Determine the system equation of motion in termsof the variable shown in the fi gure

When a 3-kg collar is placed upon the pan which is attached to the spring of unknown constant, the additional static deflection of the pan is observed to be 42 mm. Determine the spring constant k in N /m, lb /in., and lb /ft.

Determine the natural frequency of the spring-mass system in both radians per second and cycles per second (Hz).

For the system of Prob. 8 /2, determine the position x of the mass as a function of time if the mass is released from rest at time t = 0 from a position 2 inches to the left of the equilibrium position. Determine the maximum velocity and maximum acceleration of the mass over one cycle of motion.

For the system of Prob. 8 /2, determine the position x as a function of time if the mass is released at time t = 0 from a position 2 inches to the right of the equilibrium position with an initial velocity of 9 in. /sec to the left. Determine the amplitude C and period \(\tau\) of the motion.

For the spring-mass system shown, determine the static deflection \(\delta_{\text {st }}\), the system period \(\tau\), and the maximum velocity \(v_{\max }\) which result if the cylinder is displaced 100 mm downward from its equilibrium position and released from rest.

The cylinder of the system of Prob. 8/5 is displaced 100 mm downward from its equilibrium position and released at time t = 0. Determine the position y, velocity v, and acceleration a when t = 3 s. What is the maximum acceleration?

Determine the natural frequency in cycles per second for the system shown. Neglect the mass and friction of the pulleys. Assume that the block of mass m remains horizontal.

The vertical plunger has a mass of 2.5 kg and is supported by the two springs, which are always in compression. Calculate the natural frequency \(f_{n}\) of vibration of the plunger if it is deflected from the equilibrium position and released from rest. Friction in the guide is negligible.

Determine the period \(\tau\) for the system shown. The cable is always taut, and the mass and friction of the pulley are to be neglected.

In the equilibrium position, the 30-kg cylinder causes a static deflection of 50 mm in the coiled spring. If the cylinder is depressed an additional 25 mm and released from rest, calculate the resulting natural frequency \(f_{n}\) of vertical vibration of the cylinder in cycles per second (Hz).

For the cylinder of Prob. 8 /10, determine the vertical displacement x, measured positive down in millimeters from the equilibrium position, in terms of the time t in seconds measured from the instant of release from the position of 25 mm added deflection.

Determine the natural frequency in radians per second for the system shown. Neglect the mass and friction of the pulleys.

An old car being moved by a magnetic crane pickup is dropped from a short distance above the ground. Neglect any damping effects of its worn-out shock absorbers and calculate the natural frequency \(f_{n}\) in cycles per second (Hz) of the vertical vibration which occurs after impact with the ground. Each of the four springs on the 1000-kg car has a constant of 17.5 kN/m. Because the center of mass is located midway between the axles and the car is level when dropped, there is no rotational motion. State any assumptions.

The 4-oz slider oscillates in the fixed slot under the action of the three springs, each of stiffness k = 0.5 lb /in. If the initial conditions at time t = 0 are \(x_{0}=0.1 \text { in. and } \dot{x}_{0}=0.5 \mathrm{in} . / \mathrm{sec}\), determine the position and velocity of the slider at time t = 2 sec. What is the system period?

During the design of the spring-support system for the 4000-kg weighing platform, it is decided that the frequency of free vertical vibration in the unloaded condition shall not exceed 3 cycles per second. (a) Determine the maximum acceptable spring constant k for each of the three identical springs. (b) For this spring constant, what would be the natural frequency \(f_{n}\) of vertical vibration of the platform loaded by the 40-Mg truck?

Calculate the natural frequency \(f_{n}\) of vibration if the mass is deflected from its equilibrium position and released from rest. Each pair of springs is connected by an inextensible cable. Evaluate your results for m = 15 kg, \(k_{1}=225 \mathrm{~N} / \mathrm{m}, \text { and } k_{2}=150 \mathrm{~N} / \mathrm{m}\).

Replace the springs in each of the two cases shown by a single spring of stiffness k (equivalent spring stiffness) which will cause each mass to vibrate with its original frequency.

With the assumption of no slipping, determine the mass m of the block which must be placed on the top of the 6-kg cart in order that the system period be 0.75 s. What is the minimum coefficient \(\mu_{s}\) of static friction for which the block will not slip relative to the cart if the cart is displaced 50 mm from the equilibrium position and released?

If both springs are unstretched when the mass is in the central position shown, determine the static deflection \(\delta_{\mathrm{st}}\) of the mass. What is the period of oscillatory motion about the position of static equilibrium?

An energy-absorbing car bumper with its springs initially undeformed has an equivalent spring constant of 3000 lb/in. If the 2500-lb car approaches a massive wall with a speed of 5 mi/hr, determine (a) the velocity v of the car as a function of time during contact with the wall, where t = 0 is the beginning of the impact, and (b) the maximum deflection \(x_{\max}\) of the bumper.

A 90-kg man stands at the end of a diving board and causes a vertical oscillation which is observed to have a period of 0.6 s. What is the static deflection \(\delta_{\mathrm{st}}\) at the end of the board? Neglect the mass of the board.

Shown in the figure is a model of a one-story building. The bar of mass m is supported by two light elastic upright columns whose upper and lower ends are fixed against rotation. For each column, if a force P and corresponding moment M were applied as shown in the right-hand part of the figure, the deflection \(\delta\) would be given by \(\delta=P L^{3} / 12 E I\), where L is the effective column length, E is Young’s modulus, and I is the area moment of inertia of the column cross section with respect to its neutral axis. Determine the natural frequency of horizontal oscillation of the bar when the columns bend as shown in the figure.

Calculate the natural circular frequency \(\omega_{n}\) of the system shown in the figure. The mass and friction of the pulleys are negligible.

The slider of mass m is confined to the horizontal slot shown. The two springs each of constant k are linear. Derive the nonlinear equation of motion for small values of y, retaining terms of order \(y^{3}\) and larger. Both springs are unstretched when y = 0. Neglect friction.

Determine the value of the damping ratio \(\zeta\) for the simple spring-mass-dashpot system shown.

The period \(\tau_{d}\) of damped linear oscillation for a certain 1-kg mass is 0.3 s. If the stiffness of the supporting linear spring is 800 N/m, calculate the damping coefficient c.

Viscous damping is added to an initially undamped spring-mass system. For what value of the damping ratio \(\zeta\) will the damped natural frequency \(\omega_{d}\) be equal to 90 percent of the natural frequency of the original undamped system?

The addition of damping to an undamped spring-mass system causes its period to increase by 25 percent. Determine the damping ratio \(\zeta\).

Determine the value of the viscous damping coefficient c for which the system shown is critically damped.

The 8-lb body of Prob. 8 /25 is released from rest a distance \(x_{0}\) to the right of the equilibrium position. Determine the displacement x as a function of time t, where t = 0 is the time of release.

The figure represents the measured displacement-time relationship for a vibration with small damping where it is impractical to achieve accurate results by measuring the nearly equal amplitudes of two successive cycles. Modify the expression for the viscous damping factor \(\zeta\) based on the measured amplitudes \(x_{0}\) and \(x_{N}\) which are N cycles apart.

A linear harmonic oscillator having a mass of 1.10 kg is set into motion with viscous damping. If the frequency is 10 Hz and if two successive amplitudes a full cycle apart are measured to be 4.65 mm and 4.30 mm as shown, compute the viscous damping coefficient c.

Determine the damping ratio \(\zeta\) for the system shown. The system parameters are m = 4 kg, k = 500 N /m, and c = 100 N? s /m. Neglect the mass and friction of all pulleys, and assume that the cord remains taut throughout a motion cycle.

Further design refinement for the weighing platform of Prob. 8 /15 is shown here where two viscous dampers are to be added to limit the ratio of successive positive amplitudes of vertical vibration in the unloaded condition to 4. Determine the necessary viscous damping coefficient c for each of the dampers.

Derive the differential equation of motion for the system shown in terms of the variable \(x_{1}\). Neglect friction and the mass of the linkage.

The system shown is released from rest from an initial position \(x_{0}\). Determine the overshoot displacement \(x_{1}\). Assume translational motion in the x-direction.

Determine the equation of motion for the system in terms of the variable x. The cables remain taut at all times, and the pulleys turn independently. Neglect friction and the mass of the pulleys. Additionally, determine expressions for the natural circular frequency \(\omega_{n}\) and the damping ratio \(\zeta\).

The mass of a given critically damped system is released at time t = 0 from the position \(x_{0}>0\) with a negative initial velocity. Determine the critical value \(\left(\dot{x}_{0}\right)_{c}\) of the initial velocity below which the mass will pass through the equilibrium position.

The mass of the system shown is released from rest at \(x_{0}=6 \text { in. }\) when t = 0. Determine the displacement x at t = 0.5 sec if (a) c = 12 lb-sec/ft and (b) c = 18 lb-sec/ft.

Derive the equation of motion for the system shown in terms of the displacement x. The masses are coupled through the light connecting rod ABC which pivots about the smooth bearing at point O. Neglect all friction, consider the rollers on \(m_{2} \text { and } m_{3}\) to be light, and assume small oscillations about the equilibrium position. State the system natural circular frequency \(\omega_{n}\) and the viscous damping ratio \(\zeta\) for \(m_{1}=15 \mathrm{kg}, m_{2}=12 \mathrm{kg}, m_{3}=8 \mathrm{kg}, k_{1}=400 \mathrm{N} / \mathrm{m}\), \(k_{2}=650 \mathrm{N} / \mathrm{m}, k_{3}=225 \mathrm{N} / \mathrm{m}, c_{1}=44 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}, c_{2}=36 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}, c_{3}=52 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}\), a = 1.2 m, b = 1.8 m, and c = 0.9 m.

The owner of a 3400-lb pickup truck tests the action of his rear-wheel shock absorbers by applying a steady 100-lb force to the rear bumper and measuring a static deflection of 3 in. Upon sudden release of the force, the bumper rises and then falls to a maximum of \(\frac{1}{2} \text { in }\). below the unloaded equilibrium position of the bumper on the first rebound. Treat the action as a one-dimensional problem with an equivalent mass of half the truck mass. Find the viscous damping factor \(\zeta\) for the rear end and the viscous damping coefficient c for each shock absorber assuming its action to be vertical.

The 2-kg mass is released from rest at a distance \(x_{0}\) to the right of the equilibrium position. Determine the displacement x as a function of time.

Develop the equation of motion in terms of the variable x for the system shown. Determine an expression for the damping ratio \(\zeta\) in terms of the given system properties. Neglect the mass of the crank AB and assume small oscillations about the equilibrium position shown.

Investigate the case of Coulomb damping for the block shown, where the coefficient of kinetic friction is \(\mu_{k}\) and each spring has a stiffness k/2. The block is displaced a distance \(x_{0}\) from the neutral position and released. Determine and solve the differential equation of motion. Plot the resulting vibration and indicate the rate r of decay of the amplitude with time.

A viscously damped spring-mass system is excited by a harmonic force of constant amplitude \(F_{0}\) but varying frequency \(\omega\). If the amplitude of the steady-state motion is observed to decrease by a factor of 8 as the frequency ratio \(\omega / \omega_{n}\) is varied from 1 to 2, determine the damping ratio \(\zeta\) of the system.

Determine the amplitude X of the steady-state motion of the 10-kg mass if (a) c = 500 N?s/m and (b) c = 0.

The 30-kg cart is acted upon by the harmonic force shown in the fi gure. If c = 0, determine the range of the driving frequency \(\omega\) for which the magnitude of the steady-state response is less than 75 mm.

If the viscous damping coefficient of the damper in the system of Prob. 8 /47 is c = 36 N? s /m, determine the range of the driving frequency \(\omega\) for which the magnitude of the steady-state response is less than 75 mm.

If the driving frequency for the system of Prob. 8 /47 is \(\omega=6 \mathrm{rad} / \mathrm{s}\), determine the required value of the damping coefficient c if the steady-state amplitude is not to exceed 75 mm.

A spring-mounted machine with a mass of 24 kg is observed to vibrate harmonically in the vertical direction with an amplitude of 0.30 mm under the action of a vertical force which varies harmonically between \(F_{0} \text { and }-F_{0}\) with a frequency of 4 Hz. Damping is negligible. If a static force of magnitude \(F_{0}\) causes a deflection of 0.60 mm, calculate the equivalent spring constant k for the springs which support the machine.

The block of weight W = 100 lb is suspended by two springs each of stiffness k = 200 lb/ft and is acted upon by the force F = 75 cos 15t lb where t is the time in seconds. Determine the amplitude X of the steady-state motion if the viscous damping coefficient c is (a) 0 and (b) 60 lb-sec/ft. Compare these amplitudes to the static spring deflection \(\delta_{\mathrm{st}}\).

An external force \(F=F_{0} \sin \omega t\) is applied to the cylinder as shown. What value \(\omega_{c}\) of the driving frequency would cause excessively large oscillations of the system?

A viscously damped spring-mass system is forced harmonically at the undamped natural frequency \(\left(\omega / \omega_{n}=1\right)\). If the damping ratio \(\zeta\) is doubled from 0.1 to 0.2, compute the percentage reduction \(R_{1}\) in the steady-state amplitude. Compare with the result \(R_{2}\) of a similar calculation for the condition \(\omega / \omega_{n}=2\). Verify your results by inspecting Fig. 8/11.

The 4-lb body is attached to two springs, each of which has a stiffness of 6 lb /in. The body is mounted on a shake table which vibrates harmonically in the horizontal direction with an amplitude of 0.5 in. and a frequency ƒ which can be varied. Power to the shake table is turned off when electrical contact is made at A or B. Determine the maximum value of the frequency ƒ at which the shake table may be operated without turning the power off as it starts from rest and increases its frequency gradually. Damping may be neglected. The equilibrium position is centered between the fixed contacts.

It was noted in the text that the maxima of the curves for the magnification factor M are not located at \(\omega / \omega_{n}=1\). Determine an expression in terms of the damping ratio \(\zeta\) for the frequency ratio at which the maxima occur.

The motion of the outer frame B is given by \(x_{B}=b \sin \omega t\). For what range of the driving frequency \(\omega\) is the amplitude of the motion of the mass m relative to the frame less than 2b?

The 20-kg variable-speed motorized unit is restrained in the horizontal direction by two springs, each of which has a stiffness of 2.1 kN/m. Each of the two dashpots has a viscous damping coefficient c = 58 N?s/m. In what ranges of speeds N can the motor be run for which the magnification factor M will not exceed 2?

A single-cylinder four-stroke gasoline engine with a mass of 90 kg is mounted on four stiff spring pads, each with a stiffness of \(30\left(10^{3}\right) \mathrm{kN} / \mathrm{m}\), and is designed to run at 3600 rev /min. The mounting system is equipped with viscous dampers which have a large enough combined viscous damping coefficient c so that the system is critically damped when it is given a vertical displacement and then released while not running. When the engine is running, it fires every other revolution, causing a periodic vertical displacement modeled by 1.2 cos \(\omega t\) mm with t in seconds. Determine the magnification factor M and the overall damping coefficient c.

When the person stands in the center of the floor system shown, he causes a static deflection \(\delta_{\mathrm{st}}\) of the floor under his feet. If he walks (or runs quickly!) in the same area, how many steps per second would cause the floor to vibrate with the greatest vertical amplitude?

The instrument shown has a mass of 43 kg and is spring-mounted to the horizontal base. If the amplitude of vertical vibration of the base is 0.10 mm, calculate the range of frequencies \(f_{n}\) of the base vibration which must be prohibited if the amplitude of vertical vibration of the instrument is not to exceed 0.15 mm. Each of the four identical springs has a stiffness of 7.2 kN/m.

Derive the equation of motion for the inertial displacement \(x_{i}\) of the mass of Fig. 8 /14. Comment on, but do not carry out, the solution to the equation of motion.

Attachment B is given a horizontal motion \(x_{B}=b \cos \omega t\). Derive the equation of motion for the mass m and state the critical frequency \(\omega_{c}\) for which the oscillations of the mass become excessively large.

Attachment B is given a horizontal motion \(x_{B}=b \cos \omega t\). Derive the equation of motion for the mass m and state the critical frequency \(\omega_{c}\) for which the oscillations of the mass become excessively large. What is the damping ratio \(\zeta\) for the system?

A device to produce vibrations consists of the two counter-rotating wheels, each carrying an eccentric mass \(m_{0}=1 \mathrm{kg}\) with a center of mass at a distance e = 12 mm from its axis of rotation. The wheels are synchronized so that the vertical positions of the unbalanced masses are always identical. The total mass of the device is 10 kg. Determine the two possible values of the equivalent spring constant k for the mounting which will permit the amplitude of the periodic force transmitted to the fixed mounting to be 1500 N due to the imbalance of the rotors at a speed of 1800 rev/min. Neglect damping.

The seismic instrument shown is attached to a structure which has a horizontal harmonic vibration at 3 Hz. The instrument has a mass m = 0.5 kg, a spring stiffness k = 20 N/m, and a viscous damping coefficient \(c=3 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}\). If the maximum recorded value of x in its steady-state motion is X = 2 mm, determine the amplitude b of the horizontal movement \(x_{B}\) of the structure.

The equilibrium position of the mass m occurs where y = 0 and \(y_{B}=0\). When the attachment B is given a steady vertical motion \(y_{B}=b \sin \omega t\), the mass m will acquire a steady vertical oscillation. Derive the differential equation of motion for m and specify the circular frequency \(\omega_{c}\) for which the oscillations of m tend to become excessively large. The stiffness of the spring is k, and the mass and friction of the pulley are negligible.

Derive and solve the equation of motion for the mass m in terms of the variable x for the system shown. Neglect the mass of the lever AOC and assume small oscillations.

The seismic instrument is mounted on a structure which has a vertical vibration with a frequency of 5 Hz and a double amplitude of 18 mm. The sensing element has a mass m = 2 kg, and the spring stiffness is k = 1.5 kN/m. The motion of the mass relative to the instrument base is recorded on a revolving drum and shows a double amplitude of 24 mm during the steady-state condition. Calculate the viscous damping constant c.

Derive the expression for the energy loss E over a complete steady-state cycle due to the frictional dissipation of energy in a viscously damped linear oscillator. The forcing function is \(F_{0} \sin \omega t\), and the displacement-time relation for steady-state motion is \(x_{P}=X \sin (\omega t-\phi)\) where the amplitude X is given by Eq. 8/20. (Hint: The frictional energy loss during a displacement dx is \(c \dot{x} d x\), where c is the viscous damping coefficient. Integrate this expression over a complete cycle.

Determine the amplitude of vertical vibration of the car as it travels at a velocity v = 40 km / h over the wavy road whose contour may be expressed as a sine or cosine function with a double amplitude 2b = 50 mm. The mass of the car is 1800 kg and the stiffness of each of the four car springs is 35 kN/ m. Assume that all four wheels are in continuous contact with the road, and neglect damping. Note that the wheelbase of the car and the spatial period of the road are the same at L = 3 m, so that it may be assumed that the car translates but does not rotate. At what critical speed \(v_{c}\) is the vertical vibration of the car at its maximum?

The light rod and attached small spheres of mass m each are shown in the equilibrium position, where all four springs are equally precompressed. Determine the natural frequency \(\omega_{n}\) and period \(\tau\) for small oscillations about the frictionless pivot O.

A uniform rectangular plate pivots about a horizontal axis through one of its corners as shown. Determine the natural frequency \(\omega_{n}\) of small oscillations.

The thin square plate is suspended from a socket (not shown) which fits the small ball attachment at O. If the plate is made to swing about axis A-A, determine the period for small oscillations. Neglect the small offset, mass, and friction of the ball.

If the square plate of Prob. 8/73 is made to oscillate about axis B-B, determine the period of small oscillations.

The 20-lb spoked wheel has a centroidal radius of gyration k = 6 in. A torsional spring of constant \(k_{T}=160 \mathrm{lb}-\mathrm{ft} / \mathrm{rad}\) resists rotation about the smooth bearing. If an external torque of form \(M=M_{0} \cos \omega t\) is applied to the wheel, what is the magnitude of its steady-state angular displacement? The moment magnitude is \(M_{0}=8 \mathrm{lb}-\mathrm{ft}\) and the driving frequency is \(\omega=25 \mathrm{rad} / \mathrm{sec}\).

The uniform rod of length l and mass m is suspended at its midpoint by a wire of length L. The resistance of the wire to torsion is proportional to its angle of twist \(\theta\) and equals \((J G / L) \theta\) where J is the polar moment of inertia of the wire cross section and G is the shear modulus of elasticity. Derive the expression for the period \(\tau\) of oscillation of the bar when it is set into rotation about the axis of the wire.

The uniform sector has mass m and is freely hinged about a horizontal axis through point O. Determine the equation of motion of the sector for large-amplitude vibrations about the equilibrium position. State the period \(\tau\) for small oscillations about the equilibrium position if r = 325 mm and \(\beta=45^{\circ}\).

The assembly of mass m is formed from uniform and slender welded rods and is freely hinged about a horizontal axis through O. Determine the equation of motion of the assembly for large-amplitude vibrations about the equilibrium position. State the period \(\tau\) for small oscillations about the equilibrium position if r = 325 mm and \(\beta=45^{\circ}\).

The thin-walled cylindrical shell of radius r and height h is welded to the small shaft at its upper end as shown. Determine the natural circular frequency \(\omega_{n}\) for small oscillations of the shell about the y-axis.

Determine the system equation of motion in terms of the variable \(\theta\) shown in the figure. Assume small angular motion of bar OA, and neglect the mass of link CD.

The uniform rod of mass m is freely pivoted about a horizontal axis through point O. Assume small oscillations and determine an expression for the damping ratio \(\zeta\). For what value \(c_{cr}\) of the damping coefficient c will the system be critically damped?

The mass of the uniform slender rod is 3 kg. Determine the position x for the 1.2-kg slider such that the system period is 1 s. Assume small oscillations about the horizontal equilibrium position shown.

The triangular frame of mass m is formed from uniform slender rod and is suspended from a socket (not shown) which fits the small ball attachment at O. If the frame is made to swing about axis A-A, determine the natural circular frequency \(\omega_{n}\) for small oscillations. Neglect the small offset, mass, and friction of the ball. Evaluate for l = 200 mm.

If the triangular frame of Prob. 8 /83 is made to oscillate about axis B-B, determine the natural circular frequency \(\omega_{n}\) for small oscillations. Evaluate for l = 200 mm and compare your answer with that of Prob. 8 /83.

The uniform rod of mass m is freely pivoted about point O. Assume small oscillations and determine an expression for the damping ratio \(\zeta\). For what value \(c_{cr}\) of the damping coefficient c will the system be critically damped?

The mechanism shown oscillates in the vertical plane about the pivot O. The springs of equal stiffness k are both compressed in the equilibrium position \(\theta=0\). Determine an expression for the period \(\tau\) of small oscillations about O. The mechanism has a mass m with mass center G, and the radius of gyration of the assembly about O is \(k_{O}\).

When the motor is slowly brought up to speed, a rather large vibratory oscillation of the entire motor about O-O occurs at a speed of 360 rev/min, which shows that this speed corresponds to the natural frequency of free oscillation of the motor. If the motor has a mass of 43 kg and radius of gyration of 100 mm about O-O, determine the stiffness k of each of the four identical spring mounts.

The system of Prob. 8 /35 is repeated here with the added information that link AOB now has mass \(m_{3}\) and radius of gyration \(k_{O}\) about point O. Ignore friction and derive the differential equation of motion for the system shown in terms of the variable \(x_{1}\).

Determine the value \(m_{\text {eff }}\) of the mass of system (b) so that the frequency of system (b) is equal to that of system (a). Note that the two springs are identical and that the wheel of system (a) is a solid homogeneous cylinder of mass \(m_{2}\). The cord does not slip on the cylinder.

The system of Prob. 8 /43 is repeated here. If the crank AB now has mass m2 and a radius of gyration kO about point O, determine expressions for the undamped natural frequency \(\omega_{n}\) and the damping ratio \(\zeta\) in terms of the given system properties. Assume small oscillations. The damping coefficient for the damper is c.

The two masses are connected by an inextensible cable which passes securely over the periphery of the cylindrical pulley of mass \(m_{3}\), radius r, and radius of gyration \(k_{O}\). Determine the equation of motion for the system in terms of the variable x. State the critical driving frequency \(\omega_{c}\) of the block B which will result in excessively large oscillations of the assembly. Evaluate for \(m_{1}=25 \mathrm{kg}, m_{2}=10 \mathrm{kg}\), \(m_{3}=15 \mathrm{kg}, k_{1}=450 \mathrm{N} / \mathrm{m}, k_{2}=300 \mathrm{N} / \mathrm{m}\), r = 300 mm, and \(k_{O}=200 \mathrm{mm}\).

The lower spring of Prob. 8 /91 is replaced by a damper. Determine the equation of motion for the system in terms of the variable x. State the value of the viscous damping coefficient c which will give a damping ratio = 0.2. Evaluate for \(m_{1}=25 \mathrm{kg}, m_{2}=10 \mathrm{kg}\), \(m_{3}=15 \mathrm{kg}, k = 450 N /m, r = 300 mm, and \(k_{O}=200 \mathrm{mm}\).

The uniform solid cylinder of mass m and radius r rolls without slipping during its oscillation on the circular surface of radius R. If the motion is confined to small amplitudes \(\theta=\theta_{0}\), determine the period \(\tau\) of the oscillations. Also determine the angular velocity \(\omega\) of the cylinder as it crosses the vertical centerline. (Caution: Do not confuse \(\omega\) with \(\dot{\theta}\) or with \(\omega_{n}\) as used in the defining equations. Note also that \(\theta\) is not the angular displacement of the cylinder.)

The cart B is given the harmonic displacement \(x_{B}=b \sin \omega t\). Determine the steady-state amplitude ? of the periodic oscillation of the uniform slender bar which is pinned to the cart at P. Assume small angles and neglect friction at the pivot. The torsional spring is undeformed when \(\theta=0\).

The assembly of Prob. 8 /40 is repeated here with the additional information that body ABC now has mass \(m_{4}\) and a radius of gyration \(k_{O}\) about its pivot at O, about which it is balanced. If a harmonic torque \(M=M_{0} \cos \omega t\) is applied to body ABC, determine the equation of motion for the system in terms of the variable x. State the critical frequency \(\omega_{c}\) of the harmonic torque which will result in an excessively large system response. Evaluate \(\omega_{c}\) for \(m_{1}=15 \mathrm{kg}\), \(m_{2}=12 \mathrm{kg}, m_{3}=8 \mathrm{kg}, m_{4}=6 \mathrm{kg}, k_{1}=400 \mathrm{N} / \mathrm{m}\), \(k_{2}=650 \mathrm{N} / \mathrm{m}, k_{3}=225 \mathrm{N} / \mathrm{m}, c_{1}=44 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}, c_{2}=36\mathrm{N} \cdot \mathrm{s} / \mathrm{m}, c_{3}=52 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}\), a = 1.2 m, b = 1.8 m, c = 0.9 m, and \(k_{O}=0.75 \mathrm{m}\). What is the damping ratio \(\zeta\) for these conditions?

The elements of the “swing-axle” type of independent rear suspension for automobiles are depicted in the figure. The differential D is rigidly attached to the car frame. The half-axles are pivoted at their inboard ends (point O for the half-axle shown) and are rigidly attached to the wheels. Suspension elements not shown constrain the wheel motion to the plane of the figure. The weight of the wheel– tire assembly is W = 100 lb, and its mass moment of inertia about a diametral axis passing through its mass center G is \(1 \text { lb-ft-sec }\). The weight of the half-axle is negligible. The spring rate and shock-absorber damping coefficient are k = 50 lb/in. and c = 200 lb-sec/ft, respectively. If a static tire imbalance is present, as represented by the additional concentrated weight w = 0.5 lb as shown, determine the angular velocity \(\omega\) which results in the suspension system being driven at its undamped natural frequency. What would be the corresponding vehicle speed v? Determine the damping ratio \(\zeta\). Assume small angular deflections and neglect gyroscopic effects and any car frame vibration. In order to avoid the complications associated with the varying normal force exerted by the road on the tire, treat the vehicle as being on a lift with the wheels hanging free.

The 1.5-kg bar OA is suspended vertically from the bearing O and is constrained by the two springs each of stiffness k = 120 N/m and both equally precompressed with the bar in the vertical equilibrium position. Treat the bar as a uniform slender rod and compute the natural frequency \(f_{n}\) of small oscillations about O.

The light rod and attached sphere of mass m are at rest in the horizontal position shown. Determine the period \(\tau\) for small oscillations in the vertical plane about the pivot O.

A uniform rod of mass m and length l is welded at one end to the rim of a light circular hoop of radius l. The other end lies at the center of the hoop. Determine the period \(\tau\) for small oscillations about the vertical position of the bar if the hoop rolls on the horizontal surface without slipping.

The spoked wheel of radius r, mass m, and centroidal radius of gyration \(\bar{k}\) rolls without slipping on the incline. Determine the natural frequency of oscillation and explore the limiting cases of \(\bar{k}=0\) and \(\bar{k}=r\).

Determine the period \(\tau\) for the uniform circular hoop of radius r as it oscillates with small amplitude about the horizontal knife edge.

The length of the spring is adjusted so that the equilibrium position of the arm is horizontal as shown. Neglect the mass of the spring and the arm and calculate the natural frequency \(f_{n}\) for small oscillations.

The body consists of two slender uniform rods which have a mass \(\rho\) per unit length. The rods are welded together and pivot about a horizontal axis through O against the action of a torsional spring of stiffness \(k_{T}\). By the method of this article, determine the natural circular frequency \(\omega_{n}\) for small oscillations about the equilibrium position. The spring is undeformed when \(\theta=0\), and friction in the pivot at O is negligible.

By the method of this article, determine the period of vertical oscillation. Each spring has a stiffness of 6 lb/in., and the mass of the pulleys may be neglected.

The homogeneous circular cylinder of Prob. 8/93, repeated here, rolls without slipping on the track of radius R. Determine the period \(\tau\) for small oscillations.

The disk has mass moment of inertia \(I_{O}\) about O and is acted upon by a torsional spring of constant \(k_{T}\). The position of the small sliders, each of which has mass m, is adjustable. Determine the value of x for which the system has a given period \(\tau\).

The uniform slender rod of length l and mass \(m_{2}\) is secured to the uniform disk of radius l/5 and mass \(m_{1}\). If the system is shown in its equilibrium position, determine the natural frequency \(\omega_{n}\) and the maximum angular velocity for small oscillations of amplitude \(\theta_{0}\) about the pivot O.

The assembly shown consists of two sheaves of mass \(m_{1}=35 \mathrm{kg} \text { and } m_{2}=15 \mathrm{kg}\), outer groove radii \(r_{1}=525 \mathrm{mm} \text { and } r_{2}=250 \mathrm{mm}\), and centroidal radii of gyration \(\left(k_{O}\right)_{1}=350 \mathrm{mm} \text { and }\left(k_{O}\right)_{2}=150 \mathrm{mm}\). The sheaves are fitted to a central shaft at O with bearings which allow them to rotate independently of each other. Attached to the central shaft is a carriage of mass \(m_{3}=25 \mathrm{kg}\). Each sheave has an inextensible cable wrapped securely within its outer groove. Each cable is attached to a spring at one end and to a fixed support at the other end. The springs have stiffnesses \(k_{1}=800 \mathrm{N} / \mathrm{m} \text { and } k_{2}=650 \mathrm{N} / \mathrm{m}\). By the method of this article, determine the equation of motion for the system in terms of the variable x and state the period \(\tau\) for small vertical oscillations about the equilibrium position. Neglect friction in the bearings at O.

Derive the natural frequency \(f_{n}\) of the system composed of two homogeneous circular cylinders, each of mass M, and the connecting link AB of mass m. Assume small oscillations.

The rotational axis of the turntable is inclined at an angle \(\alpha\) from the vertical. The turntable shaft pivots freely in bearings which are not shown. If a small block of mass m is placed a distance r from point O, determine the natural frequency \(\omega_{n}\) for small rotational oscillations through the angle \(\theta\). The mass moment of inertia of the turntable about the axis of its shaft is I.

The assembly of Prob. 8 /95 is repeated here without the applied harmonic torque. By the method of this article, determine the equation of motion of the system in terms of x for small oscillations about the equilibrium configuration if \(c_{1}=c_{2}=c_{3}=0\).

The ends of the uniform bar of mass m slide freely in the vertical and horizontal slots as shown. If the bar is in static equilibrium when \(\theta=0\), determine the natural frequency \(\omega_{n}\) of small oscillations. What condition must be imposed on the spring constant k in order that oscillations take place?

The 12-kg block is supported by the two 5-kg links with two torsion springs, each of constant \(k_{T}=500 \mathrm{N} \cdot \mathrm{m} / \mathrm{rad}\), arranged as shown. The springs are sufficiently stiff so that stable equilibrium is established in the position shown. Determine the natural frequency \(f_{n}\) for small oscillations about this equilibrium position.

The front-end suspension of an automobile is shown. Each of the coil springs has a stiffness of 270 lb/in. If the weight of the front-end frame and equivalent portion of the body attached to the front end is 1800 lb, determine the natural frequency \(ƒ_{n}\) of vertical oscillation of the frame and body in the absence of shock absorbers. (Hint: To relate the spring deflection to the deflection of the frame and body, consider the frame fixed and let the ground and wheels move vertically.)

The system shown features a nonlinear spring whose resisting force F increases with deflection from the neutral position according to the graph shown. Determine the equation of motion for the system by the method of this article.

The semicircular cylindrical shell of radius r with small but uniform wall thickness is set into small rocking oscillation on the horizontal surface. If no slipping occurs, determine the expression for the period \(\tau\) of each complete oscillation.

A hole of radius R/4 is drilled through a cylinder of radius R to form a body of mass m as shown. If the body rolls on the horizontal surface without slipping, determine the period \(\tau\) for small oscillations.

The quarter-circular sector of mass m and radius r is set into small rocking oscillation on the horizontal surface. If no slipping occurs, determine the expression for the period \(\tau\) of each complete oscillation.

Determine the natural frequency \(ƒ_{n}\) of the inverted pendulum. Assume small oscillations, and note any restrictions on your solution.

Determine the period of small oscillations for the uniform sector of mass m. The torsional spring of modulus \(k_{T}\) is undeflected when the sector is in the position shown.

The 0.1-kg projectile is fi red into the 10-kg block which is initially at rest with no force in the spring. The spring is attached at both ends. Calculate the maximum horizontal displacement X of the spring and the ensuing period of oscillation of the block and embedded projectile.

The uniform circular disk is suspended by a socket (not shown) which fi ts over the small ball attachment at O. Determine the frequency of small motion if the disk swings freely about (a) axis A-A and (b) axis B-B. Neglect the small offset, mass, and friction of the ball.

Determine the natural frequency \(ƒ_{n}\) for small oscillations of the semicircular slender rod about the equilibrium position if (a) r = 150 mm, (b) r = 300 mm, and (c) r = 600 mm. Motion takes place in a vertical plane and friction in the pivot at O is negligible.

The triangular frame is constructed of uniform slender rod and pivots about a horizontal axis through point O. Determine the critical driving frequency \(\omega_{c}\) of the block B which will result in excessively large oscillations of the assembly. The total mass of the frame is m.

A linear oscillator with mass m, spring constant k, and viscous damping coefficient c is set into motion when released from a displaced position. Derive an expression for the energy loss Q during one complete cycle in terms of the amplitude \(x_{1}\) at the start of the cycle. (See Fig. 8 /7.)

Calculate the damping ratio of the system shown if the weight and radius of gyration of the stepped cylinder are W = 20 lb and \(\bar{k}=5.5 \mathrm{in}\)., the spring constant is k = 15 lb/in., and the damping coefficient of the hydraulic cylinder is c = 2 lb-sec/ft. The cylinder rolls without slipping on the radius r = 6 in. and the spring can support tension as well as compression.

Determine the value of the viscous damping coefficient c for which the system is critically damped. The cylinder mass is m = 2 kg and the spring constant is k = 150 N/m. Neglect the mass and friction of the pulley.

The seismic instrument shown is secured to a ship’s deck near the stern where propeller-induced vibration is most pronounced. The ship has a single three-bladed propeller which turns at 180 rev/ min and operates partly out of water, thus causing a shock as each blade breaks the surface. The damping ratio of the instrument is \(\zeta=0.5\), and its undamped natural frequency is 3 Hz. If the measured amplitude of A relative to its frame is 0.75 mm, compute the amplitude \(\delta_{0}\) of the vertical vibration of the deck.

The square plate of side length 2b pivots about its center point O against the action of four springs, each of stiffness k. Each spring is attached to the corner of the plate at one end and to the center of a slotted collar at the other end. As the plate rotates, a rod slides through the smooth slot in the collar and deflects the spring. Determine the frequency \(ƒ_{n}\) of small oscillations about the equilibrium configuration.

An experimental engine weighing 480 lb is mounted on a test stand with spring mounts at A and B, each with a stiffness of 600 lb/in. The radius of gyration of the engine about its mass center G is 4.60 in. With the motor not running, calculate the natural frequency \(\left(f_{n}\right)_{y}\) of vertical vibration and \(\left(f_{n}\right)_{\theta}\) of rotation about G. If vertical motion is suppressed and a light rotational imbalance occurs, at what speed N should the engine not be run?

The uniform bar of mass M and length l has a small roller of mass m with negligible bearing friction at each end. Determine the period \(\tau\) of the system for small oscillations on the curved track. The radius of gyration of the rollers is negligible.

A 200-kg machine rests on four floor mounts, each of which has an effective spring constant k = 250 kN/m and an effective viscous damping coefficient c = 1000 N? s /m. The floor is known to vibrate vertically with a frequency of 24 Hz. What would be the effect on the amplitude of the absolute machine oscillation if the mounts were replaced with new ones which have the same effective spring constant but twice the effective damping coefficient?

The mass of a critically damped system having a natural frequency \(\omega_{n}=4 \mathrm{rad} / \mathrm{s}\) is released from rest at an initial displacement \(x_{0}\). Determine the time t required for the mass to reach the position \(x=0.1 x_{0}\).

The uniform sector of Prob. 8 /77 is repeated here with m = 4 kg, r = 325 mm, and \(\beta=45^{\circ}\). If the sector is released from rest with \(\theta_{0}=90^{\circ}\), plot the value of for the time period \(0 \leq t \leq 6 \mathrm{s}\). Friction in the pivot at O results in a resistive torque of magnitude \(M=c \dot{\theta}\), where the constant c = 0.35 N?m? s /rad. Compare your large-angle results with those for the small-angle approximation of sin \(\theta \cong \theta\) and state the value of \(\theta\) when t = 1 s for both large-angle and small-angle cases. (Note: The solution to this problem is of considerable difficulty and involves elliptic integrals. A numerical solution utilizing appropriate mathematics software is recommended.

The mass of the system shown is released with the initial conditions \(x_{0}=0.1 \mathrm{m} \text { and } \dot{x}_{0}=-5 \mathrm{m} / \mathrm{s}\) at t = 0. Plot the response of the system and determine the time(s) (if any) at which the displacement x = ?0.05 m

Shown in the figure are the elements of a displacement meter used to study the motion \(y_{B}=b \sin \omega t\) of the base. The motion of the mass relative to the frame is recorded on the rotating drum. If \(l_{1}=1.2 \mathrm{ft}, l_{2}=1.6 \mathrm{ft}, l_{3}=2 \mathrm{ft}, W=2 \mathrm{lb}, c=0.1 \mathrm{lb}-\mathrm{sec} / \mathrm{ft} \text {, and } \omega=10 \mathrm{rad} / \mathrm{sec}\), determine the range of the spring constant k over which the magnitude of the recorded relative displacement is less than 1.5b. It is assumed that the ratio \(\omega / \omega_{n}\) must remain greater than unity.

The 4-kg mass is suspended by the spring of stiff- ness k = 350 N/m and is initially at rest in the equilibrium position. If a downward force F = Ct is applied to the body and reaches a value of 40 N when t = 1 s, derive the differential equation of motion, obtain its solution, and plot the displace- ment y in millimeters as a function of time during the first second. Damping is negligible.

Plot the response x of the 50-lb body over the time interval \(0 \leq t \leq 1\) second. Determine the maximum and minimum values of x and their respective times. The initial conditions are \(x_{0}=0\) and \(0 \leq t \leq 1\)

Determine and plot the response x as a function of time for the undamped linear oscillator subjected to the force F which varies linearly with time for the first \(\frac{3}{4}\) second as shown. The mass is initially at rest with x = 0 at time t = 0.

The 4-kg cylinder is attached to a viscous damper and to the spring of stiffness k=800N/m. If the cylinder is released from rest at time t = 0 from the position where it is displaced a distance y = 100 mm from its equilibrium position, plot the displacement y as a function of time for the first second for the two cases where the viscous damping coefficient is (a) c=124 \(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}\) and (b) \(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}\).