Proceedings of the Tomsk Technological Institute of Emperor Nicholas II.

Vol. XXVI, 1912, No. 2.

Theory of Elastic Suspension on Leaf Springs with Shackles.

The most commonly used method in Europe of elastic suspension for railway rolling stock is the method of suspension by means of leaf springs, the middle part of which rests on axleboxes, while the ends, by means of hinged links, usually called shackles, are connected to the frame of the vehicle. In this arrangement the shackles are given various inclinations and arbitrary lengths; thus, for example, on locomotives, generally speaking, comparatively long vertical shackles are used; on freight cars, on the contrary, short shackles are almost always placed with an outward inclination from the spring, while in passenger cars one may encounter not only both of the indicated types, but also shackles with an inward inclination.

From the standpoint of the smoothness of the vehicle’s motion, it is not immaterial what inclination and what length are assigned to the shackles, which, just like the dimensions of the spring, have a substantial influence on the action of the suspension.

The influence of the inclination of the shackles on the action of spring suspension was first noted by the engineer Féraud, who proposed a new suspension system with internal shackles. Nevertheless, the question of the advantages of one or another method of suspension on leaf springs has still not been clarified theoretically with sufficient completeness.

Usually, in calculating spring suspension, no attention is paid to the influence of the inclination and length of the shackles on the flexibility of the spring, and only a correction is introduced for the influence of the inclination of the shackles on the magnitude of the stress in the leaves of the spring.

Thus, not only is there no sufficient criterion, founded on theoretical considerations, concerning the various systems of suspension on leaf springs, but the calculation of spring suspension itself is far from satisfying all the necessary requirements.

When calculating the deflection of a spring under load by formulas in which the influence of the shackles on the flexibility …

springs, results are obtained that do not always correspond to the actual deflection of the spring—a fact very often observed in practice.

In view of all this, the author set himself the aim of filling the above-mentioned gap and of determining theoretically the dependence of the deflection, vertical flexibility, and oscillations of a system of spring suspension, consisting of a leaf spring and the shackles, on the elements of this system and on the external forces acting upon it.

In a leaf spring connected with shackles, the vertical components \(P\) of the forces applied to its ends and bending the spring are not equal to the vertical forces of the whole carriage \(Q\), acting on the spring brackets (see Fig. 1, in which the case of outer shackles is represented by solid lines, and the case of inner shackles by dotted lines).

If we denote the angle of inclination of the chord \(AO\) of half the axis of the main leaf of the spring to the horizontal line by \(\alpha\), and the inclination of the shackle axis with respect to the vertical by \(\beta\), then the component \(S\) of the weight of the carriage, directed along the shackle, will be equal to

\[ S=\frac{Q}{\cos\beta}, \]

while the component \(T\), directed along the chord \(AO\), if the second component \(P\) is vertical, will be

\[ T=S\frac{\sin\beta}{\cos\alpha}=Q\frac{\tg\beta}{\cos\alpha}. \]

Taking the sum of the moments of the forces applied to the end of the spring with respect to the point \(O\), coinciding with the mean axis of the main leaf of the spring, we obtain the equation:

\[ P\,\overline{AO}\cos\alpha=\overline{AO}\cos(\beta-\alpha)\frac{\cos\beta}{Q}, \]

in which the positive values of the angles \(\alpha\) and \(\beta\) are measured for the left half of the spring (Fig. 1) in the direction of clockwise motion, and for the right half in the direction opposite to the motion of the clock hand. From this equation we determine the magnitude \(P\):

\[ P=Q\,\frac{\cos\alpha\cos\beta+\sin\alpha\sin\beta}{\cos\alpha\cos\beta} =Q(1+\tg\alpha\tg\beta) \tag{1} \]

Thus, the ratio of \(P\) to \(Q\) depends on the angle of inclination of the axis of the shackle and on the angle of inclination of the chord of half the axis of the main leaf of the spring.

It is not difficult, on the basis of equation (1), to observe that \(P=Q\) when \(\alpha=0\) or \(\beta=0\), i.e., either when the shackle is in a vertical position, or when the spring is completely straightened. In all other cases \(P>Q\) when the signs of \(\operatorname{tg}\alpha\) and \(\operatorname{tg}\beta\) are the same, and \(P<Q\) when their signs are opposite. In other words, with external shackles (i.e., when the horizontal projections of the axes of the shackles do not coincide with a part of the horizontal projection of the axis of the spring, but lie on its continuation), before it is completely straightened \(P>Q\), and after it is straightened \(P<Q\); with internal shackles (i.e., when the horizontal projections of their axes coincide with a part of the horizontal projection of the axis of the spring), before the spring is straightened \(P<Q\), and after it is straightened \(P>Q\).

Thus, one can foresee in advance that the settlement of some carriage on its springs, if the latter are connected to its frame by means of inclined shackles, will not be proportional to the increase of its load; consequently, the flexibility of the spring suspension (usually defined as a constant quantity depending only on the dimensions of the spring: the ratio of the deflection of the spring to its load) will be a variable quantity, depending on the change of load and on the inclination of the shackles, and therefore will be expressed by the ratio

\[ \frac{\Delta S}{\Delta Q}, \]

i.e., by the ratio of the infinitely small vertical displacement of the carriage frame to the corresponding infinitely small increase of the load falling on the end of the shackle.

The component \(T\), directed along the chord \(AO\), will also exert some influence on the deflection of the spring; but the influence of \(T\) on the deflection of the spring will be a small quantity of higher order in comparison with the influence of those forces which we have taken, just as the influence of the arm of action of the force \(P\) bending the spring, which is not taken into account in the basic formulas used for calculating springs, is likewise not taken into account.

These formulas are derived under the assumption of small values of the deflections of springs; thus, using for our analysis the basic formulas derived by Phillips*), we shall henceforth remain within the limits of accuracy of these formulas.

To determine the analytical dependence between \(P\) and \(Q\), it is necessary to know the law of variation of \(\operatorname{tg}\alpha\) and \(\operatorname{tg}\beta\) as a function of

*) Mémoire sur les ressorts en acier. M. Phillips.

the length and deflection of the spring, and also on the length and position of the shackles; therefore it is necessary first of all to determine the curve along which the end of the axis of the main leaf of the spring moves when it is straightened.

It is usually assumed that, when the spring is straightened, its ends describe arcs of a circle whose center coincides with the middle of the axis of the main leaf. Such an assumption, generally speaking, does not correspond to reality, since the axis of the main leaf, which when the spring is fully straightened is a straight line, in other positions will represent a certain curve, the form of which depends on the construction of the spring.

We shall consider the question under discussion with respect to a triangular leaf spring (otherwise called a full leaf spring), in which the thickness of all the leaves is the same, the magnitude of the steps is the same, and the ends of the leaves have the correct form determined by theory. We shall assume here that the axis of the main leaf of the spring is bent along the arc of a circle, as is always done in practice, so that the deflection of the spring \(y\) when it is straightened is determined by the formula \(y=\dfrac{6PL^3}{ibEh^3}\), where \(i\), \(b\), and \(h\) are the number, width, and thickness of the leaves of the spring, \(L\) is half the length of the main leaf of the spring, \(P\) is the load on each end, and \(E\) is the coefficient of elasticity of the metal of the spring. When a triangular leaf spring is straightened, as the theory of springs shows, the curvature of the axis of the main leaf is the same at all its points, i.e., the axis of the main leaf always represents an arc of a circle.

Let us denote by \(2L\) and \(R\) the length and radius of curvature of the main leaf of the spring, by \(m\) the length of the shackle, and by \(2l\) the distance between the centers of the eyes of the brackets by means of which the ends of the shackles are connected with the frame of the carriage (Fig. 2). Let the point \(O\), coinciding with the middle of the axis of the main leaf, be taken as the origin of coordinates, and let the coordinate axes be directed along the straight lines \(OX\) and \(OY\), of which the first is tangent to the axis of the main leaf at the point \(O\), and the second passes through its center of curvature.

With the adopted notation we may write:

\[ \operatorname{tg}\alpha=\frac{AD}{OD}=\frac{y}{x}, \]

but since the central angle \(OYA=2\alpha\) and since the length of the arc \(OA=L\), we have

\[ \alpha=\frac{L}{2R}\quad \text{and}\quad y=x\,\operatorname{tg}\frac{L}{2R} \tag{2} \]

In turn, \(R\) depends on \(x\) and \(y\), namely

\[ (2R-y)y=x^{2},\quad \text{whence}\quad 2R=\frac{x^{2}+y^{2}}{y}. \]

Consequently, the equation of the curve along which, when a triangular leaf spring is straightened, the end point of the axis of the root leaf moves, will have the following form:

\[ \frac{Ly}{x^{2}+y^{2}}=\operatorname{arctg}\frac{y}{x}. \tag{3} \]

We shall show that this curve, with a sufficient degree of accuracy for the purposes of the present investigation, may be replaced by an arc of a circle described from some center \(F\) located on the axis of abscissas.

Let the distance of the point \(F\) along the axis of abscissas from the origin of coordinates be equal to \(x\). If the curved trajectory of the end of the spring can be replaced, with a sufficient degree of accuracy, by an arc of a circle, then the ratio of the segments \(FA\) and \(FK\) must be equal to \(1\)—also with a sufficient degree of accuracy—as the ratio of the radii of curvature vectors of the curve nearly coinciding with the arc of a circle.

We form the equation expressing this condition, obtaining

\[ L-x=\sqrt{AD^{2}+(OD-x)^{2}}. \]

But since \(AD=R\left(1-\cos\frac{L}{R}\right)\) and \(OD=R\sin\frac{L}{R}\), the equation, after substitution of these values, will take the following form:

\[ L-x=R\sqrt{\left(1-\cos\frac{L}{R}\right)^{2}+\left(\sin\frac{L}{R}-\frac{x}{R}\right)^{2}}. \tag{4} \]

Squaring both sides of this equation, we obtain the value of \(\frac{x}{R}\):

\[ \frac{x}{R}= \frac{2\left(1-\cos\frac{L}{R}\right)-\frac{L^{2}}{R^{2}}} {2\left(\sin\frac{L}{R}-\frac{L}{R}\right)}. \]

We expand $\cos\dfrac{L}{R}$ and $\sin\dfrac{L}{R}$ in series and substitute their values into the expression found for $\dfrac{x}{R}$; we obtain

\[ \frac{x}{R}= \frac{ 2\left(1-1+\dfrac{L^2}{2\cdot R^2}-\dfrac{L^4}{2\cdot3\cdot4\cdot R^4} +\dfrac{L^6}{2\cdot3\cdot4\cdot5\cdot6\cdot R^6}-\cdots\right)-\dfrac{L^2}{R^2} }{ 2\left(\dfrac{L}{R}-\dfrac{L^3}{2\cdot3\cdot R^3} +\dfrac{L^5}{2\cdot3\cdot4\cdot5\cdot R^5}-\cdots-\dfrac{L}{R}\right) }, \]

whence

\[ \frac{x}{R}= \frac{ -\dfrac{1}{12}\dfrac{L^4}{R^4} +\dfrac{1}{360}\dfrac{L^6}{R^6} -\dfrac{1}{20160}\dfrac{L^8}{R^8} +\cdots }{ -\dfrac{1}{3}\dfrac{L^3}{R^3} +\dfrac{1}{60}\dfrac{L^5}{R^5} -\dfrac{1}{2520}\dfrac{L^7}{R^7} +\cdots }. \]

Dividing the numerator of the obtained expression by the denominator, we get:

\[ \frac{x}{R} = \frac{1}{4}\frac{L}{R} +\frac{1}{240}\frac{L^3}{R^3} +\frac{1}{16800}\frac{L^5}{R^5} +\frac{1}{1008000}\frac{L^7}{R^7} +\cdots . \tag{5} \]

Considering the expression obtained, one may conclude that, if $\dfrac{L}{R}$ is a proper fraction, then the value of $\dfrac{x}{L}$ will be less than the sum $S$ of the terms of the infinitely decreasing geometric progression:

\[ \frac{1}{4}\frac{L}{R},\quad \frac{1}{240}\frac{L^3}{R^3},\quad \frac{1}{14400}\frac{L^5}{R^5},\quad \frac{1}{864000}\frac{L^7}{R^7},\ldots \]

and at the same time will be greater than $\dfrac{1}{4}\dfrac{L}{R}$.

Computing the value of $S$, we obtain

\[ S= \frac{\dfrac{1}{4}\dfrac{L}{R}} {1-\dfrac{1}{60}\dfrac{L^2}{R^2}}. \]

Thus, if we take $x=\dfrac{1}{4}L$, then we shall make an error whose magnitude will be no greater than

\[ \frac{\dfrac{1}{4}\dfrac{L}{R}} {1-\dfrac{1}{60}\dfrac{L^3}{R^3}} -\frac{1}{4}\frac{L}{R}, \]

that is, the error will be less than

\[ \frac{\dfrac{1}{240}-\dfrac{L^3}{R^3}}{1-\dfrac{1}{60}\dfrac{L^2}{R^2}}. \tag{6} \]

If the value \(\dfrac{L}{R}=1\), then, in determining the position of the center \(F\), we obtain an error of less than \(\dfrac{1}{236}\), taking as the abscissa of the center \(x=\dfrac{1}{4}L\); for values \(\dfrac{L}{R}<1\) the error will be still more insignificant. As regards the maximum deflections of springs encountered in practice, their limits, generally speaking, very rarely exceed those values for which \(L=R\).

Thus, with more than sufficient accuracy for calculating spring suspension, we may replace the curve described by the end of a deflecting triangular leaf spring by an arc of a circle whose center is determined by the coordinates (Fig. 2):

\[ x=\frac{1}{4}L \quad \text{and} \quad y=0 \tag{7} \]

and whose radius is \(r=\dfrac{3}{4}L\).

If the spring suspension were arranged on an incomplete leaf spring, i.e., on a spring for which the half-length of the shortest leaf is not equal to the length of the step, and the spring itself contains one or several leaves of the same length as the first, then determining the trajectory of the end of the spring analytically would present a very complicated problem. But experimentally there is no difficulty in determining this trajectory; it is only necessary, while producing the deflection of the spring, to determine, together with the deflection arrow \(y\), also the distance between the axes of the spring ears—\(2x\); the values \(x\) and \(y\) will be the coordinates of the trajectory of the end of the spring in the coordinate-axis system we have adopted (Fig. 2).

Expressing the values \(\operatorname{tg}\alpha\) and \(\operatorname{tg}\beta\) through the coordinates of the curve along which the end of the spring moves when it is straightened, and through the data relating to the spring suspension: the length of the shackle \(m\) and the distance between the centers of the bracket hinges—\(2l\), we obtain the values:

\[ \operatorname{tg}\alpha=\frac{y}{x-\dfrac{1}{4}L}, \]

a quantity depending only on the elements of the spring and

\[ \operatorname{tg}\beta = \frac{l-\mathrm{L}+\frac{3}{4}\mathrm{L}-x} {\sqrt{m^2-\left(l-\mathrm{L}+\frac{3}{4}\mathrm{L}-x\right)^2}}, \]

a quantity depending, in addition, on the dimensions of the suspensions and their arrangement.

If we denote the difference \(l-\mathrm{L}\) by \(n\), then we obtain

\[ \operatorname{tg}\alpha \operatorname{tg}\beta = \frac{ \dfrac{y}{\mathrm{L}}\left(\dfrac{3}{4}-\dfrac{x}{\mathrm{L}}+\dfrac{n}{\mathrm{L}}\right) }{ \left(\dfrac{x}{\mathrm{L}}+\dfrac{1}{4}\right) \sqrt{ \left(\dfrac{m}{\mathrm{L}}\right)^2 - \left(\dfrac{3}{4}-\dfrac{x}{\mathrm{L}}+\dfrac{n}{\mathrm{L}}\right)^2 } }, \tag{8} \]

in which the values of \(n\) are positive for suspension from the outer suspensions and negative for suspension from the inner suspensions.

Let us determine the quantity \(\dfrac{x}{\mathrm{L}}\) from the equation of the circle by which the curve of displacement of the end of the spring may be replaced:

\[ x^2+y^2-\frac{9}{16}\mathrm{L}^2=0. \]

Discarding terms with powers of \(\dfrac{y}{\mathrm{L}}\) higher than the third order, we obtain:

\[ \frac{x}{\mathrm{L}}=\frac{3}{4}-\frac{2}{3}\left(\frac{y}{\mathrm{L}}\right)^2 \]

After substituting this quantity into equation (8), we obtain:

\[ \operatorname{tg}\alpha \operatorname{tg}\beta = \frac{ \dfrac{y}{\mathrm{L}} \left[ \dfrac{n}{\mathrm{L}}+\dfrac{2}{3}\left(\dfrac{y}{\mathrm{L}}\right)^2 \right] }{ \left[ 1-\dfrac{2}{3}\left(\dfrac{y}{\mathrm{L}}\right)^2 \right] \sqrt{ \left(\dfrac{m}{\mathrm{L}}\right)^2 - \left[ \dfrac{n}{\mathrm{L}}+\dfrac{2}{3}\left(\dfrac{y}{\mathrm{L}}\right)^2 \right]^2 } }. \]

Consequently, to determine the relationships between \(P\) and \(Q\), we have the following expression:

\[ P = Q \left\{ 1 + \frac{\dfrac{y}{L}\left[\dfrac{n}{L}+\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2\right]} {\left[1-\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2\right] \sqrt{\left(\dfrac{m}{L}\right)^2- \left[\dfrac{n}{L}+\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2\right]^2}} \right\} \tag{9} \]

The dependence between the vertical load on the end of the spring \(P\), its deflection ordinate \(y\), the dimensions of the spring \(L, i, b, h\), and the ordinate of the initial camber \(y_0\) is expressed as follows:

\[ y_0 - y = \frac{6PL^3}{Eibh^3} \tag{10} \]

Substituting into equation (9) the value of \(P\) determined from the last expression, we obtain an equation relating two variable quantities \(Q\) and \(y\):

\[ Q = \frac{ Eibh^3\left(\dfrac{y_0}{L}-\dfrac{y}{L}\right) }{ 6L^2 \left\{ 1+ \frac{ \dfrac{y}{L}\left[\dfrac{n}{L}+\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2\right] }{ \left[1-\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2\right] \sqrt{\left(\dfrac{m}{L}\right)^2- \left[\dfrac{n}{L}+\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2\right]^2} } \right\} } \tag{11} \]

By means of this equation it is always possible, for a given \(y\), to find \(Q\), and consequently one can construct the curve of dependence between \(Q\) and \(y\), by means of which the deflection ordinate of the spring is easily determined for a given value of the weight \(Q\) falling on one end of the spring.

Examining equation (11), it may be observed that when \(y=0\) the quantities \(m\) and \(n\) have no influence on the value of \(Q\); consequently, complete straightening of the spring always occurs under one and the same load, regardless of whether the suspension is on external or internal shackles, or even without them at all. Thus, for \(y=0\) we always obtain

\[ Q_0 = P_0 = \frac{Eibh^3y_0}{6L^3} \tag{12} \]

If \(n \to 0\), then for positive values of \(y\), \(P>Q\), while for negative values of \(y\), \(P<Q\).

S. P. Gomelya.

If, however, \(n<0\), then there may be three values of \(y\) (of which two are equal in magnitude but opposite in sign) for which \(P=Q\). These values are determined from the equation:

\[ \frac{\dfrac{y}{L}\left[\dfrac{n}{L}+\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2\right]} {\left[1-\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2\right] \sqrt{\left(\dfrac{m}{L}\right)^2-\left[\dfrac{n}{L}+\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2\right]^2}} =0 \tag{13} \]

from which we obtain: \(y_1=0\); \(y_2=\sqrt{-\dfrac{2}{3}nL}\) and \(y_3=-\sqrt{-\dfrac{2}{3}nL}\),

where \(y_1\) corresponds to the complete straightening of the spring, while \(y_2\) and \(y_3\) are those magnitudes of deflection for which the axes of the spring shackles have a vertical position. As regards the relation between \(P\) and \(Q\), we have \(P>Q\) if the deflection arrow of the spring \(y\) satisfies the inequality \(y_0>y>y_2\), and also if \(y_1>y>y_3\). In all other positions, i.e., when \(y_2>y>y_1\) and when \(y<y_3\), we obtain \(Q>P\).

When the spring entering into the suspension system under consideration is an elastic beam bending according to a law other than that of a laminated triangular spring, the dependence between the magnitude of the load \(Q\) acting on the end of the shackle and the deflection arrow of the spring \(y\) can in general be expressed by the equation:

\[ Q=-\frac{\displaystyle\int_{y_0}^{y}\varphi(y)\,dy}{1+\tg\alpha\,\tg\beta} \tag{14} \]

where \(-\displaystyle\int_{y_0}^{y}\varphi(y)\,dy=P\) represents the dependence between the deflection arrow of the spring \(y\) and the magnitude of the vertical load \(P\) on its ends (when the spring is straightened without shackles), while \(\tg\alpha\) and \(\tg\beta\) are determined by taking into account the trajectory of the end of the spring.

If the suspension were arranged on an incomplete spring, then, to determine the dependence between \(P\) and \(y\), one could use the formula derived by Phillips:

\[ y_0-y=\frac{2P}{E\,i\,b\,h^3}\left[2L^3+i^2e^3\right], \tag{15} \]

in which \(e\) represents the magnitude of the stage of the spring; the other letters have their former meanings.

Theory of Suspension on Leaf Springs with Shackles

For a non-complete spring containing several leaves identical with the first, one could use the formula derived by Rey and Vallot:

\[ y_0-y=\frac{4PL^3}{Ebh^3}\left[\frac{2L^3+(L-L_1)^3}{2iL^3+i'(L-L_1)^3}\right], \tag{16} \]

in which \(L_1\) is the half-length of the prismatic part of the last leaf and \(i'\) is the number of leaves of the same length as the first (inclusive).

To determine the trajectory of the end of the spring, one may also use a graphical method, constructing, by Phillips’s method, for various values of \(P\), the exact form of the axis of the main leaf of the spring; obviously, this method is applicable to any form of spring.

If the trajectory of the end of the spring can, with a sufficient degree of accuracy, be replaced by an arc of a circle described from some point lying on the \(X\)-axis, and if the radius of this circle is \(r=(1-k)L\), then the following equation serves for determining \(Q\):

\[ Q= -\frac{\displaystyle \int_{y_0}^{y}\varphi(y)\,dy} {\displaystyle 1+ \left\{ \frac{ \displaystyle \frac{y}{L}\frac{n}{L} +\frac{1}{2(1-k)}\left(\frac{y}{L}\right)^2 }{ \displaystyle \left[ 1+\frac{1}{2(1-k)}\left(\frac{y}{L}\right)^2 \right] \sqrt{ \left(\frac{m}{L}\right)^2 - \left[ \frac{n}{L} +\frac{1}{2(1-k)}\left(\frac{y}{L}\right)^2 \right]^2 } } \right\} } \tag{17} \]

In the case where the relation between \(y\) and \(P\) is expressed graphically in the form of a diagram, and also when the trajectory of the end of the spring has been constructed, the relation between \(Q\) and \(y\) can be obtained by means of a fairly simple construction.

As an illustration, let us give this construction for a triangular leaf spring with shackles, whose axis in bending assumes, as was mentioned, the form of an arc of a circle, and whose ends, when it is straightened, describe curves that may also, with a sufficient degree of accuracy, be replaced by arcs of circles.

Let \(OX\) and \(OY\) (Fig. 3) represent the coordinate axes; from the point \(x=\frac{1}{4}L\) and \(y=0\) we describe, with radius \(\frac{3}{4}L\), an arc of a circle corresponding to the trajectory of the end of the spring during its straightening, and also draw the vertical straight line \(VW\), along which the center \(C\) of the shackle eye, connected by means of a hanger with the vehicle frame, moves.

The value \(Q\) is determined graphically on the basis of the proportion:

\[ \frac{Q}{1}=\frac{P}{1+\operatorname{tg}\alpha\,\operatorname{tg}\beta}. \]

The quantity \(1+\operatorname{tg}\alpha\,\operatorname{tg}\beta\) for any \(y\) is obtained graphically; for this purpose, from point \(A\), as from a center, we draw through point \(C\) the straight line \(VW\), along which the end of the shackle moves, with radius equal to the length of the shackle \(m\). The intersection of the straight line \(CA\) with the \(X\)-axis at point \(R\) determines the segment \(OR\), representing the quantity \(1+\operatorname{tg}\alpha\,\operatorname{tg}\beta\) on the scale on which the segment \(OD\) is equal to one unit.

Now, if on the ordinate axis we lay off the segment \(OP'\) equal to the abscissa \(yP\) and, joining the points \(P'\) and \(R\) by a straight line, draw through point \(D\) the line \(Q'D\) parallel to \(P'R\), then we obtain the segment \(OQ'\), which represents the magnitude of the vertical load \(Q\) transmitted by the vehicle frame to the end of the spring shackle at a spring deflection \(y\).

If we lay off the obtained segment \(OQ'\) along the line \(yP\) to the right from \(y\), we obtain the point \(Q\), belonging to the sought curve \(y_0QQ_0Q_1\), expressing the dependence between \(Q\) and \(y\).

For any other leaf spring for which a graphical dependence between \(P\) and \(y\) is available, the construction of the dependence between \(Q\) and \(y\) will be entirely analogous to that given above. As regards the scale for measuring the abscissas of the curve of dependence of \(Q\) and \(y\), it is determined simply if the magnitude of the load \(P_0=Q_0\) that straightens the spring is known: in this case the unit of scale will be the quantity

\[ \frac{\overline{OQ_0}}{P_0}, \]

where, for a triangular leaf spring, \(P_0\) is computed from formula (12).

The dependence between the amount of lowering of the vehicle frame and the magnitude of the load falling on the end of the spring is expressed as the algebraic sum of two quantities: the deflection of the spring and the relative displacement of the projections of the centers of the shackle hinges onto the vertical. Analytically this dependence may be represented in the following form:

\[ S=y+m\cos\beta, \]

or

\[ S=y+L_1\sqrt{\left(\frac{m}{L}\right)^2-\left[\frac{n}{L}+\frac{2}{3}\left(\frac{y}{L}\right)^2\right]^2}, \tag{18} \]

where \(S\) is the ordinate of the center of the shackle hinge connected, by means of a hanger, with the vehicle frame, when the load on the end of the sus-

the suspension is equal to \(Q\); \(y\) is the ordinate of the end of the spring axis under the same load; \(\beta\) is the angle of inclination of the suspension to the vertical corresponding to this load, and \(m\) is the length of the suspension between the centers of its hinges.

The graphical dependence between \(S\) and \(Q\) can be obtained quite simply (Fig. 3) if, downward from the ordinate \(y\) of some point of the curve expressing the dependence between \(Q\) and \(y\), one lays off the magnitude of the projection of \(m\) on the \(y\)-axis, i.e. \(m\cos\beta\). Connecting a series of points constructed in this manner by a continuous curve, we obtain the curve \(y_0 S_0 S_0 S_1\), expressing the dependence between \(S\) and \(Q\).

Knowing the dependence between the magnitude of the load at the end of the suspension and the magnitude of the lowering of the carriage frame under the influence of this load, we can determine the magnitude of the flexibility \(F\) of the suspension system as a function of the deflection arrow of the spring \(y\).

As was already mentioned, the flexibility of the system is a variable quantity and is expressed by the ratio \(F=\dfrac{ds}{dQ}\).

Let us determine the value of \(F\) for a triangular leaf spring. Differentiating equation (18), we obtain:

\[ ds=dy+d\left[ L\sqrt{\left(\frac{m}{L}\right)^2- \left[ \frac{n}{L}+\frac{2}{3}\left(\frac{y}{L}\right)^2 \right]^2} \right]. \tag{19} \]

and consequently,

\[ F=\frac{ds}{dQ} = \frac{dy}{dQ} + \frac{d}{dy} \left\{ L\sqrt{\left(\frac{m}{L}\right)^2- \left[ \frac{n}{L}+\frac{2}{3}\left(\frac{y}{L}\right)^2 \right]^2} \right\} \frac{dy}{dQ}. \]

Differentiating equation (11), in turn, we obtain:

\[ \frac{dQ}{dy} = \left| \frac{ E i b h^3\left(\frac{y_0}{L}-\frac{y}{L}\right) }{ 6L^2 \left\{ 1+ \frac{ \frac{y}{L}\left[ \frac{n}{L}+\frac{2}{3}\left(\frac{y}{L}\right)^4 \right] }{ \left[ 1-\frac{2}{3}\left(\frac{y}{a}\right)^2 \right] \sqrt{ \left(\frac{m}{L}\right)^2- \left[ \frac{2n}{3L}+\left(\frac{y}{L}\right)^2 \right]^2 } } \right\}_y } \right|'. \]

Thus, the flexibility of the suspension system on triangular leaf springs with suspensions is expressed by the following equality:

\[ \mathrm{F}=\frac{ds}{dQ}= \frac{ 1+\left\{\mathrm{L}\sqrt{\left(\frac{m}{\mathrm{L}}\right)^2-\left[\frac{n}{\mathrm{L}}+\frac{2}{3}\left(\frac{y}{\mathrm{L}}\right)^2\right]^2}\right\}'y }{ \left\{ \dfrac{ Eibh^3\left(\dfrac{y_0}{\mathrm{L}}-\dfrac{y}{\mathrm{L}}\right) }{ 6\mathrm{L}^2\left\{ 1+\dfrac{ \dfrac{y}{\mathrm{L}}\left[\dfrac{n}{\mathrm{L}}+\dfrac{2}{3}\left(\dfrac{y}{\mathrm{L}}\right)^2\right] }{ \left[1-\dfrac{2}{3}\left(\dfrac{y}{\mathrm{L}}\right)^2\right] \sqrt{\left(\dfrac{m}{\mathrm{L}}\right)^2-\left[\dfrac{n}{\mathrm{L}}+\dfrac{2}{3}\left(\dfrac{y}{\mathrm{L}}\right)^2\right]^2} } \right\}} \right\}'y } \tag{20} \]

If we denote the flexibility of the spring \(\dfrac{y}{P}\) by \(f\), and, moreover,

\[ \left. \begin{aligned} y_0-y &\quad \text{by } A,\\[6pt] \dfrac{\dfrac{y}{\mathrm{L}}\left[\dfrac{n}{\mathrm{L}}+\dfrac{2}{3}\left(\dfrac{y}{\mathrm{L}}\right)^2\right]} {1-\dfrac{2y}{3\mathrm{L}}\left(\dfrac{y}{\mathrm{L}}\right)^2} &\quad \text{by } B,\\[6pt] \sqrt{\left(\frac{m}{\mathrm{L}}\right)^2-\left[\frac{n}{\mathrm{L}}+\frac{2}{3}\left(\frac{y}{\mathrm{L}}\right)^2\right]^2} &\quad \text{by } C, \end{aligned} \right\} \tag{21} \]

then the derivatives of these quantities will be:

\[ \left. \begin{aligned} A'&=-1;\\[6pt] B'&= \frac{ \left(\dfrac{n}{\mathrm{L}^2}+\dfrac{4}{3}\dfrac{y}{\mathrm{L}^2}\right) \left[1-\dfrac{2}{3}\left(\dfrac{y}{\mathrm{L}}\right)^2\right] +\dfrac{y}{\mathrm{L}}\left[\dfrac{n}{\mathrm{L}}+\dfrac{2}{3}\left(\dfrac{y}{\mathrm{L}}\right)^2\right]\dfrac{4}{3}\dfrac{y}{\mathrm{L}^2} }{ \left[1-\dfrac{2}{3}\left(\dfrac{y}{\mathrm{L}}\right)^2\right]^2 };\\[10pt] C'&= -\frac{ \dfrac{4ny}{3\mathrm{L}^3}+\dfrac{8y^3}{9\mathrm{L}^4} }{ \sqrt{\left(\dfrac{m}{\mathrm{L}}\right)^2-\left[\dfrac{n}{\mathrm{L}}+\dfrac{2}{3}\left(\dfrac{y}{\mathrm{L}}\right)^2\right]^2} }; \end{aligned} \right\} \tag{21 bis} \]

and the quantity \(F\) will have the following form:

\[ \mathrm{F}= \frac{ f(1+\mathrm{L}C')(B+C)^2 }{ A'C^2+A'BC-AB'C+ABC' } \tag{22} \]

With the aid of equation (22) one can determine the flexibility of the lifting system under consideration for any prescribed deflection value of the spring—

Theory of suspension on leaf springs with hangers

of the spring, if the deflection arrow \(y\) and the flexibility of the spring \(f\) are known.

We obtain the flexibility of the system when the spring is fully straightened, \(F_0\), if in equation (22) we substitute the value \(y = 0\); in this case

\[ F_0 = \frac{f\left(m^2-n^2\right)} {L^2\left[ \dfrac{m^2-n^2}{L^2} + \dfrac{y_0}{L}\dfrac{n}{L^2} \sqrt{\dfrac{m^2-n^2}{L^2}} \right]}, \]

or

\[ F_0 = \frac{f} {1+\dfrac{y_0}{L}\dfrac{n}{\sqrt{m^2-n^2}}}. \tag{23} \]

Considering the expression (22) obtained by us, we conclude that the magnitude of the flexibility of the suspension system on a triangular leaf spring with hangers depends not only on the elements characterizing the spring \(L\), \(i\), \(b\), \(h\), \(y_0\), but also on the elements of the spring suspension \(m\) and \(n\); moreover, the magnitude \(F\) varies as a function of the deflection of the spring \(y\) within very wide limits.

From expression (22) it can be seen that the flexibility of the suspension system under consideration becomes infinite when \(C = 0\) and \(A = 0\), i.e. when

\[ \frac{m^2}{L^2} - \left[ \frac{n}{L} + \frac{2}{3}\left(\frac{y}{L}\right)^2 \right]^2 = 0 \quad \text{and} \quad y_0-y=0. \tag{24} \]

Indeed, for the values of \(y\) that reduce equations (24) to zero, the denominator of expression (22) is a finite quantity, while the numerator becomes infinite, since \(C'=\infty\).

The values of \(y\) satisfying equations (24) and making \(F\) infinite correspond, as is not difficult to see, to the horizontal position of the hanger axes, for which the quantity \(Q\) has an infinitely small value. When, for the horizontal position of the hanger axes, \(y\) is not equal to \(y_0\), then

\[ F = \frac{ f\left[ \dfrac{n}{L} + \dfrac{2}{3}\left(\dfrac{y}{L}\right)^2 \right] } { \left(\dfrac{y_0}{y}-1\right) \left[ 1-\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2 \right] }. \tag{25} \]

Expression (25) shows that the greater the ratio \(\dfrac{y_0}{y}\), the smaller the flexibility of the system in the horizontal position of the hangers.

The flexibility of the suspension system under consideration, \(F\), may also become equal to zero.

Indeed, \(F = 0\) if \(y\) satisfies, separately, each of the following two equations:

\[ 1 + LC' = 0 \]

and

\[ B + C = 0. \]

Substituting the values of \(B\), \(C\), and \(C'\) (21 and 21bis), we obtain the following two independent equations:

\[ \frac{4}{3}\frac{y}{L}\left[\frac{n}{L}+\frac{2}{3}\left(\frac{y}{L}\right)^2\right] = \sqrt{ \left(\frac{m}{L}\right)^2 - \left[ \frac{n}{L}+\frac{2}{3}\left(\frac{y}{L}\right)^2 \right]^2 } \tag{26} \]

and

\[ - \frac{ \frac{y}{L}\left[\frac{n}{L}+\frac{2}{3}\left(\frac{y}{L}\right)^2\right] }{ 1-\frac{2}{3}\left(\frac{y}{L}\right)^2 } = \sqrt{ \left(\frac{m}{L}\right)^2 - \left[ \frac{n}{L}+\left(\frac{y}{L}\right)^2 \right]^2 }, \tag{27} \]

which may be represented in the following form:

\[ \operatorname{ctg}\beta = \frac{4y}{3L} \quad \text{and} \quad \operatorname{tg}\beta\,\operatorname{tg}\alpha = -1, \]

where \(\beta\) and \(\alpha\) have the same meanings as in equation (1).

The equation \(\operatorname{ctg}\beta = \dfrac{4y}{3L}\) corresponds to that position of the suspension system in which the axis of the shackle is normal to the trajectory of the end of the spring.

Indeed, (Fig. 4)

\[ \operatorname{ctg}\beta=\frac{A'D'}{DF}, \quad \text{or} \quad \operatorname{ctg}\beta= \frac{y}{ \sqrt{\left(\frac{3}{4}L\right)^2-y^2} }, \]

which, with the accuracy with which the problem is being investigated, corresponds to the equation

\[ \operatorname{ctg}\beta=\frac{4y}{3L}. \]

The second equation (27) corresponds to that position of the spring and the shackles in which the directions of the axes of the shackles coincide with the direction of the chord \(OA\) of half the axis of the main leaf of the spring and in which \(Q\) is equal to infinity.

To determine from equation (26) those values of \(Q\) for which the flexibility of the system is equal to zero, we solve this equation with respect to \(\dfrac{y}{L}\); for this purpose we square both sides of it; we obtain an equation of the third degree with respect to \(\left(\dfrac{y}{L}\right)^2\):

\[ \frac{16}{9}\left(\frac{n}{L}\right)^2 +\frac{64}{81}\left(\frac{y}{L}\right)^6 +\frac{16}{9}\frac{n}{L}\left(\frac{y}{L}\right)^4 = \frac{m^2-n^2}{L^2} -\frac{4}{3}\frac{n}{L}\left(\frac{y}{L}\right)^2 -\frac{4}{9}\left(\frac{y}{L}\right)^4; \]

then we substitute the found values of \(\dfrac{y}{L}\) into equation (11), which thereby takes the following form:

\[ Q= \frac{ f\left(y_0-y\right)\left[12-8\left(\dfrac{y}{L}\right)^2\right] }{ \left[21-8\left(\dfrac{y}{L}\right)^2\right] } \tag{28} \]

The values \(y\) corresponding to the quantities \(Q\) for which the flexibility of the system \(F\) is equal to zero may be determined graphically. For this purpose, from the point \(F\) (Fig. 4), with radii \(\dfrac{3}{4}L-m\) and \(\dfrac{3}{4}L+m\), we describe arcs concentric with the arc of the circle replacing the trajectory of the end of the spring.

The intersection of these arcs with the straight line \(VW\), along which the center of the eye of the shackle connected with the hanger moves, gives us the points 1, 2, 3, 4. If we connect these points by radii with the center \(F\), then the intersection of these radii (or their extensions) with the arc of the trajectory of the end of the spring gives us the positions of the end of the spring \(A'_1\), \(A'_2\), \(A'_3\), and \(A'_4\), for which the flexibility of the system is equal to zero, in accordance with equation (26).

It is not difficult to observe that, if \(m>n\), then the flexibility of the system can become 0 only in two positions of the end of the spring: \(A'_1\) and \(A'_4\).

To determine the magnitudes of the deflection of the spring when, according to the second equation (27), the flexibility of the system \(F\) is equal to zero, we square both sides of this equation; we obtain an equation of the fourth degree with respect to \(\left(\dfrac{y}{L}\right)^2\), whose roots may be determined by Ferrari’s method.

The values \(S\) satisfying the condition \(\operatorname{tg}\alpha \operatorname{tg}\beta=-1\) differ very little from the values \(S\) satisfying the condition \(\operatorname{ctg}\beta=\dfrac{4y}{3L}\).

Thus, in calculating the suspension system under consideration, one may use, in order to determine the deflections of the spring for which \(F=0\), the following construction, analogous to the preceding one.

From the point \(F\) (Fig. 4), as from a center, we describe circles with radii \(\dfrac{3}{4}L+m\) and \(\dfrac{3}{4}L-m^*\), and the points of intersection of these circles with the straight line \(VW\), along which the center of the suspension eye connected with the hanger moves, we join by straight lines with the midpoint \(O\) of the axis of the master leaf. The intersections of these straight lines with the circle of the trajectory of the end of the spring, \(A_1, A_2, A_3,\) and \(A_4\), determine the required values of the deflections, equal to the ordinates of these intersection points. It is clear that, both in this case and in the preceding one, not all solutions found by the construction have real values.

Let us now see how the flexibility of the suspension system under consideration, at complete straightening of the spring, will vary depending on \(y_0\), \(m\), and \(n\).

From expression (23) for \(F_0\) it can be seen that the values of \(F_0\) are greater for negative \(n\) than for positive \(n\); that is, at complete straightening of the spring, the flexibility of the system with internal shackles is greater (all other conditions being equal) than the flexibility of the system with external suspension.

The angle of inclination of the shackles at complete straightening of the spring and the deflection arrow of its camber \(y_0\) also exert their influence on the flexibility \(F_0\), which, when these elements increase, increases for internal shackles and decreases for external ones.

If \(n=0\), or if \(m\) is very large in comparison with \(n\) (as, for example, with long vertical earrings), then, as formula (23) shows, the method of suspending the spring does not affect \(F_0\): in this case \(F_0=f\).

The flexibility of the suspension system \(F_0\) at straightening of the spring becomes infinite when the denominator of expression (23) becomes zero, i.e., when the values \(m\) and \(n\) satisfy the equation:

\[ 1+\frac{y_0}{L}\frac{n}{\sqrt{m^2-n^2}}=0 \tag{29} \]

Solving this equation with respect to \(\dfrac{n}{m}\), we find those ratios between \(m\) and \(n\) for which the flexibility of the suspension system under consideration—

* In an exact construction, the length of the radius should be smaller by a certain amount, the maximum value of which for a triangular leaf spring is approximately equal to \(\dfrac{m}{20}\) (when \(\alpha\) approaches the value \(-90^\circ\)).

...of suspension \(F_0\) becomes infinitely large (with internal shackles), whence we obtain

\[ \frac{n}{m}=-\frac{1}{\sqrt{\left(\frac{y_0}{L}\right)^2+1}} . \tag{30} \]

In exactly the same way, from expression (23) we obtain, for the stiffness of the suspension system \(F_0\) when the spring is fully straightened, the value \(0\), when its denominator tends to infinity, i.e. when \(\frac{n}{m}=1\) (with external shackles).

As for the dependence between the stiffness of the suspension system of a triangular leaf spring with shackles and the magnitude of the load at the end of the shackle, it can be established by means of two equations:

\[ \begin{aligned} F&=\frac{f(1+LC')(B+C)^2}{A'C^2+A'BC-AB'C+ABC'},\\ \text{and}\qquad Q&=\frac{AC}{f(C+B)} \end{aligned} \tag{31} \]

where \(A\), \(B\), \(C\), \(A'\), \(B'\), and \(C'\) have the values indicated by equations (21) and (21 bis). To eliminate \(\frac{y}{L}\) from these equations and reduce them to the form \(F=f(Q)\) is not possible without a considerable lowering of the accuracy of the solution; therefore, in order to determine the dependence between \(F\) and \(Q\), it is perhaps most convenient to use formulas (31) and, with their aid, to construct the curve of dependence between \(F\) and \(Q\).

When the spring is fully straightened, the dependence between \(F_0\) and \(Q_0\) is expressed by a formula easily obtained from expression (23):

\[ F_0=\frac{fL}{L+fQ_0\frac{n}{\sqrt{m^2-n^2}}}. \tag{32} \]

Graphically, the curve of dependence between \(F\) and \(Q\) is obtained as follows: first it is necessary to construct the first derivative with respect to \(Q\) from the curve expressing the dependence between \(S\) and \(Q\). To construct this derivative, we draw a series of tangents to points of this curve (Fig. 5) and on the ordinates \(S_1\) corresponding to these...

to the points of tangency, we lay off the quantities \(E_1 F_1\), proportional to the tangents of the angles made by these tangents with the abscissa axis \(Q\). In this way we obtain a series of points belonging to the curve of dependence between \(F\) and \(Q\), which we connect by the continuous curve \(F_1 F F_0\).

Now let us determine the duration of the period of oscillation of the carriage frame on the elastic suspension system under consideration. Let us first consider small oscillations, within whose limits the flexibility of the system may be regarded as constant, equal to the mean value of the flexibility over the period of oscillation.

We set up the differential equation of motion of the center of gravity of the mass \(\dfrac{2Q}{g}\), resting on the suspension system consisting of the triangular leaf spring under consideration with hangers:

\[ \frac{Q}{g}\frac{d^2 s}{dt^2}=Q-(Q+s\varphi) \tag{33} \]

where \(\varphi\) is the quantity reciprocal to the flexibility of the suspension system, i.e.

\[ \varphi=\frac{dQ}{ds_1}; \]

\(s\) is the lowering of the center of gravity of the oscillating mass from its mean position \(s_1\), corresponding to the equilibrium position of the load on the suspension system under consideration, and \(t\) is the time of oscillation of the carriage frame. After certain transformations of equation (33) we obtain:

\[ \frac{d^2 s}{dt^2}=-s\frac{dQ}{ds_1}\frac{g}{Q}. \]

Solving this differential equation, we obtain:

\[ \left(\frac{ds}{dt}\right)^2=-\frac{g s^2}{Q}\frac{dQ}{ds_1}+u^2 \]

\[ \frac{ds}{dt}=\pm\sqrt{u^2-\frac{g}{Q}\frac{dQ}{ds_1}s^2}, \tag{34} \]

where \(u\) is the constant of integration, equal to the maximum velocity of oscillation of the carriage frame over the whole period.

Then we find:

\[ dt=\frac{ds}{\sqrt{u^2-\frac{g}{Q}\frac{dQ}{ds_1}s^2}}. \]

Whence, after integrating within the limits

\[ \text{from } s\max = +u\sqrt{\frac{Q}{g}\frac{ds_1}{dQ}} \quad \text{to} \quad s\min = -u\sqrt{\frac{Q}{g}\frac{ds_1}{dQ}}, \]

we obtain \(T\)—the time of a complete oscillation of the mass resting on the suspension system under consideration,

\[ T = 2\sqrt{\frac{Q}{g}\frac{ds_1}{dQ}} \left[ \arc \Sin \frac{s}{u} \sqrt{\frac{g}{Q}\frac{ds_1}{dQ}} \right]_{s=0}^{s=u\sqrt{\frac{Q}{g}\frac{ds_1}{dQ}}} \]

or

\[ T=\pi\sqrt{\frac{QF}{g}}. \tag{35} \]

Substituting for \(F\) and \(Q\) their values (31), we obtain an expression for \(T\) in terms of the values \(y_0, y, L, m\), and \(n\).

\[ T=\pi\sqrt{ \frac{(1+LC')(B+C)AC} {g(A'C^2+A'BC-AB'C+ABC')} }. \tag{36} \]

where the values \(A, B, C, A', B'\), and \(C'\) are determined by formulas (21) and (21 bis).

From equation (36) it is evident that the time of a complete oscillation of the body frame on an elastic suspension made of leaf springs with shackles does not depend on the flexibility of the spring; moreover, the quantity \(T\) becomes zero for the same values of the variables for which \(F\) becomes \(0\), i.e. when \(1+LC'=0\) or when \(B+C=0\).

With complete straightening of the spring, the time of a complete oscillation of the body frame is expressed in the following manner:

\[ T_0=\pi\sqrt{ \frac{y_0} {g\left(1+\frac{y_0}{L}\frac{n}{\sqrt{m^2-n^2}}\right)} }. \tag{37} \]

Thus, with complete straightening of the spring, the time of oscillation of the system under consideration \(T_0\) becomes zero or infinity for the same relations between \(n\) and \(m\) that make the values of \(F\) become zero and infinity. Thus, if

\[ \frac{n}{m} = \frac{1} {\sqrt{\left(\frac{y_0}{L}\right)^2+1}}, \]

then \(T_0\) is equal to infinity; if, however, \(\frac{n}{m}=1\), then \(T_0\) is equal to zero.

In order to obtain an expression for \(T\) for large amplitudes of oscillation, over the course of which the flexibility of the system varies within such wide limits that it is impossible to replace it by some mean value, we set up the equation of motion of the center of gravity of the mass suspended on the spring system under consideration, of the following form:

\[ \frac{Q}{g}\frac{d^2s}{dt^2}=Q_s-Q_1 \tag{38} \]

where \(Q=\psi(s_1)\) and \(Q_s=\psi(s_1+s)\), and the relation between \(s_1\) and \(Q\) is expressed by means of equations (11) and (18).

It should be noted, however, that the function \(\psi\), which determines the analytic relation between \(Q\) and \(s\), is very complicated, so that the differential equation (38) presents insurmountable difficulties for integration. Nevertheless, graphically the question of the time of a complete oscillation of the carriage frame on the elastic system under consideration is solved quite simply.

Indeed, having the curve of the dependence between \(Q\) and \(s\), it is not difficult to construct the curve expressed by the equation:

\[ \frac{d^2s}{dt^2}=\frac{g}{Q}(Q_s-Q_1) \]

For this purpose, taking the point of the curve corresponding to the value \(Q\) as the origin of coordinates parallel to the former coordinate axes, we proportionally change the abscissae \(Q_s\) of the curve obtained in this way in the ratio \(g\) to \(Q\); in this manner we obtain the curve of accelerations \(\frac{d^2s}{dt^2}\). Knowing the curve of accelerations, we can construct the curve of velocities \(\frac{ds}{dt}\), as an integral curve expressing the change of the areas between the axis of ordinates and the curve \(\frac{d^2s}{dt^2}\). For this purpose, drawing a series of horizontal lines intersecting the acceleration curve, we set off on each, to the right of the axis of ordinates, a segment expressing the square content of the above-indicated areas from the origin of coordinates \(O_s\) to the line of the segment.

In exactly the same way, having the curve \(\frac{ds}{dt}\), one also constructs the curve of paths traversed by the oscillating point.

To determine the time of a complete oscillation of the vehicle frame, we construct the curve of variation of \(t\), as an integral curve of the line expressed by equation 1:

\[ \frac{ds}{dt}. \]

Now, if we set ourselves some half-amplitude of the oscillation, then with the aid of the indicated curves we can determine the magnitude of the second half of the amplitude (not equal to the first), the maximum velocity of the oscillation, and the time of a complete oscillation of the vehicle frame.

However, in the majority of cases, with a sufficient degree of accuracy for purposes of calculation, one may use, in determining the oscillation time of the vehicle frame \(T\), formula (35), since for the most part the direction of the tangent to the curve of dependence between \(S\) and \(Q\) at the point \(Q\), characterizing the flexibility of the system, will not differ significantly from the mean flexibility of the system over the period of a complete oscillation of the vehicle frame.

As is seen from formula (35), for this case the time of a complete oscillation \(T\) is proportional to \(\sqrt{QF}\).

This quantity can be constructed graphically as the geometric mean between \(Q\) and \(F\).

Let \(E y_0\) (Fig. 5) represent the abscissa of some point of the curve expressing the dependence between \(Q\) and \(F\); describing from the point \(E\) an arc of a circle with radius equal to the ordinate of the point \(F\) of the curve, we cut the axis of abscissas with it so as to obtain the segment \(EI\), adjacent to the segment \(y_0E\), representing the abscissa of the point of the curve. If the distance \(y_0I\), equal to \(Q + F\), is divided in half and from the midpoint \(G\) we describe a circle with radius

\[ \frac{Q+F}{2}, \]

then the intersection with it of a straight line parallel to the axis of ordinates and passing through the point of the curve \(F\) that we have taken gives us the segment \(ET\), equal to \(\sqrt{QF}\), and, consequently, it will serve, on a known scale, as the measure of the magnitude \(T\).

Having constructed a series of such points, we obtain a curve of dependence between \(T\) and \(Q\), which in essence is the most important characteristic of the operation of the suspension system. On the basis of the form of this curve one may judge how advantageous the operation of the suspension system is in the sense of smoothness of oscillations.

The scale for the curve of dependence between \(T\) and \(Q\) can easily be determined by comparing the ordinate of this curve at complete straightening of the spring with its actual value \(T_0\), determ—

... determined by formula (37); if the length of this ordinate is equal to \(\overline{Q_0T_0}\), then the unit of scale for the measurement will be

\[ \frac{Q_0T_0}{T_0}. \]

Figure 6 gives the construction of the characteristic curves for a leaf spring with inner shackles.

The dotted lines show the values of the flexibility and of the time of a complete oscillation of the leaf spring, which are usually and incorrectly applied also to the suspension system consisting of a leaf spring with shackles. From the drawing it may be seen that not only can the absolute magnitudes of the ordinates of the curves of flexibility and of the time of a complete oscillation for the spring and for the system considered by us differ considerably, but even the form of the curves is quite different: the flexibility of the spring is a constant quantity, whereas the flexibility of the system has its maximum near the position of complete straightening of the spring; the time of a complete oscillation of the spring increases together with the load, whereas the time of a complete oscillation of the system has its maximum at a load slightly exceeding the load that straightens the spring.

In conclusion, let us summarize those results at which we have arrived in considering the theory of elastic suspension on leaf springs with shackles.

1) The ends of a full, or triangular, leaf spring, when it is straightened and then further deflected, describe certain curves which, with a sufficient degree of accuracy for calculation, may be replaced by arcs of a circle; the centers of these circles lie on the tangent drawn to the middle of the axis of the main leaf and at distances from it \(\pm \frac{1}{4}L\), while their radii are equal to \(\frac{3}{4}L\), where \(2L\) is the length of the main leaf.

2) The formulas for determining the deflection arrow and the time of a complete oscillation of a leaf spring are incorrectly applied also to a suspension system consisting of a leaf spring with shackles, as a result of which results may sometimes be obtained that do not at all correspond to reality.

3) The flexibility of the suspension system under consideration is incorrectly regarded as a constant quantity.

4) The relation between the vertical load \(Q\), falling on the end of the shackle, and the deflection arrow \(y\) of a triangular leaf spring is expressed by formula (11):

$$ Q = \frac{ E i b h^3 \left(\dfrac{y_0}{L} - \dfrac{y}{L}\right) }{ 6L^2 \left\{ 1+ \dfrac{ \dfrac{y}{L}\left|\dfrac{n}{L}+\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2\right| }{ \left|1-\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2\right| \sqrt{ \dfrac{m^2}{L^2} - \left[\dfrac{n}{L}+\dfrac{2}{3}\left(\dfrac{y}{L}\right)^2\right]^2 } } \right\} }. $$

5) The flexibility of a suspension system on leaf springs with shackles is a variable quantity, which changes within very wide limits, depending not only on the dimensions of the spring, but also on the elements of the suspension: the length of the shackles and the distance between their fixed hinges.

6) The flexibility of a suspension system on triangular leaf springs \(F\) with shackles is determined by formula (22):

$$ F = \frac{ f(1+LC')(B+C)^2 }{ A'C^2 + A'BC - AB'C + ABC'' }, $$

in which the values of the quantities: \(A\), \(B\), \(C\), \(A'\), \(B'\), and \(C'\) are determined by formulas (21) and (21 bis).

7) When the triangular leaf spring is fully straightened, the flexibility \(F_0\) of the suspension system under consideration is expressed by formula (23):

$$ F_0 = \frac{ f }{ 1+\dfrac{y_0}{L}\dfrac{n}{\sqrt{m^2-n^2}} }. $$

8) When the spring is fully straightened, the flexibility of the system under consideration is greater with inner shackles than with outer shackles.

9) The time of a complete oscillation of the vehicle frame \(T\) in the suspension system under consideration for a triangular leaf spring, with a small amplitude of oscillations, is expressed by formula (36):

$$ T = \pi \sqrt{ \frac{ (1+LC')(B+C)AC }{ g(A'C^2 + A'BC - AB'C + ABC') } }, $$

in which the values of \(A\), \(B\), \(C\), \(A'\), \(B'\), and \(C'\) are given by equalities (21) and (21bis).

10) When the triangular leaf spring is fully straightened, the time of a complete oscillation of the system \(T_0\) is obtained from formula (37):

$$ T_0 = \pi \sqrt{ \frac{ y_0 }{ g\left(1+\dfrac{y_0}{L}\dfrac{n}{\sqrt{m^2-n^2}}\right) } }. $$

11) With complete straightening of the triangular leaf spring, the oscillations of the carriage frame are slower with inner earrings than with outer earrings.

12) The flexibility of the suspension system on a triangular leaf spring with suspension links tends to infinity under complete straightening of the spring, when the following relation between \(m\) and \(n\) (30) holds:

\[ \frac{n}{m}=-\frac{1}{\sqrt{\left(\frac{y_0}{L}\right)^2+1}}. \]

13) The flexibility of the suspension system under consideration tends to zero when the axis of the suspension link coincides with the direction of the chord of half the axis of the root leaf, or when the axis of the suspension link is normal to the trajectory of displacement of the end of the spring.

14) When the flexibility of the suspension system under consideration tends to zero or to infinity, the time of a complete oscillation of the carriage frame likewise tends to zero or to infinity.

Literature on the Question of Leaf Springs.

1) “Mémoire sur les ressorts” Par M. Phillips. Annales des mines 1852.

2) “Étude sur les ressortes de suspension et de traction à lames étagées.” Albert de Fierlant 1889.

3) “Note sur l’établissement des ressorts à lames, employés dans le matériel des chemins de fer” Par MM L. Rey et H. Valot. Mémoires de la société des ingénieurs civiles 1881.

4) “Note sur les ressorts de suspension” Par M. Rey. Mémoires de la société des ingénieurs civiles 1876.

5) “Note sur l’amélioration de la suspension des voitures de chemins de fer par l’application en dedans de menottes des ressorts à lames” par M. Féraud. Mémoires de la société des ingénieurs civiles 1881.

6) “Tableaux graphiques pour le calcul des ressorts à lames employés dans le matériel des chemins de fer” Par H. Chevalier. Mémoires de la société des ingénieurs civiles 1887.

7) “Fonctionnement des organes de la suspension dans les locomotives.” Revue général des chemins der fer 1905—6.

8) “Traité d’exploitation des chemins de fer” Par A. Flamanche, A. Huberti et A. Stévart. T. III, “Suspension.”

9) “On the suspension springs of passenger cars” E. Holstein. XIV Consultative Congress of Engineers of the Traction Service, 1892.

10) “Spring suspension” Galakhov.

“Improvement of the running of cars and semi-elliptic springs” Galakhov. Bulletin of the Saratov Branch of the Imperial Russian Technical Society 1909.

11) “On elliptic springs of passenger cars” R. Nagel. Bulletin of the Society of Technologists 1909.

To the article by S. P. Gomel: “Theory of Elastic Suspension on Leaf Springs with Shackles.”

Fig. 3.

Fig. 4.

Fig. 1.

Fig. 2.

Table I.

For the article by S. P. Gomel: “Theory of Elastic Suspension on Leaf Springs with Shackles.”

Fig. 5

Plate II

Fig. 6

Visible labels in the diagram: \(0\), \(c_k\), \(k\), \(k'\), \(x\), \(x_0\), \(f\), \(f_0\), \(F^\circ\), \(F_0\), \(V\), \(v\), \(A^\circ\), \(a\), \(S^\circ\).