Response Statistics of Three-Degree-of-Freedom Nonlinear System to Narrow-Band Random Excitation

The principal resonance of a 3-DOF nonlinear system to narrow-band random external excitations is investigated. The method of multiple scales is used to derive the equations for modulation of amplitude and phase. The behavior, stability and bifurcation of steady-state responses are studied by means of qualitative analysis. The effects of damping, detuning, and excitation intensity on responses are analyzed. The theoretical analyses are verified by numerical results. Both theoretical analyses and numerical simulations show that when the intensity of the random excitation increases, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. Under some conditions, co-existence of two kinds of stable steady-state solutions, saturation and jump phenomena may occur. The stationary probability density function of responses for the co-existence case is obtained approximately.     

窄带随机噪声作用下单自由度非线性碰撞系统的响应

研究了单自由度非线性单边碰撞系统在窄带随机噪声激励下的次共振响应问题。用Zhuravlev变换将碰撞系统转化为速度连续的非碰撞系统,然后用随机平均法得到了关于慢变量的随机微分方程。在没有随机扰动情形,得到了系统响应幅值满足的代数方程;在有随机扰动的情形下,给出了系统响应稳态矩计算的迭代公式。讨论了系统阻尼项、非线性项、随机扰动项和碰撞恢复系数等参数对于系统响应的影响。理论计算和数值模拟表明,系统响应幅值将在激励频率接近于次共振频率时达到最大。而当激励频率逐渐偏离次共振频率时,系统响应迅速衰减。     

Response statistic of strongly non-linear oscillator to combined deterministic and random excitation

The principal resonance of a van der Pol-Duffing oscillator to the combined excitation of a deterministic harmonic component and a random component has been investigated. By introducing a new expansion parameter , the method of multiple scales is adapted for the strongly non-linear system. Then the method of multiple scales is used to determine the equations of modulation of response amplitude and phase. The behavior and the stability of steady-state response are studied by means of qualitative analysis. The effects of damping, detuning, bandwidth, and magnitudes of random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increases, the non-trivial steady-state solution may change from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady-state solutions. Random jump may be observed under some conditions. The results obtained in the paper are adapted for a strongly non-linear oscillator that complement previous results in the literature for the weakly non-linear case.     

随机窄带参激下二自由度非线性系统

研究了带平方二自由度非线性系统在随机窄带参数激励下，用多尺度法分离了系统的快变项，讨论了系统的各参数对响应的影响。在一定条件下，系统具有两个均方响应值，具有跳跃现象和饱和现象，数值模拟表明提出的方法是有效的。     

Stochastic responses of viscoelastic system with real-power stiffness under randomly disordered periodic excitations

The stochastic dynamic responses of viscoelastic systems with real-power exponents of stiffness term subjects to randomly disordered periodic excitations are studied. The assumed viscoelastic damping depends on the past history of motion via convolution integrals over an exponentially decaying kernel function. The multiple scales method is used to derive the stochastic different equations of modulation of amplitude and phase. The changes of the shape of resonance curves are obtained with real-power exponents of stiffness term and viscoelastic parameters, and then, the numerical simulation method was used to verify the accuracy of the theoretical analysis results. Theoretical analysis and numerical simulations show that as the intensity of the random excitation increases, the steady-state solution changes from a limit cycle to a diffused limited cycle. Under some conditions, the system may have two steady-state solutions and the phenomenon of jumps will happen to them under the random excitations.     

确定性谐和与随机噪声联合作用下二自由度系统的响应

研究了二自由度非线性系统在确定性谐和与随机噪声联合激励下的主共振响应。用多尺度法分离了系统的快变项 ,讨论了系统的阻尼项、随机项等对系统响应的影响。在一定条件下 ,系统具有两个均方响应值和跳跃现象 ,饱和现象也存在。数值模拟表明本文提出的方法是有效的     

窄带随机噪声激励下多自由度非线性系统的响应

在随机振动的研究中，研究较多的是系统在宽带噪声作用下的响应问题，对于非线性系统特别是多自由度非线性系统在窄带随机噪声作用下的响应问题则研究较少。本文研究了三自由度非线性系统在窄带随机噪声激励下的主共振响应和稳定性问题。用多尺度法分离了系统的快变项，给出了系统响应的振幅和相位角满足的方程。用摄动法讨论了系统随机项对系统响应的影响。当随机扰动较小时，在一定的参数范围内，对应于不同的初值，系统具有两个均方响应值，随机饱和现象也存在。当随机扰动增大时，系统可从一个大的响应突跳为一个小的响应，或从一个小的响应突跳为一个大的响应，即存在随机跳跃现象。数值模拟表明本文提出的方法是有效的。     

谐和与窄带随机噪声联合作用下Duffing系统的参数主共振

研究了Ｄｕｆｉｎｇ振子在谐和与窄带随机噪声联合激励下的参数主共振响应和稳定性问题．用多尺度法分离了系统的快变项，并求出了系统的最大Ｌｙａｐｕｎｏｖ指数．本文还分析了失稳及跳跃现象，及系统的阻尼项、非线性项、随机项、确定性参激强度对系统响应的影响．数值模拟表明本文提出的方法是有效的．     

Response statistics of two-degree-of-freedom non-linear system to narrow-band random excitation

The principal resonance of two-degree-of-freedom non-linear system to narrow-band random external excitation is investigated. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response are studied by means of qualitative analysis. The effects of damping, detuning, bandwidth, and magnitudes of deterministic and random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increases, the nontrivial steady state solution may be changed from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady state solutions, saturation and jumps may exist.     

窄带噪声作用下二自由度非线性系统的响应

研究二自由度非线性系统在窄带随机噪声激励下的主共振响应,用多尺度法分离了系统的快变项,讨论了系统的阻尼项、随机项等对系统响应的影响。在一定条件下,系统具有两个均方响应值和跳跃现象,饱和现象也存在。数值模拟表明文中所提出的方法是有效的。     

Response of nonlinear oscillator under narrow-band random excitation

IntroductionThestudyoftheresponseofnonlinearsystemstonarrow_bandrandomexcitationofconsiderableimportance.Forexample ,theexcitationofsecondarysystemwouldbeanarrow_bandrandomprocessiftheprimarysystemcouldbemodeledasasingle_degree_of_freedomsystemwithlightdampingsubjecttowide_bandexcitation .Inthetheoryofnonlinearrandomvibration ,mostresultsobtainedsofarareattributedtotheresponseofnonlinearoscillatorstowide_bandrandomexcitation .Incomparison ,resultsontheeffectofnarrow_bandexcitationonnonlinearos…     

Visco-elastic systems under both deterministic harmonic and random excitation

IntroductionForlinearviscoelasticsystemsunderbothadditiveandmultiplicativebroad_bandexcitationexcitations,Ariaratnam[1]studiedthestochasticstabilityofthesystembyusingthemethodofstochasticaveragingprocedure .Itwasshownthatthevisco_elasticforcecontributedtowarddamping ,hence ,stabilityofthesystem .However,thestiffnesseffectofthevisco_elasticcomponentwasnotfullyaccountedfor.FurthermoreAriaratnam[2 ]studiedthestochasticstabilityofthesystembutthemodelislinear.Inthetheoryofnonlinearrandomvibration…     

谐和与随机噪声联合作用下非线性系统的响应

研究了Duffing振子在谐和与随噪声联合激励下的响应和稳应性问题。用谐波平均法分析了系统在确定性谐和激励和随机激励联合作用下的响应,用随机平均法讨论了随机扰动项对系统晌应的影响。在一定条件下,系统具有两个均方响应值和跳跃现象。数值模拟表明本文提出的方法是有效的。     

One-to-Two Internal Resonance in Two-Degrees-of-Freedom Nonlinear System with Narrow-Band Excitations

Response of two-degrees-of-freedom nonlinearsystem to narrow-band random parametric excitation isinvestigated. The method of multiple scales is used todetermine the equations of modulation of amplitude andphase. The effect of detunings and amplitude areanalyzed. Theoretical analyses and numerical simulationsshow that the nontrivial steady-state solution may changeform a limit cycle to a diffused limit cycle as theintensity of the random excitation increase. Under someconditions, the system may have two steady-statesolutions.     

A unified approach to two types of evolutionary random response problems in engineering

Summary Response of structures to earthquake excitations and response of vehicles to road undulations are two typical evolutionary random problems in engineering. Both kinds of the evolutionary random excitations can be regarded as evolved from stationary random excitations, though through two utterly different ways. The former one may be obtained by filtering a stationary random process through a linear time-dependent system, while the latter one may result from nonlinear transformations of the argument of a stationary random process. However, the response problems due to both types of excitations have much in common. By introducing the concept of “evolutionary frequency response”, the expressions of the response evolutionary spectra for both problems can be obtained in a unified, concise way, similar to the input/output PSD relationship in a stationary random problem. For both the evolutionary random problems, the solution procedures are all the same, but the expressions for evolutionary frequency responses are different from each other. Moreover, the evolutionary frequency responses may be interpreted as transient responses of the system subject to certain deterministic evolutionary harmonic excitations. In this sense, an evolutionary random response problem can be reduced to a deterministic response problem. Based on the complex modal analysis, a unified approach to these two response problems is derived here. The method can be applied to any linear time-invariant systems, whether they are symmetrical or not, and whether they are classically damped or not. And the method might be hopefully applied to nonlinear systems, if the statistical linearization technique is accompanied. To the knowledge of the authors, this unified approach to two types of evolutionary random response problems is the first time reported in literature. Received 28 May 1996; accepted for publication 8 January 1997     

Stochastic P-bifurcation analysis of a fractional smooth and discontinuous oscillator via the generalized cell mapping method

The smooth and discontinuous oscillator with fractional derivative damping under combined harmonic and random excitations is investigated in this paper. The short memory principle is introduced so that the evolution process of the oscillator with fractional derivative damping can be described by the Markov chain. Then the stochastic generalized cell mapping method is used to obtain the steady-state probability density functions of the response. The stochastic response and bifurcation of the oscillator with fractional derivative damping are discussed in detail. We found that both the smoothness parameter, the noise intensity, the amplitude and frequency of the harmonic force can induce the occurrence of stochastic P-bifurcation in the system. Monte Carlo simulation verifies the effectiveness of the method we adopt in the paper.     

Bifurcation and Chaos Analysis of Stochastic Duffing System Under Harmonic Excitations

The Chebyshev polynomial approximation is applied to the dynamic response problem of a stochastic Duffing system with bounded random parameters, subject to harmonic excitations. The stochastic Duffing system is first reduced into an equivalent deterministic non-linear one for substitution. Then basic non-linear phenomena, such as stochastic saddle-node bifurcation, stochastic symmetry-breaking bifurcation, stochastic period-doubling bifurcation, coexistence of different kinds of steady-state stochastic responses, and stochastic chaos, are studied by numerical simulations. The main feature of stochastic chaos is explored. The suggested method provides a new approach to stochastic dynamic response problems of some dissipative stochastic systems with polynomial non-linearity.     

Partition method and experimental validation for impact dynamics of flexible multibody system

The impact problem of a flexible multibody system is a non-smooth, high-transient, and strong-nonlinear dynamic process with variable boundary. How to model the contact/impact process accurately and efficiently is one of the main difficulties in many engineering applications. The numerical approaches being used widely in impact analysis are mainly from two fields: multibody system dynamics (MBS) and computational solid mechanics (CSM). Approaches based on MBS provide a more efficient yet less accurate analysis of the contact/impact problems, while approaches based on CSM are well suited for particularly high accuracy needs, yet require very high computational effort. To bridge the gap between accuracy and efficiency in the dynamic simulation of a flexible multibody system with contacts/impacts, a partition method is presented considering that the contact body is divided into two parts, an impact region and a non-impact region. The impact region is modeled using the finite element method to guarantee the local accuracy, while the non-impact region is modeled using the modal reduction approach to raise the global efficiency. A three-dimensional rod-plate impact experiment is designed and performed to validate the numerical results. The principle for how to partition the contact bodies is proposed: the maximum radius of the impact region can be estimated by an analytical method, and the modal truncation orders of the non-impact region can be estimated by the highest frequency of the signal measured. The simulation results using the presented method are in good agreement with the experimental results. It shows that this method is an effective formulation considering both accuracy and efficiency. Moreover, a more complicated multibody impact problem of a crank slider mechanism is investigated to strengthen this conclusion.     

On statistical quasi-linearization

In carrying out the statistical linearization procedure to a non-linear system subjected to an external random excitation, a Gaussian probability distribution is assumed for the system response. If the random excitation is non-Gaussian, however, the procedure may lead to a large error since the response of bother the original non-linear system and the replacement linear system are not Gaussian distributed. It is found that in some cases such a system can be transformed to one under parametric excitations of Gaussian white noises. Then the quasi-linearization procedure, proposed originally for non-linear systems under both external and parametric excitations of Gaussian white noises, can be applied to these cases. In the procedure, exact statistical moments of the replacing quasi-linear system are used to calculate the linearization parameters. Since the assumption of a Gaussian probability distribution is avoided, the accuracy of the approximation method is improved. The approach is applied to non-linear systems under two types of non-Gaussian excitations: randomized sinusoidal process and polynomials of a filtered process. Numerical examples are investigated, and the calculated results show that the proposed method has higher accuracy than the conventional linearization, as compared with the results obtained from Monte Carlo simulations.     

组合系统在限带白噪声激励下的均方响应

本文考察组合系统在限带白噪声激励下的均方响应。通过离散化和复模态分析,求得系统在限带白噪声激励下均方响应的闭式解。本方法通用于宽带激励与窄带激励情形.