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摘要: 提出了基于雙偏心誤差齒輪副的驅動齒面與齒背面（雙齒面）無負載傳動誤差計算模型，建立與時變側隙計算公式的等價關系，從理論上證明了基于雙齒面傳動誤差的側隙測量方法。通過實驗方法測量不同負載力矩、不同初始嚙合面的雙面傳動誤差并獲得相應載荷下的初始回差。基于雙齒面傳動誤差實驗曲線，實現了對齒輪副整個大周期側隙的連續測量與預測。結果表明，連續側隙曲線與機械滯后回差法測量結果吻合良好，而側隙預測較好地反應了側隙值變化范圍和變化趨勢。同時，側隙連續測量方法及側隙預測均證明了理論模型的正確性，提高了側隙測量效率并獲得了更全面的側隙數據，對齒輪傳動的非線性研究、消隙控制以及齒輪精度研究等均具有指導意義和參考價值。Abstract: Transmission error in gear system and backlash are important factors that affect the accuracy of precision transmission systems. The main sources of the complete cycle transmission errors and periodic backlash are eccentric errors due to imperfections in machining and assembling. Therefore, analyzing the transmission error in gear system and backlash under the effect of eccentricity error is necessary. Scholars from around the world have conducted extensive research on the transmission error of the gear system. They observed that the back-side contact of the gear tooth generally happens in high-speed light-load conditions, or when alternate load torque occurs or anti-backlash gear has been used. However, the existing literature does not evidently show the calculation method of the transmission error of the back-side tooth mesh. Therefore, analyzing the transmission error of the back-side tooth mesh is of great significance. The present work develops a calculation model of drive- and back-side (two-sided) transmission error of an involute gear pair with eccentricities to construct the equivalence of time-varying backlash calculation formula to demonstrate the continuous measurement of gear backlash based on two-sided transmission error that can also be realized in theory. The experimental transmission error and the corresponding hysteresis were acquired under various load torques and different initial conditions, such as initial position of the gear unit. The continuous backlash curve of the gear pair was obtained from the two-sided transmission curves, and the backlash of the whole cycle could be predicted. The result of the experiment shows that the continuous measurement curve agrees with the mechanical hysteresis method, and the prediction of backlash perfectly reflects the variation range and trend of backlash. Moreover, both the continuous measurement method of backlash and prediction of backlash demonstrate that the model is practicable and have higher efficiency, and the overall data retain instructional significance, reference value for nonlinear research, anti-backlash control, and gear accuracy of gear transmission.
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圖 1 主動輪偏心誤差下嚙合線增量計算模型。（a）嚙合齒面為驅動齒面；（b）嚙合齒面為齒背面
Figure 1. Engagement increment calculation model with eccentric error of pinion: (a) drive-side meshing; (b) back-side meshing
圖 2 從動輪偏心誤差下嚙合線增量計算模型。（a）嚙合齒面為驅動齒面；（b）嚙合齒面為齒背面
Figure 2. Engagement increment calculation model with eccentric error of driven gear: (a) drive-side meshing; (b) back-side meshing
圖 3 傳動誤差及側隙測量實驗裝置圖
Figure 3. Experimental device for transmission error and backlash measurement
圖 4 驅動齒面傳動誤差測量起始位置示意圖。（a）初始接觸齒面為驅動齒面；（b）初始接觸齒面為齒背面
Figure 4. Schematic of the initial position of the drive-side transmission error measurement: (a) drive-side contact in initial position; (b) back-side contact in initial position
圖 5 齒背面傳動誤差測量起始位置示意圖。（a）初始接觸齒面為驅動齒面；（b）初始接觸齒面為齒背面
Figure 5. Schematic of the initial position of the back-side transmission error measurement: (a) drive-side contact in initial position; (b) back-side contact in initial position
圖 6 驅動齒面傳動誤差測量且初始時刻驅動齒面接觸。（a）傳動誤差對比曲線；（b）各負載與其初始回差對比
Figure 6. Drive-side transmission error measurement with drive-side contact at initial position: (a) comparison of transmission error curves; (b) comparison of each load torque with its corresponding hysteresis
圖 7 驅動齒面傳動誤差測量且初始時刻齒背面接觸。（a）傳動誤差對比曲線；（b）各負載與其初始回差對比
Figure 7. Drive-side transmission error measurement with back-side contact at initial position: (a) comparison of transmission error curves; (b) comparison of each load torque with its corresponding hysteresis
圖 8 齒背面傳動誤差測量且初始時刻驅動齒面接觸。（a）傳動誤差對比曲線；（b）各負載與其初始回差對比
Figure 8. Back-side transmission error measurement with drive-side contact at initial position: (a) comparison of transmission error curves; (b) comparison of each load torque with its corresponding hysteresis
圖 9 齒背面傳動誤差測量且初始時刻齒背面接觸。（a）傳動誤差對比曲線；（b）各負載與其初始回差對比
Figure 9. Back-side transmission error measurement with back-side contact at initial position: (a) comparison of transmission error curves; (b) comparison of each load torque with its corresponding hysteresis.
圖 10 雙面傳動誤差與側隙連續曲線
Figure 10. Two-sided transmission error and continuous backlash curve
圖 11 齒背面傳動誤差實驗及擬合曲線
Figure 11. Experimental and fitting curve of back-side transmission error
圖 13 機械滯后回差法測量原理
Figure 13. Measurement principle of mechanical return backlash method
表 1 齒輪副基本參數
Table 1. Basic parameters of gear pair
System parameters Value Tooth number of driving gear 45 Tooth number of driven gear 80 Module/mm 2 Pressure angle/(°) 20 Tooth width/mm 20 Addendum circle diameter of driving gear/mm 94 Dedendum circle diameter of driving gear/mm 85 Addendum circle diameter of driven gear/mm 164 Dedendum circle diameter of driven gear/mm 155 Driving speed/(r·min?1) 10 Loading torque/(N?m) 0 2.5 5 7.5 10 
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表 2 各部分理論剛度
Table 2. Theoretical stiffness of each part
Part Theoretical stiffness Gear 20 N?(μm?mm)?1 Shaft 289.44 N?m?(°)?1 Spline 9.768×107?P1/3 N?m?1 Coupling 1.87×104 N?m?rad?1
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表 3 不同負載力矩下，各部分變形數據
Table 3. Deformation data of various parts under different loads
Part Theoretical deformation/(10?3 °) 2.5 N?m 5 N?m 7.5 N?m 10 N?m Gear 0.06 0.11 0.17 0.22 Shaft 8.64 17.27 25.91 34.55 Spline 1.06 1.69 2.21 2.68 Coupling 7.66 15.32 22.98 30.64 Theoretical total deformation 17.42 34.39 51.27 68.09 Experimental total deformation Drive-side 15 30 48 65 Back-side 15 31 50 75
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