Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions

• Published in: International journal for numerical methods in engineering Vol. 95; no. 7; pp. 608 - 630
• Main Authors: Wu, Jinglai, Luo, Zhen, Zhang, Yunqing, Zhang, Nong, Chen, Liping
• Format: Journal Article
• Language: English
• Published: Chichester Blackwell Publishing Ltd 17.08.2013; Wiley; Wiley Subscription Services, Inc
• Subjects: Algebra; Chebyshev approximation; Chebyshev inclusion functions; DAEs; Dynamical systems; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Inclusions; interval method; Intervals; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematical models; Mathematics; Mechanical systems; multibody dynamics; Nonlinear dynamics; Ordinary differential equations; Physics; Sciences and techniques of general use; Solid dynamics (ballistics, collision, multibody system, stabilization...); Solid mechanics; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Uncertainty; multibody dynamics; Constraint; Differential equation; Supply; Solid dynamic; N body system; Probability distribution; Dynamical system; Modeling; Interval arithmetic; Uncertain system; DAEs; Linkage mechanism; Interval matrix; Pendulum; Monotonic function; Lower bound; Vibration; Probabilistic approach; Chebyshev polynomial; Chebyshev inclusion functions; Newton Raphson method; interval method; Upper bound; Mechanical system; Algebraic equation; Truncated shape; Non linear effect; Time integration
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.4525

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.