Technical Brief

The Transfer Matrix Metamodel for Dynamic Systems With Arbitrary Time-Variant Inputs

[+] Author and Article Information
Gordon J. Savage

Department of Systems Design Engineering,
University of Waterloo,
Waterloo, ON N2L 3G1, Canada
e-mail: gjsavage@uwaterloo.ca

Young Kap Son

Department of Mechanical and Automotive Engineering,
Andong National University,
1375 Gyeongdong-ro,
Andong-si, Gyeongsangbuk-do 36729, South Korea
e-mail: ykson@anu.ac.kr

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 18, 2017; final manuscript received July 22, 2017; published online August 30, 2017. Assoc. Editor: Gary Wang.

J. Mech. Des 139(10), 104502 (Aug 30, 2017) (6 pages) Paper No: MD-17-1143; doi: 10.1115/1.4037630 History: Received February 18, 2017; Revised July 22, 2017

This paper addresses the problem of mapping a vector of input variables (corresponding to discrete samples from a time-varying input) to a vector of output variables (discrete samples of the time-dependent response). This mapping is typically performed by a mechanistic model. However, when the mechanistic model is complex and dynamic, the computational effort to iteratively generate the response for design purposes can be burdensome. Metamodels (or, surrogate models) can be computationally efficient replacements, especially when the input variables have some amplitude and frequency bounds. Herein, a simple metamodel in the form of a transfer matrix is created from a matrix of a few training inputs and a corresponding matrix of matching responses provided by simulations of the dynamic mechanistic model. A least-squares paradigm reveals a simple way to link the input matrix to the columns of the response matrix. Application of singular value decomposition (SVD) introduces significant computational advantages since it provides matrices whose properties give, in an elegant fashion, the transfer matrix. The efficacy of the transfer matrix is shown through an investigation of a nonlinear, underdamped, double mass–spring–damper system. Arbitrary excitations and selected sinusoids are applied to check accuracy, speed and robustness of the methodology. The sources of errors are identified and ways to mitigate them are discussed. When compared to the ubiquitous Kriging approach, the transfer matrix method shows similar accuracy but much reduced computation time.

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Fig. 1

Schematic of a lumped-parameter mechanical system with two masses

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Fig. 2

Training data for case 1: (a) training excitations and (b) training responses

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Fig. 3

Comparison of responses from an arbitrary input: (a) responses and (b) differences

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Fig. 4

Training data for case 2: (a) training excitations and (b) training responses

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Fig. 5

Comparison of mechanistic and metamodels using fSA(t): (a) responses and (b) differences

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Fig. 6

Difference for different initial conditions and frequency ranges: (a) initial conditions and (b) frequency ranges



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