1 Establishment of vibration analysis physical model It can be found that in the coordinate system shown, all lateral vibrations in the x direction are not coupled with lateral vibration and circumferential vibration in the y direction, and vibration and gear wheels in the x direction The tooth engagement is independent. Since we mainly study the influence of the gear mesh excitation on the vibration performance of the reduction gearbox, the lateral vibration of the gear x direction can be ignored, so that the vibration model is reduced from 31 degrees of freedom to 19 degrees of freedom.
The expression of the mass matrix after the reduction of the vibration problem is as shown in Equation 1, and the stiffness matrix is ​​as shown in Equation 2.
Where, Ii,j are the moments of inertia corresponding to degrees; Mi,j are the masses corresponding to degrees; KT is the torque of the gearbox of the gearbox when the large pulley has a unit angle KTP is the torque added to overcome the resistance of the belt when the large pulley has a unit angle; K11, K11, K11! K11 is the force acting on the relevant coordinates of the high-speed on-axis gear, and R1, R21, R22 and R3 are the pitch circle radius of the high-speed shaft gear, the intermediate shaft large gear and the pinion and low-speed shaft gear respectively; KV1 KV2 is the meshing stiffness of the high-speed and low-speed gears respectively. KT1 is the torque required to be added to the gear when the high-speed shaft gear has a unit angle; K21, K21, K21, K21, K21 are the center of the intermediate shaft gear The force applied to the corresponding coordinates when the unit is laterally deformed, K22, K22, K22! , K22, K22 are the forces applied to the corresponding coordinates when the center of rotation of the intermediate shaft pinion has a unit lateral deformation; KT21, KT22 are the intermediate shaft and the small gear has a unit angle, respectively, the torque applied to the corresponding coordinates ; K3, K3 are the forces applied to the corresponding coordinates when the center of rotation of the low-speed gear has a lateral displacement; Kb12, Kb12, Kb22, Kb22, Kb31, Kb31 are the unit displacements of the high, medium and low shaft bearings. The force applied to the corresponding coordinates against the elasticity of the shaft; Kb1, Kb2, Kb3, the force exerted by the bearing on the shaft when there is a unit displacement at the high, medium and low speed shaft bearings.
By analyzing the damping characteristics of the system, a damping matrix similar to the stiffness matrix structure shown in Equation 2 can be obtained, as long as Kvi in ​​Equation 2 is replaced by Cvi, Kbi is replaced by Cbi, and Knij is replaced by Cnkij.
In order to save space, we omit the specific expression of the damping matrix.
Here, Cvi represents the meshing damping of the gear pair i (i = 1, 2), which can be calculated by Equation 3; Cbi is the damping of the bearing on the axis i (i = 1, 2, 3), which can be calculated by Equation 4, and Cnkij is the axis deformation. The structural damping is generally smaller than the bearing damping and gear damping, and can be ignored when making vibration analysis of the gearbox.
From the above analysis, the vibration of the double arc gear reduction box is a variable stiffness, variable damping, and multiple degrees of freedom with elastic and damping coupling.
3 Analysis of vibration characteristics of two-stage split-type double-arc gear reducer The fixed-frequency and vibration mode of the reducer can be obtained by using the above equation to obtain the first eight-order natural frequency of the gear-shaft vibration system of the reducer, and the third-order mode . It can be seen that each natural frequency only corresponds to a few degrees of freedom of vibration response. When running smoothly, the vibration excitation of the reduction gearbox is mainly a change in the meshing rigidity.
The speed of the low speed shaft of the above reduction gearbox is 9r/min. According to the gear parameters, the time taken for the low speed gear to complete an axial joint engagement (the period of the low speed stiffness change) is about 0.03s, and the high speed gear completes an axial direction. The time taken for the engagement of the circumference is about 0.00625 s. From the spectrum analysis of the vibration response signal, the basic period of the response signal is the same. The vibration condition of the gears on each axis is analyzed. It is found that the vibration of the low-speed shaft is mainly caused by the low-speed meshing excitation; the intermediate shaft large gear is less affected by the change of the meshing stiffness, and is mainly caused by the high-speed meshing excitation; the pinion is subject to the meshing stiffness change. The influence is large, mainly the result of low-speed meshing excitation; the vibration of high-speed shaft is mainly caused by high-speed meshing excitation, but it is also obviously affected by low-speed meshing excitation. It can be seen that the meshing state of the low speed grade is the main factor affecting the vibration characteristics of the gearbox.
When performing the vibration diagnosis of the gearbox, we judge the fault condition of the internal gears and bearings of the gearbox by detecting the vibration signal at the casing bearing. Therefore, it is meaningful to analyze the vibration response of the bearing and the lateral vibration of the gear and the lateral vibration of the center of the gear rotation, which is meaningful for correct gear fault diagnosis. A large number of vibration analysis examples show that the vibration response at the bearing is the best with the lateral vibration of the center of rotation of the gear, and the consistency with the circumferential vibration of the gear is poor.
The vibration characteristics of the large pulley can be found from Equation 3, and the large pulley is relatively independent of the circumferential vibration of the high-speed gear; therefore, its vibration response should be relatively independent. It can be seen from the time-domain response curve of the circumferential vibration of the pulley that the vibration is an approximate sinusoidal curve, and its energy is concentrated at about 75 Hz, which is the natural frequency of the vibration system composed of the pulley and the high-speed gear. Since the influence of structural damping is not considered, the vibration waveform is not attenuated.
The expression of the mass matrix after the reduction of the vibration problem is as shown in Equation 1, and the stiffness matrix is ​​as shown in Equation 2.
Where, Ii,j are the moments of inertia corresponding to degrees; Mi,j are the masses corresponding to degrees; KT is the torque of the gearbox of the gearbox when the large pulley has a unit angle KTP is the torque added to overcome the resistance of the belt when the large pulley has a unit angle; K11, K11, K11! K11 is the force acting on the relevant coordinates of the high-speed on-axis gear, and R1, R21, R22 and R3 are the pitch circle radius of the high-speed shaft gear, the intermediate shaft large gear and the pinion and low-speed shaft gear respectively; KV1 KV2 is the meshing stiffness of the high-speed and low-speed gears respectively. KT1 is the torque required to be added to the gear when the high-speed shaft gear has a unit angle; K21, K21, K21, K21, K21 are the center of the intermediate shaft gear The force applied to the corresponding coordinates when the unit is laterally deformed, K22, K22, K22! , K22, K22 are the forces applied to the corresponding coordinates when the center of rotation of the intermediate shaft pinion has a unit lateral deformation; KT21, KT22 are the intermediate shaft and the small gear has a unit angle, respectively, the torque applied to the corresponding coordinates ; K3, K3 are the forces applied to the corresponding coordinates when the center of rotation of the low-speed gear has a lateral displacement; Kb12, Kb12, Kb22, Kb22, Kb31, Kb31 are the unit displacements of the high, medium and low shaft bearings. The force applied to the corresponding coordinates against the elasticity of the shaft; Kb1, Kb2, Kb3, the force exerted by the bearing on the shaft when there is a unit displacement at the high, medium and low speed shaft bearings.
By analyzing the damping characteristics of the system, a damping matrix similar to the stiffness matrix structure shown in Equation 2 can be obtained, as long as Kvi in ​​Equation 2 is replaced by Cvi, Kbi is replaced by Cbi, and Knij is replaced by Cnkij.
In order to save space, we omit the specific expression of the damping matrix.
Here, Cvi represents the meshing damping of the gear pair i (i = 1, 2), which can be calculated by Equation 3; Cbi is the damping of the bearing on the axis i (i = 1, 2, 3), which can be calculated by Equation 4, and Cnkij is the axis deformation. The structural damping is generally smaller than the bearing damping and gear damping, and can be ignored when making vibration analysis of the gearbox.
From the above analysis, the vibration of the double arc gear reduction box is a variable stiffness, variable damping, and multiple degrees of freedom with elastic and damping coupling.
3 Analysis of vibration characteristics of two-stage split-type double-arc gear reducer The fixed-frequency and vibration mode of the reducer can be obtained by using the above equation to obtain the first eight-order natural frequency of the gear-shaft vibration system of the reducer, and the third-order mode . It can be seen that each natural frequency only corresponds to a few degrees of freedom of vibration response. When running smoothly, the vibration excitation of the reduction gearbox is mainly a change in the meshing rigidity.
The speed of the low speed shaft of the above reduction gearbox is 9r/min. According to the gear parameters, the time taken for the low speed gear to complete an axial joint engagement (the period of the low speed stiffness change) is about 0.03s, and the high speed gear completes an axial direction. The time taken for the engagement of the circumference is about 0.00625 s. From the spectrum analysis of the vibration response signal, the basic period of the response signal is the same. The vibration condition of the gears on each axis is analyzed. It is found that the vibration of the low-speed shaft is mainly caused by the low-speed meshing excitation; the intermediate shaft large gear is less affected by the change of the meshing stiffness, and is mainly caused by the high-speed meshing excitation; the pinion is subject to the meshing stiffness change. The influence is large, mainly the result of low-speed meshing excitation; the vibration of high-speed shaft is mainly caused by high-speed meshing excitation, but it is also obviously affected by low-speed meshing excitation. It can be seen that the meshing state of the low speed grade is the main factor affecting the vibration characteristics of the gearbox.
When performing the vibration diagnosis of the gearbox, we judge the fault condition of the internal gears and bearings of the gearbox by detecting the vibration signal at the casing bearing. Therefore, it is meaningful to analyze the vibration response of the bearing and the lateral vibration of the gear and the lateral vibration of the center of the gear rotation, which is meaningful for correct gear fault diagnosis. A large number of vibration analysis examples show that the vibration response at the bearing is the best with the lateral vibration of the center of rotation of the gear, and the consistency with the circumferential vibration of the gear is poor.
The vibration characteristics of the large pulley can be found from Equation 3, and the large pulley is relatively independent of the circumferential vibration of the high-speed gear; therefore, its vibration response should be relatively independent. It can be seen from the time-domain response curve of the circumferential vibration of the pulley that the vibration is an approximate sinusoidal curve, and its energy is concentrated at about 75 Hz, which is the natural frequency of the vibration system composed of the pulley and the high-speed gear. Since the influence of structural damping is not considered, the vibration waveform is not attenuated.
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