Path integral solution of vibratory energy harvesting systems

A transition Fokker-Planck-Kolmogorov(FPK) equation describes the procedure of the probability density evolution whereby the dynamic response and reliability evaluation of mechanical systems could be carried out. The transition FPK equation of vibratory energy harvesting systems is a four-dimensional nonlinear partial differential equation. Therefore, it is often very challenging to obtain an exact probability density. This paper aims to investigate the stochastic response of vibration energy harvesters(VEHs)under the Gaussian white noise excitation. The numerical path integration method is applied to different types of nonlinear VEHs. The probability density function(PDF)from the transition FPK equation of energy harvesting systems is calculated using the path integration method. The path integration process is introduced by using the GaussLegendre integration scheme, and the short-time transition PDF is formulated with the short-time Gaussian approximation. The stationary probability densities of the transition FPK equation for vibratory energy harvesters are determined. The procedure is applied to three different types of nonlinear VEHs under Gaussian white excitations. The approximately numerical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulation(MCS).     

Variations in steepness of the probability density function of beam random vibration

Dynamic behaviour of a beam, subjected to stationary random excitation, has been investigated for the situation in which the response is different from the model of a Gaussian random process. The study was restricted to the case of symmetric non-Gaussian probability density functions of beam vibrations. There are two possible causes of deviations of the system response from the Gaussian model: the first, nonlinear behaviour, concerns the system itself and the second is external when the excitation is not Gaussian. Both cases have been considered in the paper. To clarity the conclusions for each case and to avoid interference of these different types of system behaviour, two beam structures, clamped-clamped and cantilevered, have been studied. A numerical procedure for prediction of the nonlinear random response of a clamped-clamped beam under the Gaussian excitations was based on a linear modal expansion. Monte Carlo simulation was undertaken using Runge–Kutta integration of the generalised coordinate equations. Probability density functions of the beam response were analysed and approximated making use of different theoretical models. An experimental study has been carried out for a linear system of a cantilevered beam with a point mass at the free end. A pseudo-random driving signal was generated digitally in the form of a Fourier expansion and fed to a shaker input. To generate a non-Gaussian excitation a special procedure of harmonic phase adjustment was implemented instead of the random choice. In so doing, the non-Gaussian kurtosis parameter of the beam response was controlled.     

简支梁非线性响应的三维路径积分法分析

采用基于Gauss-Legendre积分公式的三维路径积分法,分析了在过滤高斯白噪声激励下的简支梁非线性随机振动响应的概率密度函数;联立一阶滤波方程与简支梁一阶模态的振动模型,得到在过滤高斯白噪声激励下的简支梁随机振动模型,基于Gauss-Legendre积分公式的积分法和短时高斯近似法求解响应的概率密度函数值。结果表明,三维路径积分法计算值与蒙特卡洛模拟值符合良好,即使在尾部区域也符合良好。三维路径积分法比等效线性化法的计算精度更高。     

非高斯随机激励下滞迟系统响应与首次穿越失效

针对由有界噪声、泊松白噪声和高斯白噪声共同构成的非高斯随机激励,通过Monte Carlo数值模拟方法研究了此激励作用下双线性滞迟系统和Bouc-Wen滞迟系统这两类经典滞迟系统的稳态响应与首次穿越失效时间。一方面,分析了有界噪声和泊松白噪声这两种分别具有连续样本函数和非连续样本函数的非高斯随机激励,在不同激励参数条件下对双线性滞迟系统和Bouc-Wen滞迟系统的稳态响应概率密度、首次穿越失效时间概率密度及其均值的不同影响;另一方面,揭示了在这类非高斯随机激励荷载作用下,双线性滞迟系统的首次穿越失效时间概率密度将出现与Bouc-Wen滞迟系统的单峰首次穿越失效时间概率密度截然不同的双峰形式。     

Transient stochastic response of quasi integerable Hamiltonian systems

The approximate transient response of quasi integrable Hamiltonian systems under Gaussian white noise excitations is investigated. First, the averaged Ito equations for independent motion integrals and the associated Fokker-Planck-Kolmogorov (FPK) equation governing the transient probability density of independent motion integrals of the system are derived by applying the stochastic averaging method for quasi integrable Hamiltonian systems. Then, approximate solution of the transient probability density of independent motion integrals is obtained by applying the Galerkin method to solve the FPK equation. The approximate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coefficients. The transient probability densities of displacements and velocities can be derived from that of independent motion integrals. Three examples are given to illustrate the application of the proposed procedure. It is shown that the results for the three examples obtained by using the proposed procedure agree well with those from Monte Carlo simulation of the original systems.     

Transient stochastic response of quasi-partially integrable Hamiltonian systems

The approximate transient response of multi-degree-of-freedom (MDOF) quasi-partially integrable Hamiltonian systems under Gaussian white noise excitation is investigated. First, the averaged Itô equations for first integrals and the associated Fokker–Planck–Kolmogorov (FPK) equation governing the transient probability density of first integrals of the system are derived by applying the stochastic averaging method for quasi-partially integrable Hamiltonian systems. Then, the approximate solution of the transient probability density of first integrals of the system is obtained from solving the FPK equation by applying the Galerkin method. The approximate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coefficients. The transient probability densities of displacements and velocities can be derived from that of first integrals. One example is given to illustrate the application of the proposed procedure. It is shown that the results for the example obtained by using the proposed procedure agree well with those from Monte Carlo simulation of the original system.     

An equivalent polynomial approximation technique in nonlinear structural dynamics

Summary A new technique is proposed to obtain an approximate probability density for the response of a general nonlinear system under Gaussian white noise excitations. In this new technique, the original nonlinear system is replaced by another equivalent nonlinear system, structured by the polynomial formula, for which the exact solution of stationary probability density function is obtainable. Since the equivalent nonlinear system structured in this paper originates directly from certain classes of real nonlinear mechanical systems, the technique is applied to some very challenging nonlinear systems in order to show its power and efficiency. The calculated results show that applying the technique presented here can yield exact stationary solutions for the nonlinear oscillators. This is obtained by using an energy-dependent system, and for a nonlinearity of a more complex type. A more accurate approximate solution is then available, and is compared with the approximation. Application of the technique is illustrated by examples.     

非Gauss分布随机波浪力对海洋平台的非线性作用

为避免求解决定Maikov过程转移概率密度的Fokker—Planck方程，基于尺度分离的假设导出了一组描述非线性海洋平台受非Gauss分布随机波浪载荷作用所产生响应的矩量的常微分方程组。矩量方程清楚地反映出分别对应随机载荷和结构响应的两种不同统计特性的相互关系。由于矩量方程不依赖载荷的概率分布的具体细节，以它来模拟随机激励作用下的非线性系统将免于Monte Carlo方法所面临的正确模拟载荷概率分布的困难任务。将摄动法用于矩量方程可使线性化不再需要，这样就不会因为线性化而产生不可预料的误差。     

基于FPK方程的减摆器的非线性阻尼测试方法

受高斯白噪声外激的一阶非线性动力学方程能通过求解对应的FPK方程得到精确稳态解．本文基于这一结果导出减摆器非线性阻尼力与系统速度输出的概率结构的关系,将动力学系统中非线性阻尼力参数的测试问题转化测量系统的概率结构,并通过仿真进行了验证．     

Nonstationary probability densities of strongly nonlinear single-degree-of-freedom oscillators with time delay

The approximate nonstationary probability density of a nonlinear single-degree-of-freedom (SDOF) oscillator with time delay subject to Gaussian white noises is studied. First, the time-delayed terms are approximated by those without time delay and the original system can be rewritten as a nonlinear stochastic system without time delay. Then, the stochastic averaging method based on generalized harmonic functions is used to obtain the averaged Itô equation for amplitude of the system response and the associated Fokker–Planck–Kolmogorov (FPK) equation governing the nonstationary probability density of amplitude is deduced. Finally, the approximate solution of the nonstationary probability density of amplitude is obtained by applying the Galerkin method. The approximate solution is expressed as a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. The proposed method is applied to predict the responses of a Van der Pol oscillator and a Duffing oscillator with time delay subject to Gaussian white noise. It is shown that the results obtained by the proposed procedure agree well with those obtained from Monte Carlo simulation of the original systems.     

Non-stationary response of a van der Pol-Duffing oscillator under Gaussian white noise

This paper investigates the probability density evolution process of a van der Pol-Duffing oscillator under Gaussian white noise. A path integration method is employed with the Gauss–Legendre integration scheme. In the path integration method, the short-time Gaussian approximation scheme is used for computing the one-step transition probability density. Two cases are considered with slight nonlinearity or strong nonlinearity in displacement. The stationary and non-stationary responses of the oscillator are studied. Compared with the simulation result, the path integration method can present a satisfactory probability density function (PDF) solution for each case. Different probability density evolution processes are observed correspondingly. In the case of slight nonlinearity, the PDF undergoes a clockwise motion around the origin. The peak region gradually expands and the PDF eventually forms a circle. By contrast, the strong nonlinearity drives the oscillator to oscillate around the limit cycle. In such a case, the PDF rapidly forms a circle. The circle keeps its shape and develops until the oscillator becomes stationary. More complicated phenomena can be studied by the adopted path integration method.     

Response analysis of randomly excited nonlinear systems with symmetric weighting Preisach hysteresis

An approximate method for analyzing the response of nonlinear systems with the Preisach hysteresis of the non-local memory under a stationary Gaussian excitation is presented based on the covariance and switching probability analysis. The covariance matrix equation of the Preisach hysteretic system response is derived. The cross correlation function of the Preisach hysteretic force and response in the covariance equation is evaluated by the switching probability analysis and the Gaussian approximation to the response process. Then an explicit expression of the correlation function is given for the case of symmetric Preisach weighting functions. The numerical result obtained is in good agreement with that from the digital simulation. The project supported by the National Natural Science Foundation of China (19972059) and Zhejiang Provincial Natural Science Foundation (101046)     

Stochastic responses of Duffing-Van der Pol vibro-impact system under additive and multiplicative random excitations

The paper is devoted to an averaging approach to study the responses of Duffing-Van der Pol vibro-impact system excited by additive and multiplicative Gaussian noises. The response probability density functions (PDFs) are formulated analytically by the stochastic averaging method. Meanwhile, the results are validated numerically. In addition, stochastic bifurcations are also explored.     

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.     

Stochastic averaging of quasi-integrable and non-resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A stochastic averaging method for predicting the response of quasi-integrable and non-resonant Hamiltonian systems to combined Gaussian and Poisson white noise excitations is proposed. First, the motion equations of a quasi-integrable and non-resonant Hamiltonian system subject to combined Gaussian and Poisson white noise excitations is transformed into stochastic integro-differential equations (SIDEs). Then $n$ -dimensional averaged SIDEs and generalized Fokker–Plank–Kolmogrov (GFPK) equations for the transition probability densities of $n$ action variables and $n$ - independent integrals of motion are derived by using stochastic jump–diffusion chain rule and stochastic averaging principle. The probability density of the stationary response is obtained by solving the averaged GFPK equation using the perturbation method. Finally, as an example, two coupled non-linear damping oscillators under both external and parametric excitations of combined Gaussian and Poisson white noises are worked out in detail to illustrate the application and validity of the proposed stochastic averaging method.     

基于概率统计方法的微弱GPS信号捕获算法

微弱GPS信号的信噪比较低,需采用更长的相干和非相干积分时间增强信噪比,提出一种从概率统计角度分析相干和非相干积分运算过程的方法。首先给出了复数下变频信号输入的相关运算表达式;然后推导了相干积分的相关输出数学期望和方差,计算出单元捕获失败的概率密度和概率表达式;最后,用Matlab对GPS信号的相干和非相干积分进行仿真。仿真结果验证了捕获失败概率表达式的结论,即由于高斯近似和非相干平均产生的性能损失,非相干积分的捕获性能比相干积分弱3.5 dB。     

A finite-element method for analysis of a non-linear system under stochastic parametric and external excitation

A finite-element method approach is developed to solve the Fokker-Planck-Kol-mogorov equation for the probability density function of stationary response of a general non-linear system subject to both parametric and external Gaussian white noise excitation. N-dimensional shape functions are expanded by a one-dimensional shape function in global coordinate. As the domain of the density function is usually infinite, adaptive grid generation is adopted for the estimation of a finite range of the finite-element mesh. Several examples with existing exact solutions are used to illustrate the validity of the method. Results are also compared with those obtained by the Gaussian closure method, the cumulant-neglect method, equivalent external excitation and Monte Carlo simulation.     

Numerical investigation of a response probability density function of stochastic vibroimpact systems with inelastic impacts

The paper is devoted to numerical calculations of a response probability density function of stochastic vibroimpact systems excited by additive Gaussian white noise. An impact in the considered systems is modeled as a classical, inelastic one against motionless barrier(s) with values of a restitution coefficient both very small and close to unity. Certain empirical formulas are derived based on the results of numerical simulation.     

不完全测量系统鲁棒SGQKF的传递对准滤波器设计和稳定性分析

针对复杂环境下运载体观测信息不完全测量并且存在随机干扰不确定的传递对准问题,研究了不完全测量随机不确定系统的鲁棒稀疏网格求积分(H_∞-SGQKF)的高斯逼近滤波算法。基于非线性离散系统的最优贝叶斯滤波框架和间断观测滤波算法以及不确定性扰动噪声下的H_∞范数的鲁棒SGQKF算法,给出了不完全测量的稀疏网格求积分的高斯逼近滤波算法;通过非线性系统随机稳定性理论,分析并给出了系统估计误差和估计误差方差有界的充分条件,同时给出了系统稳定的不完全测量时的丢包率临界值,证明间断观测条件下的不完全测量H_∞-SGQKF算法是稳定的。通过传递对准仿真试验和某型激光捷联式惯性导航系统的跑车试验对该算法进行了验证。结果表明,该方法比未采用鲁棒的不完全测量的稀疏网格求积分滤波的传递对准精度有所提高,说明不完全测量的鲁棒稀疏网格求积分滤波算法是正确的、稳定的,并且具有鲁棒性能。