Stochastically Perturbed Linear Gyroscopic Systems

ABSTRACT

Dynamic stability of linear conservative gyroscopic systems under stochastic parametric excitations of small intensity is examined. Conditions for mean square stability of dynamic response are obtained. Results are shown to depend only on those values of the excitation spectral density near twice the natural frequencies and the combination frequencies of the system. These results are applied to the problem of flow induced vibration in a supported pipe conveying fluid with pulsating velocity. The effects of mean flow velocity and virtual mass on the extent of parametric instability regions are then discussed.     

Dynamic analysis of piezoelectric energy harvester under combination parametric and internal resonance: a theoretical and experimental study

In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

     

Parametric resonance,stability and heteroclinic bifurcation in a nonlinear oscillator with time-delay: Application to a quarter-car model

This work examines dynamical behavior of a nonlinear oscillator with symmetric potential that models a quarter-car forced by the road profile under parametric excitation. The parametric resonance of a harmonically excited nonlinear quarter-car model with position and velocity time-delayed active control are investigated. We focus on the influence of delay and parametric excitation in the system. The influence of parametric excitation, time-delay and feedback gain parameters on the stability of the steady state response are investigated. By means of Melnikov's method, conditions for onset of chaos resulting from heteroclinic bifurcation is derived analytically and numerically.     

Nonlinear dynamics of a viscoelastic beam traveling with pulsating speed,variable axial tension under two-frequency parametric excitations and internal resonance

This paper investigates nonlinear combined parametric transverse vibrations of a traveling viscoelastic beam. The combined parametric excitations originate from the time dependency of axial velocity as well as axial tension. Two parametric excitations are enforced into the system amid the internal resonance. Two-frequency parametric resonance is assumed to be comprised of combination parametric resonance of first two modes due to the time dependency of axial velocity, and the principal parametric resonance of first mode due to the variable tension in the axial direction in the presence of internal resonance for viscoelastic beam is considered for the first time. The higher-order integro-partial differential equation of motion is solved through direct method of multiple scales. Continuation algorithm is employed to explore the stability and various bifurcations of the nonlinear dynamic system. Focus has been made to study the effect of variations of fluctuating tension component, fluctuating velocity component independently and when combined, internal and parametric frequency detuning parameters and damping on the system response. Frequency response equilibrium curves are complex and unique in shapes which are embodied with various bifurcations. Such steady-state behavior is not seen in the existent literature. With variation in fluctuating velocity component, the number of steady-state nontrivial equilibrium curves increases to three and with variation in fluctuating axial tension, they become four. In this process, significant changes in stability, number and position of various bifurcations like supercritical and subcritical pitchfork, Hopf and saddle node are observed. Unlike the previous study, the shape, stability and bifurcations of equilibrium curves under the combined effect of axial velocity and tension closely match with the case of fluctuating axial tension component. The effect of variation in internal and parametric frequency detuning parameter is more realized for second mode compared to first mode. A comparison of the present work with a previous one where axial tension is variable reveals many qualitative and quantitative similarities and dissimilarities. But when compared with earlier work where axial velocity is constant, significant dissimilarities are surfaced. The system displays a wide ranging dynamic behavior including stable periodic, quasiperiodic and unstable chaotic behavior. The numerical computation depicts various nonlinear characteristics and oscillatory behaviors which are not found so far in the existent literature.     

Nonlinear dynamics of a viscoelastic sandwich beam with parametric excitations and internal resonance

Nonlinear dynamical behaviors of an axially accelerating viscoelastic sandwich beam subjected to three-to-one internal resonance and parametric excitations resulting from simultaneous velocity and tension fluctuations are investigated. The direct method of multiple scales is adopted to obtain a set of first-order ordinary differential equations and associated boundary conditions. The frequency and amplitude response curves along with their stability and bifurcation are numerically studied. A great number of dynamic behaviors are presented in the form of phase portraits, time traces, Poincaré sections, and FFT power spectra. Due to modal interaction, various periodic, quasiperiodic, and chaotic behaviors are displayed, depending on the initial conditions. The largest Lyapunov exponent is carried out to determine the midly chaotic response by the convergent form of exponents. Numerical results show various oscillatory behaviors indicating the influence of internal resonance and coupled effects of fluctuating axial velocity and tension.     

Nonlinear vibrations of axially moving multi-supported strings having non-ideal support conditions

In this study, nonlinear vibrations of an axially moving multi-supported string have been investigated. The main difference of this study from the others is in that there are non-ideal supports allowing minimal deflections between ideal supports at both ends of the string. Nonlinear equations of the motion and boundary conditions have been obtained using Hamilton’s Principle. Dependence of the equations of motion and boundary conditions on geometry and material of the string have been eliminated by non-dimensionalizing. Method of multiple scales, a perturbation technique, has been employed for solving the equations of motion. Axial velocity has been assumed a harmonically varying function about a constant value. Axially moving string has been investigated in three regions. Vibrations have been examined for three different cases of the velocity variation frequency. Stability has been analyzed and stability boundaries have been established for the principal parametric resonance case. Effects of the non-ideal support conditions on stability boundaries and vibration amplitudes have been investigated.

     

Theoretical and Experimental Study on the Parametrically Excited Vibration of Mass-Loaded String

The parametrically excited lateral vibration of a mass-loaded string is investigated in this paper. Supposing that the mass at the lower end of the string is subjected to a vertical harmonic excitation and neglecting the higher-order vibration modes, the equation of motion for the mass-loaded string can be represented by a Mathieu's equation with cubic nonlinearity. Based on the stability criterion for Mathieu's equation, the critical conditions inducing parametric resonance are clarified. Theoretical analysis shows that when the natural frequency f _s of the string lateral vibration and the vertical excitation frequency f satisfy f _s= (n/2)f, n= 1, 2, 3, ..., parametric resonance occurs in the case of no damping. For a damped system, parametric resonance most likely occurs when f is close to 2f _s, and depends on the damping of the system and the vertical excitation. The critical excitation has been derived at different frequencies. If the natural frequency of the mass vertical vibration happens to be twice that of the string lateral vibration, the parametric resonance may occur due to a small disturbance. Numerical simulations show that the lateral vibration of the string does not increase infinitely at parametric resonance because the parametric excitation is self-tuned due to the coupling between the vertical and lateral vibrations. Finally, the theoretical results are supported by some experimental work.     

主动控制压电旋转悬臂梁的参数振动稳定性分析

在工程实际中旋转机械由于制造和加工误差,装配的不均匀性等原因,往往会脉动运行,这将使得机械系统发生参数振动. 当脉动参数满足一定关系时,这种参数振动将会失稳,进而影响机械结构的正常运转. 本文针对这一问题,引入压电材料对 脉动旋转悬臂梁系统的振动进行控制,研究主动控制悬臂梁系统的参数振动优化设计问题,采用 Hamilton 变分原理与一阶 Galerkin 离散相结合的方法,建立了受速度反馈传感器主动控制的压电旋转悬臂梁的一阶近似线性控制方程. 运用多尺度方法,得到了压电旋转悬臂梁系统在发生1/2亚谐波参数共振时稳定性边界的控制方程,并利用直接分析方法验证了解析摄动解的正确性. 将摄动解中临界阻尼比和轮毂角速度脉动幅值的无量纲参数作为评价系统稳定性能的指标. 通过数值算例,分析了轮毂半径、轮毂角速度平均值和脉动幅值、梁长以及速度传感器的反馈增益系数对系统稳定性区域的影响. 研究结果表明,梁长、轮毂半径、脉动幅值会降低系统稳定性,反馈增益系数可以提高系统稳定性,而轮毂角速度平均值与系统稳定性之间有非单调的关系. 为进一步设计压电旋转机械结构提供了理论依据.      

On the Homoclinic Orbits in a Class of Two-Degree-of-Freedom Systems under the Resonance Conditions

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周期变速旋转运动电流变夹层梁的参激振动

采用多尺度法对周期变速旋转运动电流变夹层梁的动力稳定性进行了研究. 假设电流变夹层梁绕固定轴线做随时间变化的简谐周期运动,将变速度转动梁作为一个时变参激振动系统,分析了不同结构和控制参数对失稳区域的影响. 仿真结果表明,改变外加控制电场强度的大小和梁的结构参数,可改变旋转电流变夹层梁发生动力失稳的临界角速度和失稳区域. 故在一定的条件下,可以通过控制作用于电流变夹层梁的电场强度来调节旋转运动柔性梁的振动特性,提高结构的动力稳定性.      

Heteroclinic connections in the 1:4 resonance problem using nonlinear transformation method

In this paper, the method of nonlinear time transformation is applied to obtain analytical approximation of heteroclinic connections in the problem of stability loss of self-oscillations near 1:4 resonance. As example, we consider the case of parametric and self-excited oscillator near the 1:4 subharmonic resonance. The method uses the unperturbed heteroclinic connection in the slow flow to determine conditions under which the perturbed heteroclinic connection persists. The results show that for small values of damping, the nonlinear time transformation method can predict well both the square and clover heteroclinic connection near the 1:4 resonance. The analytical finding is confirmed by comparisons to the results obtained by numerical simulations.     

轴向变速黏弹性Poynting-Thompson梁参数共振稳定性

本文研究了轴向变速黏弹性梁的组合参数共振和主参数共振稳定性．梁的材料黏弹性本构关系由Poynting-Thompson模型描述．使用多尺度法渐近展开求解,导出了其可解性条件．根据Routh-Hurwitz准则给出了组合参数共振和主参数共振稳定性条件．考虑Poynting-Thompson模型退化到Kelvin-Voigt模型的情况．通过数值算例对两个模型进行了失稳边界的比较．     

Visco-elastic systems under both deterministic and bound random parameteric excitation

The principal resonance of a visco-elastic systems under both deterministic and random parametric excitation was investigated. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response were studied by means of qualitative analysis. The contributions from the visco-elastic force to both damping and stiffness can be taken into account. The effects of damping, detuning, bandwidth, and magnitudes of deterministic and random excitations were analyzed. The theoretical analysis is verified by numerical results. Foundation item: the National Natural Science Foundation of China (10072049) Biography: XU Wei (1957∼), Professor, Doctor (E-mail: weixu@nwpu.edu.cn)     

Parametric Resonance of a Two-Dimensional System Under Bounded Noise Excitation

The dynamic stability of a two-dimensional system under bounded noise excitation with a narrow band characteristic is studied through the determination of the pth moment Lyapunov exponent and the Lyapunov exponent. The case when the system is in primary parametric resonance in the absence of noise is considered and the effect of noise on the parametric resonance is investigated. The partial differential eigenvalue problem governing the moment Lyapunov exponent is established. For small amplitudes of the bounded noise, a method of singular perturbation is applied to determine analytical expansions of the moment Lyapunov exponents and Lyapunov exponents, which are shown to be in excellent agreement with those obtained using numerical approaches.     

参数激励下非线性受控系统的动力学和稳定性

研究两自由度参数激励系统的非线性动力学与控制问题．利用Lagrange方程建立含反馈控制的参激捅及其驱动机构组成的系统动力学方程，以多尺度方法获得一阶近似控制方程．然后，对系统受一阶摸态参激主共振与一、二阶模态间3：1内共振联合作用下的幅额响应及其稳定性，以及反馈参数对系统稳态行为的影响作了详细分析．结果表明，响应的稳定域位置和大小取决于位移反馈，位移立方反馈改变了系统的非线性程度，速度反馈类似于阻尼，可使系统呈现自激振动特性．     

Effect of rotary inertia on stability of axially accelerating viscoelastic Rayleigh beams

The dynamic stability of axially moving viscoelastic Rayleigh beams is presented. The governing equation and simple support boundary condition are derived with the extended Hamilton’s principle. The viscoelastic material of the beams is described as the Kelvin constitutive relationship involving the total time derivative. The axial tension is considered to vary longitudinally. The natural frequencies and solvability condition are obtained in the multi-scale process. It is of interest to investigate the summation parametric resonance and principal parametric resonance by using the Routh-Hurwitz criterion to obtain the stability condition. Numerical examples show the effects of viscosity coefficients, mean speed, beam stiffness, and rotary inertia factor on the summation parametric resonance and principle parametric resonance. The differential quadrature method (DQM) is used to validate the value of the stability boundary in the principle parametric resonance for the first two modes.     

Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances

In this paper, the nonlinear planar vibration of a pipe conveying pulsatile fluid subjected to principal parametric resonance in the presence of internal resonance is investigated. The pipe is hinged to two immovable supports at both ends and conveys fluid at a velocity with a harmonically varying component over a constant mean velocity. The geometric cubic nonlinearity in the equation of motion is due to stretching effect of the pipe. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of mean flow velocity, resulting in a three-to-one internal resonance. The analysis is done using the method of multiple scales (MMS) by directly attacking the governing nonlinear integral-partial-differential equations and the associated boundary conditions. The resulting set of first-order ordinary differential equations governing the modulation of amplitude and phase is analyzed numerically for principal parametric resonance of first mode. Stability, bifurcation, and response behavior of the pipe are investigated. The results show new zones of instability due to the presence of internal resonance. A wide array of dynamical behavior is observed, illustrating the influence of internal resonance.     

Principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string

To investigate the principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string, the method of multiple scales is applied directly to the nonlinear partial differential equation that governs the transverse vibration of the string. To derive the governing equation, Newton‘s second law, Lagrangean strain, and Kelvin‘s model are respectively used to account the dynamical relation, geometric nonlinearity and the viscoelasticity of the string material. Based on the solvability condition of eliminating the secular terms, closed form solutions are obtained for the amplitude and the existence conditions of nontrivial steady-state response of the principal parametric resonance. The Lyapunov linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions in the principal parametric resonance. Some numerical examples are presented to show the effects of the mean transport speed, the amplitude and the frequency of speed variation.     

Nonlinear parametric vibrations of orthotropic cylindrical shells interacting with a pulsating fluid flow

The paper addresses the dynamic interaction of an orthotropic cylindrical shell with the fluid flowing inside. Its velocity has a constant component and low-amplitude pulsations. A method to calculate the characteristics of the parametric vibrations of the shell when the velocity of the fluid is close to critical is proposed. The amplitude–frequency characteristics of the shell–fluid system at fundamental parametric resonance are determined     

Stability of axially accelerating viscoelastic beams: asymptotic perturbation analysis and differential quadrature validation

An asymptotic perturbation method is proposed to investigate stability of an axially accelerating viscoelastic beam. The material time derivative is used in the viscoelastic constitutive relation. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The stability condition can be determined via the asymptotic perturbation method. The differential quadrature scheme is developed to solve numerically the equation of axially accelerating viscoelastic beams with simple supports. The stability boundaries are numerically located in the summation parametric resonance and the principal parametric resonance. Numerical examples show the effects of the beam viscoelasticity and the mean axial speed. The numerical calculations validate the analytical results in the principal parametric resonance.