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Vibrations

Compute a value of the damping coefficient c such that the steady state response ampplitude of the system in the figure is 0.01 m.

Find the equation of motion (EOM) in terms of K, l, m, and/or for the following systems. Show your free body diagram.

Using the Laplace transform method, calculate the response of the system of example for the overdamped case. Plot the response for m=1 kg, k=100 N/m, and z=1.5.

Write the equation of motion in matrix form for the system shown below.

Two subway cars of the figure have a 2000 kg mass each and are connected by a coupler. The coupler can be modeled as a spring of stiffness k=280,000 N/m. Write the equation of motion and calculate the natural frequencies and mode shapes.

Calculate the solution to xdoubledot + 2xdot + 2x = t-pi for x(0)=1 and xdot(0)=1, and plot the response.

Given the following EOM: 5xdoubdot + 100x=3coswt a) Calculate the maximum amplitude (0-peak) of vibration due only to the forcing function when w=2.0 rad/sec. Is the vibration response in or out of phase with the forcing function? Explain your answer. b) Calculate the maximum amplitude of vibration due only to the forcin function when w=6.0 rad/sec. Is the vibration response in or out of phase with the forcing function? Explain your answer.

Consider the disk in the figure below connected to two springs. Use the energy method to calculate the system's natural frequency of oscillation for small angles theta(t).

Find the equation of motion (EOM) in terms of K, l, m, and/or for the following systems. Show your free body diagram.

Using the Laplace transform method, calculate the response of the system of example for the overdamped case. Plot the response for m=1 kg, k=100 N/m, and z=1.5.

Write the equation of motion in matrix form for the system shown below.

Two subway cars of the figure have a 2000 kg mass each and are connected by a coupler. The coupler can be modeled as a spring of stiffness k=280,000 N/m. Write the equation of motion and calculate the natural frequencies and mode shapes.

Calculate the solution to xdoubledot + 2xdot + 2x = t-pi for x(0)=1 and xdot(0)=1, and plot the response.

Given the following EOM: 5xdoubdot + 100x=3coswt a) Calculate the maximum amplitude (0-peak) of vibration due only to the forcing function when w=2.0 rad/sec. Is the vibration response in or out of phase with the forcing function? Explain your answer. b) Calculate the maximum amplitude of vibration due only to the forcin function when w=6.0 rad/sec. Is the vibration response in or out of phase with the forcing function? Explain your answer.

Consider the disk in the figure below connected to two springs. Use the energy method to calculate the system's natural frequency of oscillation for small angles theta(t).

Determine the equation of motion (EOM) for the following systems A, B, and C. A) What is the natural frequency in terms of E, I, L, and M? consider the beam mass to be negligible. B) What is the natural frequency in terms of E, I, L, and M? Consider the beam mass to be negligible. C) What is the natural frequency in terms of G, J (polar area MOI), L, and Jp (wheel polar mass MOI)? Consider the rod mass MOI to be negligible.

Using the solution of equation in the form x(t)=Bsinwt + Ccoswt calculate the values of B and C in terms of the initial conditions x0 and v0.

Plot x(t) for a damped system of natural frequency wn=2 rad/s and initial conditions x0=1mm, v0=1mm, for the folloiwng values of the damping ratio: 0.01, 0.2, 0.1, 0.4, and 0.8. For each of the values of the damping ratio, is the system underdamped, critically damped, or overdamped? What happened to the magnitude of the maximum amplitude of the response as the damping ratio increased in this example? Explain your answer.

A 2-kg mass connected to a spring of stiffness 10^3 N/m has a dry sliding friction force (Fd) of 3 N. As the mass oscillates, its amplitude decreases 20 cm. How long does this take?

A harmonic force of maximum value of 25 N and frequency of 180 cycles/min acts on a machine of 25 kg mass. Design a support system for the machine (i.e., choose c, k) so that only 10% of the force applied to the machine is transmitted to the base supporting the machine.

Calculate the reponse of an underdamped system to the excitation given in the figure. Plot of a pules input of the form f(t)=F0sint.

Calculate the vectors u1 and u2 for the previous problem.

Using a word processor (not hand written), explain how to solve the seawall example problem from week 8. Introduce your explanation by clearly stating your goals for the problem.

Using the solution of equation in the form x(t)=Bsinwt + Ccoswt calculate the values of B and C in terms of the initial conditions x0 and v0.

Plot x(t) for a damped system of natural frequency wn=2 rad/s and initial conditions x0=1mm, v0=1mm, for the folloiwng values of the damping ratio: 0.01, 0.2, 0.1, 0.4, and 0.8. For each of the values of the damping ratio, is the system underdamped, critically damped, or overdamped? What happened to the magnitude of the maximum amplitude of the response as the damping ratio increased in this example? Explain your answer.

A 2-kg mass connected to a spring of stiffness 10^3 N/m has a dry sliding friction force (Fd) of 3 N. As the mass oscillates, its amplitude decreases 20 cm. How long does this take?

A harmonic force of maximum value of 25 N and frequency of 180 cycles/min acts on a machine of 25 kg mass. Design a support system for the machine (i.e., choose c, k) so that only 10% of the force applied to the machine is transmitted to the base supporting the machine.

Calculate the reponse of an underdamped system to the excitation given in the figure. Plot of a pules input of the form f(t)=F0sint.

Calculate the vectors u1 and u2 for the previous problem.

Using a word processor (not hand written), explain how to solve the seawall example problem from week 8. Introduce your explanation by clearly stating your goals for the problem.

Using a word processor (not hand written), explain the step by step procedure for solving the in class problem about the manufacturing system with a triangular base motion from the cam. Explain how to set up the problem and how to solve for the response.

Reproduce the figure for the various time steps indicated.

A vibrating mass of 300 kg, mounted on a massless support by a spring of stiffness 40,000 N/m and a damper of unknown damping coefficient, is observed to vibrate with a 10-mm amplitude while the support vibration has a maximum amplitude of only 2.5 mm (at resonance). Calculate the damping constant and the amplitude of the force on the base.

Given the following equation of motion for the system in the figure: xdoubdot + 50 xdot + 2000x = F(t) , solve for the steady state system response (Xp(t)) when F(t)=400 sin(62t).

To familiarize yourself with the nature of the forced response, plot the solution of a forced response of equation with w=2 rad/s, given by equation for a variety of values of the initial conditions and wn as given in the following chart.

A manufacturer makes a cantilevered leaf spring from steel (E= 2x10^11 N/m^2) and sizes the spring so that the decive has a specific frequency. Later, to save weight, the spring is made of aluminum (E=7.1x10^10 N/m^2). Assuming that the mass of the spring is much smaller than that of the device the spring is attached to, determine if the frequency increases or decreases and by how much.

It is important to understand the difference between area moment of inertia and mass MOI and when to use each. Do the following: A) From 1 C) above, if the steel rod has a radius=2 in, and L=6 in, what is the torsional stiffness of the rod, Krod? B) from 1 C) above, if the radius of the aluminum wheel is 4 inches, and the wheel thickness is t=0.5 in, what is Jp? C) Using these values, determine the undamped natural frequency of the system.

In your own words describe the difference between linear and non-linear systems.

Determine the Fourier series representation of the sawtooth curve illustrated in the figure.

Reproduce the figure for the various time steps indicated.

A vibrating mass of 300 kg, mounted on a massless support by a spring of stiffness 40,000 N/m and a damper of unknown damping coefficient, is observed to vibrate with a 10-mm amplitude while the support vibration has a maximum amplitude of only 2.5 mm (at resonance). Calculate the damping constant and the amplitude of the force on the base.

Given the following equation of motion for the system in the figure: xdoubdot + 50 xdot + 2000x = F(t) , solve for the steady state system response (Xp(t)) when F(t)=400 sin(62t).

To familiarize yourself with the nature of the forced response, plot the solution of a forced response of equation with w=2 rad/s, given by equation for a variety of values of the initial conditions and wn as given in the following chart.

A manufacturer makes a cantilevered leaf spring from steel (E= 2x10^11 N/m^2) and sizes the spring so that the decive has a specific frequency. Later, to save weight, the spring is made of aluminum (E=7.1x10^10 N/m^2). Assuming that the mass of the spring is much smaller than that of the device the spring is attached to, determine if the frequency increases or decreases and by how much.

It is important to understand the difference between area moment of inertia and mass MOI and when to use each. Do the following: A) From 1 C) above, if the steel rod has a radius=2 in, and L=6 in, what is the torsional stiffness of the rod, Krod? B) from 1 C) above, if the radius of the aluminum wheel is 4 inches, and the wheel thickness is t=0.5 in, what is Jp? C) Using these values, determine the undamped natural frequency of the system.

In your own words describe the difference between linear and non-linear systems.

Determine the Fourier series representation of the sawtooth curve illustrated in the figure.

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