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I have a question about Multi DOF vibrating systems. For free vibration of undamped MDOF systems, we have the equations of motion as :

[itex]M[/itex] [itex]\ddot{q}[/itex] + [itex]K[/itex] [itex]{q}[/itex] = [itex]{0}[/itex] (1)

Where,

[itex]M[/itex] - n x n mass matrix

[itex]K[/itex] - n x n stiffness matrix

[itex]{q}[/itex] - n x 1 vector of generalized coordinates

Most vibrations book try to obtain the eigenvalue problem by assuming the solution to (1) as

[itex]{q}[/itex] = [itex]{Q}[/itex] [itex]e^{j \omega t}[/itex] (2)

For a 2-DOF system, (2) is

[itex]q_{1}[/itex] = [itex]Q_{1}[/itex] [itex]e^{j \omega t}[/itex] (scalar eqn, 1st component of [itex]{q}[/itex])

[itex]q_{2}[/itex] = [itex]Q_{2}[/itex] [itex]e^{j \omega t}[/itex] (scalar eqn, 2nd component of [itex]{q}[/itex])

My question is why do we assume the same exponent for all components of [itex]{q}[/itex]? Why not assume (for a 2 DOF system) the solution as

[itex]q_{1}[/itex] = [itex]Q_{1}[/itex] [itex]e^{j \omega_{1} t}[/itex]

[itex]q_{1}[/itex] = [itex]Q_{1}[/itex] [itex]e^{j \omega_{2} t}[/itex]

ie, [itex]{q}[/itex] = [itex]W[/itex] [itex]Q[/itex], where [itex]W[/itex] is a diagonal matrix containing the exponent terms?

Thanks,

yogesh