A Lightweight Model for Gear Mesh Dynamics Incorporating Variable Mesh Stiffness

Variable stiffness of the gear tooth mesh for a pair of spur gears is computed using an accurate lightweight mathematical formulation. This is used to simulate gear dynamic behavior. Gear eigenfrequencies are calculated for the SDOF system and correlated with gear physical properties and the effect of stiffness variation during a mesh cycle is studied.


Introduction
The study of gear vibration has traditionally treated problems such as the determination of the primary resonant speeds of given gear mechanisms, or calculating the corresponding load factors [1][2][3][4][5].However these models are typically either too simplistic in terms of how stiffness is modelled, or employ complex FEA analysis that is resource intensive.In this paper a novel accurate and yet computationally lightweight SDOF model for gear dynamics is developed on the basis of recently developed analytical and numerical models for calculating gear geometry, contact and kinematics [6][7][8][9][10][11][12][13] and mechanical response of the elastic tooth mesh, in consideration of various tooth geometries [14][15][16][17][18][19][20][21][22][23][24].The implications of the varying tooth stiffness during a mesh cycle on the overall dynamic response are studied and a correlation is made between the gear properties and the tooth stiffness distribution based on the results of numerous simulations.

Modelling
The general dynamic equations governing the torsional behavior of a single gear pair are: ( ) where J is the moment of inertia, ω is the angular velocity, A is the instantaneous mesh contact point, O is the gear centre of rotation, Fn is the mesh normal force, Fs is the mesh friction force and T is the external torque acting on the gear shaft.
By introducing the stiffness coefficients K and friction torque coefficients Fr into the previous equations, we obtain: where: ( ) ( ) where ( ) Fr Fr θ = and after a series of calculations, we obtain for the system of dynamic equations the eigenvalues: where . The torsional eigenfrequencies of the gear stage are then found as functions of K : For the purposes of this analysis let us consider steady operating conditions, i.e. conditions where the gear speed and the external loads are either constant or changing slowly.Assuming such conditions, it can be argued without significant error for a time interval spanning the execution of a few mesh cycles that: where 1 ω is the mean angular of gear 1.
Because the coefficients ( ) Fr Fr θ = , ( ) Fr Fr θ = are all periodic in terms of 1 θ , with a period of 1 2π z , Eq. ( 10) allows the previous coefficients to be formulated as ( ) Fr Fr t = , ( ) Fr Fr t = , which are also periodic in terms of t with a period of . This stimulates a biased vibration of the gear stage, similar to a vibration produced by external periodic excitation.It follows that this vibration has a frequency , which is of course the tooth mesh frequency.
Looking at Fig. 1, which shows the calculated variation of the stiffness and friction torque coefficients, one can observe almost step-like changes in values of magnitudes K , 1 Fr and 2 Fr , which occur near the highest and lowest points of single tooth contact and at the pitch point.During each mesh cycle these abrupt changes cause vibrations that are superimposed on the predominant vibration of frequency z f .These vibrations manifest themselves in the eigenfrequencies of the gear pair and quickly tend to decay (as in Fig. 2).Similar eigenvibrations are also stimulated by external excitations, such as those which occur during start-up.

Results and Discussion
In Fig. 1 it can be readily observed that for the most part of the duration of a tooth pair mesh cycle pair stiffness K takes values in two very narrow regions which may in fact be accurately represented by only two characteristic values u K (lower stiffness limit) and o K (upper stiffness limit) respectively.For a given gear material (in this case steel) each one of the two stiffness limits can be considered to be a function of m (module), 1 z (number of pinion teeth), 12 i (gear pair transmission ratio) and t b (tooth width), hence: ( ) , , , With the application of Eq. ( 11) on several geometries it is concluded that: where functions 0 c and 1 c are extracted as statistical regressions.
After completing a series of simulations for gear pair geometries within the specifications: ( ) where m and t b are given in mm.For a single stage of involute spur gears in compliance with the specifications used previously for the stiffness coefficient limits regressions the limits , e u f and , e o f are determined by means of Eq. ( 9) and Eqs. ( 16)-( 17) as: Two test cases A, B were simulated for a pair of standard involute spur gears with a transmission ratio of 23/50 and the results shown in Fig. 2. In both cases the predominant vibration is in frequency

Conclusion
A SDOF torsional model was presented for the simulation of gear dynamics, using an accurate formulation for mesh contact kinematics and variable mesh stiffness.Simulation results were analysed, showing the influence of the stiffness on the resulting vibrations for two test cases.

Figure 1 .
Figure 1.Calculated stiffness (black line) and friction torque coefficients (blue and red lines) at different % positions in the mesh cycle.

Figure 2 .
Figure 2. Simulated dynamic transmission error during low speed rotation vs time (s) (case A, left) and after torsional impulse at zero rotational speed (0.05s interval shown), showing vibration attenuation (case B, right).
case A is 1.917 Hz and in case B is 575 Hz.Immediately after 0 t = follows a dampened transitional response in a system eigenfrequency, which results from the sudden application of the external load on the rotating gears.Measurements on the data of Fig. 2 reveal the eigenfrequencies 1 92 4 f = ± Hz, 2 100 5 f = ± Hz and in complete agreement with the previous measurements, thus confirming the preceding theoretical analysis.CasesA and B in Fig.2, besides the main vibration component in tooth meshing frequency, also reveal an initial eigenvibration in frequency

f and 3 f
no longer have a visible influence on the dynamic response.