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Dynamic Response of MDOF Structures
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Equilibrium of forces condition gives the equations of motions:
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Example: 2-DOF, identical mass & stiffness Free vibration
How do we solve this set of coupled equations?
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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
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Natural frequencies of the 2-DOF system.
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1 2
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Example: 5-DOF, identical mass & stiffness Free vibration
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Method for Uncoupling the Equations of Motion
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Orthogonality Properties of Modeshapes
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Scaling the Modeshapes
Arbitrary Set one modeshape coefficient equal to unity (ex: coeff. at roof level = 1.0) Mass normalized
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Repeat the 2-DOF example
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Forced Vibration modal representation
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How to find the initial conditions in the modal space?
Premultiply by
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Including damping
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MDOF Examples
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find modes
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rad/sec
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Mass-normalized modeshapes
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71 72
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Example
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