Engineer-Mechanic P. I. Gorsunov.

DEFORMATION OF HELICAL CYLINDRICAL SPRINGS OF THE SIMPLEST TYPE ^1)

A spring is an elastic body made for considerable changes of shape, with the aim of accumulating energy under the action of a load, and capable of converting the accumulated energy, in a practically reversible form, after the action of the load has ceased.

As a component part, the spring is used in various mechanisms for: 1) producing motion by virtue of recoil, 2) measuring forces, 3) softening shocks and impacts.

Being very widespread parts of modern machine building, springs very often perform responsible work. However, despite the broad use of springs and their responsible function, and also despite the fact that in our time ideas about the deformation of springs differ considerably from the old views, we still do not have exhaustive data for their calculation.

In order partially to fill this gap, in 1929, in the materials-testing laboratory of the MIMESKh, special investigations of springs and spring wire were carried out. The results of these investigations were processed and published in the book Collection of Works of the Materials-Testing Laboratory (Novyi Agronom, 1931). The experimental data obtained in this investigation on the magnitudes of deformations under different loads made it possible to compare the calculation formulas of the modern theory of helical springs with one another and, most importantly, in respect to the agreement of the deformations determined from them with those found experimentally. This latter comparison showed a large one-sided discrepancy between the values calculated by the characteristic formulas and the experimental deformations (Table No. 5, p. 110). The indicated situation, and the unclear state of the theory of helical springs concerning the influence of the fixing moment on the axial deformation of springs, call for elucidation. The development of the question of the deformation of springs of the simplest type, with an evaluation of the influence of all components of the external forces from the standpoint of the elementary theory of the strength of materials, constitutes the aim of the present article.

^1) By springs of the simplest type are meant springs with a circular cross-section of the coil.

I. Determination of the Total Potential Energy of Deformation of Springs

Suppose that a helical spring \(AB\), with a circular cross section of the coil, is subjected to the action of an axial force \(P\) (Fig. 1). Let us isolate a transverse section of the spring coil; we shall take its center \(O\) as the origin of rectangular coordinates. We shall direct the \(X\)-axis perpendicular to the section taken, or in the direction of the tangent to the helical line of the spring at the point \(O\). The axes \(Z\) and \(Y\) are mutually perpendicular to the \(X\)-axis and lie in the horizontal and vertical planes. We transfer the acting force \(P\) to the center of the section under consideration, \(O\), and resolve it along the coordinate axes \(X\), \(Y\), and \(Z\). Then from the equation of projection of the force we obtain:

Projection of the force \(P\) on the \(X\)-axis:

\[ P_x = P \cdot \operatorname{Sn}\alpha \]

on the \(Y\)-axis:

\[ P_y = P \cdot \operatorname{Cos}\alpha \]

on the \(Z\)-axis:

\[ P_z = O \]

The moment of the force \(P\), which stretches the spring, with respect to the point \(O\), is equal to \(P \cdot R\) and acts in the plane containing the axis of the spring and the point \(O\). In Fig. 1 this moment is represented in the form of the vector \(OM\), lying in a plane perpendicular to the axis of the spring. Resolving the vector \(OM\) into the two directions \(X\) and \(Y\), we obtain:

With respect to the \(X\)-axis, the twisting moment

\[ M_{x1} = P \cdot R \cdot \operatorname{Cos}\alpha \]

With respect to the \(Y\)-axis, the bending moment

\[ M_{y1} = P \cdot R \cdot \operatorname{Sn}\alpha \]

Here \(\alpha\) is the pitch angle of the coil, or the angle formed by the \(X\)-axis with the plane perpendicular to the axis of the spring.

Fig. 1.

If we draw a line along the generatrix of the free spring and begin to compress (stretch) it, we shall note that the vertical straight line deviates to the right or to the left, depending on how the spring is wound. If one of the supporting washers of the press is made rotatable and this washer is provided with degree divisions, it is possible to observe and determine the angle of rotation of the spring relative to the longitudinal axis. Such a deformation process will hereafter be called free compression (extension) of the spring.

Let us draw a straight line along the generatrix of the spring and begin to compress (stretch) it in such a way that all points lying on the indicated straight line could not leave it but would remain on it all the time; i.e., we reproduce constrained compression—without an angle of rotation. The case of constrained compression, obviously, occurs with non-rotating supporting washers of the spring, i.e., in the presence of a moment of re—

attachment, applied in a plane perpendicular to the axis of the spring.

Let \(ON\) (Fig. 1) be the vector of the fixing moment applied to the spring perpendicular to its axis. The relation between the positive direction of the moment \(M\) and the direction of the vector \(ON\) is the same as that between the rotation and translation of a right-hand screw. Resolving the vector \(ON\) along the directions of the axes \(X\) and \(Y\), we obtain the magnitudes of the bending and twisting moments acting in section \(O\):

\[ OF = My_2 = M.\Cos\alpha \]

\[ OC = Mx_2 = M.\Sn\alpha \]

The total magnitudes of the bending and twisting moments acting in section \(O\), in the case of non-free compression of the spring, will be:

\[ My = -My_1 + My_2 = -P.R.\Sn\alpha + M.\Cos\alpha \]

\[ Mx = Mx_1 + Mx_2 = P.R.\Cos\alpha + M\Sn\alpha \]

The minus sign before \(My_1\) is taken owing to the opposite direction of the vector \(OE\) with respect to the vector \(OF\).

Thus, we see that in the section under consideration of a coil of the spring there must arise a compound stress. The force \(Px\), applied at the center of gravity of the section, produces tension (compression). The force \(Py\), lying in the plane of the section and passing through the center of gravity of the section, produces shear. The moment \(Mx\), whose plane is perpendicular to the axis \(X\), produces torsion. The moment \(My\), whose plane is perpendicular to the axis \(Y\), produces bending.

Let us make the assumptions that are fundamental in the theory of helical springs.

a) The load acts along the axis of the spring.
b) The acting load is transmitted uniformly to each coil of the spring.
c) The diameter of the wire and of the spring are the same along the entire length.
d) The helix situated on a cylindrical surface intersects all generators at a constant angle \(\alpha\).

On the basis of the adopted assumptions, let us isolate an infinitely small element of any coil of the spring, with length along the axis equal to \(dS\) (Fig. 2).

Considering this element as a curved bar, we write the expression for the potential energy from the bending moment \(My\), from the normal force \(Px\), and from the shearing force \(Py\).

\[ U=\int_{0}^{s}\left(\frac{My^{2}}{2E.F.\varepsilon.R}+\frac{Px^{2}}{2.E.F}-\frac{My.Px}{E.F.R}+\frac{K.Py^{2}}{2.G.F}\right)dS \tag{a} \]

where \(F\) is the area of the transverse section of the coil.

1) Timoshenko. Strength of Materials, Part 2, 1932, p. 45.

\(\varepsilon\)—the distance of the neutral axis from the center of gravity of the section.
\(K\)—the coefficient allowing for the nonuniform distribution of shear stresses.
\(E\) and \(G\)—the moduli of normal and angular elasticity.

Fig. 2.

Usually the influence of the angle of rise of the helical line and of the fixing moment on the deformation of the spring is neglected, i.e. it is assumed that \(\sin\alpha = 0\), \(\cos\alpha = 1\), and \(M = 0\). In that case it is admitted that, in any transverse section of the spring turn, there acts only the twisting moment \(P.R\). If one further neglects, as is usually done, the curvilinearity of the turn, then under such assumptions the magnitude of the potential energy, determined by expression \((a)\), is not reflected in the calculation formulas for the deformation of the spring.

The authors of the theory of helical springs, who in their derivations use the elementary laws of strength of materials, in determining the stress due to the twisting moment apply the equation

\[ \tau = \frac{M}{W_0}, \]

which is valid, with certain assumptions, for straight cylindrical bars. In reality the element of the spring is twisted under different conditions; this to a considerable extent explains the discrepancy between experimental data and theoretical data derived on the basis of the elementary laws of strength of materials. If, during twisting, the transverse section \(ab\) (Fig. 2) rotates relative to the section \(ck\), then the displacement of point \(b\) relative to point \(c\) will be the same as that of point \(a\) relative to point \(k\). But the length \(ka\) is less than the length \(cb\). Consequently, both the relative shear and the stress on the inner side of the spring turn will be greater than on the outer side. Let us take this remark into account by the following reasoning, referring to Fig. 2.

\[ \varphi \cdot r = \gamma (R-r)\,d\beta \ldots \ldots \ldots \ldots \ldots \ldots \ldots (б) \]

where \(\gamma\) is the relative shear;

\(\varphi\)—the angle of twist,

but

\[ \varphi = \frac{M.R.d\beta}{G.I_0} \ldots \ldots \ldots \ldots \ldots \ldots \ldots (в) \]

Substitution of \((в)\) into \((б)\) gives the stress for the fiber \(ka\):

\[ \tau_{ka} = \frac{M.R.r}{I_0(R-r)} \]

By analogy we shall write:

\[ \tau_{bc}=\frac{M.R.r}{I_0(R+r)} \]

From the last expressions for the stresses in the fibers \(ka\) and \(cb\), it is seen that, when applying the formula derived for cylindrical rods with a rectilinear axis to a cylindrical rod with a curvilinear axis, it is necessary to introduce correction factors respectively

\[ \frac{R}{R-r}\quad \text{and}\quad \frac{R}{R+r}. \]

Let us assume that the law of distribution of tangential stresses over the cross-section under torsion of a cylindrical rod with a curvilinear axis will be the same as under torsion of a rod with a rectilinear axis. Keeping in mind only what has been set forth and taking into account the correction factors for the fibers \(ka\) and \(cb\), we shall replace the equation:

\[ \tau=\frac{M}{W_0} \]

by the following equation

\[ \tau=\frac{M.r}{I_0}\sqrt{A} \tag{f} \]

where

\[ \sqrt{A}= \frac{\dfrac{R}{R-r}+\dfrac{R}{R+r}}{2} = \frac{R^2}{R^2-r^2} \tag{g} \]

where \(A\) will be the mean correction factor, taking into account the curvature of the element under consideration.

The introduction of the mean value of the correction factor cannot be rigorously proved on the basis of the elementary theory of strength of materials. But it can be proved by the well-known method according to which all consequences flowing from the adopted assumption must find complete confirmation in experiment. A carefully conducted experimental verification will undoubtedly confirm the correctness of the proposed proposition concerning the correction factor. This assertion of ours is based partly on the detailed logical justification already given, and also on the investigations of Rover¹ and Wahl².

Taking the value of the stress in the cross-section of the spring coil according to expression (f) and taking into account the expression

(a), let us write the equation for the total potential energy of deformation of the spring:

\[ U=\int_{0}^{S'}\left[ \frac{A M_x^2}{2GI_0} +\frac{M_y^2}{2EF\varepsilon R} -\frac{M_y\cdot P_x}{EFR} +\frac{P_x^2}{2EF} +\frac{K\cdot P_y^2}{2G\cdot I_0} \right]\,dS \tag{e} \]

From the projection of the element of the turn \(dS\) onto the plane perpendicular to the axis of the spring (Figs. 1 and 2) it follows that:

\[ dS=\frac{R\,d\beta}{\operatorname{Cos}\alpha}. \]

Since the spring has \(n\) turns, under the adopted assumption the total change of the angle \(d\beta\) will be from \(0\) to \(2\Pi n\); and therefore, substituting the value of \(dS\) into equation (e), we obtain the expression for the total potential energy of deformation of the spring in the following final form:

\[ U=\frac{\Pi nR}{\operatorname{Cos}\alpha} \left[ \frac{A\cdot M_x^2}{I_0G} +\frac{M_y^2}{EF\varepsilon R} -\frac{2M_y\cdot P_x}{EFR} +\frac{P_x^2}{EF} +\frac{KP_y^2}{G\cdot I_0} \right] \tag{1} \]

If the curvilinearity of the spring turn is neglected, then equation (1) reduces to the following form:

\[ U=\frac{\Pi nR}{\operatorname{Cos}\alpha} \left[ \frac{A\cdot M_x^2}{I_0G} +\frac{M_y^2}{EI} +\frac{P_x^2}{E\cdot F} +\frac{K\cdot P_y^2}{G\cdot I_0} \right] \tag{1-a} \]

2. Determination of the fixing moment at the end of the spring.

To determine the fixing moment at the end of the spring, we form the partial derivative of the expression for the total potential energy of deformation of the spring with respect to \((M)\) and apply the principle of least work.

\[ \frac{\partial U}{\partial M} =0= \frac{\Pi nR}{\operatorname{Cos}\alpha} \left[ \frac{2A\cdot M_x\,\dfrac{\partial M_x}{\partial M}}{I_0G} +\frac{2M_y\,\dfrac{\partial M_y}{\partial M}}{E\cdot F\varepsilon R} -\frac{2P_x\,\dfrac{\partial M_y}{\partial M}}{E\cdot F\cdot R} \right] \]

According to the preceding:

\[ M_x=P\cdot R\cdot \operatorname{Cos}\alpha+M\cdot \operatorname{Sn}\alpha; \qquad \frac{\partial M_x}{\partial M}=\operatorname{Sn}\alpha \]

\[ M_y=-P\cdot R\cdot \operatorname{Sn}\alpha+M\cdot \operatorname{Cos}\alpha; \qquad \frac{\partial M_y}{\partial M}=\operatorname{Cos}\alpha \]

Substituting the values given and solving the equation with respect to \(M\), we obtain:

\[ M=\pm \frac{ P.R.Sn\,2\alpha\left[\dfrac{A}{I_0G}-\dfrac{1}{EF.\varepsilon R}-\dfrac{1}{E.F.R^2}\right] }{ 2\left[\dfrac{A.Sn^2\alpha}{I_0G}+\dfrac{\operatorname{Cos}^2\alpha}{E.F.\varepsilon R}\right] } \tag{2} \]

The distance of the neutral axis from the center of gravity is expressed by the following equation:

\[ \varepsilon = R\frac{m}{1+m} \]

where

\[ m=\frac{1}{4}\left(\frac{d}{D}\right)^2+ \frac{1}{8}\left(\frac{d}{D}\right)^4+ \frac{5}{64}\left(\frac{d}{D}\right)^6+\ldots+ \]

if \(d\) is the diameter of the spring wire, \(D\) the diameter of a coil of the spring.

In the expression for \(m\), beginning with the second term, we neglect the terms because of their smallness. Then the expression for the distance \(\varepsilon\) will have the following form:

\[ \varepsilon=\frac{D.d^2}{2(4D^2+d^2)} \]

From expression (g) we have:

\[ A=\left(\frac{D^2}{D^2-d^2}\right)^2 = \frac{C^4}{C^4-2C^2+1} \cong \frac{C^2}{C^2-2} \]

In the last transformation \(C=\dfrac{D}{d}\) has been introduced—the ratio of the coil diameter to the diameter of the spring wire—and unity has been omitted in the denominator, owing to its small influence on the numerical result.

Applying substitutions of the values found for \(\varepsilon\) and \(A\) in expression (2), and using the following relations:

\[ I_0=\frac{F.d^2}{8};\qquad G=\frac{E}{2(1+m)} \]

we obtain, in the following final form, the equation for determining the fixing moment:

\[ M=- \frac{ P.R.Sn\,2\alpha \left[ \dfrac{C^2}{C^2-2} - \dfrac{2C^2+1}{2(1+m)C^2} \right] }{ 2\left[ \dfrac{C^2}{C^2-2}Sn^2\alpha + \dfrac{4C^2+1}{4(1+m)}\operatorname{Cos}^2\alpha \right] } \tag{2-a} \]

Here the minus sign indicates the opposite direction of the moment.

From equation (2a) it follows that the magnitude of the fixing moment varies depending on the angle of rise of the helical line of the spring and depending on the ratio of the coil diameter to the diameter of the spring wire. For \(\alpha = O\) and \(\alpha = \dfrac{\pi}{2}\), the magnitude of the fixing moment becomes zero.

Let us now find for what value of the angle \(\alpha\) the fixing moment will have its maximum value. By substituting:

\[ \frac{C^2}{C^2 - 2} = A;\quad \frac{2C^2 + 1}{2(1 + m)C^2} = B;\quad \frac{4C^2 + 1}{4(1 + m)C^2} = D \]

we bring equation (2a) to the form:

\[ M = -\frac{P . R . \operatorname{tg}\alpha\,(A - B)} {A . \operatorname{tg}^2 \alpha + D}\ .\ .\ .\ .\ (2b) \]

We form the first derivative of (2b) with respect to \(\alpha\) and set it equal to zero.

\[ \frac{dM}{d\alpha} = -\frac{ P . R(A-B) . \operatorname{Sec}^2 \alpha (A\operatorname{tg}^2\alpha + D)\alpha + 2A\operatorname{tg}\alpha . \operatorname{Sec}^2\alpha\, P . R(A-B) . \operatorname{tg}\alpha } {(A\operatorname{tg}^2\alpha + D)^2} = 0 \]

or

\[ \operatorname{Sec}^2 \alpha \,(A\operatorname{tg}^2\alpha - D) = 0 \]

whence:

\[ \operatorname{tg}\alpha = \pm \sqrt{\frac{D}{A}} \]

For \(c = 4\) and \(m = 0.3\); \(\alpha \simeq 39^\circ 40\),

„ \(c = 14\) and \(m = 0.3\); \(\alpha \simeq 41^\circ 10\).

If, as limiting values of springs used in practice, we take springs for which \(C = 4\) and \(C = 14\), then at \(\alpha = 39^\circ 40'\) and \(\alpha = 41^\circ 10'\) the fixing moment reaches the maximum value

\[ M_{max_{c=4}} = 0.187 . P . R \]

\[ M_{max_{c=14}} = 0.137 . P . R \]

Thus, when a spring is stretched or compressed, in the case where free untwisting is impossible, fixing moments arise at its ends, the maximum value of which amounts to from 0.137 to 0.187 of the principal twisting moment \(P . R\).

Figure 3 gives a graphical representation of the dependence of \(M\) on \(\alpha\) and \(C\).

3. Determination of the angle of unwinding of the spring.

To determine the angle of rotation \(\varphi\) of any cross section \(mn\) of a turn of the spring (Fig. 1), relative to the axis of the spring, let us add to the moment \(M\), acting in a plane perpendicular to the axis \(AB\), a fictitious moment \(Mf\). The work of the elastic forces of the spring due to the action of the moment \(M\), taking into account the fictitious moment \(Mf\), may be represented in the following form:

\[ U=\int_0^s \frac{(M+Mf)^2\cdot dS}{2EI} \]

Fig. 3.

Applying Castigliano’s theorem and putting, in the final result, \(Mf=0\), with the angle \(\alpha\beta\) varying from 0 to \(2\pi n\), we find the required angle of rotation:

\[ \varphi=\frac{2\pi n\cdot R\cdot M}{EI} \]

or

\[ \varphi=-\frac{2\pi n\cdot P\cdot R^2\cdot \tg\alpha\,(A-B)} {EI\,(A\tg^2\alpha+D)} \tag{3} \]

The minus sign in the last expression indicates the opposite direction of the unwinding angle.

The law of variation of the unwinding angle \(\varphi\), as a function of \(\alpha\) and \(C\), is evidently the same as the law of variation of the fixing moment with respect to the indicated quantities.

4. Determination of the axial deformation of the spring.

Taking the partial derivative of expression (1) with respect to the force \(P\), we obtain the vertical displacement of point \(A\) in Fig. (1):

\[ \frac{\partial U}{\partial P} =f= \frac{\pi\cdot n\cdot R}{\Cos\alpha} \left[ \frac{2AM_x\,\dfrac{\partial M_x}{\partial P}}{I_0G} + \frac{2M_y\,\dfrac{\partial M_y}{\partial P}}{E\cdot F\cdot e\cdot R} - \right. \]

\[ -\frac{ 2\left(\frac{\partial M_y}{\partial P}P_x+\frac{\partial P_x}{\partial P}M_y\right) }{E.F.R} +\frac{2P_x\frac{\partial P_x}{\partial P}}{E.F} +\frac{K.2P_y\frac{\partial P_y}{\partial P}}{GF} \right] \]

The partial derivatives will be:

\[ \frac{\partial M_x}{\partial P} = R.\operatorname{Cos}^{2}\alpha + \frac{\partial M}{\partial P}\operatorname{Sn}\alpha; \qquad \frac{\partial P_x}{\partial P} = \operatorname{Sn}\alpha \]

\[ \frac{\partial M_y}{\partial P} = -R.\operatorname{Sn}\alpha + \frac{\partial M}{\partial P}\cdot \operatorname{Cos}\alpha; \qquad \frac{\partial P_y}{\partial P} = \operatorname{Cos}\alpha \]

The quantity \(\dfrac{dM}{dp}\operatorname{Sn}\alpha\) may be neglected in comparison with \(R\operatorname{Cos}\alpha\), since for \(C=7\) and \(\alpha=10^\circ\) this quantity is equal to \(0.009R\), while the quantity \(R.\operatorname{Cos}\alpha\) is equal to \(0.985R\).

Then, carrying out the corresponding transformations when substituting the values given above into the formula for \((f)\), we obtain:

\[ f= \frac{2\Pi.n.P.R^{3}}{\operatorname{Cos}\alpha} \left[ \frac{A.\operatorname{Cos}^{2}\alpha}{I_0G} + \frac{\operatorname{Sn}^{2}\alpha}{E.F.\varepsilon.R} + \frac{3\operatorname{Sn}^{2}\alpha}{E.F.R^{2}} + \frac{K.\operatorname{Cos}^{2}\alpha}{G.F.R^{2}} - \right. \]

\[ \left. - \frac{ \operatorname{Sn}2\alpha\frac{\partial M}{\partial P} }{ 2E.F.R^{3} } \left(1+\frac{R}{\varepsilon}\right) \right] + 2\Pi.n.R^{2}M.\operatorname{Sn}\alpha \left[ \frac{A}{I_0G} - \frac{1}{EF\varepsilon R} - \right. \]

\[ \left. - \frac{1}{EF.R^{2}} + \frac{\operatorname{Ctg}\alpha\frac{\partial M}{\partial P}}{EF\varepsilon.R^{2}} \right] \tag{4} \]

Neglecting the magnitude of the fixing moment, i.e. considering the case of free extension (compression), in which \(M=0\), while \(\varphi\)—the angle of unwinding of the spring—is not equal to zero, the formula for \((f)\) will take the following form:

\[ f= \frac{2\Pi.n.P.R^{3}}{\operatorname{Cos}\alpha} \left[ \frac{A\operatorname{Cos}^{2}\alpha}{I_0G} + \frac{\operatorname{Sn}^{2}\alpha}{E.F.\varepsilon.R} + \frac{3.\operatorname{Sn}^{2}\alpha}{EFR^{2}} + \frac{K.\operatorname{Cos}^{2}\alpha}{G.F.R^{2}} \right] \]

Applying the same substitutions as in the case of transforming expression (2), from the last expression we obtain:

\[ f= \frac{16.\Pi.n.PR^{3}}{F.d^{2}G} \cdot \operatorname{Cos}\alpha \left[ \frac{c^{2}}{c^{2}-2} + \frac{k}{2c^{2}} + \frac{c^{2}+1}{(1+m)c^{2}}\operatorname{tg}^{2}\alpha \right]. \]

The quantity \(\dfrac{16\pi.n.P.R^{3}}{F.d^{2}.G}\) represents the magnitude of deforma-

of the spring according to Reuleaux’s formula. Therefore we introduce the following substitution:

\[ \frac{16\Pi \cdot n \cdot P R^{3}}{F \cdot d^{2} \cdot G}=f_R \]

We take the value of the correction coefficient for the nonuniformity of the distribution of shear stresses over the section to be \( \frac{4}{3} \).

Then the formula for determining the axial deformation of the spring in the case of its free extension (compression) will take the following form:

\[ f=f_R \cdot \operatorname{Cos}\alpha \left[ \frac{c^{2}}{c^{2}-2} +\frac{2}{3c^{2}} +\frac{c^{2}+1}{(1+m)c^{2}}\, tg^{2}\alpha \right] \tag{4-a} \]

If we take \(\operatorname{Cos}\alpha=\) and \(tg^{2}\alpha=0\), then we obtain:

\[ f=f_R \cdot \left[ \frac{c^{2}}{c^{2}-2} +\frac{2}{3c^{2}} \right] \tag{4-b} \]

In the last expression, the first term accounts for the influence of curvature, and the second for the influence of the shearing force on the deformation of the spring.

Let us now consider the case of constrained deformation of the spring. From equation (4) we obtained the magnitude of the axial deformation of the spring for \(M=0\). Let us determine from this equation the magnitude of the axial deformation of the spring due to the influence of the fixing moment. For this purpose, from equation (4) we write out the terms containing \(M\) and \(\frac{dM}{dp}\)

\[ f_M= -\frac{2\Pi \cdot n \cdot P \cdot R^{3}}{\operatorname{Cos}\alpha} \cdot \frac{\operatorname{Sn}2\alpha\, \dfrac{\partial M}{\partial P}} {2E \cdot F \cdot R^{3}} \left(1+\frac{R}{e}\right) + \]

\[ +2\Pi \cdot n \cdot R^{2}M \cdot \operatorname{Sn}\alpha \left[ \frac{A}{I_0G} -\frac{1}{EF \cdot eR} -\frac{1}{E \cdot F \cdot R^{2}} +\frac{c_0 tg\alpha\, \dfrac{\partial M}{\partial P}} {E \cdot F \cdot eR^{2}} \right]. \]

Transforming the last equation with the aid of the substitution of the corresponding quantities used by us in deriving equations (2) and (4), we obtain:

\[ f_M= -\frac{f_R \cdot \operatorname{Sn}\alpha(2c^{2}+1)} {2R(1+m)c^{2}} \cdot \frac{\partial M}{\partial P} + \frac{f_R \cdot M \cdot \operatorname{Sn}\alpha}{P \cdot R} \left[ A-\frac{2c^{2}+1}{2(1+m)c^{2}} + \right. \]

\[ \left. +\frac{4c^{2}+1}{4(1+m)c^{2}} \operatorname{Cotg}\alpha \, \frac{\partial M}{\partial P} \right] \]

Let us denote:

\[ \frac{2c^2+1}{2(1+m)c^2}=B;\qquad \frac{4c^2+1}{4(1+m)c^2}=D \]

In determining the maximum value of the fixing moment, the equation for it was reduced to the following form:

\[ M=-\frac{P\cdot R\cdot \operatorname{tg}\alpha\,(A-B)}{A\operatorname{tg}^2\alpha+D} \]

therefore:

\[ \frac{\partial M}{\partial P}=-\frac{R\cdot \operatorname{tg}\alpha\,(A-B)}{A\operatorname{tg}^2\alpha+D} \]

By substituting the values given above into the equation for \((f_M)\), we shall have:

\[ f_M= \frac{f_R\cdot \operatorname{Sn}\alpha\cdot \operatorname{tg}\alpha} {A\operatorname{tg}^2\alpha+D} \left[ B-\frac{(A-B)\operatorname{tg}^2\alpha}{A\operatorname{tg}^2\alpha+D} \right] \left[ A-B \right] \tag{5} \]

On the basis of equations (4a) and (5) we obtain the value of the axial deformation of the spring for the case \(M\ne O\) and \(\varphi=O\), in the following final form:

\[ f=f_R\cdot \operatorname{Cos}\alpha \left[ A+\frac{2}{3c^2}+\frac{(c^2+1)\operatorname{tg}^2\alpha}{2(1+m)c^2} \right] - \]

\[ -\frac{f_R\cdot \operatorname{Sn}\alpha\cdot \operatorname{tg}\alpha} {A\operatorname{tg}^2\alpha+D} \left[ B-\frac{(A-B)\operatorname{tg}^2\alpha}{A\operatorname{tg}^2\alpha+D} \right] \left[ A-B \right] \tag{6} \]

Here the minus sign before the second term is taken on the basis of equation (2).

If, at the end of the spring, one applies a moment in the direction shown in Fig. 1, as determined by equation (2), then in this case the plus sign must be taken before the second term. Thus, the fixing moment arising at the end of the spring in the case of constrained extension (compression) increases the stiffness of the spring. Consequently, the direct application of a moment at the end of the spring may change its stiffness either toward an increase or toward a decrease, depending on the direction of the applied moment.

From equation (6) it is seen that the influence of the fixing moment on the axial deformation of the spring depends on the angle of rise of the helical line and on the ratio of the coil diameter to the wire diameter of the spring. Let us determine at what value of the angle of rise \(\alpha\) the influence of the fixing moment will have a maximum value. To simplify the solution of the problem posed, we shall write equation (5) in the following form:

\[ f_M=\frac{f_R\cdot Sn\,\alpha\cdot tg\,\alpha\cdot(A-B)\cdot B}{A\,tg^2\alpha+D} -\frac{f_R\,Sn\,\alpha\cdot tg^3\alpha\,(A-B)^2\cdot A}{(A\,tg^2+D)^2} \]

We shall neglect the second term of the last equation as small in comparison with the first. Indeed, for \(C=8\) and \(\alpha=10^\circ\), the second term amounts to \(0.012\) of the first term. Then the last equation for determining the maximum will have the following form:

\[ y=\frac{Sn\,\alpha\cdot tg\,\alpha\,(AB-B^2)}{A\,tg^2\alpha+D} \]

Proceeding in the general way, we shall have:

\[ \frac{dy}{d\alpha}=0= \frac{ \left(Cos\,\alpha\cdot tg\,\alpha+\frac{Sn\,\alpha}{Cos\,\alpha}\right)(A\,tg^2\alpha+D) -2A\cdot tg^2\alpha\,\frac{Sn\,\alpha}{Cos\,\alpha} }{ (A\,tg^2\alpha+D)^2 } \]

or

\[ A\cdot Sn^3\alpha\cdot Cos^2\alpha -A\,Sn^3\alpha +D\,Sn\,\alpha\cdot Cos^4\alpha +D\,Sn\,\alpha\cdot Cos^2\alpha=0 \]

Replacing \(Cos\,\alpha\) in the last equation by \(Sn\,\alpha\), after transformation we obtain:

\[ Sn^4\alpha(D-B)-3D\,Sn^2\alpha+2D=0. \]

Let us introduce the following substitution \(Sn^2\alpha=Z\) and solve the equation with respect to \(Z\).

\[ Z=\frac{3D\pm\sqrt{D^2+8BD}}{2(D-B)} \]

Since \(B>D\), we take the minus sign before the radical. Then:

\[ Sn\,\alpha=\sqrt{\frac{3D-\sqrt{D^2+8BD}}{2(D-B)}} \]

From the last expression we determine the value of the angle

\[ \text{For } C=4 \text{ and } m=0.3;\quad \alpha\cong51^\circ30' \]

\[ \text{For } C=14 \text{ and } m=0.3;\quad \alpha\cong52^\circ20'. \]

To judge the magnitude of the error that may occur when using formula (4a), or how much the stiffness of the spring increases due to the moment of restraint at the end, a graph is given in Fig. 4. Here, along the axis of abscissas are plotted the angles of rise \(\alpha\), and along the axis of ordinates the percentage of the axial deformation determined by Reuleaux’s formula. The graph is constructed for two values, namely, for \(C=4\) and for \(C=14\).

From the given graph it is evident that for the majority of springs used in practice, for which \(C=6\div11\), and \(\alpha=4^\circ\div15^\circ\),

the influence of the fixing moment on the axial deformation is very small, and the error from applying formula (4a) will amount to a tenth of a percent. But for \(C=14\) and \(\alpha=52^\circ 20'\) the error will be equal to 2.3%. And for \(C=4\) and \(\alpha=51^\circ 30'\) the error will have the maximum value, 4.6%.

5. Brief survey of the formulas of the modern theory of helical coil springs.

Authors estimate differently the influence of the components of the external forces. Therefore the calculation formulas for deformation ultimately have a variety of forms. Since our investigation has been carried out taking into account all the influencing factors, then, depending on which of them are neglected, we can obtain from expressions (1) and (1a) all the formulas derived on the basis of the elementary theory of strength of materials.

Fig. 4.

Turning to expressions (1a) and (4a), taking into account deformation only from torsion without the correction coefficient \(A\), and neglecting the angle of rise \(\alpha\), we obtain Reuleaux’s formula (Reuleaux (1)).

\[ f_R=\frac{8\cdot n\cdot P\cdot D^3}{G\cdot d^4} \tag{7} \]

Neglecting the curvilinearity of the coil and the fixing moment \(M\), i.e. considering the case \(M=0\); taking the correction coefficients \(A=1\) and \(K=1\); from (1a), by taking the partial derivative with respect to \(P\), we obtain Lecornu’s formula (Lecornu)\(^2\).

\[ f=\frac{2\Pi\cdot n\cdot P\cdot R^3}{\operatorname{Cos}\alpha} \left[ \frac{\operatorname{Cos}^2\alpha}{I_0 G} + \frac{\operatorname{Sn}^2\alpha}{I\cdot E} + \frac{\operatorname{Cos}^2\alpha}{G\cdot F\cdot R^2} + \frac{\operatorname{Sn}^2\alpha}{E\cdot F\cdot R^2} \right] \tag{8} \]

If in the last formula we neglect the influence of the longitudinal (with respect to the coil) and transverse forces, we obtain Zacharias’s formula (Zacharias)\(^3\).

1) Reuleaux. Konstruktion Berechnung der wichtigsten. Feder arten; W. W. Comp. 1857.
2) Lecornu. Cours de Mechanique 1918 p. 79.
3) Zacharias. V. D. I. Bd. 55. 1911, S. 1801.

\[ f=\frac{2\pi\cdot n\cdot P\cdot R^3}{\operatorname{Cos}\alpha} \left[ \frac{\operatorname{Cos}^2\alpha}{I_0G} + \frac{\operatorname{Sn}^2\alpha}{IE} \right]. \tag{9} \]

Neglecting the deformation due to the normal and shearing forces, and also not taking into account the curvature of the coil, but allowing for the influence of the fixing moment, from expressions (1a) and (4) we obtain Perry’s formula¹:

\[ f= \frac{2\pi\cdot n\cdot P\cdot R^3}{\operatorname{Cos}\alpha} \left[ \frac{\operatorname{Cos}^2\alpha}{I_0G} + \frac{\operatorname{Sn}^2\alpha}{IE} \right] + \frac{\pi\cdot n\cdot R^2 M\cdot \operatorname{Sn}2\alpha}{\operatorname{Cos}\alpha} \left[ \frac{1}{I_0G} - \frac{1}{IE} \right]. \tag{10} \]

F. Lender,² in order to obtain more accurate results in the case of large changes in load, proceeding from Perry’s basic assumptions, introduced a method of approximate integration. As a result, after a whole series of transformations and neglecting the squares of the quantities

\[ \frac{H}{L} \quad\text{and}\quad \frac{H}{L}-\frac{32P\cdot R^2}{G\cdot \pi\cdot d^4}, \]

he gave the formula for the axial deformation of the spring, for the case \(M=O\) and \(\varphi\ne O\), in the following form:

\[ f= \left[ 1-\frac{2}{3E}\cdot G\frac{H\cdot(2H-f)}{L^2} \right]f. \tag{11} \]

And for the case \(M\ne O\) and \(\varphi=O\), under the assumption that \(\operatorname{Sn}\alpha=O\); \(\operatorname{Cos}\alpha=1\),

\[ f= \left\{ 1- \left(1-\frac{2G}{E}\right) \frac{2H-f}{L} \left[ \frac{H}{3} + \left(\frac{E}{2G}-1\right)\frac{H-f}{2} \right] \right\}f. \tag{12} \]

In Lender’s formulas: \(H\) is the total height of the spring before deformation; \(L\) is the total length of the spring, and \(f\) is the deformation of the spring, determined by Reuleaux’s formula.

Pilgram’s formula³ is derived on the basis of the general principles of the theory of elasticity. In the derivation, the influence of the fixing moment \(M\) and of the pitch angle \(\alpha\) on the axial deformation of the spring was not taken into account. As the equation of the contour of the section, instead of a circle, an equation of the fourth degree was adopted. The quantity

\[ \frac{3}{32}\cdot\frac{d^2}{D^2}, \]

as very small in comparison with unity, was omitted in the derivation. The derivation was given for the following relation: \(R>6d\). In the final result the formula is reduced to the form:

\[ f=\frac{16.n.P.R^{2}}{G.d^{4}}\sqrt{16R^{2}-d^{2}} \tag{13} \]

Grodsky’s formula\(^1\) was derived on the basis of Kirchhoff’s theory\(^2\) on the equilibrium of an infinitely thin rod under deformation. In the derivation, only the change in the height of the spring and the increase in radius were taken into account. All the previously noted factors affecting the deformation of the spring were not taken into account. In its final form, Grodsky proposed the following formula:

\[ f=\frac{AL}{2Sn\,\alpha}+L\sqrt{\frac{A^{2}}{4.Sn^{2}\alpha}-\frac{64PR^{2}}{E\Pi.d^{4}Sn\,\alpha}} \tag{14} \]

where

\[ A=\left[2G+(E-3G).Sn^{2}\alpha\right]\frac{1}{E} \]

Analyzing Grodsky’s work, Engineer Sazhin\(^3\), proceeding from the same principles, supplemented Grodsky’s work by deriving a formula taking into account the fixing moment \(M\). In the final result, the formula for the case \(M\ne O\) and \(\varphi=O\) is given in the following form:

\[ P=\frac{G.I_{0}\left[h\alpha.\Pi.(R+\Delta R)-(h-\varepsilon)\right]\sqrt{L^{2}-(h-\varepsilon)^{2}}}{(R+\Delta R)^{2}.L.\sqrt{L^{2}-(h-\varepsilon)^{2}}} \tag{15} \]

and for the case \(M=O\) and \(\varphi\ne O\)

\[ P=\frac{\Pi^{2}I_{0}R^{3}}{L^{3}}\left[\varepsilon.G+2E\Pi.\frac{H.\Delta R}{\sqrt{L^{2}-(h-\varepsilon)^{2}}}\right] \tag{16} \]

Here \(\Delta R\) is the increment of the radius of the spring; \(h\) is the height of a half-turn; \(L=\Pi R\) is the length of a half-turn, and \(\varepsilon=\dfrac{f}{2n}\) is the compression of the spring over the length of a half-turn. The author neglects in formula (16) the influence of the term:

\[ 2E\Pi\frac{H.\Delta R}{\sqrt{L^{2}-(\varepsilon-\varepsilon)^{2}}} \]

Then formula (16) is reduced to the following final form:

\[ f=\frac{f_{R}}{Cos^{3}\alpha} \tag{17} \]

Göhner’s formula (Göhner)\(^4\) was derived on the basis of his previous works: “Distribution of tangential stresses in the transverse section

\(^1\) Gorodsky. Theory of gun carriages, part 1. 1906.
\(^2\) G. Kirchhoff. Mathematische phisik, 1897.
\(^3\) T. E. vol. 18, p. 429.
\(^4\) Göhner. V. D. I. 1932. S. 269.

“helical springs”¹ and “Stress distribution in a sector of an annular bar with loading in the end sections”², and also on the basis of the above-mentioned work of Pilgrim. The influence of the fixing moment was not taken into account. The derivation is made for two cases: 1) taking account of the helix angle, 2) without taking account of the helix angle. The curvature of the coil is taken into account in both cases. In the final result the formulas are given in the following form:

\[ f=\frac{2\pi n \cdot P \cdot R^3}{C} \tag{18} \]

For the case taking account of the helix angle,

\[ C=C_0\left[1+\frac{K\dfrac{d^2}{D^2}\cos^4\alpha}{1-\dfrac{d^2}{D^2}}\right]. \]

For the case without taking account of the helix angle,

\[ C=C_0\left[1+\frac{K\dfrac{d^2}{D^2}}{1-\dfrac{d^2}{D^2}}\right], \]

and where \(C_0=\dfrac{\pi d^4}{32}G\); for a circular cross-section of the coil, \(K=\dfrac{3}{16}\).

6) The position of the formula we have derived for the axial deformation of a spring among the other formulas presented.

Analyzing expression (6), we determined the magnitude of the influence of the fixing moment on the axial deformation of springs. And since it was established that this influence is very insignificant, for comparison of the formula we derived with the other previously presented formulas, we shall take expression (4a). We carry out the comparison of the calculated data according to expression (4a) with the experimental data and with the calculations according to other formulas as follows:

From the cited work of M.I.M.E.S.H. we borrow in full table No. 5. This table gives experimental data for 55 different springs and compares them with results determined by the most characteristic formulas, namely: Rölo, Peltrem, Zacharias, and Sazhin. Using the data of the principal dimensions of the 55 springs from the indicated table, we calculated the deformation according to expression (4a), and the results, with the deviation in percent, are given in table No. 1.

Table No. 1

Spring No.	Diameter, P/mm	Diameter, n/mm	\(D/d\)	Number of turns	Angle of inclination, in degrees and minutes	Load, kg	Experimental data, \(f\) in mm	Reuleaux, \(f\) in mm	Reuleaux, difference in %	Sazhina, \(f\) in mm	Sazhina, difference in %	Tsaharik, \(f\) in mm	Pilgrim, \(f\) in mm	Pilgrim, difference in %	Formula 4-a, \(f\) in mm	Formula 4-a, difference in %
1	41.2	6.1	6.8	20.5	87	25.0	15.38	15.81	−2.7	16.38	−6.1	16.15	16.04	−4.1	16.85	−10.0
2	42.3	6.44	6.6	22.0	7.45	25.0	16.5	14.75	+11.9	15.29	+7.9	14.30	14.81	+11.4	15.72	+4.7
3	43.7	6.78	6.4	21.0	—	—	13.5	12.45	+8.4	12.85	+5.1	12.58	12.63	+6.9	13.34	+1.2
4	—	—	—	—	—	50.0	27.0	24.90	+8.4	25.70	+5.1	25.16	25.26	+6.9	26.70	+1.12
5	—	—	—	—	—	75.0	40.87	37.35	+9.0	38.55	+6.0	37.74	37.89	+7.9	40.04	+2.0
6	41.78	6.38	6.5	19.5	8.44	25.0	14.50	13.25	+9.4	13.75	+5.5	12.98	13.18	+10.0	14.2	+2.07
7	43.7	6.73	6.5	21.0	7.51	25.0	13.75	12.75	+7.8	13.15	+4.1	12.50	13.15	+4.6	13.65	+0.65
8	42.6	6.20	6.9	19.5	9.27	25.0	16.25	16.20	+0.3	17.00	−4.4	15.95	16.08	+1.1	17.25	−6.15
9	31.1	4.20	7.4	8.5	10.30	22.5	14.38	13.65	+5.3	14.37	+0.1	13.45	12.24	+7.5	14.46	−0.56
10	29.3	5.84	6.0	10.75	13.40	50.0	7.5	6.06	+23.6	6.62	+13.3	6.87	6.00	+25.0	6.73	+10.3
11	30.8	6.00	5.1	12.5	—	50.0	8.25	7.45	+10.7	7.95	+3.8	7.3	7.47	+10.4	7.70	+2.6
12	30.0	5.90	5.1	11.0	13.71	50.0	7.5	6.45	+16.3	7.00	+7.1	6.17	7.07	+6.1	7.31	+1.77
13	30.6	5.80	5.3	10.75	13.14	100.0	16.75	14.74	+13.6	16.10	+4.0	14.24	14.64	+14.4	16.71	+0.24
14	31.4	5.90	5.3	12.0	11.59	50.0	9.25	8.32	+11.2	8.90	+3.9	8.35	8.28	+11.7	9.20	+0.54
15	24.8	3.40	7.3	13.0	10.14	15.0	14.5	14.5	0	15.24	−4.6	14.35	14.46	+0.3	15.40	−6.20
16	23.8	3.40	7.0	9.0	12.28	15.0	9.5	8.85	+7.3	9.52	−0.2	8.0	8.67	+9.6	9.57	−0.74
17	23.6	3.40	6.9	9.25	12.15	15.0	9.63	8.80	+9.4	9.42	+2.2	8.6	8.65	+11.3	9.54	+0.94
18	23.6	3.40	6.9	9.25	12.15	30.0	19.0	17.60	+8.0	18.84	+0.8	—	17.30	+9.8	19.00	0
19	24.6	3.46	7.1	12.25	13.41	15.0	12.25	12.56	−2.5	13.56	−9.7	11.93	12.38	−1.1	13.57	−10.7
20	8.2	0.96	8.5	6.5	14.16	1.5	5.83	4.47	+30.4	4.92	+18.5	4.35	4.60	+26.7	4.75	+18.5
21	9.2	1.18	7.8	7.0	10.56	1.125	3.25	3.18	+2.2	3.24	+0.3	3.23	2.36	+2.73	3.34	−2.77
22	9.2	1.80	5.1	7.0	—	2.25	6.13	6.36	−3.60	6.48	−5.4	6.25	6.67	−8.4	6.50	−6.1
23	17.5	1.64	10.5	10.0	9.39	1.5	9.0	8.40	+7.1	8.79	+2.4	8.14	8.41	+7.0	8.8	+2.2
24	16.9	1.68	9.9	10.0	8.59	1.5	7.38	6.74	+9.5	7.01	+5.3	6.72	6.66	+10.8	7.10	+3.8
25	14.2	1.64	9.7	18.0	10.06	1.5	7.63	7.56	+0.9	7.95	−4.0	7.47	7.51	+1.6	8.00	−4.85
26	17.1	1.60	10.6	9.0	8.45	1.5	8.0	7.80	+2.6	8.10	−1.2	7.70	7.77	+3.0	8.13	−1.62
27	17.7	1.76	10.0	9.0	9.26	1.5	6.87	5.82	+18.0	6.07	+13.2	5.73	5.86	+17.2	6.02	+12.35

No.	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
28	13.5	1.64	8.2	5.5	11.40	2.24	3.33	2.89	+15.0	3.02	+8.5	2.83	2.90	+14.8	3.10	+6.87
29	13.4	1.70	7.9	5.5	12.15	4.5	5.5	4.82	+14.1	5.05	+8.2	4.69	4.83	+13.9	5.12	+6.90
30	12.0	1.64	7.3	7.5	10.12	3.75	4.25	4.40	−3.4	4.62	−8.0	4.30	4.38	−3.0	4.67	−10.0
31	12.4	1.68	7.4	7.0	10.56	7.5	7.5	8.26	−8.2	8.75	−14.3	8.15	8.19	−8.0	8.77	−16.91
32	9.7	1.64	5.9	17.0	14.51	7.5	9.5	9.40	+1.1	10.40	−8.7	0.00	9.34	+1.7	10.33	−8.75
33	15.2	1.82	8.4	6.5	11.22	3.75	6.0	5.42	+10.7	5.70	+5.0	5.38	5.34	+12.4	5.72	+4.67
34	14.5	2.16	6.7	9.0	14.54	15.0	13.25	12.10	+9.5	13.00	+1.9	11.50	11.72	+13.1	13.24	−0.07
35	14.7	1.80	8.2	3.5	10.39	7.5	6.0	5.45	+10.1	5.65	+5.8	5.34	5.38	+11.5	5.73	+4.6
36	16.0	2.00	8.0	3.5	14.18	10.5	6.63	6.41	−3.4	7.05	−6.0	6.23	6.39	+3.8	6.87	−3.16
37	19.5	2.00	9.8	4.75	12.31	10.5	18.0	17.00	+5.9	18.30	+1.6	16.60	16.97	+6.1	17.81	+1.05
38	19.3	2.10	9.2	6.00	12.15	5.25	10.0	8.88	+12.6	9.00	+11.1	8.22	8.39	+19.2	9.29	+7.10
39	19.1	2.10	9.5	6.5	12.52	15.0	17.5	16.80	+4.2	18.10	−3.3	16.40	16.86	+3.8	17.51	−0.06
40	19.4	2.06	9.4	17.0	8.19	3.75	18.0	18.9	−4.8	19.50	−7.7	18.6	17.80	+1.1	19.58	−8.78
41	18.1	2.12	8.5	11.0	12.31	7.5	17.5	17.60	−0.85	18.30	−4.4	16.60	17.05	+2.6	18.46	−5.39
42	16.2	2.10	7.7	6.0	11.42	7.5	8.0	6.61	+22.0	6.9	+13.8	6.48	6.59	+21.4	7.03	+12.10
43	15.3	2.06	7.5	6.0	13.40	7.5	6.75	6.05	+11.6	6.65	+3.1	5.90	6.03	+11.9	6.45	+4.45
44	14.3	2.00	7.2	6.5	11.12	10.5	8.0	8.05	−0.6	8.54	−6.3	7.91	8.03	+1.8	8.52	−6.50
45	15.2	2.10	7.2	7.0	10.49	5.25	4.25	3.70	+15.0	3.90	+9.0	3.64	3.32	+18.0	3.93	+7.50
46	12.1	1.98	6.1	6.5	12.31	10.5	5.25	4.67	+12.4	5.03	+4.4	4.60	4.65	+12.9	5.10	+2.86
47	15.1	1.96	7.7	9.45	8.51	5.25	8.00	7.60	+5.30	7.88	+1.5	7.55	7.58	+5.5	8.01	−0.12
48	19.0	2.64	7.2	6.5	11.58	15.0	9.5	8.97	+5.90	9.60	−1.0	8.76	8.91	+6.6	9.58	−0.84
49	19.9	2.64	7.5	7.0	12.21	15.0	10.5	9.65	+8.8	10.10	+3.8	9.40	11.25	−6.8	10.25	+2.38
50	19.0	2.80	6.8	8.5	12.05	30.0	20.0	18.30	+9.3	19.60	+2.0	17.50	17.81	+12.3	19.70	+1.5
51	16.6	3.02	5.6	6.0	11.06	15.0	2.75	2.74	+0.4	2.91	−5.5	2.69	2.73	+0.7	3.04	−10.5
52	25.2	3.44	7.3	12.0	12.02	30.0	28.75	27.00	+6.5	28.80	−0.2	26.40	26.91	+6.8	28.78	−0.1
53	25.5	3.54	7.2	13.0	10.49	30.0	28.75	27.30	+5.3	28.80	−0.2	25.90	26.78	+7.4	28.90	−0.5
54	24.7	3.48	7.1	12.5	11.39	30.0	15.5	25.80	−1.2	27.50	−7.3	24.30	24.60	+3.7	27.55	−7.9
55	19.1	2.44	7.8	8.0	10.20	5.25	6.25	5.56	+12.4	5.77	+8.6	4.14	5.54	+12.7	5.86	+6.2

For a clearer representation, Fig. 5 gives a graph constructed from the data of Table No. 1. Here the number of springs is plotted along the ordinate axis, and the classes, or property of the springs, along the abscissa axis. The concentration of each class represents the deviation in percent from 0–3%; from 3–6%, etc. The results of comparing the formulas of Zacharias and Wahlström, as giving a small deviation in comparison with Reuleaux’s formula, are not shown on the graph.

Fig. 5.

From a comparison of the results of No. 1 and the graph it can be seen that the best agreement of the calculated deformations with the experimental data is obtained by formula (4a). Here almost identical deviation is observed both in the positive and in the negative direction. A plus–minus difference of 3% occurs for 26 springs, or for 47% of the total number. A difference of 6% occurs for 36 springs, or for 66% of the total number. A difference of 9% occurs for 46 springs, or 84% of the total number of tested springs. Thus, the proposition advanced earlier concerning the mean value of the correction coefficient for the curvature of the spring coil is proved by objective experimental data. For more convenient use of formulas (4a) and (4в), Fig. 5a gives a graph. From this graph, knowing \(C\) and \(\alpha\), one can determine the coefficient \(\psi\), by which it is necessary to multiply the result obtained by Reuleaux’s formula in order to obtain the deformation according to formulas (4a) and (4в).

7. Testing of springs.

With the means and materials that were at the disposal of the author and of the mechanical laboratory of S.M.M.I., it was not possible to manufacture the necessary number of characteristic springs for the investigation. Therefore, 6 springs were made for compression testing. Three springs were made with the same \(C\), but with different angle of rise. The remaining three with the same—

angle of pitch, but with different ratios of the coil diameter to the diameter of the spring wire. The desire to carry out experiments with a larger number of springs was achieved by taking some springs, not made specially for our investigation but well selected, in finished form.

Fig. 5a.

The springs were made of steel wire with \(d = 5\) mm and with \(d = 4.20\) mm. The chemical composition of the wire is given in Table No. 2.

The springs were wound on a lathe in the cold state. They were quenched in oil at a temperature of \(800^\circ\) C and tempered to \(300^\circ\) C.

Table No. 2

\(d\) in mm	Content of chemical elements in the wire	Carbon	Manganese	Silicon	Phosphorus	Sulfur
5.00	Content of chemical elements in the wire	0.74	0.68	0.23	0.037	0.047
4.20	Content of chemical elements in the wire	0.70	0.71	0.25	0.04	0.041

Pieces of the same wire were subjected to analogous heat treatment in order to determine the mechanical properties of the metal. The test was carried out on a 2-ton pendulum machine by Amsler Laffou with Martens mirror apparatus. The results of the tests are given in Table No. 3.

Table No. 3

\(d\) of wire in mm	Mechanical characteristics	Proportional limit in kg/mm²	Temporary resistance in kg/mm²	Elongation in % at \(l=10d\)	Reduction of cross-sectional area in %	Modulus of elasticity in kg/mm²
5.00	Mechanical characteristics	48.3	98.4	16.0	29.4	21800
4.20	Mechanical characteristics	47.8	87.2	18.2	31.6	21600

In addition to those made for the experiment, 20 different springs were selected. The selection was carried out by comparing hardness according to Rockwell.

The diameter of the wire, the diameter of the coil, and the pitch of the spring were measured on an Amsler Laffon comparator, with an accuracy of up to 0.02 mm. The measurement of the diameter of the coil and of the wire diameter of the spring was carried out, as indicated in Fig. 6, and in two mutually perpendicular directions. As for the pitch of the spring, since it is very difficult to make it the same during manufacture, we made the measurement, as indicated by the brackets in Fig. 6, along the entire length of the spring. Thus the diameter of the coil and of the spring wire were determined as the mean value of four measurements. The pitch of the spring was determined as the mean value from twice the number of turns—measurements. The results of the measurements are brought together in Table No. 4, where the pitch of the spring has been recalculated into the angle of rise. The angle of rise is given in degrees and minutes, and the resulting seconds were omitted because of their small influence.

Fig. 6.

¹ According to data from the chemical laboratory of the Siberian Institute of Metals.

In Table 4, springs 1 to 20 are selected, and 21 to 26 are manufactured.

Table No. 4

Spring Nos.	Wire diameter in mm	Coil diameter in mm	Number of coils of the spring	Pitch angle in deg. and min.	Ratio of coil diam. to wire diameter	Rockwell hardness
1	2.35	19.48	7.00	6.20	8.3	42
2	3.03	26.77	8.50	5.20	8.8	41
3	3.03	27.05	8.50	7.40	8.9	40
4	3.00	34.30	6.00	6.40	11.4	43
5	3.21	28.49	5.60	6.00	8.9	40
6	3.00	33.81	10.00	6.10	11.3	45
7	3.42	37.07	13.00	8.00	10.8	41
8	3.57	35.76	14.10	9.10	10.0	44
9	3.43	37.99	17.25	5.40	11.9	46
10	3.55	28.52	12.00	5.20	8.0	43
11	4.10	34.10	5.75	6.50	8.3	45
12	4.10	34.10	12.00	5.00	8.3	43
13	4.31	44.45	10.00	5.00	10.3	42
14	4.90	32.00	6.00	4.50	6.6	47
15	4.92	31.59	13.10	12.10	6.4	42
16	4.58	37.92	6.00	5.10	8.3	40
17	7.10	59.87	10.00	6.00	8.3	47
18	7.00	59.01	6.6	5.00	8.5	45
19	6.50	69.50	5.20	6.10	10.7	40
20	9.00	83.00	8.50	5.40	9.2	43
21	5.00	40.94	8.00	5.20	8.2	44
22	5.00	40.94	12.50	8.00	8.2	44
23	5.00	40.94	13.00	12.20	8.2	44
24	4.20	21.00	11.00	8.20	5.0	44
25	4.20	30.06	6.40	8.20	7.3	45
26	4.20	42.48	9.35	8.20	10.1	45

Rockwell hardness testing was carried out on all springs for the purpose of checking the finished springs. The test itself was performed on the end coils, at the places where they had been ground by an emery wheel. The load was set at 150 kg, and a diamond indenter was used. In Table 4 the Rockwell hardness is given as the average of four determinations. For the specially manufactured springs the average hardness of all observations is given, owing to their insignificant deviations.

The eccentricity of the coils, both of manufactured and of selected springs, was checked by simple application to a plate. The springs listed in Table 4 had a gap between individual coils and the plate of not more than 0.5 mm.

The manufactured and selected springs had a bearing surface amounting to 0.6 to 0.75 of the annular area (Fig. 6). To eliminate the influence of imperfect contact of the spring with the table, end caps were made. In these caps the springs were set strictly along the axis by means of guide rods, and the bearing surfaces were at the same time checked for parallelism. In such a co-

the ends of the spring were cast with lead in caps. The manufactured and selected springs, the principal dimensions of which are given in Table 4, with cast ends are shown in part in Fig. 7.

Fig. 7.

The bending along its axis, which always occurs when springs are compressed, was eliminated in the springs tested by a small number of coils.

The number of complete coils was determined by simple counting, and the number of incomplete ones by measuring with a protractor the protruding part of the coil from the cap.

To establish the degree of accuracy in determining the axial deformation of a spring, let us analyze Reuleaux’s formula, as the more convenient one. According to this formula, as is known:

\[ f=\frac{8\cdot n\cdot P\cdot D^3}{G\cdot d^4} \]

Let us determine the relative errors of the quantities entering into the formula.

Bearing in mind loading by weights, then

\[ \frac{\Delta P}{F}=0.01 \]

On the basis of the measurements,

\[ \frac{\Delta D}{D}=0.0007 \]

\[ \frac{\Delta d}{d}=0.006 \]

We take:

\[ \frac{\Delta n}{n}=0.01;\quad \frac{\Delta G}{G}=0.01 \]

The relative error \(\frac{\Delta f}{f}\) will be determined as follows:

\[ \frac{\Delta f}{f} = \frac{\Delta n}{n} + \frac{\Delta P}{P} + 3\frac{\Delta D}{D} + \frac{\Delta G}{G} + 4\frac{\Delta d}{d} \]

Substituting the values, we obtain:

\[ 100\frac{\Delta f}{f} = 100(0.01+3.0.0007+0.01+0.01+4.006) = 5.6\%. \]

The obtained result for the relative error of the axial deformation of the spring, with modestly chosen values of the quantities entering into Rayleigh’s formula, requires a measurement accuracy of \(1/20\) mm.

Proceeding from the established necessary accuracy of reading, we found it possible to use the following accessible and sufficiently accurate method. \(^{1}\) For measuring the deformation by this method only one sighting tube is required. It is installed directly at the place where the deformation is measured (Fig. 8). The optical axis \(AB\), intersecting at the point \(O\) the horizontal axis of rotation of the tube, passes through the point \(B\), whose displacement we shall measure. The tube is connected with the spring being tested by means of a special coupling \(D\), fitted firmly onto the objective of the tube, and by a bracket \(Q\), fastened to the spring cap by clamping screws. The displacement of the spring carries along the objective of the tube; as a result, the optical axis of the tube passes from the position \(OC\) to the position \(OC_{1}\). If \(CC_{1}\) is a millimeter scale, then the axial deformation of the spring will be determined as follows:

\[ \overline{BB_{1}} = \frac{\overline{CC_{1}}\cdot \overline{OB}}{\overline{CO}} \]

In our case \(\overline{OB}=8\) cm; \(\overline{CO}=708\) cm, whence:

\[ \overline{BB_{1}}=0.0113\,\overline{CC_{1}} \]

Thus, with full possibility of making a reading on the scale to \(0.5\) mm, the reading of the deformation will be \(0.00565\) mm, which considerably exceeds the necessary degree of accuracy.

\(^{1}\) M. Shulgin. Zh. T. F., vol. II, nos. 9–10, p. 1089.

The spring being tested was fastened at its lower end to a rigid crosspiece \(G\) (Fig. 8). Its upper cap, fixed in the bracket \(Q\), was connected by means of the nut \(X\) to the rod \(Y\). The rod passes through a long guide hole in the crosspiece \(G\). At the point where the guide hole is traversed, the rod and the hole itself were well finished and, during the test, lubricated. As a result of this arrangement, the bracket \(Q\), and consequently

Fig. 8.

the spring as well, could rotate freely during deformation about the longitudinal axis, since the load was transmitted to the free end of the rod by weights and remained suspended.

For the simultaneous measurement of the axial deformation and the angle of twist of the spring, the tube on the rod \(H\) was made rotatable by means of ball bearings \(L\) and a well-made guide tube. With the aid of rollers and weights, the tube during the experiment was pressed against the single-sided prisms of the driver of the bracket \(Q\), in the vertical and horizontal directions. The tube was so balanced that the forces required to move the tube in the two directions did not exceed 30 g. Replacing the scale with a sheet of millimeter paper, we simultaneously determined the axial and angular deformation of the spring.

The angle of untwisting \(\varphi\) was determined from the following relation according to Fig. 8,

\[ \overline{NF}\cdot \varphi_1 \simeq \overline{MF} \simeq \overline{EF}\cdot \varphi, \]

whence we obtain:

\[ \varphi=\frac{\overline{NF_1}\cdot \overline{K_1K}}{\overline{EF}\,(FK+\overline{NF})}. \]

In our case we had:

\[ \overline{NF}=12.2\text{ cm};\qquad \overline{EF}=7\text{ cm};\qquad \overline{KF}=700\text{ cm}. \]

Then

\[ \varphi=0.00244\,\overline{KK_1}. \]

When reading on the scale in units of \(0.5\) mm, the angular deformation will be expressed as \(0.00122\,\overline{KK_1}\).

The displacement of the pointer of the bracket \(Q\) and of the rod of the coupling \(D\) does not take place along straight lines, as has been assumed, but along arcs. The error resulting from this will be insignificant. Indeed, when reading on the scale for one load of \(200\) mm, the angle will not exceed \(2^\circ\). And \(\tg 2^\circ=0.03492\), whereas the arc of a circle, for a radius equal to 1, is \(0.03491\). This indicates that the error under such an assumption will be negligibly small.

Below is given the observation log for the deformation of spring No. 6. The data from the observation logs were processed as follows. From the column of differences of readings on the vertical scale, the mean reading was determined. This mean reading was multiplied by their number and inserted into the above dependence between the scale reading and the actual deformation of the spring.

The mean vertical reading from the log given is equal to 130. The number of readings is equal to 20. Then \(f=0.0113\cdot 20\cdot 130=29.4\) mm.

The mean horizontal reading is equal to \(0.85\) mm.

Then \(\varphi=0.00244\cdot 20\cdot 0.85=0.0415\).

Observation log for the deformation of the spring.

No. of experiment	Load in kg	Increase of load in kg	Reading on the vertical scale	Difference of readings on the vertical scale	Reading on the horizontal scale	Difference of readings on the horizontal scale
0	0	0	630.0	0	125.0
1	0.31	0.31	500.0	130.0	123.5	1.5
2	0.62	—	368.0	132.0	122.5	1.0
3	0.93		230.0	138.0	121.5	1.0
4	1.24		103.0	127.0	120.0	1.0
			657.0		140.0
5	1.55		529.0	128.0	139.5	0.5
6	1.86		400.0	129.0	139.0	0.5

Test No.	Load, kg	Load increment, kg	Reading on the vertical scale	Difference of readings on the vertical scale	Reading on the horizontal scale	Difference of readings on the horizontal scale
7	2.17		268.0	132.0	137.5	1.0
8	1.48		130.0	138.0	136.5	1.0
			650.0		133.5
9	2.79		525.0	125.0	133.0	0.5
10	3.10		397.0	128.0	132.0	1.0
11	3.51		263.0	134.0	131.5	0.5
12	3.72		130.0	133.0	130.5	1.0
			662.0		143.0
13	4.03		527.0	135.0	142.5	0.5
14	4.34		400.0	127.0	141.0	1.5
15	4.65		270.0	130.0	140.0	1.0
16	4.96		135.0	135.0	139.0	1.0
			642.0		138.0
17	5.27		516.0	126.0	147.0	1.0
18	5.58		379.0	137.0	146.5	0.5
19	5.89		250.0	129.0	146.0	0.5
20	6.20		127.0	123.0	145.5	0.5

In the case where the horizontal readings, owing to insignificant loads, were close to 0.5 mm, they were recorded not for each loading, but for several—3, 4, and 5.

The data processed in this way are placed in Table No. 5, respectively in the columns: “experimental data for \(M = O\)” and “experimental data for the angle of untwisting \(\varphi\).”

The experimental setup for the restrained compression of springs was arranged according to the same basic scheme shown in Fig. 8. Only in this case no horizontal reading was taken. The untwisting of the spring relative to the longitudinal axis was eliminated by the fact that the bracket \(Q\) was rigidly connected to a bar freely moving vertically along two guide rods, installed on the crosspiece \(G\) at a distance of 30 cm from one another. The guide rods, as well as the holes of the bar, which were of sufficient length, were well finished and were lubricated at the moment of testing.

The processing of the experimental data was carried out in the same way as for the case of tests of free compression of springs. The results of the experimental data are placed in Table No. 5, in the column “experimental data for \(M \ne O\).”

Table No. 5 also contains the results of the theoretical determination of the deformation by the formulas: in the case of restrained compression (6), in the case of free compression (4a), and for the angle of untwisting (3). The calculation was performed for the indicated load \(P\), with the adoption of \(G = 0.385\,E\). With the average

\[ E = 21700\,\frac{kg}{mm^2}; \qquad G = 8350\,\frac{kg}{mm^2}. \]

In the same table are written out the expres-

Table No. 5

Spring No.	Load \(P\), kg	Experimental, at \(M=0\), mm	Calculation according to formula 4-a, mm	Difference, %	Experimental, at \(M\ne 0\), mm	Calculation according to formula 6, mm	Difference, %	Experimental for \(\varphi\), rad.	Calculation according to formula 3, rad.	Difference, %
1	5.20	8.32	8.72	+4.8	8.29	8.63	+5.4	0.0220	0.0221	+0.4
2	8.15	15.70	15.79	+0.56	15.63	15.78	+0.95	0.0306	0.0183	−7.5
3	8.04	16.38	16.00	−2.3	16.97	16.98	0.0	0.0996	0.0405	+2.3
4	6.14	17.03	17.54	+3.0	16.90	17.523	+3.5	0.0290	0.0296	+2.0
5	9.00	10.53	10.06	+4.07	11.00	10.95	−0.45	0.0194	0.0192	−1.0
6	6.20	29.40	28.97	−1.5	28.70	28.93	+0.8	0.0415	0.0409	−1.4
7	6.50	30.00	30.28	+0.93	29.06	30.25	+4.0	0.0440	0.0431	−2.0
8	10.00	38.75	38.05	−1.85	38.30	38.00	−0.79	0.0640	0.0610	−4.7
9	6.40	33.00	32.42	−1.75	31.96	32.40	+1.4	0.0412	0.0413	+0.24
10	13.50	26.03	24.96	−4.10	26.00	24.93	−4.1	0.0408	0.0440	+7.8
11	15.80	13.07	12.55	−3.23	12.00	12.53	+4.4	0.0230	0.0215	−6.5
12	14.60	24.20	24.21	−0.00	24.40	24.20	−0.82	0.0321	0.0321	0.0
13	10.00	25.30	25.03	−1.10	25.02	25.00	0.0	0.0243	0.0240	−1.2
14	28.00	10.00	10.14	+1.4	10.30	10.12	−1.8	0.0143	0.0147	−2.8
15	29.40	20.02	21.02	+5.0	20.46	21.00	+2.6	0.0790	0.0796	+0.76
16	23.80	17.40	17.47	+0.4	17.20	17.46	+1.7	0.0250	0.0243	−2.8
17	47.00	37.50	37.15	−0.93	37.22	37.11	−0.3	0.0267	0.0258	−3.4
18	40.00	24.05	23.02	−3.3	23.46	23.00	−2.0	0.0149	0.0154	+3.3
19	21.00	29.03	20.25	+1.1	22.23	20.20	−0.1	0.0140	0.0143	+2.1
20	50.00	36.37	36.38	0.0	36.00	36.31	+0.86	0.0203	0.0218	+7.4
21	20.80	18.00	18.23	+1.3	17.50	18.19	+3.9	0.0256	0.0237	−7.5
22	20.80	28.10	27.65	−1.6	27.90	27.60	−1.07	0.0443	0.0391	−11.7
23	20.80	31.00	29.87	−3.55	30.46	29.79	−2.2	0.0705	0.0673	−5.3
24	20.80	6.76	7.05	+5.00	6.93	7.02	+1.8	0.0160	0.0154	+2.5
25	20.80	10.67	11.23	+5.25	10.40	11.20	−7.7	0.0173	0.0175	+1.7
26	10.00	22.50	22.66	+0.74	22.30	22.64	+1.4	0.0230	0.0250	+8.7

… expressed in percent, the differences between the magnitudes of the theoretical deformation and that obtained from experiment are computed in relation to the latter.

From consideration of the data in Table 5 it can be seen that, for the majority of the springs tested, there is good agreement between the calculated deformations and the experimental ones. For the case of free compression, the difference for a large number of springs does not exceed 5%, either in the positive or in the negative direction. The same situation is also observed in the case of constrained compression of the spring. Somewhat larger deviations were obtained for the angle of unwinding of the spring.

The deviations obtained of the experimental deformations from the calculated ones may be said to be insignificant, if one takes into account the relative errors in the calculated deformations due to the inaccurate measurement and determination of the quantities entering into the formulas. In addition, it is necessary to note that, in the case of free compression of the spring, the experimental data agree better with the theoretical data than in the case of constrained compression. As is evident, however, there is no sharp difference in the magnitudes of the experimental deformations between these two cases. The presence

the discrepancies obtained from all the formulas cannot be attributed to insufficient accuracy of the formulas we have obtained. This discrepancy is, rather, the result of a combination of causes: the quality of the spring-wire material; the ratios of the principal dimensions of the springs, under which the assumptions made in deriving the calculation formulas have a more significant influence; the conditions of fastening, etc. From all that has been set forth here, and also on the basis of the preceding comparison, it is natural to conclude that the formulas we have derived give the best and a good convergence of the calculated deformations with the experimental ones.

8. Testing of springs at elevated temperature.

A whole series of machines, or complex parts, whose action is inseparably connected with the use of springs, operate under conditions of high and elevated temperatures. It is obvious that here the behavior of the spring material under stress will be different than at normal room temperature. Therefore, deformation tests at normal temperature, for those springs which operate under high-temperature conditions, are insufficient. Consequently, in order to judge correctly the relation between the force and the deformation of a spring at high temperatures, it is necessary to carry out a test. Guided by these considerations, and by the absence in the literature of sufficient data on the very complex change in the elastic properties of metals and the measurement of spring deformation at high temperatures, we give below a description and the results of tests of 16 springs.

The springs tested were heated in an electric resistance furnace (Fig. 9). The temperature was measured by a calibrated thermocouple \(d\). The measurement of deformations was carried out in the same way as in the preceding test.

Fig. 9.

The springs subjected to testing were those which, in the preceding test, had shown good agreement of the experimental deformations with the calculated ones. Deformation measurements were made at steady temperatures of \(50^\circ\text{C}\), and then at intervals of \(50^\circ\text{C}\) up to \(250^\circ\text{C}\). The temperature increase lasted 20 min.

The experimental data obtained were processed by the method given above, and their results are presented in Table No. 6.

Comparison of the experimental results shows that when the temperature is raised to \(50^\circ\text{C}\), the elastic properties of the spring change both in the direction of increase and in the direction of decrease. This occurs, evidently, owing to a change in the internal stresses. A further increase in temperature continuously lowers the elastic

Table No. 6

Specimen No.	Load in kg	Deflection at 15–20° C	Deflection at 50° C	Difference in %	Deflection at 200° C	Difference in %	Deflection at 150° C	Difference in %	Deflection at 100° C	Difference in %	Deflection at 250° C	Difference in %
2	8.15	15.70	15.73	+0.2	15.80	+0.6	16.00	+1.9	16.48	+4.9	17.00	+8.3
3	8.04	16.38	16.40	+0.12	16.70	+1.9	16.93	+4.0	17.10	+4.5	17.30	+5.6
6	6.20	29.40	29.00	−1.4	29.40	0.0	30.10	+2.4	30.56	+4.0	30.98	+5.4
7	6.50	30.00	30.30	+1.0	30.40	+1.33	30.96	+3.2	31.12	+3.7	31.42	+4.7
8	10.00	38.75	38.80	+0.13	39.00	+0.64	39.55	+2.0	40.09	+3.2	40.64	+4.0
9	6.40	33.00	33.00	+0.0	33.40	+1.20	33.78	+2.36	34.08	+3.3	34.20	+3.6
12	14.60	24.20	24.50	+1.2	24.60	+1.65	24.83	+2.6	25.00	+3.3	25.30	+4.5
13	10.00	25.30	25.25	−0.2	25.80	+2.0	25.96	+2.6	26.10	+3.2	26.94	+6.5
14	28.00	10.00	10.60	+6.0	10.80	+8.0	10.86	+8.6	11.50	+15.0	—	—
16	23.80	17.40	17.50	+0.6	17.60	+1.15	18.00	+3.44	18.20	+4.6	18.43	+5.9
17	47.00	37.50	36.80	−1.9	37.50	0.0	38.73	+3.30	38.80	+3.5	39.00	+4.0
19	21.00	20.03	21.00	+4.8	21.10	+5.5	21.54	+7.7	21.80	+9.0	—	—
20	50.00	36.37	38.00	+1.8	38.20	+2.30	38.40	+2.8	38.60	+6.15	38.84	+6.8
21	20.80	18.00	18.20	+1.1	18.50	+2.67	18.70	+3.9	18.84	+4.7	19.00	+5.6
22	20.80	28.10	27.80	−1.06	28.36	+0.93	28.56	+1.6	28.74	+2.18	28.80	+6.0
26	10.00	22.50	22.70	+0.9	22.85	+1.55	22.96	+2.0	23.10	+2.66	23.96	+6.55

properties of the spring. The decrease in the elastic properties of the spring at a temperature of 150° C is so insignificant that it indicates the full possibility, when calculating springs operating in this temperature range, of neglecting the influence of the latter. When springs operate at an ambient temperature of 150°–250° C, with accurate determination of the principal dimensions and stiffness of the springs and with the use of the formulas we have derived, it is necessary to take into account the decrease in the elastic properties of the spring. Changes in the elastic properties of the spring at higher temperatures require further investigation with greater necessity.

General Conclusions

1. Formula (6) has been obtained for determining the axial deformation of a spring, taking into account the influence of all components of the external forces.

2. Formula (4a) has been obtained for determining the axial deformation of a spring under free extension (compression); according both to our experimental data and to the experimental data given in Table No. 1, it provides the best agreement between the calculated deformations and the experimental ones.

3. Formula (3) has been obtained for determining the angular deformation of a spring and has been verified experimentally.

4. It has been established, by both theoretical and experimental investigation, that the influence of the moment of restraint of the ends on the axial deformation of springs used in practice is very small and amounts to tenths of a percent.

5. Tests of springs carried out at elevated temperature have established that the elastic properties of the spring from \(15^\circ\mathrm{C}\) to \(150^\circ\mathrm{C}\) remain practically unchanged.

For reviewing the manuscripts I express my deep gratitude to Prof. G. V. Trapeznikov.

1. Ing. Arch. Bd. 1, 1930, S. 619. ↩↩↩

2. Ibid., Bd. 2, 1931, S. 381. ↩↩↩

3. T. E. T. 18, p. 428. ↩