Research Papers

Frequency-Domain Identification of Linear Time-Periodic Systems Using LTI Techniques

[+] Author and Article Information
Matthew S. Allen

Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI 53706msallen@engr.wisc.edu

J. Comput. Nonlinear Dynam 4(4), 041004 (Aug 24, 2009) (6 pages) doi:10.1115/1.3187151 History: Received December 28, 2007; Revised December 31, 2008; Published August 24, 2009

A variety of systems can be faithfully modeled as linear with coefficients that vary periodically with time or linear time-periodic (LTP). Examples include anisotropic rotor-bearing systems, wind turbines, and nonlinear systems linearized about a periodic trajectory. Many of these have been treated analytically in the literature, yet few methods exist for experimentally characterizing LTP systems. This paper presents a set of tools that can be used to identify a parametric model of a LTP system, using a frequency-domain approach and employing existing algorithms to perform parameter identification. One of the approaches is based on lifting the response to obtain an equivalent linear time-invariant (LTI) form and the other based is on Fourier series expansion. The development focuses on the preprocessing steps needed to apply LTI identification to the measurements, the postprocessing needed to reconstruct the LTP model from the identification results, and the interpretation of the measurements. This elucidates the similarities between LTP and LTI identification, allowing the experimentalist to transfer insight between the two. The approach determines the model order of the system and the postprocessing reveals the shapes of the time-periodic functions comprising the LTP model. Further postprocessing is also presented, which allows one to generate the state transition and time-varying state matrices of the system from the output of the LTI identification routine, so long as the measurement set is adequate. The experimental techniques are demonstrated on simulated measurements from a Jeffcott rotor mounted on an anisotropic flexible shaft supported by anisotropic bearings.

Copyright © 2009 by American Society of Mechanical Engineers
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Figure 1

(Left) illustration of a rotor on an anisotropic shaft and bearings; (right) lumped parameter schematic of the LTP system

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Figure 2

Noise contaminated time response of a LTP system with circles showing the first lifted response yL1

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Figure 3

Responses of mass in X and Y: (top) DFT of time response and (bottom) DFT of two of the lifted responses

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Figure 4

Responses of turntable Xtt and Ytt: (top) DFT of time response and (bottom) DFT of two of the lifted responses

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Figure 5

(Top) DFT of X-component of the residue vector for mode 1. Circles show all 50 DFT coefficients, points indicate the coefficients that were retained. (Bottom) Plot of residue for mode 1. Lines show the real (solid) and imaginary (dashed) parts of the residue found by AMI, while red circles show the reconstruction of the real and imaginary parts using the indicated coefficients.

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Figure 6

Sample coefficients of the time-varying system matrix A(t): (solid) actual coefficient as a function of shaft angle and (circles) coefficient estimated from noise contaminated response data




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