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Research Papers

A Linear Feedback Control Framework for Optimally Locating Passive Vibration Isolators With Known Stiffness and Damping Parameters

[+] Author and Article Information
Jihyun Lee, Amir H. Ghasemi

Department of Mechanical Engineering,
University of Michigan–Ann Arbor,
2350 Hayward,
Ann Arbor, MI 48109

Chinedum E. Okwudire

Department of Mechanical Engineering,
University of Michigan–Ann Arbor,
2350 Hayward,
Ann Arbor, MI 48109
e-mail: okwudire@umich.edu

Jeffrey Scruggs

Department of Civil and Environmental Engineering,
University of Michigan–Ann Arbor,
2350 Hayward,
Ann Arbor, MI 48109

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received January 26, 2016; final manuscript received September 4, 2016; published online October 27, 2016. Assoc. Editor: Patrick S. Keogh.

J. Vib. Acoust 139(1), 011006 (Oct 27, 2016) (11 pages) Paper No: VIB-16-1051; doi: 10.1115/1.4034771 History: Received January 26, 2016; Revised September 04, 2016

This paper investigates the problem of optimally locating passive vibration isolators to minimize unwanted vibration caused by exogenous disturbance forces. The stiffness and damping parameters of the isolators are assumed to be known, leaving the isolator locations, which are nonlinearly related to system states, as unknown optimization variables. An approach for reformulating the nonlinear isolator placement problem as a linear time-invariant (LTI) feedback control problem, by linking fictitious control forces to fictitious measured outputs using a nonzero feedforward term, is proposed. Accordingly, the isolator locations show up within a static output feedback gain matrix which can be optimized, using methods from optimal control theory, to minimize the H2 and/or H norms of transfer functions representing unwanted vibration. The proposed framework also allows well-established LTI control theories to be applied to the analyses of the optimal isolator placement problem and its results. The merits of the proposed approach are demonstrated using single and multivariable case studies related to isolator placement in precision manufacturing machines. However, the framework is applicable to optimal placement of passive isolators, suspensions, or dampers in automotive, aerospace, civil, and other applications.

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Figures

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Fig. 1

Model of generalized passive isolation system

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Fig. 2

Standard LTI feedback control configuration

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Fig. 3

Schematic of Mori Seiki's NN 1000 DCG five-axis ultraprecision machine tool used for simulations and experiments

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Fig. 4

Plot of ||Gz∞w∞|| versus az showing local optimum at az* = −116.8 mm

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Fig. 5

FRFs of NN 1000 obtained from (a) simulations and (b) experiments for az = 0 and az = −116 mm

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Fig. 6

Position error measured from encoder for az = 0 and az = −116 mm during motion of y-axis from rest to 1000 mm/min. Position error is due to residual vibration.

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Fig. 7

FRF magnitudes from the inputs to outputs for cases 1 and 2 of Example 2 compared with the nominal case

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Fig. 8

Pareto curves of cases 1 and 2 compared with the fixed γ2 and γ values of the nominal case

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Fig. 9

Pole locations of cases 1 and 2 for optimal solutions (of Example 2) and nominal case: (a) without stability constraints and (b) with α = 4 rad/s constraint imposed

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Fig. 10

Planar model of isolated machine

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