chapter22

chapter22 - 628 CHAPTER 22 VlBRATlONS PROBLEMS 22—1. When...

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Unformatted text preview: 628 CHAPTER 22 VlBRATlONS PROBLEMS 22—1. When a ZD-lb weight is suspended from a spring, the spring is stretched a distance of 4 in. Determine the natural frequency and the period of vibration for a 10—lb weight attached to the same spring. 22—2. A spring has a stiffness of 600 N/m. If a 4—kg block is attached to the spring. pushed 50 mm above its equilibrium position. and released from rest. determine the equation which describes the block’s motion. Assume that positive displacement is measured downward. 22—3. When a 3kg block is suspended from a spring. the spring is stretched a distance of 60 mm. Determine the natural frequency and the period of vibration for a 0.2-kg block attached to the same spring. *22—4. An 8-kg block is suspended from a spring having a stiffness k = 80 N/m. If the block is given an upward velocity of 0.4 m/s when it is 90 mm above its equilibrium position. determine the equation which describes the motion and the maximum upward displacement of the block measured from the equilibrium position. Assume that positive displacement is measured downward. 22—5. A 2-lb weight is suspended from a spring having a stiffness k = 2 lb/in. If the weight is pushed l in. upward from its equilibrium position and then released from rest, determine the equation which describes the motion. What is the amplitude and the natural frequency of the vibration? 2%. A 6—lb weight is suspended from a spring having a stiffness k = 3lb/in. If the weight is given an upward velocity of 20 ft/s when it is 2 in. above its equilibrium position. determine the equation which describes the motion and the maximum upward displacement of the weight. measured from the equilibrium position. Assume positive displacement is downward. I22—7. A spring is stretched 175 mm by an 8-kg block. If the block is displaced 100 mm downward from its equilibrium position and given a downward velocity of 1.50 m/s, determine the differential equation which describes the motion. Assume that positive displacement is measured downward. Use the Runge—Kutta method to determine the position of the block. measured from its unstretched position. when I = 0.22 3. (See Appendix B.) Use a time increment of A! = 0.02 3. *22—8. If the block in Prob. 22-7 is given an upward velocity of 4 m/s when it is displaced downward a distance of 60 mm from its equilibrium position, determine the equation which describes the motion. What is the amplitude of the motion? Assume that positive displacement is measured downward. 22—9. Determine the frequency of Vibration for the block. The springs are originally compressed A. k k mm mm nun &._.._M Prob. 22—9 22—10. A pendulum has a 0.4-m-long cord and is given a tangential velocity of 0.2 m/s toward the vertical from a position 6 = 0.3 rad. Determine the equation which describes the angular motion. 22—11. A platform. having an unknown mass, is supported by four springs, each having the same stiffness it. When nothing is on the platform, the period of vertical vibration is measured as 2.35 s; whereas if a 3-kg block is supported on the platform, the period of vertical vibration is 5.23 s. Determine the mass of a block placed on the (empty) platform which causes the platform to vibrate vertically with a period of 5.62 5. What is the stiffness k of each of the springs? Prob. 22—1 1 *22—12. If the lower end of the 30-kg slender rod is displaced a small amount and released from rest, determine the natural frequency of vibration. Each spring has a stiffness of k = 500 N/m and is unstretched when the rod is hanging vertically. k Prob. 22—12 22—13. The body of arbitrary shape has a mass m, mass center at G, and a radius of gyration about G of kc. If it is displaced a slight amount 0 from its equilibrium position and released, determine the natural period of vibration. Prob. 22—13 PROBLEMS 62 9 22—14. Determine to the nearest degree the maximum angular displacement of the bob if it is initially displaced 0 = 0.2 rad from the vertical and given a tangential velocity of 0.4 m/s away from the vertical. Prob. 22—14 22—15. The semicircular disk weighs 20 lb. Determine the natural period of vibration if it is displaced a small amount and released. Prob. 22—15 *22—16. The square plate has a mass m and is suspended at its corner by the pin 0. Determine the natural period of vibration if it is displaced a small amount and released. Prob. 22—16 630 CHAPTER 22 VIBRATIONS 22—17. The disk has a weight of 10 lb and rolls without slipping on the horizontal surface as it oscillates about its equilibrium position. If the disk is displaced, by rolling it counterclockwise 0.4 rad, determine the equation which describes its oscillatory motion when it is released. k = 1001b/ft Prob. 22-17 22—18. The pointer on a metronome supports a 0.4-1b slider A, which is positioned at a fixed distance from the pivot O of the pointer. When the pointer is displaced, a torsional spring at O exerts a restoring torque on the pointer having a magnitude M = (1.26) lb'ft, where 6 represents the angle of displacement from the vertical, measured in radians. Determine the natural period of vibration when the pointer is displaced a small amount 8 and released. Neglect the mass of the pointer. \k =1.21b-ft/rad Prob. 22—18 22—19. The block has a mass m and is supported by a rigid bar of negligible mass. If the spring has a stiffness k, determine the natural period of vibration for the block. Prob. 22—19 *22—20. The disk, having a weight of 15 lb, is pinned at its center 0 and supports the block A that has a weight of 3 lb. If the belt which passes over the disk is not allowed to slip at its contacting surface, determine the natural period of vibration of the system. Prob. 22-20 PROBLEMS 6 31 22—21. While standing in an elevator, the man holds a pendulum which consists of an 18-in. cord and a 0.5-lb bob. If the elevator is descending with an acceleration a = 4 ft/sz, determine the natural period of Vibration for small amplitudes of swing. V/ Prob. 22—23 *22—24. The bar has a length l and mass m. It is supported at its ends by rollers of negligible mass. If it is given a small displacement and released, determine the natural frequency of vibration. Prob. 22-21 22—22. The 50-lb spool is attached to two springs. If the spool is displaced a small amount and released, determine the natural period of vibration.The radius of gyration of the spool is k0 = 1.5 ft. The spool rolls without slipping. Prob. 22—24 22—25. The 25-lb weight is fixed to the end of the rod assembly. If both springs are unstretched when the assembly is in the position shown, determine the natural period of vibration for the weight when it is displaced slightly and released. Neglect the size of the block and the mass of the rods. {‘76 in.~>l«i6 in!» k = 21b/in. k = 2 lb/in. Prob. 22—22 22—23. Determine the natural frequency for small oscillations of the 10-lb sphere when the rod is displaced a slight distance and released. Neglect the size of the sphere and the mass of the rod. The spring has an unstretched length of 1 ft. Prob. 22—25 636 CHAPTER 22 VIBRATIONS PROBLEMS, 22-26. Solve Prob. 22—13 using energy methods. 22—27. Solve Prob. 22—15 using energy methods. *22—28. Solve Prob. 22—16 using energy methods. 22—29. Solve Prob. 22—20 using energy methods. 22—30. The uniform rod of mass m is supported by a pin at A and a spring at B. If the end B is given a small downward displacement and released, determine the natural period of vibration. Prob. 22—30 22-31. Determine the differential equation of motion of the 3-kg block when it is displaced slightly and releasedThe surface is smooth and the springs are originally unstretched. Prob. 22-31 >1‘22—32. Determine the natural period of vibration of the 10-lb semicircular disk. Prob. 22—32 22—33. The 7-kg disk is pin-connected at its midpoint. Determine the natural period of Vibration of the disk if the springs have sufficient tension in them to prevent the cord from slipping on the disk as it oscillates. Hint: Assume that the initial stretch in each spring is 60. This term will cancel out after taking the time derivative of the energy equation. k = 600 N/m . k = 600 N/m Prob. 22—33 22—34. The 5-lb sphere is attached to a rod of negligible mass and rests in the horizontal position. Determine the natural frequency of vibration. Neglect the size of the sphere. In 1ft 4 Prob. 22—34 PROBLEMS 6 3 7 22—35. The bar has a mass of 8 kg and is suspended from two 22—38. Determine the natural frequency of Vibration of springs such that when it is in equilibrium, the springs make an the 20-lb disk. Assume the disk does not slip on the angle of 45° with the horizontal as shown. Determine the inclined surface. natural period of Vibration if the bar is pulled down a short distance and released. Each spring has a stiffness of k = 40 N/m. 101b/in.. 1 1" i“ Prob. 22—38 Pmb' 22—35 22—39. If the disk has a mass of 8 kg. determine the natural .9246. Determine the natural period of Vibration Of the frequency of vibrationThe springs are originally unstretched. 3-kg sphere. Neglect the mass of the rod and the size of the sphere. g300 mm ~47 300 mm Prob. 22—36 Prob. 22—39 22—37. The slender rod has a weight of 4lb/ft. If it is *22—40. Determine the differential equation of motion of supported in the horizontal plane by a ball-and—socketjoint the 3-kg spool. Assume that it does not slip at the surface at A and a cable at B, determine the natural frequency of of contact as it oscillates. The radius of gyration of the spool vibration when the end B is given a small horizontal about its center of mass is kG = 125 mm. displacement and then released. k:400N/m ~———-I/I///////////.——~ 1.5 ft Prob. 22—37 fl Prob. 22—40 PROBLEMS 22-41. Use a block-and—spring model like that shown in Fig. 22—1401 but suspended from a vertical position and subjected to a periodic support displacement of 5 = 50 cos wot, determine the equation of motion for the system, and obtain its general solution. Define the displacement y measured from the static equilibrium position of the block when t = 0. 22—42. The 20-lb block is attached to a spring having a stiffness of 201b/ft. A force F = (6 cos 2t) lb, where I is in seconds, is applied to the block. Determine the maximum speed of the block after frictional forces cause the free vibrations to dampen out. ZF=6c0521 0:01.10 Prob. 22—42 22—43. A 4—lb weight is attached to a spring having a stiffness k = 101b/ft. The weight is drawn downward a distance of 4 in. and released from rest. If the support moves with a vertical displacement 6 = (0.5 sin 4:) in, where t is in seconds, determine the equation which describes the position of the weight as a function of time. PROBLEMS 649 *22~44. If the block is subjected to the impressed force F = F0 cos wot, show that the differential equation of motion is j} + (k/m)y = (FD/m) cos (not, where y is measured from the equilibrium position of the block. What is the general solution of this equation? F = F0 cos war Prob. 22—44 22—45. The light elastic rod supports a 4-kg sphere. When an lS-N vertical force is applied to the sphere, the rod deflects 14 mm. If the wall oscillates with harmonic frequency of 2 Hz and has an amplitude of 15 mm, determine the amplitude of vibration for the sphere. Prob. 22—45 650 CHAPTER 22 VIBRATIONS 22—46. Use a block-and-spring model like that shown in Fig. 22-l4a, but suspended from a vertical position and subjected to a periodic support displacement 5 = 60 sin wot, determine the equation of motion for the system, and obtain its general solution. Define the displacement y measured from the static equilibrium position of the block when I = 0. 22—47. A S-kg block is suspended from a spring having a stiffness of 300 N/ m. If the block is acted upon by a vertical force F = (7 sin St) N, where [is in seconds, determine the equation which describes the motion of the block when it is pulled down 100 mm from the equilibrium position and released from rest at t = 0. Assume that positive displacement is downward. .. k = 300N/m F=7sin8t Prob. 22—47 *22—48. The 4-kg circular disk is attached to three springs, each spring having a stiffness k = 180 N/m. If the disk is immersed in a fluid and given a downward velocity of 0.3 m/s at the equilibrium position, determine the equation which describes the motion. Assume that positive displacement is measured downward, and that fluid resistance acting on the disk furnishes a damping force having a magnitude F = (60M) N, wherevis in m/s. Prob. 22—48 22—49. The instrument is centered uniformly on a platform P, which in turn is supported by four springs, each spring having a stiffness k = 130 N/m. If the floor is subjected to a vibration (00 2 7 Hz, having a vertical displacement amplitude 50 = 0.17 ft, determine the vertical displacement amplitude of the platform and instrument. The instrument and the platform have a total weight of 18 lb. Prob. 22—49 22—50. The 450-kg trailer is pulled with a constant speed over the surface of a bumpy road, which may be approximated by a cosine curve having an amplitude of 50 mm and wave length of 4 m. If the two springs s which support the trailer each have a stiffness of 800 N/ m, determine the speed 11 which will cause the greatest vibration (resonance) of the trailer. Neglect the weight of the wheels. L m~i~2m~—l Prob. 22—50 22—51. Determine the amplitude of vibration of the trailer in Prob. 22—50 if the speed 1) = 15 km/h. F2m—+~—2m—~i Prob. 22—51 *22—52. The electric motor turns an eccentric flywheel which is equivalent to an unbalanced 0.25—lb weight located 10 in. from the axis of rotation. If the static deflection of the beam is 1 in. because of the weight of the motor, determine the angular velocity of the flywheel at which resonance will occur‘ The motor weights 150 lb. Neglect the mass of the beam. PROBLEMS 6 5 1 22—53. What will be the amplitude of steady-state vibration of the motor in Prob. 22—52 if the angular velocity of the flywheel is 20 rad/s? 22—54. Determine the angular velocity of the flywheel in Prob. 22—52 which will produce an amplitude of vibration of 0.25 in. Probs. 22—53/54 22—55. The engine is mounted on a foundation block which is spring-supported. Describe the steady-state vibration of the system if the block and engine have a total weight of 1500 lb and the engine, when running, creates an impressed force F = (SOsin21)1b, where t is in seconds. Assume that the system vibrates only in the vertical direction, with the positive displacement measured downward, and that the total stiffness of the springs can be represented as k = 2000 lb/ft. Prob. 22—52 Prob. 22—55 652 CHAPTER 22 VIBRATIONS *22—56. Determine the rotational speed (0 of the engine in Prob. 22—55 which will cause resonance. Prob. 22-56 22—57. The block, having a weight of 12 lb, is immersed in a liquid such that the damping force acting on the block has a magnitude of F = (0.7lvl) lb, where v is in ft/s. If the block is pulled down 0.62. ft and released from rest, determine the position of the block as a function of time. The spring has a stiffness of k = 53lb/ft. Assume that positive displacement is downward. Prob. 22—57 22-58. A 7-lb block is suspended from a spring having a stiffness of k = 75 lb/ ft. The support to which the spring is attached is given simple harmonic motion which may be expressed as 6 = (0.15 sin 2:) ft. where tis in seconds. If the damping factor is c/cc = 0.8, determine the phase angle d) of forced vibration. 22—59. Determine the magnification factor of the block, spring. and dashpot combination in Prob. 22-58. *22—60. The bar has a weight of 6 1b. If the stiffness of the spring is k = 8 ib/ft and the dashpot has a damping coefficient (2 = 60 lb - s/ft. determine the differential equation which describes the motion in terms of the angle 6 of the bar’s rotation. Also, what should be the damping coefficient of the dashpot if the bar is to be critically damped? l~ ~——2 ft + 3ft i Prob. 22—60 22-61. A block having a mass of 7 kg is suspended from a spring that has a stiffness k = 600 N/m. If the block is given an upward velocity of 0.6 m/s from its equilibrium position at t = 0, determine its position as a function of time. Assume that positive displacement of the block is downward and that motion takes place in a medium which furnishes a damping force F = (50M) N, where v is in m/s. 22—62. The damping factor, c/c(., may be determined experimentally by measuring the successive amplitudes of vibrating motion of a system. If two of these maximum displacements can be approximated by x, and x2, as shown in Fig. 22—17, show that the ratio In xl/xz = 2w(c/c(.)/ V1 ~ (c/c(,)2.The quantity 1n xl/xg is caiied the logarithmic decrement. 22—63. Determine the differential equation of motion for the damped vibratory system shown.What type of motion occurs? (=200N-s/m *22—64. The 20-kg block is subjected to the action of the harmonic force F = (90 cos 6t) N, where I is in seconds, Write the equation which describes the steady-state motion. PROBLEMS 653 22—66. The 10-kg block-spring-damper system is continually damped. If the block is displaced to x = 50 mm and released from rest, determine the time required for it to return to the position x = 2 mm. Prob. 22—64 22—65. Draw the electrical circuit that is equivalent to the mechanical system shown. What is the differential equation which describes the charge q in the circuit? Prob. 22—65 Prob. 22—66 ...
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This note was uploaded on 04/16/2008 for the course CE 325 taught by Professor Docwong during the Spring '08 term at USC.

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chapter22 - 628 CHAPTER 22 VlBRATlONS PROBLEMS 22—1. When...

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