Stochastic Dynamic Response Research Articles

Deterministic and stochastic thermomechanical nonlinear dynamic responses of functionally graded sandwich plates

Functionally graded composites are commonly used under extreme mechanical and thermal loads wherein a minimal degree of uncertainty might significantly deviate the structural responses. Meanwhile, there is a lack of studies that consider the effects of inevitable source-uncertainty, especially in the stochastic dynamics of composites. This paper, for the first time, investigates both deterministic and stochastic dynamics of functionally graded sandwich plates with four different layering configurations under thermomechanical loads. Euler-Lagrange equations are derived based on the third-order shear deformation theory and Hamilton principle. Closed-form solutions for fundamental frequencies are obtained by introducing a stress function. Dynamic responses, phase-space plots, backbone curves, and forced response curves are numerically solved and graphically illustrated leading to notable insights into the stability and softening/hardening behaviors of the plate dynamics. A large number of random configurations are generated and analyzed resulting in stochastic dynamic bounds of dynamic responses, backbone curves, and forced response curves. The study shows that the plates have dynamic hardening behaviors, and the existence of thermal loads makes the plate random responses more diverse in stochastic environments. This work provides a good understanding of the deterministic and stochastic dynamics of the composites, which are crucially important for practical engineering applications.

Seismic time-history response and system reliability analysis of slopes considering uncertainty of multi-parameters and earthquake excitations

In practical geotechnical engineering of slopes, it is well-known that the uncertainty of soil properties and ground motions need to be considered for the safety of the project. The effect of uncertain parameters and stochastic ground motions on seismic responses as well as dynamic failure probabilities of slopes are analyzed and discussed in this study. A recently proposed generalized probability density evolution method (GPDEM) is formulated to investigate the system reliability of complex slopes with consideration to uncertainty in multiple slope parameters and in ground motions. Firstly, the fundamental theory of dynamic stability, GPDEM, dynamic system reliability, generation of random samples and stochastic ground motion model are described. Then, the efficiency and accuracy of proposed GPDEM are verified and a novel evaluating index is presented. Four complex slopes, each with multi-parameter uncertainties, are examined in separate study cases. Each study case uses GPDEM to study the random dynamic responses as well as probability responses of slopes while considering deterministic ground motion. The efficiency and accuracy of GPDEM used in complex slopes are validated by comparing with the Monte Carlo method (MCM). In addition, a novel evaluating index called the average large failure probability Pff, is proposed to describe the failure probability of the slopes, and can take the probability time histories intoaccount. Finally, the seismic system reliability of complex slopes is evaluated using the time-history method, and randomness in earthquakes, parameters and their coupling are investigated. The stochastic dynamic responses and probability responses of deterministic parameters subjected to stochastic earthquake excitations, and random parameters subjected to stochastic earthquake excitations are also discussed and are compared based on study case 3. The results demonstrate the high accuracy and validity of the GPDEM. Furthermore, the probability of failure and the degree of safety for the slopes can be directly estimated through probability analysis. In addition, the results also indicate that randomness in ground motions have a much larger influence than randomness in slope parameters, but randomness in slope parameters cannot be completely neglected.

An Efficient Approach for Stochastic Vibration Analysis of High-Speed Maglev Vehicle-Guideway System

The high-speed maglev vehicle-guideway coupled system (MVGCS) is a complex system, whose random vibration characteristics have not been well studied due to a limited number of examples. To address this issue, a new efficient approach is proposed for the random vibration analysis of the MVGCS, which combines the probability density evolution method and multi-time step method with multiple random loads considered. The random model established for 10-degree-of-freedom maglev vehicles and guideway is time-dependent, considering two different supporting conditions. The Monte Carlo method is used to assess the accuracy and efficiency of the proposed approximate approach, and the random model is verified through comparison with available results. The stochastic dynamic responses of the vehicles, guideway, and electromagnetic levitation forces, including the mean values and standard deviations, are determined in a case study. The results show that the proposed method is feasible for the dynamic analysis of maglev systems with a reasonably good efficiency in computation. Furthermore, critical parametric analyses involving vehicle speed, irregularity, and cut-off wavelength are performed with the results discussed.

Stochastic dynamic response and seismic reliability analysis of nuclear power plant’s vertical retaining wall based on plastic failure

Vertical retaining wall is usually built at the water intake in front of the pump station and is widely used in the protection of intake and voidance structures of nuclear power plants (NPPs). It has great practical significance for disaster prevention to gauge the plastic failure of vertical retaining wall accurately. In this paper, a generalized plastic model is introduced to obtain the residual deformation of vertical retaining wall during earthquake and the numerical simulation results based on plastic model reflect the plastic failure mode of the vertical retaining wall, which is consistent with the shaking table test results. In addition, the seismic excitation has great uncertainty, and in order to reflect the uncertainty, a set of fully nonstationary stochastic seismic loading samples are generated with spectral representation random function method. As a recently developed dynamic reliability methodology, the probability density evolution method (PDEM) is incorporated to the generalized plastic model to perform stochastic seismic response analysis of a vertical retaining wall. Plentiful probabilistic results of vertical retaining wall, such as evolution of probability density functions (PDF) and second-order statistics are acquired in this study. Finally, the dynamic reliability of vertical retaining wall under two earthquake levels are obtained based on the dip angle, which can more directly reflect the failure probability and degree of vertical retaining wall. The results of this study indicate that the geotechnical structures can show a more objective failure mode under earthquakes based on the plastic model, and it is more reliable to evaluate the safety of structures from the perspective of probability combined with PDEM.

Stochastic dynamical response of a gear pair under filtered noise excitation

In this study, the stochastic dynamic response of a spur gear pair model under the excitation of filtered noise is investigated. The spur gear pair is modeled as a single degree of freedom (SDOF) system in which nonlinear and non-smooth backlash and time-varying mesh stiffness as well as stochastic excitation are concurrently considered. Four cases are addressed, based on how the noise is incorporated in the loading terms. A first order filter is employed to generate various filtered noises with the same energy but different power spectrum. The combination of the SDOF gear model and the shaping filter leads to a (3D) Markov model. The numerical path integration (PI) technique is adopted to obtain the probabilistic response of the gear model using an adaptive time-stepping method in order to increase the accuracy of the time integration. A 4D system is also considered by applying a second order filter to model a narrow-band noise. The results are verified by comparing with Monte Carlo simulations. The effect of the noise spectrum on the probabilistic response is evaluated for different loading cases. The response PDF is of key importance in relation to reliability assessment of gear systems.

A reliability evaluation method for RV reducer by combining multi-fidelity model and Bayesian updating technology

RV (rotating vector) reducer is a two-stage transmission mechanism composed of involute planetary gear transmission and cycloid gear transmission with small tooth difference. It is widely used in the transmission mechanism of industrial robots and CNC machine tools. Nonlinear dynamics of gear system is always an important and difficult problem, the randomness of manufacturing and assembly makes the dynamic response become more complex. In this paper, the stochastic dynamic response analysis and dynamic reliability estimation method is proposed for RV reducer. In this method, firstly, considering the uncertainties of manufacturing error and assembly error, the dynamic transmission performance is analyzed respectively using physical prototype test, virtual prototype simulation and theoretical model. Secondly, multifidly model and Bayesian theory are combined to estimate the statistical distribution and distribution parameters of the dynamic transmission error. Finally, the reliability of RV reducer is estimated, and the effects of uncertainties from manufacturing error and assembly error on the transmission accuracy are discussed.

Dynamic reliability analysis of a floating offshore wind turbine under wind-wave joint excitations via probability density evolution method

Floating offshore wind turbine (FOWT) towers are dynamically sensitive to wind and wave excitations. Since the wave tends to be harsher with the increase of wind speed, FOWT towers are likely to experience the most severe vibration operating at the cut-out wind speed. In the present study, the short-term dynamic reliability of a spar-type FOWT is evaluated based on the probability density evolution method (PDEM). For this purpose, an integrated coupled dynamics model for the FOWT is firstly established by incorporating the multibody dynamics with the finite element (FE) method. Next, the conditional joint probability distribution of the significant wave height and peak spectral wave period at the cut-out wind speed is constructed based on the copula model. Then, the stochastic dynamic response and reliability of the FOWT can be analyzed via PDEM. The numerical example of reliability analysis of a 5-MW spar-type FOWT operating at the cut-out wind speed is carried out, in which the long-term met-ocean data at a South China Sea site is utilized. Simulation results show that the reliability of the FOWT for normal operation is less than 0.2 when the acceleration at the tower top is adopted as the failure criterion.

Detection of Damage in OperatingWind Turbines by Signature Distances

Wind turbines operate in the atmospheric boundary layer and are subject to complex random loading. This precludes using a deterministic response of healthy turbines as the baseline for identifying the effect of damage on the measured response of operating turbines. In the absence of such a deterministic response, the stochastic dynamic response of the tower to a shutdown maneuver is found to be affected distinctively by damage in contrast to wind. Such a dynamic response, however, cannot be established for the blades. As an alternative, the estimate of blade damage is sought through its effect on the third or fourth modal frequency, each found to be mostly unaffected by wind. To discern the effect of damage from the wind effect on these responses, a unied method of damage detection is introduced that accommodates different responses. In this method, the dynamic responses are transformed to surfaces via continuous wavelet transforms to accentuate the effect of wind or damage on the dynamic response. Regions of signicant deviations between these surfaces are then isolated in their corresponding planes to capture the change signatures. The image distances between these change signatures are shown to produce consistent estimates of damage for both the tower and the blades in presence of varying wind eld proles.

Research on Cracking Mechanism of Coastal Bridges' Piers Scoured by Sea Wave

Liu, Y.; Zhu, Z.-W.; Liu, F., and Xu, J., 2020. Research on cracking mechanism of coastal bridges' piers scoured by sea wave. In: Wan, Z. (ed.), Coordinated Sustainable Development in Coastal Areas: Environment and Economy. Journal of Coastal Research, Special Issue No. 109, pp. 159–166. Coconut Creek (Florida), ISSN 0749-0208.In this paper, the nonlinear dynamics of cracked coastal bridges' cylindrical piers impacted by sea waves are studied. The constitutive model of the materials is proposed where the nonlinearity of the cement's stiffness is considered. The dynamic characteristics of the system subjected to harmonic excitation are analyzed, and the effect of the system's parameters to the stability is discussed. Moreover, the stochastic dynamic responses of the system subjected to random excitation are obtained. The theoretical and numerical results show that there is random Hopf bifurcation occurs in the system, which causes the system's motion switch between the slight oscillation and large-amplitude vibration.

Gaussian process assisted stochastic dynamic analysis with applications to near-periodic structures

This paper characterizes the stochastic dynamic response of periodic structures by accounting for manufacturing variabilities. Manufacturing variabilities are simulated through a probabilistic description of the structural material and geometric properties. The underlying uncertainty propagation problem has been efficiently carried out by functional decomposition in the stochastic space with the help of Gaussian Process (GP) meta-modelling. The decomposition is performed by projected the response onto the eigenspace and involves a nominal number of actual physics-based function evaluations (the eigenvalue analysis). This allows the stochastic dynamic response evaluation to be solved with low computational cost. Two numerical examples, namely an analytical model of a damped mechanical chain and a finite-element model of multiple beam-mass systems, are undertaken. Two key findings from the results are that the proposed GP based approximation scheme is capable of (i) capturing the stochastic dynamic response in systems with well-separated modes in the presence of high levels of uncertainties (up to 20%), and (ii) adequately capturing the stochastic dynamic response in systems with multiple sets of identical modes in the presence of 5–10% uncertainty. The results are validated by Monte Carlo simulations.

Wheel–rail stochastic dynamics and rail wear analysis of small radius curved sections of a tram line based on generalized probability density evolution

Compared with ordinary railways, the curve radius of tram lines tends to be smaller, with minimum values of only 30 m. Therefore, wheel–rail interaction is more intense and complicated in sections of small radius tram line curves. Using a stochastic variable sample set based on a generalized probability density evolution method, the stochastic variable–spectrum representation method was used to generate a time-domain sample set of stochastic track irregularities. By inputting the stochastic set of track irregularities into a tram-track coupled dynamic system model, the stochastic dynamic response of the coupled dynamic system can be obtained. Moreover, by substituting the stochastic dynamic system response into the generalized probability density evolution formula, the process of probability density evolution of each evaluation index can be obtained by the finite difference method. Finally, the dynamic response of the tram-track coupled dynamic system can be evaluated by the probability distribution of each index. By setting a series of specific groove rail wear values, a tram-track coupled dynamic analysis was carried out, and compared with the specification requirements, vehicle safety limits under different wear values were obtained. This research has great engineering value for guiding the routine maintenance of small radius curve sections of trams.

The analyses of dynamic response and reliability for failure-dependent stochastic micro-resonator with thermoelastic coupling effects

A crucial measure for the design of high-performance micro-resonators is to consider the randomness of structural parameters when analyzing the structural system reliability. In this work, the stochastic dynamic response analysis and subsequently, a dynamic reliability assessment of the random micro-resonators are originally presented, where the thermoelastic coupling effects are freshly incorporated in the models proposed. The dynamic characteristics equation of the deterministic micro-resonator is firstly established based on the finite element method. The random dynamic characteristics of the resonator are then solved by implementing the left and right eigenvectors and the block Lanczos algorithm, and the random temperature field and structural random dynamic stress are also tackled. Afterwards, the overall structural reliability is investigated with a comprehensive consideration of the strength failure and frequency resonance failure, in which the Copula function is used for describing the dynamic correlation between two failure modes. Finally, the feasibility and rationality of the method put forward are demonstrated via a practically motivated example.

Multi-scale stochastic dynamic response analysis of offshore risers with lognormal uncertainties

This paper presents a multi-scale stochastic dynamic analysis method for offshore structures. The uncertainties in the structural material parameters, such as mass density and Young's modulus, are considered. They are assumed to be lognormal distributions and represented by using the Karhunen–Loeve (KL) and Polynomial Chaos (PC) expansions. Since the variance of the output responses is unknown, the output vibration response is represented by using PC expansion. The multi-scale stochastic analysis is conducted with PC expansions of different orders representing responses at different DOFs defined as three categories, namely, important, less important and the least important ones. Iterated Order Reduced (IOR) model reduction technique is employed to remove the PC coefficients of slave DOFs. Two numerical examples are taken to verify the accuracy and efficiency of the proposed method for the multi-scale stochastic dynamic response analysis of offshore risers. The response statistics such as mean value and variance can be obtained from the proposed method. The results are compared with those from Monte Carlo Simulation (MCS) and Stochastic Finite Element Method (SFEM). Results demonstrate that the computational demand for uncertainty evaluation is greatly reduced, and the accuracy of the results is maintained.

Stochastic dynamic response analysis of submerged multi-body structure considering the uncertainty of connection gap

The aim of this study is to determine the stochastic dynamic characteristics of the system due to the uncertainty of the gap contact between the mooring cable and the structural body of the submerged multi-body structure. The sag effect caused by the light, flexible, and low damping characteristics of the mooring cable and its dynamic elongation under the action of the flow field and the impact of the gap connection is considered. The sag effect is corrected using the equivalent elastic modulus method; the stochastic generalized non-deterministic dynamic model of the system considering the dynamic elongation and clearance contact of the mooring cable is established; and the Newmark-β method and the interval algorithm are used to solve the problem. The calculation results show that the mean and peak values of the kinematic parameters of the structure are larger than the ideal articulated state; the dynamic tension is changed randomly; and the deflection angle of the gap-contact state is greater than that of the articulated state, which may lead to the instability of the system. Therefore, when studying the concrete structure in water, we should consider the contact state of the joint and analyze its stochastic dynamic characteristics.

Stochastic dynamic analysis of composite plate with random temperature increment

During the service life, laminated composites may be subject to some random thermal environment. Quantification of the uncertainty in static and dynamic response of the composites under such condition is still a challenging issue. This work presents a stochastic dynamic response analysis of a graphite-epoxy composite plate using generalized polynomial chaos (gPC) expansion due to random mean temperature increment. A stochastic finite element method (SFEM) based on the first-order shear deformation theory (FSDT) is used to describe the free and forced vibration response of the graphite-epoxy composite plate under a uniform distribution of the temperature throughout the plate. Newmark’s time integration scheme is used to predict the time-dependent displacement response under dynamic loading. The collocation-based non-intrusive gPC expansion method is used for stochastic dynamic analysis of the graphite-epoxy composite plate. The increment in the temperature is considered as an uncertain parameter and presented by the truncated gPC expansion. The stochastic system response of the plate is projected to the deterministic solver by using the stochastic Galerkin method. The statistical response of eigen frequencies and dynamic displacements of the composite plate at incremental random mean temperature are investigated, and are compared with the results of the Monte Carlo simulation. The numerical studies show a reduction in amplitude of the dynamic mean displacements with the increment in the time and it increases with the increment in the random mean temperature. The characteristics of loading have also significantly influenced the uncertainty in the time-dependent displacement response.

Phase driven study for stochastic linear multi-dofs dynamic response

Abstract This work addresses the computation of dynamic responses of stochastic linear systems using polynomial chaos expansion. As is now well known, polynomial chaos does not offer an accurate representation of dynamic response around resonances when the responses are evaluated for several frequency values. A new parametrization of the frequency response function is then proposed: instead of considering the frequency as the main parameter, a “total phase” parameter is defined and used to define the dynamical system to be solved. It is shown via two applications that this approach offers very accurate results when conjugated to polynomial chaos with low degree.

Probability density evolution analysis of a shear-wall structure under stochastic ground motions by shaking table test

In this study, the accurate dynamic response analysis and dynamic reliability assessment upon a 12-storey shear-wall structure induced by stochastic seismic excitations are implemented through shaking table test combined with the probability density evolution method. To generate the stochastic seismic excitations, this study decomposes the ground motion process as a superposition of two contributions: the first one is a fully non-stationary counterpart modeled by a known evolutionary power spectrum using the dimension-reduction spectral representation; the second one is a corrective term adjusting the first counterpart in order to make it spectrum compatible. Through this treatment, the dimension-reduction simulation of the non-stationary stochastic ground motion processes satisfying the provision imposed by certain seismic codes is accomplished. Making the generated stochastic ground motion processes as the input stimulus of the shaking table test, the stochastic dynamic response analysis and the dynamic reliability assessment of the structural components as well as the whole shear-wall structure can then be carried out adopting the probability density evolution method. The typical studies sufficiently demonstrate the superiority of the proposed method and provide a powerful reference for the reliability-based seismic optimization design with regard to the cast-in-place high-rise shear-wall structures.

Stochastic dynamic response analysis and reliability assessment of non-linear structures under fully non-stationary ground motions

In this paper, stochastic dynamic response analysis and reliability assessment of nonlinear structures under fully non-stationary ground motions is investigated based on the probability density evolution method (PDEM). To handle this problem, a new method is proposed for points selection in high-dimensional random-variate space for PDEM, which is of paramount importance to the tradeoff of accuracy and efficiency. First, the original high-dimensional space is decomposed into several two-dimensional orthogonal subspaces. Then, each subspace is partitioned via the Voronoi cells and the representative points and assigned probabilities in each subspace are specified accordingly. Finally, the representative points and assigned probabilities in each subspace are randomly paired to formulate the representative points and assigned probabilities in high-dimension. A numerical example is studied to validate the proposed method, which indicates that the proposed method is of efficiency and accuracy for high-dimensional stochastic dynamic problem of structures. The problem, which needs to be further investigated, is also pointed out.

A modified reliability model for lithium-ion battery packs based on the stochastic capacity degradation and dynamic response impedance

The reliability assessment of battery packs is an important topic in the reliability design of electric vehicles. To improve the accuracy of the reliability analysis, a modified reliability model for lithium-ion battery packs is developed that couples multiphysics models, degradation models, and a multistate system reliability model. For the degradation models, a stochastic capacity degradation model and a dynamic response impedance (DRI) model are established based on experimental data: a) the stochastic capacity degradation characteristics of cells are investigated, and b) the dynamic response characteristics of the impedance with degradation and changing temperature are studied. It is found that Weibull distribution is slightly more appropriate than normal distribution when the temperature is 25 °C, then reversed with increasing temperature. The algorithm for distribution selection is proposed to describing this concept drift phenomenon. Then, the influence of DRI on the thermal performance is analyzed. Finally, reliability assessment of battery pack and several calculation examples are implemented for comparative analysis. The results show that the fading rate of cells in battery pack are accelerated when DRI is incorporated. Thus, multiphysics-based system reliability model with considering the stochastic capacity degradation and DRI enables more scientific and accurate reliability evaluation of battery pack.

An advanced pseudo excitation method and application in analyzing stochastic wind-induced response of slender bridge tower

Based on the theory of pseudo excitation method, a so-called stochastic pseudo excitation method is proposed to acquire the sampling results of stochastic dynamic response of uncertain bridge structure directly, which cannot be achieved by the classical pseudo excitation method. In order to reduce the computational cost of the dynamic finite element analysis of the samples to gain the probability density function of structural responses, the response surface method was adopted to handle the sampling results. The stochastic pseudo excitation method combing response surface method is verified by comparing with Monte Carlo method in good agreement. As an example, the stochastic response of a super-high slender bridge tower subjected to the Davenport fluctuating wind excitations is evaluated taking into account the uncertainties of material properties and wind speed. Numerical results show that the effect of uncertainties of structure or excitations plays a constant role in the stochastic response at the different levels of external excitations.