# Finite Element Analysis of the Stiffness Characteristics of Leaf Spring

Leaf springs are elastic components widely used in automobiles, and stiffness is an important physical parameter. Therefore, before the product is trial-produced, how to calculate its actual stiffness more accurately has become a common concern. Traditional calculation methods, such as “common curvature method” and “concentrated load method”, have certain limitations, and it is often necessary to add empirical correction coefficients to adjust the calculation results. With the development of computers, the finite element method has gradually been applied to the design of leaf springs due to its high accuracy, good convergence, and convenient use. Zou Hairong et al. applied the finite element method to analyze the abnormal fracture problem of a certain graded stiffness leaf spring, and proposed improvement measures to avoid such fracture. Hu Yumei et al. applied Ansys software to analyze the static strength characteristics of the leaf spring of a certain automobile rear suspension, and gave the stress distribution of the leaf spring under different loads. The calculation results are in good agreement with the test. Gu Antao discussed the general process of applying the finite element method to design leaf springs and gave a design example.

One of the biggest advantages of the finite element method is that it can simulate the actual working state of the design object, so it can partially replace the experiment and guide the precise design. Automotive leaf springs have nonlinear and hysteresis characteristics. When applying the finite element method for analysis, large deformation and contact need to be considered, that is, geometric nonlinearity and state nonlinearity need to be considered at the same time. This will make the calculation not easy to converge, so higher solving skills and analysis strategies are required.

This paper uses Nastran’s nonlinear analysis module to analyze the stiffness characteristics of a leaf spring, and discusses the effect of friction on its performance. The analysis process and results can provide references for the design of similar products.

2 Calculation method of leaf spring stiffness

The traditional calculation methods include “common curvature method” and “concentrated load method”. In addition, domestic scholar Guo Konghui proposed a calculation method called the master piece analysis method for the inherent defects in the common curvature method, and Tian Guangyu et al. proposed an improved concentrated load method for the inherent defects of the concentrated load method. The starting point of these methods is to regard the leaves of the leaf spring as a cantilever beam of equal cross-section. The friction between the leaves of the leaf spring and the large deformation characteristics during the deformation of the leaf spring are not considered. The classical beam formula is used to calculate the first leaf End deflection, and then obtain the stiffness of the leaf spring.

2.1 Common curvature method

The common curvature method was proposed by Parsilowski of the former Soviet Union. The basic assumption is that after the leaf spring is loaded, each blade has the same curvature in any section, that is, the entire leaf spring is regarded as a variable section beam. The formula for calculating the stiffness of the symmetrical leaf spring is as follows:

2.2 Concentrated load method

The basic assumption of the concentrated load method is that the leaves of the leaf spring only contact each other at the ends, that is, it is assumed that there is only one contact point at the end between the ith piece and the i-1th piece, and the contact force is Pi, and at the contact point The deflection of two adjacent blades is equal. Among them, P1 is the external load on the first piece. Therefore, there are n-1 unknown forces in the system P2, P3,…, Pn, n-1 equations can be obtained by equal deflection at the contact point, and the unknown forces P2, P3,…, Pn, and then calculate the end deflection of the first piece based on the load received by the first piece, and then the stiffness of the leaf spring can be obtained. The formula for calculating the stiffness of the leaf spring is as follows:

2.3 The main film analysis method

The common curvature method assumes that each blade has the same curvature on any cross section after the leaf spring is loaded. There is an obvious inconsistency in this assumption, that is, there is no concentrated bending moment at the free end of each leaf. It cannot be equal to the curvature of the same section of the previous one. For this reason, the main film analysis method has made the following assumptions.

a. Each leaf spring is divided into a constrained part and a non-constrained part. The definitions of the constrained part and the non-constrained part of the i-th leaf spring are shown in Figure 1.

b. The leaves of the leaf spring are free to deform downward in the unconstrained part, and the constrained part conforms to the assumption of common curvature, that is, the curvature of each section in this section is the same as that of the previous one.

Based on the above assumptions, the formula for calculating the stiffness of the leaf spring is as follows:

The meaning of each symbol in the formula is the same as above, where an+2=an+1=l1.

2.4 Improved concentrated load method

The concentrated load method assumes that each leaf of the leaf spring is in contact with each other only at the ends, but in fact the points in the leaf spring may also be in contact with each other. Based on this idea, the improved concentrated load method proposes the following assumptions.

a. There are not only interactions between the end points, but also several contact points. As shown in Figure 2, there are Ni contact points between the i-th slice and the i-1th slice. Record these points to The distance of the symmetry plane of the leaf spring is lij, j=1, 2,…, Ni.

b. The interaction between the i-th slice and the i-1th slice is concentrated only at the preset Ni contact points, denoted as Pi1, Pi2, PiNi, as shown in Figure 2.

Similar to the concentrated load method, there are total unknown forces in the system. From the equal deflection at the contact points, one equation can be obtained, and the magnitude of each unknown force can be obtained by solving this equation system. According to the force on the first piece, the end deflection of the first piece can be obtained, and then the stiffness of the leaf spring can be obtained.

Unlike the concentrated load method, the results calculated by this method cannot guarantee that each unknown force is greater than or equal to zero (that is, there can only be pressure between the contact points). For this reason, iterative algorithms are needed to solve this problem.

The common point of the above calculation methods is that the leaves of the leaf spring are approximately equivalent to a cantilever beam, and the contact between the leaves is simulated by different methods. In fact, the leaf spring has large deformation characteristics when it is working, and there is a certain deviation in the linear cantilever beam simulation, and the contact simulation method between the leaf springs is also rough. Using the finite element method to calculate the leaf spring stiffness can overcome the above shortcomings and make the calculation more accurate. Moreover, the working stiffness of variable section springs, less leaf springs and gradual stiffness leaf springs can be obtained very well, which has practical significance.