Influence of Cast-Iron Tooth Dimensions on Its Relative Bending Strength

I. V. Rodionov

Problems and aims of the investigation

In calculating the teeth of gear wheels for bending, the tooth is regarded as a beam fixed at one end and loaded by a force at the free end \[1–14\]. Depending on the adopted scheme of load distribution among the teeth in mesh, the magnitude and point of application of the force on the tooth-beam are determined, the critical section is established, and calculation formulas are compiled.

The generally accepted approach is as follows.

A parabola is inscribed in the tooth profile. The parabolic body separated from the tooth represents a beam of uniform resistance to bending. The total pressure of tooth on tooth (acting along the line of pressure) is transferred to the vertex of the parabola and resolved into two components: a circumferential force, bending the tooth, and a force compressing the tooth.

Existing methods for calculating gear wheels for strength under bending stresses, being based on the initial equation—the elementary formula of strength of materials—take into account, to one degree or another, a number of factors affecting the normal operation of the tooth: the circumferential speed of the wheel, manufacturing accuracy, the nature of the load, the influence of the fillet at the tooth root on the occurrence of local stresses, the tooth form, the influence of the radial component of the force on the tooth, etc.

Thus, the initial equation in calculating a tooth for bending may be represented in general form as:

\[ M_i = c W \sigma_i, \]

where
\(M_i\) — bending moment,
\(W\) — section modulus of the critical section,
\(\sigma_i\) — stress in the critical section due to the force bending the tooth,
\(c\) — a coefficient which, to one extent or another (depending on the method), takes into account the factors indicated above and in essence refines the stress.

Using the obtained calculation formulas, teeth of gear wheels of different dimensions are calculated, i.e., the same calculation formulas are applied to the calculation of both small and large teeth. It follows from this that no method takes into account the influence of tooth dimensions on its relative volumetric strength. This follows from the fact that it is customary to consider the initial relationship between the quantities characterizing the bending phenomenon of the tooth-beam as a relationship reflecting the law of mechanical similarity.

But it is known that the law of mechanical similarity is not universal. Experimental data available in the literature indicate the presence of su-

…of a substantial influence of absolute dimensions on the strength of a specimen or a part, both under static and under alternating loads [15; 16; 17; 18].

In the second volume of ESM [2, 10] it is recommended, for critical cases when bending strength determines the permissible load on teeth, to choose the safety factor as the product of partial coefficients. Moreover, the coefficient taking into account the influence of tooth dimensions on its strength is given according to the data of the differential method for selecting the overall safety factor [19], which, as is known, refer to specimens of circular dimensions \((d = 10—65\ \text{mm})\). At the same time, the most reliable strength characteristics can be obtained only as a result of direct tests of the part itself. We know of no studies of the influence of tooth dimensions on its relative strength; this question has not been covered in the literature.

Clarification of the influence of tooth dimensions on its volumetric strength is of special importance today, when the rapid growth in the mechanization of labor-intensive processes requires the creation of powerful units in whose kinematic chains large-module gear transmissions occupy one of the leading places. The considerations set forth above compelled the author to carry out an investigation of the influence of the geometrical dimensions of a tooth on its volumetric strength under the action of a static load. It is quite obvious that the conditions of static tests differ in character from the conditions of operation of a tooth in a transmission. Nevertheless, the objective material obtained as a result of the investigation makes it possible to refine the physical picture of tooth operation, to recommend correction coefficients for the calculation formula, and may prove useful for further investigation of the influence of tooth dimensions on its relative strength under conditions of transmission operation. In addition, as practice shows, tooth breakages are a very common type of failure during transmission operation. One of the causes of breakage, apart from fatigue in bending, may also be short-term overloads, when in the critical section the normal bending stress reaches its limiting value, which leads to failure of the tooth. Therefore, an additional calculation of teeth according to static characteristics, especially for transmissions operating under possible impact loads (automobiles, tanks, crushing machines, cranes, etc.), is also necessary.

The solution of the problem posed required clarifying whether the conditions following from the law of mechanical similarity are satisfied for geometrically similar teeth under the action of a static load.

On the law of mechanical similarity

The discovery of the law of mechanical similarity belongs to the well-known Russian scientist, professor of the Petersburg Technological Institute V. L. Kirpichev. On November 5, 1874, at a meeting of the Russian Physical Society, V. L. Kirpichev reported on the law of elastic similarity he had discovered, as the theorem “On similarity in elastic phenomena.” In the same year, in issue 9 (volume VI) of the Journal of the Russian Chemical Society and Physical Society at the Imperial St. Petersburg University, he published an article in which a proof of the law was given, based on an analysis of the general differential equations of equilibrium in the theory of elasticity [20]. V. L. Kirpichev formulated the law of similarity as follows: “It is commonly said that geometrically similar bodies are not equally strong; this is indeed true if the bodies’ own weight is taken into account. But very often the influence of a body’s own weight on the change in its shape may be neglected; then it turns out that two bodies made of one and the same material, which were similar before the application to them of external forces, remain similar after their action, if the forces are distributed

in a similar manner over the surface of both bodies, and the magnitudes of the corresponding forces per unit surface are identical in both bodies. In this case all the internal forces of the first body will be equal to the corresponding forces of the second, i.e., both bodies will be equally strong.”

The law of mechanical similarity formulated by Professor V. L. Kirpichev was undeservedly attributed to the German professor F. Kick, and in some works was called the Barba–Kick law. The reason for this was the work of the professor of the German Higher Technical School in Prague, F. Kick, “The Law of Proportional Resistances and Its Application,” dating from 1885. The priority of Professor V. L. Kirpichev was pointed out by engineer A. Karmishensky in his article “A Discovery Ascribed to a Foreign Scientist,” published in the newspaper Leningradskaya Pravda on August 13, 1948.

Further development of the question of mechanical similarity is associated with the works of Soviet scientists, and chiefly with Corresponding Member of the Academy of Sciences of the Ukrainian SSR N. N. Davidenkov [15; 22; 23].

N. N. Davidenkov gives a more precise definition of mechanical similarity, indicating that two geometrically similar bodies made of one and the same material will be mechanically similar if the stressed states (stress tensors) and deformations (strain tensors) at all corresponding points of both bodies are identical.

From the law of mechanical similarity it follows that, for two geometrically similar bodies made of one and the same material, under the action of a static load in the absence of body forces (neglecting the action of the bodies’ own weight), the similarity criteria can be represented in general form as:

\[ \frac{P}{l^2} = \mathrm{const} \quad \text{and} \quad \sigma = \mathrm{const}\;(\text{or } \tau = \mathrm{const}), \]

where
\(P\) — force,
\(l\) — characteristic dimension,
\(\sigma\) — normal stress,
\(\tau\) — shear stress.

The principal parameter determining the dimensions of an involute tooth is the module of engagement \(m\).

Consequently, for two geometrically similar teeth, which can be represented by one and the same drawing but at different scales, the ratio between the forces acting on the teeth will take the form:

\[ \frac{P_2}{P_1} = \left(\frac{m_2}{m_1}\right)^2 . \]

For the following discussion, two concepts are established:

\[ \lambda_{\text{theoretic}}=\left(\frac{m_2}{m_1}\right)^2 \]
— the theoretical transition scale of mechanical similarity with respect to forces;

\[ \lambda_{\text{experimental}}=\frac{P_2}{P_1} \]
— the experimental transition scale of mechanical similarity with respect to forces.

When the law of mechanical similarity is satisfied, the following equality must hold:

\[ \lambda_{\text{theoretic}}=\lambda_{\text{experimental}}. \]

Experiments with Cast-Iron Teeth

In setting up the experiments, using Hooke’s law, our aim was to determine whether the conditions of mechanical similarity are observed for geometrically similar teeth under the action of a static load, and thereby to obtain an objective reflection of the influence of tooth dimensions on its relative strength. For cast gear wheels, which are used in open transmissions, the calculation of teeth for bending plays a decisive role in determining the load-carrying capacity of such transmissions. Therefore, we began studies of the influence of tooth dimensions on its relative strength with geometrically similar cast teeth made of cast iron SCh 12—28. It is known that, when the material is in a brittle state, failure occurs after the stage of small deformations, which more or less accurately follow Hooke’s law. Therefore, in judging the strength of such a material as cast iron, it is necessary to study only one limiting (dangerous) state—the transition from elastic deformation to fracture [24; 25]. It should be noted that, in practice, testing brittle materials in bending acquires especially great importance [26].

If it is assumed that the stress states are mechanically similar and that fracture is determined by the values of the maximum stresses, then it is obvious that, in geometrically similar involute teeth, fractures will occur under similar conditions. As practice shows, for such brittle materials as cast iron, the first and second theories of strength are fully applicable [27; 28]. Therefore it must be assumed that the conclusion we have drawn is fully justified. Thus, if we know the forces that fracture two similar teeth made of a brittle material, then with a sufficient degree of accuracy, using Hooke’s law, we can judge whether the conditions of mechanical similarity are observed under the action of a static load.

Normal and shortened teeth with \(m = 5;\ 10;\ 15\) and \(20\) mm, geometrically similar and equivalent in profile, were subjected to bending tests. The model was a cast sector with two teeth. The tests were carried out on a Krause-system machine.

In order to exclude, to some extent, the influence of the nonuniformity of the material structure in cast models of different size, the influence of the casting skin, of the endings of the tooth-beam, and of the method of applying the load, in addition to experiments on cast-iron models made of SCh 12—28, experiments were carried out on cast models made of aluminum AL 26, as well as on geometrically similar cantilever beams of rectangular shape. As a result of the investigation, a qualitative picture was obtained of the influence of tooth dimensions on its relative strength, from which it follows:

1) Geometrically similar cast teeth of involute profile, made of one and the same material, are not of equal strength, which indicates a deviation from the law of mechanical similarity under the action of a static load.

2) With increasing tooth dimensions, the relative strength decreases.

3) The deviation from the law of mechanical similarity obeys a regularity clearly expressed in graphs plotted in the coordinates “\(\sqrt{\lambda}—m\)” (Fig. 1).

Machined cast-iron gear wheels are widely used both in open transmissions and in closed ones, owing to the good resistance of modified and alloyed cast irons to contact stresses. The maximum load-carrying capacity of gear transmissions in which one or both wheels are cast iron is limited not by the endurance of the working surfaces of the teeth, but by bending strength. Therefore, the study of the question of the influence of tooth dimensions on its strength acquires special significance for machined cast-iron gear wheels.

To obtain a comparative picture of the influence of the dimensions of a machined tooth on its relative strength, two grades of cast iron were selected as the material for the models: SCh 21–40 and MCh 28–48. The model was a toothed sector (Fig. 2), part of a complete gear wheel, \(m = 5;\ 10;\)

Fig. 1

14 and 20 mm (\(\alpha_{on}=20^\circ;\ h=2.2\,m;\ z=20;\ \psi=\dfrac{b}{m}=6\)). The gear wheels were made to accuracy class 3 and were cut with worm hobs from MI3. The machining conditions in the final operations were kept identical.

Fig. 2

The casting of blanks for each batch (SCh 21–40 and MCh 28–48) was carried out from one ladle of a single melt, in order to obtain a more uniform material structure for teeth of different sizes.

The bending tests of the teeth were carried out on a universal testing machine—a 50-ton press of the Denison system. In doing so, the following requirements were observed:

a) full contact of the punch with the tooth along a straight line (the tooth edge) was ensured;

b) the force on the tooth was transmitted along the line of pressure;

c) the model together with the machine crosshead constituted an elastic system which reproduced, to some extent, the operating picture of the “wheel–key–shaft” assembly.

Fig. 3

On the basis of the considerations set forth above, the tooth was loaded through the punch by a static force, which was gradually increased up to the force that fractured the tooth. The displacement of the contact line of the punch with the tooth (deflection of the tooth along the normal) was measured by an indicator fixed to the machine bed.

The results of tests with geometrically similar machined teeth made of SCh 21–40 and MCh 28–48 fully confirmed the qualitative picture obtained as a result of experiments with cast teeth:

1) At the moment of fracture, \(\lambda_{\text{experimental}}\) does not coincide with \(\lambda_{\text{theoretical}}\) (Fig. 3), moreover

\[ \lambda_{\text{experimental}} < \lambda_{\text{theoretical}}. \]

2) With an increase in tooth dimensions, the bending strength limit decreases (Fig. 4). Over the range of \(m\) from 5 to 20 mm, and taking \(m = 5\) mm as the unit of comparison, the decrease in strength amounts to:

\[ \begin{array}{ll} \text{for SCh 21–40} & \text{about } 40\%,\\ \text{for MCh 28–48} & \text{about } 30\%. \end{array} \]

Fig. 4.

Bending strength limit \(\sigma_{bu}\), kg/cm\(^2\), of geometrically similar machined teeth of involute profile.

Module of engagement \(m\) (in mm).

Brinell hardness.

\[ \alpha_0 = 20^\circ,\qquad h = 2.2m,\qquad \psi=\frac{b}{m}=6,\qquad z=20. \]

Curve labels: Ch24-40; MCh28-48.

Fig. 5.

Square root of the transition scale of mechanical similarity with respect to forces, \(\sqrt{\lambda}\).

\[ \sqrt{\lambda}_{\text{test}}=\sqrt{\frac{p_{nt}}{p_s}}=\sqrt{\frac{p_t}{p_s}} \]

Geometrically similar machined teeth of involute profile.

\[ \alpha=20^\circ,\qquad \psi=6 \]

Theoretical curve.

Module of engagement \(m\) (in mm).

Displacement of the punch contact line with the tooth (deflection along the normal) for geometrically similar machined teeth (in mm).

Fig. 6.

Module of engagement \(m\) (in mm).

\[ \alpha_0=20^\circ,\qquad h=2.2m,\qquad \psi=\frac{b}{m}=6,\qquad z=20. \]

Curve labels: Ch24-40; MCh28-48.

1. The deviation from the law of similarity obeys a regularity expressed in the graphs “\(\sqrt{\lambda} - m\)” by a straight line (Fig. 5).

2. From the graph in the coordinates “tooth deflection along the normal—\(m\)” (Fig. 6) it follows that, at the moment of failure, the deflections of geometrically similar teeth obey the law of similarity, whereas similarity is not observed with respect to loads.

To determine the influence of tooth length on its strength, additional tests were carried out with a tooth \(m = 20\) mm for tooth length \(B = 60 \div 90\) mm. The tests showed that doubling the tooth length does not have a significant effect on its strength.

Explanation of the Experimental Results

The observed phenomenon of a decrease in the relative strength of a tooth with an increase in its dimensions can be explained as follows.

A tooth is a straight-axis bar of complex outline and, owing to the nonuniformity of the stressed state in the dangerous section, may be regarded as a specimen with a notch. A notch qualitatively changes the mechanical properties of the material in the part [25; 29]. For brittle materials, whose limiting dangerous state is determined by the magnitude of the greatest principal normal stress, the nonuniformity of the stressed state creates conditions for the formation of a brittle crack and thereby lowers the average strength of the notched specimen. Fig. 7 presents the pattern of the trajectories of the principal normal stresses (\(\sigma_1\) and \(\sigma_2\)) in the tooth section according to Bukharinov’s data [30]. The stress concentration reaches its maximum value at the place where the involute passes into the body of the rim. The greatest principal stresses \(\sigma_1\) are directed normal to the surface of the fillet.

Fig. 7

Experiments show that the initial crack (Fig. 2) arises in the region of the transition curve, normal to its contour, and propagates from one fillet to the other with curvature near the axis of symmetry of the tooth.

Comparing the pattern of the trajectories of the principal stresses according to Bukharinov’s data with the character of tooth fracture in the experiments, one may conclude that the stressed state in the region of the fillet determines the initiation and character of propagation of the brittle crack across the tooth section. In the region of the transition curve, one can identify a “dangerous volume,” the material of which is in the most highly stressed state. The brittle crack ari-

arises when the principal normal stress $\sigma_1$ reaches its limiting value. It may be assumed that the occurrence of a brittle crack is influenced by the inhomogeneity of the material in the “dangerous volume” and by defects on the tooth surface in the region of the transition curve. As the dimensions of the tooth increase, the “dangerous volume” increases, and with it the probability of material inhomogeneity and surface flaws. Consequently, as the dimensions of the tooth increase, conditions are created for the earlier appearance of a brittle crack. The inhomogeneity of the material, as the dimensions of the tooth increase, also increases throughout the entire volume that includes the whole dangerous section from one fillet to the other. These circumstances lead to a decrease in the average strength of the tooth with increasing dimensions. Thus, the influence of the dimensions of a tooth made of a brittle material on its strength is quite explicable by the “statistical theory of brittle strength” developed by T. A. Kontorova and Ya. I. Frenkel [31; 32].

As indicated above, doubling the tooth length (for a given $m$) has a less significant effect on its strength than increasing all dimensions when passing to a geometrically similar tooth of another module. This phenomenon may be explained as follows: when only the tooth length is doubled, the dangerous volume also doubles, whereas when all dimensions of the tooth are doubled it increases fourfold.

It follows from this that the inhomogeneity of the material, when all dimensions of the tooth are increased, manifests itself to a greater degree than when only the length is changed.

At the same time, let us note that, after the formation of a brittle crack, the subsequent fracture of the tooth is fully explicable by the theory of the “internal resources” of energy proposed by Moroz and Shurakov in the work “The Problem of the Strength of Case-Hardened Steel” [33]. The crack plays the role of a sharp notch, and its appearance causes a sharp, dynamically characterized redistribution of stresses in the dangerous section. The resistance of the tooth drops instantaneously, as a result of which part of the potential energy accumulated by the tooth in the process of elastic deformation is released and is expended on the subsequent dynamic bending of the tooth. The phenomenon observed in the experiments—the arrest of a brittle crack on small teeth ($m = 5; 10\ \text{mm}$) and sudden fracture with rebound on large ones ($m = 14$ and $20\ \text{mm}$)—confirms the proposition expressed by N. N. Davidenkov [33] that the theory of the “internal resources” of energy is fully applicable for explaining the scale effect.

Indeed, if the deformation energy is proportional to the volume of the tooth, while the energy of brittle fracture of the core depends on the area of the dangerous section, then, as the size of the tooth increases, the ratio of the potential deformation energy to the fracture energy of the core increases, and thus an excess of energy is created, which is expended on the rebound of the tooth.

Coefficient of Unequal Strength

The validity of the initial formula that underlies the existing methods for calculating gear wheels evidently holds in the case of an isotropic material and a uniform triangular law of stress distribution in the dangerous section of the tooth-beam. In this case the law of mechanical similarity will also be satisfied.

In reality, the material of the teeth is inhomogeneous; the law of stress distribution is violated, which, with a change in tooth dimensions, leads to the inequality:

\[ M_u < c W \sigma_u . \]

To preserve equality, it is necessary to introduce into the right-hand side a correction factor that reflects, to some extent, the influence of the tooth dimensions on its strength.

Then the strength equation takes the form

\[ M_u = c W \sigma_u K_n, \]

where \(K_n\) is the nonuniform-strength factor, accounting for the decrease in strength with increasing geometric dimensions of the tooth, and is less than unity:

\[ K_n < 1. \]

The nonuniform-strength factor may be represented as the product of 2 factors:

\[ K_n = K_{ng}\cdot K_{nv}, \]

\(K_{ng}\) is the factor accounting for the decrease in relative strength of a series of geometrically similar teeth (for a given \(\varphi = \dfrac{b}{m} = \mathrm{const}\)), and

\(K_{nv}\) is the factor accounting for the influence of tooth length on strength when passing from one series of geometrically similar teeth to another (i.e., when \(\psi\) changes).

Since experiments have established that doubling the tooth length does not have a substantial influence on its strength, within the limits

\[ \psi = \frac{b}{m} = 6 \div 12 \div 15, \]

we recommend taking

\[ K_{nv}=1, \]

then

\[ K_n = K_{ng} = \frac{\lambda_{\mathrm{exper}}}{\lambda_{\mathrm{theor}}}: \]

the nonuniform-strength factor is equal to the ratio of the experimental transition scale of mechanical similarity in forces to the theoretical one.

The straight-line regularity established as a result of the investigation for the deviation from the law of mechanical similarity on the graphs “\(\sqrt{\lambda} - m\)” (Fig. 8) makes it possible to represent \(K_n\) as a function of the angle of inclination of the experimental curve in the following form:

\[ K_n = \frac{\lambda_{\mathrm{exper}}}{\lambda_{\mathrm{theor}}} = m_1^2 \left[ \frac{1 + (m_i - m_1)\operatorname{tg}\Theta_0}{m_i} \right]^2, \]

where \(m_1\) is the tooth module adopted as the unit of comparison,

\(m_i\) is the tooth module for which the value \(K_n\) is determined,

\[ \operatorname{tg}\Theta_0 = \frac{\sqrt{\lambda_0}-1}{m_i-m_1} \]

is the tangent of the angle of inclination of the experimental curve “\(\sqrt{\lambda} - m\)” to the \(m\)-axis.

For the materials we tested (with \(m_1 = 5\) mm), the values of \(\operatorname{tg}\Theta_{0\,\mathrm{avg}}\) are given in the table.

Table

The mean values of the coefficient of nonuniform strength may be represented in a form convenient for practical use:

for

\[ \text{SCh }21—40 \quad K_n=\left(0.76+\frac{1.18}{m_i}\right)^2, \]

for

\[ \text{MCh }28—48 \quad K_n=\left(0.82+\frac{0.92}{m_i}\right)^2. \]

In design calculations it is recommended, as a first approximation, to take \(K_n=1\), and after determining \(m_i\), to refine the values of \(K_n\) according to the formulas given above and correct the calculation.

Fig. 8

In verification calculations, \(K_n\) should be determined from the known \(m_i\) and introduced into the right-hand side of the calculation formula:

\[ M_i=cW\sigma_u K_n. \]

Introducing \(K_n\) into the right-hand side will make it possible to refine the choice of the material strength safety factor, which will permit correction of the allowable stress toward an increase; ultimately this will lead to a reduction in the dimensions of the tooth and, consequently, of the transmission as well.

The author has made the first attempt to establish the influence of tooth dimensions on its relative strength under the action of a static load. The author believes that, because of its importance, this question will receive further development, as a result of which values of \(K_n\) will be established for a number of other materials and the results obtained will be refined with reference to operating transmissions.

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