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In this study, the primary resonance of a hyperelastic thin-walled cylindrical shell subjected to external excitation is examined. The incompressible Moony–Rivlin model, nonlinear Donnell’s shell theory, and Hamilton principle are employed to extract the fundamental equations in three directions. The Galerkin approach is employed to reduce the governing equation into ordinary differential equations form. Then, by neglecting the membrane inertia, three equations are reduced to one equation to analyze the transverse vibrations of the hyperelastic shells. Natural frequencies of different asymmetric modes are used to define the fundamental mode. Then, the multiple scales method is applied to evaluate the frequency–amplitude curves for various parameters. In addition, the model’s accuracy is validated with those presented in the literature for a cylindrical shell and the comparison shows good arrangement results. In the simulation part, the effects of the amplitude of excitation, the ratio of thickness–radius, and the ratio of radius–length on the primary resonances of shells are evaluated. The simulation plots show that for small values of radius-to-length ratios, increasing its value causes to decrease in the stiffness of the thin hyperelastic shell. Also, by increasing the radius–length ratios from 0.3 to 1.18 the hardening of the shell is decreased, and by increasing the radius–length ratios from 1.18 to 2 the hardening behavior is increased.

Modal discretization is commonly applied for dynamic analysis of non-linear continuum system. Considering the possible coupling effect between modes is necessary to obtain accurate results. In this case, the system may become increasingly complex, as the number of adopted modes can be a lot. Such complexity will lead to the difficulty of solution finding. This paper proposes a generic technique to simplify the governing functions by making non-linear stiffness matrix symmetric. The symmetric non-linear stiffness matrix is constructed by utilizing the mode shape vectors. The proposed procedure can theoretically guarantee non-linear stiffness matrix symmetric. The incremental harmonic balance (IHB) method is served as the main tool for finding solutions of systems. Dynamic analysis of axially moving beam and generalized suspension bridge are presented in this study for illustration. Results proved that the neighboring modes are critical during the resonance of target mode, which suggests the necessity of including sufficient modes for non-linear dynamic analysis. By applying the proposed technique, it is found that calculating time of IHB method can greatly shortened, especially for the case included modes becomes large. Results show that the time consumption with using the proposed method can be half of that not using it, when more than 5 modes are considered in the calculation.

The primary resonance behavior of a rotating ferromagnetic functional gradient cylindrical shell in a magnetic field, temperature field, and excitation force is investigated. Based on the physical neutral surface deformation theory and the Donnell theory, considering the effect of geometric nonlinearity, expressions of strain energy and kinetic energy of the shell and the work of forces are given, respectively. Applying the Hamilton principle, the magneto–thermo–elastic equation of a functional gradient cylindrical shell is derived by considering the magnetization effect of ferromagnetic metal. The question is discretized by Galerkin method and solved by the multi-scale method to obtain the amplitude–frequency response equation. The stability of the solution is discriminated by using the Lyapunov theory. Through numerical examples, the response curves of the system under different parameters are plotted, and the parameter ranges corresponding to multi-valued solution regions and single-valued solution regions are determined. The effects of parameter changes on the dynamic response and stability of system are analyzed. The results show that a coupling mechanism between temperature field, magnetic field, and excitation force affects the response and stability of the system, and the change of parameters have a significant effect on the vibration characteristics and stability. The dynamics model established in this paper is a theoretical reference for investigation on the multi-physics field coupling dynamic behaviors of structures.

This paper investigates the effect of double-frequency excitation on a fractional cerebral aneurysm model at the circle of Willis, where two kinds of external excitation are caused by the heart and arrhythmia respectively. Firstly, numerical analysis is performed on the proposed fractional model by using bifurcation diagrams and phase diagrams, and it is found that the periodic solution windows and chaotic solution windows switch more frequently than that resulted from single-frequency excitation, as a consequence of simultaneous resonances. Then, by using the method of multiple scales, the primary resonance response equation is obtained, and the effects of the system parameters on the amplitude of the fractional cerebral aneurysm model is analytically investigated. It is shown that double-frequency excitation changes the vibration patterns of the system under single-frequency excitation, and makes the primary resonance curve of the system deviate. Finally, the flexibility of using the amplitude-frequency equation to estimate the amplitude of the resonant solutions is verified numerically.

This paper investigates the nonlinear vibration of a size-dependent doubly clamped nanoresonator based on modified indeterminate couple-stress theory and Euler–Bernoulli beam theory. Surface effects, dispersion Casimir force, and fringing field effects are considered in the nonlinear model. The electrostatic actuation is a combination of DC and AC voltages and imposed on the nanobeam through one electrode. The governing differential equation of motion is derived using the extended Hamilton’s principle and discretized to a nonlinear ODE using Galerkin’s procedure. The multiple time scale method is applied to the reduced-order model in order to obtain the nanobeam frequency-response curves analytically under small AC voltage loads. The influences of the mentioned parameters are investigated on the primary resonance characteristics of the nanoresonator. It is shown that the application of non-classical continuum theory results in a softening effect on the dynamic response of the system near primary resonance. Moreover, it is concluded that the influence of surface energy on the system dynamic behavior depends on the value of DC voltage load.

The primary resonance and 1/3 subharmonic resonance of a quasi-zero stiffness (QZS) vibration system under base excitation with load mismatch are studied in this research. The incremental harmonic balance (IHB) method is applied to obtain highly accurate solutions involving more dynamic behaviors. The effect of the offset displacement mainly caused by overloading on the primary resonance and displacement transmissibility is investigated. The results indicate that the system exhibits a softening characteristic under certain conditions. Although the isolation performance of the QZS system deteriorates, it still outperforms the equivalent linear system for excitation amplitudes that are not too large. The parametric analysis of the 1/3 subharmonic resonance shows that the response is unbounded, and interesting dynamic behaviors can be observed, such as the jump phenomenon. Moreover, the 1/3 subharmonic resonance can be avoided by applying a larger damping or reducing the excitation amplitude to a lower level.

This paper predicts the nonlinear relative motion of a cantilever with a tip mass and magnets based on a distributed-parameter model. The Kelvin viscoelastic model is used to account for the damping in the cantilever. Under the harmonic base excitation, the governing equation is deduced via a coordinate transformation linked to its static equilibrium. To qualitatively validate those results captured from the approximate methods, the finite difference method and the multi-scale method are respectively employed to determine the natural frequency and the steady-state response of forced vibration. It is analytically demonstrated that those modes uninvolved in a certain resonance actually have no effect on the response amplitude of the stable steady-state motion. The effects of forcing amplitude, viscoelastic damping, and tip mass on the steady-state response are detailed via the amplitude–frequency response curves. For the first time, the current works theoretically illustrated the conversion from the hardening-type behavior to the softening-type versus the augmenting magnetic force, as well as the opposing effect of different tip masses on the first mode and the higher modes.

In this paper, the midspan deflection of a beam bridge with vehicles passing through the bridge successively is investigated. The midspan deflection can be modeled as the vibration trace of smooth-and-discontinuous (SD) oscillator by considering the mode of the first order and up-and-down vibration. The nonlinear behaviors of the established model are studied and presented. KAM (Kolmogorov–Arnold–Moser) structures on the Poincaré section are constructed for the driven system without dissipation with generic KAM curve and a series of resonant points and the surrounding island chains connected by chaotic orbits. Introducing a series of complete elliptic integrals of the first and the second kind, the response curves of the system are detected, to which the effects of parameters are revealed. The relevant dynamics is depicted under external excitation exhibiting period leading to chaos. The efficiency of the bifurcation diagrams obtained in this paper is demonstrated via numerical simulations.

In this paper, the vibration and primary resonance of electrostatically actuated microbridges are investigated, with the effects of electrostatic actuation, axial stress, and mid-plane stretching considered. Galerkin's decomposition method is adopted to convert the governing nonlinear partial differential equation to a nonlinear ordinary differential equation. The homotopy perturbation method (a special case of homotopy analysis method) is then employed to find the analytic expressions for the natural frequencies of predeformed microbridges, by which the effects of the voltage, mid-plane stretching, axial force, and higher mode contribution on the natural frequencies are studied. The primary resonance of the microbridges is also investigated, where the microbridges are predeformed by a DC voltage and driven to vibrate by an AC harmonic voltage. The methods of homotopy perturbation and multiple scales are combined to find the analytic solution for the steady-state motion of the microbeam. In addition, the effects of the design parameters and damping on the dynamic responses are discussed. The results are shown to be in good agreement with the existing ones.