Vibrations of Multi Degree of Freedom Systems A Two Degree of Freedom System: Equation of Motion:

In Matrix Form: Or, Solution in general, Substituting in the equation: Rearranging, where, X: Mode shapes,  : Natural Frequencies For the non-trivial solution, we set

For the above example, We find the eigenvalues (modes, natural frequencies), And the eigenvectors (mode shapes),

Solution is then written in general as: With initial conditions specified as:

Unknowns A i and  i are found from: For the above problem:

Solution for the above example, Or,

Principal Coordinates: Find coordinates, called principal coordinates p 1, p 2,…,p n, such that the equations are uncoupled, i.e. let x = Pp, where Then the equations uncouple to become

Viscous Damping (Proportional Damping) The equations of motion for a multi-degree of freedom system with viscous damping where, C =  K +  M = Proportional Damping Matrix ,  = constants Equation with principal coordinates:

In standard form: Solution:

General Viscous Damping The equations of motion for a multi-degree of freedom system with viscous damping where, C = General Damping Matrix Solution is obtained from the equation: where,

Solution is then: where,  and  are eigenvalues and eigenvectors of matrix C j : Constants (to be found from IC) Review Example 6.17, p. 342

Forced Vibrations of Multi-Degree-of-Freedom Systems Undamped Response for Harmonic Force (Excitation) Solution is obtained from: where,

Damped Response for General Excitation where, We do the transformation using the principal coordinates Equation is converted to: where,

With the transformation the equations are uncoupled as where g i (t) is obtained by: P T F Solution is then obtained using the convolution integral: