder, which helps to support the hatch while it closes and holds it up in the open position. The mechanism is an inverted slider-crank linkage, where the car acts as the base link. There are two identical linkages one on each side of the car. ssume that the angular velocity of the driver (link ) is a constant 0.8 rad/s clockwise and the angular acceleration is 0. ompute the linear velocity of point D, and the angular velocity and angular acceleration of link for the position shown - 1 -
D.0" 1.60" 0.91" 1ω 5.7 1.50" 15 Problem.9 The wedge mechanism has three links and three sliding joints. Link slides on link and link 1. Link slides on link and link 1. ompute the degrees of freedom of the mechanism. Determine the relationship among β, δ, and γ which permits the mechanism to move. ssume that link is the input link so that r, r 1, ṙ 1, and r 1, are known. Derive expressions for r, ṙ, r in terms of the variables given. γ δ r β r 1 x - -
Supplemental Exercise Problems for hapter Problem S.1 Design a four-bar linkage to move a coupler containing the line through the three positions shown. The moving pivot (circle point) of one crank is at and the fixed pivot (center point) of the other crank is at *. Draw the linkage in position 1, and use Grashof s equation to identify the type of four-bar linkage designed. Position 1 1 is horizontal, and positions and are vertical. = 6 cm. Y (-, ) 1 (, ) 1 (6,.) * (0, 0) X Problem S. Design a four-bar linkage to move its coupler through the three positions shown below using points and as moving pivots. = cm. What is the Grashof type of the linkage generated? Y 60 (,.) 50 (, 0.85) 1 (0, 0) 1 X - -
Problem S. Synthesize a four-bar mechanism in position that moves its coupler through the three positions shown below if points * and D* are center points. Position 1 1 and position are horizontal. = cm. Y D* (.0,.6) * (., 1.6) X 5 (.7, - 0.7) Problem S. 1 1 (0.7, - 1.8) Synthesize a four-bar mechanism in position that moves its coupler through the three positions shown below. Point is a circle point, and point * is a center point. Position 1 1 and position are horizontal. = cm. Y * (.,.5) (.7,.5) 5 (.0, 1.0) X 1 1 - -
Problem S.5 Design a four-bar linkage to move the coupler containing line segment through the three positions shown. The moving pivot for one crank is to be at, and the fixed pivot for the other crank is to be at *. Draw the linkage in position 1 and determine the classification of the resulting linkage (e.g., crank rocker, double crank). Positions 1 1 and are horizontal, and position is vertical. =.0 in. Y (0.0, 1.5) 1 1 (-1.0, 0.5) * X (-0.5, -.0) Problem S.6 Design a four-bar linkage to move a coupler containing the line through the three positions shown. The moving pivot (circle point) of one crank is at and the fixed pivot (center point) of the other crank is at *. Draw the linkage in position and use Grashof s equation to identify the type of four-bar linkage designed. Position 1 1 is horizontal, and positions and are vertical. = in. Y (0, ) 1 1 1 * (0, 0) (, 0) (, 0) X - 5 -
Problem S.7 Design a four-bar linkage to generate the function y=e x -x for values of x between 0 and 1. Use hebyshev spacing with three position points. The base length of the linkage must be 0.8 cm. Use the following angle information: θ 0 = 65 φ 0 = 0 θ = 60 φ = 0 Problem S.8 Use Freudenstein's equation to design a four-bar linkage to generate approximately the function y=1/x, 1 x. Use three precision points with hebyshev spacing. θ is to be proportional to x and φ is to be proportional to y. Let θ 1 = 5, φ 1 = 15, θ = 90, φ = -90, and the base length be in. ccurately sketch the linkage at the three precision point positions. Problem S.9 device characterized by the input-output relationship φ = x1+ xsin θ is to be used to generate (approximately) the function φ = θ (θ and φ both in radians) over the range 0 θ π /. Use hebyshev spacing and determine the values of x 1 and x which will allow the device to best approximate the function. Problem S.10 mechanical device characterized by the input-output relationship φ = a1 8. + atan θ + a is to be used to generate (approximately) the function y= x over the range 0 x π / where x, y, φ, and θ are all in radians. ssume that the device will be used over the ranges 0 θ π / and 0 φ π /. Exterior constraints on the design require that the parameter a 1 = 1. Determine: a) The number of precision points required to complete the design of the system. b) Use hebyshev spacing, and determine the values for the unknown design variables which will allow the device to approximate the function. c) ompute the error generated by the device for x = π/6. Problem S.11. (15 points) mechanical device characterized by the following input-output relationship: φ= 6. p.* π + p cosθ+ p 1 7-6 -
is to be used to generate (approximately) the function y= x over the range 0 x π where x, y, φ, and θ are all in radians. ssume that the device will be used over the ranges 0 θ π and 0 φ π. The problem is such that you may pick any value for any one of the design variables and solve for the other two; i.e., pick a value for one of p 1, p, p and solve for the other two. Determine: 1) The number of precision points required to complete the design of the system. ) Use hebyshev spacing, and determine the values for the unknown design variables which will allow the device to approximate the function. ) ompute the error generated by the device for x = π/6. Problem S.1 Design a crank-rocker mechanism such that with the crank turning at constant speed, the oscillating lever will have a time ratio of advance to return of :. The lever is to oscillate through an angle of 80 o, and the length of the base link is to be in. Problem S.1 Design a crank-rocker mechanism which has a base length of 1.0, a time ratio of 1.5, and a rocker oscillation angle of 0. The oscillation is to be symmetric about a vertical line through O. Specify the length of each of the links. - 7 -
Problem S.1 Determine the two -bar linkages cognate for the drag-link mechanism shown. The dimensions are MQ = 1 m, M = Q = m, = m, and angles and both equal 5. Notice that the cognates will also be drag-link mechanisms. Draw the cognates in the position for θ = 180. θ Q 5 M - 8 -
Supplemental Exercise Problems for hapter 5 No supplemental problems were developed for hapter 5. Problem S6.1 Supplemental Exercise Problems for hapter 6 ssume that s is the cam-follower displacement and θ is the cam rotation. The rise is 1.0 cm after β degrees of rotation, and the rise begins at a dwell and ends with a constant-velocity segment. The displacement equation for the follower during the rise period is n i s= h i θ β i= 0 a) If the position, velocity, and acceleration are continuous at θ = 0, and the position and velocity are continuous at θ = β, determine the n required in the equation, and find the coefficients i which will satisfy the requirements. b) If the follower is a radial flat-faced follower and the base circle radius for the cam is 1 cm, determine whether the cam will have a cusp corresponding to θ = 0. Show all equations used. 1.0 cm S Slope = 1/β Dwell β θ Problem S6. ssume that s is the cam-follower displacement and θ is the cam rotation. The rise is h after β radians of rotation, and the rise begins and ends at a dwell. The displacement equation for the follower during the rise period is n i s= h i θ β i= 0 t, dynamic effects are important, so displacement, velocity, and acceleration must be continuous there. t only displacement and velocity are important. To satisfy these conditions, determine the number of terms required and the corresponding values for the coefficients i if the cam velocity is constant. If h is 1 cm and β is 90, determine the minimum base circle radius to avoid cusps if the cam is used with a flat-faced follower. - 9 -
h S Β Dwell Dwell Α β θ, rad Problem S6. Resolve Problem 6.1 if the cam velocity is rad/s constant. Note: Use the minimum number of i possible. If h is cm and β is 60, determine the minimum base circle radius to avoid cusps if the cam is used with a flat-faced follower. Problem S6. For the cam displacement schedule given, h is the rise, β is the angle through which the rise takes place, and s is the displacement at any given angle θ. The displacement equation for the follower during the rise period is 5 i s= h a i θ β i= 0 Determine the required values for a 0... a 5 such that the displacement, velocity, and acceleration functions are continuous at the end points of the rise portion. If h is 1 cm and β = π /, determine the minimum base circle radius to avoid cusps if the cam is used with a flat-faced follower. h S Β Dwell Dwell Α β θ, rad Problem S6.5 Resolve Problem 6.1 if the cam will be used for a low-speed application so that acceleration effects can be neglected. Determine the number of terms required and the corresponding values for the constants ( i ) if the displacement and velocity are continuous at the end points of the rise (points and ) and if the cam velocity is constant. If h is 1 cm and β is 90, determine the minimum base circle radius to avoid cusps if the cam is used with a flat-faced follower. - 50 -
Problem S6.6 Lay out a cam profile using a harmonic follower displacement profile for both the rise and return. The cam is to have a translating, flat-faced follower which is offset in the clockwise direction by 0. in. ssume 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 0 to 0. The cam will rotate clockwise, and the base circle radius is to be in. Lay out the cam profile using 0 plotting intervals. - 51 -
Supplemental Exercise Problems for hapter 7 Problem S7.1 In the manipulator shown, the joint axes at and are oriented along the z and m axes, respectively. In the position to be analyzed, link lies in a plane parallel to the XY plane and points along a line parallel to the Y axis, and link is perpendicular to link. For the position to be analyzed, link is vertical (parallel to Z). The joint between link and the frame at is a cylindrical joint, and that at is a revolute joint. Determine 1 v /1, 1 a /1, and 1 α. Z X s θ φ m Y θ = 90, θ = 1rad / s, θ = 0rad / s, φ = 1rad / s, φ = 0rad / s, s = 0in/ s, s= 1in/ s, = 1in, = 1in Supplemental Exercise Problems for hapter 8 No supplemental problems were developed for hapter 8. Supplemental Exercise Problems for hapter 9 No supplemental problems were developed for hapter 9. - 5 -
Supplemental Exercise Problems for hapter 10 Problem S10.1 n alternative gear train is shown as a candidate for the spindle drive of a gear hobbing machine. The gear blank and the worm gear (gear 9) are mounted on the same shaft and rotate together. If the gear blank is to be driven clockwise, determine the hand of the hob. Next determine the velocity ratio (ω / ω5) to cut 75 teeth on the gear blank. Finally, select gears and 5 which will satisfy the ratio. Gears are available which have all of the tooth numbers from 15 to 0. Idler Gear Right-Handed Single Tooth Worm 5 6 (5T) 9 Gear blank 7 (180T) To motor 105T Hob (1T) Problem S10. simple gear reduction unit is to be used to generate the gear ratio.10599. Make up a table of possible gear ratios where the maximum number of teeth on all gears is 100. Identify the gear set that most closely approximates the desired ratio. Note that this can be done most easily with a computer program. What is the error? Problem S10. Resolve Problem 10.0 when N = 80T, N = 0T, N = 0T and N 5 = 10. - 5 -
Problem S10. In the mechanism shown, let the input be gear and assume that all of the gear tooth numbers (N, N, N, N 5, N 6, N 7, and N 8 ) are known. Derive an expression for the angular velocity of gear 8. ω 8 ω 8 7 6 5 Problem 10.5 In Problem S10., assume that ω 50 = rpm, N = 60, N = 0, N = 50, N 5 = 10, N 6 = 10, N 7 = 0, and N 8 = 80. Determine the angular velocity of gear 8. - 5 -
Supplemental Exercise Problems for hapter 11 Problem S11.1 Find the torque T 1 for a coefficients of friction µ of 0.0 and 0.. onsider friction at the slider only and neglect the masses of the links. T 1 ω 0 = in =. in 0.5" 1.0" 0 lb - 55 -
Supplemental Exercise Problems for hapter 1 Problem S1.1 uniform rectangular plate is suspended from a rail by means of two bogies as shown. The plate is connected to the bogies by means of frictionless hinge joints at and at. t time t=0 the pin of joint breaks, allowing the plate to swing downward. Write the equations of motion of the plate as it starts to move. Hence find its initial angular acceleration and the initial acceleration of point. You may assume that the rollers which support point are frictionless and that they remain in contact with the rail. You may also assume that the angular displacement from the initial position is small. The moment of inertia of a uniform rectangular plate with sides a and b about an axis normal to its plane passing through its centroid is m(a+b)/1, where m is the mass of the plate. b b G a a - 56 -
Supplemental Exercise Problems for hapter 1 Problem S1.1 For the mechanism and data given, determine the shaking force and its location relative to point. Draw the shaking force vector on the figure. 1ω = 10 rad/s W 1α = rad/s W =.0 lb I G = 0.1085 lb-s -in W =.0 lb W =.85 lb G 5 = in = 1 in G = 6 in G G Problem S1. For the engine given in Problem 1.11, 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. The smallest horizontal shaking force. The smallest vertical shaking force Problem S1. Resolve Problem 1.16 for the following values for the phase angles and offset distances. φ1= 0 φ = 180 φ= 90 φ = 70 a = 0 a = a a = a a = a 1 Problem S1. The four-cylinder V engine shown below has identical cranks, connecting rods, and pistons. The rotary masses are perfectly balanced. Derive an expression for the shaking forces and shaking moments for the angles and offset values indicated. The V angle is ψ = 60, and the phase angle is φ = 180. re the primary or secondary shaking forces balanced? What about the primary and - 57 -
ylinders and ylinders 1 and secondary shaking moments? Suggest a means for balancing any shaking forces or couples which result. y X 1 L ψ φ 1 θ φ 1 = 0 φ = 180 R a Problem S1.5 Resolve Problem S1. if the V angle is ψ = 90. Estimate the optimum value for ψ. Problem S1.6 ssume that a seven-cylinder in-line engine is being considered for an engine power plant. The engine is characterized by the following phase angles: φ = 0, φ = 90, φ = 180, φ = 180, φ = 0, φ = 10, and φ = 0 1 5 6 7 a= 0, a = aa, = aa, = aa, = aa, = 5a, and a = 6a 1 5 6 7 Determine the type of unbalance which exists in the engine. - 58 -