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**I. Proposal of the problem**

It is generally considered that the large-cycle error of the gear is caused by the combination of geometric eccentricity and motion eccentricity, which is an error of one cycle of the gear. When the gear hobbing machine was used as a gear machining experiment, the authors found that the large cycle error of gears produced was a "double-peak" phenomenon (see Figure 1, the figure shows the number of teeth from the starting point), rather than the gear described in the general literature. Turn into a periodic sine wave pattern.

Fig. 1 Gear large cycle error

Since the coaxiality of the workpiece (see Figure 2) is very accurate, and the measurement reference is consistent with the machining reference, the "double peak" error is not related to errors such as installation eccentricity and inconsistent reference.

Figure 2 tooth blank specimen

During the test, the workpiece is mounted on the machine tool and the outer radial runout of the workpiece is measured. The measurement result is shown in Figure 3. It can be seen from Fig. 3 that the maximum radial runout occurs at the diameter of the workpiece. Since the roundness of the workpiece is very good, and the maximum value of the radial runout is basically at the radial position, the diameter shown in Fig. 3 can be judged. When the runout is actually one rotation of the table, the rotation axis of the table is reflected in the measurement position.

Fig. 3 Radial runout measurement of workpiece outer circle

**Second, the experimental conditions**

1. Processing machine: Y3150E hobbing machine.

2. Measuring instrument: 891E gear testing center, measuring error 0.001mm.

3. Machined workpiece: m=2, Z=73 spur gear (see Figure 2).

4. Adjustment and processing schematic (see Figure 4).

Figure 4 processing adjustment

5. Installation: double top.

**Third, experimental theory analysis**

The so-called workbench axis shake error, which is the amount of shaking of the actual axis of the hobbing machine rotary table near the theoretical axis, can be described by the Fourier series. Since the indexing worm gear and the workpiece are centered on the axis of the table, the wobbling of the axis of the table will cause the following two errors:

1. Caused by the pulsation caused by the center distance between the tool and the workpiece Ao error [1], the axis error vector of the worktable axis can be expressed as:

In the formula â€”â€”Working axis sway error eaâ€”The error value an described by P-order Fourier seriesâ€”The error amplitude Î¸n of each orderâ€”The initial phase angle of each order error (Ï†)â€”turning unit vector Ï†â€”turning angle nâ€”order of Fourier series (n=1, 2, ..., P)

due to The existence of the pulsation between the center distance between the workpiece and the tool is essentially similar to the installation eccentricity, but its frequency component is much more complicated than the installation eccentricity.

Fig. 5 is a schematic diagram of transmission error formation in a fan gear, where P is a theoretical meshing point; Indicates the left and right meshing lines; Î± is the gear meshing angle; Ï† is the fan gear rotation angle. Because of There are left and right tooth profile transmission errors as follows:

Left tooth profile transmission error:

Right tooth profile transmission error:

From the above equation, the radial error and tangential error are:

Fig. 5 Schematic diagram of transmission error when fan gear

Two special cases are discussed below:

1) When n=0 (equal to the eccentricity of the table installation)

=aocosÎ¸o (Ï†)

At this time, the axis motion error trajectory is a circle, and the center of the circle is the theoretical center of rotation as shown in Fig. 6a. The transmission error caused by this situation is exactly the same as the installation eccentricity.

At this point, the transmission error is:

Î´faL=aocosÎ¸osin(Ï†+Î±)

Î´faR=-aocosÎ¸osin(Ï†-Î±)

2) When n=1 (ie similar to the previous experiment)

=[aocosÎ¸o+a1cos(Î¸1+Ï†)] (Ï†)

For simplification, remove the influence of installation eccentricity, set a0 = 0 and do not consider the initial phase angle. then:

=a1cosÏ† (Ï†)

The axis motion error trajectory is still a circle, but the center of the circle has moved to the X axis as shown in Figure 6b. At this time, the transmission error is:

Radial and tangential errors are:

Figure 6 Axis shaking error trajectory

It can be seen that quadratic error components appear in transmission error, radial error, and tangential error.

2. Caused by the error of the pitch worm and the worm center pitch Af pulsation According to the reference [1], [2], after deduction, due to the pulsation of the Af caused uneven indexing worm wheel rotation, resulting in workpiece circle pitch radius error:

In the formula, Roâ€”workpiece pitch circle radius Rfâ€”index worm gear pitch circle radius left tooth profile transmission error: Î´ffL=cosÎ±âˆ«2Ï€oÎ´RdÏ†

Right tooth profile transmission error: Î´ffR=-cosÎ±âˆ«2Ï€oÎ´RdÏ†

Radial error: Î´fr=0

Tangential error: Î´ft=2âˆ«2Ï€oÎ´RdÏ†

It can be seen that the Af error at this time is similar to the eccentricity of motion.

Discuss two cases separately 1) When n=1, that is, the left-hand tooth profile transmission error as shown in the experiment: Î´ffL=CcosÎ±{-aocosÎ¸ocosÏ†-

Right tooth profile transmission error: Î´ffR=-Î´ffL

Radial error: Î´fr=0

Tangential error: Î´ft=-2C{a0cosÎ¸ocosÏ†+

Transmission error contains secondary error components.

2) When n>1

From the above equation, we can see that when n>1, the magnitudes of Î´ffL, Î´ffR, and Î´ft are . When n increases, the amplitude will attenuate rapidly, which means that the spindle oscillation error is caused by the transmission error and tangential error caused by indexing the worm gear channel. When n is small, that is, when the large cycle error is significant, it is high. The magnitude of the secondary error decays rapidly and has little effect on transmission error and tangential error.

3. Integration of Ao and Af pulsation errors due to The above-mentioned two errors are finally reflected in the comprehensive effect of the workpiece transmission error, which is their linear superposition. which is:

Left transmission error Î´fL = Î´faL + Î´ffL

Right transmission error Î´fR = Î´faR + Î´ffR

The transmission error after synthesis will present a very complicated situation.

**Fourth, experimental verification**

A series of measurements and tooth cutting tests were performed on tooth blanks with different moduli and different numbers of teeth (according to Figure 2). Tests have shown that:

1. The radial runout and phase of the measured gear blank are in good agreement with the measurement result of Z=73 tooth blank shown in FIG. 3 .

2. The large period error of measuring gears after gear cutting is also characterized by a "double peak", and the phase is also very consistent with that shown in Figure 1.

Taking processing of Z=73 tooth blank as an example, referring to FIG. 4 , it can be seen that the measuring point lags the processing area by about 90Â° (ie, the corresponding processing 73 teeth are about 18 teeth), and the corresponding relationship is:

When Z measurement <18 teeth Z processing = 54.75 + Z When Z measurement> 18 teeth Z processing = Z measurement - 18.25

From Fig. 3 and Fig. 4, it can be seen that when the external radial runout of the test specimen is increased from 0 to +1 (corresponding to the 10 tooth position), the right tooth surface of the processing corresponding to the tooth is continuously thinned to 65 teeth. Lowest; Similarly, when the outer radial runout increases from 0 to +1 (corresponding to the 50 tooth position), the processing area corresponding to the processing of each tooth's right tooth surface continues to thin, to 32 teeth is the lowest, which is exactly the error curve of Figure 1 The two low-lying areas correspond. In the same way, two high areas of the error curve can also be analyzed. Since there are two high points and two low points in a continuous error curve, it can be confirmed that there is a second harmonic in the error curve, which means that the axis error of the worktable will cause the gear to be â€œmultimodalâ€ (ie, multiple times The large period error of harmonics. Consistent with theoretical analysis.

**V. Conclusion**

In summary, the sloshing of the rotary axis of the table of the gear hobbing machine will affect the large period error of the gear, and the low-frequency component of this axis wobble will affect the accumulated error of the weekly section. When n=1, the double-peak characteristic pattern as shown in FIG. 1 is caused. In the design, manufacture, and inspection of the gear machining machine, it is necessary to pay attention to detecting and controlling the accuracy of the axis shake of the worktable.

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