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
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Fig. 1

Analytical model of the air spring

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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)

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Fig. 3

Vibration model of a body supported by an air spring

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Fig. 4

Experimental setup

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Fig. 5

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

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Fig. 6

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

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Fig. 7

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

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Fig. 8

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

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Fig. 9

A vibration model of the air spring

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Fig. 10

Pascal's principle




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