Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 5169018, 12 pages

http://dx.doi.org/10.1155/2016/5169018

## Efficient Method for Calculating the Composite Stiffness of Parabolic Leaf Springs with Variable Stiffness for Vehicle Rear Suspension

State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130025, China

Received 7 October 2015; Revised 28 January 2016; Accepted 2 February 2016

Academic Editor: Reza Jazar

Copyright © 2016 Wen-ku Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The composite stiffness of parabolic leaf springs with variable stiffness is difficult to calculate using traditional integral equations. Numerical integration or FEA may be used but will require computer-aided software and long calculation times. An efficient method for calculating the composite stiffness of parabolic leaf springs with variable stiffness is developed and evaluated to reduce the complexity of calculation and shorten the calculation time. A simplified model for double-leaf springs with variable stiffness is built, and a composite stiffness calculation method for the model is derived using displacement superposition and material deformation continuity. The proposed method can be applied on triple-leaf and multileaf springs. The accuracy of the calculation method is verified by the rig test and FEA analysis. Finally, several parameters that should be considered during the design process of springs are discussed. The rig test and FEA analytical results indicate that the calculated results are acceptable. The proposed method can provide guidance for the design and production of parabolic leaf springs with variable stiffness. The composite stiffness of the leaf spring can be calculated quickly and accurately when the basic parameters of the leaf spring are known.

#### 1. Introduction

The development of lightweight technology and energy conservation has resulted in the extensive application of leaf springs with variable stiffness on cars. Leaf springs with variable stiffness are one of the focal points in automobile leaf springs because of their advantages over traditional leaf springs [1–5]. Parabolic leaf springs are springs that have leaves with constant widths but have varying cross-sectional thicknesses along the longitudinal direction following a parabolic law. This type of leaf spring has many advantages, such as lightness of weight; mono-leaf springs have been applied in car parts. However, the reliability of multileaf springs cannot be fully guaranteed because of manufacturing limitations; therefore, this type of spring has not been widely applied on cars. Double-leaf parabolic springs with variable stiffness are the simplest form of multileaf springs with mono-leaf springs being applied in car parts. However, the reliability of multileaf springs cannot be fully guaranteed because of manufacturing limitations; therefore, this type of spring has not been widely applied on cars. Double-leaf parabolic springs with variable stiffness are the simplest form of multileaf springs with variable stiffness and are also the final generation of this type of multileaf spring. At present, a second main spring is added under the first main spring to protect the latter for safety considerations, thus forming a triple-leaf spring with variable stiffness.

Stiffness is an important design parameter for leaf springs with variable stiffness. This parameter can be calculated using three methods, namely, formula method, FEA method, and rig test. The formula and FEA methods are preferred over the rig test because of the high manpower and time requirements of the latter. The formula method is commonly used in calculating leaf spring stiffness. In [1], the main and auxiliary springs are modeled as multileaf cantilever beams, and an efficient method for calculating the nonlinear stiffness of a progressive multileaf spring is developed and evaluated. In [2], the stiffness of the leaf spring is calculated by an in-house software based on mathematical calculations using the thickness profile of the leaves. In [3], the deformations of the main and auxiliary springs are considered simultaneously to calculate the stiffness of the leaf spring with variable stiffness by the method of common curvature. In [4], a force model of one type of leaf spring with variable stiffness is established, and a curvature-force hybrid method for calculating the properties of such a leaf spring is developed. In [5, 6], an equation for calculating the stiffness of a leaf spring with large deflection is derived. The FEA method is also frequently used to calculate the leaf spring stiffness. In [7–9], leaf spring stiffness is calculated using the FEA method, and the result is verified by a rig test. In [10], the finite element method was used to calculate the stiffness of the parabolic leaf spring, and the effects of materials, dimensions, load, and other factors on the stiffness are discussed. In [11], an explicit nonlinear finite element geometric analysis of parabolic leaf springs under various loads is performed. The vertical stiffness, wind-up stiffness, and roll stiffness of the spring are also calculated. The FEA method is more commonly used than the formula method in the calculation of leaf spring stiffness. However, the former is more complex than the latter and requires specialized finite element software as a secondary step. Engineers who use this method should also have some work experience. By contrast, designers can easily master the formula method in calculating the stiffness of the leaf spring because it is simple and does not require special software and computer aid. Thus, deriving a stiffness calculation formula for leaf springs is significant. However, the type and size of leaf springs vary. Thus, no uniform formula can be applied to all types of leaf springs. Various equations can only fit a particular type of leaf spring. Current studies on calculating the stiffness of parabolic leaf springs with variable stiffness mostly considered the FEA method. Hence, a simple composite stiffness calculation formula should be derived for this method.

The main spring bears the load alone when the load is small in parabolic leaf springs with variable stiffness in the form of main and auxiliary springs. Thereafter, the main and auxiliary springs come into contact because of the increased load. Finally, the main and auxiliary springs bear the load together when the main and auxiliary springs are under full contact. Therefore, the stiffness of the spring changes in three stages. During the first stage, the stiffness of the spring is equal to the stiffness of the main spring before the main spring and secondary springs come into contact. During the second stage, stiffness increases when the main and auxiliary springs come into contact. During the third stage, the stiffness of the spring has a composite stiffness generated by both the main and auxiliary springs when the springs are under full contact. Changes in the stiffness of the leaf spring during stage two are nonlinear, and the stiffness is difficult to calculate. Three calculation equations are currently used [12–14] to calculate the stiffness of mono-leaf parabolic springs. However, only integral equations are available for calculating the composite stiffness of parabolic leaf spring with variable stiffness [12, 13]. However, the direct calculation of the composite stiffness of a taper-leaf spring using integral equations is quite challenging, and these equations cannot be applied during the design of a leaf spring. In this paper, a simple and practical equation for calculating the composite stiffness of parabolic leaf springs with variable stiffness is derived. The equation can be used not only for calculating the composite stiffness but also for designing such a type of leaf spring.

In this paper, firstly, the leaf spring assembly model was simplified. A simple model of double-leaf parabolic spring with variable stiffness (referred to as double-leaf spring model) was considered. The difficulties in the calculation of the composite stiffness of double-leaf spring were analyzed, and the equation for calculating the composite stiffness was derived using the method of material mechanics. Thereafter, the superposition principle was used to derive the equation for triple-leaf springs. This equation can be extended to calculate the stiffness of multileaf springs. The correctness of the equation was verified by a rig test and finite element simulation. Finally, the parameters that should be considered when using the equations for leaf spring design were discussed.

#### 2. Model of Two-Level Parabolic Leaf Spring with Variable Stiffness and Its Simplification

##### 2.1. Leaf Spring Assembly Model

The leaf spring studied in this paper is a parabolic leaf spring with variable stiffness and unequal arm length. A triple-leaf spring model is analyzed (Figure 1). The triple-leaf spring consists of two main springs and an auxiliary spring. The stiffness of the spring is designed as a two-level variable stiffness after the requirements for a comfortable ride under different loads were considered. The first-level stiffness is the stiffness of the main spring, and the second-level stiffness is a composite stiffness determined by the stiffness of both the main and the auxiliary springs. The rear shackle is designed as an underneath shackle that makes the length of the auxiliary spring significantly less than that of the main spring because of the limited installation space for the spring, security (leaf spring reverse bend), and vehicle performance requirements. Thus, the composite stiffness of the multileaf spring is not a simple stiffness superposition of multiple parabolic spring leaves with equal lengths.