Asymptotic response of a two DOF elastoplastic system under harmonic excitation. Internal resonance case

The study of a two DOF elastoplastic system is formulated in a suitable phase space, velocity and force, in which an originally multi-valued restoring force is represented by a proper function. The asymptotic response can thus be studied using the Poincaré map concept and avoiding approximate analytical techniques. On account of the peculiarity of this hysteretic system, which has a well-defined yielding point, its dynamics is studied in a reduced dimension phase space using an efficient numerical algorithm. It is shown that the asymptotic response is always periodic with the period of the driven frequency and is always stable. Thus the response of the oscillator is described by its frequency response curves at various intensities of the excitation. The results presented refer to a system with two linear frequencies in a ratio of 1 : 3. The response is highly complex with numerous peaks corresponding to higher harmonics. The effect of coupling in conditions of internal resonance is a strong modification of the frequency response curves and of the oscillation shape of the structure.     

Periodic and Non-Periodic Oscillations of a Class of Hysteretic Two Degree of Freedom Systems

The equations governing the response of hysteretic systems to sinusoidal forces, which are memory dependent in the classical phase space, can be given as a vector field over a suitable phase space with increased dimension. Hence, the stationary response can be studied with the aids of classical tools of nonlinear dynamics, as for example the Poincaré map. The particular system studied in the paper, based on hysteretic Masing rules, allows the reduction of the dimension of the phase space and the implementation of efficient algorithms. The paper summarises results on one degree of freedom systems and concentrates on a two degree of freedom system as the prototype of many degree of freedom systems. This system has been chosen to be in 1:3 internal resonance situation. Depending on the energy dissipation of the elements restoring force, the response may be more or less complex. The periodic response, described by frequency response curves for various levels of excitation intensity, is highly complex. The coupling produces a strong modification of the response around the first mode resonance, whereas it is negligible around the second mode. Quasi-periodic motion starts bifurcating for sufficiently high values of the excitation intensity; windows of periodic motions are embedded in the dominion of the quasi-periodic motion, as consequence of a locking frequency phenomenon.     

Parametric resonance of a single-degree-of-freedom system with double bilinear hysteresis

A simple pendulum with a hinge of double bilinear hysteretic restraining moment-rotation characteristic under parametric excitation is studied. In contrast with a linear system with viscous damping, a double bilinear hysteretic system leads, in general, to finite response under parametric resonance. The response curves and the conditions under which unbounded response results are given. Further, it is shown that unlike the bilinear hysteretic system, a double bilinear hysteretic system may be shock excited into parametric resonance even when the exciting frequency is outside the parametrically resonant frequency range.     

轴向运动复合材料圆柱壳的非线性振动研究

采用Runge–Kutta法和多尺度法对轴向运动分层复合材料薄壁圆柱壳的非线性振动特性进行了研究。首先根据层合壳理论建立轴向运动分层复合材料薄壁圆柱壳的波动方程,利用Galerkin法对方程进行离散,得到相互耦合模态方程组。然后应用Runge –Kutta法分析了不同参数条件下的幅频特性曲线,得到了系统由于固有频率接近所导致的内共振现象,以及系统呈现软特性等非线性特性。最后采用多尺度法进行了系统1:1内共振时的近似解析分析,对系统在不同参数下的振动研究表明,激振力幅值、阻尼、速度等参数对位移响应幅值、共振区间、模态间的耦合度及系统软特性程度均有影响,其结论与数值计算结果一致,并同时对解的稳定性进行了研究。     

Nonlinear coupling of transverse modes of a fixed–fixed microbeam under direct and parametric excitation

Tuning of linear frequency and nonlinear frequency response of microelectromechanical systems is important in order to obtain high operating bandwidth. Linear frequency tuning can be achieved through various mechanisms such as heating and softening due to DC voltage. Nonlinear frequency response is influenced by nonlinear stiffness, quality factor and forcing. In this paper, we present the influence of nonlinear coupling in tuning the nonlinear frequency response of two transverse modes of a fixed–fixed microbeam under the influence of direct and parametric forces near and below the coupling regions. To do the analysis, we use nonlinear equation governing the motion along in-plane and out-of-plane directions. For a given DC and AC forcing, we obtain static and dynamic equations using the Galerkin’s method based on first-mode approximation under the two different resonant conditions. First, we consider one-to-one internal resonance condition in which the linear frequencies of two transverse modes show coupling. Second, we consider the case in which the linear frequencies of two transverse modes are uncoupled. To obtain the nonlinear frequency response under both the conditions, we solve the dynamic equation with the method of multiple scale (MMS). After validating the results obtained using MMS with the numerical simulation of modal equation, we discuss the influence of linear and nonlinear coupling on the frequency response of the in-plane and out-of-plane motion of fixed–fixed beam. We also analyzed the influence of quality factor on the frequency response of the beams near the coupling region. We found that the nonlinear response shows single curve near the coupling region with wider width for low value of quality factor, and it shows two different curves when the quality factor is high. Consequently, we can effectively tune the quality factor and forcing to obtain different types of coupled response of two modes of a fixed–fixed microbeam.

     

Three-to-one internal resonance in multiple stepped beam systems

In this study, the vibrations of multiple stepped beams with cubic nonlinearities are considered. A three-to-one internal resonance case is investigated for the system. A general approximate solution to the problem is found using the method of multiple scales （a perturbation technique）. The modulation equations of the amplitudes and the phases are derived for two modes. These equations are utilized to determine steady state solutions and their stabilities. It is assumed that the external forcing frequency is close to the lower frequency. For the numeric part of the study, the three-to-one ratio in natural frequencies is investigated. These values are observed to be between the first and second natural frequencies in the cases of the clamped-clamped and clamped-pinned supports, and between the second and third natural frequencies in the case of the pinned-pinned support. Finally, a numeric algorithm is used to solve the three-to-one internal resonance. The first mode is externally excited for the clamped-clamped and clamped-pinned supports, and the second mode is externally excited for the pinned-pinned support. Then, the amplitudes of the first and second modes are investigated when the first mode is externally excited. The amplitudes of the second and third modes are investigated when the second mode is externally excited. The force-response, damping-response, and .frequency- response curves are plotted for the internal resonance modes of vibrations. The stability analysis is carried out for these plots.     

基于多尺度方法的平动圆柱贮箱航天器刚——液耦合动力学研究

以充液航天器为工程背景,借助多尺度方法研究刚--液耦合动力学系统非线性动力学特性.利用多维模态方法,将描述横向外激励下圆柱贮箱中液体非线性晃动的自由边界问题转换为液体模态系数相互耦合的有限维非线性常微分方程组.推导液体晃动产生的作用于贮箱壁的晃动力和晃动力矩的解析表达式,进而建立航天器刚体部分平动和液体晃动耦合的非线性动力学方程组.应用多尺度方法对刚--液耦合系统的动力学特性进行解析分析,通过固有频率的特征方程求解耦合系统固有频率,推导外激励频率接近耦合系统第一阶固有频率时液体晃动稳态解的幅值频率响应方程.结合数值方法,研究了液体晃动稳态解的幅值频率响应曲线和激励--幅值响应曲线.结果表明, 随充液比变化,液体晃动稳态解的幅值频率响应曲线会发生软、硬弹簧特性转换现象和"跳跃"现象;幅值频率响应曲线的软、硬弹簧特性转换点受重力加速度和弹簧刚度系数影响;以上所得研究结果表明,考虑非线性效应时的刚--液耦合系统动力学特性与传统的线性系统模型所显示的动力学特性具有本质区别.本文的研究工作对进一步分析充液航天器刚--液耦合非线性动力学特性具有重要参考价值.      

Frequency lock-in during nonlinear vibration of an airfoil coupled with van der Pol Oscillator

The frequency lock-in during the nonlinear vibration of a turbomachinery blade is modeled using a spring-mounted airfoil coupled with a van der Pol Oscillator (VDP) oscillator. The proposed reduced-order model uses the nonlinear VDP oscillator to represent the oscillatory nature of wake dynamics caused by the vortex shedding. The damping term in the VDP oscillator is assumed to be nonlinear. The coupled equations governing the pitch and plunge motion of an airfoil are used to approximate the vibration of a turbomachinery blade. Springs having cubic-order nonlinearity for their stiffnesses are used to mount the airfoil. The unsteady lift acting on the blade is modeled using a self-excited nonlinear wake oscillator. The model for wake dynamics takes into account the influence of blade inertia. The nonlinear coupled three degrees of freedom (dof) aeroelastic system is studied for instability resulting in the frequency lock-in phenomenon. The equations are transformed into non-dimensional form, and then the frequencies of the coupled system are plotted to demonstrate the frequency lock-in. Further, the method of multiple scales is used to derive modulation equations which represent the amplitude and phase of the oscillation. The results obtained using the method of multiple scales are compared with direct numerical solutions to verify the present modeling method. The steady-state amplitudes of the response are plotted against the detuning parameter, which represents the frequency response curve. Further, the sensitivity of non-dimensional parameters such as coupling coefficients, mass ratio, reduced velocity, static unbalance, structural damping coefficient and the ratio of uncoupled pitch and plunge natural frequencies on the frequency response is investigated. The study revealed that parameters such as mass ratio, reduced velocity, structural damping coefficient, and coupling coefficients have a stronger influence in suppressing the amplitude of vibration. Meanwhile, parameters such as the frequency ratio, static unbalance, reduced velocity, and mass ratio significantly affect the range of frequency in which the lock-in phenomenon happens. Further, linear perturbation analysis is done to understand the qualitative effect of the system parameters such as coupling coefficients, mass ratio, frequency ratio, and static unbalance on the range of lock-in.     

Dynamic characteristics of a beam and distributed spring-mass system

This paper studies dynamic characteristics of a beam with continuously distributed spring-mass which may represent a structure occupied by a crowd of people. Dividing the coupled system into several segments and considering the distributed spring-mass and the beam in each segment being uniform, the equations of motion of the segment are established. The transfer matrix method is applied to derive the eigenvalue equation of the coupled system. It is interesting to note from the governing equations that the vibration mode shape of the uniformly distributed spring-mass is proportional to that of the beam at the attached regions and can be discontinuous if the natural frequencies of the spring-masses in two adjacent segments are different. Parametric studies demonstrate that the natural frequencies of the coupled system appear in groups. In a group of frequencies, all related modes have similar shapes. The number of natural frequencies in each group depends on the number of segments having different natural frequencies. With the increase of group order, the largest natural frequency in a group monotonically approaches the natural frequency of corresponding order of the bare beam from the upper side, whereas the others monotonically move towards those of the independent spring-mass systems from the lower side. Numerical results show that the frequency coupling between the beam and the distributed spring-mass mainly occurs in the low order of frequency groups, especially in the first group. In addition, vibratory characteristics of the coupled system can be approximately represented by a series of discrete multi-degrees-of-freedom system. It also demonstrates that a beam on Winkler elastic foundation and a beam with distributed solid mass are special cases of the proposed solution.     

Autoparametric interaction of a liquid surface in a rectangular tank with an elastic support structure under 1:1 internal resonance

Autoparametric interaction of a liquid free surface in a rectangular tank with an elastic support structure, which is subjected to vertical excitation, is investigated. When the natural frequency of the structure is equal to the lowest natural frequency of liquid sloshing, this system is categorized as an autoparametric system with an internal resonance ratio 1:1. The structure is elastically supported so there is a higher possibility that the 1:1 internal resonance can be observed. The nonlinear theoretical analysis is conducted for a fluid assumed to be perfect in a tank with a finite liquid depth. The equations of motion for the first three sloshing modes are derived employing Galerkin’s technique and considering both the nonlinearity of the fluid motion, and the viscous damping effect. Then the theoretical frequency response curves for the harmonic oscillations of the structure and sloshing are determined using van der Pol’s method. The frequency response curves show that high amplitudes of the structure’s vibrations facilitate the liquid sloshing. Furthermore, the influence of the internal detuning parameter is investigated by showing the frequency response curves and bifurcation sets. Hopf bifurcations may occur followed by amplitude-modulated motions. The theoretical results are in quantitative agreement with the experimental data.     

Amplitude modulated motions in a two degree-of-freedom system with quadratic nonlinearities under parametric excitation: experimental investigation

We conducted an experimental investigation of amplitude modulated response of a two degree-of-freedom mechanical structure with quadratic nonlinearities to parametric excitation. The linear natural frequencies of the system were tuned so that they were approximately in the ratio of two-to-one, and the excitation frequency was in principal parametric resonance with the first mode. We observed periodically amplitude-modulated motions and chaotically amplitude-modulated motions.     

Nonlinear vibrations of a circular cylindrical shell with multiple internal resonances under multi-harmonic excitation

The nonlinear response of a water-filled, thin circular cylindrical shell, simply supported at the edges, to multi-harmonic excitation is studied. The shell has opportune dimensions so that the natural frequencies of the two modes (driven and companion) with three circumferential waves are practically double than the natural frequencies of the two modes (driven and companion) with two circumferential waves. This introduces a one-to-one-to-two-to-two internal resonance in the presence of harmonic excitation in the spectral neighbourhood of the natural frequency of the mode with two circumferential waves. Since the system is excited by a multi-harmonic point-load excitation composed by first and second harmonics, very complex nonlinear dynamics is obtained around the resonance of the fundamental mode. In fact, at this frequency, both modes with two and three circumferential waves are driven to resonance and each one is in a one-to-one internal resonance with its companion mode. The nonlinear dynamics is explored by using bifurcation diagrams of Poincaré maps and time responses.     

Parametric study of a two degree-of-freedom cylinder subject to vortex-induced vibrations

This numerical work is an attempt to build accurate and continuous response surfaces of two degree-of-freedom vortex-induced vibrations (VIV) of flexibly mounted cylinders for a wide range of transverse and in-line natural frequencies. We consider both the structure and the flow to be two-dimensional. The structure has a low mass damping, with the transverse and in-line mass ratios as well as the transverse and in-line damping coefficients being equal. The goal is to capture the sensitivity of the response to the change in the natural frequencies of the structure. The system is studied for a wide range of transverse natural frequency within the synchronization region. The extent of variation of the in-line natural frequency is chosen to be larger than the one of the transverse natural frequency in order to favor multi-modal responses. No preferred frequencies are emphasized within the intervals of study. The numerical technique uses a multi-element stochastic collocation method coupled to a spectral element based deterministic solver.     

Nonlinear modal coupling in a T-shaped piezoelectric resonator induced by stiffness hardening effect

The nonlinear modal coupling in a T-shaped piezoelectric resonator, when the former two natural frequencies are away from 1:2, is studied. Experimentally sweeping up the exciting frequency shows that the horizontal beam exhibits a nonlinear hardening behavior. The first primary resonance of the vertical beam, owing to modal coupling, exhibits an abrupt amplitude increase, namely the Hopf bifurcation. The frequency comb phenomenon induced by modal coupling is measured experimentally. A Duffing-Mathieu coupled model is theoretically introduced to derive the conditions of the modal coupling and frequency comb phenomenon. The results demonstrate that the modal coupling results from nonlinear stiffness hardening and is strictly dependent on the loading range and sweeping form of the driving voltage and the frequency of the piezoelectric patches.

     

Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system

Thermo-mechanical vibrations of a simply supported spring-mass-beam system are investigated analytically in this paper. Taking into account the thermal effects, the nonlinear equations of motion and internal/external boundary conditions are derived through Hamilton’s principle and constitutive relations. Under quasi-static assumptions, the equations governing the longitudinal motion are transformed into functions of transverse displacements, which results in three integro-partial differential equations with coupling terms. These are solved using the direct multiple-scale method, leading to closed-form solutions for the mode functions, nonlinear natural frequencies and frequency–response curves of the system. The influence of system parameters on the linear and nonlinear natural frequencies, mode functions, and frequency–response curves is studied through numerical parametric analysis. It is shown that the vibration characteristics depend on the mid-plane stretching, intra-span spring, point mass, and temperature change.     

Dynamics of a nonlinear microresonator based on resonantly interacting flexural-torsional modes

A novel microresonator operating on the principle of nonlinear modal interactions due to autoparametric 1:2 internal resonance is introduced. Specifically, an electrostatically actuated pedal-microresonator design, utilizing internal resonance between an out-of-plane torsional mode and a flexural in-plane vibrating mode is considered. The two modes have their natural frequencies in 1:2 ratio, and the design ensures that the higher frequency flexural mode excites the lower frequency torsional mode in an autoparametric way. A Lagrangian formulation is used to develop the dynamic model of the system. The dynamics of the system is modeled by a two degrees of freedom reduced-order model that retains the essential quadratic inertial nonlinearities coupling the two modes. Retention of higher-order model for electrostatic forces allows for the study of static equilibrium positions and static pull-in phenomenon as a function of the bias voltages. Then for the case when the higher frequency flexural mode is resonantly actuated by a harmonically varying AC voltage, a comprehensive study of the response of the microresonator is presented and the effects of damping, and mass and structural perturbations from nominal design specifications are considered. Results show that for excitation levels above a threshold, the torsional mode is activated and it oscillates at half the frequency of excitation. This unique feature of the microresonator makes it an excellent candidate for a filter as well as a mixer in RF MEMS devices.     

Steady-state response of a binon-linear hysteretic system

The steady-state responses are analyzed for a binon-linear hysteretic system using the method of slowly varying parameters. It is shown that the triple-valued and quintuple-valued response curves exist in this system in certain parameter ranges. The stability is studied for the steady-state responses. The results obtained show that the unstable region of response is divided into two regions in certain parameter range, and the distance between the two regions goes to infinity as the value of non-linearity parameter goes to zero.     

Drift response of a bilinear hysteretic system to periodic excitation under sustained load effects

This paper presents an investigation of response characteristics for hysteretic systems idealized as a bilinear hysteretic model subjected to period excitations composed of a harmonic function and a sustained load. It is shown that the displacement solution can exhibit a drift sequence persistently repeated at a frequency identical to the excitation frequency in the case of zero post-yielding stiffness. The periodic-like drift sequence is further classified into three major types according to their different hysteretic looping behaviors. An approximate solution approach based on the method of weighted residuals is proposed to analyze the drift amplitude per response cycle. The assumed response shape is composed of two concatenated harmonic functions each with a frequency slightly detuned from the excitation frequency. The method is accompanied with a subsequent first-order analysis to obtain a closed-form approximation for the drift response. Good response predictions of the proposed solution method are demonstrated through both undamped and damped drift-frequency analyses.     

Non-linear response of a magneto-elastic translating beam with prismatic joint for higher resonance conditions

The non-linear response of a magneto-elastic translating beam having prismatic joint for higher resonance conditions is studied. A periodically varying transverse magnetic field is applied to the system. Two frequencies of prismatic motion and oscillating transverse magnetic field are implemented to the system. The method of multiple scales as one of the perturbation techniques is used to derive two first order ordinary differential equations that govern the time variation of the amplitude and phase of the response. Then a stability analysis is conducted for subharmonic resonance and simultaneous resonance conditions. A parametric study is performed to investigate the effect of magnetic field strength, amplitude of prismatic motion, damping and payload mass on the frequency response curves for both the resonance conditions. The catastrophic failure of the system may occur due to the presence of saddle-node and pitchfork bifurcations. The results obtained by method of multiple scales are compared with those obtained by numerically integrating the reduced equations and are found to be in good agreement. The developed results can be applied to control the vibration of a beam with prismatic joint subjected to magnetic field for third order subharmonic resonance and simultaneous resonance conditions.     

Non-linear interactions in the flexible multi-body dynamics of cable-supported bridge cross-sections

An elastic section model is proposed to analyze some characteristic issues of the cable-supported bridge dynamics through an equivalent planar multi-body system. The quadratic non-linearities of the four-degree-of-freedom model essentially describe the geometric coupling which may strongly characterize the dynamic interactions of the bridge deck and a pair of identical suspension cables (hangers or stays). The linear modal solution shows that the flexural and torsional modes of the deck (global modes) typically co-exist with symmetric or anti-symmetric modes of the cables (local modes). The combinations of parameters which realize remarkable 2:1:1 internal resonance conditions among one of the global modes (with higher natural frequency) and two local modes (with lower and close natural frequencies) are obtained by virtue of a multiparameter perturbation method. The non-linear response of the resonant systems shows that the global deck motion – directly forced at primary resonance by an external harmonic load – can parametrically excite the local cable motion, when the deck vibration amplitude overcomes the critical value at which a period-doubling bifurcation occurs. The relevant effects of both viscous damping and internal detuning on the instability boundaries are parametrically investigated. All the internal resonance conditions as well as the critical vibration amplitudes are expressed as an explicit, though asymptotically approximate, function of the structural parameters.