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lecture2210

# lecture2210 - 4 Does this single degree of freedom equation...

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ME 563 Mechanical Vibrations Lecture #22 Forced Response of Continuous Systems Using Modal Approach

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String Forced Response 1 Consider the string equation of motion: If we assume a solution of the form, , we can turn the partial differential equation into an ordinary differential equation of motion. To solve this ordinary differential equation, we can use the free response (eigenvalue solution) to decouple the individual modal equations from one another.
Recall Normal Coordinates 2 If we normalize the spatial functions, Y n (x) , as follows: and we expand the solution as follows: then we multiply by one function Y r (x) at a time to uncouple the modes of vibration from one another.

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3
Analogy to Discrete Case

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Unformatted text preview: 4 Does this single degree of freedom equation remind you of anything? Both of these sets of equations describe how forcing functions (in space and time) excite uncoupled modes of vibration. M r [ ] ˙ ˙ p { } + C r [ ] ˙ p { } + K r [ ] p { } = Ψ [ ] T f { } Example f(x,t) 5 If we assume a forcing function, then we can arrive at the following set of equations: Example 6 The meaning of this solution, is that only modes like not are excited by a spatial forcing function like . Also, modes of vibration with natural frequencies, ω r , close to the forcing frequency, , are excited the most. Note the analogy to FRFs… Numerator Denominator...
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lecture2210 - 4 Does this single degree of freedom equation...

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