Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions

Abstract

SUMMARYThis study proposes a new uncertain analysis method for multibody dynamics of mechanical systems based on Chebyshev inclusion functions The interval model accounts for the uncertainties in multibody mechanical systems comprising uncertain‐but‐bounded parameters, which only requires lower and upper bounds of uncertain parameters, without having to know probability distributions. A Chebyshev inclusion function based on the truncated Chebyshev series, rather than the Taylor inclusion function, is proposed to achieve sharper and tighter bounds for meaningful solutions of interval functions, to effectively handle the overestimation caused by the wrapping effect, intrinsic to interval computations. The Mehler integral is used to evaluate the coefficients of Chebyshev polynomials in the numerical implementation. The multibody dynamics of mechanical systems are governed by index‐3 differential algebraic equations (DAEs), including a combination of differential equations and algebraic equations, responsible for the dynamics of the system subject to certain constraints. The proposed interval method with Chebyshev inclusion functions is applied to solve the DAEs in association with appropriate numerical solvers. This study employs HHT‐I3 as the numerical solver to transform the DAEs into a series of nonlinear algebraic equations at each integration time step, which are solved further by using the Newton–Raphson iterative method at the current time step. Two typical multibody dynamic systems with interval parameters, the slider crank and double pendulum mechanisms, are employed to demonstrate the effectiveness of the proposed methodology. The results show that the proposed methodology can supply sufficient numerical accuracy with a reasonable computational cost and is able to effectively handle the wrapping effect, as cosine functions are incorporated to sharpen the range of non‐monotonic interval functions. Copyright © 2013 John Wiley & Sons, Ltd.

A trigonometric interval method for dynamic response analysis of uncertain nonlinear systems

“…Recently, there have been some published papers which consider the uncertainty in the analysis or optimization of multi-body systems [31][32][33][34]. The proposed method in this paper can also be extended for use in uncertain multi-body dynamics systems governed using differential algebraic equations (DAEs).…”

A new interval uncertain optimization method for structures using Chebyshev surrogate models

“…Similar to the Taylor series, the Chebyshev series can also be used to expand the continuous function, with the Chebyshev polynomials to replace the power function in the Taylor expansion. Wu and et al [33,34] has shown that the Chebyshev polynomials have higher approximation accuracy than the Taylor polynomials under the same orders.…”

“…It can be found that under the same numerical cost which indicates the same number of terms of the Taylor expansion and Chebyshev expansion, the latter will lead to higher accuracy for most problems. A detailed comparison between the Taylor expansion and the Chebyshev expansion can be found in [33,34], which demonstrated the advantages of the Chebyshev method, including in the dynamics problems.…”

“…To this end, the Chebyshev series [32] are used to approximate these coefficients of the Taylor inclusion (polynomial) function, so as to develop a Chebyshev surrogate model. This model can be constructed by evaluating function values at specific interpolation points rather than the high-order derivatives, to improve computational efficiency [33,34]. After obtaining the Chebyshev surrogate model for the Taylor inclusion function, the interval arithmetic can be used to directly calculate the bounds of the inclusion function in the inner loop, without requiring the inner loop optimization.…”

Direct reliability-based design optimization of uncertain structures with interval parameters

“…The interval objective and constraint functions were transformed into deterministic ones by prescribing their acceptable possibility levels, and the resulting deterministic model was further transformed into an unconstrained singleobjective one by weighting and penalty function methods, which was then solved by deterministic algorithms. Wu et al (2013;2015a) proposed the high-order Taylor inclusion function to compress overestimation in interval arithmetic and utilized the Chebyshev surrogate model to approximate the highorder coefficients of the Taylor inclusion function. They further integrated the Chebyshev inclusion function and an interval bisection algorithm to avoid the inner layer optimization.…”