Design information for engineers, such as spring calculation formulas,
which are the basis of spring design, can be found here.
Selecting the shape to get the required spring load or deflection with a limited volume and estimating the position and magnitude of the maximum stress generated in the spring are important when designing a leaf spring. The formulas shown in general materials can be used for relatively simple spring shapes, but since there are various shapes and applications of leaf springs in reality, we will introduce calculation formulas for flat springs by shapes and applications here.
The simplest flat spring is a cantilever spring with a rectangular cross-section.
When the fixed end is A and the free end is B, and the load P is applied to point B, the calculation formula is as follows.
Here, I is the second moment of area.
It is expressed by
, and when is large, it is
Thus, the calculation formula for when is large is
ν represents the Poisson’s ratio, and in the case of steel, ν ≈ 0.3. The maximum stress is at the fixed end and is represented by the formula below.
Table 2 shows the longitudinal elasticity modulus (E) value of the main flat spring materials.
Table 1. Symbols and units used in calculations
|Symbol||Meaning of Symbol||Unit|
|l||Distance From The Fulcrum To The Load Point||mm|
|E||Longitudinal Elastic Modulus||N/mm2|
|I||Second Moment of Area||mm4|
|P||Load (force) Applied To The Spring||N|
|δ||Deflection At Load Point||mm|
Table 2.Longitudinal elastic modulus：E（N/m㎡）
|Spring Steel Material||206×103|
|Phosphor Bronze Wire||98×103|
|Beryllium Copper Wire||127×103|
As shown in Figure 2, when the plate thickness of the flat spring is fixed and the plate width changes linearly, the deflection of the free endis as per the following formula.
To calculate B in the formula, use the following two formulas according to the plate thickness.
If the plate is thick, use the formula below.
If the plate is thin, use the following formula:
Also, the value in the formula can be calculated from Figure 3 by β = b1 / b.
As shown in Figure 4, the deflection at the free end of a flat spring with a fixed plate thickness and a joggled plate width is as per the following formula.
Here,、are the deflection and deflection angle of the joggled part A due to P, and is the deflection at the free end of the cantilever with length l2 and plate width .
When the vertical load (P) acts on the free end with the shape as shown in Figure 5 in which the center of the plate thickness is a straight line and the center line of the plate width is an arc, the deflection (δφ) at any position φ is as per the following formula.
Here, C represents the torsional strength of the plate.
Generally, Castigliano’s theorem is used to calculate the deflection when a load is applied to a cantilever spring whose center line of plate thickness is an arc. The calculation results using this theorem are shown below.
When the vertical load (P) and the horizontal load (W) are simply applied to the position of the central angle on the arc-shaped flat spring shown in Figure 6, deflection in the y-direction and deflection in the x-direction at the central angle position are as follows.
Deflection due to P
Deflection due to W:
In Figure 7, δy and δx 、are as follows.
In Figure 8, δy and δx are as follows.
The maximum stress due to P always occurs at the fixed end.
The maximum stress due to W occurs at point A in Figure 8 for and at the fixed end for .
The circular spring shown in Figure 9 is vertically symmetrical, and so the total deflection is twice the deflection of the shape in Figure 8.
In the flat spring with the combination of the half-circle and the quarter-circle shown in Figure 10, the deflection is as per the following formula.
The maximum stress occurs at the fixed end can be calculate with the following formula:
The deflection of the free end of the shape shown on the left side of Figure 11 is as per the following formula.
In the case of an arc that is constrained in the horizontal direction as shown on the right side of Figure 11, the deflection can be calculated with the following formula.
In both cases, the maximum stress is calculated with the following formula.
As shown in Figure 12, when the straight-line part (AB) and the circular arc part (BD) are combined, one end (D) is fixed, and the vertical load (P) or the horizontal load (W) acts on the other end (A), and will be as follows.
If W acts, the formula will be as below.
Here, in the formula represents
The maximum stress occurs at the fixed end when , and at the point C when .
The spring in Figure 13 is a combination of the two springs in Figure 12, and the deflection is times the deflection obtained in formula 23.
As shown in Figure 14, the deflection of the A end of the spring with a straight line part and an arced part is as per the following formula.
Here, and .
The maximum bending stress occurs at point C can be calcumated with the following formula.
In the case of , if , the maximum stress occurs at the fixed end, and when and , the formula is as follows.
In the case of the shape shown in Figure 15, it is possible to calculate the deflection of part A by dividing the AC part and the CD part, doubling the deflection of formula 25 and calculating the deflection of each formula before combining them.
As shown in Figure 16, the straight part is fixed, and so when a load is applied to the A end of the arc part, the vertical deflection and the horizontal deflection of the A end are as per the following formula when the P load acts as .
When the W load acts, the formula is as follows.
For the shape in Figure 17, when the load (P) acts, the formula is as follows.
When W acts, the formula is as follows.
It is represented here as .
For a spring with the shape shown in Figure 18 that combines an arc with a small curvature radius and a straight line, the deflection that excludes the radius of the arc is expressed by the following formula.
The maximum stress occurs at the BC part when , can be calculated with the following formula.
If , the maximum stress occurs at the fixed end, with the formula as follows.
The shape of the flat spring is often a complex combination of an arced part and a straight part. The formulas introduced so far can be used for the shape of the thin leaf springs. The shapes and calculation formulas shown below can be used as practical concepts.
The shape in Figure 19 is considered to be a combination of the two shapes in Figure 13, and the deflection can be calculated by doubling that of Formula 28.
In the shape of Figure 20, both ends are similar to Figure 10, and the stress formula can be shown with Formula 19. The deflection on one side with respect to the axis of symmetry is an added part from Formula 18, and the deflection on one side is given by the formula below.
The deflection formula is:
A flat spring with non-linear characteristics, as shown in Figure 21, is achieved by adopting an adhesive part structure where the fixed adhesive position changes in sequence due to deflection.
The formula for a flat spring with non-linear characteristics is as follows.
Flat springs are found in applications such as load measuring equipment as the one shown in Figure 22. One end is fixed and the other end can move laterally but cannot rotate. In this case, assuming that the axial load (P) is smaller than the buckling load, the deflection and stress due to the lateral load (Q) are expressed by the following formulas.
If P is larger than the buckling load, the formula above is multiplied by the coefficients and determined by . Here, is Euler’s buckling load and . Here, the coefficients and are as per the following formulas.
When the deflection is large, changes to . Figure 24 shows the calculation result with this effect added.
The horizontal axis of the figure shows , and the vertical axis shows , . represents the bending stiffness of the plate, and when is large, . As is clear from Figure 24, when the value of is small, that is, when the load P is small, and are close to 1, and when , and . Therefore, in the case of this degree of deformation, it is deemed unnecessary to handle it as a large deflection in terms of the spring’s practical application.
Figures 25 and 26 show the approximate results when the deflection of the flat spring with a trapezoidal cantilever is large. The horizontal axis is and the vertical axis shows the rate of decrease in deflection or stress with as a parameter. This just needs to be applied to the formula .