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.
The bifurcation behavior of an articulated loading platform subjected to harmonic excitation is investigated by the incremental harmonic balance (IHB) method. The platform is modeled as a single-degree-of-freedom (SDOF) non-linear system with piecewise non-linear restoring force characteristics. The elements of the Jacobian matrix and the residue vector arising in the IHB formulations are derived in closed form. The path-following procedure using the arc length continuation method is used to trace the response curves and bifurcation diagrams. The periodic solutions and the subharmonic solutions obtained by the IHB method compare very well with the numerically integrated solutions. The bifurcation points also compare well with the numerically obtained results. The system exhibits chaotic motion through a sequence of period doubling bifurcations. Isolated period 3 solutions are also present. The Lyapunov exponents are computed and the initial condition map corresponding to coexistent attractors are obtained by the interpolated cell mapping (ICM) method.
Periodic motions of the nonlinear system representing the escape equation with cosine and sine parametric excitations and external harmonic excitations are obtained by the incremental harmonic balance (IHB) method. The system contains quadratic stiffness terms. The Jacobian matrix and the residue vector for the type of nonlinearity with parametric excitation are explicitly derived. An are length path following procedure is used in combination with Floquet theory to trace the response diagram and to investigate the stability of the periodic solutions. The system undergoes chaotic motion for increase in the amplitude of the harmonic excitation which is investigated by numerical integration and represented in terms of phase planes, Poincare sections and Lyapunov exponents. The interpolated cell mapping (ICM) method is used to obtain the initial condition map corresponding to two coexisting period 1 motions. The periodic motions and bifurcation points obtained by the IHB method compare very well with results of numerical integration. (C) 2000 Elsevier Science Ltd. All rights reserved.
Periodic oscillations and bifurcations of a two-dimensional airfoil in plunge and pitching motions with cubic pitching stiffness in incompressible flow is investigated using the incremental harmonic balance (IHB) method. The bifurcations are obtained with the parametric continuation technique and the stability of the periodic motions is investigated using the Floquet theory. The autonomous non-linear system of the airfoil undergoes initial Hopf bifurcation leading to limit cycle oscillation as the airspeed parameter is increased. Further increase in the airspeed causes symmetry breaking, saddle-node and period-doubling bifurcations leading to chaos. The frequency of the limit cycle oscillation is also determined in the IHB method.
The periodic motions of a non-linear geared rotor-bearing system are investigated by the incremental harmonic balance (IHB) method. A path following procedure using arc length continuation technique is used to trace the bifurcation diagrams. The system exhibits a period doubling route and a quasiperiodic route to chaos in different regions of excitation frequency. The chaotic motions are investigated by numerical integration and the Lyapunov exponents are computed. The periodic solutions and subharmonic solutions obtained by the IHB method compare very well with those obtained by numerical integration.
The bifurcation behavior of an articulated loading platform subjected to harmonic excitation is investigated by the incremental harmonic balance (IHB) method. The platform is modeled as a single-degree-of-freedom (SDOF) non-linear system with piecewise non-linear restoring force characteristics. The elements of the Jacobian matrix and the residue vector arising in the IHB formulations are derived in closed form. The path-following procedure using the arc length continuation method is used to trace the response curves and bifurcation diagrams. The periodic solutions and the subharmonic solutions obtained by the IHB method compare very well with the numerically integrated solutions. The bifurcation points also compare well with the numerically obtained results. The system exhibits chaotic motion through a sequence of period doubling bifurcations. Isolated period 3 solutions are also present. The Lyapunov exponents are computed and the initial condition map corresponding to coexistent attractors are obtained by the interpolated cell mapping (ICM) method.
