Competing dynamics of an inextensible beam: Inertial vs. geometric nonlinearities

Competing dynamics of an inextensible straight beam with a weakly constrained end (i.e., end mass+end spring/stiffness), involved with both inertial nonlinearity (favoring softening) and geometric nonlinearity (favoring hardening), is fully investigated through the lens of asymptotic analysis. Besides monotonic dynamics (either softening or hardening), the current paper focuses on non-monotonic dynamics i.e., mixed hardening/softening (H/S) dynamics. It turns out that the H/S transition curve (or separatrix) plays a key role in locating the above two distinct nonlinear dynamics, i.e., monotonic or non-monotonic one, with the former found away from H/S transition while the latter located in the vicinity of H/S separatrix. In both scenarios, the inextensible beam’s dominant nonlinear dynamics are found to critically depend on the beam’s end mass and end stiffness.