Coefficient of Unequal Strength of Steel Teeth in Bending

I. V. Rodionov

Research objectives and experimental procedure

As a result of investigating the influence of the dimensions of a tooth made of cast iron of grades SCh12—28, SCh21—40, MSch28—48, and of aluminum AL 26 on its relative strength under the action of a static load, we established that geometrically similar teeth are not of equal strength [1]. It could be assumed that one of the reasons for the deviation from the law of mechanical similarity is the brittle material’s imperfect obedience to Hooke’s law. This circumstance prompted us to proceed to an experimental study of the mechanical similarity of geometrically similar teeth made of steel. Two grades of steel were adopted as the material for the models: St 6 and steel 45 \((\sigma_b \geq 60\ \text{kg}/\text{mm}^2;\ \sigma_s \geq 30\ \text{kg}/\text{mm}^2)\). The model was a toothed sector—a part of a complete gear wheel with module \(m = 5,\ 10,\ 14,\) and \(20\) mm. The blanks for the gear wheels were made by forging. In order to equalize the structure, relieve internal stresses obtained during hot working, and reduce deformation during subsequent machining, the forgings were normalized. It is known that the direction of the material fibers in a stamped blank has a substantial influence on the bulk strength of gears. The maximum dynamic strength of a tooth is achieved when the blank from a bar is upset on the end. This provision was taken into account in manufacturing the gear-wheel blanks. The gear wheels were made to accuracy class 3 and were cut with MIZ worm cutters. The machining regime in the final operations was kept the same; special attention was paid to this when cutting the steel wheels. Gear wheels with \(m = 5,\ 10,\ 14\) mm were made from St 6, and from steel 45—wheels with

\[ m = 5\ \text{and}\ 20\ \text{mm}\left(\alpha_{0n}=20^\circ;\ h=2.2m;\ z=20;\ \psi=\frac{b}{m}=6\right). \]

In the present investigation we set ourselves the task of determining the influence of the dimensions of a steel tooth on its strength, using Hooke’s law; therefore the tests were carried out within the limits of elastic deformations.

From the law of mechanical similarity [2] it follows that if two geometrically similar teeth, made of one and the same material, are subjected to external forces whose magnitudes are proportional to the squares of the modules

\[ \left(\frac{P_i}{m_i^2}=\text{const}\right) \]

and which are distributed in a similar manner over their surfaces, then: 1) the teeth will be of equal strength; consequently, \(\sigma=\text{const}\); 2) corresponding points of similar teeth will receive displacements parallel in direction and, in magnitude, proportional to \(m\); 3) having changed their shape under the action of the forces, the teeth will remain similar, and the ratio of their corresponding dimensions will retain the same value that it had before the action of the forces.

It follows from this that, in bending tests within the limits of elastic deformations of two geometrically similar teeth, the following may be taken as the deformation characteristic:

a) the displacement of the contact line of the punch with the tooth—\(s\), in which case

\[ s_1 \parallel s_2 \quad \text{and} \quad \frac{s_1}{m_1}=\frac{s_i}{m_i}, \]

or

b) the angle of bending of the tooth

\[ \beta_1=\beta_i=\mathrm{const}. \]

To verify the propositions stated, the normal pressures acting on geometrically similar teeth were determined at equal bending angles.

The bending tests of the teeth were carried out on a universal testing press, the diagram of which is shown in Fig. 1. The working table of the machine (\(P\)) has, on its lower plane, a circular groove in which a disk is fastened

Fig. 1

to which the punch (\(P\)) is attached. The punch is connected to the disk by a screw pair. In order to satisfy the conditions of similarity in the application of the load, four geometrically similar punches were specially made, corresponding to \(m=5,\ 10,\ 14\), and \(20\) mm. The model (Z.S.) was mounted on the crosshead—the movable support (Tr.), which is suspended on the frame of the machine by means of two tie-rods (\(T\)) arranged along the diagonal of the support. Along the second diagonal the crosshead is connected to the machine frame by means of springs (Pr.). Owing to this arrangement, the crosshead can undergo certain displacements in the longitudinal and transverse directions and thus performs the functions of a spherical bearing. When the punch came into contact with the tooth of the model, possible misalignments were eliminated, and the model together with the crosshead assumed the working position. The angles of bending of the teeth were measured by the mirror method, from the deflection on the screen of the projection of a luminous filament. For this purpose we assembled an apparatus according to the following scheme (Fig. 1). In a fixed stand (Sh) on the frame of the machine there was mounted an optical system consisting of a fixed tube (N.T.) carrying the objective, and a movable tube (P.T.) carrying a small lamp with an “S”-shaped thin filament connected to a 4.5 V transformer. On the end face of the tooth, by means of a liquid …

with glass a mirror (3) was glued in a special holder. Through the lens, the light beam, by means of a movable tube, was focused on the mirror, from which it was reflected onto a screen installed at a distance of 4420 mm from the center line of the punch. From the deflection of the projection of the luminous filament on the screen, the double angle of bending of the tooth was measured.

The analysis we carried out of the possible errors in measuring the angles of bending of the teeth and of the causes producing them made it possible to conclude that this method is reliable and that the readings are objective.

The calibration of the scale was performed on a tooth with \(m = 5\) mm. The tooth was loaded within the range of elastic deformations up to a load from 0 to 1500 kg, which amounts, for steel St 6, to \(0.78 P_s\), and for steel 45, to \(0.74 P_s\), where \(P_s\) is the calculated load at the yield limit. At every 500 kg a mark of the bending angle was made on the scale. The data are given in Table 1.

Table 1

Experiments with teeth of modules \(m = 10,\ 14\), and \(20\) mm were carried out in the reverse order. The tooth was loaded by a normal force and, at the bending angles \(\beta_1,\ \beta_2,\ \beta_3\), read from the scale, the loads were measured. When the tooth was unloaded, the projection of the luminous filament on the scale smoothly returned to its initial position, which indicates the accuracy of this method. The measurement of the angles and the measurement of the loads had first been worked out on a considerable number of experiments. For the final experiments, three teeth of each module were selected, with geometrically similar bearing surfaces, which were subjected to finish milling and alignment on a plate. Three experiments were carried out with each tooth.

Results of the experiments

Table 2 gives the mean values of the quantities obtained from the experiments.

From the data of the experiments with geometrically similar steel teeth it follows:

1) at identical tooth bending angles \((\beta_i = \beta_3)\), \(\lambda_{\text{experimental}}\) do not coincide with \(\lambda_{\text{theoretical}}\) (Fig. 2), and

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

In the range \(m = 5 \div 20\) mm, and with \(m = 5\) mm taken as the unit of comparison, the ratio

\[ \frac{\lambda_{\text{experimental}}}{\lambda_{\text{theoretical}}} \]

is of the order of 75%;

2) the deviation from the law of mechanical similarity obeys a regularity clearly expressed on the graphs “\(\sqrt{\lambda} - m\)” (Fig. 3) by a straight line. Thus, the investigations show that the influence of the tooth dimensions on its relative strength within the limits of elastic deformations is manifested also for geometrically similar steel teeth, and the regularity of the deviation from the law of similarity has the same character as for geometri-

Table 2

Vertical axis of Fig. 2:

Experimental scale of similarity
according to force:

\[ \lambda_{\mathrm{test}} = \frac{P_{ni}}{P_{n5}} = \frac{P_i}{P_5}; \qquad \lambda_m= \left(\frac{m_i}{m_5}\right)^2 \]

Text inside the graph:

\[ \lambda_{\mathrm{test}} = \frac{P_{ni}}{P_{n5}} = \frac{P_i}{P_5} \]

for geometrically similar machined teeth of involute profile.

\[ \alpha=20^\circ;\quad h=2.2m;\quad z=20;\quad \varphi=\frac{b}{m}=6. \]

Normal pressure, kg.
St. 6.
St. 45.
$\beta$.
Limit of elastic resistance.
$m=5$.
$\beta$—tooth bend angle in scale units.

\[ \lambda_{\mathrm{theoretical}}=\left(\frac{m_i}{m_5}\right)^2 \]

\[ \lambda_{\mathrm{test}} \]

Horizontal axis:

Gear module $m$ (in mm)

Fig. 2

geometrically similar teeth made of a brittle material—cast iron. This makes it possible to establish the coefficient of unequal strength \((K_n)\), which takes into account the decrease in strength with an increase in the geometrical dimensions of the tooth in the same form as for cast-iron teeth [1], namely:

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

where \(m_1\) is the module of the tooth adopted as the unit of comparison,
\(m_i\) is the module of the tooth for which the value \(K_n\) is determined.

\[ \tg\Theta_0=\frac{\sqrt{\lambda_0}-1}{m_i-m_5} \]

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

For \(m_1=5\) mm,

\[ K_n=25\left[\frac{1+(m_i-5)\tg\Theta_0}{m_i}\right]^2 . \]

For St. 6 and steel 45, experiments established the mean value \(\tg\Theta_0=0.168\).

Fig. 3

For practical use, for St. 6 and steel 45 it may be recommended that

\[ K_n=\left[0.84+\frac{0.8}{m_i}\right]^2 . \]

Explanation of the Results

Using the research data of N. P. Shchapov, L. S. Moroz, and S. S. Shurakov, as well as the considerations of Ya. B. Fridman, the author makes a first attempt to explain the influence of the dimensions of a steel tooth on its relative strength within the range of elastic deformations.

N. P. Shchapov [3] investigated the influence of a hard surface layer on the static and impact strength of metal. As a result of such technological operations as nitriding, carburizing, rolling, and surface hardening, the resistance of a part to wear is increased; at the same time, this is accompanied by a loss of plasticity in the surface zone, where a very brittle layer of metal is formed. Specimens of square cross section made of rimmed steel of the St. 5 type were tested. The brittle surface layer was produced by carburizing without subsequent heat treatment. It was established experimentally that, in static bending, as the dimensions of the specimens increase, the ultimate strength decreases. N. P. Shchapov explained this phenomenon by the possibility of development of plastic deformation in the subsurface layer of the metal, which, in his opinion, is too small to produce a noticeable change in the shape of the specimen, but sufficient to make the formation of a brittle crack inevitable.

As a result of an extensive investigation of the causes of failures of heavily loaded carburized gears, L. S. Moroz and S. S. Shurakov [4] established “the complete absence of any relation between the ductile properties of the core and tooth breakages.” This led them to undertake a special study of questions of strength of carburized products. As a result of carburizing, a very brittle layer of metal is formed on the surface of the specimen. The core of the specimen possesses ductile properties. In bending, the carburized layer undergoes only elastic deformations, whereas in the subsurface zone local plastic deformation develops, which reduces the strength of the surface layer, creates conditions for the formation of a brittle crack, and, as a result, reduces the strength of the specimen as a whole. With increasing hardness of the core they observed an increase in the breaking load.

Ya. B. Fridman points to the role of local plastic deformation, writing: “The majority of the discrepancies observed between the conclusions of strength of materials and the actual behavior of metals are explained precisely by the influence of local plastic deformation” [5].

In our investigations, steel involute teeth cut with hob cutters were subjected to bending tests. The forgings were normalized before machining. After cutting, the gears were not subjected to any additional heat treatment. The process of cutting the teeth is associated with considerable plastic deformation of the metal, as a result of which the surface layers of the tooth material are work-hardened, their yield point is increased, and their plasticity is reduced.

Let us determine the influence of the dimensions of a tooth with a work-hardened surface layer on the magnitude of the stresses in the subsurface layer within the range of elastic deformations. Suppose that, at some instant of bending, the stresses at corresponding points \(A\) (Fig. 4) of the transition curves of two geometrically similar teeth are equal in magnitude. Let us take the depth of the work-hardened layer to be the same for both teeth. Along the height of the critical section of each tooth, select an elementary volume of material. Since the teeth are geometrically similar, what has been said above can be represented by a single drawing, where \(Aa\) is the stress at the corresponding point of the transition curve; \(O_1a\) and \(O_2a\) are the laws of distribution of stresses in the critical sections; and I—I and II—II are the axes of symmetry of the teeth.

From consideration of Fig. 4 it becomes obvious that, for equal stress at point \(A\) of the transition curve, the stresses in the subsurface layer are not equal to one another; moreover, with increasing tooth dimensions they increase:

\[ O_1 b'=\sigma_{n1}>O_2 c'=\sigma_{n2}. \]

Hence it may be concluded that, with increasing tooth dimensions, the intensity of development, and consequently also the volume of subsurface

layers encompassed by local plastic deformation. This leads, in turn, to a reduction in the strength of the work-hardened layer, to the possibility of the formation of a brittle crack and, as a result, to a reduction in the strength of the tooth. The decrease in load observed by us in the course of testing with an increase in tooth dimensions, as against the load according to the law of similarity at identical bending angles, confirms the correctness of these considerations.

Fig. 4

The development of local plastic deformation of the surface layers of the tooth material is, evidently, possible also in the absence of a work-hardened layer. In this case, the different stress gradient in the critical section for the small and the large tooth will also condition a different volume of surface layers encompassed by plastic deformation, which will lead to a reduction in strength with an increase in the tooth.

In confirmation of our thought we shall cite the considerations of Ya. B. Fridman. He points out that in solid bodies one always observes not a purely plastic, but an elastic-plastic deformation. Considering static tensile tests of notched specimens, he writes: “In plastic metals, up to fracture, a cardinal change in the stressed state is observed; the surface layers pass earlier into the plastic region, and the notched specimen behaves as if consisting of materials with different moduli. The surface layers, located in the plastic region, have a small, and the inner layers—a large ‘modulus,’ equal to the modulus of elasticity.” And further… “In bending, which causes nonuniformity of the stressed and deformed state even in smooth specimens, the relation between volume and nonuniformity is changed in comparison with tension in favor of nonuniformity. Therefore, in plastic bending of notched specimens, the displacement of the maximum stresses from the surface into the interior of the specimen is more difficult. And indeed, Matteas found that a number of materials (Fe with 0.4% C, steel 50, etc.), which under tension

—with a notch gave an increase in resistance to failure; under bending, specimens with a notch gave a reduction in strength of 25–38%” [5].

Thus, it may be assumed that, within the range of elastic deformations, the development of local plastic deformation in the surface layers of the material reduces strength as the dimensions of the tooth increase. A hard surface skin (work hardening, cementation, surface hardening) affects only the degree of development of local plastic deformation, which in turn lowers the strength of the surface layer and thereby creates conditions for the occurrence of a brittle crack. The brittle crack plays the decisive role in the resistance of the tooth, and its appearance releases part of the accumulated potential energy of elastic deformation, at the expense of which the subsequent failure of the core occurs, dynamic in character.

Modern methods of surface strengthening (cementation, high-frequency-current hardening, etc.) make it possible to obtain tooth surfaces that are highly resistant to wear. Therefore, at the present time, questions of the bending strength of teeth acquire very great importance in solving the problem of increasing the load-carrying capacity and service life of gear transmissions. The solution of this problem requires an in-depth study of the factors affecting tooth strength.

As the author’s investigation has shown, the influence of the geometrical dimensions of a tooth on its relative strength under static bending is substantial. The established coefficient of unequal strength of teeth makes it possible, in a simple form, to take this influence into account and to correct the allowable stress upward, which ultimately will lead to a reduction in tooth dimensions and to an increase in the load-carrying capacity of transmissions.

A further task should be the investigation of the influence of tooth dimensions on its relative fatigue strength under bending.

REFERENCES

1. Rodionov I. V. The influence of the dimensions of a cast-iron tooth on its relative strength under bending. Izv. TPI, vol. 75.

2. Kirpichev V. L. On similarity in elastic phenomena. Journal of the Russian Chemical Society and of the Physical Society at the Imperial St. Petersburg University, vol. VI, issue 9, physical part, first section, 1874, pp. 151–155.

3. Shapov N. P. The influence of a hard, brittle surface layer on the static and impact strength of metal. In: Friction and Wear in Machines. Publishing House of the Academy of Sciences of the USSR, vol. 1, pp. 113–132, Moscow–Leningrad.

4. Moroz L. S., Shurakov S. S. The Problem of the Strength of Cemented Steel. Leningrad, 1947. Ministry of Transport Machine Building. Central Order of Lenin Scientific Research Institute.

5. Fridman Ya. B. Mechanical Properties of Metals. Oborongiz, Moscow, 1946.