1. Mechanical Vibrations Multi Degrees of Freedom System
Philadelphia University
Engineering Faculty
Mechanical Engineering Department
Professor Adnan Dawood Mohammed

2. Multi DOF system Multi-DOF systems are so similar to two-DOF.
Equations of motion:
They are obtained using:
Vector mechanics (Newton or D’ Alembert)
Hamilton's principles
Lagrange's equations
[M] is the Mass matrix
[K] is the Stiffness matrix
[C] is the Damping matrix

3. Un-damped Free Vibration: the eigenvalue problem
Equation of motion:
in terms of the generalized D.O.F. qi
Write the matrix equation as:

4. Assuming harmonic motion:

6. Consider the multi-story building shown in figure. The
Example:
Consider the multi-story building shown in figure. The
Equations of motion can be written as:
Pre-multiply by the inverse of mass matrix
The characteristic equation from the determinant of the above matrix is

7. The eigenvectors can be found from Eqn
The eigenvectors can be found from Eqn.(b) by substituting the above values of
l. The adjoint matrix from Eqn. (b) is
Substituting l1 into Eqn. (e) we obtain:
Here each column is already normalized to unity and the first eigenvector is

8. Similarly when l =l 2 = 0.5(k/m) the adjoint matrix gives;
Normalizing to Unity;
The second eigenvector from either column is;