# lecture910 - ME 563 Mechanical Vibrations Lecture #9 Second...

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ME 563 Mechanical Vibrations Lecture #9 Second and Fourth Order Continuous Differential Equations

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Continuous Systems 1 If mechanical systems have continuously distributed mass and elasticity (strings, membranes, beams, plates, shells, etc.), then it is necessary to use differential equations of motion that take into account the continuity in the system. Assume that: - Material properties are continuous (density, modulus, etc.) - Small motions meaning geometric nonlinearities are absent - Linear elastic behavior of the material
String Example 2 Consider the string shown below subjected to a distributed force that varies with time, f(x,t). The string displaces in the transverse (vertical) direction at a position x from the left end by y(x,t) as a function of time.

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String Example 3 The free body diagram is drawn using the assumption that tension, density, force, displacement, and slope are continuously varying functions.
String Example 4 Only the first two terms of the Taylor series expansion for the

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lecture910 - ME 563 Mechanical Vibrations Lecture #9 Second...

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