1. Dynamic Response of MDOF Structures

2. Equilibrium of forces condition gives the equations of motions:

3. Example: 2-DOF, identical mass & stiffness Free vibration
How do we solve this set of coupled equations?

4. By observation, the solution should be harmonic.
Substituting into the equations of motion:
Need to find non-trivial solution  coefficient matrix has to be singular

5. Natural frequencies of the 2-DOF system.

7. 1 2

11. Example: 5-DOF, identical mass & stiffness Free vibration

14. Method for Uncoupling the Equations of Motion

16. Orthogonality Properties of Modeshapes

18. Scaling the Modeshapes
Arbitrary Set one modeshape coefficient equal to unity (ex: coeff. at roof level = 1.0) Mass normalized

19. Repeat the 2-DOF example

22. Forced Vibration
modal representation

25. How to find the initial conditions in the modal space?
Premultiply by

26. Including damping

27. MDOF Examples

29. find modes

30. rad/sec

35. Mass-normalized modeshapes

46. 71 72

48. Example