Dynamic Analysis of Bevel Gear System in Main Retarder

The main retarder of vehicle is gear rotor system, its vibration property is complex. The whole transfer matrix method is a new algorithm to study, relationship of the displacements and forces are established considering gyroscopic moment and coupling vibration. The meshing point between rotors is treated as coupling element, balance equations are estalished on deformation compatibility condition. The globe status vectors including forces and displacements of all shafts were integrated into a whole transfer matrix. So it can be programming and has high numerical stability and accuracy. The influence of the meshing effect on vibration of the system can be systematically studied. The status vectors such as displacements, deflection angles, torsion angles, shearing forces, bending moments and torques are no longer independent. The vibration property of the geared rotor system was calculated and analyzed. The numerical results revealed that many coupling critical rational speeds of the system are derived. vibration of main retarder can be improved. Dynamic Analysis Program. The subroutine program is based on Gauss-Jordan elimination method. of accurate quantitative calculation are almost coincident with experimental data. The lower critical rotational speeds are sensitive to length of the shaft. Vibration of the gear rotor system can be undercontrol.


Introduction
Vehicle power train is typical gear rotor system. Vibration characteristics of it is different from single rotor in [1]. Bending and torsional deformations are coupled due to gear meshing. There is a distance between the meshing force and the shaft, so lateral response by torsional excitation of geared rotors system and the influence of torsional-lateral coupling on the stability behavior of it were studied in [2,3]. Coupled bending torsional vibration of rotors using finite element and a modified transfer matrix method for the coupling lateral and torsional vibrations of symmetric rotor-bearing systems are visible in [4,5]. Many coupling critical rotational speeds are generated and must be considered. Resonance could occur when working speed approaching them. It's important to study the coupling critical rotational speeds of the system. General methods of vibration calculating are the finite element method and the transfer matrix method. The former is an approximate method, it is suitable to study block body. Dynamic analysis of gear rotor system by finite elements was published in [6,7,8]. The latter is an accurate algorithm and suitable to study dynamics of rotor. Now based on the whole transfer matrix method, dynamic equations of the geared rotor systems are deduced by D'Alembert principle, Lagrange's equations, machinery system dynamics, nonlinear dynamics and geared system dynamics. The research on vibration of the vehicle power transmission system is synthesized here. components of the wheel center. External forces along forward direction of the axes are plus. Balance of shearing forces on the wheel is shown in fig. 1. Q L is shearing force acting on left of the wheel. Fig. 1 balance of shearing forces on the wheel Fig. 2 balance of torques on the wheel

Balance of torques on the wheel
Balance of torques on the wheel are shown in fig. 2. I p  2  is the inertia torque, torque generated by meshing Fig. 4 balance of bending moments In fig.3, o, o 1 and o 1 are inertia central principal axes of the revolving wheel, I p is rotating inertia of the wheel around o-axis, I d is rotating inertia of the wheel around o 1 -axis or o 1 -axis considering three Euler angles.

Fig. 3 inertia moments of the wheel
Rotation angle around Y-axis is p and angular speed is p (t). Rotation angle around o 1 -axis is p and angular speed is   p (t), Rotation angle around o-axis is  and angular speed is .  is deflection angel of the wheel,  is torsion angel of it. At the beginning of the precession, p and  p are infinitesimal, cos p 1, cos p 1, sin p p , sin p p . In accordance with the theorem of angular momentum, I d  p is momentum along o 1 -axis, I d   p is momentum along o 1 -axis, I p is momentum along o-axis.
x and M g y are inertia moments of the wheel, It is synchronous whirling of small deflection at the beginning of the precession, .
Balance of bending moments on the wheel is shown in fig. 4. A is the driving wheel, its direction of rotation is minus, and the direction of revolution is minus too when it is doing synchronous forward precession, The critical rotational speeds of spur gear rotor systems are listed in table 1. The lower critical rotational speeds are sensitive to length of shaft. The boldface numbers are new critical rotational speeds due to meshing effect. If transfer matrix of the shaft is false, uncoupling critical rotational speeds of the gear rotor system will not be coincidence with critical rotational speeds of single rotor. The analysis of gear shaft bearing system are available in [12].

Vibration of the orthogonal helical bevel gear-rotor system
The vehicle main retarder is a orthogonal helical bevel gear rotor system for power transmission shown in fig.   6. Dynamic of the spiral bevel geared rotor-bearing system has been analyzed in [13] . The coupling critical rotational speeds of the system were calculated based on the whole transfer matrix method here. The mechanical model is established in Fig. 7. Direction from intersection point to the big end of the driving wheel is Z-axes forward, direction from intersection point to the big end of the driven wheel is Y-axes forward. Direction of displacement or force along the coordinate axis forward is plus, angular displacement, moment and torque should be according to the right-hand rule.
A is the driving wheel, R A is the base radius of it,  1 is the cone-apex angle of it. B is the driven wheel, R B is the base radius of it,  2 is the cone-apex angle of it. = 1 + 2 =90,  n is the normal pressure angle,  is the helical angle,  is deflection angel of the rotor,  is torsion angel of it. R m =(R+r)/2 is the radius of the pitch circle at middle point. Helical angle of driving wheel is plus, it is clockwise from small end to big end looks from conical top, driven wheel is opposite.
the status vectors of the wheels can be listed,