Self-Excited Oscillators with Asymmetric Nonlinearities and One-to-Two Internal Resonance

An analysis is presented on the dynamics of asymmetric self-excited oscillators with one-to-two internal resonance. The essential behavior of these oscillators is described by a two degree of freedom system, with equations of motion involving quadratic nonlinearities. In addition, the oscillators are under the action of constant external loads. When the nonlinearities are weak, the application of an appropriate perturbation approach leads to a set of slow-flow equations, governing the amplitudes and phases of approximate motions of the system. These equations are shown to possess two different solution types, generically, corresponding to static or periodic steady-state responses of the class of oscillators examined. After complementing the analytical part of the work with a method of determining the stability properties of these responses, numerical results are presented for an example mechanical system. Firstly, a series of characteristic response diagrams is obtained, illustrating the effect of the technical parameters on the steady-state response. Then results determined by the application of direct numerical integration techniques are presented. These results demonstrate the existence of other types of self-excited responses, including periodically-modulated, chaotic, and unbounded motions.     

Free Vibration in a Class of Self-Excited Oscillators with 1:3 Internal Resonance

Free vibration of a two degree of freedom weakly nonlinear oscillator is investigated. The type of nonlinearity considered is symmetric, it involves displacement as well as velocity terms and gives rise to self-excited oscillations in many engineering applications. After presenting the equations of motion in a general form, a perturbation methodology is applied for the case of 1:3 internal resonance. This yields a set of four slow-flow nonlinear equations, governing the amplitudes and phases of approximate motions of the system. It is then shown that these equations possess three distinct types of solutions, corresponding to trivial, single-mode and mixed-mode response of the system. The stability analysis of all these solutions is also performed. Next, numerical results are presented by applying this analysis to a specific practical example. Response diagrams are obtained for various combinations of the system parameters, in an effort to provide a complete picture of the dynamics and understand the transition from conditions of 1:3 internal resonance to non-resonant response. Emphasis is placed on identifying the effect of the linear damping, the frequency detuning and the stiffness nonlinearity parameters. Finally, the predictions of the approximate analysis are confirmed and extended further by direct integration of the averaged equations. This reveals the existence of other regular and irregular motions and illustrates the transition from phase-locked to drift response, which takes place through a Hopf bifurcation and a homoclinic explosion of the averaged equations.     

Interaction of free and forced nonlinear normal modes in two-DOF dissipative systems under resonance conditions

The resonance dynamics of a dissipative spring-mass and of a dissipative spring-pendulum system is studied. Internal resonance case is considered for the first system; both external resonances and simultaneous external and internal resonance are studied for the second one. Analysis of the systems resonance behavior is made on the base of the concept of nonlinear normal vibration modes (NNMs) by Kauderer and Rosenberg, which is generalized for dissipative systems. The multiple time scales method under resonance conditions is applied. The resulting equations are reduced to a system with respect to the system energy, arctangent of the amplitudes ratio and the difference of phases of required solution in the resonance vicinity. Equilibrium positions of the reduced system correspond to nonlinear normal modes; in energy dissipation case they are quasi-equilibriums. Analysis of the equilibrium states of the reduced system permits to investigate stability of nonlinear normal modes in the resonance vicinity and to describe transfer from unstable vibration mode to stable one. New vibration regimes, which are called transient nonlinear normal modes (TNNMs) are obtained. These regimes take place only for some particular levels of the system energy. In the vicinity of values of time, corresponding to these energy levels, the TTNM attract other system motions. Then, when the energy decreases, the transient modes vanish, and the system motions tend to another nonlinear normal mode, which is stable in the resonance vicinity. The reliability of the obtained analytical results is confirmed by numerical and numerical-analytical simulations.     

Internal-external resonance of a curved pipe conveying fluid resting on a nonlinear elastic foundation

In this study, the forced vibration of a curved pipe conveying fluid resting on a nonlinear elastic foundation is considered. The governing equations for the pipe system are formed with the consideration of viscoelastic material, nonlinearity of foundation, external excitation, and extensibility of centre line. Equations governing the in-plane vibration are solved first by the Galerkin method to obtain the static in-plane equilibrium configuration. The out-of-plane vibration is simplified into a constant coefficient gyroscopic system. Subsequently, the method of multiple scales (MMS) is developed to investigate external first and second primary resonances of the out-of-plane vibration in the presence of three-to-one internal resonance between the first two modes. Modulation equations are formed to obtain the steady state solutions. A parametric study is carried out for the first primary resonance. The effects of damping, nonlinear stiffness of the foundation, internal resonance detuning parameter, and the magnitude of the external excitation are investigated through frequency response curves and force response curves. The characteristics of the single mode response and the relationship between single and two mode steady state solutions are revealed for the second primary resonance. The stability analysis is carried out for these plots. Finally, the approximately analytical results are confirmed by the numerical integrations.     

Response of a non-linear system with strong damping to multifrequency excitations

Summary In this paper, a new asymptotic calculation method is presented for a class of nonlinear nonautonomous vibration systems with multiple external periodic interferences. Simple calculation formulae of an asymptotic solution for resonance and off-resonance vibration are derived. The paper is concerned with a vibration system representing a class of nonlinear oscillators. Consequently, the calculation of a class of nonlinear oscillators is routinized. Two different Duffing equations are verified which shows that the results are completely in accordance with the solutions of references [3, 4, 6]. The derivation of solutions of Duffing equations becomes easier and simpler. In addition, some errors in reference [6] are pointed out.     

内共振条件下轴向运动梁磁弹性超谐共振研究

以载流导线激发的磁场中轴向运动梁为研究对象,同时考虑外激励力作用,推导出梁的磁弹性非线性振动方程．通过位移函数的设定和伽辽金积分法,将非线性振动方程离散为常微分方程组．采用多尺度法得到系统的近似解析解．应用Matlab 和Mathematica 软件求解幅频响应方程,并对稳态解进行稳定性判定．通过具体算例得到前两阶假设模态的响应幅值随不同参数的变化规律．结果发现：系统在内共振条件下发生超谐波共振时,二阶假设模态幅值明显小于一阶;随着外激励的增大,多值解区域范围明显缩小;随着电流强度增加,振动幅值减小,表明载流导线能够起到控制共振的作用．     

Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system

Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration absorber system are studied for resonant excitations. The method of averaging is used to obtain first-order approximations to the response of the system. A complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. The locked pendulum mode of response is found to bifurcate to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mistuned away from the exact internal resonance condition. The software packages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super-and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.     

Dynamics of a mass–spring–beam with 0:1:1 internal resonance using the analytical and continuation method

The nonlinear harmonic response of an autoparametric system comprised of a linear oscillator with a vertically attached flexural beam is investigated and the capability of the beam as a vibration absorber is assessed. A weak torsional spring is used for constraining the rotation of the beam giving rise to an almost non-flexural rotational mode with a low frequency. The system parameters are also tuned to enforce the zero-to-one-to-one internal resonance condition. The Lagrange’s formulation accompanied by the assumed-mode method is used to derive the discretized equations based on the order three nonlinear Euler–Bernoulli beam theory. An analytical solution is developed based on the method of multiple scales where the generalized coordinate corresponding to the non-flexural rotational mode is approximated by higher order perturbation expansion than the other coordinates, due to much larger contribution of the non-flexural rotation to the response. Comprehensive response and bifurcation analysis are performed using analytical and direct numerical solutions. The results are obtained for vertically-aligned and also initially inclined beams and various interesting behaviors are recognized for different non-dimensional system parameters. Different types of bifurcations such as the Pitch-fork, Hopf, Period-doubling and symmetry breaking bifurcations are observed in the solution of slow-flow equations and some of them are found to be beneficial for vibration absorption in a limited range of excitation amplitudes and frequencies.     

轴向运动正交各向异性板的超谐波共振及稳定性

研究了轴向运动正交各向异性条形薄板在线载荷作用下的超谐波共振问题。通过哈密顿原理导出了几何非线性下正交各向异性条形板的非线性振动方程。运用伽辽金积分法,推得了关于时间变量的量纲归一化非线性振动微分方程组。应用多尺度法求解三阶超谐波共振问题,得到了稳态运动下一阶、二阶、三阶共振形式的共振幅值响应方程。利用Liapunov方法推得不同共振形式稳态解的稳定性判据,并据此分析不同参数对系统稳定性的影响。绘制了振幅特性变化曲线图和与之对应的激发共振多解临界点曲线图,分析系统参数对共振的影响,并预测系统进入非线性共振区域的临界条件。得出激励在特定位置区间时可激发系统的超谐波共振,随着激励幅值的增加,上稳定解支减小,下稳定解支增加,且一阶模态振幅大于二阶、三阶振幅。     

Vibration reduction of a nonlinear spring pendulum under multi external and parametric excitations via a longitudinal absorber

The vibration of a ship pitch-roll motion described by a non-linear spring pendulum system (two degrees of freedom) subjected to multi external and parametric excitations can be reduced using a longitudinal absorber. The method of multiple scale perturbation technique (MSPT) is applied to analyze the response of this system near the simultaneous primary, sub-harmonic and internal resonance. The steady state solution near this resonance case is determined and studied applying Lyapunov’s first method. The stability of the system is investigated using frequency response equations. Numerical simulations are extensive investigations to illustrate the effects of the absorber and some system parameters at selected values on the vibrating system. The simulation results are achieved using MATLAB 7.0 programs. Results are compared to previously published work.     

Three-to-One Internal Resonances in Hinged-Clamped Beams

The nonlinear planar response of a hinged-clamped beam to a primary excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load and a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact due to a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear partial-differential equation and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of primary resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of primary resonance of the first mode, only two-mode solutions are possible, whereas for the case of primary resonance of the second mode, single- and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. A shooting technique is used to calculate limit cycles of the modulation equations and Floquet theory is used to ascertain their stability. The limit cycles correspond to periodically modulated motions of the beam. The limit cycles are found to undergo cyclic-fold bifurcations and period-doubling bifurcations, leading to chaos. The chaotic attractors may undergo boundary crises, resulting in the destruction of the chaotic attractors and their basins of attraction.     

Modal interaction and bifurcations in two degree of freedom duffing oscillators

A methodology is first presented for analyzing long time response of periodically exited nonlinear oscillators. Namely, a systematic procedure is employed for determining periodic steady state response, including harmonic and superharmonic components. The stability analysis of the located periodic motions is also performed, utilizing results of Froquet theory. This methodology is then applied to a special class of two degree of freedom nonlinear oscillators, subjected to harmonic excitation. The numberical results presented in the second part of this study illustrate effects caused by the interaction of the modes as well as effects of the nonlinearities on the steady state response of these oscillators. In addition, sequences of bifurcations are analyzed for softening systems, leading to unbounded response of the model examined. Finally, the importance of higher harmonics on the response of systems with strongly nonlinear characteristics is investigated.     

Vibration and stability of an aircraft tail under simultaneous primary-combined and internal resonance

This paper adds a negative velocity feedback to the dynamical system of twin-tail aircraft to suppress the vibration. The system is represented by two coupled second-order nonlinear differential equations having both quadratic and cubic nonlinearities. The system describes the vibration of an aircraft tail subjected to both multi-harmonic and multi-tuned excitations. The method of multiple time scale perturbation is adopted to solve the nonlinear differential equations and obtain approximate solutions up to the third order approximations. The stability of the proposed analytic solution near the simultaneous primary, combined and internal resonance is studied and its conditions are determined. The effect of different parameters on the steady state response of the vibrating system is studied and discussed by using frequency response equations. Some different resonance cases are investigated numerically     

Nonlinear vibration modes and instability of a conceptual model of a spar platform

Spar floating platforms have been largely used for deepwater drilling, oil and natural gas production, and storage. In extreme weather conditions, such structures may exhibit a highly nonlinear dynamical behavior due to heave-pitch coupling. In this paper, a 2-DOF model is used to study the heave and pitch dynamical response in free and forced vibration. Special attention is given to the determination of the nonlinear vibration modes (NNMs). Nonsimilar and similar NNMs are obtained analytically by direct application of asymptotic and Galerkin-based methods. The results show important NNM features such as instability and multiplicity of modes. The NNMs are used to generate reduced order models consisting of SDOF nonlinear oscillators. This allows analytic parametric studies and the derivation of important features of the system such as its frequency-amplitude relations and resonance curves. The stability is analyzed by the Floquet theory. The analytical results show a good agreement with the numerical solution obtained by direct integration of the equation of motion. Instability analyses using bifurcation diagrams and Mathieu charts are carried out to understand the fundamental mechanism for the occurrence of unstable coupled heave-pitch resonant motions of floating structures in waves and to study the dependencies of the growth rate of unstable motions on physical parameters.     

Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams

The nonlinear planar response of a hinged-clamped beam to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of static axial loads, resulting in a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive three sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the cases of (a) principal parametric resonance of either the first or the second mode, and (b) a combination parametric resonance of the additive type of these modes. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of principal parametric resonance of the first mode or combination parametric resonance of the additive type, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single, and two-mode solutions are possible. The trivial and two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. For some excitation parameters, we found complex responses including period-doubling bifurcations and blue-sky catastrophes.     

气动载荷下旋转运动圆板磁弹性主共振分析

基于Kirchhoff薄板理论与哈密顿原理,建立旋转运动导电圆板的磁-气动弹性非线性动力学方程．根据电磁场基本原理得到旋转运动圆板所受电磁力表达式,同时采用一种简化的气动模型以描述作用于板上的气动载荷．基于贝塞尔函数形式振型函数的选取,应用伽辽金法得到旋转圆板的磁气动弹性轴对称非线性振动微分方程．应用多尺度法推导出主共振下系统的幅频响应方程,并依据Lyapunov方法得到系统稳态运动稳定性判据．通过算例,得到周边夹之约束下圆板主共振的幅频特性曲线图,以及振幅随磁感应强度和激励力幅值的变化曲线图;阐述了不同参数对系统共振幅值的影响规律,并对解的稳定性进行了分析．     

Vibration suppression in multi-tool ultrasonic machining to multi-external and parametric excitations

Ultrasonic machining （USM） is of particular interest for the machining of non-conductive, brittle materials such as engineering ceramics. In this paper, a multi-tool technique is used in USM to reduce the vibration in the tool holder and have reasonable amplitude for the tools. This can be done via dynamic absorbers. The coupling of four nonlinear oscillators of the tool holder and tools representing ultrasonic cutting process are investigated. This leads to a four-degree-of-freedom system subjected to multi-external and multi-parametric excitation forces. The aim of this work is to control the tool holder behavior at simultaneous primary, sub-harmonic and internal resonance condition. Multiple scale perturbation method is used to obtain the solution up to the second order approximations. The different resonance cases are reported and studied numerically. The stability of the system is investigated by using both phase-plane and frequency response techniques. The effects of the different parameters of the tools on the system behavior are studied numerically. Comparison with the available published work is reported.     

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.     

Analytical study of the nonlinear behavior of a shape memory oscillator: Part II—resonance secondary

In this work, the response of a single-degree-of-freedom shape memory oscillator subjected to the excitation harmonic has been investigated. Equation of motion is formulated assuming a polynomial constitutive model to describe the restitution force of the oscillator. Here the method of multiple scales is used to obtain an approximate solution to the equations of the motion describing the modulation equations of amplitude and phase, and to investigate theoretically its stability. This work is presented in two parts. In Part I of this study we showed the modeling of the problem where the free vibration of the oscillator at low temperature is analyzed, where martensitic phase is stable. Part I also presents the investigation dynamics of the primary resonance of the pseudoelastic oscillator. Part II of the work is focused on the study in the secondary resonance of a pseudoelastic oscillator using the model developed in Part I. The analysis of the system in Part I as well as in Part II is accomplished numerically by means of phase portraits, Lyapunov exponents, power spectrum and Poincare maps. Frequency-response curves are constructed for shape memory oscillators for various excitation levels and detuning parameter. A rich class of solutions and bifurcations, including jump phenomena and saddle-node bifurcations, is found.     

Analytical study of the nonlinear behavior of a shape memory oscillator: Part I—primary resonance and free response at low temperatures

In this work, the response of a single-degree-of-freedom shape memory oscillator subjected to the excitation harmonic has been investigated. Equation of motion is formulated assuming a polynomial constitutive model to describe the restitution force of the oscillator. Here the method of multiple scales is used to obtain an approximate solution to the equations of the motion describing the modulation equations of amplitude and phase, and to investigate theoretically its stability. This work is presented in two parts. In Part I of this study we showed the modeling of the problem where the free vibration of the oscillator at low temperature is analyzed, where martensitic phase is stable. Part I also presents the investigation dynamics of the primary resonance of the pseudoelastic oscillator. Part II of the work is focused on the study in the secondary resonance of a pseudoelastic oscillator using the model developed in Part I. The analysis of the system in Part I as well as in Part II is accomplished numerically by means of phase portraits, Lyapunov exponents, power spectrum and Poincare maps. Frequency-response curves are constructed for shape memory oscillators for various excitation levels and detuning parameter. A rich class of solutions and bifurcations, including jump phenomena and saddle-node bifurcations, is found.