Abstract
In this paper, a simplified model is studied to predict analytically the vibration from the helical gear system due to an axial excitation of helical gears. The simplified model describes gear, shaft, bearing, and housing. In order to obtain the axial force of helical gears, the mesh stiffness is calculated in the load deflection relation. The axial force is obtained from the solution of the equation of motion, using the mesh stiffness. It is used as a longitudinal excitation of the shaft, which in turn drives the gear housing through the bearing. In this study, the shaft is modeled as a rod, while the bearing is modeled as a parallel spring and damper only supporting longitudinal forces. The gear housing is modeled as a clamped circular plate with viscous damping. For the modeling of this system, transfer matrices for the rod and bearing are used, using a spectral method with four pole parameters. The model is validated by finite element analysis. Using the model, parameter studies are carried out. As a result, the linearized dynamic shaft force due to the gear excitation in the frequency domain was proposed. Out-of-plan displacement from the forced vibrating circular plate and the renewed mode normalization constant of the circular plate were also proposed. In order to control the axial vibration of the helical gear system, the plate was more important than the shaft and the bearing. Finally, the effect of the dominant design parameters for the gear system can be investigated by this model.
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Abbreviations
- a :
-
The radius of the circular plate
- C :
-
Damping
- e :
-
Tooth error
- E :
-
Modulus of elasticity
- F :
-
Force
- I m :
-
Modified Bessel functions of the first kind of orderm
- J m :
-
Bessel functions of the first kind of orderm
- j :
-
Complex number
- J :
-
Mass moment of inertia
- h :
-
Plate thickness
- K :
-
Stiffness
- k:
-
Wave number
- l :
-
The length of the shaft
- M :
-
Mass
- R b :
-
Base circle radius
- t :
-
Time
- T :
-
Torque
- U :
-
Displacements
- δ:
-
Kronecker delta
- ρ:
-
Density
- ν:
-
Poisson’s ratio
- ξ:
-
Damping coefficient
- βb :
-
Base helix angle
- θ:
-
Rotation angle
- 1:
-
Driving gear
- 2:
-
Driven gear
- B:
-
Bearing
- G:
-
Gear
- P:
-
Plate
- S:
-
Shaft
- •:
-
Dot, time derivative
- -:
-
Average
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Park, C.I. Vibration from a shaft-bearing-plate system due to an axial excitation of helical gears. J Mech Sci Technol 20, 2105–2114 (2006). https://doi.org/10.1007/BF02916327
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DOI: https://doi.org/10.1007/BF02916327