Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions

Jinglai Wu; Zhen Luo; Nong Zhang; Liping Chen

doi:10.1002/nme.4525

journal article Jun 26, 2013

Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions

Jinglai Wu Zhen Luo

Nong Zhang Liping Chen

International Journal for Numerical Methods in Engineering Vol. 95 No. 7 pp. 608-630 · Wiley

View at Publisher Save 10.1002/nme.4525

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.

Topics

No keywords indexed for this article. Browse by subject →

References

48

[1]

10.1115/1.3079784

[2]

Isukapalli SS (1999)

[3]

10.1016/j.cma.2008.05.004

[4]

10.1016/j.compstruc.2005.10.001

[5]

10.1016/b978-0-444-88406-0.50006-6

[6]

10.1002/nme.636

[7]

10.1007/s00158-009-0461-6

[8]

10.1115/1.2888303

[9]

The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations

Dongbin Xiu, George Em Karniadakis

SIAM Journal on Scientific Computing 10.1137/s1064827501387826

[10]

Fuzzy sets

L.A. Zadeh

Information and Control 10.1016/s0019-9958(65)90241-x

[11]

BedeB StefaniniL.Numerical solution of interval differential equations with generalized Hukuhara differentiability.Proceedings of the IFSA/EUSFLAT Conference Lisbon Portugal July 20–24 2009.

[12]

10.1016/j.ins.2006.08.021

[13]

10.1016/j.fss.2008.10.002

[14]

10.1016/j.nahs.2009.06.013

[15]

10.1016/s0165-0114(02)00045-3

[16]

10.1016/j.cma.2004.03.019

[17]

10.1016/j.cma.2007.03.024

[18]

10.1016/j.cma.2009.06.001

[19]

10.1023/a:1004687620019

[20]

10.1016/s0377-0427(00)00342-3

[21]

10.1016/j.jlap.2004.07.008

[22]

Li F "An uncertain multidisciplinary design optimization method using interval convex models" Engineering Optimization (2012)

[23]

10.1016/j.ymssp.2006.05.007

[24]

10.1002/nme.281

[25]

Jackson KR "Some recent advances in validated methods for IVPs for ODEs" Applied Mathematics and Computation (2002)

[26]

10.1023/a:1026437523641

[27]

10.1016/j.apnum.2006.10.006

[28]

10.1002/nme.1891

[29]

10.1002/nme.2248

[30]

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

Jinglai Wu, Liping Chen, Zhen Luo

Applied Mathematical Modelling 10.1016/j.apm.2012.09.073

[31]

10.1007/s11044-006-9007-5

[32]

10.1007/s11044-006-9008-4

[33]

Abramowitz M (1972)

[34]

10.1115/1.2803257

[35]

10.1115/1.2803258

[36]

A Method of Computation for Structural Dynamics

Nathan M. Newmark

Journal of the Engineering Mechanics Division 10.1061/jmcea3.0000098

[37]

10.1115/1.2389231

[38]

A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method

J. Chung, G. M. Hulbert

Journal of Applied Mechanics 10.1115/1.2900803

[39]

10.1007/978-1-4471-0249-6

[40]

Neumaier A "The wrapping effect, ellipsoid arithmetic, stability and confidence regions" Computing (1993)

[41]

Makino K "Suppression of the wrapping effect by Taylor model‐based verified integrators: long‐term stabillization by shrink wrapping" International Journal of Differential Equations and Applications (2005)

[42]

10.1016/s0096-3003(98)10083-8

[43]

10.2140/pjm.1961.11.777

[44]

Li Q (2003)

[45]

Rivlin TJ (1981)

[46]

Fox L (1968)

[47]

10.4028/www.scientific.net/amm.152-154.1555

[48]

10.1016/s0370-2693(96)01349-4

Cited By

206

Interval modeling and constraint model updating for friction dynamics of shrouded turbine blades

Jinyan Li, Zhiyu Shi · 2026

Aerospace Science and Technology

Hybrid uncertainty propagation for mechanical dynamics problems via polynomial chaos expansion and Legendre interval inclusion function

Liqun Wang, Chengyuan Guo · 2025

Mechanical Systems and Signal Proce...

Efficient uncertainty computation method for solving mechanical dynamic systems with a large-scale of interval parameters

Jinglai Wu, Yupeng Duan · 2024

Acta Mechanica Sinica

A bivariate subinterval method for dynamic analysis of mechanical systems with interval uncertain parameters

Xin Jiang, Zhengfeng Bai · 2023

Communications in Nonlinear Science...

A novel interval model updating framework based on correlation propagation and matrix-similarity method

Baopeng Liao, Rui Zhao · 2022

Mechanical Systems and Signal Proce...

A bivariate Chebyshev polynomials method for nonlinear dynamic systems with interval uncertainties

Tonghui Wei, Feng Li · 2021

Nonlinear Dynamics

An interval uncertainty propagation method using polynomial chaos expansion and its application in complicated multibody dynamic systems

Liqun Wang, Guolai Yang · 2021

Nonlinear Dynamics

Non-probabilistic interval process method for analyzing two-stage straight bevel gear system with uncertain time-varying parameters

Wassim Lafi, Fathi Djemal · 2020

Proceedings of the Institution of M...

Dynamic computation for rigid–flexible multibody systems with hybrid uncertainty of randomness and interval

Jinglai Wu, Liang Luo · 2019

Multibody System Dynamics

Non-probabilistic interval process analysis of time-varying uncertain structures

Baizhan Xia, Lifang Wang · 2018

Engineering Structures

Dynamic response analysis of structure under time-variant interval process model

Baizhan Xia, Yuan Qin · 2016

Journal of Sound and Vibration

Dynamic response analysis on torsional vibrations of wind turbine geared transmission system with uncertainty

Sha Wei, Jingshan Zhao · 2015

Renewable Energy

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

Jinglai Wu, Zhen Luo · 2015

Computers & Structures

A new sampling scheme for developing metamodels with the zeros of Chebyshev polynomials

Jinglai Wu, Zhen Luo · 2014

Engineering Optimization

Metrics

206

Citations

48

References

Details

Published: Jun 26, 2013
Vol/Issue: 95(7)
Pages: 608-630
License: View

Authors

J

Jinglai Wu

National Engineering Research Center for CAD Huazhong University of Science and Technology Wuhan Hubei 430074 China; School of Electrical, Mechanical and Mechatronic Systems The University of Technology Sydney NSW 2007 Australia

Z

Zhen Luo

School of Electrical, Mechanical and Mechatronic Systems The University of Technology Sydney NSW 2007 Australia

N

Nong Zhang

School of Electrical, Mechanical and Mechatronic Systems The University of Technology Sydney NSW 2007 Australia

L

Liping Chen

National Engineering Research Center for CAD Huazhong University of Science and Technology Wuhan Hubei 430074 China

Cite This Article

Jinglai Wu, Zhen Luo, Nong Zhang, et al. (2013). Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions. International Journal for Numerical Methods in Engineering, 95(7), 608-630. https://doi.org/10.1002/nme.4525

Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions

You May Also Like