Nonlinear Responses of Inextensible Cantilever and Free-Free Beams Undergoing Large Deflections

A theoretical and computational model has been developed for the nonlinear motion of an inextensible beam undergoing large deflections for cantilevered and free-free boundary conditions. The inextensibility condition was enforced through a Lagrange multiplier which acted as a constraint force. The Rayleigh-Ritz method was used by expanding the deflections and the constraint force in modal series. Lagrange’s Equations were used to derive the equations of motion of the system, and a 4th order Runge-Kutta solver was used to solve them. Comparisons for the cantilevered beam were drawn to experimental and computational results previously published and show good agreement for responses to both static and dynamic point forces. Some physical insights into the cantilevered beam response at the 1st and 2nd resonant modes were obtained. The free-free beam condition was investigated at the 1st and 3rd resonant modes and the nonlinearity (primarily inertia) was shown to shift the resonant frequency significantly from the linear natural frequency and lead to hysteresis in both modes.