Probability Density Function Of Response Research Articles

Dynamic Response of Asymmetric and Nonlinear Packaging System under Random Excitation

This paper focuses on the dynamic response of three asymmetric and nonlinear packaging systems (ANPSs) of practical transport package under random excitation. Next to presenting the displacement probability density functions (PDFs) of ANPSs derived through FPK equations, the influences of the excitation grades and the characteristic parameters (such as the damping, the nonlinear stiffness, and the strain) of packaging cushion materials on the response are discussed in detail. Then, the generalized PDFs of the response peaks are defined and examined by three common distributions. The investigation shows that “inverse excitation cushion factors” (IECFs) have a significance on effecting the displacement PDFs approaching different distributions. Most PDFs of the unilateral response peaks approach Rayleigh distributions and present non-Gaussian characters. The present methods are verified through the validation numerical solutions. Furthermore, the application of the scheme for fatigue damage evaluation of the transport package is carried out.

Addressing the curse of dimensionality in stochastic dynamics: a Wiener path integral variational formulation with free boundaries.

A Wiener path integral variational formulation with free boundaries is developed for determining the stochastic response of high-dimensional nonlinear dynamical systems in a computationally efficient manner. Specifically, a Wiener path integral representation of a marginal or lower-dimensional joint response probability density function is derived. Due to this a priori marginalization, the associated computational cost of the technique becomes independent of the degrees of freedom (d.f.) or stochastic dimensions of the system, and thus, the 'curse of dimensionality' in stochastic dynamics is circumvented. Two indicative numerical examples are considered for highlighting the capabilities of the technique. The first relates to marine engineering and pertains to a structure exposed to nonlinear flow-induced forces and subjected to non-white stochastic excitation. The second relates to nano-engineering and pertains to a 100-d.f. stochastically excited nonlinear dynamical system modelling the behaviour of large arrays of coupled nano-mechanical oscillators. Comparisons with pertinent Monte Carlo simulation data demonstrate the computational efficiency and accuracy of the developed technique.

A Study of Noise Impact on the Stability of Electrostatic MEMS

Abstract Noise-induced motions are a significant source of uncertainty in the response of micro-electromechanical systems (MEMS). This is particularly the case for electrostatic MEMS where electrical and mechanical sources contribute to noise and can result in sudden and drastic loss of stability. This paper investigates the effects of noise processes on the stability of electrostatic MEMS via a lumped-mass model that accounts for uncertainty in mass, mechanical restoring force, bias voltage, and AC voltage amplitude. We evaluated the stationary probability density function (PDF) of the resonator response and its basins of attraction in the presence noise and compared them to that those obtained under deterministic excitations only. We found that the presence of noise was most significant in the vicinity of resonance. Even low noise intensity levels caused stochastic jumps between co-existing orbits away from bifurcation points. Moderate noise intensity levels were found to destroy the basins of attraction of the larger orbits. Higher noise intensity levels were found to destroy the basins of attraction of smaller orbits, dominate the dynamic response, and occasionally lead to pull-in. The probabilities of pull-in of the resonator under different noise intensity level are calculated, which are sensitive to the initial conditions.

Identification of nonlinear aerodynamic damping of wind-excited flexible structures by curve-fitting non-Gaussian response probability density function

This study presents a new approach for identification of nonlinear aerodynamic damping of wind-excited flexible structures from stochastic response time history that can be modeled as the output of a single-degree-of-freedom system under a white noise excitation. The amplitude-dependent aerodynamic damping ratio is described as a polynomial function of vibration displacement or velocity. Based on the equivalent nonlinear equation approach, the probability density function (PDF) of displacement can be given in terms of damping model parameters, which generally follows a hardening non-Gaussian distribution. It is proposed to identify the damping model parameters by curve-fitting displacement PDF determined from the response data using a nonlinear least-square optimization technique. Firstly, the stochastic response of a tall building with a known nonlinear aerodynamic model is simulated and the performance of the proposed approach is investigated. This method is then applied to a square-shaped tall building model and a bridge deck section using wind tunnel measured data. Finally, it is used for a traffic-signal-support structure using field measurement response data. The results are also compared with those from other existing approaches. The proposed approach permits determination of nonlinear aerodynamic damping from single stochastic response history and is very easy to use in practice.

Probabilistic analysis on parametric random vibration of a marine riser excited by correlated Gaussian white noises

A path integration method is used to discuss the stochastic response of a marine riser excited parametrically and externally by correlated Gaussian white noises. On the basis of the nonlinear vibration equation of a marine riser, the first-order vibration mode is studied and its governing equation is formulated. The response probability density function (PDF) is further developed by a path integration method. During the solution procedure, the Gaussian closure method and the short-time Gaussian approximation method are used to develop the transition probability density function. The results of different time intervals are compared with the results of Monte-Carlo simulation (MCS) to determine the adequate time interval. After that, the path integration method with a Gauss–Legendre scheme is used to obtain the stationary PDF solution of displacement and velocity. The path integration solutions (PIS) are compared with those of the equivalent linearization method (EQL) and MCS. Five demonstrative cases are analyzed to evaluate the effectiveness of the path integration method. The influences of nonlinearity and excitations are considered. It shows that the results of PIS coincide better with the results of MCS than those of EQL. The analysis shows that parametric excitation causes the softening behavior of the PDF distribution compared with those of EQL. The increase of excitation correlation results in the nonsymmetrical PDF distribution of response. The increase of nonlinearity in displacement can lead the PDF of displacement to showing a hardening behavior.

On reliability analysis method through rotational sparse grid nodes

This study aims to develop a rotational sparse grid (R-SPGR) method for statistical moment evaluation and structural reliability analysis with enhanced accuracy and efficiency. The optimal rotational angle in the proposed R-SPGR method is determined by an optimization approach, in which the objective function is constructed in terms of discrepancy between the marginal moments of each input uncertain variables calculated from these rotational sparse grid nodes and their exact values. The R-SPGR nodes allow capturing more information of exact probability distribution for the system response, which is considered critically important to the calculation of statistical moments accurately. Following this, the probability density function (PDF) and failure probability of system response can be efficiently determined by using the saddlepoint approximation technique. To demonstrate the effectiveness of the proposed method, four benchmark examples, one structural analysis example and one practical example of industrial robot are provided here, in which the results obtained from the proposed R-SPGR method are compared with those calculated from the conventional sparse grid (SPGR) method, the maximum entropy problem with fractional moments (ME-FM) method and Monte Carlo simulation (MCS) method. From these illustrative examples involving a wide range of complexity, it is demonstrated that the proposed R-SPGR method has fairly high accuracy and efficiency for structural reliability analysis.

A general frame for uncertainty propagation under multimodally distributed random variables

Uncertainty propagation under multimodally distributed random variables, called multimodal distribution propagation for short, is a challenging problem due to the complicate probability density function of the random variables. In this paper, a general frame based on a new finite mixture model (λMM) constructed by derivative lambda probability density function and polynomial chaos expansion method is put forward to efficiently solve the multimodal distribution propagation problem. First, λMM with high accuracy and extensive applicability is proposed to represent the arbitrary unimodal distributions and multimodal distributions. Second, a pseudo EM method proved strictly is proposed to estimate the parameters of this λMM. Third, the statistical moments of the response by polynomial chaos expansion method is derived mathematically. Since the new mixture model can be decomposed into several unimodal probability distributions, the multimodal distribution propagation can be successfully converted into several unimodal distribution propagations, which further can be easily solved by polynomial chaos expansion method. Finally, the maximum entropy principle is adopted to evaluate the probability density function of the result of every unimodal distribution propagation, which is then superimposed into the final probability density function of the system response. The proposed frame only requires the first four statistical moments to evaluate the probability density function and avoid calculating the higher statistical moments, especially for the situation that the probability distribution of the system response is multimodal. Four examples are presented to verify the high accuracy and efficiency of the proposed general frame for uncertainty propagation.

Functional Series Expansions and Quadratic Approximations for Enhancing the Accuracy of the Wiener Path Integral Technique

A novel Wiener path integral (WPI) technique is developed for determining the response of stochastically excited nonlinear oscillators. This is done by employing functional series expansions in conjunction with quadratic approximations. The technique can be construed as an extension and enhancement in terms of accuracy of the standard (semiclassical) WPI solution approach where only the most probable path connecting initial and final states is considered for determining the joint response probability density function (PDF). In contrast, the technique developed herein accounts also for fluctuations around the most probable path, thus yielding an increased accuracy degree. An additional significant advantage of the proposed enhancement as compared to the most probable path approach relates to the fact that low-probability events (e.g., failure probabilities) can be estimated directly in a computationally efficient manner by determining only a few points of the joint response PDF. Specifically, the normalization step in the standard approach, which requires the evaluation of the joint response PDF over its entire effective domain, is circumvented. The performance of the technique is assessed in several numerical examples pertaining to various oscillators exhibiting diverse nonlinear behaviors, where analytical results are set vis-à-vis pertinent Monte Carlo simulation data.

Probabilistic analysis of fatigue crack growth using efficient surrogate model

In this paper probabilistic analysis of a fatigue crack growth process governed by the Walker law is performed. Sensitivity of its parameters to uncertainties related to the material properties is considered. Based on statistical analysis of experimental data, they are modelled as independent normal random variables. The effect of these uncertainties on the fatigue crack growth behaviour of Centre Cracked Plate specimen is assessed using Monte-Carlo simulations and surrogate model based on polynomial chaos expansion. An efficient truncation scheme allows to discard the high order interactions having a weak effect on the mechanical response and consequently reduce the number of finite elements calls when identifying the unknown coefficients of the polynomial chaos expansion. The statistical moments namely the mean and the standard deviation in addition to the probability density function of the mechanical response are derived with a good accuracy. The obtained results show that the fatigue crack growth life is significantly affected by the uncertainties on the material properties since its coefficient of variation reaches 21% and that it follows a log-normal distribution law.

Stationary probability densities of generalized Maxwell-type viscoelastic systems under combined harmonic and Gaussian white noise excitations

Viscoelasticity has been widely studied over the past few decades as a combination of the viscous effects of energy dissipation and the elastic effects of energy storage. As an application, viscoelastic dampers are commonly used to suppress the dynamic behaviour of structures. However, in an actual physical environment, it is difficult to explore the stationary probability densities of random responses for viscoelastic systems. For this purpose, we present a numerical method to investigate the dynamic behaviour of a generalized Maxwell-type viscoelastic system under harmonic and Gaussian white noise excitations. Using approximate equivalent and stochastic averaging, we establish an averaged Ito differential equation for the amplitude of the system. For primary external resonance, we solve the reduced Fokker–Planck–Kolmogorov equation by using the successive over-relaxation technique combined with a finite difference method. Through Monte Carlo simulations, we verify the applicability, accuracy and efficiency of the proposed methodology and demonstrate that viscoelasticity has a significant impact on the dynamic behaviour of the viscoelastic systems. This work reveals the remarkable influences of viscoelasticity, excitation intensities, linear and nonlinear damping and linear and nonlinear stiffness on the probability density functions of system responses, which can guide the selection of materials, stiffnesses and structures in viscoelastic damper design.

Sequential approximate reliability-based design optimization for structures with multimodal random variables

For practical engineering design problems, random variables tend to follow multimodal probability distributions when working at multiple operating conditions. For example, the structural fatigue stress of a steel bridge carrying both highway and railway traffic obeys the bimodal distribution. The existing popular reliability-based design optimization (RBDO) methods are mainly used to treat random variables with only unimodal distributions, which, therefore, tends to result in relatively large computational errors when multimodal random variables are involved. In this paper, a sequential approximate RBDO method is firstly proposed for engineering design involving multimodal random variables. The probability density function (PDF) of the response function is firstly calculated to assess the reliability of each probabilistic constraint, due to the existence of multimodal random variables. Then, a deterministic optimization problem is established to calculate candidate design points, based on the approximation that the response PDF demonstrates only transitional deformation in the optimization process. Three numerical examples and one engineering application of a thermal laptop design are presented to demonstrate the effectiveness of the proposed method.

Stochastic response determination of nonlinear structural systems with singular diffusion matrices: A Wiener path integral variational formulation with constraints

The Wiener path integral (WPI) approximate semi-analytical technique for determining the joint response probability density function (PDF) of stochastically excited nonlinear oscillators is generalized herein to account for systems with singular diffusion matrices. Indicative examples include (but are not limited to) systems with only some of their degrees-of-freedom excited, hysteresis modeling via additional auxiliary state equations, and energy harvesters with coupled electro-mechanical equations. In general, the governing equations of motion of the above systems can be cast as a set of underdetermined stochastic differential equations coupled with a set of deterministic ordinary differential equations. The latter, which can be of arbitrary form, are construed herein as constraints on the motion of the system driven by the stochastic excitation. Next, employing a semi-classical approximation treatment for the WPI yields a deterministic constrained variational problem to be solved numerically for determining the most probable path; and thus, for evaluating the system joint response PDF in a computationally efficient manner. This is done in conjunction with a Rayleigh-Ritz approach coupled with appropriate optimization algorithms. Several numerical examples pertaining to both linear and nonlinear constraint equations are considered, including various multi-degree-of-freedom systems, a linear oscillator under earthquake excitation and a nonlinear oscillator exhibiting hysteresis following the Bouc–Wen formalism. Comparisons with relevant Monte Carlo simulation data demonstrate a relatively high degree of accuracy.

REPLICATION-BASED EMULATION OF THE RESPONSE DISTRIBUTION OF STOCHASTIC SIMULATORS USING GENERALIZED LAMBDA DISTRIBUTIONS

Due to limited computational power, performing uncertainty quantification analyses with complex computational models can be a challenging task. This is exacerbated in the context of stochastic simulators, the response of which to a given set of input parameters, rather than being a deterministic value, is a random variable with unknown probability density function (PDF). Of interest in this paper is the construction of a surrogate that can accurately predict this response PDF for any input parameters. We suggest using a flexible distribution family -- the generalized lambda distribution -- to approximate the response PDF. The associated distribution parameters are cast as functions of input parameters and represented by sparse polynomial chaos expansions. To build such a surrogate model, we propose an approach based on a local inference of the response PDF at each point of the experimental design based on replicated model evaluations. Two versions of this framework are proposed and compared on analytical examples and case studies.

The uncertainty importance measure of slope stability based on the moment-independent method

It is well known that there are many uncertainties in slope engineering, which have great impacts on slope stability models. However, the uncertainty importance measure is rarely applied in slope stability analysis. In this paper, the moment-independent method of the uncertainty importance measure is first developed to analyze slope stability, in which the low deviation sequence of Sobol is used to simulate the geotechnical parameter samples, and a nonparametric method of the kernel density estimate is employed to estimate the probability density functions of the output responses. In addition, the Gaussian copula is applied to construct the joint distribution of the shear strength parameters in the uncertainty importance measure of the correlation parameters. Two examples are shown to indicate that the impacts of geotechnical parameters on slope stability models are greatly different. The importance sequences obtained by the moment-independent method are in good agreement with those obtained by the variance-based Monte Carlo method. The importance measure indexes clearly state that the uncertainty of the geotechnical parameters significantly affects slope stability models, and the influence of the negative correlation between the shear strength parameters on the uncertainty importance measure should not be neglected.

Probabilistic Response and Stochastic Bifurcation in a Turbulent Swirling Flow

Abstract Stochastic dynamics in a turbulent swirling flow are reported in this paper via the probability density functions (PDFs) of responses with the generalized cell mapping (GCM) method. Based on the short-time Gaussian approximation (STGA) procedure, the influence generated by the time average and the amplitude of the fluctuation to the turbulent flow on the probabilistic responses are demonstrated. We observe that the shapes of the steady-state PDFs change from two peaks to the single peak with the change of system parameters, indicating that the rotation to shear ratio will change from two stable states into one stable state, while the torque difference of the propellers in the von-Karman turbulence experimental setup becomes large or changes in a wide range. That is to say, the stochastic P-bifurcation phenomena occur. The evolutionary mechanism of the transient response is revealed with the global portraits. Furthermore, the idea of block matrix is devoted to solving the storage problem due to the amount of image cells for the STGA procedure in high dimensional system. Monte Carlo (MC) simulations are in good agreement with the proposed strategy.

Direct probability integral method for stochastic response analysis of static and dynamic structural systems

This paper proposes a unified and efficient direct probability integral method (DPIM) to calculate the probability density function (PDF) of responses for linear and nonlinear stochastic structures under static and dynamic loads. Firstly, based on the principle of probability conservation, the probability density integral equation (PDIE) equivalent to the probability density differential equation is derived for stochastic system. We highlight that, for time dependent stochastic system the PDIE is satisfied at each time instant. Secondly, the novel DPIM is proposed to solve PDIE directly by means of the point selection technique based on generalized F discrepancy and the smoothing of Dirac delta function. Moreover, the difference and connection among the DPIM, the existing probability density evolution method and probability transformation method are examined. Finally, four typical examples for stochastic response analysis, including the linear and nonlinear systems subjected to static and dynamic loads, demonstrate the high computational efficiency and accuracy of proposed DPIM.

Application to nonlinear mechanical systems with dry friction: hard bifurcation in SD oscillator

The stochastic model approach of a nonlinear non-smooth dynamical system with the probable occurrence of stochastic P-bifurcations is devoted. The response probability density functions (PDF) for the stationary measure of a smooth and discontinuous oscillator under moving loads belt frictions is constructed. The appearance of abrupt changes and unpredictable events illustrate the complexity of the system. The stationary measure varies continuously with system’s parameters and describes various kinds of catastrophic events. In light of these facts, the behaviour of the “stochastic attractors” is examined through the stationary solution of the PDF. According to Zeeman, in the phenomenological approach in the presence of noise, “a change in character of the density function as a parameter is varied is known as p- bifurcation”. Numerous new events unique to non-smooth systems are observed under slight variation of system’s parameters. Discontinuous bifurcations are defined as the “hard bifurcations” that were the subject of Catastrophe theory. Peaks numbers increase as coefficient of friction µ (or smoothness parameter α) increases. Numerical simulations are presented that provide insights into the dynamics of these oscillators.

Dynamics analysis of multi-field coupled piezoelectric energy harvester under random excitation

Automaatic fully mechanized mining face requires highly equipment condition monitoring. Complex environment of coalmine means the difficulty of wiring and power supply of wired monitoring. The multi-field coupled piezoelectric energy harvester (MPEH) could power wireless monitoring nodes by capture vibration energy from mining equipment. This paper makes a complete investigation of the distance effect of magnet, noise spectrum density and damping on the dynamic response characteristics of the MPEH under random excitation. Taking white Gaussian noise as input excitation, a dynamic model of the MPEH with random excitation in multiple fields is established. The probability density function of dynamic response under different system parameters is obtained by FPK equation, and the stationary mean square value and power spectral density function of system response are given by the ignored high order cumulant truncation method. In order to verify the accuracy of analytical results, the Monte Carlo numerical analysis is presented. Results show that the decrease of the magnetic distance will cause the moving of peak power spectrum frequency to low frequency. Reducing system damping ratio, narrowing magnetic distance or increasing excitation spectral density can enhance the probability of large period vibration and increase the average power of the system.

An advanced algorithm to study the effect of uncertainties on the stochastic performance of high-pier bridge under earthquake

A hybrid method combining the stochastic pseudo excitation method (SPEM) and the response surface method (RSM) was presented to handle the effect of both random characteristics of power spectral density of excitation and uncertain parameters. Based on basic principle of pseudo excitation method (PEM), an optimized algorithm was proposed to be SPEM. The effects of the uncertainties of structure and excitations could be considered through SPEM, thus directly acquiring the whole sampling results of random vibration of uncertain structural system. To reduce the cost of computing the total number of required finite element analysis simulations, RSM was adopted to hold calculated sampling results to gain the probability density function (PDF) of random vibration. The proposed method can consider both the effects of random characteristics of power spectrum of random excitations and the uncertainties of target system. Neverthless, Since the proposed method is based on the pseudo excitation method, the SPEM-RSM can handle the stochastic vibration analysis of determine and uncertain structure. In order to verify the accuracy and efficiency of proposed method, taking a single degree of freedom linear uncertain oscillator and a pier construction as two verification cases, the PDF of responses computed by the hybrid method was verified by Monte Carlo method (MCM), and the efficiency of the proposed method was presented. Then, stochastic responses of the railway continuous bridge subjected to earthquake were calculated, several uncertainties are involved in the computation. In addition, sensitivity analysis was solved for uncertain parameters.

Stochastic Dynamic Analysis of an Offshore Wind Turbine Structure by the Path Integration Method

Stochastic dynamic analysis of an offshore wind turbine (OWT) structure plays an important role in the structural safety evaluation and reliability assessment of the structure. In this paper, the OWT structure is simplified as a linear single-degree-of-freedom (SDOF) system and the corresponding joint probability density function (PDF) of the dynamic response is calculated by the implementation of the path integration (PI) method. Filtered Gaussian white noise, which is obtained from the utilization of a second-order filter, is considered as horizontal wind excitation and used to excite the SDOF system. Thus, the SDOF model and the second-order linear filter model constitute a four-dimensional dynamic system. Further, a detailed three-dimensional finite element model is applied to obtain the natural frequency of the OWT and the efficient PI method, which is modified based on the fast Fourier transform (FFT) convolution method, is also utilized to reduce the execution time to obtain the PDF of the response. Two important parameters of wind conditions, i.e., horizontal mean wind speed and turbulence standard deviation, are investigated to highlight the influences on the PDF of the dynamic response and the reliability of the OWT.