Thermo-mechanical instability in vibration absorbers

We analyze a problem of the thermo-mechanical instability caused by small changes of a viscous damping in vibration absorbers. The nonlinear coupling between the oscillations and temperature takes place due to a linear thermal dependence of the coefficient of energy dissipation. This provides typical phase–amplitude frequency patterns inherent in unstable regimes. While the damping coefficient decreases with the increase in the temperature, the effect of bifurcated oscillations can be exhibited brightly as some abnormal operating regimes. The vibration absorber appears as a complex dynamical system, behaving strongly upon the ambient temperature. Typical thermo-mechanical instability patterns are traced in detail within a parametric analysis along an approach closed to the Lie method. This study would explain some unwanted dynamical effects accompanying the utilizing of vibration absorbers.     

Nonlinear vibration of a two-mass system with nonlinear stiffnesses

An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of the proposed approach is twofold. First, it introduces the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment a nonlinear differential system by linearization coupled with Newton’s method and harmonic balance method. New and accurate higher-order analytical approximate solutions for the nonlinear system are established. After solving the nonlinear differential equation, the displacement of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, this higher-order Newton–harmonic balance (NHB) method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation unlike the classical harmonic balance method which results in complicated algebraic equations requiring further numerical analysis. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and the classical harmonic balance methods. Two examples of nonlinear two-degree-of-freedom mass–spring system are analyzed and verified with published result, exact solutions and numerical integration data.     

Resonance analysis of the finite-damping nonlinear vibration system under random disturbances

In this paper we propose a new technique for obtaining an approximate probability density for the resonance and off-resonance response of the finite-damping nonlinear vibration system under both random disturbances and deterministic excitation.     

Thermo-mechanical nonlinear dynamics of a buckled axially moving beam

The thermo-mechanical nonlinear dynamics of a buckled axially moving beam is numerically investigated, with special consideration to the case with a three-to-one internal resonance between the first two modes. The equation of motion of the system traveling at a constant axial speed is obtained using Hamilton??s principle. A closed form solution is developed for the post-buckling configuration for the system with an axial speed beyond the first instability. The equation of motion over the buckled state is obtained for the forced system. The equation is reduced into a set of nonlinear ordinary differential equations via the Galerkin method. This set is solved using the pseudo-arclength continuation technique to examine the frequency response curves and direct-time integration to construct bifurcation diagrams of Poincaré maps. The vibration characteristics of the system at points of interest in the parameter space are presented in the form of time histories, phase-plane portraits, and Poincaré sections.     

Frequency analysis of nonlinear free vibration of a cross string

This work concerns nonlinear free vibration of a cross string under large amplitude. The equations governing the nonlinear vibration of the cross string are derived at first from the Hamilton principle, and they take the form of Duffing equation. Then the perturbation method is used to solve the nonlinear coupled natural frequency of the cross string. The nonlinear natural frequency not only has the characteristic of nonlinearity, but also reflects the coupled characteristic, i.e., the natural frequency of the cross string varying with that of its constituent strings. The results show that the overall effect on the cross string is somehow averaged due to the nonlinearity of each constituent string, i.e., the natural frequencies of the cross string contain both the linear natural frequencies of the constituent strings and the nonlinear parts that depend upon the vibration amplitude, the diameter of one constituent string, the length ratio of the two strings, etc., but the contribution of each constituent string to the natural frequency is in different proportions.     

Nonlinear harmonic vibration analysis of a plate-cavity system

Nonlinear harmonic oscillation of a plate-cavity system is analytically studied in this paper. Von-Karman theory is used to model a rectangular plate backed by an air cavity. Coupled nonlinear differential equations of system are analytically derived using Galerkin’s approach. The Multiple Scales Method (MSM) is then employed to solve the corresponding nonlinear equations. Primary, secondary, and combinational resonance conditions are taken into account and the corresponding closed-form frequency-amplitude relationships are derived. A parametric study is carried out and effects of different parameters on the frequency responses are investigated.     

Free vibration analysis of nonlinear resilient impact dampers

In the present study, free vibration of a vibratory system equipped with an impact damper, which incorporates the Hertzian contact theory, is investigated. A nonlinear model of an impact damper is constructed using spring, mass, and viscous damper. To increase accuracy of the solution, deformation of the impact damper during the collision with and the main mass is considered. The governing coupled nonlinear differential equations of a cantilever beam equipped with the impact damper are solved using the parameter expanding perturbation method. Contact durations, which are obtained using the presented method, are compared with previous results. Gap sizes of the impact dampers are classified to two main parts. The effects of selecting the gap sizes regarding to the discussed classification are investigated on the application of the impact dampers. Based on types of collision between colliding masses, the so-called “effectiveness” is defined. Finally, it is shown variation of the damping inclination with the gap size is similar to variation of the effectiveness.     

Nonlinear vibration analysis of geometrically nonlinear shell structures

The nonlinear vibration analysis of a geometrically nonlinear shell structure is investigated in this study. In general, when the shell structure is subjected to excessive loadings, the large deformation of the shell structures must be considered, and the governing equation of the shell structure becomes nonlinear since the stiffness matrix of the governing equation is related to the deflection. Therefore, the natural frequency of the shell structure is varied with respect to the time which is quite different from that of the linear structures. In order to solve the nonlinearity of the governing equations of the shell structures, the well known Newton-Raphson iteration procedure in conjunction with Newmark scheme is adopted to perform the frequency analysis of the nonlinear-shell structures. Incidentally, the natural frequencies for various curvatures of the shell structures are also investigated from the practical engineering point of view.     

Continuation analysis of a nonlinear rotor system

Nonlinear Dynamics - Nonlinearities in rotating systems have been seen to cause a wide variety of rich phenomena; however, the understanding of these phenomena has been limited because numerical...     

Stochastic averaging for nonlinear vibration energy harvesting system

A stochastic averaging technique for the nonlinear vibration energy harvesting system to Gaussian white noise excitation is developed to analytically evaluate the mean-square electric voltage and mean output power. By introducing the generalized harmonic transformation, the influence of the external circuit on the mechanical system is equivalent to a quasi-linear stiffness and a quasi-linear damping with energy-dependent coefficients, and then the equivalent nonlinear system with respect to the mechanical states is completely established. The Itô stochastic differential equation with respect to the mechanical energy of the equivalent nonlinear system is derived through the stochastic averaging technique. Solving the associated Fokker–Plank–Kolmogorov equation yields the stationary probability density of the mechanical states, and then the mean-square electric voltage and mean output power are analytically obtained through the approximate relation between the electric quantity and the mechanical states. The agreements between the analytical results and those from the moment method and from Monte Carlo simulations validate the effectiveness of the proposed technique.     

Bifurcation of piecewise-linear nonlinear vibration system of vehicle suspension

A kinetic model of the piecewise-linear nonlinear suspension system that consists of a dominant spring and an assistant spring is established. Bifurcation of the resonance solution to a suspension system with two degrees of freedom is investigated with the singularity theory. Transition sets of the system and 40 groups of bifurcation diagrams are obtained. The local bifurcation is found, and shows the overall character- istics of bifurcation. Based on the. relationship between parameters and the topological bifurcation solutions, motion characteristics with different parameters are obtained. The results provides a theoretical basis for the optimal control of vehicle suspension system parameters.     

The nonlinear vibration analysis of a clamped-clamped beam by a reduced multibody method

The nonlinear response characteristics for a dynamic system with a geometric nonlinearity is examined using a multibody dynamics method. The planar system is an initially straight clamped-clamped beam subject to high frequency excitation in the vicinity of its third natural mode. The model includes a pre-applied static axial load, linear bending stiffness and a cubic in-plane stretching force. Constrained flexibility is applied to a multibody method that lumps the beam into N elements for three substructures subjected to the nonlinear partial differential equation of motion and N-1 linear modal constraints. This procedure is verified by d'Alembert's principle and leads to a discrete form of Galerkin's method. A finite difference scheme models the elastic forces. The beam is tuned by the axial force to obtain fourth order internal resonance that demonstrates bimodal and trimodal responses in agreement with low and moderate excitation test results. The continuous Galerkin method is shown to generate results conflicting with the test and multibody method. A new checking function based on Gauss' principle of least constraint is applied to the beam to minimize modal constraint error.     

Suppression of nonlinear vibration system described by nonlinear differential equations using passive controller

In this paper, the linear absorber is proposed to reduce the vibration of a nonlinear dynamical system at simultaneous primary resonance and the presence of 1:1 internal resonance. This leads to a two-degree-of-freedom system subjected to external excitation force. The method of multiple scales perturbation technique is applied throughout to determine the analytical solution up to first-order approximations. The stability of the system near the one of the worst resonance case is studied using the frequency response equations. The effects of the different system and absorber parameters on the behavior of the main system are studied numerically. For validity, the numerical solution is compared with the analytical solution and gets a good agreement. Effectiveness of the absorber ( \(E_{a})\) is about 800 for the nonlinear vibrating system. The simulation results are achieved using MATLAB programs. At the end of the work, the comparison with the available published work is reported.     

Chaotic vibration of a nonlinear full-vehicle model

The present work investigates the chaotic responses of a nonlinear seven degree-of-freedom ground vehicle model. The disturbances from the road are assumed to be sinusoid and the time delay between the disturbances is investigated. Numerical results show that the responses of the vehicle model could be chaotic. With the bifurcation phenomenon detected, the chaotic motion is confirmed with the dominant Lyapunov exponent. The results can be useful in dynamic design of a vehicle.     

转子—非线性支承系统振动响应的优化计算

本文用一种新的优化方案计算装有非线性弹性支承－挤压油膜阻尼器的转子系统的振动响应。首先根据转子系统的结构特点，建立其无量纲形式的非线性运动微分方程；然后由微分方程构造－控制目标函数，最后对此目标函数进行优化计算，求得转子系统的振动响应。     

Accurate modeling and analysis of a typical nonlinear vibration isolator with quasi-zero stiffness

Nonlinear Dynamics - A typical quasi-zero stiffness (QZS) vibration isolator composed of two lateral springs and a vertical spring has been widely studied previously, aiming to widen the frequency...     

Nonlinear vibration analysis of vortex-induced vibrations in overhead power lines with nonlinear vibration absorbers

Nonlinear Dynamics - Vortex-induced vibrations are one of the major factors in fatigue failure of power transmission lines and can be mitigated using vibration absorbers in the form of Stockbridge...     

Nonreciprocity of energy transfer in a nonlinear asymmetric oscillator system with various vibration states

The nonreciprocity of energy transfer is constructed in a nonlinear asymmetric oscillator system that comprises two nonlinear oscillators with different parameters placed between two identical linear oscillators. The slow-flow equation of the system is derived by the complexification-averaging method. The semi-analytical solutions to this equation are obtained by the least squares method, which are compared with the numerical solutions obtained by the Runge-Kutta method. The distribution of the ...     

Numerical analysis of the nonlinear vibration of axisymmetric liquid bridges

The vibration of axisymmetric liquid bridges is analyzed numerically in the framework of the one-dimensional approximation. Nonlinear effects on both the interface deformation and velocity field are studied. The solutions of the Lee, average, and Cosserat models are compared. For the Cosserat model, the results are compared with those obtained from an asymptotic analysis performed in the Plateau–Rayleigh stability limit. In addition, some results concerning the influence of the viscosity on the linear vibrations are reported.