1

where the ellipsis refers to terms of third and higher degree in the spatial partial derivatives of u and v. Then, when all terms are dropped that are of third and higher degree in the spatial partial derivatives of u and v, the strain energy becomes

P = Tq I —dx H--—- I I — ) dx + — [ (—^ dx +

dx J

Assuming ù (0) = û (¿o) — 0, one finds that the first term vanishes. Since perturbations of the transverse deflections are the unknowns in which we are most interested, and since perturbations of the longitudinal displacements are uncoupled from these and involve oscillations with much higher frequency, we will not need the last term. This leaves only the second term. As before, noting that e0 1 and dropping the ' and subscripts for convenience, one obtains the potential energy for a vibrating string,

as found in vibration texts. In terms of the mode shapes as represented in Eq. (2.83) the total potential energy can then be written as

0 0

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