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.     

Chaotic Oscillation of Saddle Form Cable-Suspended Roofs under Vertical Excitation Action

The purpose of this paper is to investigate the dynamic behaviour of saddle form cable-suspended roofs under vertical excitation action. The governing equations of this problem are system of nonlinear partial differential and integral equations. We first establish a spectral equation, and then consider a model with one coefficient, i.e., a perturbed Duffing equation. The analytical solution is derived for the Duffing equation. Successive approximation solutions can be obtained in likely way for each time to only one new unknown function of time. Numerical results are given for our analytical solution. By using the Melnikov method, it is shown that the spectral system has chaotic solutions and subharmonic solutions under determined parametric conditions.     

Stochastic averaging of energy harvesting systems

A stochastic averaging method is proposed for nonlinear energy harvesters subjected to external white Gaussian noise and parametric excitations. The Fokker–Planck–Kolmogorov equation of the coupled electromechanical system of energy harvesting is a three variables nonlinear parabolic partial differential equation whose exact stationary solutions are generally hard to find. In order to overcome difficulties in solving higher dimensional nonlinear partial differential equations, a transformation scheme is applied to decouple the electromechanical equations. The averaged Itô equations are derived via the standard stochastic averaging method, then the FPK equations of the decoupled system are obtained. The exact stationary solution of the averaged FPK equation is used to determine the probability densities of the displacement, the velocity, the amplitude, the joint probability densities of the displacement and velocity, and the power of the stationary response. The effects of the system parameters on the output power are examined. The approximate analytical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulations.     

Application of a modified Lindstedt–Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation

An analytical approach is developed for the nonlinear oscillation of a conservative, two-degree-of-freedom (TDOF) mass-spring system with serial combined linear–nonlinear stiffness excited by a constant external force. The main idea of the proposed approach lies in two categories, the first one is the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation. Another is the treatment a quadratic nonlinear oscillator (QNO) by the modified Lindstedt–Poincaré (L-P) method presented recently by the authors. The first-order and second-order analytical approximations for the modified L-P method are established for the QNOs with satisfactory results. After solving the nonlinear differential equation, the displacements of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, the modified L-P 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. 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 classical harmonic balance methods. Two examples of nonlinear TDOF mass-spring systems excited by a constant external force are selected and the approximate solutions are verified with the exact solutions derived from the Jacobi elliptic function and also the numerical fourth-order Runge–Kutta solutions.     

Dynamic buckling of a single-degree-of-freedom system with variable mass

In this paper dynamic buckling of the single-degree-of-freedom system with variable mass is analyzed. In the system the mass variation is slow and is a function of slow variable time. Due to mass variation the impact force acts. The motion of the system is described with a nonlinear ordinary differential equation with time variable parameters. A new approximate analytic criterion of dynamic buckling for the non-autonomous systems which have the conservation law of energy type is developed. The conservation law is formed applying the Noetherian approach. The suggested method allows the determination of dynamic buckling load without solving the corresponding nonlinear differential equation of motion. For this value of dynamic load the motion of the system becomes unbounded. The obtained analytic value is compared with the numeric one. It shows a good agreement.     

On numerical solution to fractional non-linear oscillatory equations

In this article, the multi-step differential transform method (MsDTM) is applied to give approximate solutions of nonlinear ordinary differential equation such as fractional-non-linear oscillatory and vibration equations. The results indicate that the method is very effective and sufficient for solving nonlinear differential equations of fractional order.     

Bifurcations and chaos of an inclined cable

The nonlinear behavior of an inclined cable subjected to a harmonic excitation is investigated in this paper. The Galerkin’s method is applied to the partial differential governing equations to obtain a two-degree-of-freedom nonlinear system subjected to harmonic excitation. The nonlinear systems in the presence of both external and 1:1 internal resonances are transformed to the averaged equations by using the method of averaging. The averaged equations are numerically examined to obtain the steady-state responses and chaotic solutions. Five cascades of period-doubling bifurcations leading to chaotic solutions, 3-periodic solutions leading to chaotic solution, boundary crisis phenomena, as well as the Shilnikov mechanism for chaos, are observed. In order to study the global dynamics of an inclined cable, after determining the averaged equations of motion in a suitable form, a new global perturbation technique developed by Kova?i? and Wiggins is used. This technique provides analytical results for the critical parameter values at which the dynamical system, through the Shilnikov type homoclinic orbits, possesses a Smale horseshoe type of chaos.     

Nonlinear dynamical behavior of shallow cylindrical reticulated shells

By using the method of quasi-shells,the nonlinear dynamic equations of three-dimensional single-layer shallow cylindrical reticulated shells with equilateral tri- angle cell are founded.By using the method of the separating variable function,the transverse displacement of the shallow cylindrical reticulated shells is given under the conditions of two edges simple support.The tensile force is solved out from the compati- ble equations,a nonlinear dynamic differential equation containing second and third order is derived by using the method of Galerkin.The stability near the equilibrium point is discussed by solving the Floquet exponent and the critical condition is obtained by using Melnikov function.The existence of the chaotic motion of the single-layer shallow cylin- drical reticulated shell is approved by using the digital simulation method and Poincarémapping.     

Nonlinear analysis of chatter vibration in a cylindrical transverse grinding process with two time delays using?a?nonlinear time transformation method

A nonlinear dynamic system of cylindrical transverse grinding process is studied in this paper. The system consists of a grinding wheel and a workpiece, which are connected to the base by spring-damper elements, interacting with nonlinear normal forces. This two DOF model includes two time delays originated from the regenerative effects of the workpiece and the grinding wheel. Bifurcation points are located using a numerical algorithm by which we can find all the eigenvalues in a given rectangular region on the complex plane for the delayed differential equations. Supercritical bifurcation has been found for some sets of system parameter values. The amplitudes of the limit cycles are predicted using a nonlinear time transformation method, which is similar to the harmonic balance approach in that a periodic solution is approximated by a Fourier series. However, the main difference is that a nonlinear time ? is introduced in the Fourier series rather than the physical time t. The analytical solutions of stable limit cycles up to the third harmonics are compared with numerical simulations for the retarded system. It is shown that the proposed method gives accurate approximate solutions.     

基于黎曼几何的非线性振子动力学分析递推解析算法

基于黎曼几何和变分原理,推导了黎曼流形上非线性耗散动力系统的二阶微分动力学方程,并运用流形收缩的概念将动力学方程离散化,进而建立了相应的递推求解格式。选取3个自治非线性阻尼振子系统,分别采用递推解析算法和龙格库塔法求解微分动力学方程,并比较分析了不同的时间步长下两种算法的计算耗时。结果表明,与龙格库塔法相比,基于黎曼几何的递推算法不仅能得到每一时步的解析表达式,而且计算耗时短,计算效率高。基于黎曼流形的动力学方程递推算法为非线性动力学系统的解析求解提供了新思路。     

Nonlinear responses of a cable-beam coupled system under parametric and external excitations

This paper analytically investigates the nonlinear responses of a cable-beam coupled system under the combined effects of internal and external resonance. The cable is considered a geometric nonlinearity, and the beam is considered as Euler–Bernoulli model, but it is coupled by fixing it at one end. The coupled nonlinear differential equations are formulated by using the Hamilton principle. The spatial problem is solved by using Galerkin’s method to simplify the governing equations to a set of ordinary differential equations. Applying the multiple time scales method to the ordinary differential equations, the first approximate solutions and solvability condition are derived. The effects of the cable sag to span ratio, mass ratio, and stiffness ratio on the nonlinear responses are investigated. The results show good agreement between the analytical and numerical solutions especially near the external resonance frequency.     

Methods of Identification of Nonlinear Mechanical Vibrating Systems

Methods for determination of the dynamic characteristics and parameters of mechanical vibrating systems by processing experimental data on controlled vibrations are presented. These methods are intended for construction of mathematical models of objects to be identified and classed as parametric and nonparametric methods. The quadrature formulas of the nonparametric-identification method are derived by inverting the integral parameters of approximate analytical solutions of nonlinear differential equations. The parametric-identification method involves setting up and solving systems of linear algebraic equations in the sought-for inertia, stiffness, and dissipation parameters by integrating experimental processes using special weighting functions. Depending on the type of the nonlinearity of the vibrating system and the method of representing experimental processes, the weighting functions can be oriented toward displacement, velocity, or acceleration gauges. The results of studies made mainly at the Institute of Mechanics of the National Academy of Sciences of Ukraine are presented     

Construction of approximate analytical solutions to strongly nonlinear damped oscillators

An analytical approximate method for strongly nonlinear damped oscillators is proposed. By introducing phase and amplitude of oscillation as well as a bookkeeping parameter, we rewrite the governing equation into a partial differential equation with solution being a periodic function of the phase. Based on combination of the Newton’s method with the harmonic balance method, the partial differential equation is transformed into a set of linear ordinary differential equations in terms of harmonic coefficients, which can further be converted into systems of linear algebraic equations by using the bookkeeping parameter expansion. Only a few iterations can provide very accurate approximate analytical solutions even if the nonlinearity and damping are significant. The method can be applied to general oscillators with odd nonlinearities as well as even ones even without linear restoring force. Three examples are presented to illustrate the usefulness and effectiveness of the proposed method.     

输液管的非线性振动、分叉与混沌——现状与展望

围绕输液管的非线性振动、分叉与混沌问题，对近几年来的主要研究工作加以综述并提出预测，其中包括运动方程中非线性项的归纳与讨论、非线性动力分析的一些现代计算方法、定常流速下输液管的分叉与混沌行为、振荡流速下输液管的参数共振以及今后值得进一步研究的某些问题．     

Modeling reactive transport using exact solutions for first-order reaction networks

Operator splitting is often used for solving advection-dispersion-reaction (ADR) equations. Each operator can be solved separately using an algorithm appropriate to its mathematical behavior. Although a lot of research has been done in operator splitting for solving ADR equations, numerical approaches for the reaction operator are computationally expensive. To meet the convergence criteria of ODE (ordinary differential equation) or DAE (differential algebraic equation) solvers, a transport time step has to be subdivided into a large number of reaction time steps. Additional computation effort is also required to reduce the splitting error. In this paper, we develop exact solutions of various first-order reaction networks for the reaction operator and couple those solutions with numerical solutions of the transport operator. The reactions are treated as local phenomena and simulated using exact solutions that we develop, while advection and dispersion are treated as global processes and simulated numerically. The proposed method avoids the numerical error from the reaction operator and requires a single-step calculation to solve the reaction operator. Compared to conventional operator-splitting methods, the proposed method offers both computational efficiency and simulation accuracy.     

Remarks on the Perturbation Methods in Solving the Second-Order Delay Differential Equations

The paper presents a study on the validity of perturbation methods, suchas the method of multiple scales, the Lindstedt–Poincaré method and soon, in seeking for the periodic motions of the delayed dynamic systemsthrough an example of a Duffing oscillator with delayed velocityfeedback. An important observation in the paper is that the method ofmultiple scales, which has been widely used in nonlinear dynamics, worksonly for the approximate solutions of the first two orders, and givesrise to a paradox for the third-order approximate solutions of delaydifferential equations. The same problem appears when theLindstedt–Poincaré method is implemented to find the third-orderapproximation of periodic solutions for delay differential equations,though it is effective in seeking for any order approximation ofperiodic solutions for nonlinear ordinary differential equations. Apossible explanation to the paradox is given by the results obtained byusing the method of harmonic balance. The paper also indicates thatthese perturbation methods, despite of some shortcomings, are stilleffective in analyzing the dynamics of a delayed dynamic system sincethe approximate solutions of the first two orders already enable one togain an insight into the primary dynamics of the system.     

A new multistage spectral relaxation method for solving chaotic initial value systems

In this paper, we present a new pseudospectral method application for solving nonlinear initial value problems (IVPs) with chaotic properties. The proposed method, called the multistage spectral relaxation method (MSRM) is based on a novel technique of extending Gauss–Siedel type relaxation ideas to systems of nonlinear differential equations and using the Chebyshev pseudo-spectral methods to solve the resulting system on a sequence of multiple intervals. In this new application, the MSRM is used to solve famous chaotic systems such as the such as Lorenz, Chen, Liu, Rikitake, Rössler, Genesio–Tesi, and Arneodo–Coullet chaotic systems. The accuracy and validity of the proposed method is tested against Runge–Kutta and Adams–Bashforth–Moulton based methods. The numerical results indicate that the MSRM is an accurate, efficient, and reliable method for solving very complex IVPs with chaotic behavior.     

A mixed modal-differential quadrature method for free and forced vibration of beams in contact with fluid

A simple and accurate mixed modal-differential quadrature formulation is proposed to study the dynamic behavior of beams in contact with fluid. Both free and forced vibration problems are considered. The proposed mixed methodology uses the modal technique for the structural domain while it applies the differential quadrature method (DQM) to the fluid domain. Thus, the governing partial differential equations of the beam and fluid are reduced to a set of ordinary differential equations in time. In the case of forced vibration, the Newmark time integration scheme is employed to solve the resulting system of ordinary differential equations. The proposed formulation, in general, combines the simplicity of the modal method and high accuracy and efficiency of the DQM. Its application is shown by solving some beam-fluid interaction problems. Comparisons with analytical solutions show that the present method is very accurate and reliable. To demonstrate its efficiency, the test problems are also solved using the finite element method (FEM). It is found that the proposed method can produce better accuracy than the FEM using less computational time. The technique presented in this investigation is general and can be used to solve various fluid-structure interaction problems.     

A New Method for Approximate Analytical Solutions to Nonlinear Oscillations of Nonnatural Systems

This paper deals with nonlinear oscillations of a conservative,nonnatural, single-degree-of-freedom system with odd nonlinearity. Bycombining the linearization of the governing equation with the method ofharmonic balance, we establish approximate analytical solutions for thenonlinear oscillations of the system. Unlike the classical harmonicbalance method, the linearization is performed prior to proceeding withharmonic balancing thus resulting in linear algebraic equations insteadof nonlinear algebraic equations. Hence, we are able to establish theapproximate analytical formulas for the exact period and periodicsolution. These approximate solutions are valid for small as well aslarge amplitudes of oscillation. Two examples are presented toillustrate that the proposed formulas can give excellent approximateresults.     

轮对非线性动力学系统蛇行运动的解析解

在对铁路车辆系统的极限环幅值和非线性临界速度进行分析时通常采用数值方法, 不便于研究其随系统参数的变化规律. 轮对系统保留了影响车辆系统动力学性能的几个关键要素: 如轮轨几何非线性约束、轮轨接触蠕滑关系和悬挂系统等, 可以反映铁路车辆系统蛇行运动的本质特性. 轮对系统自由度少、参数少, 可以采用解析方法进行分析. 本文选取合适的特征量把轮对非线性动力学方程无量纲化, 得到了带有小参数的两自由度微分方程; 采用多尺度方法对该方程进行了解析求解; 给出了轮对系统极限环幅值的解析表达式并对其稳定性进行了判定; 给出了轮对系统的分岔速度解析表达式, 并进而获得系统的非线性临界速度的解析表达式. 在对得到的解析解用数值结果进行验证后, 用得到的解析解进行了系统参数影响分析. 传统的分岔图计算方法(如降速法、路径跟踪法等)需对微分方程进行大量数值积分计算方可求解系统的非线性临界速度值, 而通过本文获得的解析表达式可直接给出系统的非线性临界速度值和极限环幅值, 便于研究轮对系统动力学特性随参数的变化规律,进行快速方案比对和筛选, 为转向架结构优化设计提供参考.