Example_III_1_03 - Example 3 free response of a fixed/fixed string Find the free vibration solution of a fixed/fixed homogeneous string having a

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Example 3 - free response of a fixed/fixed string Find the free vibration solution of a fixed/fixed homogeneous string having a constant tension of T and mass/length of ! if the string is released from rest with an initial deformation shown to the right. x = 0 x=L A u 0 (x) x=L/2 EOM: T " 2 u " x 2 = # " 2 u " t 2 (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: TT t ( ) d 2 x ( ) dx 2 = #" x ( ) d 2 T t ( ) dt 2 $ T d 2 x ( ) / dx 2 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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Example_III_1_03 - Example 3 free response of a fixed/fixed string Find the free vibration solution of a fixed/fixed homogeneous string having a

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