example_III_1_03

# 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

This preview shows pages 1–2. Sign up to view the full content.

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,

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This document was uploaded on 12/23/2011.

### Page1 / 6

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online