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Dynamics of spatial rigid–flexible multibody systems with uncertain interval parameters

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Abstract

A non-intrusive computation methodology is proposed to study the dynamics of rigid–flexible multibody systems with a large number of uncertain interval parameters. The rigid–flexible multibody system is meshed by using a unified mesh of the absolute nodal coordinate formulation (ANCF). That is, the flexible parts are meshed by using the finite elements of the ANCF, while the rigid parts are described via the ANCF reference nodes (ANCF-RNs). Firstly, the interval differential-algebraic equations are directly transformed into the nonlinear interval algebraic equations by using the generalized-alpha algorithm. Then, the Chebyshev sampling methods, including Chebyshev tensor product sampling method and Chebyshev collocation method, are used to transform the nonlinear interval algebraic equations into sets of nonlinear algebraic equations with deterministic sampling parameters. The proposed computation methodology is non-intrusive because the original generalized-alpha algorithm is not amended. OpenMP directives are also used to parallelize the solving process of these deterministic nonlinear algebraic equations. To circumvent the interval explosion problem and maintain computation efficiency, the scanning method is used to determine the upper and lower bounds of the deducted Chebyshev surrogate models. Finally, two numerical examples are studied to validate the proposed methodology. The first example is used to check the effectiveness of the proposed methodology. And the second one of a complex rigid–flexible robot with uncertain interval parameters shows the effectiveness of the proposed computation methodology in the dynamics analysis of complicated spatial rigid–flexible multibody systems with a large number of uncertain interval parameters.

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Abbreviations

\(\mathbf{q}\) :

Nodal coordinates

\(\mathbf{M}\) :

System mass matrix

\(\mathbf{S}^{kr}\) :

Transformation matrix of ANCF-RN

\(\varvec{\upomega }\) :

ANCF-RN angular velocity

\(\varvec{\upalpha }\) :

ANCF-RN angular acceleration

\(\mathbf{T}\) :

External torque

\(\varvec{\Phi }\) :

System constraint equations

\(\mathbf{Q}\) :

External force vector

\(\mathbf{V}\) :

Solution vector of deterministic dynamics systems

\(\varvec{\upzeta }\) :

Deterministic system parameter vector

\({[}\varvec{\varsigma }{]}\) :

Interval parameter vector

\({[}\hat{{\mathbf{V}}}{]}\) :

Interval solution vector of uncertain dynamics system

\(\phi _j \) :

Chebyshev interpolation points

\(\mathbf{U}\) :

Chebyshev polynomial vector

\(C_p (x)\) :

Chebyshev polynomial of degree p

\(\hat{{C}}_{a_1 a_2 \ldots a_k } \left( \mathbf{x} \right) \) :

k-dimensional Chebyshev polynomial

\(\varvec{\Psi }\) :

CTP sample vectors

\(\hat{\mathbf{K}}\) :

CTP surrogate coefficient matrix

\(\hat{\mathbf{U}}\left( \mathbf{x} \right) \) :

CTP polynomial vector

\(\hat{\varvec{\Psi }}\) :

CCM sample vectors

\(\hat{\hat{\mathbf{K}}}\) :

CCM surrogate coefficient matrix

\(\hat{\hat{\mathbf{U}}}\left( \mathbf{x} \right) \) :

CCM polynomial vector

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Acknowledgments

This research was supported in part by National Natural Science Foundations of China under Grants 11290151 and 11472042. The research was also supported in part by the Beijing Higher Education Young Elite Teacher Project under Grant YETP1201.

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  1. MOE Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Zhe Wang, Qiang Tian & Haiyan Hu

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  1. Zhe Wang
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  2. Qiang Tian
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Correspondence to Qiang Tian.

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Wang, Z., Tian, Q. & Hu, H. Dynamics of spatial rigid–flexible multibody systems with uncertain interval parameters. Nonlinear Dyn 84, 527–548 (2016). https://doi.org/10.1007/s11071-015-2504-4

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