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Research Papers

Extended Divide-and-Conquer Algorithm for Uncertainty Analysis of Multibody Systems in Polynomial Chaos Expansion Framework

[+] Author and Article Information
Mohammad Poursina

Assistant Professor
Mem. ASME
Department of Aerospace
and Mechanical Engineering,
University of Arizona,
Tucson, AZ 85721
e-mail: poursina@email.arizona.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 24, 2014; final manuscript received September 3, 2015; published online November 16, 2015. Assoc. Editor: Dan Negrut.

J. Comput. Nonlinear Dynam 11(3), 031015 (Nov 16, 2015) (9 pages) Paper No: CND-14-1221; doi: 10.1115/1.4031573 History: Received September 24, 2014; Revised September 03, 2015

In this paper, an advanced algorithm is presented to efficiently form and solve the equations of motion of multibody problems involving uncertainty in the system parameters and/or excitations. Uncertainty is introduced to the system through the application of polynomial chaos expansion (PCE). In this scheme, states of the system, nondeterministic parameters, and constraint loads are projected onto the space of specific orthogonal base functions through modal values. Computational complexity of traditional methods of forming and solving the resulting governing equations drastically grows as a cubic function of the size of the nondeterministic system which is significantly larger than the original deterministic multibody problem. In this paper, the divide-and-conquer algorithm (DCA) will be extended to form and solve the nondeterministic governing equations of motion avoiding the construction of the mass and Jacobian matrices of the entire system. In this strategy, stochastic governing equations of motion of each individual body as well as those associated with kinematic constraints at connecting joints are developed in terms of base functions and modal terms. Then using the divide-and-conquer scheme, the entire system is swept in the assembly and disassembly passes to recursively form and solve nondeterministic equations of motion for modal values of spatial accelerations and constraint loads. In serial and parallel implementations, computational complexity of the method is expected to, respectively, increase as a linear and logarithmic function of the size.

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Figures

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Fig. 1

The schematics of consecutive bodies with associated handles and spatial constraint loads

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Fig. 2

Assembling two consecutive bodies to form a new body

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Fig. 3

Assembly and disassembly sweeps in a binary tree framework to form and solve the stochastic equations of motion

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Fig. 4

Planar double pendulum system with the uncertainty in the location of the mass center of the second pendulum

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Fig. 5

Comparing the mean value of states of the system from 2000 MC simulations with those from the first-, second-, and third-order PCE simulations

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Fig. 6

Comparing the standard deviation of states of the system from 2000 MC simulations with those from the first-, second- and third-order PCE simulations

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Fig. 7

Comparing the standard deviation of q2 from 2000 MC simulations with that of the PCE simulations for 7–10 s

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