Non-linear responses of suspended cables to primary resonance excitations

We investigate the non-linear forced responses of shallow suspended cables. We consider the following cases: (1) primary resonance of a single in-plane mode and (2) primary resonance of a single out-of-plane mode. In both cases, we assume that the excited mode is not involved in an autoparametric resonance with any other mode. We analyze the system by following two approaches. In the first, we discretize the equations of motion using the Galerkin procedure and then apply the method of multiple scales to the resulting system of non-linear ordinary-differential equations to obtain approximate solutions (discretization approach). In the second, we apply the method of multiple scales directly to the non-linear integral-partial-differential equations of motion and associated boundary conditions to determine approximate solutions (direct approach). We then compare results obtained with both approaches and discuss the influence of the number of modes retained in the discretization procedure on the predicted solutions.     

NON-LINEAR DYNAMIC ANALYSIS OF THE TWO-DIMENSIONAL SIMPLIFIED MODEL OF AN ELASTIC CABLE

The non-linear behavior of an elastic cable subjected to a harmonic excitation is investigated in this paper. Using Garlerkin's method and method of multiple scales, the discrete dynamical equations and a set of first order non-linear differential equations are obtained. A numerical simulation is used to obtain the steady state response and stable solutions. Finally the coupled dynamic features between the out-planar pendulation and the in-planar vibration of an elastic cable are analyzed.     

A comparison of three perturbation methods for non-linear hyperbolic waves

Simple wave solutions of non-linear hyperbolic equations are studied by using the method of renormalization, the analytic method of characteristics, and the method of multiple scales. It is shown that the results of the method of renormalization depend on whether the potential function or the velocity is normalized. This arbitrariness does not occur when using either the analytic method of characteristics or the method of multiple scales. However, special consideration must be given in determining the potential from the velocity obtained by the analytic method of characteristics. No such consideration is needed when the method of multiple scales is applied. The first term obtained for the potential by the method of multiple scales contains a cumulative term in addition to a non-cumulative term. This first-order term is shown to yield the equal area rule for shock waves, and the slope of an equipotential line is the arithmetic mean of the slope of the characteristic in the unperturbed medium and the slope of the characteristic at the point under consideration.     

On the influence of Poisson's ratio on circular plate natural frequencies

Nayfeh and Kluwick [1] studied simple-wave solutions of non-linear hyperbolic equations comparatively by using three methods: namely, (1) the method of renormalization, (2) the analytic method of characteristics and (3) the method of multiple scales. The method of multiple scales was found to be superior to the other two, and yielded correct results. However, the method seems to involve difficulties in respect to conventional concepts of partial differentiation. In this context, this note presents another method—the Poincaré-Lighthill-Kuo method—which provides not only correct end results but also an elegant procedure.     

LARGEST LYAPUNOV EXPONENT AND ALMOST CERTAIN STABILITY ANALYSIS OF SLENDER BEAMS UNDER A LARGE LINEAR MOTION OF BASEMENT SUBJECT TO NARROWBAND PARAMETRIC EXCITATION

The first order approximate solutions of a set of non-liner differential equations, which is established by using Kane's method and governs the planar motion of beams under a large linear motion of basement, are systematically derived via the method of multiple scales. The non-linear dynamic behaviors of a simply supported beam subject to narrowband random parametric excitation, in which either the principal parametric resonance of its first mode or a combination parametric resonance of the additive type of its first two modes with or without 3:1 internal resonance between the first two modes is taken into consideration, are analyzed in detail. The largest Lyapunov exponent is numerically obtained to determine the almost certain stability or instability of the trivial response of the system and the validity of the stability is verified by direct numerical integration of the equation of motion of the system.     

Analytical method for steady state vibration of system with localized non-linearities using convolution integral and Galerkin method

The analytical method using transfer function or impulse response is very effective for analyzing non-linear systems with localized non-linearities. This is because the number of non-linear equations can be reduced to that of the equations with respect to points connected with the non-linear element. In the present paper, analytical method for the steady state vibration of non-linear system including subharmonic vibration is proposed by utilizing convolution integral and the impulse response. The Galerkin method is introduced to solve the non-linear equations formulated by the convolution integral, and then the steady state vibration is obtained. An advantage of the present method is that stability or instability of the steady state vibration can be discriminated by the transient analysis from convolution integral. The three-degree-of-freedom mass-spring system is shown as a numerical example and the proposed method is verified by comparing with the result by Runge-Kutta-Gill method.     

STATISTICAL SEISMIC RESPONSE ANALYSIS AND RELIABILITY DESIGN OF NONLINEAR STRUCTURE SYSTEM

Industrial structure systems may have non-linearity, and are also sometimes exposed to the danger of earthquake. In the design of such system, these factors should be accounted for from the viewpoint of reliability. This paper proposes a method to analyze seismic response and reliability design of a complex non-linear structure system under random excitation. The actual random excitation is represented to the corresponding Gaussian process for the statistical analysis. Then, the non-linear system is subjected to this random process. The non-linear structure system is modelled by substructure synthesis method (SSM) procedure. The non-linear equations are expanded sequentially. Then, the perturbed equations are solved in probabilistic method. Several statistical properties of a random process that are of interest in random vibration applications are reviewed in accordance with the non-linear stochastic problem. The system performance condition in the design of system is that responses caused by random excitation be limited within safe bounds. Thus, the reliability of the system is considered according to the crossing theory. Comparing with the results of the numerical simulation proved the efficiency of the proposed method.     

The effect of the number of nodal diameters on non-linear interactions in two asymmetric vibration modes of a circular plate

In order to investigate the effect of the number of nodal diameters on non-linear interactions in asymmetric vibrations of a circular plate, a primary resonance of the plate is considered. The plate is assumed to have an internal resonance in which the ratio of the natural frequencies of two asymmetric modes is three to one. The response of the plate is expressed as an expansion in terms of the linear, free oscillation modes, and its amplitude is considered to be small but finite, and the method of multiple scales is used. In view of the corrected solvability conditions for the responses, it has been found that in order for the modes to interact, the ratio of the numbers of nodal diameters of two modes must be either three to one or one to one. In this study the one-to-one case, in which the modes have the same number of nodal diameters, is examined. The non-linear governing equations are reduced to a system of autonomous ordinary differential equations for amplitude and phase variables by means of the corrected solvability conditions. The steady state responses and their stability are determined by using this system. The result shows very complicated interactions between two modes by telling existence of non-vanishing amplitudes of the mode not directly excited.     

THE DYNAMIC ANALYSIS OF A BEAM-MASS SYSTEM DUE TO THE OCCURRENCE OF TWO-COMPONENT PARAMETRIC RESONANCE

The objective of this paper is an analytical and numerical study of the dynamics of a beam--mass system. Special attention is given to the phenomena arising due to the motion of the attached mass and modal interactions produced by the existence of multi-component, specifically two-component, parametric resonance under primary resonance. The two-component parametric resonance occurs when the sums or the differences among internal frequencies are the same, or close, as the dimensionless speed parameter of the moving mass. The effects of two-component parametric resonance post on dynamic condition are investigated. Resonance generated by more than two-component modes are neglected due to its remote probability of occurrence in nature.The mechanics of the problem is Newtonian. Based on the assumption that when the moving mass is set in motion the mass is assumed to be rolling on the beam, the mechanics, including the effects due to friction and convective accelerations, of the interface between the moving mass and the beam are determined.Based on the Bernoulli-Euler beam theory, the coupled non-linear equations of motion of an inextensible beam with an attached moving mass are derived. By employing Galerkin procedure, the partial differential equations which describe the motion of a beam-mass system are reduced to an initial-value problem with finite dimensions. The method of multiple time scales is applied to consider the solutions and the occurrence of internal resonance of the resulting multi-degree-of-freedom beam--mass system with time dependent coefficients.     

A dynamic Lagrangian frequency-time method for the vibration of dry-friction-damped systems

A new frequency-time domain procedure, the dynamic Lagrangian mixed frequency-time method (DLFT), is proposed to calculate the non-linear steady state response to periodic excitation of structural systems subject to dry friction damping. In this formulation, the dynamic Lagrangians are defined as the non-linear contact forces obtained from the equations of motion in the frequency domain, with the adjunction of a penalization on the difference between the interface displacements calculate by the non-linear solver in the frequency domain and those calculated in the time domain from the non-linear contact forces, thus accounting for Coulomb friction and non-penetration conditions. The dynamic Lagrangians allow one to solve for the non-linear forces between two points in contact without using artifacts such as springs. The new DLFT method is thus particularly well suited to handling finite element models of structures in frictional contact, as it does not require a special model for the contact interface. Dynamic Lagrangians are also better suited to frequency-domain friction problems than the traditional time-domain method of augmented Lagrangians. Furthermore, a reduction of the non-linear system to relative interface displacements is introduced to decrease the computation time. The DLFT method is validated for a beam in contact with a flexible dry friction element connected to ground, for frictional constraints that feature two-dimensional relative motion. Results are also obtained for a large-scale structural system with a large number of one-dimensional dry-friction dampers. The DLFT method is shown to be accurate and fast, and it does not suffer from convergence problems, at least in the examples studied.     

On solutions of the fifth-order dispersive equations with porous medium type non-linearity

In this work, we focus on obtaining the exact solutions of the fifth-order semi-linear and non-linear dispersive partial differential equations, which have the second-order diffusion-like (porous-type) non-linearity. The proposed equations were not studied in the literature in the sense of the exact solutions. We reveal solutions of the proposed equations using the classical Riccati equations method. The obtained exact solutions, which can play a key role to simulate non-linear waves in the medium with dispersion and diffusion, are illustrated and discussed in details.     

Response of a bar constrained by a non-linear spring to a harmonic excitation

An analysis is presented of the longitudinal response of a bar constrained by a non-linear spring to a harmonic excitation. The method of multiple scales is used to determine equations describing the evolution of the amplitudes and phases with damping, non-linearity and the cases of primary, subharmonic, superharmonic, combination and ultrasubharmonic resonances. These equations are used to determine the steady state responses and their stability.     

APPROXIMATE ANALYTICAL SOLUTIONS FOR PRIMARY CHATTER IN THE NON-LINEAR METAL CUTTING MODEL

This paper considers an accepted model of the metal cutting process dynamics in the context of an approximate analysis of the resulting non-linear differential equations of motion. The process model is based upon the established mechanics of orthogonal cutting and results in a pair of non-linear ordinary differential equations which are then restated in a form suitable for approximate analytical solution. The chosen solution technique is the perturbation method of multiple time scales and approximate closed-form solutions are generated for the most important non-resonant case. Numerical data are then substituted into the analytical solutions and key results are obtained and presented. Some comparisons between the exact numerical calculations for the forces involved and their reduced and simplified analytical counterparts are given. It is shown that there is almost no discernible difference between the two thus confirming the validity of the excitation functions adopted in the analysis for the data sets used, these being chosen to represent a real orthogonal cutting process. In an attempt to provide guidance for the selection of technological parameters for the avoidance of primary chatter, this paper determines for the first time the stability regions in terms of the depth of cut and the cutting speed co-ordinates.     

The effect of spin on the non-linear resonant motions of a dynamical system

The motions of a two degree of freedom mechanical oscillator in a state of internal resonance due to the non-linear coupling between its modes are analyzed by the method of multiple scales. The system is connected by a motor to a vertical shaft driven at a constant spin rate relative to inertial space. It is shown that the non-linear resonance phenomenon can effectively be controlled by properly changing the spin rate of the motor. In addition, the transition curves that separate the non-linear resonant and the non-resonant motions of the system are also determined analytically by a straightforward perturbation method. The analytical expression for the transition curves is used in connection with the multiple scale analysis to yield a refined approximation for the main characteristics of the non-linear resonant motion.     

ASYMMETRIC NON-LINEAR FORCED VIBRATIONS OF FREE-EDGE CIRCULAR PLATES. PART 1: THEORY

In this article, a detailed study of the forced asymmetric non-linear vibrations of circular plates with a free edge is presented. The dynamic analogue of the von Kàrmàn equations is used to establish the governing equations. The plate displacement at a given point is expanded on the linear natural modes. The forcing is harmonic, with a frequency close to the natural frequency ω_kn of one asymmetric mode of the plate. Thus, the vibration is governed by the two degenerated modes corresponding to ω_kn, which are one-to-one internally resonant. An approximate analytical solution, using the method of multiple scales, is presented. Attention is focused on the case where one configuration which is not directly excited by the load gets energy through non-linear coupling with the other configuration. Slight imperfections of the plate are taken into account. Experimental validations are presented in the second part of this paper.     

Parametrically excited non-linear multidegree-of-freedom systems with repeated natural frequencies

A method for analyzing multidegree-of-freedom systems having a repeated natural frequency subjected to a parametric excitation is presented. Attention is given to the ordering of the various terms (linear and non-linear) in the governing equations. The analysis is based on the method of multiple scales. As a numerical example involving a parametric resonance, panel flutter is discussed in detail in order to illustrate the type of results one can expect to obtain with this analysis. Some of the analytical results are verified by a numerical integration of the governing equations.     

有界噪声激励下带有时滞反馈的随机Mathieu-Duffing系统的响应

研究了参数激励下带有时滞反馈的随机Mathieu-Duffing方程的主参数共振响应问题.运用多尺度方法分离了系统的快慢变量.分析了系统的分岔性质,发现调谐参数、时滞、时滞项的系数以及非线性项的强度等都可以影响系统的分岔行为,适当选择这些参数可以改变系统的分岔响应.同时,还讨论了非零解的稳定性,得到了非零解稳定的充要条件,而且发现在随机激励的带宽较小时,系统的多解现象仍然存在,分岔和跳跃现象仍会发生,数值模拟验证了理论推导的有效性. 关键词： 随机Mathieu-Duffing系统 多尺度 稳定性 分岔     

Vibration suppression of rotating beams using time-varying internal tensile force

A new strategy for vibration suppression of a rotating beam using a time-increasing internal tensile force is proposed in this paper. Nonlinear coupled longitudinal and bending equations of motion are derived in non-dimensional form using the Hamilton principle. The first-order analytical solution of the equations of motion is obtained using the Galerkin technique combined with the multiple scales method (MSM). Numerical simulations are then performed for various increasing rates of the internal tensile force and performance of the vibration suppression strategy is studied. A very close agreement between the simulation results obtained by the numerical integration and the first-order analytical solution is achieved. Forced vibrations of the system for input excitations of either a sinusoidal or a random function with white noise time history are considered. The simulation results and dynamic performance of the suppressed system for an externally excited rotating beam show an interesting phenomenon of the form of remarkable effectiveness of the proposed vibration reduction strategy.     

Response of non-linear non-conservative systems of two degrees of freedom to transient excitations

This paper deals with the approximate analysis of non-linear non-conservative systems of two degrees of freedom subjected to transient excitations. By using a transformation of the co-ordinates, the governing differential equations of the system are brought into a form to which the method of averaging of Krylov and Bogoliubov can be applied. The response of a representative spring-mass-damper system to typical pulses like a blast pulse and a half sinusoidal pulse is determined. The validity of the approach is demonstrated by comparison of the approximate solutions with numerical results obtained on a digital computer.     

A COMPARISON OF THREE TECHNIQUES USING STEADY STATE DATA TO IDENTIFY NON-LINEAR MODAL BEHAVIOR OF AN EXTERNALLY EXCITED CANTILEVER BEAM

Non-linear system identification is used to generate models of modes in physical structures. Analysis of the theoretical non-linear model of a cantilevered beam is used to predict the inputs to the physical system that will produce responses suitable for enhanced parameter estimation, thereby improving the model. Three identification techniques are described and applied to both numerical and experimental data: the first is based on the continuous-time differential equation model of the system, the second uses relationships generated by the method of harmonic balance, and the third is based on fitting steady state response data to the amplitude and phase modulation equations resulting from a multiple time scales analysis. The performance of each method improves as the non-linearities in the system become more pronounced. The benefits and limitations of the methods are discussed.