22-1. A spring has a stiffness of 600 N/m. If a 4-kg block

22-1. 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.

22-2. When a 2-kg block is suspended from a spring, the spring is stretched a distance of 40 mm. Determine the frequency and the period of vibration for a 0.5-kg block attached to the same spring.

22-3. A spring is stretched 200 mm by a 15-kg block. If the block is displaced 100 mm downward from its equilibrium position and given a downward velocity of 0.75 m/s, determine the equation which describes the motion. What is the phase angle? Assume that positive displacement is downward.

*22-4. When a 20-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.

22-5. When a 3-kg 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.

22-6. 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.

22-7. A 2-lb weight is suspended from a spring having a stiffness k = 2 Ib/in. If the weight is pushed 1 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?

*22-8. A 6-lb weight is suspended from a spring having a stiffness k = 3 Ib/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.

22-9. A 3-kg block is suspended from a spring having a stiffness of k = 200 N/m. If the block is pushed 50 mm upward from its equilibrium position and then released from rest, determine the equation that describes the motion. What are the amplitude and the natural frequency of the vibration? Assume that positive displacement is downward.

22-10. Determine the frequency of vibration for the block.The springs are originally compressed A.

22-11. The semicircular disk weighs 20 lb. Determine the natural period of vibration if it is displaced a small amount and released.

*22-12. The uniform beam is supported at its ends by two springs A and B. each having the same stiffness A'.When nothing is supported on the beam, it has a period of vertical vibration of 0.83 s. If a 50-kg mass is placed at its center, the period of vertical vibration is 1.52 s. Compute the stiffness of each spring and the mass of the beam.

22-13. The body of arbitrary shape has a mass m. mass center at G. and a radius of gyration about G of kG. If it is displaced a slight amount 0 from its equilibrium position and released, determine the natural period of vibration.

The connecting rod is supported by a knife edge at A and the period of vibration is measured as \(\tau_{A}=3.38 \mathrm{\ s}\). It is then removed and rotated \(180^{\circ}\) so that it is supported by the knife edge at B. In this case the period of vibration is measured as \(\tau_{B}=3.96 \mathrm{\ s}\). Determine the location d of the center of gravity G. and compute the radius of gyration kG.

22-15. The thin hoop of mass m is supported by a knife-edge. Determine the natural period of vibration for small amplitudes of swing.

*22-16. A block of mass m is suspended from two springs having a stiffness of k\ and k2. arranged a) parallel to each other, and b) as a series. Determine the equivalent stiffness of a single spring with the same oscillation characteristics and the period of oscillation for each case.

22-17. The 15-kg block is suspended from two springs having a different stiffness and arranged a) parallel to each other, and b) as a series. If the natural periods of oscillation of the parallel system and series system are observed to be 0.5 s and 1.5 s, respectively,determine the spring stiffnesses k{ and k2.

22-18. The pointer on a metronome supports a 0.4-lb 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 = (1.20) lb ft, where 0 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 0 and released. Neglect the mass of the pointer.

22-19. The 50-kg block is suspended from the 10-kg pulley that has a radius of gyration about its center of mass of 125 mm. If the block is given a small vertical displacement and then released, determine the natural frequency of oscillation.

*22-20. A uniform board is supported on two wheels which rotate in opposite directions at a constant angular speed. If the coefficient of kinetic friction between the wheels and board is//, determine the frequency of vibration of the board if it is displaced slightly, a distance x from the midpoint between the wheels, and released.

22-21. If the 20-kg block is given a downward velocity of 6 m/s at its equilibrium position, determine the equation that describes the amplitude of the block's oscillation.

22-22. The bar has a length / 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.

22-23. 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 kG = 1.5 ft.The spool rolls without slipping.

*22-24. The cart has a mass of m and is attached to two springs, each having a stiffness of = k2 = k, unstretched length of /0, and a stretched length of / when the cart is in the equilibrium position. If the cart is displaced a distance of .t = .t0 such that both springs remain in tension (a0 < / - /0), determine the natural frequency of oscillation.

22-25. The cart has a mass of m and is attached to two springs, each having a stiffness of k\ and k2. respectively. If both springs are unstretched when the cart is in the equilibrium position shown, determine the natural frequency of oscillation.

22-26. A flywheel of mass nu which has a radius of gyration about its center of mass of k(), is suspended from a circular shaft that has a torsional resistance of M = CO. If the flywheel is given a small angular displacement of 0 and released, determine the natural period of oscillation.

22-27. If a block D of negligible size and of mass m is attached at C and the bell crank of mass M is given a small angular displacement of 0. the natural period of oscillation is T|. When D is removed, the natural period of oscillation is r2. Determine the bell cranks radius of gyration about its center of mass, pin B. and the springs stiffness A'.The spring is unstretched at 0 = 0, and the motion occurs in the horizontal plane.

22-28. The platform AB when empty has a mass of 400 kg. center of mass at G,. and natural period of oscillation 71 = 2.38 s. If a car. having a mass of 1.2 Mg and center of mass at G2, is placed on the platform, the natural period of oscillation becomes 72 = 3.16 s. Determine the moment of inertia of the car about an axis passing through G2.

22-29. A wheel of mass m is suspended from three equal-length cords. When it is given a small angular displacement of 0 about the z axis and released, it is observed that the period of oscillation is r. Determine the radius of gyration of the wheel about the z axis.

22-30. Determine the differential equation of motion of the 3-kg block when it is displaced slightly and released.The surface is smooth and the springs arc originally unstrctched.

22-31. Determine the natural period of vibration of the pendulum. Consider the two rods to be slender, each having a weight of 8 Ib/ft.

*22-32. 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.

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 80. This term will cancel out after taking the time derivative of the energy equation.

22-34. The machine has a mass m and is uniformly supported by four springs, each having a stiffness k. Determine the natural period of vertical vibration.

22-35. Determine the natural period of vibration of the 3-kg sphere. Neglect the mass of the rod and the size of the sphere.

*22-36. The slender rod has a mass m and is pinned at its end O. When it is vertical, the springs are unstretched. Determine the natural period of vibration.

22-37. Determine the natural frequency of vibration of the 20-lb disk. Assume the disk does not slip on the inclined surface.

22-38. If the disk has a mass of 8 kg, determine the natural frequency of vibration.The springs are originally unstretched.

22-39. The semicircular disk has a mass m and radius r. and it rolls without slipping in the semicircular trough. Determine the natural period of vibration of the disk if it is displaced slightly and released. Hint: J0 ^ wr.

*22-40. The gear of mass m has a radius of gyration about its center of mass O of ka.Thc springs have stiffnesses of and k2. respectively, and both springs are unstretched when the gear is in an equilibrium position. If the gear is given a small angular displacement of 0 and released, determine its natural period of oscillation.

22-41. If the block is subjected to the periodic force F = F0 cos col. show that the differential equation of motion is y + (k/m)y = <F0/m) cos col. where y is measured from the equilibrium position of the block. What is the general solution of this equation?

22-42. The block shown in Fig. 22 15 has a mass of 20 kg. and the spring has a stiffness k = 600 N/m. When the block is displaced and released, two successive amplitudes are measured as x{ = 150 mm and x, = 87 mm. Determine the coefficient of viscous damping, c.

A 4-lb weight is attached to a spring having a stiffness k = 10 lb/ft. The weight is drawn downward a distance of 4 in. and released from rest. If the support moves with a vertical displacement \(\delta=(0.5\sin4t)\) in., where t is in seconds, determine the equation which describes the position of the weight as a function of time.

*22-44. A 4-kg block is suspended from a spring that has a stiffness of k = 600N/in. The block is drawn downward 50 mm from the equilibrium position and released from rest when i = 0. If the support moves with an impressed displacement of 8 = (10 sin 4/) mm. where t is in seconds, determine the equation that describes the vertical motion of the block. Assume positive displacement is downward.

22-45. Use a block-and-spring model like that shown in Fig. 22-13*7, but suspended from a vertical position and subjected to a periodic support displacement 8 = 8() sin co0i. 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 7 = 0.

A \(5-\mathrm{kg}\). block is suspended from a spring having a stiffness of \(300 \mathrm{~N} / \mathrm{m}\). If the block is acted upon by a vertical force \(F=(7 \sin 8 t) \mathrm{N}\), where t is in seconds, determine the equation which describes the motion of the block when it is pulled down \(100 \mathrm{~mm}\) from the equilibrium position and released from rest at t=0. Assume that positive displacement is downward.

22-47. The electric motor has a mass of 50 kg and is supported by four springs, each spring having a stiffness of 100 N/m. If the motor turns a disk D which is mounted eccentrically. 20 mm from the disks center, determine the angular velocity oj at which resonance occurs. Assume that the motor only vibrates in the vertical direction.

*22-48. The 20-lb block is attached to a spring having a stiffness of 20 lb/ft. A force F = (6 cos 2/) lb. where t 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.

22-49. The light elastic rod supports a 4-kg sphere. When an 18-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.

22-50. 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 a) = 7 Hz, having a vertical displacement amplitude 80 =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.

The uniform rod has a mass of m. If it is acted upon by a periodic force of \(F=F_0\ \sin\ \omega t\), determine the amplitude of the steady-state vibration.

*22-52. Using a block-and-spring model, like that shown in Fig. 22-13a, but suspended from a vertical position and subjected to a periodic support displacement of 8 = 80 cos corf, 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.

22-53. The fan has a mass of 25 kg and is fixed to the end of a horizontal beam that has a negligible mass. The fan blade is mounted eccentrically on the shaft such that it is equivalent to an unbalanced 3.5-kg mass located 100 mm from the axis of rotation. If the static deflection of the beam is 50 mm as a result of the weight of the fan, determine the angular velocity of the fan blade at which resonance will occur. Hint: See the first part of Example 22.8.

22-54. In Prob. 22-53. determine the amplitude of steady-state vibration of the fan if its angular velocity is 10 rad/s.

22-55. What will be the amplitude of steady-state vibration of the fan in Prob. 22-53 if the angular velocity of the fan blade is 18 rad/s? Hint: See the first part of Example 22.8.

*22-56. The small block at A has a mass of 4 kg and is mounted on the bent rod having negligible mass. If the rotor at 13 causes a harmonic movement 8H = (0.1 cos 15/) m, where / is in seconds, determine the steady-state amplitude of vibration of the block.

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 weighs 150 lb. Neglect the mass of the beam.

22-58. What will be the amplitude of steady-state vibration of the motor in Prob. 22-57 if the angular velocity of the flywheel is 20 rad / s?

22-59. Determine the angular velocity of the flywheel in Prob. 22-57 which will produce an amplitude of vibration of 0.25 in.

*22-60. 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 = (50 sin 2/) lb, where / 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.

22-61. Determine the rotational speed co of the engine in Prob. 22-60 which will cause resonance.

22-62. The motor of mass M is supported by a simply supported beam of negligible mass. If block A of mass m is clipped onto the rotor, which is turning at constant angular velocity of determine the amplitude of the steady-state vibration. Hint: When the beam is subjected to a concentrated force of P at its mid-span, it deflects 8 = PL-/48EI at this point. Here E is Young's modulus of elasticity, a property of the material, and / is the moment of inertia of the beam's cross-sectional area.

22-63. A block having a mass of 0.8 kg is suspended from a spring having a stiffness of 120 N/m. If a dashpot provides a damping force of 2.5 N when the speed of the block is 0.2 m/s. determine the period of free vibration.

*22-64. The block, having a weight of 15 lb, is immersed in a liquid such that the damping force acting on the block has a magnitude of F = (0.81 v \) lb. where v is the velocity of the block in ft/s. If the block is pulled down 0.8 ft and released from rest, determine the position of the block as a function of time. The spring has a stiffness of k = 40 Ib/ft. Consider positive displacement to be downward.

22-65. A 7-lb block is suspended from a spring having a stiffness of k = 75 Ib/ft. The support to which the spring is attached is given simple harmonic motion which may be expressed as 8 = (0.15 sin 21) ft. where i is in seconds. If the damping factor is c/cc = 0.8, determine the phase angle </> of forced vibration.

22-66. Determine the magnification factor of the block, spring, and dashpot combination in Prob. 22-65.

22-67. 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 / = 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 = (50|v|) N. where v is in m/s.

*22-68. 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. Consider positive displacement to be measured downward, and that fluid resistance acting on the disk furnishes a damping force having a magnitude F = (601 v\) N, where v is the velocity of the block in m/s.

22-69. If the 12-kg rod is subjected to a periodic force of F = (30 sin 6/) N. where t is in seconds, determine the steady-state vibration amplitude 0max of the rod about the pin B. Assume 0 is small.

22-70. The damping factor, c/cc, 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. 22-16. show that the ratio In (.V|/.v2) = 2tt(c/cc)/VI {c/cL)2. The quantity In (.r|/*2) is called the logarithmic decrement.

22-71. If the amplitude of the 50-lb cylinders steady-vibration is 6 in., determine the wheels angular velocity oj.

*22-72. The 10-kg block-spring-damper system is damped. If the block is displaced to .v = 50 mm and released from rest, determine the time required for it to return to the position jc = 2 mm.

22-73. The 20-kg block is subjected to the action of the harmonic force F (90cos 60 N, where t is in seconds. Write the equation which describes the steady-state motion.

22-74. A bullet of mass w has a velocity of v0 just before it strikes the target of mass M. If the bullet embeds in the target, and the vibration is to be critically damped, determine the dashpots critical damping coefficient, and the springs' maximum compression. The target is free to move along the two horizontal guides that are nested" in the springs,

22-75. A bullet of mass m has a velocity v just before it strikes the target of mass M. If the bullet embeds in the target, and the dashpots damping coefficient is 0 < c cc. determine the springs' maximum compression. The target is free to move along the two horizontal guides that arc nested" in the springs.

*22-76. Determine the differential equation of motion for the damped vibratory system shown. What type of motion occurs? Take k = 100 N/m,c = 200 N s/m.w = 25 kg.

22-77. Draw the electrical circuit that is equivalent to the mechanical system shown. Determine the differential equation which describes the charge q in the circuit.

22-78. 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?

22-79. Draw the electrical circuit that is equivalent to the mechanical system shown. Determine the differential equation which describes the charge q in the circuit.