Geometrically nonlinear free vibration of Euler-Bernoulli shallow arch

Abstract

Abstract The purpose of this work is to investigate the geometrical non-linearity in free vibrations of the Euler-Bernoulli shallow arch with clamped ends. The nonlinear governing equilibrium equation of the shallow arch is obtained after the Euler Bernoulli theory and Won Karman geometrical nonlinearity assumptions. The initial curvature of the arch is not due to the axial displacement of the beam but is due to the geometric of the beam itself. Taking into account a harmonic motion, the kinetic and total strain energy are discretized into a series of finite functions, which are a combination of the linear modes calculated before and the coefficients of contribution. The discretized expressions are derived by applying a Hamilton principle energy and spectral analysis. A cubic nonlinear algebraic system is obtained and solved numerically using an approximation method (the so-called second formulation) is applied to resolve various nonlinear vibration problems. To illustrate the effect of the curvature on the fundamental nonlinear mode and nonlinear frequency, the corresponding backbone curves, nonlinear amplitude vibration, and curvature of the arch are presented for the first modes shapes.