Modelling of Stiffness of Conical Springs

The variable stiffness of the spring during operational loads should ensure high softness of the system at surface loads and significant stiffness when exposed to concentrated forces or forces of high intensity. For such exploitation conditions, a construction minimum is a biconical spring consisting of two conical springs differing in stiffness coefficients, but made from a single piece of wire.

The differentiation of the stiffness coefficient value should be forced by the selec­tion of a suitable geometry of each part of the spring, which has been schematically shown in Fig. 8.27.

In the course of designing the shape of such a spring, it should be ensured that the coils of the lower cone are first settled on a hard base, therefore ensuring the exhaustion of the border of the largest soft deformations. At the same time, the upper cone should deform slightly, providing increased stiffness in the second stage of the system operation.

Therefore, let us assume the following: when the lower cone of the spring will be completely compressed X1 = H, the upper cone will deform for only about 10 % of its initial height H; that is, 12 = 0.1 H. The stiffness of the upper cone is therefore defined by the following dependency:


k1 = 10 ; (8.130)


Fig. 8.27 Calculation scheme of a biconical spring with nonlinear characteristics

whereas the stiffness of the lower cone


P the force loading the spring and H half of the height of the biconical spring.

For serially connected conical springs, the total deformation therefore amounts to


kc = 53 ki, (8.132)



kcH + 0,1 H = 1,1 H,

whereas the stiffness of the biconical spring as a system of two conical springs with given stiffness coefficients k1 and k2 amounts to:

which gives

and finally

From this relation, it results that the total stiffness of the kc system is smaller than the component k1 and k2 stiffnesses. However, it should be noted that this formula ceases to have effect when the deformation reaches a value of 1.1 H; that is, when the lower spring cone and 10 % of the upper cone completely settles, then only the upper spring cone will be compressed. Therefore, the stiffness will increase from kc to k2 = 10P/H (Fig. 8.28).

Designing the shape of the discussed spring involves the selection of a suitable equation describing the form of the spiral of wound coils (Fig. 8.29). Therefore, knowing the stiffness coefficients of each conical part, the following equation of the spiral was used:

2nRm-1dR = Cdk,


C constant determined from border conditions,

R the intermediate radius of the spiral determined for different angles between this radius and the smallest radius of the spiral R1, m the coefficient of the changes in the value of the radius R,

2n Rm-1dR = Cd k,





= Ca + Cb


and then, taking into account the border conditions, we obtain a system of equa­tions, which allows to determine the constant values C and C1,


n the number of coins of a single spring cone

Therefore, the sought equation of the intermediate radius R has the form:

This formula shows that depending on the selection of the parameter m, the increase of the value of the radius R will be variable for the same angle a. This will obviously condition the change of the stiffness of a given conical spring. Because the torsion deformation of the cross section of the wire is expressed by the equation:

dR _ PR

dS = GJO ’


Jo = r4/2 polar moment of inertia of the cross section, and

ds = Rda the length of the springs section,

therefore, elementary deflection of the spring is equal to

, PR3

dk = —— da.


We exclude from this equation the part that corresponds to the stiffness coeffi­cient k and introduce the following auxiliary function n(R):


dR C

da 2nRm~1 ’

therefore, determining the values of parameters — for k1 and k2 requires solving the equation:

k = pj g(R)dR, (8.148)


k = P -4mn – TR—+2dR, (8.149)

Gr4 (R – – R—

which gives

k = rU. (8Л5°)

Gr#R—) 0 R-+2dR

Assuming for the analysis the number of coils of the spring equal to n = 2, the dimensions of the coils R1 and R2 as well as module G and radius r as for Bonnell springs, the solution of this equation has been shown in Fig. 8.30. On the axis of

Fig. 8.30 Stiffness of the conical springs of variable geometry of the spiral: for — = 0 the function k = f(m) is not specified

ordinates of the chart, the previously calculated values of the stiffness coefficients k2 and k2 have been marked. On the axis of abscissae, values m2 = -1 and m1 = 1.8 corresponding to these points have been found. In this way, the two basic equations describing the geometry of the spiral of each of the conical springs have been obtained.

By entering to these equations, the arc values of the angle are in the range 0 ( a ( 4л; the actual shape of the designed springs has been obtained. Their sketches are shown in Fig. 8.31, and the geometry of the whole spring is shown in

Fig. 8.32.

Fig. 8.32 Side view of the designed spring