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Chapter 8: Problem 9

Consider the system of Problem \(6.15,7.5\) and 8.1. Use modal analysis to determine the steady state response of the system when \(F(t)=F_{0} \cos \Omega t\).

Short Answer

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#Short Answer Question# Describe the steps involved in determining the steady-state response of a system subjected to an applied force \(F(t) = F_0 \cos{\Omega t}\) using modal analysis.

Step by step solution

01

Determine the system's natural frequencies and mode shapes

To determine the system's natural frequencies and mode shapes, first find the system's mass and stiffness matrices. Then, solve the eigenvalue problem formed by these matrices to find the natural frequencies and mode shapes.
02

Compute the response of each mode to the applied force

For each mode, compute the response to the applied force \(F(t) = F_0 \cos{\Omega t}\) using the mode shape and natural frequency. This can be done by solving a single-degree-of-freedom problem for each mode. For the single-degree-of-freedom problem, the equation of motion is: $$ m \ddot{x} + kx = F(t) $$ where \(m\) is the effective mass, \(k\) the effective stiffness, and \(x\) the modal coordinate. Transform this equation into a frequency domain equation using the Laplace Transform, and then solve for the modal coordinate, \(X(\omega)\).
03

Transform the individual modal responses back to the time domain

For each modal coordinate \(X(\omega)\), determine the inverse Laplace Transform to obtain the individual modal responses in the time domain. This will provide the response of each mode in the time domain as a function of the applied force, \(F(t)\).
04

Sum the individual modal responses to find the total response

To find the total response of the system in the time domain, sum the individual modal responses. This will provide the desired steady-state response of the system when subjected to the applied force, \(F(t)\). The steady-state response of the system can be presented in the form: $$ x(t) = \sum_{i=1}^n x_i(t) $$ where \(x_i(t)\) is the response of the \(i\)-th mode, and \(n\) is the number of modes. Following these steps will allow you to determine the steady-state response of the system when subjected to the applied force \(F(t) = F_0 \cos{\Omega t}\) using modal analysis.

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Most popular questions from this chapter

An automobile is traveling at constant speed \(v_{0}\) along a flat road when it encounters as speed bump described by \(y(\xi)=-h_{0}[1-\cos (2 \pi \xi / \lambda)]\) where \(\lambda\) is the wavelength of the bump and \(h_{0}\) is the height. Determine the motion of the car as it traverses the bump. Assume that the frame of the car may be modeled as a uniform rod of mass \(m\) and length \(L\) and the combined stiffness of the tires and suspension in both the front and the back is \(k\).

An automobile is traveling at constant speed \(v_{0}\) over a buckled road. Determine the motion of the car if the buckle is described as \(y(x)=h_{0} \sin (2 \pi x / \lambda)\) where \(\lambda\) is the wavelength of the buckle and \(h_{0}\) is its rise. Assume that the frame of the car may be modeled as a uniform rod of mass \(m\) and length \(L\) and the combined stiffness of the tires and suspension in both the front and the back is \(k\).

Consider the floating platform of Problems \(6.24\) and \(7.18\) when \(m_{b}=m_{a} / 2\) and \(k=\rho_{f} g R^{2} / 2\). Determine the response of the system if \(F(t)=F_{0} \sin \Omega t\).

Consider the linked system of Problems \(6.22\) and \(7.16\) when \(k_{1}=k_{2}=k\) and \(2 m_{2}\) \(=m_{1}=m\). Determine the response of the system if the upper link is subjected to a (temporally) symmetric triangular pulse of magnitude \(F_{0}\) and duration \(\tau^{2}=m / k\).
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