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Technical Brief

Sources and Propagation of Nonlinearity in a Vibration Isolator With Geometrically Nonlinear Damping

[+] Author and Article Information
J. C. Carranza

Departamento de Engenharia Mecânica,
Universidade Estadual Paulista (UNESP),
Ilha Solteira,
São Paulo 15385-000, Brazil
e-mail: carranzacamilo@gmail.com

M. J. Brennan

Professor
Departamento de Engenharia Mecânica,
Universidade Estadual Paulista (UNESP),
Ilha Solteira,
São Paulo 15385-000, Brazil
e-mail: mjbrennan0@btinternet.com

B. Tang

Associate Professor
Institute of Internal Combustion Engine,
Dalian University of Technology,
Dalian 116023, China
e-mail: btang@dlut.edu.cn

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 20, 2015; final manuscript received October 26, 2015; published online December 10, 2015. Editor: I. Y. (Steve) Shen.

J. Vib. Acoust 138(2), 024501 (Dec 10, 2015) (6 pages) Paper No: VIB-15-1023; doi: 10.1115/1.4031997 History: Received January 20, 2015; Revised October 26, 2015

In this paper, the behavior of a single degree-of-freedom (SDOF) passive vibration isolation system with geometrically nonlinear damping is investigated, and its displacement and force transmissibilities are compared with that of a linear system. The nonlinear system is composed of a linear spring and a linear viscous damper which are connected to a mass so that the damper is perpendicular to the spring. The system is excited by a harmonic force applied to the mass or a displacement of the base in the direction of the spring. The transmissibilities of the nonlinear isolation system are calculated using analytical expressions for small amplitudes of excitation and by using numerical simulations for high amplitude of excitation. When excited with a harmonic force, the forces transmitted through the spring and the damper are analyzed separately by decomposing the forces in terms of their harmonics. This enables the effects of these elements to be studied and to determine how they contribute individually to the nonlinear behavior of the system as a whole. For single frequency excitation, it is shown that the nonlinear damper causes distortion of the velocity of the suspended mass by generating higher harmonic components, and this combines with the time-varying nature of the damping in the system to severely distort the force transmitted though the damper. The distortion of the force transmitted through the spring is much smaller than that through the damper.

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References

Harris, C. M. , and Piersol, A. G. , 2002, Shock and Vibration Handbook, 5th ed., McGraw-Hill, New York, Chap. 30.
Snowdon, J. C. , 1979, “ Vibration Isolation: Use and Characterization,” J. Acoust. Soc. Am., 66(5), pp. 1245–1274. [CrossRef]
Rivin, E. I. , 2003, Passive Vibration Isolation, ASME Press, New York, Chap. 3.
Ibrahim, R. A. , 2008, “ Recent Advances in Nonlinear Passive Vibration Isolators,” J. Sound Vib., 314(3–5), pp. 371–452. [CrossRef]
Ruzicka, J. E. , and Derby, T. F. , 1971, Influence of Damping in Vibration Isolation, The Shock and Vibration Information Center, Washington DC.
Ravindra, B. , and Mallik, A. K. , 1994, “ Performance of Non-Linear Vibration Isolators Under Harmonic Excitation,” J. Sound Vib., 170(3), pp. 325–337. [CrossRef]
Lang, Z. Q. , Jing, X. J. , Billings, S. A. , Tomlinson, G. R. , and Peng, Z. K. , 2009, “ Theoretical Study of the Effects of Nonlinear Viscous Damping on Vibration Isolation of SDOF Systems,” J. Sound Vib., 323(1–2), pp. 352–365. [CrossRef]
Laalej, H. , Lang, Z. Q. , Daley, S. , Zazas, I. , Billings, S. A. , and Tomlinson, G. R. , 2012, “ Application of Non-Linear Damping to Vibration Isolation: An Experimental Study,” Nonlinear Dyn., 69(1–2), pp. 409–421. [CrossRef]
Andersen, D. , Starosvetsky, Y. , Vakakis, A. , and Bergman, L. , 2012, “ Dynamic Instabilities in Coupled Oscillators Induced by Geometrically Nonlinear Damping,” Nonlinear Dyn., 67(1), pp. 807–827. [CrossRef]
Tang, B. , and Brennan, M. J. , 2012, “ A Comparison of the Effects of Nonlinear Damping on the Free Vibration of a Single-Degree-of-Freedom System,” ASME J. Vib. Acoust., 134(2), p. 024501. [CrossRef]
Tang, B. , and Brennan, M. J. , 2013, “ A Comparison of Two Nonlinear Damping Mechanisms in a Vibration Isolator,” J. Sound Vib., 332(3), pp. 510–520. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

An isolation system with geometrically nonlinear damping. (a) Physical arrangement with a damper oriented perpendicular to the spring. (b) Equivalent system. The system is either force-excited with y(t)=0 or base-excited with fe(t)=0 and its movement is restricted to the vertical direction.

Grahic Jump Location
Fig. 2

Force and displacement transmissibility for (a) a SDOF system with linear damping and (b) low amplitude excitation of the nonlinear isolator when F̂e=0.4ζ and Ŷ=0.4ζ, with ζh=10 (dB value ref. unity)

Grahic Jump Location
Fig. 3

(a) Force and (b) displacement transmissibility for the nonlinear isolator with high amplitude excitation when ζh=10, and the linear isolator with ζ=0.1. The inside figure in (a) corresponds to the total transmitted force when the nonlinear isolator is excited by F̂e=0.5 at Ω = 0.5 (dB value ref. unity).

Grahic Jump Location
Fig. 4

Force transmissibility for each harmonic of the nonlinear system with ζh=10 in the nondimensional frequency range Ω=0.1−2.0, when the excitation forces are (a) F̂e=0.1, (b) F̂e=0.2, (c) F̂e=0.3, and (d) F̂e=0.5 (dB value ref. unity)

Grahic Jump Location
Fig. 5

Illustration of the propagation of nonlinearity through the spring and the damper when the system is excited by a harmonic force F̂e=0.5 at Ω=0.5. (a) Nondimensional displacement of the mass; (b) nondimensional stiffness as a function of time; (c) force transmitted through the spring; (d), (e), and (f) corresponding spectra; (g) nondimensional velocity of the mass; (h) nondimensional nonlinear damping coefficient as a function of time; (i) force transmitted through the damper; (j), (k), and (l) corresponding spectra. ×, multiplication; ⊗, convolution.

Grahic Jump Location
Fig. 6

Frequency spectrum of (a) and (b) the nondimensional force transmitted through the spring (a) and the damper (b), and (c) the force transmissibility when the nonlinear isolator is excited with harmonic forces F̂=0.1,0.2,0.3, and 0.5 at the excitation frequency Ω=0.5

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