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Structural Health Monitoring with Piezoelectric Wafer Active Sensors, 2nd Edition by Victor Giurgiutiu

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image (100)

Proof of first line of Eq. (99):

image (101)

A similar proof applies to the second line of Eq. (99). It can be easily verified that orthogonality properties similar to Eq. (99) also apply to the interval [0,2π]image, i.e.,

02πsinpϕsinqϕdϕ=πδpq02πcospϕcosqϕdϕ=πδpq (102)

image (102)

For vibrations and waves analysis, one can apply a similar argument to harmonic ...

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