On the Testing of Transmissions with Intersecting Axes on Closed-Loop Test Rigs
Abstract Generated abstract
The paper addresses the design and analytical treatment of closed-loop test rigs for gear transmissions with intersecting axes, focusing on ball transmissions and bevel-type meshes. It compares two rig layouts, one for a 90 degree shaft-intersection angle and another adaptable to 30 and 60 degrees, and describes their loading, drive, reconfiguration, and measurement arrangements. The authors derive relations for determining mesh efficiency and shaft torques under both directions of rotation, showing that the two-housing scheme used for the 90 degree case leads to a simpler quadratic efficiency equation and more accurate determination when structural constraints permit.
Full Text
ON THE QUESTION OF TESTING TRANSMISSIONS WITH INTERSECTING AXES ON CLOSED-LOOP TEST RIGS
A. E. BELYAEV, Yu. S. SEMENOV, T. Yu. ISHCHENKO, V. M. ISHCHENKO
(Presented by the scientific seminar of the Department of Applied Mechanics)
In view of the known advantages of closed-loop rigs over open-loop ones [1, 2], they have become widely used in tests of gear transmissions. In creating a rig intended for testing ball transmissions with intersecting axes, developed at the Department of Applied Mechanics of TPI, certain difficulties arose in deriving analytical relationships for determining the efficiency of the transmissions. These difficulties are due to the fact that, unlike closed-loop rigs for testing cylindrical transmissions, in rigs for testing transmissions with intersecting axes it is impossible to limit oneself to two pairs of gears; at least four pairs are necessary. In addition, since the rigs are intended for testing a new type of transmission, it was necessary to provide for the possibility of studying the influence of such factors as variation of the angle of intersection of the axes and variation of the transmission ratio, which caused additional difficulties in the structural design of the test stand.
The purpose of the present article is to show the variants proposed by the authors for solving these questions.
An analysis of possible rig layouts showed that structurally implementing a change in the angle of intersection of the axes $\varphi$ (with different transmission ratios, for example: $i = 2 \div 6$) in one rig over wide limits $(0 \div 90^\circ)$ is practically impossible. In this connection, three values of the angle $\varphi$ were selected $(90^\circ, 60^\circ,$ and $30^\circ)$, which were implemented on two separate rigs: one for $\varphi = 90^\circ$, the other for $\varphi = 30$ and $60^\circ$. The reason for this was that $\varphi = 90^\circ$ is the most widespread case in machine-building practice, while angles of $30^\circ$ and $60^\circ$ are conveniently implemented in one rig; moreover, these values of $\varphi$ are intermediate between $\varphi = 0$ (transmissions with parallel axes) and $\varphi = 90^\circ$.
The layouts of the rigs are shown in Fig. 1. As can be seen from Fig. 1, the rigs are made in different structural designs. The rig for $\varphi = 90^\circ$ (Fig. 1, a) consists of two identical reducers, each of which includes two pairs of the transmissions being tested. This makes it possible to obtain a high compactness of the rig and, as shown below, to avoid, in determining the efficiency, a complex analytical dependence that has no rigorous solution.
In the rig shown in Fig. 1, b (for $\varphi = 30^\circ$ and $60^\circ$), it proved structurally more convenient to place each of the tested transmissions in a separate...
...in a one-piece housing, since otherwise the configuration of the housing becomes very complex.
Since the operating principle of closed-loop rigs for testing gear transmissions is fairly widely known in the technical literature, and the conventional designations of the parts (with the exception of ball transmissions) in Fig. 1 correspond to the standard ones, we shall confine ourselves only to a brief description of the rigs.
Fig. 1. Diagrams of rigs for testing transmissions with intersecting axes:
a) at an axis-intersection angle \(\varphi = 90^\circ\); b) at axis-intersection angles \(\varphi = 30^\circ\) and \(60^\circ\).
Loading of the closed loop of the rigs is carried out by the method of Prof. V. N. Kudryavtsev [1], using loads \(Q\) (12 and 16, Fig. 1, a) and 1, b, respectively), applied to the levers of the housings of the loading gear reducers. The latter are suspended in supports on ball bearings and can rotate relative to the axes of the supports. The diagrams of the forces acting on the elements of the transmissions are shown in Fig. 1 (the force \(P\), with the corresponding subscript, is applied to the wheel; the force \(P'\), with the corresponding subscript, to the pinion shaft).
The drive of the tested wheels of the rigs is effected by means of direct-current electric motors 1, which makes it possible to vary the speed...
rotation over wide ranges. The power of electric motor 1 is expended only to cover the various losses in the rig. To measure the torque, and consequently also the power consumed, the stator of the electric motor is suspended on rolling bearings.
In the rig shown in Fig. 1, b, the housings of the gearboxes are adapted both for transmissions with an angle \(\varphi = 30^\circ\) and for transmissions with \(\varphi = 60^\circ\). The bearings of the pinion shaft of each gearbox are placed in a special cup; on the opposite ends of each gearbox, seats are bored for cups of the same diameter: one for the case \(\varphi = 30^\circ\), the other for \(\varphi = 60^\circ\). Therefore, when replacing a mesh with an axis-intersection angle of \(30^\circ\) by one of \(60^\circ\), it is sufficient to move the cup with the pinion shaft from one end of the gearbox housing to the opposite end; the position of the gear relative to the gearbox housing does not change. The pinion shafts are identical for both types of mesh; only the gears differ.
The gearboxes and electric motors of the rig (Fig. 1, b) are mounted on two plates 14 and 15. When the rig is reconfigured for testing transmissions from one angle \(\varphi\) to another, the universal joints that connect, by means of intermediate shafts, pinion shafts 3, 4 and 9, 10 are disconnected from the latter; in the gearboxes, the corresponding replacement of the transmissions for another angle \(\varphi\) is carried out; then plates 14 and 15 (together with the gearboxes and electric motor fastened to them) are rotated on the frame of the rig by \(90^\circ\) (clockwise relative to the position shown in Fig. 1, b), and pinion shafts 3, 4 and 9, 10 are again connected by means of the same universal joints and intermediate shafts.
The rigs make it possible to test spherical ball and involute bevel meshing for contact and bending strength and wear resistance, and also make it possible to determine the efficiency of the mesh.
To determine the forces in the transmissions being tested and the efficiency of the meshes, it is necessary to have relations that make it possible to obtain the efficiency of the transmissions and the magnitudes of the torques on the shafts.
As is known, in driving gear wheels the forces in the mesh are directed opposite to their rotation, while in driven wheels they are directed along the rotation. According to the force diagrams in the closed loop (Fig. 1), when the electric-motor shaft rotates clockwise (\(\omega_1\)), the driving gear wheels are 3, 5, 8, 10 (Fig. 1, a) and 2, 4, 8, 10 (Fig. 1, b), while the driven wheels are 4, 6, 9, 11 (Fig. 1, a) and 3, 5, 9, 11 (Fig. 1, b). The direction of circulation of the power flow (\(N_{\mathrm{c}}\)) in the closed loop is indicated by arrows in Fig. 1. When the direction of rotation is reversed, the driving and driven wheels change places, and the directions of circulation of the power flow are reversed.
If the efficiencies of speed-reducing and speed-increasing transmissions are taken to be the same, as is done when determining the efficiency of involute transmissions on closed-loop rigs, then formulas can be obtained for determining the efficiency of the transmissions of the closed loop, as well as the torques on the shafts for different directions of rotation (Table 1).
Since, in the rig shown in Fig. 1, a, the transmissions being tested are mounted in only two gearbox housings and the angular velocities of the input shaft 5 and output shaft 9 of the loading gearbox are identical, then when determining the efficiency of one transmission \((\eta)\), irrespective of the direction of rotation of electric-motor shaft 1, we obtain one and the same quadratic equation
\[ M_{13}\cdot i\cdot \eta^2 + M_1\cdot \eta - M_{12}\cdot i = 0, \tag{1} \]
which is easily solved, taking into account the fact that \(\eta\) can only be a positive quantity.
In the setup shown in Fig. 1, b), each transmission is made as a separate reducer; the input shaft 7 and output shaft 9 of the loading reducer have different angular velocities, and therefore, when determining \(\eta\), we obtain (depending on the direction of rotation of the shaft of electric motor 1) different equations.
For clockwise rotation (\(\omega_1\), Fig. 1b)
\[ M_{16}\cdot i\cdot \eta^4 - M_1\cdot \eta^3 + M_1\cdot i\cdot \eta^2 - M_{16}\cdot i = 0, \tag{2} \]
for counterclockwise rotation
\[ M_{16}\cdot i\cdot \eta^4 + M'_1\cdot i\cdot \eta^2 - M'_1\cdot \eta - M_{16}\cdot i = 0. \tag{3} \]
Equations 2 and 3 do not have an exact solution; they can be solved only approximately (for example, with the aid of Cardano’s equation). The following procedure may also be proposed. In testing transmissions, at the moment of determining the efficiency, the torque \(M\) is measured not only for the adopted direction of rotation, but also for the opposite one; then equations 2 and 3 can be solved simultaneously. The result of such a solution is given in Table 1.
Table 1
| Angles of intersection of axes | Expression for efficiency |
|---|---|
| \(\varphi = 90^\circ\) | \(\displaystyle \eta=\frac{-M_1+\sqrt{M_1^2+4M_1^{\prime 2}\cdot i^2}}{2M_{16}\cdot i}\) |
| \(\varphi = 30\) and \(60^\circ\) | \(\displaystyle \eta=\frac{(M_1-M'_1)\cdot i+\sqrt{\left[(M_1-M'_1)\cdot i\right]^2+4M_1\cdot M'_1\cdot \cos^2\varphi}}{2M_1\cdot \cos\varphi}\) |
| Torques on the shafts when the motor shaft \(\omega_1\) rotates | Torques on the shafts when the motor shaft \(\omega_1\) rotates |
|---|---|
| Clockwise | Counterclockwise |
| \(\varphi = 90^\circ\) | \(\varphi = 90^\circ\) |
| \(\displaystyle M_2=\frac{M_{12}\cdot i}{(1+\eta^2)\cdot \eta}\) \(\displaystyle M_4=M_5=\frac{M_{12}}{1+\eta^2}\) \(\displaystyle M_7=\frac{M_{12}\cdot i\cdot \eta}{1+\eta^2}\) \(\displaystyle M_9=M_{10}=\frac{M_{12}\cdot \eta^2}{1+\eta^2}\) |
\(\displaystyle M_2=\frac{M_{12}\cdot i\cdot \eta^3}{1+\eta^2}\) \(\displaystyle M_4=M_5=\frac{M_{12}\cdot \eta^2}{1+\eta^2}\) \(\displaystyle M_7=\frac{M_{12}\cdot i\cdot \eta}{1+\eta^2}\) \(\displaystyle M_9=M_{10}=\frac{M_{12}}{1+\eta^2}\) |
| \(\varphi = 30\) and \(60^\circ\) | \(\varphi = 30\) and \(60^\circ\) |
| \(\displaystyle M_3=M_4=\frac{M_{16}}{(i-\eta\cos\varphi)\cdot \eta}\) \(\displaystyle M_6=M_7=\frac{M_{16}\cdot i}{i-\eta\cos\varphi}\) \(\displaystyle M_9=M_{10}=\frac{M_{16}\cdot \eta}{i-\eta\cos\varphi}\) \(\displaystyle M_{12}=M_{13}=\frac{M_{16}\cdot i-\eta^2}{i-\eta\cos\varphi}\) |
\(\displaystyle M_3=M_4=\frac{M_{16}\cdot \eta^3}{i\cdot \eta-\cos\varphi}\) \(\displaystyle M_6=M_7=\frac{M_{16}\cdot i\cdot \eta}{i\cdot \eta-\cos\varphi}\) \(\displaystyle M_9=M_{10}=\frac{M_{16}}{i\cdot \eta-\cos\varphi}\) \(\displaystyle M_{12}=M_{13}=\frac{M_{16}\cdot i}{(i\cdot \eta-\cos\varphi)\cdot \eta}\) |
In formulas 1, 2, 3 and in the formulas presented in Table 1, the subscripts of the moments on the shafts correspond to the designations of the shafts in Fig. 1, and \(M_1\) is the moment on the shaft of electric motor 1. When determining \(\eta\) for \(\varphi = 30^\circ\) and \(60^\circ\), \(M_1\) is for clockwise rotation, while \(M_1'\) is for counterclockwise rotation. \(M_{12}\) and \(M_{16}\) are the moments on the housings of the loading reducers, produced by weights 12 and 16 (Fig. 1, a) and 1, b), respectively), relative to the axes of the supports from which the housings of the loading reducers are suspended; \(i\) is the gear ratio from the pinion shaft to the wheel.
The moments on the wheels of the closed loop, with the exception of wheels 11 and 2 (Fig. 1, a) and 1, b), respectively), are equal to the moments on the corresponding shafts. The moments on wheels 11 and 2, since they are located next to electric motor 1, are equal to the algebraic sum of the moments of shaft 2 or 13 (Fig. 1, a) and Fig. 1, b), respectively) and the moment \(M_1\) on the motor shaft.
On the basis of the analysis carried out, it may be concluded that, unless considerations of a structural nature prevail, installations for testing transmissions with intersecting axes should preferably be made according to the scheme of Fig. 1, a) (with two reducer housings), since in this case the efficiency of the transmission is determined with greater accuracy.
REFERENCES
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V. N. Kudryavtsev. Closed-loop installations for testing gear transmissions. Moscow, Vestnik mashinostroeniya, No. 10, 1951.
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Yu. S. Semenov. A closed-loop installation for testing variable-load gear transmissions. Izvestiya TPI, vol. 96, part 1, Tomsk State University Press, Tomsk, 1959.