Theoretical modelling and experimental analysis of the vertical stiffness of a convoluted air spring including the effect of the stiffness of the bellows

Abstract

A new refined model including a model of the bellows to describe the vertical stiffness of a convoluted air spring is developed. The new model of a convoluted air spring incorporates the structural parameters including the effective area, the rate of change in the effective area, the effective volume, the rate of change in the effective volume and the stiffness of the bellows. Analytical models of the structural parameters of a convoluted air spring are established using a geometrical analysis approach. Two convoluted air springs (type 1B5002 and type 2B5281) are designed and manufactured for testing purposes. An experimental set-up is designed in order to carry out identification of the structural parameters and to test the vertical static stiffness and the vertical dynamic stiffness. The analytical models of the structural parameters are validated by experiments, which provide good design guidance and improvement in the convoluted air spring at the design stage. Tests on the vertical static stiffness and the dynamic stiffness indicate that the stiffness of the bellows should be considered; furthermore, the stiffness of the bellows exhibits amplitude-dependent chacteristics and frequency-dependent characteristics. In a different approach from the previous studies, a model of the bellows is developed in this paper and is composed of a smooth friction model and a fractional derivative Zener model in parallel to represent the viscoelastic properties of the bellows. It is concluded that the simulation results of the proposed new model are in good agreement with the experimental data. The new refined model may be an effective tool for predicting the mechanical behaviours of a convoluted air spring before a prototype is constructed. The proposed refined model can also be practically utilized as a guide for designing the parameters and for matching the air suspension of the vehicle.

Mathematical Modeling and Characteristic Analysis of the Vertical Stiffness for Railway Vehicle Air Spring System

“…In order to obtain an analytical solution of the air spring by using the approximate analysis approach, the following simplifications and assumptions are required [5][6][7]:…”

“…Shi et al [4] built a finite element model of an air spring to study the static elastic characteristics and utilized sensitivity analysis method to study the impact of different parameters on the elastic characteristic. Li et al [5,6] and Chen et al [7] deduced an analytical formula to study air spring vertical stiffness, and the influence of geometric parameters on vertical stiffness was analyzed. Liu and Lee [8] derived a theoretical model based on the energy conservation equation and gas state equation, to investigate the dynamic stiffness and overall equivalent damping of air spring.…”

An Air Spring Vibration Isolator Based on a Negative-Stiffness Structure for Vehicle Seat

“…It can be shown that this spring is compressed by approximately x from its uncompressed state. By applying the first law of thermodynamics, the relationship between the force F b and the pressure on the air bellows is as follows [34] F b = (P b − P a ).A b (11) where P b is the absolute pressure in the bellows-type air spring, and A b is the effective area of the spring. The stiffness characteristic of the bellows-type air spring can be defined as the derivative of this force with respect to the height x according to Equation ( 12)…”

Theoretical modeling of the vertical stiffness of a rolling lobe air spring

“…The simplified diagram of the vertical deformation of a rolling lobe air spring is shown in Figure 3. 12,13 The symbols in Figure 3 are defined as follows: dR, S, ds, dx#, and dx$ are intermediate variables, f is the angle of the constraint of the top plate, which is defined as the angle between the symmetry axis and the constraint, and u and u + du are the angles of the diaphragm before and after deflection, respectively. Furthermore, in order to obtain an analytical solution of the geometric parameters, the diaphragm is assumed to be a circular arc, and the length of the arc is not changed during the process of vertical deformation.…”