Here and in the following, dots denote derivatives with respect to time, primes denote derivatives with re- spect to the coordinate x, and m is the mass per unit of length of the beam. The beam material is assumed to obey Hooke's law. We examine a small nonlinearity of the stress (~)-strain (e) relation, and expand it into a Maclaurin series, retaining the first two terms:
1968
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