Periodic orbits in systems with backlash: Stability, classification & observability
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Backlash, clearance or dead zone is a common feature of many mechanical systems and can undermine the performance of the system, since it has a large inﬂuence on the dynamics and control of systems. It can be caused by intended clearance necessary for assembly and operation, but may also be the result of operational wear and tear. Systems with backlash form a subclass of discontinuous mechanical systems and can be modeled as piecewise linear systems. In this work, both stiﬀness and damping are modeled with piecewise characteristics. A single and multiple degree-of-freedom model with backlash are analyzed for their harmonic periodic orbits as a function of excitation frequency and amplitude. The systems are modeled as tri-linear systems, with no stiﬀness in the backlash gap. This leads to a rigid body motion in this region. To calculate the ﬂow of the piecewise linear systems, a simulation method is used that utilizes the knowledge of the analytical solutions for linear systems. This method also allows for analytical calculation of the fundamental solution matrix. This is beneﬁcial for applying this simulation method in the multiple shooting method, which is used to calculate the periodic orbits. First, both the single and multiple degree-of-freedom system are characterized by their response diagram for a ﬁxed excitation amplitude. Here, the amplitude of both stable and unstable periodic orbits are calculated. The response diagram shows a combination of branches that is characteristic for a hardening oscillator, with multiple solutions in some frequency ranges. The periodic orbits are characterized by their number of subspace boundary crossings in excitation frequency and amplitude plane. Next to the number of boundary crossings, the periodic orbits are characterized by the maximum absolute value of the Floquet multipliers. The Floquet multipliers jump when the number of subspace boundary crossings changes, so this characterization gives the same information. However, the classiﬁcation by Floquet multipliers also distinguishes symmetric and asymmetric periodic orbits and therefore gives more information. These conclusions hold for both systems. When a system with backlash is used in practice, often only the output is measured. Information on the other states, especially the backlash gap, may however be relevant for analysis and control. Therefore, an observer is designed for the multiple degree-of-freedom system. Simulations of the observer show that it converges to an error much smaller as was expected. Yet, the convergence rate is low. Further research is needed to analyze the discrepancy between theory and simulations and to increase the performance of the observer.