Figure P13.7 shows a system being driven by base

The 0.5-kg mass shown in Figure P13.1 is attached to the frame with a spring of stiffness k = 500 N/m. Neglect the spring weight and any damping. The frame oscillates vertically with an amplitude of 4 mm at a frequency of 3 Hz. Compute the steady-state amplitude of motion of the mass.

A quarter-car representation of a certain car has a stiffness k = 2000 lb/ft, which is the series combination of the tire stiffness and suspension stiffness, and a damping constant of c = 360 lb-sec/ft. The car weighs 2000 lb. Suppose the road profile is given (in feet) by \(y(t) = 0.03 \ \sin \ \omega t\), where the amplitude of variation of the road surface is 0.03 ft, and the frequency \(\omega\) depends on the vehicle’s speed and the road profile’s period. Suppose the period of the road surface is 20 ft. Compute the steady-state motion amplitude and the force transmitted to the chassis, if the car is traveling at speeds of 20 and 50 mi/hr.

A certain factory contains a heavy rotating machine that causes the factory floor to vibrate. We want to operate another piece of equipment nearby and we measure the amplitude of the floor’s motion at that point to be 0.01 m. The mass of the equipment is 1500 kg and its support has a stiffness of \(k = 2 \times 10^4 \ \mathrm{N/m}\) and a damping ratio of \(\zeta = 0.04\). Calculate the maximum force that will be transmitted to the equipment at resonance.

An electronics module inside an aircraft must be mounted on an elastic pad to protect it from vibration of the airframe. The largest amplitude vibration produced by the airframe’s motion has a frequency of 40 Hz. The module weighs 200 N, and its amplitude of motion is limited to 0.003 m to save space. Neglect damping and calculate the percent of the airframe’s motion transmitted to the module.

An electronics module used to control a large crane must be isolated from the crane’s motion. The module weighs 2 lb. (a) Design an isolator so that no more than 10% of the crane’s motion amplitude is transmitted to the module. The crane’s vibration frequency is 3000 rpm. (b) What percentage of the crane’s motion will be transmitted to the module if the crane’s frequency can be anywhere between 2500 and 3500 rpm?

An instrument weighing 5 lb is mounted on the housing of a pump that rotates at 30 rpm. The amplitude of motion of the housing is 0.003 ft. We want no more than 10% of the housing’s motion to be transmitted to the instrument. Design a suitable isolator having negligible damping. Compute the force transmitted to the instrument.

Figure P13.7 shows a system being driven by base excitation through a damping element. Assume that the base displacement is sinusoidal: \(y(t) = Y \ \sin \ \omega t\). a. Derive the expression for X, the steady-state amplitude of motion of the mass m. b. Derive the expression for \(F_t\) , the steady-state amplitude of the force transmitted to the support.

Consider the vehicle suspension problem in Example 13.1.1. Investigate the choice of the damping constant c so that the displacement transmissibility X/Y is as small as possible for the case where the speed gives a frequency ratio of r = 2. Do this by trying different values for the damping ratio \(\zeta\) . Investigate your design’s sensitivity to a \(\pm 20 \%\) variation in r.

Consider the vehicle suspension problem in Example 13.1.1. Suppose the amplitude of variation of the road surface is 5/8 in. Determine a set of values for the suspension’s stiffness k and damping c so that the force transmitted to the chassis will be as small as possible at 40 mi/hr. For your design, calculate the transmitted force at 40 mi/hr.

Alternating-current motors are often designed to run at a constant speed, typically either 1750 or 3500 rpm. One such motor for a power tool weighs 20 lb and is to be mounted at the end of a steel cantilever beam. Static-force calculations and space considerations suggest that a beam 6 in. long, 4 in. wide, and 3/8 in. thick would be suitable. Use \(E = 3 \times 10^7 \ \mathrm{psi}\) for the modulus of elasticity of steel. Use \(\rho = 15.2 \ \mathrm{slug/ft}^3\) for the density of steel. The rotating part of the motor weighs 1 lb and has an eccentricity of 0.01 ft. (This distance can be determined by standard balancing methods or applying the analysis in this example in reverse, using a vibration test.) The damping ratio for such beams is difficult to determine but is usually very small, say, \(\zeta \leq 0.1\). Estimate the amplitude of vibration of the beam at steady state. Consider both speeds.

When a certain motor is started, it is noticed that its supporting frame begins to resonate when the motor speed passes through 900 rpm. At the operating speed of 1750 rpm the support oscillates with an amplitude of 8 mm. Determine the amplitude that would result at 1750 rpm if the support were replaced with one having one-half the stiffness.

A 500-lb motor is supported by an elastic pad that deflects 0.25 in. when the motor is placed on it. When the motor operates at 1750 rpm, it oscillates with an amplitude of 0.1 in. Suppose a 1500-lb platform is placed between the motor and the pad. Compute the oscillation amplitude that would result at 1750 rpm.

A certain pump weighs 50 lb and has a rotating unbalance. The unbalanced weight is 0.05 lb and has an eccentricity of 0.1 in. The pump rotates at 1000 rpm. Its vibration isolator has a stiffness of k = 500 lb/ft. Compute the force transmitted to the foundation if the isolator’s damping ratio is (a) \(\zeta = 0.05\) and (b) \(\zeta = 0.7\).

To calculate the effects of rotating unbalance, we need to know the value of the product \(m_{u} \epsilon\), where \(m_u\) is the unbalanced mass and \(\epsilon\) is the eccentricity. These two quantities are sometimes difficult to calculate separately, but sometimes an experiment can be performed to estimate the product \(m_{u} \epsilon\). An experiment was performed on a particular rotating machine whose mass is 75 kg. The machine’s support has negligible damping and a stiffness of k = 2500 N/m. When the machine operates at 200 rpm, the measured force transmitted to the foundation was 15 N. Estimate the value of \(m_{u} \epsilon\).

A computer disk drive is mounted to the computer’s chassis with an isolator consisting of an elastic pad. The disk drive motor weighs 3 kg and runs at 3000 rpm. Calculate the pad stiffness required to provide a 90% reduction in the force transmitted from the motor to the chassis.

Figure P13.16 shows a motor mounted on four springs (the second pair of springs is behind the front pair and is not visible). Each spring has a stiffness k = 2000 N/m. The distance D is 0.2 m. The inertia of the motor is \(I = 0.2 \ \mathrm{kg} \cdot \mathrm{m}^2\); its mass is m = 25 kg, and its speed is 1750 rpm. Because the motor mounts allow the motor to vibrate both vertically and rotationally, the system has a vertical force transmissibility \(F_{t}/F\) and a torque transmissibility \(T_{t}/T\) , where F is the vertical unbalance force and T is the torque due to the unbalance. Neglect damping in the system and obtain the vertical force transmissibility \(F_{t}/F\) and the torque transmissibility \(T_{t}/T\) .

A motor mounted on a cantilever beam weighs 20 lb and runs at the constant speed of 3500 rpm. The steel beam is 6 in. long, 4 in. wide, and 3/8 in. thick. The unbalanced part of the motor weighs 1 lb and has an eccentricity of 0.01 ft. The damping in the beam is very slight. Design a vibration absorber for this system. The available clearance for the absorber’s motion is 0.25 in. Use \(E = 3 \times 10^7 \ \mathrm{psi}\) for the modulus of elasticity of steel. Use \(\rho = 15.2 \ \mathrm{slug/ft}^3\) for the density of steel.

A motor mounted on a beam vibrates too much when it runs at a speed of 6000 rpm. At that speed the measured force produced on the beam is 60 lb. Design a vibration absorber to attach to the beam. Because of space limitations, the absorber’s mass cannot have an amplitude of motion greater than 0.08 in.

The supporting table of a radial saw weighs 160 lb. When the saw operates at 200 rpm it transmits a force of 4 lb to the table. Design a vibration absorber to be attached underneath the table. The absorber’s mass cannot vibrate with an amplitude greater than 1 in.

A certain machine of mass 8 kg with supports has an experimentally determined natural frequency of 6 Hz. It will be subjected to a rotating unbalance force with an amplitude of 50 N and a frequency of 4 Hz. a. Design a vibration absorber for this machine. The available clearance for the absorber's motion is 0.1 m. b. Let \(x_1\) be the vertical displacement of the machine. The amplitude of the rotating unbalance force is \(m_u \in \omega^2\). Plot \(X_1 / m_u \epsilon\) versus the frequency \(\omega\), and use the plot to discuss the sensitivity of the absorber to changes in the frequency \(\omega\).

The operating speed range of a certain motor is from 1500 to 3000 rpm. The motor and its mount vibrate excessively at 2100 rpm. When a vibration absorber weighing 5 lb and tuned to 2100 rpm was attached to the motor, resonance occurred at 1545 and 2850 rpm. Design a more effective absorber that will yield resonant frequencies outside the operating speed range of the motor.

Figure P13.22 shows another type of vibration absorber that uses only mass and damping, and not stiffness, to reduce vibration. The main mass is \(m_1\) and the absorber's mass is \(m_2\). Suppose the applied force f(t) is sinusoidal. a. Derive the expressions for \(X_1(j \omega) / F(j \omega)\) and \(X_2(j \omega) / F(j \omega)\). b. Use these expressions to discuss the selection of values for \(m_2\) and c to minimize the motion of mass \(m_1\). To aid in your discussion, plot \(k X_1 / F\) versus \(\omega / \sqrt{k / m_1}\) for several values of \(\zeta=c / 2 \sqrt{k m_1}\).

Figure P13.23 shows another type of vibration absorber that uses mass, stiffness, and damping to reduce vibration. The damping can be used to reduce the amplitude of motion near resonance. The main mass is \(m_1\) and the absorber's mass is \(m_2\). Suppose the applied force f(t) is sinusoidal. a. Derive the expressions for \(X_1(j \omega) / F(j \omega)\) and \(X_2(j \omega) / F(j \omega)\). b. Use these expressions to discuss the selection of values for \(m_2, k_2\), and c to minimize the motion of mass \(m_1\). To aid in your discussion, plot \(k_1 X_1 / F\) versus \(\omega / \sqrt{k_1 / m_1}\) for several values of \(\zeta=c / 2 \sqrt{k_1 m_1}\).

Find and interpret the mode ratios for the system shown in Figure P13.24. The masses are \(m_1=10 \mathrm{~kg}\) and \(m_2=30 \mathrm{~kg}\). The spring constants are \(k_1=10^4 \mathrm{~N} / \mathrm{m}\) and \(k_2=2 \times 10^4 \mathrm{~N} / \mathrm{m}\).

Find and interpret the mode ratios for the coupled pendulum system shown in Figure P13.25. Use the values \(m_1 = 1, m_2 = 4, L_1 = 2, L_2 = 5\), and k = 2.

Find and interpret the mode ratios for the torsional system shown in Figure P13.26. Use the values \(I_1 = 1, I_2 = 5, k_1 = 1\), and \(k_2 = 3\).

Find and interpret the mode ratios for the system shown in Figure P13.27.

For the roll-pitch vehicle model described in Example 13.4.2, the suspension stiffnesses are to be changed to \(k_1 = 1.95 \times 10^4 \ \mathrm{N/m}\) and \(k_2 = 2.3 \times 10^4 \ \mathrm{N/m}\). Find the natural frequencies the mode ratios, and the node locations.

A particular road vehicle weighs 4000 lb. Using the quarter-car model, determine a suitable value for the suspension stiffness, assuming that the tire stiffness is 1300 lb/in.

The vehicle model shown in Figure 13.4.2(a) has the following parameter values: \(\text{weight} = 4800 \ \mathrm{lb}, I_G = 1800 \ \mathrm{slug-ft}^2, L_1 = 3.5 \ \mathrm{ft}\), and \(L_2 = 2.5 \ \mathrm{ft}\). Design the front and rear suspension stiffnesses to achieve good ride quality.

A 125-kg machine has a passive isolation system for which \(c = 5000 \ \mathrm{N} \cdot \mathrm{m/s}\) and \(k = 7 \times 10^6 \ \mathrm{N/m}\). The rotating unbalance force has an amplitude of 100 N with a frequency of 2500 rpm. The resonant frequency of this system is 216 rad/s and is close the frequency of the disturbance. In addition, the damping ratio is small \(\left(\zeta = 0.08\right)\). Assuming that c and k cannot be changed, calculate the control gains required to give a damping ratio of \(\zeta = 0.5\) and a resonant frequency of 100 rad/s, well below the disturbance frequency.

A 20-kg machine has a passive isolation system whose damping ratio is 0.28 and whose undamped natural frequency is 13.2 rad/s. Assuming that the passive system remains in place, calculate the control gains required to give a damping ratio of \(\zeta = 0.707\) and a resonant frequency of 141 rad/s.

With the increased availability of powered wheelchairs, improved suspension designs are required for safety and comfort. One chair uses an active suspension like the one shown in Figure P13.33 for each driving wheel. An actuator exerts a force f between the tire and the vehicle. Assuming that the control algorithm is such that \(f = ?K_P(x_s ? x_w) ? K_D(\dot{x}_ s ? \dot{x}_w)\) derive the equations of motion of the system, neglecting the tire mass and tire damping.

The following model describes a mass supported by a nonlinear spring. The units are SI, so \(g = 9.81 \ \mathrm{m/s}^2\). \(5 \ddot{y} = 5g ?(900y + 1700y^3)\) a. Find the equilibrium position \(y_r\), obtain a linearized model using the equilibrium as the reference operating condition, and compute the oscillation frequency of the linearized model. b. Find the free response of the linearized model in terms of \(x(t) = y(t) ? y_r\), for the initial conditions: x(0) = 0.002 m and \(\dot{x}(0) = 0.005 \ \mathrm{m/s}\).

The following is a model of the velocity of an object subjected to cubic damping. \(m \frac{dv}{dt} = ?cv^3\) Suppose that m = 1 and c = 4. Obtain the solution in terms of the initial condition v(0).

Find the equilibria of equation (13.6.10) for \(m=1, c=12, k_1=16\), and \(k_3=-4\), and use a numerical method to solve and plot the solution for \(\dot{y}(0)=0\) and four values of\(y(0): \pm 1, \pm 1.9\), and \(\pm 2\).1. Compare the results with the stability properties predicted from the linearized model.

Use a numerical method to compute and plot the free response of equation (13.6.10) for \(m=1, c=0, k_1=2\), and \(k_3=0.1\), for \(\dot{y}(0)=0\) and two initial conditions: y(0) =10 and y(0) = 40. Compare the results with those shown in Figure 13.6.4. How does the initial condition affect the frequency of the response?

Plot the phase plane plots for the following equations and the initial conditions: \(x(0)=1, \dot{x}(0)=0\). a. \(\ddot{x}+0.1 \dot{x}+2 x=0\) b. \(\ddot{x}+2 \dot{x}+2 x=0\) c. \(\ddot{x}+4 \dot{x}+2 x=0\)

Plot the phase plane plot for the following equation with the initial conditions: \(y(0)=1, \dot{y}(0)=0\). \(\ddot{y}+2 \dot{y}+2 y+3 y^3=0\)

Plot the phase plane plot and identify the limit cycle for Van der Pol's equation (13.6.13) with \(\mu=5\) and the initial conditions: \(y(0)=1, \dot{y}(0)=0\).

Use MATLAB to obtain the root locus plot of \(5 s^2+c s+45=0\) for \(c \geq 0\).

Use MATLAB to obtain the root locus plot of the equation \(m s^2+c s+k=0\) in terms of the variable \(k \geq 0\). Use the values m = 4, c = 8. What is the smallest possible dominant time constant and the associated value of k?

Use MATLAB to obtain the root locus plot of the equation \(m s^2+c s+k=0\) in terms of the variable \(c \geq 0\). Use the values \(m=4, k=64\). What is the smallest possible dominant time constant and the associated value of c ?

Use MATLAB to obtain the root locus plot of the model \(m \ddot{x}+c \dot{x}+\) \(\left(k_1+k_2\right) x=k_2 y\) in terms of the variable \(k_2 \geq 0\). Use the values \(m=2, c=8, k_1=26\). What is the value of \(k_2\), required to give \(\zeta=0.707\)?

Use MATLAB to obtain the root locus plot of the model \(m \ddot{x}+\left(c_1+c_2\right) \dot{x}+k x=c_2 \dot{y}\) in terms of the variable \(c_2 \geq 0\). Use the values \(m=2, c_1=8, k=26\). What is the smallest possible dominant time constant and the associated value of \(c_2\) ?

Consider the two-mass model shown in Figure P13.46. Use the following numerical values: \(m_1 = m_2 = 1, c_2 = 0, k_1 = 1\), and \(k_2 = 4\). a. Use MATLAB to obtain the root locus plot in terms of the parameter \(c_1\). b. Use the root locus plot to determine the value of \(c_1\) required to give a dominant root pair having a damping ratio of \(\zeta = 0.707\). c. Use the root locus plot to determine the value of \(c_1\) required to give a dominant root that is real and has a time constant equal to 4. d. Using the value of \(c_1\) found in part (c), obtain a plot of the unit-step response.