Abstract
A nonlinear dynamic model of a spiral bevel gear train mounted on flexible shafts and bearings is proposed in this study. The finite element model of shafts is combined with a three-dimensional discrete mesh model of a spiral bevel gear pair. Bearing flexibilities are as well included in the model. Gear backlash is incorporated into the model in the form of clearance-type displacement functions and clearance nonlinearity and stiffness fluctuations of the bearings are neglected. A time-invariant mesh stiffness is assumed for the gear pair to simplify the dynamic model. Eigenvalue solution is used to predict the natural modes of the system. A multi-term Harmonic Balance Method (HBM) is employed for the solution of resulting equations of motion for periodic steady-state response. The results of HBM are validated by comparing them to the solutions obtained by direct numerical integration. Forced response of the system in the form of dynamic mesh force is studied to demonstrate the effects of static mesh force and backlash amount.
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Abbreviations
- b :
-
Half of gear backlash
- c m :
-
Mesh damping coefficient
- C :
-
Damping matrix
- e m :
-
Static transmission error
- f n :
-
Nonlinear displacement function
- F :
-
External force vector
- F m :
-
Dynamic mesh force
- F N :
-
Nonlinear restoring force vector
- h :
-
Transformation vector
- i :
-
Unit imaginary number
- J :
-
Jacobian matrix
- K :
-
Stiffness matrix
- k m :
-
Mesh stiffness
- M :
-
Mass matrix
- n :
-
Line of action (LOA) directional cosine vector
- n :
-
Directional cosine
- p :
-
Extended coordinate transformation vectors
- r :
-
Position vector of the effective mesh point
- r :
-
Harmonic index
- S :
-
Pinion/gear coordinate system
- S :
-
Nonlinear algebraic equations in vector form
- T :
-
Torque
- t :
-
Time
- U :
-
Solution vector
- u :
-
Harmonic amplitudes of displacement vector
- x :
-
Displacement vector
- β :
-
Rayleigh damping coefficient
- δ d :
-
Dynamic transmission error
- λ :
-
Directional rotational radius
- ρ :
-
Discrete time interval
- θ :
-
Rotational displacement
- ω :
-
Frequency
- b :
-
Bearing
- i :
-
Pinion (i = p) and gear (i = g)
- l :
-
Linear DOFs
- n :
-
Nonlinear DOFs
- s :
-
Shaft
- rms:
-
Root-mean-square value
- . :
-
Derivative with respect to time
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Yavuz, S.D., Saribay, Z.B., Cigeroglu, E. (2017). Nonlinear Dynamic Analysis of a Spiral Bevel Geared System. In: Di Maio, D., Castellini, P. (eds) Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics & Laser Vibrometry, Volume 8. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-54648-3_4
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DOI: https://doi.org/10.1007/978-3-319-54648-3_4
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