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An improved component modal synthesis-based nonproportional mistuning method (ICMS-NPMM) is proposed to investigate mistuned turbine blisks (MTBs) since the high-fidelity finite element models (HFEMs) involve large number of computations, which leads to low calculation efficiency. To reduce degrees of freedom and suppress the flutter of MTB, it is divided into mistuned blade structure and tuned disk structure, and the intentional mistuning is considered. Furthermore, the mistuned parameters, nonproportional mistuning, and complex loads are also considered. Firstly, the basic theory of ICMS-NPMM is investigated; secondly, the model of MTB is established via ICMS-NPMM; finally, the intentionally mistuned design of modal shape amplitudes (MSAs) is investigated via ICMS-NPMM. The results indicate that the calculation efficiency is enhanced via ICMS-NPMM relative to that of via HFEM. In addition, the sensitivity and the flutter are decreased; meanwhile, the amplitude fluctuations of MSAs are distinctly decreased and become comparatively smooth. This investigation provides an important guidance for the vibration characteristic study of complex mechanical structures in engineering practice.

Nonlinear dynamical systems with irrational and fractional nonlinear restoring forces often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational and fractional nonlinear restoring forces avoiding the conventional Taylor's expansion to retain the natural characteristics of the system. By introducing a particular dimensionless representation and a series of transformations, the two-degree-of-freedom system can be transformed into a perturbed Hamiltonian system. The extended Melnikov method is directly used to detect the chaotic threshold of the perturbed system theoretically, which overcomes the barrier caused by solving theoretical solution for the homoclinic orbit of the unperturbed system. The numerical simulations are carried out to demonstrate the complicated dynamics of the nonlinear spring-pendulum system, which show the efficiency of the criteria for chaotic motion in the system.

The smooth-and-discontinuous oscillator under constant excitations (CSD) is an unsymmetrical system with an irrational restoring force, which leads to a barrier to detect the chaotic threshold using conventional nonlinear techniques. The goal of the present work is to overcome the trouble raised by constant excitations and irrational nonlinearity to investigate the necessary and insufficient condition for chaos. A smooth approximate system, whose behaviors are topologically equivalent to the ones of the original system, is proposed to investigate the chaotic boundary. The Hamiltonian system of the approximate system is analyzed and the homoclinic orbits in analytical form are obtained. The Melnikov method is employed to determine the distance between the stable and unstable manifolds under the perturbation of damping and external forcing. Bifurcation diagrams and numerical simulations are used to reveal the motion of chaos and the period for approximate system which is in good agreement with the original system. It is worth recalling that the approach for constructing the approximate function of smoothness is presented, which puts forward a step toward the solution of the irrational nonlinearity system.

The effects of constant excitation on the recently proposed smooth-and-discontinuous (SD) oscillator are investigated, which may lead to the variation of equilibrium and the property of phase portrait. By solving a quartic algebraic equation, the transition set and bifurcation for SD oscillator under constant excitation (CSD) are presented, while the number of equilibria depends on the values of the smoothness parameter and the constant excitation. Complicated structures of Kolmogorov-Arnold-Moser (KAM) structures on the Poincare section are depicted for the driven system without dissipation. Chaotic behaviour is also demonstrated numerically for the system perturbed by both viscous-damping and external excitation. The results show that CSD is an unsymmetrical system that displays different dynamical behaviours from an SD oscillator and will enrich the range of the SD oscillator in research and application.

Nonlinear dynamical systems with an irrational restoring force often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational nonlinearity avoiding the conventional Taylor's expansion to retain the natural characteristics of the system. A series of transformations are proposed to convert the homoclinic orbits of the unperturbed system to the heteroclinic orbits in the new coordinate, which can be transformed back to the analytical expressions of the homoclinic orbits. Melnikov's method is employed to obtain the criteria for chaotic motion, which implies that the existence of homoclinic orbits to chaos arose from the breaking of homoclinic orbits under the perturbation of damping and external forcing. The efficiency of the criteria for chaotic motion obtained in this paper is verified via bifurcation diagrams, Lyapunov exponents, and numerical simulations. It is worthwhile noting that our study is an attempt to make a step toward the solution of the problem proposed by Cao Q J et al. (Cao Q J, Wiercigroch M, Pavlovskaia E E, Thompson J M T and Grebogi C 2008 Phil. Trans. R. Soc. A 366 635).