This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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: Nontrivial 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 1j ω 1 , s 3 = σ 2 +j ω 2 , and s 4 = σ 2j ω 2 Real part: decay rate Imaginary part: oscillation frequency Free Response Form...
View
Full Document
 Fall '09
 Equations, Eigenvalue, eigenvector and eigenspace, Complex number, modal vectors

Click to edit the document details