On the Generation of Harmonics by the Non-Linear Buckling of an Elastic Beam

The Euler–Bernoulli theory of beams is usually presented in two forms: (i) the linear case a small slope using Cartesian coordinates along and normal to straight undeflected position; (ii) non-linear large curvilinear deflected position, namely, arc length angle inclination. present paper starts with exact equation third form, that is, (iii) position like (i), but allowing exactly effects (ii). This form elastica shows shape superposition harmonics; thus, are equivalent generation harmonics solution for slope. In conclusion, it shown that: critical buckling load same cases because determined by fundamental mode; buckled different non-linearity adds mode. elastica, when powers cannot be neglected, illustrated first four modes cantilever, pinned, clamped lengths amplitudes.