ASYMPTOTIC GOVERNING EQUATION FOR WAVE PROPAGATION ALONG WEAKLY NON-UNIFORM EULER–BERNOULLI BEAMS

Non-uniformity in beams arises either from manufacturing imperfections or by design, and can have a singular impact on the qualitative properties of the vibratory response of the beam. To describe the mechanism causing such large changes on the dynamics of the beam, we derived asymptotically a simpler equation, in the formχss+Q (s) χ (s)=0. The coefficient function Q (s) is given by equation (52) herein in terms of the beam flexural rigidity, the mass per unit length and the tensile force applied to the beam. The equation is asymptotic to the non-uniformity of the beam, but under certain restrictions, namely of having constant tension and a constant product of the beam mass per unit length and flexural rigidity, it is an exact governing equation for wave propagation along Bernoulli–Euler beams and it has a Helmholtz-like form. The behavior of the equation is systematically explored and illustrated through numerical results.