Nonlinear Oscillations of a Suspended Gyrostat

Abstract

In this paper, the nonlinear dynamics of the disturbed Hamiltonian systems of a suspended gyrostat with five degrees of freedom are investigated in detail. The periodic motions of the torque-free symmetrical gyrostat are derived in terms of elliptic functions. The necessary conditions for the occurrence of chaotic oscillations of the disturbed, suspended gyrostat, either dissipative or conservative, are obtained via the Melnikov-Holmes-Marsden integrals. The MHM integrals built on the homoclinic orbits of the torquefree gyrostat and excitation-free spherical pendulum are utilized to establish the transversal homoclinic intersections between the stable and unstable manifolds of the associated Poincare map of the investigated system with perturbations. The results of the theoretical analyses are cross-checked with numerical simulation.

Chaotic flexural oscillations of a spinning nanoresonator

“…Equation (31) is in a form similar to the ones derived from Equation (13) (31) by dq x /dt and integrating the result yields ( dq x dt ) 2 = 4 j=0 A j (q x ) j , which is identical to Equation (31) of Kuang and Leung [15]. The integral constant A 0 can be determined from the initial generalized coordinate q x (0) and/or initial generalized velocity dq x /dt(0).…”

“…There is a tensile load acting on the nanoresonator 0 ≥ 0 yielding the van der Waals forces S < S critical defined in Equation (15). The linear stiffness α in Equation (14) becomes negative and the nonlinear stiffness β in Equation (14) is always positive.…”

“…Substitution of the given parameters into Equation (15) leads to S critical = −4.8183 × 10 −3 , 4.87291 × 10 −1 , 1.2254 MPa for 0 = 0, 40, 100 nN, respectively. It is found that β in Equation (14) is always positive for the selected first-order mode shape in Equation (11).…”

“…To better understand the essence of the symmetrical homoclinic orbits of the force-free beam-like nanoresonator, one is required to develop the particular orbits of Equation (31). The roots of the fourth-order polyno- The particular periodic orbits q x (t) of Equation (31) can be expressed in terms of the Jacobian hyperelliptic functions in Kuang and Leung [15]. Under Case (1), the particular solution called the homoclinic orbit of Equation (31) …”