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Journal of Vibration and Acoustics | Volume 131 | Issue 5 | Technical Brief
Technical Briefs

On the Displacement Transmissibility of a Base Excited Viscously Damped Nonlinear Vibration Isolator

[+] Author and Article Information
Zarko Milovanovic

Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, Novi Sad 21000, Serbia

Ivana Kovacic

Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, Novi Sad 21000, Serbiaivanakov@uns.ac.rs

Michael J. Brennan

Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UKmjb@isvr.soton.ac.uk

J. Vib. Acoust 131(5), 054502 (Sep 11, 2009) (7 pages) doi:10.1115/1.3147140 History: Received October 17, 2008; Revised March 14, 2009; Published September 11, 2009
Article
References
Figures
Citing Articles

In this article vibration isolators having linear and cubic nonlinearities in stiffness and damping terms are considered under base excitation. The influence of the system parameters on the relative and absolute displacement transmissibility is investigated. The performance characteristics of the nonlinear isolators are evaluated and compared with the performance characteristics of the linear isolators to highlight the beneficial effects in the nonlinear systems considered.
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Copyright © 2009 by American Society of Mechanical Engineers
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References

1
Harris, C. M., and Piersol, A. G., 2002, "Shock and Vibration Handbook", McGraw-Hill, New York.
2
Rivin, E. I., 2003, "Passive Vibration Isolation", ASME, New York.
3
Crede, C. E., 1995, "Vibration and Shock Isolation", John Wiley, New York.
4
Den Hartog, J. P., 1962, "Mechanical Vibrations", McGraw-Hill, New York.
5
Snowdon, J. C., 1979, “Vibration Isolation Use and Characterization,” J. Acoust. Soc. Am., 66 , pp. 1245–1279.  [CrossRef]
6
Ruzicka, J. E., and Derby, T. E., 1971, "Influence of Damping in Vibration Isolation", T.S.a.V.I. Centre, Washington, DC.
7
Thomsen, J. J., 2003, "Vibration and Stability: Advanced Theory, Analysis, and Tools", Springer-Verlag, Berlin.
8
Ravindra, B., and Mallik, A. K., 1993, “Hard Duffing-Type Vibration Isolator With Combined Coulomb and Viscous Damping,” Int. J. Non-Linear Mech., 28 (4), pp. 427–440.  [CrossRef]
9
Ravindra, B., and Mallik, A. K., 1994, “Performance of Nonlinear Vibration Isolators Under Harmonic Excitation,” J. Sound Vib., 170 , pp. 325–337.  [CrossRef]
10
Jazar, G. N., Houim, R., Narimani, A., and Golnaraghi, M. F., 2006, “Frequency Response and Jump Avoidance in a Nonlinear Passive Engine Mount,” J. Vib. Control, 12 (11), pp. 1205–1237.  [CrossRef]
11
Yang, P., Yang, J., and Ding, J., 2006, “Dynamic Transmissibility of a Complex Nonlinear Coupling Isolator,” Tsinghua Science & Technology, 11 (5), pp. 538–542.  [CrossRef]
12
Popov, G., and Sankar, S., 1995, “Modelling and Analyses of Non-Linear Orifice Type Damping in Vibration Isolators,” J. Sound Vib., 183 (5), pp. 751–764.  [CrossRef]
13
Xiong, Y. P., Xinga, J. T., and Price, W. G., 2005, “Interactive Power Flow Characteristics of an Integrated Equipment–Nonlinear Isolator–Travelling Flexible Ship Excited by Sea Waves,” J. Sound Vib., 287 , pp. 245–276.  [CrossRef]
14
Nayfeh, A., and Mook, D. T., 1979, "Nonlinear Oscillations", Wiley, New York.
15
2008, http://en.wikipedia.org/wiki/Cubic_equation.htm.
16
Brennan, M. J., Kovacic, I., Carrella, A., and Waters, T. P., 2008, “On the Jump-Up and Jump-Down Frequencies of the Duffing Oscillator,” J. Sound Vib., 318 , pp. 1250–1261.  [CrossRef]
17
Malatkar, P., and Nayfeh, A. H., 2002, “Calculation of the Jump Frequencies in the Response of a s.d.o.f. Non-Linear Systems,” J. Sound Vib., 254 (5), pp. 1005–1011.  [CrossRef]

Figures

Grahic Jump Location
+A vibration isolation system with a spring and a viscous damper subjected to base excitation
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A vibration isolation system with a spring and a viscous damper subjected to base excitation
Figure 1

A vibration isolation system with a spring and a viscous damper subjected to base excitation

Grahic Jump Location
+Regions in which the relative transmissibility has an infinite value (I), a finite maximum value (II) and does not have a maximum value (III) as a function of the nondimensional stiffness ratio γ and the nondimensional linear damping ratio ζ1; points A, B, C, and L correspond to the system parameters, which yield the relative and absolute transmissibilities shown in Figs.  3, 3; point D corresponds to (γ,ζ1)=(0,2/2) when the transmissibility curve loses its extremum
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Regions in which the relative transmissibility has an infinite value (I), a finite maximum value (II) and does not have a maximum value (III) as a function of the nondimensional stiffness ratio γ and the nondimensional linear damping ratio ζ1; points A, B, C, and L correspond to the system parameters, which yield the relative and absolute transmissibilities shown in Figs.  3, 3; point D corresponds to (γ,ζ1)=(0,2/2) when the transmissibility curve loses its extremum
Figure 2

Regions in which the relative transmissibility has an infinite value (I), a finite maximum value (II) and does not have a maximum value (III) as a function of the nondimensional stiffness ratio γ and the nondimensional linear damping ratio ζ1; points A, B, C, and L correspond to the system parameters, which yield the relative and absolute transmissibilities shown in Figs.  3, 3; point D corresponds to (γ,ζ1)=(0,2/2) when the transmissibility curve loses its extremum

Grahic Jump Location
+(a) Relative transmissibility curves for the points A, B, C, and L from Fig. 2 (for ζ2=0: A: ζ1=0.2 and γ=0.3 (red solid line); B: ζ1=0.2 and γ=−0.3 (blue dashed line); C: ζ1=0.8 and γ=0.3 (black (-..-) line); L: ζ1=0.2 and γ=0 (green dashed-dotted line)). (b) Absolute transmissibility curves for the points A, B, C, and L; dotted lines depict unstable solutions. (c) Regions (shaded) in which there is a multivalued response for ζ1=0.2 and a horizontal asymptote γa=(16/3)ζ12.
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(a) Relative transmissibility curves for the points A, B, C, and L from Fig. 2 (for ζ2=0: A: ζ1=0.2 and γ=0.3 (red solid line); B: ζ1=0.2 and γ=−0.3 (blue dashed line); C: ζ1=0.8 and γ=0.3 (black (-..-) line); L: ζ1=0.2 and γ=0 (green dashed-dotted line)). (b) Absolute transmissibility curves for the points A, B, C, and L; dotted lines depict unstable solutions. (c) Regions (shaded) in which there is a multivalued response for ζ1=0.2 and a horizontal asymptote γa=(16/3)ζ12.
Figure 3

(a) Relative transmissibility curves for the points A, B, C, and L from Fig. 2 (for ζ2=0: A: ζ1=0.2 and γ=0.3 (red solid line); B: ζ1=0.2 and γ=−0.3 (blue dashed line); C: ζ1=0.8 and γ=0.3 (black (-..-) line); L: ζ1=0.2 and γ=0 (green dashed-dotted line)). (b) Absolute transmissibility curves for the points A, B, C, and L; dotted lines depict unstable solutions. (c) Regions (shaded) in which there is a multivalued response for ζ1=0.2 and a horizontal asymptote γa=(16/3)ζ12.

Grahic Jump Location
+Regions (shaded) in which there is a multivalued response and a horizontal asymptote γa=(16/3)ζ12 for different values of the linear damping ratio: (a) ζ1=0.05, (b) ζ1=0.15, (c) ζ1=0.3, (d) ζ1=0.4
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Regions (shaded) in which there is a multivalued response and a horizontal asymptote γa=(16/3)ζ12 for different values of the linear damping ratio: (a) ζ1=0.05, (b) ζ1=0.15, (c) ζ1=0.3, (d) ζ1=0.4
Figure 4

Regions (shaded) in which there is a multivalued response and a horizontal asymptote γa=(16/3)ζ12 for different values of the linear damping ratio: (a) ζ1=0.05, (b) ζ1=0.15, (c) ζ1=0.3, (d) ζ1=0.4

Grahic Jump Location
+Regions (shaded) in which there is a multivalued response: (a) in the ζ1-γ plane with γc1 and γc2 defined by Eq. 18; and (b) in the ζ1-Ω plane with Ωc1 and Ωc2 defined by Eq. 17
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Regions (shaded) in which there is a multivalued response: (a) in the ζ1-γ plane with γc1 and γc2 defined by Eq. 18; and (b) in the ζ1-Ω plane with Ωc1 and Ωc2 defined by Eq. 17
Figure 5

Regions (shaded) in which there is a multivalued response: (a) in the ζ1-γ plane with γc1 and γc2 defined by Eq. 18; and (b) in the ζ1-Ω plane with Ωc1 and Ωc2 defined by Eq. 17

Grahic Jump Location
+The displacement transmissibility for a system with pure cubic viscous damping for different values of the nonlinear damping ratio: ζ2=0.01 (red solid line), ζ2=0.1 (blue dashed line), ζ2=0.2 (green dashed-dotted line) and ζ2=0.5 (black dotted line); (a) relative transmissibility and (b) absolute transmissibility. (For interpretation of the references to the color in this text, the reader is referred to the web version of this article).
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The displacement transmissibility for a system with pure cubic viscous damping for different values of the nonlinear damping ratio: ζ2=0.01 (red solid line), ζ2=0.1 (blue dashed line), ζ2=0.2 (green dashed-dotted line) and ζ2=0.5 (black dotted line); (a) relative transmissibility and (b) absolute transmissibility. (For interpretation of the references to the color in this text, the reader is referred to the web version of this article).
Figure 6

The displacement transmissibility for a system with pure cubic viscous damping for different values of the nonlinear damping ratio: ζ2=0.01 (red solid line), ζ2=0.1 (blue dashed line), ζ2=0.2 (green dashed-dotted line) and ζ2=0.5 (black dotted line); (a) relative transmissibility and (b) absolute transmissibility. (For interpretation of the references to the color in this text, the reader is referred to the web version of this article).

Grahic Jump Location
+Difference between the absolute transmissibilities of a linear system and a system with cubic damping T̃a for the same values of the damping ratio: ζ1=ζ2=0.1 (blue dashed line), ζ1=ζ2=0.2 (green dashed-dotted line), ζ1=ζ2=0.79 (red solid line) and ζ1=ζ2=1.5 (black dotted line)
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Difference between the absolute transmissibilities of a linear system and a system with cubic damping T̃a for the same values of the damping ratio: ζ1=ζ2=0.1 (blue dashed line), ζ1=ζ2=0.2 (green dashed-dotted line), ζ1=ζ2=0.79 (red solid line) and ζ1=ζ2=1.5 (black dotted line)
Figure 7

Difference between the absolute transmissibilities of a linear system and a system with cubic damping T̃a for the same values of the damping ratio: ζ1=ζ2=0.1 (blue dashed line), ζ1=ζ2=0.2 (green dashed-dotted line), ζ1=ζ2=0.79 (red solid line) and ζ1=ζ2=1.5 (black dotted line)

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