Modeling and analysis of an inextensible beam with inertial and geometric nonlinearities

The present study focuses on an inextensible beam and its relevant inertia nonlinearity, which are essentially distinct from the commonly treated extensible beam that is dominated by the geometric nonlinearity. Explicitly, by considering a weakly constrained or free end (in the longitudinal direction), the inextensibility assumption and inertial nonlinearity (with and without an initial curvature) are introduced. For a straight beam, a multi-scale analysis of hardening/softening dynamics reveals the effects of the end stiffness/mass. Extending the straight scenario, a refined inextensible curved beam model is further proposed, accounting for both its inertial nonlinearity and geometric nonlinearity induced by the initial curvature. The numerical results for the frequency responses are also presented to illustrate the dynamic effects of the initial curvature and axial constraint, i.e., the end mass and end stiffness.