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Research Papers

A New Cluster-Based Harmonic Balance Aided Optimization Procedure With Application to Nonlinear Vibration Absorbers

[+] Author and Article Information
V. P. Premchand

Department of Mechanical Engineering,
National Institute of Technology,
Calicut 673601, India
e-mail: premchand_p110038 me@nitc.ac.in

M. D. Narayanan

Department of Mechanical Engineering,
National Institute of Technology,
Calicut 673601, India

A. S. Sajith

Department of Civil Engineering,
National Institute of Technology,
Calicut 673601, India

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 27, 2018; final manuscript received April 10, 2019; published online May 13, 2019. Assoc. Editor: Bogdan I. Epureanu.

J. Comput. Nonlinear Dynam 14(7), 071007 (May 13, 2019) (11 pages) Paper No: CND-18-1429; doi: 10.1115/1.4043527 History: Received September 27, 2018; Revised April 10, 2019

This work presents a new strategy in design optimization of nonlinear vibration absorbers with continuous and discontinuous motions. A cluster-based harmonic balance aided optimization technique using force balance or energy balance as the basis is generalized and adapted for nonlinear systems. It is found that optimal design parameters form a cluster in the parameter space and points from the parameter space inside the cluster satisfies design considerations. One of the main disadvantages of using existing optimization methods in nonlinear systems is that the parameter regimes, which provide periodic solutions, are not known beforehand, so one has to first do bifurcation studies to arrive at periodic regimes and optimization has to be conducted in the range. Proposed method combines these two steps as it converges to periodic clusters alone. Since the method admits only periodic solutions, occurrence of conditions such as chaos and quasi periodicity can be eliminated from the dynamics of the system. The proposed method can also be used to find the optimal parameters of both linear and nonlinear dynamical systems.

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References

Watts, P. , 1883, “ On the Method of Reducing the Rolling of Ships at Sea,” Trans. Inst. Naval Archit., 24, pp. 165–190. http://mararchief.tudelft.nl/catalogue/entries/16328/
Frahm, H. , 1911, “ A Device for Damping Vibrations of Bodies,” U.S. Patent No. 989,958.
Den Hartog, J. P. , and Ormondroyd, J. , 1928, “ The Theory of Dynamical Vibration Absorber,” ASME J. Appl. Mech., 50(7), pp. 9–22.
Den Hartog, J. P. , 1985, Mechanical Vibrations, Dover Books on Engineering, McGraw Hill, New York.
Hunt, J. , and Nissen, J. C. , 1982, “ The Broadband Dynamic Vibration Absorber,” J. Sound Vib., 83(4), pp. 573–578. [CrossRef]
Soom, A. , and Lee, M. , 1983, “ Optimal Design of Linear and Nonlinear Vibration Absorbers for Damped Systems,” ASME J. Vib. Acoust., Stress, Reliab. Des., 105(1), pp. 112–119. [CrossRef]
Burton, T. D. , 1994, Introduction to Dynamic System Analysis, McGraw-Hill, New York.
Narayanan, M. D. , Narayanan, S. , and Padmanabhan, C. , 2007, “ Nonlinear System Identification Using Multiple Trials,” Nonlinear Dyn., 48(4), pp. 341–360. [CrossRef]
Balaram, B. , Narayanan, M. D. , and Rajendra, P. K. , 2012, “ Kumar. Optimal Design of Multi-Parametric Nonlinear Systems Using a Parametric Continuation Based Genetic Algorithm Approach,” Nonlinear Dyn., 67(4), pp. 2759–2777. [CrossRef]
Thothadri, M. , Casas, R. A. , Moon, F. C. , D'Andrea, R. , and Johnson, Jr., C. R. , 2003, “ Nonlinear System Identification of Multi-Degree-of-Freedom Systems,” Nonlinear Dyn., 32(3), pp. 307–322. [CrossRef]
Den Hartog, J. P. , 1930, “ Forced Vibrations With Combined Viscous and Coulomb Damping,” Philadelphia Mag., 9(59), pp. 801–817.
Fang, J. , and Wang, Q. , 2012, “ Min-Max Criterion to the Optimal Design of Vibration Absorber in a System With Coulomb Friction and Viscous Damping,” J. Nonlinear Dyn., 70(1), pp. 393–400.
Ricciardelli, F. , and Vickery, B. J. , 1999, “ Tuned Vibration Absorbers With Dry Friction Damping,” Earthquake Eng. Struct. Dyn., 28(7), pp. 707–723. [CrossRef]
Marcus, A. L. , and Roitman, N. , 2005, “ Vibration Reduction Using Passive Absorption System With Coulomb Damping,” Mech. Syst. Signal Process., 19(3), pp. 537–549. [CrossRef]
Hundal, M. S. , 1979, “ Response of a Base Excited System With Viscous and Coulomb Damping,” J. Sound Vib., 64(3), pp. 371–378.
Leine, R. I. , van Campen, D. H. , de Kraker, A. , and van den Steen, L. , 1998, “ Stick-Slip Vibrations Induced by Alternate Friction Models,” Nonlinear Dyn., 16(1), pp. 41–54. [CrossRef]
Liang, J. , and Feeny, B. , 2011, “ Balancing Energy to Estimate Damping in a Forced Oscillator With Compliant Contact,” J. Sound Vib., 330(9), pp. 2049–2061. [CrossRef]
Liang, J. , and Feeny, B. , 2006, “ Balancing Energy to Estimate Damping Parameters in Forced Oscillators,” J. Sound Vib., 295(3–5), pp. 988–998. [CrossRef]
Rao, S. S. , 2012, Engineering Optimization, Theory and Practice, 3rd ed., Wiley Interscience, New York.
Inaudi, J. A. , and Kelly, J. M. , 1995, “ Mass Damper Using Friction Dissipation Devices,” Eng. Mech., 121(1), pp. 142–149. [CrossRef]
Gordon, C. K. Y. , 1966, “ Forced Vibrations of a Two Degree of Freedom System With Combined Couloumb and Viscous Damping,” J. Acoust. Soc., 39(1), pp. 14–24. [CrossRef]
Hoffman, J. , 1992, Numerical Methods for Scientists and Engineers, McGraw-Hill, New York.

Figures

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Fig. 1

Schematic representation of SDOF

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Fig. 2

Flow chart for CBA

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Fig. 3

Schematic representation of refinement procedure, Stage-0: randomly chosen parameter set in a range. Stage-1: solution of algebraic equations with prescribed bounds, Stage-2: solution of algebraic equations with refined bounds, Stage-3: solution of differential equations using bounds from stage 2.

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Fig. 4

Sample cluster formation in parameter space

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Fig. 5

Different models of DVA: (a) model 1: linear DVA, (b) model 2: DVA with coulomb damping, and (c) model 3: DVA with viscous and coulomb damping

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Fig. 6

Parameter space for model 1: R = 3

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Fig. 7

Parameter space for model 2: R = 5

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Fig. 8

Parameter space for model 3, R = 5: (a) (ka, ca, fa), (b) (ma, ca, fa), (c) (ma, ka, ca), and (d) (ma, ka, fa)

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Fig. 9

Correlation between |Ea| and |Ed|

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Fig. 10

Stages of refinement: (a) stage 0 (initial stage, The 500 parameter sets shown are randomly selected in the prescribed range), (b) stage 1 (after the first stage of algebraic solution by defining a suitable error norm about 250 parameter sets are selected), (c) stage 2 (after the second stage of algebraic solution by about 125 parameter sets are selected), and (d) stage 3 (in the final stage differential equations are solved and parameter sets satisfying the design conditions are plotted in the parameter space. Note that this final cluster (tested by ODE) is not a shrunk subspace of stage 2; however, they are distributed in the region, given in stage 2).

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