Periodic Response and Chaos in Nonlinear Systems with Parametric Excitation and Time Delay

In this paper, the periodic motions of a nonlinear system with quadratic,cubic, and parametrically excited stiffness terms and with time-delayterms are obtained by the incremental harmonic balance (IHB) method. Theelements of the Jacobian matrix and residue vector arising in the IHBformulation are derived in closed form. A mechanism model representingthe one-mode oscillation of beams and plates is considered as anexample. A path-following algorithm with an arc-length parametriccontinuation procedure is used to obtain the response diagrams. Thesystem also exhibits chaotic motion through a cascade of period-doublingbifurcations, which is characterized by phase planes, Poincaré sectionsand Lyapunov exponents. The interpolated cell mapping (ICM) procedure isused to obtain the initial condition map corresponding to multiplesteady-state solutions.