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Abstract Quasi-periodic motions can be numerically found in piecewise-linear systems, however, their characteristics have not been well understood. To illustrate this, an incremental harmonic balance (IHB) method with two timescales is extended in this work to analyze quasi-periodic motions of a non-smooth dynamic system, i.e., a gear transmission system with piecewise linearity stiffness. The gear transmission system is simplified to a four degree-of-freedom nonlinear dynamic model by using a lumped mass method. Nonlinear governing equations of the gear transmission system are formulated by utilizing the Newton’s second law. The IHB method with two timescales applicable to piecewise-linear systems is employed to examine quasi-periodic motions of the gear transmission system whose Fourier spectra display uniformly spaced sideband frequencies around carrier frequencies. The Floquet theory is extended to analyze quasi-periodic solutions of piecewise-linear systems based on introduction of a small perturbation on a steady-state quasi-periodic solution of the gear transmission system with piecewise linearities. Comparison with numerical results calculated using the fourth-order Runge-Kutta method confirms that excellent accuracy of the IHB method with two timescales can be achieved with an appropriate number of harmonic terms. 

We measure universal temperature-independent density shifts for the thermal conductivity ${\kappa }_T$and shear viscosity $\eta$, relative to the high temperature limits, for a normal phase unitary Fermi gas confined in a box potential. We show that a time-dependent kinetic theory model enables extraction of the hydrodynamic transport times ${\tau }_{\eta }$and ${\tau }_{\kappa }$from the time-dependent free decay of a spatially periodic density perturbation, yielding the static transport properties and density shifts, corrected for finite relaxation times. Published by the American Physical Society2024