# A novel dynamic model for the spiral bevel gear drive with elastic ring squeeze film dampers

• Weitao Chen
• Siyu Chen
• Zehua Hu
• Jinyuan Tang
• Haonan Li
Original paper

## Abstract

Elastic ring squeeze film dampers (ERSFDs) show superior performances in vibration and noise control in a rotor-bearing system. Currently, the industry of spiral bevel gears is also interested in introducing ERSFDs to improve their dynamic performances. This manuscript develops a new accurate mathematical model of ERSFDs based on the generalized Reynolds equation and proposes a semi-analytical method to calculate the elastic ring deformation of ERSFDs. Moreover, an innovative numerical strategy for calculating the oil-film pressure distribution is presented, and Simpson’s rule is utilized to calculate the oil-film force of ERSFDs. Then the dynamic model of a spiral bevel gear drive supported on ERSFDs is developed by coupling the motion equations of the gear system with the oil-film force, and the dynamic characteristics of the system are studied for the first time. The new mathematical model and oil-film pressure calculation method of ERSFDs proposed in this work can be well applied to the gear system, and the ERSFD has better performance than the classical squeeze film damper in suppressing the nonlinear characteristics of systems in the speed range of 7100–8100 rpm. Besides, the mathematical model and oil-film pressure calculation method of ERSFDs proposed in this paper can be extended to all rotating machinery.

## Keywords

Spiral bevel gear drive Elastic ring squeeze film damper Dynamic characteristic Nonlinear oil-film force

## Latin Symbols

$$\beta$$

Attitude angle of the journal to the x-direction

$$\varDelta \theta$$

Angle of a subinterval in the circumferential direction, $$\varDelta \theta = 2\pi /160\left( {rad} \right)$$

$$\varDelta L$$

Width of a subinterval in the axial direction, $$\varDelta L = 40/40\left( {mm} \right)$$

$${\dot{X}}, {\dot{Y}}, {\dot{Z}}$$

Dimensionless horizontal, vertical and axial velocities

$${\dot{x}}, {\dot{y}}, {\dot{z}}$$

Horizontal, vertical and axial velocities

$$\mu$$

Oil dynamic viscosity

$$\tau$$

Dimensionless time, $$\tau = {\omega _n}t$$

$$\theta$$

Angular position

$$\varepsilon$$

Eccentricity ratio, $$\varepsilon = e/{c_1}$$

b

Center distance of elastic ring segment before and after deformation

e

Eccentricity of the journal

g

Acceleration of gravity

H

Dimensionless thickness of the fluid film

L

Width of ERSFD

$$m_e$$

Equivalent mass of the gear pair, $${m_e} = \frac{{{I_p}{I_g}}}{{{I_g}\lambda _p^2 + {I_p}\lambda _g^2}}$$

P

Dimensionless pressure distribution in the fluid film, $$P=p/{p_0}$$

p

Pressure distribution in the fluid film

r

Deformation of the elastic ring

$$R, R'$$

Radii of the elastic ring segment before and after deformation

XYZ

Dimensionless horizontal, vertical and axial coordinates

xyz

Horizontal, vertical and axial coordinates

$${{{\bar{\delta }}} _s}$$

Dimensionless static transmission error

$${{\bar{e}}_p}, {{\bar{e}}_g}$$

Dimensionless eccentricities of pinion and gear

$${\delta _0}$$

Mean static transmission error

$${\delta _d}$$

Dynamic transmission error

$${\delta _i}$$

Displacement of the ith boss

$${\delta _s}$$

Static transmission error

$${\lambda _{lz}}\left( {l = p,g} \right)$$

Directional rotation radii, $${\lambda _{lz}} = {x_{lm}}{n_{ly}} - {y_{lm}}$$

$${\varOmega _g}$$

Dimensionless circular frequency of the gear, $${\varOmega _g} = - \frac{{{N_p}}}{{{N_g}}}{\varOmega _p}$$

$${\omega _g}$$

Circular frequency of the gear, $${\omega _g} = - \frac{{{N_p}}}{{{N_g}}}{\omega _p}$$

$${\omega _n}$$

Natural frequency, $${\omega _n} = \sqrt{{k_m}/{m_e}}$$

$${\varOmega _p}$$

Dimensionless circular frequency of the pinion

$${\omega _p}$$

Circular frequency of the pinion, $${\omega _p} = \frac{{2\pi {n_p}}}{{60{\omega _n}}}$$

$${\theta _p}, {\theta _g}$$

Rotation angle of pinion and gear around z axis

$${\zeta _1}$$

Dimensionless parameter, $${\zeta _1} = \frac{{{c_e}}}{{{m_p}{\omega _n}}}$$

$${\zeta _2}$$

Dimensionless parameter, $${\zeta _2} = \frac{{{c_m}}}{{{m_p}{\omega _n}}}$$

$${\zeta _3}$$

Dimensionless parameter, $${\zeta _3} = \frac{{{c_a}}}{{{m_p}{\omega _n}}}$$

$${\zeta _4}$$

Dimensionless parameter, $${\zeta _4} = \frac{{{c_e}}}{{{m_g}{\omega _n}}}$$

$${\zeta _5}$$

Dimensionless parameter, $${\zeta _5} = \frac{{{c_m}}}{{{m_g}{\omega _n}}}$$

$${\zeta _6}$$

Dimensionless parameter, $${\zeta _6} = \frac{{{c_a}}}{{{m_g}{\omega _n}}}$$

$${\zeta _7}$$

Dimensionless parameter, $${\zeta _7} = \frac{{{c_m}}}{{{m_e}{\omega _n}}}$$

$${A_{i,j}}, {B_{i,j}}$$

Simplified coefficient for solving oil-film pressure

$${c_b}$$

Bearing damping coefficient

$${c_l}\left( {l = 1,2} \right)$$

Radial clearance of inner and outer oil-film

$${c_m}$$

Damping coefficient of the gear mesh

$${C_{i,j}}, {D_{i,j}}$$

Simplified coefficient for solving oil-film pressure

$${e_p}, {e_g}$$

Eccentricities of pinion and gear

$${f_0}$$

Dimensionless parameter

$${F_m}$$

Dynamic mesh force

$${F_{gx}}, {F_{gy}}$$

Components of the fluid film force in the horizontal and vertical directions for the gear

$${F_{i,j}}, {K_{i,j}}, {M_{i,j}}$$

Simplified coefficient for solving oil-film pressure

$${F_{px}}, {F_{py}}$$

Components of the fluid film force in the horizontal and vertical directions for the pinion

$${h_l}\left( {l = 1,2} \right)$$

Thickness of the inner and outer oil-film

$${I_p}, {I_g}$$

Moments of inertia of the pinion and gear

$${k_0}$$

Mean mesh stiffness

$${k_1}$$

Dimensionless parameter, $${k_1} = \frac{{{k_e}}}{{{m_p}\omega _n^2}}$$

$${k_2}$$

Dimensionless parameter, $${k_2} = \frac{{{k_m}}}{{{m_p}\omega _n^2}}$$

$${k_3}$$

Dimensionless parameter, $${k_3} = \frac{{{k_a}}}{{{m_p}\omega _n^2}}$$

$${k_4}$$

Dimensionless parameter, $${k_4} = \frac{{{k_e}}}{{{m_g}\omega _n^2}}$$

$${k_5}$$

Dimensionless parameter, $${k_5} = \frac{{{k_m}}}{{{m_g}\omega _n^2}}$$

$${k_6}$$

Dimensionless parameter, $${k_6} = \frac{{{k_a}}}{{{m_g}\omega _n^2}}$$

$${k_7}$$

Dimensionless parameter, $${k_7} = \frac{{{k_m}}}{{{m_e}\omega _n^2}}$$

$${k_a}, {c_a}$$

Stiffness and the damper of axial support

$${k_b}$$

Bearing stiffness

$${k_e}, {c_e}$$

Stiffness and the damper of elastic support

$${m_p}, {m_g}$$

Masses of the pinion and gear, respectively

$${n_b}$$

Elastic ring boss number

$${n_p}$$

Rotating speed of pinion (rpm)

$${N_p}, {N_g}$$

Tooth number of the pinion and gear

$${n_{gx}}, {n_{gy}}, {n_{gz}}$$

Normal vector of the mesh point for gear

$${n_{px}}, {n_{py}}, {n_{pz}}$$

Normal vector of the mesh point for pinion

$${O_b}, {O_j}$$

Geometric center of the housing and journal

$${p_0}$$

Dimensionless parameter, $${p_0} = 2\mu {R_1}/{c_1}$$

$${R_1}, {R_2}$$

$${R_3}, {R_4}$$

$${T_E}$$

External torque of the pinion

$${T_L}$$

Load torque of the gear, $$T_L={N_p}/{N_g} \times {T_E}$$

$${u_h}, {u_0}$$

Tangential velocities of oil-film at upper and lower boundaries

$${v_h}, {v_0}$$

Radial velocities of oil-film at upper and lower boundaries

$${w_h}, {w_0}$$

Axial velocities of oil-film at upper and lower boundaries

## Notes

### Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NSFC) through Grant Nos. 51535012, U1604255 and the support of the Key research and development project of Hunan province through Grant No. 2016JC2001. In addition, the authors would like to thank the anonymous reviewers for their valuable comments.

### Conflicts of interest

The authors declare that they have no conflict of interest.

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## Authors and Affiliations

• Weitao Chen
• 1
• Siyu Chen
• 1
• Zehua Hu
• 1
• Jinyuan Tang
• 1
• Haonan Li
• 1
1. 1.State Key Laboratory of High Performance Complex ManufacturingCentral South UniversityChangshaChina