Probability Density Function Of Response Research Articles

Novel study on active reliable PID controller design based on probability density evolution method and interval-oriented sequential optimization strategy

Vibration active control is a significant problem in practical engineering, but the influence of hybrid uncertainties in practical engineering cannot be ignored. For the uncertainties with adequate data, probability theory can be used while for the uncertainties with insufficient data, probability theory is limited and interval quantification can be used. Taking the widely used proportional-integral-differential (PID) control as an example, a novel study on controller design based on probability density evolution method and interval-oriented sequential optimization strategy considering interval and random uncertainties is proposed. Firstly, under a certain sample point of interval uncertainties, probability density evolution equation is utilized to calculate the probability density function of uncertain response considering only random uncertainties. In this way the probabilistic reliability corresponding to the sample point can be obtained. Then the hybrid reliability is defined as the average probability within the interval uncertainty and the adaptive dichotomy method is utilized to choose proper sample points of interval uncertainties, in which the reliability curve and the interval uncertainty axis is regarded as the judgement of convergence. Next the hybrid reliability is introduced to the optimization as a constraint and the PID gains reliable design optimization is established to balance the energy-consuming and safety. Finally, the sequential optimization strategy is adopted to improve the efficiency of optimization. In every iteration step of the optimization, the constraint of deterministic optimal design is adjusted according to the static reliability at the moment before the passage or reaching the maximum value. Three numerical examples are presented to demonstrate the accuracy and practicability of the proposed framework.

Stochastic dynamics of mechanical systems with impacts via the Step Matrix multiplication based Path Integration method

AbstractIn this work we propose the Step Matrix Multiplication based Path Integration method (SMM-PI) for nonlinear vibro-impact oscillator systems. This method allows the efficient and accurate deterministic computation of the time-dependent response probability density function by transforming the corresponding Chapman–Kolmogorov equation to a matrix–vector multiplication using high-order numerical time-stepping and interpolation methods. Additionally, the SMM-PI approach yields the computation of the joint probability distribution for response and impact velocity, as well as the time between impacts and other important characteristics. The method is applied to a nonlinear oscillator with a pair of impact barriers, and to a linear oscillator with a single barrier, providing relevant densities and analysing energy accumulation and absorption properties. We validate the results with the help of stochastic Monte-Carlo simulations and show the superior ability of the introduced formulation to compute accurate response statistics.

Uncertainty quantification and stochastic response prediction for excavator systems under extreme operating conditions

The quantification of extreme operating conditions and their corresponding response predictions is a considerable challenge regarding the operation and maintenance of electromechanical equipment, such as dynamic excavator systems. This paper proposes a novel nonstationary modulation function model based on kinematic and dynamic analyses driven by on-site measurement data acquired from excavators. Extreme conditions are simulated through Gaussian white noise and impulse functions, and harmonic functions are overlaid on them to fit uncertain excitations, thus quantifying real random excitations. A key component of this method is its detailed statistical analysis of the probability density function (PDF) of the system response via Monte Carlo simulation. The results show that, compared with the measured results, the mean error of the system response PDFs predicted by this methodology is extremely low, and the developed approach can effectively capture the heavy-tailed characteristics of random responses under extreme loads. The proposed method utilizes a limited number of physical experiments to generate statistical results for large samples, providing the dual benefit of cost-effectiveness and high accuracy.

Multi-hazard stochastic response analysis and reliability evaluation of a 5 MW monopile offshore wind turbine based on extended platform FAST-S

This study investigates the random dynamic response of a 5 MW monopile offshore wind turbine (MOWT) subjected to wind, wave and earthquake loads. Based on the aero-hydro-servo-elastic code FAST, a multi-field coupled simulation module considering earthquake, FAST-S, is developed for the fully coupled multi-hazard analysis of OWTs. Compared to the original FAST code, a seismic analysis module (SeismoDyn) and a soil-structure interaction (SSI) module (SoilDyn) have been supplemented in FAST-S. SeismoDyn is developed by modifying the Kane’s kinetic equation in the FAST, and the SoilDyn is developed through Winkler foundation model in order to consider the nonlinear SSI. Moreover, FAST-S has been validated by other recognized codes. Based on FAST-S, a fully coupled numerical analysis model of the NREL 5 MW MOWT is established, the random responses and reliability evaluation of the offshore wind turbine under wind, wave and earthquake loads are analyzed by probability density evolution method. The results show that the aerodynamic and hydrodynamic actions increase the damping of the wind turbine system, which has a certain effect on inhibiting the tower’s vibration caused by the earthquake. Therefore, compared with the coupled wind-wave-earthquake case, the sole earthquake case may be more unfavorable, and should be paid more attention to in high risk earthquake regions. The results show that the probability density functions of offshore wind turbine response under strong earthquakes obviously deviates from the typical regular distribution models. Based on the theory of equivalent extreme value event, the dynamic reliability of the offshore wind turbine is calculated. Seismic actions maybe the major factor in the design of the support structure. Nevertheless, owing to the aerodynamic and hydrodynamic damping effects of wind and wave loadings, the structural responses under wind-wave-earthquake are more moderate than ones subjected to solely earthquake. The strong seismic loads would cause significant complex variations of time-dependent probability density function of the OWT response.

An extrapolation approach within the Wiener path integral technique for efficient stochastic response determination of nonlinear systems

An extrapolation approach within the Wiener path integral (WPI) technique is developed for determining the stochastic response of diverse nonlinear dynamical systems. Specifically, the WPI technique treats the system response joint transition probability density function (PDF) as a functional integral over the space of all possible paths connecting the initial and the final states of the response vector. Further, the functional integral is evaluated, ordinarily, by considering the contribution only of the most probable path. This corresponds to an extremum of the functional integrand, and is determined by solving a functional minimization problem that takes the form of a deterministic boundary value problem (BVP). This BVP corresponds to a specific grid point of the response PDF domain. Remarkably, the BVPs corresponding to two neighboring grid points not only share the same equations, but also the boundary conditions differ only slightly. This unique aspect of the technique is exploited herein. Specifically, it is shown that solution of a BVP and determination of the response PDF value at a specific grid point can be used for extrapolating and estimating efficiently the PDF values at neighboring points without the need for considering additional BVPs. Notably, the herein developed approach enhances significantly the computational efficiency of the WPI technique without, practically, affecting the associated degree of accuracy. Two numerical examples relating to a Duffing nonlinear oscillator subjected to combined stochastic and deterministic periodic loading, and to an oscillator with asymmetric nonlinearities and fractional derivative elements are considered to demonstrate the reliability of the extrapolation approach. Juxtapositions with pertinent Monte Carlo simulation data are included as well.

A novel uncertainty quantification method for determining deformations and reliabilities of stochastic laminated composite plates with geometric nonlinearity

Stochastic finite element method (SFEM) for uncertainty quantification is widely applied for analysis of structures with intrinsic randomness. For determining the geometrically nonlinear deformations of laminated composite plates with random fields, the existing intrusive SFEMs have the limitations of low applicability, insufficient accuracy, or low efficiency. To this end, this paper proposes a novel and efficient non-intrusive SFEM incorporating direct probability integral method to achieve probability density functions (PDFs) of stochastic responses and reliabilities of laminated composite plates with geometric nonlinearity. Firstly, the von Kármán strain-displacement relation based on the third-order shear deformation theory is employed to model the geometric nonlinearity of laminated plates. The random field is discretized via Karhunen–Loève expansion. Secondly, the probability density integral equation (PDIE) is derived from the new perspective of probability conservation. The proposed non-intrusive SFEM decouples the equilibrium equation and PDIE to compute the response PDFs and reliabilities of uncertain laminated composite plates in a unified way. Moreover, the criterion which can judge the applicability of geometrically nonlinear theory is suggested for performing uncertainty quantification. Finally, comparisons of the results in terms of Monte Carlo simulation and the literature demonstrate the high accuracy and efficiency of the proposed method. For stochastic laminated plates, the response statistical moments vary nonlinearly with linear increase of load amplitude due to geometric nonlinearity, the deflection variability increases and structural reliability decreases with the increase in variability and correlation length of random field, and the stacking angle significantly affects the stochastic nonlinear deflections and reliabilities.

Dynamic reliability analysis of Aerial Building Machine under extreme wind loads using improved QBDC-based active learning

Aerial building machine (ABM) is a new comprehensive construction equipment used for the construction of high-rise buildings, and the research of its wind-induced response is essential for structural safety. However, the traditional reliability estimation for complex structures is generally computationally expensive. This study proposes a novel and time-saving active learning (AL) approach to predict the structural time-variant response and calculate the reliability under extreme wind loads based on improved Query-by-Dropout-Committee (QBDC), Deep Learning (DL) and Probability Density Evolution Method (PDEM). To establish an effective training set, the load sample pool from the turbulent wind field is constructed by the number theory method (NTM). Then a DL surrogate model, aimed to simulate the excitation-response relationship of the structure, is well-trained from data calculated by the Finite Element Method (FEM) under the workflow of the improved QBDC-based AL approach. Lastly, the structural responses predicted by the surrogate model are used to solve the Generalized Density Evolution Equation (GDEE) to obtain the Probability Density Function (PDF) of the structural response. The dynamic reliability can be gained using absorbing boundary condition. To demonstrate the feasibility of the proposed approach, an ABM in China is taken as a case study. It is found that (1) The external trusses of the platform are weak positions under the horizontal wind loads. It is suggested to append the longitudinal (windward direction) trusses to reduce the local stress and deformation. (2) Under turbulent wind loads with the average speed of 40 m/s, the maximum structural response of the ABM will exceed the threshold value and the reliability is 0.932. (3) The reliability calculation error derived by the proposed approach and the traditional PDEM is within 0.5 % in this study, while the computational efficiency of the AL approach is 77.08 % higher than that of the traditional PDEM. The novelty lies in the proposed active learning approach that can easily obtain the stochastic response and dynamic reliability of the ABM, where the analysis workflow not only has a low computational cost but also maintains a high calculation accuracy.

Probability distributions for dynamic and extreme responses of linear elastic structures under quasi-stationary harmonizable loads

The non-stationary load models based on the evolutionary power spectral density (EPSD) may lead to ambiguous structural responses. Quasi-stationary harmonizable processes with non-negative Wigner-Ville spectra are suitable for modeling non-stationary loads and analyzing their induced structural responses. In this study, random environmental loads are modeled as quasi-stationary harmonizable processes. The Loève spectrum of a harmonizable load process contains several random physical parameters. An explicit approach to calculate the probability distributions for the dynamic and extreme responses of a linear elastic structure subjected to a quasi-stationary harmonizable load is proposed. Conditioned on the specific values of the load spectral parameters, the harmonizable load process is assumed to be Gaussian. The conditional joint probability density function (PDF) of structural dynamic responses at any finite time instants and the conditional cumulative distribution function (CDF) of the structural extreme response are provided. By multiplying these two conditional probability distributions with the joint PDF of the load spectral parameters, and then integrating these two products over the parameter sample space, the joint PDF of structural dynamic responses at any finite time instants and the CDF of the structural extreme response can be calculated. The efficacy of the proposed approach is numerically validated using two linear elastic systems, which are subjected to non-stationary and non-Gaussian wind and seismic loads, respectively. The merit of the harmonizable load process model is highlighted through a comparative analysis with the EPSD load model.

Transient Response Analysis of Nonlinear Oscillators With Fractional Derivative Elements Under Gaussian White Noise Using Complex Fractional Moments

Abstract Complex fractional moment (CFM), which is defined as the Mellin transform of a probability density function (PDF), has been successfully employed to find the response PDF of a wide variety of integer-order nonlinear oscillators. In this paper, a CFM-based analysis is performed to determine the transient response PDF of nonlinear oscillators with fractional derivative elements under Gaussian white noise. First, an equivalent linear system is introduced for the purpose of deriving the Fokker–Planck (FP) equation for response amplitude. The equivalent natural frequency and equivalent damping coefficient of the system need to be determined, taking into account both the nonlinear and fractional derivative elements of the original oscillator. Moreover, to convert the FP equation into the governing equation of CFMs, these equivalent coefficients must be given in polynomial form of amplitude. This paper proposes formulas for appropriately determining the equivalent coefficients, based on an equivalent linearization technique. Then, applying stochastic averaging, the FP equation is derived from the equivalent linear system. Next, the Mellin transform converts the FP equation into coupled linear ordinary differential equations for amplitude CFMs, which are solved with a constraint corresponding to the normalization condition for a PDF. Finally, the inverse Mellin transform of the CFMs yields the amplitude PDF. The joint PDF of displacement and velocity is also obtained from the amplitude PDF. Three linear and nonlinear fractional oscillators are considered in numerical examples. For all cases, the analytical results are in good agreement with the pertinent Monte Carlo simulation results.

Non-stationary probabilistic analysis of non-linear ship roll motion due to modulated periodic and random excitations

This paper is about the study on the non-stationary probability density function (PDF) solution of the non-linear ship rolling system under modulated periodic and random excitations. The quadratic and cubic damping terms are introduced to describe the non-linear damping moment and the power terms in state variables are adopted to describe the righting moment in the ship rolling system, respectively. The relevant Fokker–Planck–Kolmogorov (FPK) equation of the ship rolling system is constructed and the non-stationary probabilistic solutions are obtained by the presented solution procedure. In the solution process, the time variable t is introduced to the traditional exponential-polynomial-closure (EPC) method to obtain the non-stationary PDF solutions of the ship rolling system. The obtained non-stationary PDF solutions of the non-linear ship rolling systems demonstrate that the presented solution procedure can give high solution accuracy even in the PDF tails when they are compared with Monte Carlo simulation (MCS). Moreover, owing to the strong system nonlinearity and the interaction of the periodic excitation and random excitation, the achieved non-stationary PDF solutions of the system responses can deviate significantly from the Gaussian PDF. It is also found that the PDFs of responses are not even symmetrical about the means of responses due to the interaction of the periodic excitation and the random excitation even if the system only contains odd nonlinear terms. In addition, the computational efficiency is increased by over five hundred times by the presented solution procedure compared to MCS.

Transient probability density of nonlinear oscillator under parametric harmonic and external modulated stochastic excitations

In this study, the exponential-polynomial-closure (EPC) method is applied to determine and study the non-stationary probability density function (PDF) of the nonlinear system simultaneously stimulated by parametric harmonic and modulated stochastic excitations. The non-stationary probabilistic responses are represented as a time-related exponential polynomial function with unknown variables. Integrating the weighted residual error can result in a group of first-order differential equations. Those equations can be solved by numerical techniques to obtain the undefined variables. Four examples are given to show that the non-stationary behaviors of the presented oscillators can be addressed by the presented EPC procedure effectively, especially in the probabilistic tails which are of particular importance in reliability analysis. Moreover, the outcomes show that the response means equal to zero and the response PDFs are symmetrical with respect to the zero means. In particular, the presented EPC procedure can heavily decrease the running effort with no sacrifice to the solution correctness when compared with MCS.

Reliability assessment of stochastic dynamical systems using physics informed neural network based PDEM

In the recent decade, the reliability analysis of a stochastic system coupled with the uncertainty related to the system’s parameter has attracted much attention. Probability density evolution method (PDEM) is one of the viable options that estimates the probability density function of the structural response by solving generalized density evolution equations (GDEEs). The advantage of PDEM is that it is derived based on the principle of probability conservation, where GDEEs are decoupled from the physical system. In general, GDEEs in PDEM are solved using a finite difference scheme in which the accuracy of the numerical solution depends on the number of temporal and spatial discretizations, leading to computationally inefficient for high-fidelity models. With this in view, this study proposes a physics-informed neural network (PINN), a novel deep learning method, based PDEM, for solving the GDEEs. PINN utilizes physical information in the form of differential equations to enhance the performance of the neural networks. This method does not need any interpolation or coordinate transformation, which is often seen in any numerical scheme, thus the computational budget is reduced. Three numerical examples are presented in this study to illustrate the proposed PINN-based PDEM, including a Van-der-Pol oscillator subjected to Gaussian white noise, a one-storey moment resisting frame coupled with a nonlinear energy sink with negative stiffness and sliding friction, and a high-rise timber building coupled with shape memory alloy-based outriggers. The first example is utilized to show the accuracy of the proposed method by comparing results with the Fokker–Planck–Kolmogorov equation and Monte Carlo simulation. The rest two examples are investigated for estimating time-dependent probability of failure. Numerical results show that the proposed PINN-based PDEM can estimate the probability of failure efficiently.

Stochastic response analysis of 3D vibro‐acoustic systems with geometrical uncertainties based on the extended finite element method

AbstractUnlike uncertainties such as loads and material parameters, geometrical uncertainties caused by errors in the geometry and differences in equipment installation locations of the vibro‐acoustic system can significantly alter the solution domain, resulting in nominally identical vibro‐acoustic systems exhibiting large response dispersions. To solve such problems, this study introduces the extended finite element method/finite element method (XFEM/FEM) and the direct probability integration method (DPIM) to construct a novel stochastic response analysis scheme for 3D vibro‐acoustic systems. Furthermore, it examines the impact of geometric uncertainties on system responses. The XFEM/FEM can accurately capture the non‐smooth solution at the geometrical interface by constructing enriched functions without modifying the original mesh, and it allows the introduction of random parameters to describe the geometrical modifications. The input probability space of the parameters is partitioned, and the Dirac function is smoothed, yielding statistical properties like the probability density function of system responses, which are derived by summing responses from each representative point. The computational performance of the proposed scheme is validated with both single and multiple uncertain geometries, while also discussing the impact of these uncertain geometries on the responses of the vibro‐acoustic system.

Dynamic responses of the end-anchored floating bridge under the combined action of wind and waves

Long floating bridges offer a highly advantageous approach for crossing extensive and deep waterbodies. However, the multifaceted marine environment emphasizes the necessity of exploring the dynamic behavior of floating bridges under diverse load conditions. This study experimentally investigates the dynamic response of an end-anchored floating bridge under the combined action of wind and waves. A truncated model was designed with a geometrical scale of 1:100, and a series of trials were conducted, including the wind-only, wave-only, and wind-wave cases. The focus of the study was on the sway, heave and roll responses. Experimental results revealed that the coupling effect of wind and waves could increase the vulnerability of floating bridges. In the wind-wave cases, the dynamic response was categorized into distinct wind-controlled and wave-controlled zones. The mean response was predominantly influenced by wind loads, while the response standard deviation was significantly affected by wave loads. An analysis of the probability density function of motion response showed that the coupling effect in high-velocity wind cases amplified motion response amplitudes and broadened the distribution range. This study significantly advances the comprehension of the coupled behavior of floating bridge motion responses in wind-wave environments, offering valuable insights for the design of floating bridges.

A decoupled approach for determination of the joint probability density function of a high-dimensional nonlinear stochastic dynamical system via the probability density evolution method

The joint probability density function (PDF) of multiple response processes of a system is a crucial topic in the fields of science and engineering. It is adopted to describe the dependent uncertainty propagation in complex stochastic dynamical systems, including the displacement and velocity of one degree of freedom (DOF), and the nonlinear probabilistic dependencies between multiple variables. However, the partial differential equations (PDEs) governing the time evolution of joint PDFs for multiple stochastic processes are often high-dimensional and coupled, making them analytically or numerically intractable for systems with high dimensionality. The probability density evolution method (PDEM) has provided a new perspective from the standpoint of random events description of the principle of preservation of probability. For an n-dimensional system, if merely the joint PDF of m processes is needed, where m is not larger than n, then it is sufficient to solve the generalized density evolution equation (GDEE), i.e., the Li–Chen equation, in m dimensions. This approach is particularly convenient for cases where n is large while m is small. However, for cases where m is not less than two, it is still challenging to avoid solving high-dimensional PDEs. This paper proposes a new method for the determination of joint PDF of multiple responses based on the principle of preservation of probability. To capture the joint PDF of m processes, the m-dimensional PDE can be transformed into m one-dimensional PDEs, significantly reducing the computational complexity of numerical implementation. The proposed method gives an enhanced version to the PDEM, enabling it to not only determine the individual probabilistic response, but also efficiently determine the dependent probabilistic distribution of multiple responses of high-dimensional stochastic dynamical systems without solving the coupling PDE. It reduces the computational cost from exponential growth of m powers to linear growth. Some numerical examples are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.

Life-Cycle Seismic Reliability Analysis of a Railway Cable-Stayed Bridge Considering Material Corrosion and Degradation

To study the life-cycle seismic reliability analysis (SRA) of cable-stayed bridges (CSBs) taking into account chloride-induced corrosion and degradation of components, an actual railway CSB with uncertainties in structural geometry and material corrosion coefficients was employed in this investigation, and time-dependent models of CSB components at different service times were studied. Based on the OpenSees batch program, we adapted a mass numerical computation to obtain time-dependent non-linear seismic response and probability density function (PDF) of response via the multiplier dimensional-reduction method (MDRM) and the maximum entropy method with fractional moments (FM-MEM). Next, the time-dependent failure possibility of every component and the association coefficient between the failure modes of different parts were acquired. In the end, the product of the conditional marginal (PCM) approach was employed to obtain the life-cycle failure possibility of the CSB system. The results showed that the system failure possibility of the CSB in a corrosive environment increases significantly with increasing servicing time.

PDEM-based seismic performance evaluation of circular tunnels under stochastic earthquake excitation

ABSTRACT Tunnels are a vital component of urban infrastructure that must be robust against seismic hazards. Given the randomness of earthquake occurrence, the seismic response of tunnel structures mut be studied by stochastic analysis methods. To this end, this study proposes a probability density evolution method (PDEM)-based framework to investigate the seismic performance of a circular tunnel under stochastic earthquake excitation. First, a suite of nonstationary earthquake motions compatible with the seismic design code was derived using a stochastic earthquake model. Then, a series of nonlinear dynamic numerical simulations were conducted for a typical circular tunnel that considers the soil-structure interaction. Finally, using the tunnel inclination angle as the performance index, the probability density function of the structural response of the tunnel was solved using the PDEM to obtain the corresponding exceedance probabilities of the tunnel under various damage states. The results show that the PDEM-based framework can be applied to evaluate the seismic performance of circular tunnels and could serve as a reference on the seismic fragility of tunnels and underground structures.

Shaping the Probability Density Function of the Output Response in a Reverberation Chamber

This paper shows that the received power and E-field in a reverberation chamber (RC) can be shaped by tuning the statistical properties of input signals. For a given probability density function (PDF) of an RC response, the Fourier transform method can be applied to find the PDF of the input signal. Numerical and measurement verifications are given to validate the theory. Limitations are also analyzed and discussed.

Identification of nonlinear aerodynamic damping of cross-wind excited slender structures in atmospheric boundary layer flow

Due to the low natural frequency and low damping of slender structures, the crosswind aero-elastic effect become dominant around the vortex-resonance wind speed and can lead to undesired crosswind responses. For understanding and describing the aero-elastic effect of slender structures, aerodynamic damping is usually adopted. This study suggested an identification procedure for crosswind aerodynamic damping of slender structures by curve-fitting the probability density function (PDF) of the crosswind response. Firstly, the aerodynamic damping is described by a generalized Van der Pol (GVDP) model, and the analytical solution of response PDF, which is a function of the parameters of GVDP model, is derived by the equivalent nonlinear equation method. Next, the proposed identification method was validated by stochastic simulation of a crosswind-excited tall building. Finally, free-vibration wind tunnel test of a slender structure with slenderness ratio equal to 12 and 20 at open flat terrain was conducted and aerodynamic damping was then quantified by the present method. It is found that the present method provides a robust and accurate identification of crosswind aerodynamic damping of slender structures in atmospheric boundary flow.

Stochastic response analysis of 3D vibro-acoustic system with acoustic impedance and modeling parameter uncertainties

A novel stochastic response analysis scheme for 3D vibro-acoustic systems is constructed based on the PUFEM/FEM and the direct probability integral method, and the effect of uncertain acoustic impedance and modeling parameters on the system response is investigated. The proposed scheme has the capability of capturing waveform within the element, the small system matrices and easy handling acoustic impedance, also can efficiently treat the propagation of parameter uncertainties through transforming the integration of the response probability density function into the summation of the response under each representative point by partitioning the input probability space and smoothing the Dirac δ function. Numerical examples show that the mean and standard deviation of the responses computed by the proposed scheme agree well with the MCS, but the agreement of the standard deviation is slightly lower at the frequency with large response dispersion. The effect of uncertain acoustic impedance on the system response is mainly at the low frequencies, where the location and amplitude of resonance peaks fluctuate significantly. Uncertain modeling parameters synthetically modify the stiffness and mass matrix, altering the intrinsic properties and response of the system, and the response at the higher frequency is more sensitive to the uncertain modeling parameters.