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Unformatted text preview: ME 563 Mechanical Vibrations Lecture #10 Single Degree of Freedom Free Response Free Response 1 When solving the homogeneous equation of motion (forcing function = 0), we are finding the free response. One way to solve for the free response is as follows:  Identify the initial conditions on all the states  Assume a solution of the form x(t)=Ae st Disc on An Incline 2 Consider the disc on an incline with no forcing function: Recall x d is called the dynamic displacement. If we make a solution of the form, , we obtain: Characteristic Equation 3 The only solutions that satisfy the equation of motion that are not trivial ( A =0) must also satisfy the characteristic equation : The solution to the homogeneous equation are then written as follows: Roots, poles, modal frequencies Because the equation is linear, the solutions superimpose Initial Conditions 4 It is important to note that the modal frequencies of the system are only a function of the mass, damping, and stiffness parameters and not a function of the initial conditions or...
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This document was uploaded on 12/23/2011.
 Fall '09

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