RELATIVE VELOCITY METHOD
EXAMPLE PROBLEM 6.19
A displacement diagram of the piston operating in a compressor was plotted in Example Problem 4.11. This diagram was converted to a displacement curve relative to time in Example Problem 6.17. Use this data to numerically gener-ate a velocity curve.
SOLUTION: 1. Determine the Time Increment between Position Data Points
The spreadsheet from Example Problem 6.17 is expanded by inserting an additional column to include the pis-ton velocity. The time increment is calculated as follows:
2. Use Equation (6.19) to Calculate Velocity Data Points
To illustrate the calculation of the velocities, a few sample calculations are shown:
= c(0.0- 0.483)
2(0.00286) d - c0.136 - 2(0.0) + 2(0.483) - 0.896
2(0.00286) d = -91.47 in.>s v12 = c(¢R13 - ¢R11)
2¢t d - c¢R2 - 2¢R13+ 2¢R11- ¢R10
12¢t d,
= c(0.896 - 1.435)
2(0.00286) d - c0.483 - 2(0.896) + 2(1.435) - 1.50
12(0.00286) d = -95.48 in.>s v9 = c(¢R10- ¢R8)
¢t d - c¢R11- 2¢R10 + 2¢R8 - ¢R7
12¢t d
= c(0.483 - 0.0)
2(0.00286) d - c0.896 - 2(0.483) + 2(0.0) - 0.136
12(0.00286) d = 142.67 in.>s v2 = c(¢R3 - ¢R1)
2¢t d - c ¢R4 - 2¢R3 + 2¢R1 - ¢R12
12¢t d
¢t = t2 - t1 = (0.00289 - 0.0) = 0.00286 s
160 CHAPTER SIX
3. Compute the Velocity Data and Plot the Velocity Curve
The results can be computed and tabulated as shown in Figure 6.45. A spreadsheet was used efficiently to perform these redundant calculations. For those who are unfamiliar with spreadsheets, refer to Chapter 8.
FIGURE 6.45 Velocity data for Example Problem 6.19
FIGURE 6.46 Velocity curve for Example Problem 6.19.
Velocity Analysis 161 These values are plotted in Figure 6.46 to form a velocity diagram relative to time. Notice that this curve is still rather rough. For accuracy purposes, it is highly suggested that the crank angle increment be reduced to 10° or 15°.
When a spreadsheet is used to generate the velocity data, even smaller increments are advisable to reduce the difficulty of the task.
PROBLEMS
General Velocity
6–1. A package is moved at a constant rate from one end of a 25-ft horizontal conveyor to the other end in 15 s.
Determine the linear speed of the conveyor belt.
6–2. A hydraulic cylinder extends at a constant rate of 2 fpm (ft/min). Determine the time required to traverse the entire stroke of 15 in.
6–3. Determine the average speed (in mph) of an athlete who can run a 4-minute mile.
6–4. Determine the average speed (in mph) of an athlete who can run a 100-m dash in 10 s.
6–5. A gear uniformly rotates 270° clockwise in 2 s.
Determine the angular velocity in rpm and rad/s.
6–6. Determine the angular velocity (in rpm) of the second, minute, and hour hand of a clock.
6–7. A servo-driven actuator is programmed to extend according to the velocity profile shown in Figure P6.7.
Determine the total displacement during this programmed move.
6–12. The drive roller for a conveyor belt is shown in Figure P6.11. Determine the linear speed of the belt when the roller operates at 30 rpm counterclockwise.
6–13. Link 2 is isolated from a kinematic diagram and shown in Figure P6.13. The link is rotating coun-terclockwise at a rate of 300 rpm. Determine the velocity of points A and B. Use ° and
.
FIGURE P6.7 Problems 7 and 8.
6–8. A servo-driven actuator is programmed to extend according to the velocity profile shown in Figure P6.7. Use a spreadsheet to generate plots of velocity versus time and displacement versus time during this programmed move.
6–9. A linear motor is programmed to move according to the velocity profile shown in Figure P6.9. Determine the total displacement during this programmed move.
6–10. A linear motor is programmed to move according to the velocity profile shown in Figure P6.9. Use a spreadsheet to generate plots of velocity versus time and displacement versus time during this programmed move.
6–11. The drive roller for a conveyor belt is shown in Figure P6.11. Determine the angular velocity of the roller when the belt operates at 10 fpm (10 ft/min).
v (in./s)
t (s) 2
1
1 2 3 4 5 6
FIGURE P6.9 Problems 9 and 10.
V
16"
ω
FIGURE P6.11 Problems 11 and 12.
A
B
2 18"
γ 8" β
FIGURE P6.13 Problems 13 and 14.
6–14. Link 2 is isolated from a kinematic diagram and shown in Figure P6.13. The link is rotating clock-wise, driving point A at a speed of 40 ft/s. Determine the velocity of points A and B and the angular veloc-ity of link 2. Use g = 50° and b = 60°.
Relative Velocity
6–15. A kinematic diagram of a four-bar mechanism is shown in Figure P6.15. At the instant shown, mm/s and mm/s. Graphically determine the relative velocity of point B with respect to point A. Also determine the angular velocity of links 2 and 4.
vB = 888 vA = 800
Relative Velocity Method—Graphical
6–19. For the compressor linkage shown in Figure P6.19, use the relative velocity method to graphically deter-mine the linear velocity of the piston as the crank rotates clockwise at 1150 rpm.
162 CHAPTER SIX
FIGURE P6.15 Problems 15 and 16.
6–16. A kinematic diagram of a four-bar mechanism is shown in Figure P6.15. At the instant shown, mm/s and mm/s. Graphically determine the relative velocity of point B with respect to point A. Also determine the angular velocity of links 2 and 4.
6–17. A kinematic diagram of a slider-crank mechanism is shown in Figure P6.17. At the instant shown, ft/s and ft/s. Graphically deter-mine the relative velocity of point A with respect to point B. Also, determine the angular velocity of link 2.
vB = 400
FIGURE P6.17 Problems 17 and 18.
6–18. A kinematic diagram of a slider-crank mechanism is shown in Figure P6.17. At the instant shown, ft/s and ft/s. Graphically determine the relative velocity of point A with respect to point B. Also, determine the angular velocity of link 2.
vB = 21
FIGURE P6.19 Problems 19, 20, 41, 52, 63, 74, 85, 96, 104, and 112.
6–20. For the compressor linkage shown in Figure P6.19, use the relative velocity method to graphically deter-mine the linear velocity of the piston as the crank rotates counterclockwise at 1775 rpm.
6–21. For the reciprocating saw shown in Figure P6.21, use the relative velocity method to graphically deter-mine the linear velocity of the blade as the crank wheel rotates counterclockwise at 1500 rpm.
Blade
.50" Crank wheel
.65"
130 3.25"
FIGURE P6.21 Problems 21, 22, 42, 53, 64, 75, 86, 97, 105, and 113.
6–22. For the reciprocating saw shown in Figure P6.21, use the relative velocity method to graphically deter-mine the linear velocity of the blade as the crank wheel rotates clockwise at 900 rpm.
6–23. For the shearing mechanism in the configuration shown in Figure P6.23, use the relative velocity method to graphically determine the linear velocity of the blade as the crank rotates clockwise at 100 rpm.
.75"
FIGURE P6.23 Problems 23, 24, 43, 54, 65, 76, 87, 98, 106, and 114.
6–24. For the shearing mechanism in the configuration shown in Figure P6.23, use the relative velocity method to graphically determine the linear velocity of the blade as the crank rotates counterclockwise at 80 rpm.
6–25. For the rear windshield wiper mechanism shown in Figure P6.25, use the relative velocity method to graphically determine the angular velocity of the wiper arm as the crank rotates counterclockwise at 40 rpm.
shown, use the relative velocity method to graphically determine the angular velocity of the water bath as the crank is driven clockwise at 75 rpm.
6–29. The device in Figure P6.29 is a drive mechanism for the agitator on a washing machine. For the configuration shown, use the relative velocity method to graphically determine the angular velocity of the segment gear as the crank is driven clockwise at 50 rpm.
Velocity Analysis 163
FIGURE P6.25 Problems 25, 26, 44, 55, 66, 77, 88, 99, 107, and 115.
6–26. For the rear windshield wiper mechanism shown in Figure P6.25, use the relative velocity method to graphically determine the angular velocity of the wiper arm as the crank rotates clockwise at 60 rpm.
6–27. The device in Figure P6.27 is a sloshing bath used to wash vegetable produce. For the configuration shown, use the relative velocity method to graph-ically determine the angular velocity of the water bath as the crank is driven counterclockwise at 100 rpm.
FIGURE P6.27 Problems 27, 28, 45, 56, 67, 78, 89, 100, 108, and 116.
6–28. The device in Figure P6.27 is a sloshing bath used to wash vegetable produce. For the configuration
2"
40
4.5"
7"
8"
FIGURE P6.29 Problems 29, 30, 46, 57, 68, 79, 90, 101, 109, and 117.
6–30. The device in Figure P6.29 is a drive mechanism for the agitator on a washing machine. For the configu-ration shown, use the relative velocity method to graphically determine the angular velocity of the segment gear as the crank is driven counter-clockwise at 35 rpm.
6–31. For the hand-operated shear shown in Figure P6.31, use the relative velocity method to graphically deter-mine the angular velocity of the handle required to have the blade cut through the metal at a rate of 3 mm/s. Also determine the linear velocity of point X.
175 mm
FIGURE P6.31 Problems 31, 32, 47, 58, 69, 80, and 91.
6–32. For the hand-operated shear shown in Figure P6.31, use the relative velocity method to graphically deter-mine the linear velocity of the blade as the handle is rotated at a rate of 2 rad/s clockwise. Also determine the linear velocity of point X.
164 CHAPTER SIX
6–33. For the foot-operated air pump shown in Figure P6.33, use the relative velocity method to graphically determine the angular velocity of the foot pedal required to contract the cylinder at a rate of 5 in./s. Also determine the linear velocity of point X.
6–38. A package-moving device is shown in Figure P6.37.
For the configuration illustrated, use the relative velocity method to graphically determine the linear velocity of the package as the crank rotates clock-wise at 65 rpm.
6–39. A package-moving device is shown in Figure P6.39.
For the configuration illustrated, use the relative velocity method to graphically determine the linear velocity of the platform as the hydraulic cylinder extends at a rate of 16 fpm.
7.5"
FIGURE P6.33 Problems 33, 34, 48, 59, 70, 81, and 92.
6–34. For the foot-operated air pump shown in Figure P6.33, use the relative velocity method to graphically determine the rate of cylinder compres-sion when the angular velocity of the foot pedal assembly is 1 rad/s counterclockwise. Also deter-mine the linear velocity of point X.
6–35. A two-cylinder compressor mechanism is shown in Figure P6.35. For the configuration shown, use the relative velocity method to graphically determine the linear velocity of both pistons as the 1.5-in.
crank is driven clockwise at 1775 rpm. Also deter-mine the instantaneous volumetric flow rate out of the right cylinder.
FIGURE P6.35 Problems 35, 36, 49, 60, 71, 82, 93, 102, 110, and 118.
6–36. A two-cylinder compressor mechanism is shown in Figure P6.35. For the configuration shown, use the relative velocity method to graphically determine the linear velocity of both pistons as the 1.5-in.
crank is driven counterclockwise at 1150 rpm. Also determine the instantaneous volumetric flow rate out of the left cylinder.
6–37. A package-moving device is shown in Figure P6.37.
For the configuration illustrated, use the relative velocity method to graphically determine the linear velocity of the package as the crank rotates clock-wise at 40 rpm.
FIGURE P6.37 Problems 37, 38, 50, 61, 72, 83, 94, 103, 111, and 119.
FIGURE P6.39 Problems 39, 40, 51, 62, 73, 84, and 95.
6–40. A package-moving device is shown in Figure P6.39.
For the configuration illustrated, use the relative velocity method to graphically determine the linear velocity of the platform as the hydraulic cylinder retracts at a rate of 12 fpm.
Relative Velocity Method—Analytical
6–41. For the compressor linkage shown in Figure P6.19, use the relative velocity method to determine the linear velocity of the piston as the crank rotates clockwise at 950 rpm.
6–42. For the reciprocating saw shown in Figure P6.21, use the relative velocity method to analytically deter-mine the linear velocity of the blade as the crank wheel rotates counterclockwise at 1700 rpm.
Velocity Analysis 165 6–43. For the shearing mechanism in the configuration
shown in Figure P6.23, use the relative velocity method to analytically determine the linear velocity of the blade as the crank rotates clockwise at 85 rpm.
6–44. For the rear windshield wiper mechanism shown in Figure P6.25, use the relative velocity method to analytically determine the angular velocity of the wiper arm as the crank rotates counterclockwise at 45 rpm.
6–45. The device in Figure P6.27 is a sloshing bath used to wash vegetable produce. For the configuration shown, use the relative velocity method to analyti-cally determine the angular velocity of the water bath as the crank is driven counterclockwise at 90 rpm.
6–46. The device in Figure P6.29 is a drive mechanism for the agitator on a washing machine. For the configura-tion shown, use the relative velocity method to analyt-ically determine the angular velocity of the segment gear as the crank is driven clockwise at 60 rpm.
6–47. For the links for the hand-operated shear shown in Figure P6.31, use the relative velocity method to analytically determine the angular velocity of the handle required to have the blade cut through the metal at a rate of 2 mm/s.
6–48. For the foot-operated air pump shown in Figure P6.33, use the relative velocity method to analyti-cally determine the rate of cylinder compression as the foot pedal assembly rotates counterclockwise at a rate of 1 rad/s.
6–49. A two-cylinder compressor mechanism is shown in Figure P6.35. For the configuration shown, use the relative velocity method to analytically determine the linear velocity of both pistons as the 1.5-in.
crank is driven clockwise at 2000 rpm. Also deter-mine the instantaneous volumetric flow rate out of the right cylinder.
6–50. A package-moving device is shown in Figure P6.37.
For the configuration illustrated, use the relative velocity method to analytically determine the linear velocity of the package as the crank rotates clock-wise at 80 rpm.
6–51. A package-moving device is shown in Figure P6.39.
For the configuration illustrated, use the relative velocity method to analytically determine the linear velocity of the platform as the hydraulic cylinder retracts at a rate of 10 fpm.
Locating Instantaneous Centers—Graphically
6–52. For the compressor linkage shown in Figure P6.19, graphically determine the location of all the instan-taneous centers.
6–53. For the reciprocating saw shown in Figure P6.21, graphically determine the location of all the instan-taneous centers.
6–54. For the shearing mechanism in the configuration shown in Figure P6.23, graphically determine the location of all the instantaneous centers.
6–55. For the rear windshield wiper mechanism shown in Figure P6.25, graphically determine the location of all the instantaneous centers.
6–56. For the produce-washing bath shown in Figure P6.27, graphically determine the location of all the instantaneous centers.
6–57. For the washing machine agitation mechanism shown in Figure P6.29, graphically determine the location of all the instantaneous centers.
6–58. For the hand-operated shear shown in Figure P6.31, graphically determine the location of all the instan-taneous centers.
6–59. For the foot-operated air pump shown in Figure P6.33, graphically determine the location of all the instantaneous centers.
6–60. For the two-cylinder compressor mechanism shown in Figure P6.35, graphically determine the location of all the instantaneous centers.
6–61. For the package-moving device shown in Figure P6.37, graphically determine the location of all the instantaneous centers.
6–62. For the package-moving device shown in Figure P6.39, graphically determine the location of all the instantaneous centers.
Locating Instantaneous Centers—Analytically
6–63. For the compressor linkage shown in Figure P6.19, analytically determine the location of all the instan-taneous centers.
6–64. For the reciprocating saw shown in Figure P6.21, analytically determine the location of all the instan-taneous centers.
6–65. For the shearing mechanism in the configuration shown in Figure P6.23, analytically determine the location of all the instantaneous centers.
6–66. For the rear windshield wiper mechanism shown in Figure P6.25, analytically determine the location of all the instantaneous centers.
6–67. For the produce-washing bath shown in Figure P6.27, analytically determine the location of all the instantaneous centers.
6–68. For the washing machine agitation mechanism shown in Figure P6.29, analytically determine the location of all the instantaneous centers.
6–69. For the hand-operated shear shown in Figure P6.31, analytically determine the location of all the instan-taneous centers.
6–70. For the foot-operated air pump shown in Figure P6.33, analytically determine the location of all the instantaneous centers.
166 CHAPTER SIX
6–71. For the two-cylinder compressor mechanism shown in Figure P6.35, analytically determine the location of all the instantaneous centers.
6–72. For the package-moving device shown in Figure P6.37, analytically determine the location of all the instantaneous centers.
6–73. For the package-moving device shown in Figure P6.39, analytically determine the location of all the instantaneous centers.
Instantaneous Center Method—Graphical
6–74. For the compressor linkage shown in Figure P6.19, use the instantaneous center method to graphically determine the linear velocity of the piston as the crank rotates counterclockwise at 1500 rpm.
6–75. For the reciprocating saw shown in Figure P6.21, use the instantaneous center method to graphically determine the linear velocity of the blade as the crank wheel rotates clockwise at 1200 rpm.
6–76. For the shearing mechanism in the configuration shown in Figure P6.23, use the instantaneous center method to graphically determine the linear velocity of the blade as the crank rotates counterclockwise at 65 rpm.
6–77. For the rear windshield wiper mechanism shown in Figure P6.25, use the instantaneous center method to graphically determine the angular velocity of the wiper arm as the crank rotates clockwise at 55 rpm.
6–78. For the produce-sloshing bath shown in Figure P6.27, use the instantaneous method to graphically determine the angular velocity of the water bath as the crank is driven clockwise at 110 rpm.
6–79. For the washing machine agitator mechanism shown in Figure P6.29, use the instantaneous center method to graphically determine the angular veloc-ity of the segment gear as the crank is driven coun-terclockwise at 70 rpm.
6–80. For the hand-operated shear in the configuration shown in Figure P6.31, use the instantaneous method to graphically determine the angular veloc-ity of the handle required to have the blade cut through the metal at a rate of 4 mm/s.
6–81. For the foot-operated air pump shown in Figure P6.33, use the instantaneous center method to graphically determine the rate of cylinder compres-sion as the foot pedal assembly rotates counter-clockwise at a rate of 0.75 rad/s.
6–82. A two-cylinder compressor mechanism is shown in Figure P6.35. For the configuration shown, use the instantaneous center method to graphically deter-mine the linear velocity of both pistons as the 1.5-in.
crank is driven counterclockwise at 2200 rpm. Also determine the instantaneous volumetric flow rate out of the right cylinder.
6–83. A package-moving device is shown in Figure P6.37.
For the configuration illustrated, use the graphical instantaneous center method to determine the linear velocity of the package as the crank rotates clockwise at 70 rpm.
6–84. A package-moving device is shown in Figure P6.39.
For the configuration illustrated, use the instanta-neous center method to graphically determine the linear velocity of the platform as the hydraulic cylinder extends at a rate of 8 fpm.
Instantaneous Center Method—Analytical
6–85. For the compressor linkage shown in Figure P6.19, use the instantaneous center method to analytically determine the linear velocity of the piston as the crank rotates clockwise at 1100 rpm.
6–86. For the reciprocating saw shown in Figure P6.21, use the instantaneous center method to analytically determine the linear velocity of the blade as the crank wheel rotates counterclockwise at 1375 rpm.
6–87. For the shearing mechanism in the configuration shown in Figure P6.23, use the instantaneous center method to analytically determine the linear velocity of the blade as the crank rotates clockwise at 55 rpm.
6–88. For the rear windshield wiper mechanism shown in Figure P6.25, use the instantaneous center method to analytically determine the angular velocity of the wiper arm as the crank rotates counterclockwise at 35 rpm.
6–89. For the produce-sloshing bath shown in Figure P6.27, use the instantaneous method to analytically determine the angular velocity of the water bath as the crank is driven counterclockwise at 95 rpm.
6–90. For the washing machine agitator mechanism shown in Figure P6.29, use the instantaneous center method to analytically determine the angular velocity of the segment gear as the crank is driven clockwise at 85 rpm.
6–91. For the hand-operated shear in the configuration shown in Figure P6.31, use the instantaneous method to analytically determine the angular velocity of the handle required to have the blade cut through the metal at a rate of 2.5 mm/s.
6–92. For the foot-operated air pump shown in Figure P6.33, use the instantaneous center method to ana-lytically determine the rate of cylinder compression as the foot pedal assembly rotates counterclockwise at a rate of 0.6 rad/s.
6–93. A two-cylinder compressor mechanism is shown in Figure P6.35. For the configuration shown, use the instantaneous center method to analytically deter-mine the linear velocity of both pistons as the 1.5-in.
crank is driven clockwise at 1775 rpm. Also deter-mine the instantaneous volumetric flow rate out of
crank is driven clockwise at 1775 rpm. Also deter-mine the instantaneous volumetric flow rate out of