Speed Factor in Calculating Gears for Endurance of the Working Surfaces of Teeth
Abstract Generated abstract
The paper addresses how peripheral speed should be accounted for when calculating high-speed gear transmissions for contact endurance of tooth working surfaces. Drawing on tests of gears with varied geometry, tooth-surface hardness, and accuracy in the range of 20 to 60 m/sec, it proposes an approximate speed factor that combines experimentally observed increases in load-carrying capacity with dynamic loading, safety factors, failure probability, tooth-tip and tooth-root strength relations, and temperature-dependent oil viscosity. The resulting formulas are intended to adjust allowable contact-stress and load-carrying-capacity coefficients from values established at 10 m/sec. The author emphasizes that the proposed relationships are a first approximation for critical high-speed drives, but notes that calculated stress levels agree closely with those found in modern ship main drives.
Full Text
Vol. 209 1976
SPEED FACTOR IN THE CALCULATION OF GEARS FOR ENDURANCE OF TOOTH WORKING SURFACES
A. E. Belyaev
(Presented by the scientific seminar of the Department of Applied Mechanics)
The works published at the present time, in which the results of experimental studies on the influence of peripheral speed on the contact strength of gear transmissions are presented, cannot in any way satisfy calculation practice. The influence of speed, noted in tests by a number of investigators, is taken into account by various speed factors [9–12] (depending on the number of revolutions of one of the gear wheels [11], or on the sum of the numbers of revolutions of the pinion \(n_1\) and the wheel \(n_2\) [12], or on the speed [10]). G. Niemann [10], who carried out the most thorough experiments in this field, proposes, for example, the speed factor shown in Fig. 1 (the hatched zone is the deviation of the experimental results from the mean value; the straight line is expressed by the equation
\[ K_0=\left(\frac{v}{10}\right)^{0.4}\, . \]
Fig. 1. Value of the speed factor according to work [10]; \(v\) is peripheral speed, m/sec.
However, the factor proposed by him, first, was found for gear transmissions in a fairly narrow range of speeds \((2 \div 32\ \text{m/sec})\), i.e., not in the range in which high-speed transmissions operate, and, second, it does not take into account the variety of factors (besides speed) that can substantially change the magnitude of the speed factor (dynamic loads, change in the load-carrying capacity of wheels made with different hardnesses of the working tooth surfaces, the safety factor for high-speed transmissions, etc.).
In order to take into account the factors indicated above, we tested a large series of gear wheels with different geometric parameters, different hardnesses of the working tooth surfaces of the pinion and wheel, and different accuracy, in the speed range \(20 \div 60\ \text{m/sec}\).
The tests we conducted [1] may serve as a basis for developing an approximate value of the speed factor \(c_v\) as applied to critical high-speed gear transmissions. The proposed speed factor can be represented by the following expression:
\[ C_v=\frac{\Omega}{\kappa_{\mathrm{dyn}}\cdot n}, \tag{1} \]
where
\(\Omega\) is an experimental coefficient characterizing the increase in the load-carrying capacity of gear transmissions with increasing speed;
\(\kappa_{\mathrm{dyn}}\) is a coefficient taking into account additional dynamic loads;
\(n\) is the safety factor in calculating high-speed gear transmissions for contact strength.
The coefficient \(\Omega\) is equal to
\[ \Omega=\frac{(K_{\mathrm{n}}^{\mathrm{e}})_v}{(K_{\mathrm{n}}^{\mathrm{e}})_{v=10}}, \]
where \((K_{\mathrm{n}}^{\mathrm{e}})_v\) is the experimental value of the coefficient of load-carrying capacity of gear transmissions at speed \(v\);
\((K_{\mathrm{n}}^{\mathrm{e}})_{v=10}\) is the same at a speed of \(10\ \mathrm{m/sec}\).
The coefficient \(\Omega\) for all transmissions tested by us (Fig. 2) in the speed range \(20\text{–}60\ \mathrm{m/sec}\) lies within the limits \(1.3\text{–}1.9\) (and reaches 2).
The safety factor \(n\) introduced into the expression for \(C_v\) depends on several factors:
- The degree of responsibility of the high-speed transmissions, \(n_{\mathrm{resp}}\);
- The probability of failure of high-speed transmissions, \(n_{\mathrm{fail}}\);
- The dimensions of the gear transmissions (diameter, face width of the gear rim, etc.), \(n_{\mathrm{dim}}\);
- The thermal intensity of high-speed transmissions, \(n_v\).
Thus, the safety factor may be represented as
\[ n=n_{\mathrm{resp}}\cdot n_{\mathrm{fail}}\cdot n_{\mathrm{dim}}\cdot n_v \]
Let us consider each of these factors.
- All the gears tested are high-speed transmissions, the failure of which is associated with major material damage. A. I. Petrusevich [7] considers that the safety factor \(n_{\mathrm{resp}}\), with respect to experimental data, should be equal to 1.2 in stresses (i.e., 1.44 in load). A similar coefficient \(n\) is also proposed in [2]. The author of [8] proposes taking the safety factor in calculations for contact strength as equal to 1.25.
Taking the value of the safety factor at a speed of \(10\ \mathrm{m/sec}\) (\(n_v=10\)) as the initial one, as has been adopted by a number of researchers, for example [10], and taking into account the very high responsibility of high-speed drives, we take the safety factor as 1.5.
- As experiments carried out at LMI and at the Kirov Plant [3] have shown, surface failures on the tooth tips of critical transmissions may be regarded (with a certain margin) as the criterion for transmission failure. Since the ratio between \(M_{\mathrm{g}}\) and \(M_{\mathrm{n}}\) (the strength of the tooth tips and roots, respectively) decreases with increasing speed (see Izvestiya TPI No. 188), we consider that this factor must necessarily also be taken into account in the safety factor.
Therefore, as a first approximation we take
\[ n_{\mathrm{fail}}= \frac{ \left(\dfrac{M_{\mathrm{g}}}{M_{\mathrm{n}}}\right)_{v=10} }{ \left(\dfrac{M_{\mathrm{g}}}{M_{\mathrm{n}}}\right)_{v} } \cdot n_{v=10}, \]
where \(n_{v=10}\) is the safety factor at a speed of \(10\ \text{m/sec}\);
\[ \left(\frac{M_{\mathrm{h}}}{M_{\mathrm{n}}}\right)_{v=10} \]
is the ratio of the load-carrying capacity of the tooth tips and roots when the transmission operates at a speed of \(10\ \text{m/sec}\);
\[ \left(\frac{M_{\mathrm{h}}}{M_{\mathrm{n}}}\right)_{v} \]
is the same when the transmission operates at a speed of \(v\ \text{m/sec}\).
In doing this, we proceed from the obvious fact that the higher the speed, the more dangerous the failure of a critical transmission.
Fig. 2. Values of the contact-stress coefficients \(C_{\mathrm{k}}\), the load-carrying-capacity coefficients \(K_{\mathrm{n}}\), and the speed coefficient \(C_v\) for gear transmissions at different speeds:
\((K_{\mathrm{n}}^{\mathrm{e}})_v\) is the experimental value of the load-carrying-capacity coefficient of gear transmissions;
\(K_{\mathrm{reg}}\) is the duty-factor coefficient [4].
- The influence of dimensions on the strength margin is taken into account in a definite way in the dynamic coefficient, while the working width of the gear rim, which affects the thermal stress of the transmission, can be taken into account by the coefficient \(n_{\delta}\) (see below).
Therefore we consider that \(n_{\mathrm{dim}} = 1\).
- Data on the influence of temperature on strength, apart from those given by N. F. Kuz’min [6] and A. I. Belyanin [5], are extremely scarce, although, in our opinion, allowance for this factor is necessary. The coefficient depends on the width of the gear rim and on the temperature of the working surfaces of the teeth, which increases with increasing speed. But with increasing spe—
increases with growth of speed, as does heat removal, owing to the more intensive supply of lubricant (in the absence of churning). Therefore, the change in the load-carrying capacity of the transmission associated with a change in the viscosity of the oil (the influence of the temperature factor) can be taken into account according to the recommendation of A. I. Belyanin,
\[ n_{\nu}=\sqrt[6]{\frac{(\nu_t)_{\vartheta=10}}{(\nu_t)_{\vartheta}}}\,; \]
where
\[ (\nu_t)_{\vartheta=10} \]
is the kinematic viscosity of the oil at the oil-inlet temperature \((T_{\mathrm{in}}\) at \(\vartheta=10)\);
\[ (\nu_t)_{\vartheta} \]
is the viscosity of the oil at the given temperature and \(\vartheta\).
To find the coefficient it is necessary to know the temperature values at different speeds. According to work [5], the temperature of gear wheels during testing at a speed of \(90\ \mathrm{m/sec}\) was \(85^\circ\mathrm{C}\) at an oil-inlet temperature \(T_{\mathrm{in}}=40^\circ\mathrm{C}\). According to our data, obtained from periodic measurements of the temperature of high-speed gear transmissions in service, it was \(\sim 65^\circ\mathrm{C}\) at a speed of \(60\ \mathrm{m/sec}\), with an oil-inlet temperature \(T_{\mathrm{in}}=35\div40^\circ\mathrm{C}\). Assuming that the temperature \((T_{\mathrm{in}})\) has a first-order effect, one can find the dependence of temperature on speed. Averaging the temperature values from various data, we construct the graph \(T=f(v)\) (Fig. 3).
Fig. 3. Change of temperature \(T_{\mathrm{in}}\) with circumferential speed
The proposed speed factor is equal to:
\[ C_{\vartheta}= \frac{\Omega}{K_{\mathrm{dyn}}\cdot n_{\vartheta=10}} \cdot \frac{\left(\dfrac{M_{\mathrm{g}}}{M_{\mathrm{n}}}\right)_{\vartheta}} {\left(\dfrac{M_{\mathrm{g}}}{M_{\mathrm{n}}}\right)_{\vartheta=10}} \cdot \sqrt[6]{\frac{(\nu_t)_{\vartheta}}{(\nu_t)_{\vartheta=10}}}. \]
The value of \(C_{\vartheta}\) in the range \(20\div60\ \mathrm{m/sec}\) is given in Fig. 2.
Thus, in calculating for contact strength by the method of [4], one can determine the allowable value of the contact-stress coefficient at any speed \([C_k]_{\vartheta}\) by the formula:
\[ [C_k]_{\vartheta}=[C_k]_{\vartheta=10}\cdot C_{\vartheta}, \]
where \([C_k]_{\vartheta=10}\) is the allowable value of the contact-stress coefficient found by the method of [4] at speeds up to \(10\ \mathrm{m/sec}\).
Calculations of transmissions show (the values \((K^{\mathrm{e}}_{\mathrm{n}})_{\vartheta}\), \([K_{\mathrm{n}}]_{\vartheta}\), \([C_k]\), \([C_k]_{\vartheta=10}=[C_k]\), \([C_k]_{\vartheta}\) are given in Fig. 2)\(*\) that the stress level obtained with allowance for the coefficient \(C_{\vartheta}\) is very close to the stress level in the main drives of modern ships.
The allowable value of the load-carrying-capacity coefficient at any speed for gear transmissions can be determined by the formula:
\(*\) 1. \(K_{\mathrm{reg}}\) was found according to [4].
2. The dependences \((K^{\mathrm{e}}_{\mathrm{n}})_{\vartheta}\) and \(\Omega\) were constructed from the results of 24 experiments.
\[ [K_{\mathrm{H}}]_{v}\,[K_{\mathrm{H}}]_{v=10}\cdot \left(\frac{\mathrm{HB}}{250}\right)^2\cdot \frac{\Omega\cdot (k_{\mathrm{dyn}})_{v=10}}{1.5\,n_{v=10}\cdot (k_{\mathrm{dyn}})_{v}}\cdot \frac{\left(\dfrac{M_{\mathrm{g}}}{M_{\mathrm{n}}}\right)_{v}} {\left(\dfrac{M_{\mathrm{g}}}{M_{\mathrm{n}}}\right)_{v=10}\cdot \sqrt{\dfrac{(\gamma_t)_{v=10}}{(\gamma_t)_{v}}}}, \tag{2} \]
where
HB is the hardness of the working surfaces of the teeth of the weakest element*).
Taking into account that, according to the data of [4], the ratio of the load-carrying capacities of the tooth heads \((M_{\mathrm{g}})\) and tooth feet \((M_{\mathrm{n}})\) at a speed \(v=10\ \text{m/s}\) does not exceed 2.4, and the value of \((k_{\mathrm{dyn}})_{v=10}\) (for the accuracy of turbine transmissions) differs very little from unity (for this reason \((k_{\mathrm{dyn}})_{v=10}\) may be omitted from formula 1), expression 2 will be equal to:
\[ [K_{\mathrm{H}}]_{v}=[K_{\mathrm{H}}]_{v=10}\cdot \left(\frac{\mathrm{HB}}{250}\right)^2\cdot \frac{0.3\,\Omega}{n_{v=10}\cdot (k_{\mathrm{dyn}})_{v}}\cdot \frac{\left(\dfrac{M_{\mathrm{g}}}{M_{\mathrm{n}}}\right)_{v}} {\sqrt[6]{\dfrac{(\gamma_t)_{v=10}}{(\gamma_t)_{v}}}}. \]
The expression given for the speed factor \(C_v\), \([K_{\mathrm{H}}]_v\), and the value \([C_k]_v\) may be regarded only as a first approximation to the solution of the problem concerning allowance for the influence of speed in the calculation of gear transmissions and the establishment of permissible values of the contact-stress coefficient for critical high-speed drives.
LITERATURE
- A. E. Belyaev. The influence of speed on the contact strength of gear transmissions. Izv. TPI, 147, Tomsk, 1966.
- I. A. Bukus. Contact strength of marine gear transmissions. Transactions of NKI, issue VII, 1954.
- R. R. Galper. Contact strength of high-speed gear transmissions with corrected gearing. LD NTP, 1964.
- V. N. Kudryavtsev. Simplified calculations of gear transmissions, Mashgiz, 1960.
- V. N. Kudryavtsev. Gear Transmissions, Mashgiz, 1957.
- N. F. Kuzmin. On the coefficient of friction in heavily loaded contact. Vestnik mashinostroeniya, No. 5, 1954.
- A. I. Petrusevich. Encyclopedic Handbook: Mechanical Engineering, vol. 2, Moscow, 1948.
- A. W. Davies. Marine Reduction Gearing, Pros. Inst., Mech., Engrs., London, vol. 170, No. 16, 1956.
- A. Fisher. A problem in the Design and Construction of Gear Loading Formular, Machinery, vol. 102, 12, 1963.
- G. Niemann, H. Rettig. Zusammenfassung und Folgerungen aus den Versuchsergebnissen, Konstruktion Heft 10, 1950.
- A. Sykes. Allowable load on helical gears, The Engineer, Nov. N 5416, 1959.
- C. Timms. Recent Development in Spur and Helical Gears, The Institut of Production Engineers, 1959.
*) Coefficient of load-carrying capacity of gear transmissions
\[ K_{\mathrm{H}}=\frac{p}{d_{\vartheta}}\cdot \frac{i\pm 1}{i}\ \text{kg/cm}^2, \]
where
\(p\) — specific load, kg/cm;
\(d\) — pitch-circle diameter of the pinion, cm;
\(i\) — transmission ratio.