lecture1210

# lecture1210 - ME 563 Mechanical Vibrations Lecture#12...

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Unformatted text preview: ME 563 Mechanical Vibrations Lecture #12 Multiple Degree of Freedom Free Response + MATLAB Free Response 1 We can solve for the homogeneous solution to a coupled set of equations in a multiple degree of freedom linear system by: - Identifying the initial conditions on all the states - Assuming a solution of the form {x(t)}={A}e st What does this last assumption imply about the response? Free Response 1 We can solve for the homogeneous solution to a coupled set of equations in a multiple degree of freedom linear system by: - Identifying the initial conditions on all the states - Assuming a solution of the form {x(t)}={A}e st What does this last assumption imply about the response? Two DOF System 2 Consider the two degree of freedom system of equations: If we make a solution of the form, , as we did for the single DOF case, we obtain: Non-trivial solutions satisfy: Ms 2 + 2 Cs + 2 K ( ) ⋅ Ms 2 + Cs + K ( ) − Cs + K ( ) 2 = Characteristic Equation 3 There are four solutions that satisfy the characteristic equation and these solutions are expressed as follows when the modal frequencies are underdamped: The solution to the homogeneous equation is then written as follows (for the first two complex conjugate roots): s 1 = σ 1 +j ω 1 , s 2 = σ 1-j ω 1 , s 3 = σ 2 +j ω 2 , and s 4 = σ 2-j ω 2 Real part: decay rate Imaginary part: oscillation frequency Free Response Form...
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lecture1210 - ME 563 Mechanical Vibrations Lecture#12...

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