A Stochastic Newmark Method for Engineering Dynamical Systems

Roy, Debasish; Dash, Mihir Kumar

doi:10.1006/jsvi.2001.3854

Cited by 20 publications

(9 citation statements)

References 15 publications

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“…The path-wise (strong) versions of stochastic Newmark methods [21,22] may also be derived based on implicit Ito-Taylor expansions of displacement and velocity vectors, as in the weak approach. However, the MSIs must be modelled so as to generate their path-wise realizations.…”

Section: Higher and Lower Order Path-wise Stochastic Newmark Methodsmentioning

confidence: 99%

“…It is known that the order of accuracy of the SH method in this case degenerates into that of the EM method. Thus, comparisons of the results obtained via HWSNM would presently be made with those via the lower order weak Newmark method (LWSNM) [33] and strong versions of the higher and lower order PSNM [21,22], referred to as HPSNM and LPSNM, respectively.…”

Section: Numerical Examplesmentioning

confidence: 99%

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A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

Roy

2006

Int. J. Numer. Meth. Engng

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SUMMARYEngineers are often more concerned with the computation of statistical moments (or mathematical expectations) of the response of stochastically driven dynamical systems than with the determination of path-wise response histories. With this in perspective, weak stochastic solutions of dynamical systems, modelled as n degrees-of-freedom (DOF) mechanical oscillators and driven by additive and/or multiplicative white noise (or, filtered white noise) processes, are considered in this study. While weak stochastic solutions are simpler and quicker to compute than strong (sample path-wise) solutions, it must be emphasized that sample realizations of weak solutions have no path-wise similarity with strong solutions. However, the statistical moment of any continuous and sufficiently differentiable deterministic function of the weak stochastic response is 'close' to that of the true response (if it exists) within a certain order of a given time step size. Computation of weak response therefore assumes great significance in the context of simulations of stochastically driven dynamical systems (oscillators) of engineering interest. To efficiently generate such weak responses, a novel class of weak stochastic Newmark methods (WSNMs), based on implicit Ito-Taylor expansions of displacement and velocity fields, is proposed. The resulting multiple stochastic integrals (MSIs) in these expansions are replaced by a set of random variables with considerably simpler and discrete probability densities. In fact, yet another simplifying feature of the present strategy is that there is no need to model and compute some of the higher-level MSIs. Estimates of error orders of these methods in terms of a given time step size are derived and a proof of global convergence provided. Numerical illustrations are provided and compared with exact solutions whenever available to demonstrate the accuracy, simplicity and higher computational speed of WSNMs vis-à-vis a few other popularly used stochastic integration schemes as well as the path-wise versions of the stochastic Newmark scheme.

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Section: Higher and Lower Order Path-wise Stochastic Newmark Methodsmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

Roy

2006

Int. J. Numer. Meth. Engng

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“…In Figs. 2-4, displacement and velocity histories of LTL-based solutions of the oscillator under weak, medium and strong intensities of additive white-noise inputs are shown and compared with a lower order stochastic Newmark method (LSNM) of comparable accuracy [18]. No deterministic and multiplicative stochastic inputs are assumed to be acting on the oscillator in these examples.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

“…A consistent time step size h =0.01 sec has been adopted in all the following numerical results. To check the accuracy and limitation of the proposed LTL technique, comparisons of results have been made with lower order stochastic Newmark method (LSNM, [18]) under different combination of periodic, additive and multiplicative loads. It may be noted that comparisons of LTL solutions with those obtained with other popular schemes, especially the stochastic Heun or Euler schemes, are not provided here as the transversal linearization schemes have already been shown to have a higher accuracy over a given time step.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

A novel stochastic locally transversal linearization (LTL) technique for engineering dynamical systems: Strong solutions

Roy

Dash

2005

Applied Mathematical Modelling

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Most available integration techniques for stochastically driven engineering dynamical systems are based on stochastic Taylor expansions of the response variables and thus require numerical modelling of multiple stochastic integrals (MSI-s). Since the latter is an extremely involved numerical task and becomes inaccurate for higher level MSI-s, these methods fail to achieve an accuracy beyond a limited order. Recently, the first author has proposed a locally transversal linearization (LTL) technique that completely avoids the use of Taylor-like expansions in the construction of the integration map [Proc. Roy. Soc. Ser. A, 457 (2001) 539; Int. J. Numer. Methods Eng. 61 (2004) 764]. A crucial step in the implementation of the LTL method is to arrive at a conditionally linearized solution which is then made to transversally intersect the non-linear solution manifold in the associated phase space. The present paper is the first part of an investigation consisting of two parts to considerably simplify and numerically expedite the generation of the conditionally linearized solution without affecting the local and global error orders of the original LTL method. In particular, the derivation of the conditionally linear form of the stochastic differential equations is done in such a way that the corresponding fundamental solution matrix (and consequently its inverse) remains unchanged during the entire integration process. In this part of the work, only strong (path wise) stochastic solutions are constructed. Through formal error estimates, it is verified that the present version of the LTL method has the same error orders as its older counterpart. Further, a host of numerical examples on stochastically driven non-linear oscillators are presented to illustrate its superior computational speed and ease 0307-904X/$ -see front matter Ó

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“…More specifically, determination of weak solutions often suffices because the first few moments of the exact and weak solutions match to within the same order of magnitude. As these low-order moments are those that are required in many engineering tasks, it is possible to employ a simplified model that is computationally efficient [11,12]. In our analysis, we compare distributions up to the fourth moment to verify our approach.…”

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confidence: 98%

Effect of Dynamical Accuracy for Uncertainty Propagation of Perturbed Keplerian Motion

Park

Fujimoto

Scheeres

2015

Journal of Guidance, Control, and Dynamics

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This paper poses the following question: How precisely must an Earth orbiter's dynamics be mapped to accurately capture the statistical evolution of its uncertainty? This question is addressed by using a simplified dynamical system that replaces the short-period variations with constants. The systems are defined and applied to the propagation of uncertainty for a non-Keplerian orbit. The simplified dynamical system is defined with two different types of approximate solutions: one is based on the Brouwer-Lyddane theory, and another is based on Lagrange planetary equations. The simplified dynamical system allows exploration of the relationship between dynamical model precision and uncertainty propagation accuracy, as well gained insight into effects due to individual variations: that is, secular, short, and long periods. Accuracy of propagation of uncertainty with the simplified dynamical system is verified with statistical methods for dynamical models including perturbations of J 2 and third-body gravity. In conclusion, the simplified dynamical system is capable of propagating uncertainty accurately and efficiently, as well as discovering that the secular variations are dominant in uncertainty mapping.

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A Stochastic Newmark Method for Engineering Dynamical Systems

Cited by 20 publications

References 15 publications

A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

A novel stochastic locally transversal linearization (LTL) technique for engineering dynamical systems: Strong solutions

Effect of Dynamical Accuracy for Uncertainty Propagation of Perturbed Keplerian Motion

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