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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. 

Abstract Riveted connections are widely used to join basic components, such as beams and panels, for engineering structures. However, accurately modeling joined structures with riveted connections can be a challenging task. In this work, an accurate linear finite element (FE) modeling method is proposed for joined structures with riveted connections to estimate modal parameters in a predictive manner. The proposed FE modeling method consists of two steps. The first step is to develop nonlinear FE models that simulate riveting processes of solid rivets. The second step is to develop a linear FE model of a joined structure with the riveted connections simulated in the first step. The riveted connections are modeled using solid cylinders with dimensions and material properties obtained from the nonlinear FE models in the first step. An experimental investigation was conducted to study accuracy of the proposed linear FE modeling method. A joined structure with six riveted connections was prepared and tested. A linearity investigation was conducted to validate that the test structure could be considered to be linear. A linear FE model of the test structure was constructed using the proposed method. Natural frequencies and corresponding mode shapes of the test structure were measured and compared with those from the linear FE model. The maximum difference of the natural frequencies was 1.63% for the first 23 out-of-plane elastic modes, and modal assurance criterion values for the corresponding mode shapes were all over 95%, which indicates high accuracy of the proposed linear FE modeling method.