CLOSED-LOOP TEST RIG FOR TESTING GEAR TRANSMISSIONS UNDER VARIABLE LOAD

Yu. S. Semenov

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

In testing individual gear pairs, reducers, and gearboxes, stands with a closed loop have become widely used. Loading of the closed loop is carried out by various methods. The simplest and most widespread is the method of loading the closed loop by the forces of elastic deformation of a shaft that is pre-twisted [1–5]. The disadvantages of this method are: the impossibility of changing the load during operation, and its decrease as the parts of the installation wear. To eliminate these disadvantages, other loading methods are used, for example, by means of a pneumatic or hydraulic device [6, 7], a differential mechanism [3, 8], or a force applied directly to the supports of the gear wheels [9]. The last method of loading a closed loop is used in the installation described below. In comparison with other methods, it is the most convenient when it is necessary to create a variable load in the closed loop.

Fig. 1

For the purpose of studying the wear of gear transmissions of mining-drill reducers, whose load varies over time within wide limits (Fig. 1), we designed and manufactured a closed-loop ...

a rig for conducting tests not only under constant, but also under variable load.

The present article contains a description of the rig and the basic relationships for determining the efficiency and torques in its links.

Description of the rig

The diagram of the rig, together with the two ER-5 mining electric-drill reduction gears connected to it for testing, is shown in Fig. 2.

Fig. 2

The housing of the loading reduction gear 5, with gear wheels 6 and 7, is suspended on ball bearings in supports 39 and can rotate relative to the axis of shaft 9 of gear wheel 7. Shaft 9 of reduction gear 5 is connected by means of toothed coupling 11 to spindle 12 of the tested reduction gear 18, containing gears 13, 15, 16, 17. Gear 17 of this reduction gear is fastened on shaft 19. On the same shaft is fastened a similar gear 21 of the second tested reduction gear 20, containing gears 21, 22, 23, and 25. Spindle 26 of reduction gear 20

by a gear coupling 27, is connected to shaft 28 of the closing reducer 29, which contains gear wheels 30 and 31. In order that the housing of the loading reducer 5 may be able to rotate freely, shaft 8 of the latter is connected to shaft 32 of the closing reducer by means of universal joints, mounted movably on the intermediate shaft 34.

When testing under a constant load, loading of the closed loop is produced by weight 37, suspended from the lever of the housing of reducer 5.

Weight 37 tends to turn the housing of reducer 5; wheel 6 thereby tends to turn relative to wheel 7, but, since the installation is closed, this rotation can take place only within the limits of the take-up of clearances and the elastic deformation of the shafts. The teeth of wheels 6 and 7, as well as of all the other gear pairs of the closed loop, are pressed against one another and are loaded by the corresponding forces, and the shafts by the corresponding torques.

The power of electric motor 1 is expended only in covering the various kinds of losses in the installation. To measure the torque and, consequently, the power expended, the stator of electric motor 1 is suspended on ball bearings (Fig. 2).

To obtain the specified peripheral speeds of the gear pairs being tested, an intermediate reducer 2 is installed between electric motor 1 and the closed loop.

In order to make it possible to apply not only a load constant in magnitude, created by weight 37, but also a load cyclically varying during the test according to any specified law, the installation has a special device.

Electric motor 40, through a worm-and-gear drive, rotates cam 43, whose profile corresponds to the required character of the load variation. Cam 43, through rod 44, presses on calibrated spring 45; the latter, through a lever system, acts on the lever of the housing of the loading reducer 5. The frequency of the load-oscillation period is changed by means of replaceable gears 41 and 42.

In order that the lower limit of the load remain unchanged, regardless of the angular position of the housing of reducer 5, it is necessary to keep unchanged the compression of spring 45 and, consequently, the position of lever 46. This is achieved by means of screw 47.

To damp the oscillations of the housing of reducer 5 caused by elastic oscillations of the shafts and by impacts in the teeth of the gear wheels, oil damper 48 is used.

The number of cycles of operation of the gear wheels is counted by revolution counter 36, connected by flexible shaft 35 to shaft 32 of the closing reducer.

To switch off the electric motors in the event of breakage of any part in the closed loop or seizure in the installation, automatic stop 38 is provided. If a breakage occurs in the closed loop, its closure is disrupted; the housing of reducer 5 will turn and press on stop 38; the latter, with its rod, will press the “stop” button of the magnetic starter (Fig. 3) and switch off the electric motors. In the event of seizure in the installation, the load on electric motor 1 will sharply increase, and its stator, which until then had been held from turning by the torque created by a certain weight on its lever, will turn and press with its lever on the stop. Minimum-voltage relays serve to protect the electric motors in the event of phase loss (Fig. 3).

Labels visible in the diagram: “Mains, 380 V”; “minimum-voltage relay”; “220 V”; “control and measuring instruments”; “start”; “stop”; “magnetic starter”; \(Д_{40}\); \(Д_1\).

Fig. 3

All the mechanisms of the rig are mounted on a bed with guideways.^1) The closing gearbox 29 and the tested gearboxes 18 and 20 are fastened on special supports. When gearboxes 18 and 20 are changed, the supports are shifted along the bed guideways; this eliminates the need each time to align shafts 9 and 28.

A considerable relative displacement of the centers of shafts 8 and 32 is undesirable, since it leads to substantial friction losses in the articulated joints 10 and 33. To ensure the coaxiality of these shafts, provision must be made for changing the mutual angular position of the connecting couplings 11 and 27. In the rig under consideration this is made possible by the special design of the toothed couplings 11 and 27, which permit an angular displacement of the connected shafts through \(2^\circ\).

On the described rig, over a period of three years the author carried out tests of the gearboxes of the ER-5 hand-held mining electric drill. In operation the rig proved sufficiently reliable and convenient, and therefore it can be recommended for the experimental investigation of gear transmissions under variable loads.

Efficiency and moments in the links of the rig

To determine the forces on the teeth of the wheels of the closed loop, it is necessary to have relations by which the efficiency of the rig and the magnitudes of the torques on the shafts can be determined. Let us determine these relations.

We shall agree to denote the efficiencies by \(\eta\), the moments by \(M\), and the numbers of teeth of the wheels by \(z\), the numerical subscript of each designation corresponding to the part or unit according to Fig. 2 to which they refer.

Let us take the case of rotation in the rig in which gear wheels 6 and 7 rotate in the direction indicated by the arrows in Fig. 2.

In this case gear wheels 6 and 30 of the rig, as well as wheels 13, 16, 21, and 23 of the tested gearboxes, operate as driving wheels; whereas gear wheels 7 and 31 of the rig and gear wheels 15, 17, 22, and 25 of the tested gearboxes operate as driven wheels.

Fig. 4

From the equilibrium condition of gearbox 5 (Fig. 4) we have:

\[ M_5 - M_8 - M_9 + M_4 = 0. \tag{1} \]

Here \(M_5 = Q \cdot l\) is the moment applied to the housing of the loading gearbox 5;

^1) In our case the bed of a lathe was used.

\(Q\) — the weight of load 37 when testing under constant load, or the compression force of spring 45 when testing under variable load;

\(l\) — the distance from the point of application of force \(Q\) to the axis of support 39;

\(M_8\) — the moment with which shaft 8 acts on gear 6;

\(M_9\) — the moment with which shaft 9 acts on gear 7;

\(M_4\) — the moment transmitted to shaft 9 by shaft of gear 4.

Let us denote by \(M_7\) the moment with which gear 7 acts on shaft 9. The direction of this moment is determined by the direction of the circumferential force \(P_7\) on gear 7 and, in the present case, coincides with the direction of moment \(M_4\); therefore,

\[ M_9 = M_7 + M_4 . \tag{2} \]

From (1) and (2) we have:

\[ M_5 - M_8 - M_7 = 0 . \tag{3} \]

Since gear 6 is the driving gear and gear 7 is the driven gear, then:

\[ M_7 = M_8 \cdot \frac{z_7}{z_6} \cdot \eta_5 , \tag{4} \]

\[ M_8 = \frac{M_7}{\eta_5} \cdot \frac{z_6}{z_7} . \tag{5} \]

From (3) and (5) we have:

\[ M_7 = \frac{M_5}{1 + \dfrac{z_6}{z_7 \cdot \eta_5}} . \tag{6} \]

Since the power of electric motor 1 is expended only on covering the losses in the installation, the moment \(M_4\) transmitted to shaft 9 from electric motor 1 by gear 4 is numerically equal to the moment of resistance to rotation caused by these losses, reduced to shaft 9.

\[ M_4 = M_9(1 - \eta_{\mathrm{zk}}). \tag{7} \]

Hence the efficiency of the closed loop is:

\[ \eta_{\mathrm{zk}} = \frac{M_9 - M_4}{M_9}, \]

\[ \eta_{\mathrm{zk}} = \frac{M_7}{M_7 + M_4}. \tag{8} \]

But

\[ M_4 = M_1 \cdot \frac{z_4}{z_3} \cdot \eta_2, \tag{9} \]

therefore

\[ \eta_{\mathrm{zk}} = \frac{M_5}{ M_5 + \dfrac{M_1 \cdot z_4 \cdot \eta_2 (z_7 \cdot \eta_5 + z_6)} {z_3 \cdot z_7 \cdot \eta_5} }. \tag{10} \]

Since \( \eta_2 \simeq \eta_5 \) and \(1-\eta_5\) is small, (10) can be written in simplified form:

\[ \eta_{\mathrm{cl}}= \frac{M_5}{M_5+\dfrac{M_1\cdot z_1 (z_7+z_6)}{z_3\cdot z_7}}. \tag{11} \]

Let us now express the moments on all shafts of the closed loop through the known moment \(M_5\).

From equations (2), (3), (4), (6), and (8), we obtain:

\[ M_8= \frac{M_5}{1+\dfrac{z_7\cdot \eta_5}{z_6}}, \tag{12} \]

\[ M_9= \frac{M_5}{1+\dfrac{z_6}{z_7\cdot \eta_5}}\cdot \frac{1}{\eta_{\mathrm{cl}}}. \tag{13} \]

The remaining moments will be expressed as follows:

\[ M_{12}= \frac{M_5}{1+\dfrac{z_6}{z_7\cdot \eta_5}}\cdot \frac{\eta_{11}}{\eta_{\mathrm{cl}}}; \tag{14} \]

\[ M_{14}= \frac{M_5}{1+\dfrac{z_6}{z_7\cdot \eta_5}}\cdot \frac{\eta_{11}\cdot \eta_{13-15}}{\eta_{\mathrm{cl}}}\cdot \frac{z_{15}}{z_{13}}; \tag{15} \]

\[ M_{19}= \frac{M_5}{1+\dfrac{z_6}{z_7\cdot \eta_5}}\cdot \frac{\eta_{11}\cdot \eta_{18}}{\eta_{\mathrm{cl}}}\cdot \frac{z_{15}\cdot z_{17}}{z_{13}\cdot z_{16}}; \tag{16} \]

\[ M_{24}= \frac{M_5}{1+\dfrac{z_6}{z_7\cdot \eta_5}}\cdot \frac{\eta_{11}\cdot \eta_{18}\cdot \eta_{21-22}}{\eta_{\mathrm{cl}}}\cdot \frac{z_{15}}{z_{13}}. \tag{17} \]

Taking into account that

\[ \eta_{\mathrm{cl}}= \eta_{11}\cdot \eta_{18}\cdot \eta_{20}\cdot \eta_{27}\cdot \eta_{29}\cdot \eta_{33}\cdot \eta_{10}\cdot \eta_{5}, \tag{18} \]

we have:

\[ M_{26}= \frac{M_5}{1+\dfrac{z_6}{z_7\cdot \eta_5}}\cdot \frac{1}{\eta_5\cdot \eta_{10}\cdot \eta_{33}\cdot \eta_{29}\cdot \eta_{27}}; \tag{19} \]

\[ M_{28}= \frac{M_5}{1+\dfrac{z_6}{z_7\cdot \eta_5}}\cdot \frac{1}{\eta_5\cdot \eta_{10}\cdot \eta_{33}\cdot \eta_{29}}. \tag{20} \]

\[ M_{32}=\frac{M_{5}}{1+\dfrac{z_{7}\cdot \eta_{5}}{z_{6}}}\cdot \frac{1}{\eta_{10}\cdot \eta_{33}}; \tag{21} \]

\[ M_{34}=\frac{M_{5}}{1+\dfrac{z_{7}\cdot \eta_{5}}{z_{6}}}\cdot \frac{1}{\eta_{10}}. \tag{22} \]

Provided that shafts 8 and 32, as well as 9, 12, 26, and 28, are coaxial, the efficiencies of gear couplings 11 and 27 and of universal joints 10 and 33 are approximately equal to unity; therefore:

\[ \eta_{3K}=\eta_{18}\cdot \eta_{20}\cdot \eta_{29}\cdot \eta_{15}, \tag{23} \]

\[ M_{12}=M_{9}=\frac{M_{5}}{1+\dfrac{z_{6}}{z_{7}\cdot \eta_{5}}}\cdot \frac{1}{\eta_{3K}}; \tag{24} \]

\[ M_{14}=\frac{M_{5}}{1+\dfrac{z_{6}}{z_{7}\cdot \eta_{5}}}\cdot \frac{\eta_{13-15}}{\eta_{3K}}\cdot \frac{z_{15}}{z_{13}}; \tag{25} \]

\[ M_{19}=\frac{M_{5}}{1+\dfrac{z_{6}}{z_{7}\cdot \eta_{5}}}\cdot \frac{\eta_{18}}{\eta_{3K}}\cdot \frac{z_{15}\cdot z_{17}}{z_{13}\cdot z_{16}}; \tag{26} \]

\[ M_{24}=\frac{M_{5}}{1+\dfrac{z_{6}}{z_{7}\cdot \eta_{5}}}\cdot \frac{\eta_{18}\cdot \eta_{21-23}}{\eta_{3K}}\cdot \frac{z_{15}}{z_{13}}; \tag{27} \]

\[ M_{26}=M_{28}=\frac{M_{5}}{1+\dfrac{z_{6}}{z_{7}\cdot \eta_{5}}}\cdot \frac{1}{\eta_{15}\cdot \eta_{29}}; \tag{28} \]

\[ M_{32}=M_{34}=M_{8}=\frac{M_{5}}{1+\dfrac{z_{7}\cdot \eta_{5}}{z_{6}}}. \tag{29} \]

With the gear units under test 18 and 20 disconnected, when gear couplings 11 and 27 are connected directly by an intermediate shaft and the coaxiality of shafts 8 and 32, 9 and 28 is maintained, we have:

\[ \eta_{3K}=\eta_{29}\cdot \eta_{15}, \tag{30} \]

where

\[ M_{28}=M_{9}\quad \text{and}\quad M_{32}=M_{8}. \]

From this, \(\eta_{5}\) and \(\eta_{29}\) can be determined; then, after connecting the gear units under test, their efficiencies can be determined and, using the expressions given above, the torques on the shafts can be calculated, after which the forces on the teeth of the gears of the closed loop are determined from known relationships.

References

1. R. M. Brumberg. Testing gear transmissions by the closed-loop method, Vestnik inzhenerov i tekhnikov No. 4, 1936.

2. V. N. Kudryavtsev. Basic principles for determining the efficiency of compound transmissions, Proceedings of the Seminar on the Theory of Machines and Mechanisms, issue 32, Publishing House of the Academy of Sciences of the USSR, 1950.

3. M. I. Lysov. Cardan transmissions of the automobile, Methods of Testing the Automobile and Its Mechanisms, issue 2, Mashgiz, 1951.

4. G. K. Trubin. Contact fatigue of the teeth of straight-tooth gears, TsNIITMASH, book 37, Mashgiz, 1950.

5. G. Niemann and H. Glaubitz, Zahnflanken-festigkeit geradverzahnter Stirnräder, VDJ—Zeitschrift Bd. 93 Nr. 6, 21. Februar. 1951.

6. A. V. Osipyan. Investigation of the efficiency of a gear transmission, Vestnik metallopromyshlennosti No. 3, 1938.

7. I. I. Dymshits. Operation of automobile gearboxes on test rigs with a closed loop, Automobile and Tractor Industry No. 10, 1954.

8. A. I. Solov’ev. New closed-loop installations for testing automobile gearboxes, Automobile and Tractor Industry No. 10, 1954.

9. V. N. Kudryavtsev. Closed-loop installations for testing gear transmissions, Vestnik mashinostroeniya No. 10, 1951.