INVESTIGATION OF LOAD CONCENTRATION ALONG THE LENGTH OF CONTACT LINES OF HELICAL GEARS

A. E. BELYAEV

(Presented by the scientific seminar of the Department of Applied Mechanics)

The load-carrying capacity of gear transmissions depends on the nature of the load distribution along the contact lines of meshing. The nonuniformity of load distribution along the length of the contact lines in helical transmissions is caused by a number of factors.

First, in transmissions where wear of the profiles occurs (for example, crane reducers), the magnitude of the latter is associated with the magnitude of the sliding velocity. In this case tooth wear will be maximal in regions remote from the pole. This circumstance is also the cause of load concentration toward the pitch line.

Second, a cause producing nonuniformity of load distribution along the length of the contact lines in helical transmissions is the variable stiffness of the tooth itself. The total stiffness of the teeth during their operation changes very little (in contrast to spur gears), as is confirmed by theoretical and experimental works [3], [7], [8], [9], [10]. The principal conclusions of these works reduce to the fact that the total specific stiffness of helical teeth does not depend on the face-width module of the teeth or on the phase of meshing, but depends only on the number, inclination, and material of the teeth.

In high-speed transmissions (marine transmissions and high-speed drives of various machines), tooth wear is either absent (except for the running-in period) or extremely small. Therefore the only cause producing nonuniformity of distribution along the length of the contact lines may be the variable stiffness of the teeth.

Different investigators have given different quantitative and qualitative evaluations of the load concentration under consideration. According to the results of investigations of rubber racks, M. D. Trbojevie [16] established that the load concentration is generally extremely small. G. Niemann [14], on the basis of investigations of stresses by the photoelasticity method, asserts that at the ends of the contact line the load is smaller than in the middle. A. Davis and N. Walker [11], on the contrary, found an increase of loads at the ends of the contact line over the average value within the range of 60–200%. In a number of published calculation methods the value of the coefficient of load concentration at the pitch point,

\[ K_p = \frac{P_p}{P_{\mathrm{av}}}, \]

was taken as equal to 1.5 [4], [5], [9].

However, recently a number of works have appeared proving that this value is overestimated. Thus, A. Sykes [15] recommends

to take the value of the load-concentration coefficient at the pitch point of meshing as \(K_p = 1.1\), while the author of [13] considers the value of the coefficient \(K_p\) to be close to 1.3.

V. N. Kudryavtsev [7] convincingly shows that the forces applied to the tooth along the contact line at different heights, and owing to the unequal compliance of the tooth at different points of the contact line \(b_1 b_2\) (Fig. 1), will have different values of the specific contact loads \(P\). The latter can be determined from the formula

Fig. 1. Variation of specific loads along the contact line: \(a\)—for an unmodified profile

\[ P = C \cdot \delta \cdot \cos \beta_0, \tag{1} \]

where \(C\) is the coefficient of specific stiffness;
\(\delta\) is the deformation measured in the end section of the plane of meshing;
\(\beta_0\) is the helix angle of the tooth on the base cylinder.

According to investigations, the greatest specific stiffness of the teeth will occur where the distances from the point of contact to the addendum circles of the gear wheels are equal. In uncorrected transmissions the maximum stiffness will be at the pitch point of meshing, and, consequently, at the pitch point of meshing the specific contact load will reach a maximum (the value of \(P\) according to formula (1) reaches its greatest value at the maximum coefficient \(C\)).

The numerical value of the coefficient \(K_p\), characterizing the load concentration at the pitch point, is recommended in [7] as equal to 1.18. Exactly the same values of the coefficient \(K_p\) are given in the works of O. A. Pyzh. The author of [1] and [2] determines the stress concentration at the pitch point by the formula

\[ K_p = \frac{48}{38.6 + \dfrac{\cos \beta}{\varepsilon_s \sin \alpha_s}} . \tag{2} \]

Calculation of a large number of variants of meshing of helical gears shows that the value of \(K_\rho\), found by the formula given, lies within the limits \(1.185\)—\(1.195\). In the general case, in calculating helical gears the author recommends taking \(K_\rho = 1.2\).

In the latest procedure for calculating gear transmissions, issued by TsNIITMASH in 1963, values of \(K_\rho = 1.2\) are also given [6].

Finally, as a result of calculations and a carefully performed experiment, the author of [12] showed that the value of the coefficient \(K_\rho\) is close to 1.2.

All that has been said above applies to unrelieved helical transmissions, in which the length of the contact line \(l_k = b_1 b_2\) (Fig. 1).

However, high-speed transmissions are almost always cut with a tool having a certain relieved portion. Depending on the number of teeth of the cut gear wheel, the height of the relieved portion on the tooth \(h_{\mathrm{fl}}\) (Fig. 1) will vary, and the length of the line of contact decreases by the amount \(b_2' b_2 + b_1' b_1\) and becomes equal to \(l_k'\). In connection with this, the portions of the contact line remote from the pole are unloaded to some extent, and the load must be concentrated to an even greater degree at the pole line (presumably the change in specific loads is illustrated by curve \(b\) in Fig. 1) than is characterized by the curve of change in specific loads \(a\).

Let us introduce the notation:

\(P_n\) — normal force;

\(\delta\) — total deformation;

\(f(x)\) — clearance function;

\(C\) — total stiffness coefficient, which is a function of the distance \(X\) from the pole line to the given contact point;

\(l_k'\) — length of the contact line.

Figure 2,a shows the contact of two surfaces (one of which is conventionally shown by the straight line \(PNK\)), with half the contact length \(l_k/2 = PN\), and the flank profile \((NI)\) is specified by an equation of the type

\[ \Delta = A x_1^b . \tag{3} \]

Let us consider the segment \(PNI\) (Fig. 3,b), consisting of the involute portion \(PN\) and the relieved portion \(NI\) of length \(l_\phi\). Under the action of a normal load, the surface under consideration is deformed by the amount \(\delta\) and assumes the position \(P'N'I\). At the same time, part of the relieved portion will come into action over the length \(z\).

We shall assume (according to [7]) that the change in the coefficient of specific stiffness is linear in character (Fig. 2,b).

Then, for the portion \(PN\) [7],

\[ C_x = C_{\mathrm{cp}} - (C_{\mathrm{cp}} - C_b)\frac{x}{l} = C_{\mathrm{cp}} - \Theta' x , \tag{4} \]

and for the portion \(NI\)

\[ C_{x_1} = C_a - (C_a - C_b)\frac{x_1}{l_\phi} = C_a - \Theta'' x_1 . \tag{5} \]

The magnitude of the deformation \(\delta\) is determined as

\[ \delta = \frac{P_n}{2(l - l_\phi + z)\cdot \overline{C}}, \tag{6} \]

where \(P_n\) is the total normal load;

\(\overline{C}\) is the mean value of the coefficient of specific stiffness.

Let us consider two segments of the contact line. For the first segment we have \(P_x = C_x \cdot \delta\), and, consequently,

\[ dP_x = d(C_x \cdot \delta)=\delta \cdot dC_x . \]

For the specified law of variation of the coefficient \(C\), \(dC_x = -\Theta' \cdot dx\). Thus,

\[ dP_x=\frac{P_n}{2(l-l_\phi+z)\cdot C}\cdot(-\Theta')\cdot dx . \tag{7} \]

Fig. 2. On determining the coefficient: \(a\)—contact lines; \(b\)—law of variation of stiffness for zones I and II; \(c\)—contact of two loaded surfaces.

After integrating equation (7), we obtain on segment I

\[ P_x=-\frac{P_n\cdot \Theta'}{2(l-l_\phi+z)\cdot C}\cdot x+B . \tag{8} \]

The constant \(B\) in equation (8) is found from the condition \(X=0\).

\[ P_x=P_{\max}=\frac{P_n C_{\mathrm{avg}}}{2(l-l_x+z)}=B . \]

And finally we find

\[ P_x=\frac{P_n}{2(l-l_\phi+z)\cdot C}\cdot(C_{\mathrm{avg}}-\Theta' x). \tag{9} \]

At the same point \(O_1\), the value of \(P_x\) is

\[ P_x=P_a=\frac{P_n}{2(l-l_\phi+z)\cdot C}\,[C_{\mathrm{cp}}-\Theta'(l-l_\phi)]. \tag{10} \]

Let us proceed to consider the law of variation of the linear load along the flanked portion (II). Taking into account that \(d\delta \ne 0\), we obtain

\[ dP_{x_1}=d[C_{x_1}(\delta-\Delta)]=(\delta-\Delta)dC_{x_1}+C_{x_1}d(\delta-\Delta). \tag{11} \]

Moreover, in the last equation

\[ dC_{x_1}=-\Theta''dx_1, \]

\[ d\Delta=-A\cdot b\cdot x_1^{b-1}dx_1. \]

Integrating expression (11), we obtain

\[ \int dP_{x_1}=-\int \delta\cdot \Theta''dx_1+\int A\cdot \Theta''\cdot x_1^b\cdot d\cdot x_1- \]

\[ -\int A\cdot b\cdot C_a\cdot x_1^{b-1}\cdot dx_1+\int A\cdot b\cdot \Theta''\cdot x_1\cdot dx_1. \tag{12} \]

Solving equation (12) and carrying out transformations, we obtain

\[ P_{x_1}=-\frac{P_n\cdot \Theta''}{2(l-l_\phi+z)}\,x_1 +A\cdot \Theta''\cdot x_1^{b+1}-A\cdot C_a\cdot x_1^b+B_1. \tag{13} \]

The constant \(B_1\) is found from the condition: at point \(O\), \(X_1=0\), therefore

\[ P_{x_1}=P_a=B_1=\frac{P_n}{2(l-l_\phi+z)\cdot C}\,[C_{\mathrm{cp}}-\Theta'(l-l_\phi)]. \tag{14} \]

Finally we obtain

\[ P_{x_1}=\frac{P_n}{2(l-l_\phi+z)\cdot C}\cdot \left[C_{\mathrm{cp}}-\Theta'\cdot(l-l_\phi)\right]- \]

\[ -\frac{P_n\cdot \Theta''}{2(l-l_\phi+z)\cdot C}\cdot x +A\cdot \Theta''\cdot x_1^{b+1}-A\cdot C_a\cdot x_1^b. \tag{15} \]

Since at the point \(x_1=z\), \(P_{x_1}=0\), it follows that

\[ \frac{P_n}{2(l-l_\phi+z)\cdot C}\,[C_{\mathrm{cp}}-\Theta'(l-l_\phi)] +A\cdot \Theta''\cdot z^{b+1}= \]

\[ =\frac{P_n\cdot \Theta''}{2(l-l_\phi+z)\cdot C}+A\cdot C_a\cdot z^b. \tag{16} \]

An analysis of the quantities entering into equation (16) shows that the terms containing \(\Theta''\) may be neglected, i.e.,

\[ \frac{P_n}{2(l-l_\phi+z)\cdot C}\,[C_{\mathrm{cp}}-\Theta'(l-l_\phi)] =AC_a\cdot z^b. \tag{17} \]

Thus, the law of variation of the linear load on the part of the flanked portion that has come into operation (the portion \(z\)) can be written as

\[ P_{x_1}=\frac{P_n}{2(l-l_\phi+z)\cdot C}[C_{\mathrm{cp}}-\Theta'(l-l_\phi)]-A\cdot C_a\cdot x_1^b. \]

Taking into account that \([C_{\mathrm{cp}}-\Theta'(l-l_\phi)]=C_a\), we finally obtain

\[ P_{x_1}=\frac{P_n}{2(l-l_\phi+z)\cdot C}\cdot C_a-A\cdot C_a\cdot x_1^b. \tag{18} \]

In solving equation (11), it was assumed that a relation of the type \(\Delta = A \cdot x_1^b\) is the equation of the projection of the flank profile onto the contact line. Meanwhile, in actual gear transmissions the parameters of the flank relate to the tooth profile in the end section. Figure 3 shows the involute tooth profile (dashed line) and the flanked portion, specified in the coordinates \(x_0\) and \(y_0\); the plotted flank was obtained by constructing the tooth profile conjugate to the tool profile. For the pinion tooth and the wheel tooth, the height of the flank was respectively \(h_{\mathrm{fl.sh}} = 0.15\,m\) and \(h_{\mathrm{fl.k}} = 0.28\,m\), where \(m\) is the module of the mesh. The equation of the flank in these coordinates is expressed as \(y_0 = a \cdot x_0^{c^*}\), where \(a\) and \(c\) have definite values for different numbers of teeth. Knowing the increments of the lengths of the contact lines corresponding to a decrease in the depth of flanking (from \(0.3m\) to \(0.01m\)), one can determine the equation of the projection of the flank profile onto the contact line. For the wheel (as for the element having the larger flank), this equation is found to be

\[ \Delta = 0.0075\,X^{1.6}. \tag{19} \]

Fig. 3. Tooth profile of the experimental wheel

Knowing the laws of variation of the linear load over both portions of the contact line, one can construct the graph of the dependence \(P_x = f(l)\). In this case, on the first portion the value of \(P_x\) is found from equation (10) for \(X = 0 \div (l - l_\phi)\) (the dependence will be linear in character), and on the second portion \(P_{x_1}\) is found from equation (18) for

\[ x_1 = (l - l_\phi) \div (l - l_\phi + z). \]

To construct the graph \(P_x = f(l)\) (Fig. 4), let us determine the quantities entering into the equations:

\[ P_n \]

is the total normal load, equal to the sum of the static and dynamic loads,

\[ P_n = P_{\mathrm{st}} + P_{\mathrm{dyn}} = 1140\ \text{kg}; \]

\[ l \]

is the length of the portion of the contact line from the pole to the addendum circle of the tooth,

\[ l = PNL = 23.8\ \text{mm}; \]

\[ l_\phi \]

is the length of the portion of the contact line passing over the flanked portion,

\[ l_\phi = 6.6\ \text{mm}. \]

The values of the stiffnesses on the various portions, of the coefficients \(\Theta'\), \(\Theta''\), and of \(z^{**}\), are given in Table 1.

Table 1

* For the wheel tooth (Fig. 3), this equation has the form \(y_{\mathrm{o.k.}} = 0.2 \cdot X_0^{1.7}\); for the tooth of the conjugate pinion, respectively, \(y_{\mathrm{o.sh}} = 0.45 \cdot X_0^{1.7}\).

** The value \(z\) was found from equation (17).

For a known length of contact lines, the magnitude of the specific contact load found without taking its nonuniformity into account is determined by

\[ P_{\mathrm{av}}=\frac{P_n}{2(l-l_{\phi}+z)}. \tag{20} \]

Fig. 4. Variation of specific loads for crowned and uncrowned tooth profiles

The concentration factor is \(K_{p\phi}=\dfrac{P_p}{P'_{\mathrm{av}}}\), where \(P_p\) is the specific contact load for the strip (i.e., the load of that zone for which the contact stresses are determined). From the constructed graph (Fig. 4) we find that \(P_{\mathrm{av}}=29.7\ \mathrm{kg/mm}\) and \(P_{xx=l=0}=35.2\ \mathrm{kg/mm}\). Consequently,

\[ K_{p\phi}=\frac{P_p}{P'_{\mathrm{av}}}=1.185. \]

References

1. A. I. Belyanin. Investigation of the load-carrying capacity of helical gears. In Gear and Worm Transmissions, edited by N. I. Kolchin, Moscow–Leningrad: Mashgiz, 1959.

2. A. I. Belyanin. Investigation of the load-carrying capacity of helical gears. In Calculation, Design, and Investigation of Transmissions, 1. Publication of the Odessa Polytechnic Institute, 1958.

3. K. I. Zablonskii. Questions of stiffness in the calculation of high-power gear transmissions and a method for their calculation. Transmissions in Mechanical Engineering, Publishing House of the Academy of Sciences of the USSR, 1953.

4. K. I. Zablonskii. Investigation of load concentration along the length of contact lines of gears. Calculation, Design, and Investigation of Transmissions, 1. Publishing House of the Odessa Polytechnic Institute, 1958.

5. Ya. G. Kistyan. Method for calculating gear engagements for strength. TsNIITMASh, Scientific and Technical Information, No. 23. Mashgiz, 1954.

6. Ya. G. Kistyan. Method for calculating gear transmissions for strength. TsNIITMASh, Book 107. Mashgiz, 1963.

7. V. N. Kudryavtsev. Gear Transmissions. Mashgiz, 1957.

8. A. I. Petrusevich. Deformation of teeth with involute normal 20° meshing. TsBR. TsNIITMASh, 1937.

9. A. I. Petrusevich. Gear transmissions. In Machine Parts, edited by N. S. Acherkan. Mashgiz, 1953.

10. I. N. Frenkel. Experimental determination of tooth stiffness taking into account the helix angle in 30° cylindrical gears. Calculation, Design, and Investigation of Transmissions, III. Publication of the Odessa Polytechnic Institute, 1959.

11. Davis A. W. Proc. Inst. Mech. Engrs, vol. 170, N 16, 1956.

12. Hayashi Kunikazu, Load Distribution on the Contact Line of Helical Gear teeth, Part 1, Fundamental concept, Bull. JSME, 1963, 6, N 22.

13. Müller L. Krytyczne spojrzenie na Zazębienie Novikowa, Przegląd mechaniczny, 1961, XX, N 15.

14. Niemann G. Maschinenelemente, II, 1960.

15. Sykes A. Allowable load on Helical Gears, The Engineer, Nov., N 5416, 1959.

16. Trbojević M. D. Engineer, vol. 204, N 5298, 1957.