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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Hrovat, D. , 1993, “ Applications of Optimal Control to Advanced Automotive Suspension Design,” ASME J. Dyn. Syst., Meas., Control, 115(2B), pp. 328–342. [CrossRef]
Yu, Y. , Naganathan, N. G. , and Dukkipati, R. V. , 2001, “ A Literature Review of Automotive Vehicle Engine Mounting Systems,” Mech. Mach. Theory, 36(1), pp. 123–142. [CrossRef]
Verros, G. , Natsiavas, S. , and Papadimitriou, C. , 2005, “ Design Optimization of Quarter-Car Models With Passive and Semi-Active Suspensions Under Random Road Excitation,” J. Vib. Control, 11(5), pp. 581–606. [CrossRef]
Swanson, D. A. , Wu, H. T. , and Ashrafiuon, H. , 1993, “ Optimization of Aircraft Engine Suspension Systems,” J. Aircr., 30(6), pp. 979–984. [CrossRef]
Ahn, Y. K. , Jin, D. S. , and Bo, S. Y. , 2003, “ Optimal Design of Engine Mount Using an Artificial Life Algorithm,” J. Sound Vib., 261(2), pp. 309–328. [CrossRef]
Preumont, A. , Horodinca, M. I. , Romanescu, I. , De Marneffe, B. , Avraam, M. , Deraemaeker, A. , Bossens, F. , and Hanieh, A. A. , 2007, “ A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform,” J. Sound Vib., 300(3), pp. 644–661. [CrossRef]
Kelly, J. M. , 1986, “ Aseismic Base Isolation: Review and Bibliography,” Soil Dyn. Earthquake Eng., 5(4), pp. 202–216. [CrossRef]
Buckle, I. G. , and Mayes, R. L. , 1990, “ Seismic Isolation: History, Application, and Performance—A World View,” Earthquake Spectra, 6(2), pp. 161–201. [CrossRef]
Lee-Glauser, G. J. , Ahmadi, G. , and Horta, L. G. , 1997, “ Integrated Passive/Active Vibration Absorber for Multistory Buildings,” J. Struct. Eng., 123(4), pp. 499–504. [CrossRef]
DeBra, D. B. , 1992, “ Vibration Isolation of Precision Machine Tools and Instruments,” CIRP Ann.—Manuf. Technol., 41(2), pp. 711–718. [CrossRef]
Rivin, E. I. , 1995, “ Vibration Isolation of Precision Equipment,” Precis. Eng., 17(1), pp. 41–56. [CrossRef]
Schellekens, P. , Rosielle, N. , Vermeulen, H. , Vermeulen, M. M. , Wetzels, S. F. , and Pril, W. , 1998, “ Design for Precision: Current Status and Trends,” CIRP Ann.—Manuf. Technol., 47(2), pp. 557–586. [CrossRef]
Karnopp, D. , 1995, “ Active and Semi-Active Vibration Isolation,” ASME J. Vib. Acoust., 117(B), pp. 177–185. [CrossRef]
Leo, D. J. , and Inman, D. J. , 1999, “ A Quadratic Programming Approach to the Design of Active–Passive Vibration Isolation Systems,” J. Sound Vib., 220(5), pp. 807–825. [CrossRef]
Li, T. H. , and Pin, K. Y. , 2000, “ Evolutionary Algorithms for Passive Suspension Systems,” JSME Int. J., Ser. C, 43(3), pp. 537–544. [CrossRef]
Singh, M. P. , and Moreschi, L. M. , 2002, “ Optimal Placement of Dampers for Passive Response Control,” Earthquake Eng. Struct. Dyn., 31(4), pp. 955–976. [CrossRef]
Ahn, Y. K. , Kim, Y. C. , Yang, B. S. , Ahmadian, M. , Ahn, K. K. , and Morishita, S. , 2005, “ Optimal Design of an Engine Mount Using an Enhanced Genetic Algorithm With Simplex Method,” Veh. Syst. Dyn., 43(1), pp. 57–81. [CrossRef]
Okwudire, C. E. , and Lee, J. , 2013, “ Minimization of the Residual Vibrations of Ultra-Precision Manufacturing Machines Via Optimal Placement of Vibration Isolators,” Precis. Eng., 37(2), pp. 425–435. [CrossRef]
Lee, J. , and Okwudire, C. E. , 2016, “ Reduction of Vibrations of Passively-Isolated Ultra-Precision Manufacturing Machines Using Mode Coupling,” Precis. Eng., 43(1), pp. 164–177. [CrossRef]
Lin, Y. , and Zhang, Y. , 1989, “ Suspension Optimization by a Frequency Domain Equivalent Optimal Control Algorithm,” J. Sound Vib., 133(2), pp. 239–249. [CrossRef]
Lin, Y. , Luo, W. , and Zhang, Y. M. , 1990, “ A New Method for the Optimization of a Vibration Isolation System,” ASME J. Vib. Acoust., 112(3), pp. 413–416. [CrossRef]
Gluck, N. , Reinhorn, A. M. , Gluck, J. , and Levy, R. , 1996, “ Design of Supplemental Dampers for Control of Structures,” J. Struct. Eng., 122(12), pp. 1394–1399. [CrossRef]
Subrahmanyan, P. K. , and Trumper, D. L. , 2000, “ Synthesis of Passive Vibration Isolation Mounts for Machine Tools a Control Systems Paradigm,” American Control Conference, (ACC), Chicago, IL, June 28–30, pp. 2886–2891.
Yang, J. N. , Lin, S. , Kim, J. H. , and Agrawal, A. K. , 2002, “ Optimal Design of Passive Energy Dissipation Systems Based on H and H2 Performances,” Earthquake Eng. Struct. Dyn., 31(4), pp. 921–936. [CrossRef]
Lopez Garcia, D. , and Soong, T. T. , 2002, “ Efficiency of a Simple Approach to Damper Allocation in MDOF Structures,” J. Struct. Control, 9(1), pp. 19–30. [CrossRef]
Zuo, L. , and Nayfeh, S. A. , 2003, “ Structured H2 Optimization of Vehicle Suspensions Based on Multi-Wheel Models,” Veh. Syst. Dyn., 40(5), pp. 351–371. [CrossRef]
Zuo, L. , and Nayfeh, S. A. , 2006, “ The Two-Degree-of-Freedom Tuned-Mass Damper for Suppression of Single-Mode Vibration Under Random and Harmonic Excitation,” ASME J. Vib. Acoust., 128(1), pp. 56–65. [CrossRef]
Bağdatlı, S. M. , Okwudire, C. E. , and Ulsoy, A. G. , 2014, “ Linear Quadratic Design of Passive Vibration Isolators,” ASME Paper No. DSCC2014-6142.
Morris, K. , and Yang, S. , 2015, “ Comparison of Actuator Placement Criteria for Control of Structures,” J. Sound Vib., 353(1), pp. 1–18. [CrossRef]
Ghasemi, A. H. , Lee, J. , and Okwudire, C. E. , 2015, “ A Control Theoretic Framework for Optimally Locating Passive Vibration Isolators to Minimize Residual Vibration,” ASME Paper No. DSCC2015-9871.
Warner, E. C. , and Scruggs, J. T. , 2015, “ Control of Vibratory Networks With Passive and Regenerative Systems,” American Control Conference, (ACC), Chicago, IL, July 1–3, pp. 5502–5508.
Scruggs, J. T. , 2007, “ Multi-Objective Optimization of Regenerative Damping Systems in Vibrating Structures,” American Control Conference, (ACC '07), New York, July 9–13, pp. 2672–2677.
Warner, E. C. , and Scruggs, J. T. , 2015, “ Regeneratively-Constrained LQG Control of Vibration Networks With Polytopic Model Uncertainty,” IEEE Conference on Control Applications (CCA), Sydney, NSW, Australia, Sept. 21–23, pp. 1498–1504.
Skogestad, S. , and Postlethwaite, I. , 2007, Multivariable Feedback Control: Analysis and Design, Vol. 2, Wiley, New York.
Asami, T. , Nishihara, O. , and Baz, A. M. , 2002, “ Analytical Solutions to H and H2 Optimization of Dynamic Vibration Absorbers Attached to Damped Linear Systems,” ASME J. Vib. Acoust., 124(2), pp. 284–295. [CrossRef]
Kim, C. J. , Oh, J. S. , and Park, C. H. , 2014, “ Modelling Vibration Transmission in the Mechanical and Control System of a Precision Machine,” CIRP Ann.—Manuf. Technol., 63(1), pp. 349–352. [CrossRef]
Braun, S. G. , Ewins, D. J. , and Rao, S. S. , 2002, Encyclopedia of Vibration, Academic Press, Cambridge, MA, pp. 1490–1491.
Dullerud, G. E. , and Paganini, F. , 2013, A Course in Robust Control Theory: A Convex Approach, Vol. 36, Springer Science and Business Media, New York.
Rivin, E. I. , 2006, “ Vibration Isolation of Precision Objects,” Sound Vib., 40(7), pp. 12–20.
Piersol, A. , and Paez, T. , 2010, Harris' Shock and Vibration Handbook, 6th ed., Vol. 39, McGraw-Hill, New York, pp. 13–39.
TMC, 2011, “ Precision Vibration Isolation Systems: Technical Background,” Technical Manufacturing Corp., Peabody, MA, accessed Jan. 21, 2016, www.techmfg.com/pdf/TMC%20Techical%20Background%202011.pdf


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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. 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. 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. 10

Planar model of isolated machine



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