Dynamics · Engineering Mechanics · Problem Companion

Chapter Twenty-Two
Vibrations

22.1 Undamped Free Vibration 15 problems
Spring–Mass Systems
  1. 22-1.A spring is stretched \(175\,\mathrm{mm}\) by an \(8\text{-kg}\) block. If the block is displaced \(100\,\mathrm{mm}\) downward from its equilibrium position and given a downward velocity of \(1.50\,\mathrm{m/s}\), determine the differential equation which describes the motion. Assume that positive displacement is downward. Also, determine the position of the block when \(t = 0.22\,\mathrm{s}\).
  2. 22-2.A spring has a stiffness of \(800\,\mathrm{N/m}\). If a \(2\text{-kg}\) block is attached to the spring, pushed \(50\,\mathrm{mm}\) above its equilibrium position, and released from rest, determine the equation that describes the block's motion. Assume that positive displacement is downward.
  3. 22-3.A spring is stretched \(200\,\mathrm{mm}\) by a \(15\text{-kg}\) block. If the block is displaced \(100\,\mathrm{mm}\) downward from its equilibrium position and given a downward velocity of \(0.75\,\mathrm{m/s}\), determine the equation which describes the motion. What is the phase angle? Assume that positive displacement is downward.
  4. 22-4.When a \(2\text{-kg}\) block is suspended from a spring, the spring is stretched a distance of \(40\,\mathrm{mm}\). Determine the frequency and the period of vibration for a \(0.5\text{-kg}\) block attached to the same spring.
  5. 22-5.When a \(3\text{-kg}\) block is suspended from a spring, the spring is stretched a distance of \(60\,\mathrm{mm}\). Determine the natural frequency and the period of vibration for a \(0.2\text{-kg}\) block attached to the same spring.
  6. 22-6.An \(8\text{-kg}\) block is suspended from a spring having a stiffness \(k = 80\,\mathrm{N/m}\). If the block is given an upward velocity of \(0.4\,\mathrm{m/s}\) when it is \(90\,\mathrm{mm}\) 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.
  7. 22-7.A pendulum has a \(0.4\text{-m}\)-long cord and is given a tangential velocity of \(0.2\,\mathrm{m/s}\) toward the vertical from a position \(\theta = 0.3\) rad. Determine the equation which describes the angular motion.
  8. 22-8.A \(2\text{-kg}\) block is suspended from a spring having a stiffness of \(800\,\mathrm{N/m}\). If the block is given an upward velocity of \(2\,\mathrm{m/s}\) when it is displaced downward a distance of \(150\,\mathrm{mm}\) from its equilibrium position, determine the equation which describes the motion. What is the amplitude of the motion? Assume that positive displacement is downward.
  9. 22-9.A \(3\text{-kg}\) block is suspended from a spring having a stiffness of \(k = 200\,\mathrm{N/m}\). If the block is pushed \(50\,\mathrm{mm}\) 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.
Rods, Pendulums & Rigid Bodies
  1. 22-10.The uniform rod of mass \(m\) is supported by a pin at \(A\) and a spring at \(B\). If \(B\) is given a small sideward displacement and released, determine the natural period of vibration. figure — add your own
  2. 22-11.The body of arbitrary shape has a mass \(m\), mass center at \(G\), and a radius of gyration about \(G\) of \(k_G\). If it is displaced a slight amount \(\theta\) from its equilibrium position and released, determine the natural period of vibration. figure — add your own
  3. 22-12.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 \(k_G\). figure — add your own
  4. 22-13.Determine the natural period of vibration of the uniform bar of mass \(m\) when it is displaced downward slightly and released. figure — add your own
  5. 22-14.A platform, having an unknown mass, is supported by four springs, each having the same stiffness \(k\). When nothing is on the platform, the period of vertical vibration is measured as \(2.35\,\mathrm{s}\), whereas if a \(3\text{-kg}\) block is supported on the platform, the period of vertical vibration is \(5.23\,\mathrm{s}\). Determine the mass of a block placed on the (empty) platform which causes the platform to vibrate vertically with a period of \(5.62\,\mathrm{s}\). What is the stiffness \(k\) of each of the springs? figure — add your own
  6. 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. figure — add your own
22.2 Energy Methods & Complex Systems 15 problems
Spring Combinations & Equivalent Stiffness
  1. 22-16.A block of mass \(m\) is suspended from two springs having a stiffness of \(k_1\) and \(k_2\) 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. figure — add your own
  2. 22-17.The \(15\text{-kg}\) block is suspended from two springs having different stiffnesses 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 \(0.5\,\mathrm{s}\) and \(1.5\,\mathrm{s}\), respectively, determine the spring stiffnesses \(k_1\) and \(k_2\). figure — add your own
  3. 22-18.The uniform beam is supported at its ends by two springs \(A\) and \(B\), each having the same stiffness \(k\). When nothing is supported on the beam, it has a period of vertical vibration of \(0.83\,\mathrm{s}\). If a \(50\text{-kg}\) mass is placed at its center, the period of vertical vibration is \(1.52\,\mathrm{s}\). Compute the stiffness of each spring and the mass of the beam. figure — add your own
  4. 22-19.The slender rod has a mass of \(0.2\,\mathrm{kg}\) and is supported at \(O\) by a pin and at its end \(A\) by two springs, each having a stiffness \(k = 4\,\mathrm{N/m}\). The period of vibration of the rod can be set by fixing the \(0.5\text{-kg}\) collar \(C\) to the rod at an appropriate location along its length. If the springs are originally unstretched when the rod is vertical, determine the position \(y\) of the collar so that the natural period of vibration becomes \(\tau = 1\,\mathrm{s}\). Neglect the size of the collar. figure — add your own
  5. 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 \(\mu\), determine the frequency of vibration of the board if it is displaced slightly, a distance \(x\) from the midpoint between the wheels, and released. figure — add your own
Pulleys, Disks & Rolling Bodies
  1. 22-21.The \(50\text{-kg}\) block is suspended from the \(10\text{-kg}\) pulley that has a radius of gyration about its center of mass of \(125\,\mathrm{mm}\). If the block is given a small vertical displacement and then released, determine the natural frequency of oscillation. figure — add your own
  2. 22-22.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. figure — add your own
  3. 22-23.The \(20\text{-kg}\) disk is pinned at its mass center \(O\) and supports the \(4\text{-kg}\) block \(A\). 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. figure — add your own
  4. 22-24.The \(10\text{-kg}\) disk is pin connected at its mass center. 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 \(\delta_O\). figure — add your own
  5. 22-25.If the disk in Prob. 22-24 has a mass of \(10\,\mathrm{kg}\), determine the natural frequency of vibration. figure — add your own
  6. 22-26.A flywheel of mass \(m\) which has a radius of gyration about its center of mass of \(k_O\) is suspended from a circular shaft that has a torsional resistance of \(M = C\theta\). If the flywheel is given a small angular displacement of \(\theta\) and released, determine the natural period of oscillation. figure — add your own
  7. 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 \(\theta\), the natural period of oscillation is \(\tau_1\). When \(D\) is removed, the natural period of oscillation is \(\tau_2\). Determine the bell crank's radius of gyration about its center of mass, pin \(B\), and the spring's stiffness \(k\). The spring is unstretched at \(\theta = 0^\circ\) and the motion occurs in the horizontal plane. figure — add your own
  8. 22-28.The platform \(AB\) when empty has a mass of \(400\,\mathrm{kg}\), center of mass at \(G_1\), and natural period of oscillation \(\tau_1 = 2.38\,\mathrm{s}\). If a car, having a mass of \(1.2\,\mathrm{Mg}\) and center of mass at \(G_2\), is placed on the platform, the natural period of oscillation becomes \(\tau_2 = 3.16\,\mathrm{s}\). Determine the moment of inertia of the car about an axis passing through \(G_2\). figure — add your own
  9. 22-29.The plate of mass \(m\) is supported by three symmetrically placed cords of length \(l\) as shown. If the plate is given a slight rotation about a vertical axis through its center and released, determine the natural period of oscillation. figure — add your own
22.3 More Vibration Systems 11 problems
Rods, Disks & Spools
  1. 22-30.Determine the differential equation of motion of the \(3\text{-kg}\) block when it is displaced slightly and released. The surface is smooth and the springs are originally unstretched. figure — add your own
  2. 22-31.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. figure — add your own
  3. 22-32.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: \(I_O = \frac{1}{2}mr^2\). figure — add your own
  4. 22-33.If the \(20\text{-kg}\) wheel is displaced a small amount and released, determine the natural period of vibration. The radius of gyration of the wheel is \(k_G = 0.36\,\mathrm{m}\). The wheel rolls without slipping. figure — add your own
  5. 22-34.Determine the differential equation of motion of the \(3\text{-kg}\) spool. Assume that it does not slip at the surface of contact as it oscillates. The radius of gyration of the spool about its center of mass is \(k_G = 125\,\mathrm{mm}\). figure — add your own
  6. 22-35.Determine the natural period of vibration of the \(3\text{-kg}\) sphere. Neglect the mass of the rod and the size of the sphere. figure — add your own
  7. 22-36.If the lower end of the \(6\text{-kg}\) slender rod is displaced a small amount and released from rest, determine the natural frequency of vibration. Each spring has a stiffness of \(k = 200\,\mathrm{N/m}\) and is unstretched when the rod is hanging vertically. figure — add your own
  8. 22-37.The gear of mass \(m\) has a radius of gyration about its center of mass \(O\) of \(k_O\). The springs have stiffnesses of \(k_1\) and \(k_2\) respectively, and both springs are unstretched when the gear is in an equilibrium position. If the gear is given a small angular displacement of \(\theta\) and released, determine its natural period of oscillation. figure — add your own
  9. 22-38.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. figure — add your own
  10. 22-39.Determine the differential equation of motion of the \(3\text{-kg}\) spool. Assume that it does not slip at the surface of contact as it oscillates. The radius of gyration of the spool about its center of mass is \(k_G = 125\,\mathrm{mm}\). figure — add your own
  11. 22-40.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. figure — add your own
22.4 Undamped Forced Vibration 20 problems
Periodic Forces & Support Motion
  1. 22-41.If the block-and-spring model is subjected to the periodic force \(F = F_0 \cos \omega t\), show that the differential equation of motion is \(\ddot{x} + (k/m)x = (F_0/m)\cos \omega t\), where \(x\) is measured from the equilibrium position of the block. What is the general solution of this equation? figure — add your own
  2. 22-42.A block which has a mass \(m\) is suspended from a spring having a stiffness \(k\). If an impressed downward vertical force \(F = F_0\) acts on the weight, determine the equation which describes the position of the block as a function of time.
  3. 22-43.The block shown in Fig. 22-15 has a mass of \(20\,\mathrm{kg}\) and the spring has a stiffness \(k = 600\,\mathrm{N/m}\). When the block is displaced and released, two successive amplitudes are measured as \(x_1 = 150\,\mathrm{mm}\) and \(x_2 = 87\,\mathrm{mm}\). Determine the coefficient of viscous damping, \(c\). figure — add your own
  4. 22-44.A \(4\text{-kg}\) block is suspended from a spring that has a stiffness of \(k = 600\,\mathrm{N/m}\). The block is drawn downward \(50\,\mathrm{mm}\) from the equilibrium position and released from rest when \(t = 0\). If the support moves with an impressed displacement of \(\delta = (10\sin 4t)\,\mathrm{mm}\), where \(t\) is in seconds, determine the equation that describes the vertical motion of the block. Assume positive displacement is downward.
  5. 22-45.Use 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 \(\delta = \delta_0 \sin \omega_0 t\), 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 \(t = 0\).
  6. 22-46.A \(5\text{-kg}\) block is suspended from a spring having a stiffness of \(300\,\mathrm{N/m}\). If the block is acted upon by a vertical force \(F = (7\sin 8t)\,\mathrm{N}\), 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. figure — add your own
  7. 22-47.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. figure — add your own
  8. 22-48.The electric motor has a mass of \(50\,\mathrm{kg}\) and is supported by four springs, each spring having a stiffness of \(100\,\mathrm{N/m}\). If the motor turns a disk \(D\) which is mounted eccentrically, \(20\,\mathrm{mm}\) from the disk's center, determine the angular velocity \(\omega\) at which resonance occurs. Assume that the motor only vibrates in the vertical direction. figure — add your own
  9. 22-49.The light elastic rod supports a \(4\text{-kg}\) sphere. When an \(18\,\mathrm{N}\) vertical force is applied to the sphere, the rod deflects \(14\,\mathrm{mm}\). If the wall oscillates with harmonic frequency of \(2\,\mathrm{Hz}\) and has an amplitude of \(15\,\mathrm{mm}\), determine the amplitude of vibration for the sphere. figure — add your own
  10. 22-50.Find the differential equation for small oscillations in terms of \(\theta\) for the uniform rod of mass \(m\). Also show that if \(c < \sqrt{mk}/2\), then the system remains underdamped. The rod is in a horizontal position when it is in equilibrium. figure — add your own
  11. 22-51.The \(40\text{-kg}\) block is attached to a spring having a stiffness of \(800\,\mathrm{N/m}\). A force \(F = (100 \cos 2t)\,\mathrm{N}\), where \(t\) is in seconds, is applied to the block. Determine the maximum speed of the block for the steady-state vibration. figure — add your own
  12. 22-52.The fan has a mass of \(25\,\mathrm{kg}\) 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 \(3.5\text{-kg}\) mass located \(100\,\mathrm{mm}\) from the axis of rotation. If the static deflection of the beam is \(50\,\mathrm{mm}\) as a result of the weight of the fan, determine the angular velocity of the fan blade at which resonance will occur. figure — add your own
  13. 22-53.In Prob. 22-52, determine the amplitude of steady-state vibration of the fan if its angular velocity is \(10\,\mathrm{rad/s}\).
  14. 22-54.What will be the amplitude of steady-state vibration of the fan in Prob. 22-52 if the angular velocity of the fan blade is \(18\,\mathrm{rad/s}\)?
  15. 22-55.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 \(\omega\), 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 \(\delta = PL^3/48EI\) at this point. Here \(E\) is Young's modulus of elasticity and \(I\) is the moment of inertia of the beam's cross-sectional area. figure — add your own
  16. 22-56.The electric motor turns an eccentric flywheel which is equivalent to an unbalanced \(0.125\text{-kg}\) mass located \(250\,\mathrm{mm}\) from the axis of rotation. If the static deflection of the beam is \(25\,\mathrm{mm}\) because of the mass of the motor, determine the angular velocity of the flywheel at which resonance will occur. The motor has a mass of \(75\,\mathrm{kg}\). Neglect the mass of the beam. figure — add your own
  17. 22-57.What will be the amplitude of steady-state vibration of the motor in Prob. 22-56 if the angular velocity of the flywheel is \(20\,\mathrm{rad/s}\)?
  18. 22-58.Determine the angular velocity of the flywheel in Prob. 22-56 which will produce an amplitude of vibration of \(6\,\mathrm{mm}\).
  19. 22-59.The \(450\text{-kg}\) 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 \(50\,\mathrm{mm}\) and wavelength of \(4\,\mathrm{m}\). If the two springs \(s\) which support the trailer each have a stiffness of \(800\,\mathrm{N/m}\), determine the speed \(v\) which will cause the greatest vibration (resonance) of the trailer. Neglect the weight of the wheels. figure — add your own
  20. 22-60.Determine the amplitude of vibration of the trailer in Prob. 22-59 if the speed \(v = 15\,\mathrm{km/h}\).
22.5 Viscous Damped Vibration 18 problems
Damped Free & Forced Vibration
  1. 22-62.The spring system is connected to a crosshead that oscillates vertically when the wheel rotates with a constant angular velocity of \(\omega\). If the amplitude of the steady-state vibration is observed to be \(400\,\mathrm{mm}\) and the springs each have a stiffness of \(k = 2500\,\mathrm{N/m}\), determine the two possible values of \(\omega\) at which the wheel must rotate. The block has a mass of \(50\,\mathrm{kg}\). figure — add your own
  2. 22-63.The spring system is connected to a crosshead that oscillates vertically when the wheel rotates with a constant angular velocity of \(\omega = 5\,\mathrm{rad/s}\). If the amplitude of the steady-state vibration is observed to be \(400\,\mathrm{mm}\), determine the two possible values of the stiffness \(k\) of the springs. The block has a mass of \(50\,\mathrm{kg}\). figure — add your own
  3. 22-64.A \(3.5\text{-kg}\) block is suspended from a spring having a stiffness of \(k = 1250\,\mathrm{N/m}\). The support to which the spring is attached is given simple harmonic motion which can be expressed as \(\delta = (0.045\sin 2t)\,\mathrm{m}\), where \(t\) is in seconds. If the damping factor is \(c/c_c = 0.8\), determine the phase angle \(\phi\) of forced vibration.
  4. 22-65.Determine the magnification factor of the block, spring, and dashpot combination in Prob. 22-64.
  5. 22-66.A block having a mass of \(7\,\mathrm{kg}\) is suspended from a spring that has a stiffness \(k = 600\,\mathrm{N/m}\). If the block is given an upward velocity of \(0.6\,\mathrm{m/s}\) 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 \(F = (50|v|)\,\mathrm{N}\), where \(v\) is in m/s.
  6. 22-67.The \(20\text{-kg}\) block is subjected to the action of the harmonic force \(F = (90 \cos 6t)\,\mathrm{N}\), where \(t\) is in seconds. Write the equation which describes the steady-state motion. figure — add your own
  7. 22-68.Two identical dashpots are arranged parallel to each other, as shown. Show that if the damping coefficient \(c < \sqrt{mk}\), then the block of mass \(m\) will vibrate as an underdamped system. figure — add your own
  8. 22-69.The damping factor, \(c/c_c\), 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 \(x_1\) and \(x_2\) as shown in Fig. 22-16, show that \(\ln(x_1/x_2) = 2\pi(c/c_c)/\sqrt{1 - (c/c_c)^2}\). The quantity \(\ln(x_1/x_2)\) is called the logarithmic decrement. figure — add your own
  9. 22-70.The \(4\text{-kg}\) circular disk is attached to three springs, each spring having a stiffness \(k = 180\,\mathrm{N/m}\). If the disk is immersed in a fluid and given a downward velocity of \(0.3\,\mathrm{m/s}\) 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 \(F = (60v)\,\mathrm{N}\), where \(v\) is the velocity of the block in \(\mathrm{m/s}\). figure — add your own
  10. 22-71.Determine the differential equation of motion for the damped vibratory system shown. What type of motion occurs? Take \(k = 100\,\mathrm{N/m}\), \(c = 200\,\mathrm{N\cdot s/m}\), \(m = 25\,\mathrm{kg}\). figure — add your own
  11. 22-72.If the \(12\text{-kg}\) rod is subjected to a periodic force of \(F = (30 \sin 6t)\,\mathrm{N}\), where \(t\) is in seconds, determine the steady-state vibration amplitude \(\theta_{\max}\) of the rod about the pin \(B\). Assume \(\theta\) is small. figure — add your own
  12. 22-73.A bullet of mass \(m\) has a velocity of \(\mathbf{v}_0\) 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 dashpot's 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. figure — add your own
  13. 22-74.A bullet of mass \(m\) has a velocity \(\mathbf{v}_0\) just before it strikes the target of mass \(M\). If the bullet embeds in the target, and the dashpot's damping coefficient is \(0 < c < c_c\), determine the springs' maximum compression. The target is free to move along the two horizontal guides that are "nested" in the springs. figure — add your own
  14. 22-75.Determine the differential equation of motion for the damped vibratory system shown. What type of motion occurs? Take \(k = 100\,\mathrm{N/m}\), \(c = 200\,\mathrm{N\cdot s/m}\), \(m = 25\,\mathrm{kg}\). figure — add your own
22.6 Electrical Circuit Analogs 3 problems
Circuit Equivalents
  1. 22-76.Draw the electrical circuit that is equivalent to the mechanical system shown. Determine the differential equation which describes the charge \(q\) in the circuit. figure — add your own
  2. 22-77.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? figure — add your own
  3. 22-78.Draw the electrical circuit that is equivalent to the mechanical system shown. Determine the differential equation which describes the charge \(q\) in the circuit. figure — add your own

Note: Problem 22-61 is referenced in the text but no problem statement is provided in the source material. Problems 22-11, 22-34/22-39, 22-38/22-40, and 22-71/22-75 appear as printed in the original text.

The chapter review (pages 14–15) summarizes undamped free vibration, energy methods, undamped forced vibration, viscous damped free vibration, viscous damped forced vibration, and electrical circuit analogs.