Research Papers

An Approximate Formula to Calculate the Restoring and Damping Forces of an Air Spring With a Small Pipe

[+] Author and Article Information
Toshihiko Asami

e-mail: asami@eng.u-hyogo.ac.jp

Yasutaka Yokota

Graduate Student

Tomohiko Ise

Assistant Professor

Itsuro Honda

Department of Mechanical Engineering,
University of Hyogo
2167 Shosha, Himeji
Hyogo 671-2280, Japan

Hiroya Sakamoto

Tokkyokiki Corporation
10-133 Minamihatsushima-cho,
Amagasaki, Hyogo 660-0833, Japan

Contributed by the Design Engineering Division of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received April 25, 2012; final manuscript received January 17, 2013; published online July 1, 2013. Assoc. Editor: Philip Bayly.

J. Vib. Acoust 135(5), 051029 (Jul 01, 2013) (9 pages) Paper No: VIB-12-1117; doi: 10.1115/1.4023820 History: Received April 25, 2012; Revised January 17, 2013

This paper proposes a simple expression for calculating the restoring and damping forces of an air spring equipped with a small pipe. Air springs are commonly used in railway vehicles, automobiles, and various vibration isolators. The air spring discussed in this study consists of two tanks connected by a long pipe. Using a pipe instead of an orifice enables flexibility in the arrangement of the two tanks. In addition, this makes it possible to manufacture a thin air spring. A vertical translational oscillating system, which consists of a single mass supported by this type of air spring, looks like a single-degree-of-freedom (SDOF) system. However, it may have two resonance points. In this paper, we propose a vibratory model of a system supported by the air spring. With the proposed model it is possible to correctly reproduce the two resonance points of a system consisting of a single mass supported by this type of air spring. In our analysis, assuming that the vibration amplitude is small and the flow through the pipe is laminar, we derive the spring constant and damping coefficient of an air spring subjected to a simple harmonic motion. Then, we calculate the frequency response curves for the system and compare the calculated results with the experimental values. According to the experiment, there is a remarkable amplitude dependency in this type of air spring, so the frequency response curves for the system change with the magnitude of the input amplitude. It becomes clear that the calculation results are in agreement with the limit case when the input amplitude approaches zero. We use a commercially available air spring in this experiment. Our study is useful in the design of thin air spring vibration isolators for isolating small vibrations.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Tokita, Y., and Morimura, M., 1992, “Vibration Control Handbook,” Vol. 1, FUJI Techno Systems Co., Ltd., pp. 413–420 (in Japanese).
Kuneida, M., 1958, “Theory and Experiment on Vertical Vibration of Rolling,” Railway Tech. Res. Rep., 3(6), pp. 11–44, (in Japanese).
Naoteru, O. D. A., and Nishimura, S., 1969, “Vibration of Air Suspension Vehicles and Their Design,” Trans. Jap. Soc. Mech. Eng., 35(273), pp. 996–1002, (in Japanese). [CrossRef]
Koyanagi, S., 1983, “Optimum Design Methods of Air Spring Suspension Systems,” Trans. Jap. Soc. Mech. Eng., 49(439), pp. 410–415, (in Japanese). [CrossRef]
Koyanagi, S., 1986, “The Influences of Nonlinearities on Air Spring Vibration Isolation Characteristics,” Trans. Jap. Soc. Mech. Eng., 52(480), pp. 2084–2089, (in Japanese). [CrossRef]
Asami, T., Yokota, Y., Ise, T., Honda, I., and Sakamoto, H., 2013, “Theoretical and Experimental Analysis of the Non-Linear Characteristics of an Air Spring With an Orifice,” ASME J. Vib. Acoust., 135(1), p. 011012. [CrossRef]
Fujita, T., Okimura, H., Yamada, T., Inoue, N., Endoh, S., and Kagawa, T., 1997, “Affection of Connecting Conduit to Characteristics of Air Spring With Subtank,” Trans. Jap. Soc. Mech. Eng., 63(610), pp. 1920–1926, (in Japanese). [CrossRef]
Takahasshi, M., and Wakui, S., 2009, “Improvement of Isolated Table Using Auxiliary Tank,” J. Jap. Soc. Precision Eng., 75(4), pp. 542–547, (in Japanese). [CrossRef]
Berg, M., 2000, “A Three-Dimensional Airspring Model With Friction and Orifice Damping,” Vehicle Syst. Dyn., 33, pp. 528–539.
Docquier, N., Fisette, P., and Jeanmart, H., 2007, “Multiphysic Modelling of Railway Vehicles Equipped With Pneumatic Suspensions,” Vehicle Syst. Dyn., 45(6), pp. 505–524. [CrossRef]
Bruni, S., Vinolas, J., Berg, M., Polach, O., and Stichel, S., 2011, “Modelling of Suspension Components in a Rail Vehicle Dynamics Context,” Vehicle Syst. Dyn., 49(7), pp. 1021–1072. [CrossRef]
Facchinetti, A., Mazzola, L., Alfi, S., and Bruni, S., 2010, “Mathematical Modelling of the Secondary Airspring Suspension in Railway Vehicles and Its Effect on Safety and Ride Comfort,” Vehicle Syst. Dyn., 48, pp. 429–449. [CrossRef]
Kat, C. J., and Els, P. S., 2009, “Interconnected Air Spring Model,” Math. Comput. Model., Dyn. Syst., 15(4), pp. 353–370. [CrossRef]
Debra, D. B., 1984, “Design of Laminar Flow Restrictors for Damping Pneumatic Vibration Isolators,” CIRP Ann. Manuf. Tech., 33(1), pp. 351–356. [CrossRef]
Bachrach, B. I., and Rivin, E., 1983, “Analysis of a Damped Pneumatic Spring,” J. Sound Vib., 86(2), pp. 191–197. [CrossRef]
Quaglia, G., and Sorli, M., 2001, “Air Suspension Dimensionless Analysis and Design Procedure,” Vehicle Syst. Dyn., 35(6), pp. 443–475. [CrossRef]
Zhao, T. S., and Cheng, P., 1996, “Experimental Studies on the Onset of Turbulence and Frictional Losses in an Oscillatory Turbulent Pipe Flow,” Int. J. Heat Fluid Flow, 17(4), pp. 356–362. [CrossRef]
Jeung-Hoon, L., and Kwang-Joon, K., 2007, “Modeling of Nonlinear Complex Stiffness of Dual-Chamber Pneumatic Spring for Precision Vibration Isolations,” J. Sound Vib., 301(3–5), pp. 909–926. [CrossRef]
Eickhoff, B. M., Evans, J. R., and Minnis, A. J., 1995, “A Review of Modelling Methods for Railway Vehicle Suspension Components,” Vehicle Syst. Dyn., 24, pp. 469–496. [CrossRef]
Toyofuku, K., Yamada, C., Kagawa, T., and Fujita, T., 1999, “Study on Dynamic Characteristic Analysis of Air Spring With Auxiliary Chamber,” JSAE Rev., 20, pp. 349–355. [CrossRef]
Uchida, S., 1956, “The Pulsating Viscous Flow Superposed on the Steady Laminar Motion of Incompressible Fluid in a Circular Pipe,” Zeitschrift fur Angewandte Mathematik und Physik, 7(5), pp. 403–422. [CrossRef]
Shibata, K., Misaji, K., and Kato, H., 1993, “Vibration Characteristics of Rubber (Nonlinear Vibration Characteristics Depending on Frequency and Amplitude of Displacement),” Trans. Jap. Soc. Mech. Eng., 92(1833), pp. 2408–2414, (in Japanese). [CrossRef]


Grahic Jump Location
Fig. 1

Analytical model of the air spring

Grahic Jump Location
Fig. 2

Calculation results of the pressure in the tanks and the flow rate in the circular tube (ya = 0.1 mm, Vb = 546 mL)

Grahic Jump Location
Fig. 3

Vibration model of a body supported by an air spring

Grahic Jump Location
Fig. 4

Experimental setup

Grahic Jump Location
Fig. 5

Comparison between experimental and theoretical values for various pipe lengths (d = 4.0 mm, X0 = 0.1 mm, Vb = 546 mL)

Grahic Jump Location
Fig. 6

Comparison between experimental and theoretical values for various input amplitude (d = 4.0 mm, L = 1.0 m, Vb = 546 mL)

Grahic Jump Location
Fig. 7

Critical oscillation amplitude for laminar flow versus Reynolds number Reω (Zhao and Cheng [17])

Grahic Jump Location
Fig. 8

Vibration response of supported mass and fluid in the pipe (d = 4.0 mm, X0 = 0.1 mm, Vb = 546 mL)

Grahic Jump Location
Fig. 9

A vibration model of the air spring

Grahic Jump Location
Fig. 10

Pascal's principle



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In