example_III_2_01 - Example 2 - free response of a simply...

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Example 2 - free response of a simply supported bending beam Find the general form of the free vibration solution of a simply supported homogeneous bending beam having a flexural rigidity EI, cross-sectional area A and mass/volume ! . EI " 4 u " x 4 + # A " 2 u " t 2 = 0 (1) Solution form The above is a homogeneous, partial differential equation in terms of independent variable of x and t. Will write solution in a “separable form” of a product of functions of x and t as u x , t ( ) = " x ( ) T t ( ) (2) If we substitute (2) into (1) we find: EIT t ( ) d 4 x ( ) dx 4 + A x ( ) d 2 T t ( ) dt 2 = 0 $ EI A d 4 x ( ) / dx 4 x ( ) = $ d 2 T t ( ) / dt 2 T t ( ) (3) The left hand side of (3) is strictly a function of x, and the right hand side of (3) is strictly a function of time, t. This equation of left side with right side must be true for ALL x and for ALL t. In order for this equality to the true, then both sides of the equation must be CONSTANT in both x and t; that is,
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This document was uploaded on 12/23/2011.

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example_III_2_01 - Example 2 - free response of a simply...

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