Effects of Pitting Shapes on Spur Gear Mesh Stiffness: A MATLAB Analysis

Abstract

The mathematical exhibiting of the vibrant analysis of gear has become essential with increased demand for high-speed mechanism. There are many numerical methods available for finding out the spur gear mesh stiffness like FEA methods and some numerical methods. In this present paper comparative study of effects of pitting shapes on spur gear mesh, stiffness is done using MATLAB Software. The pitting shapes considered are circular, Rectangular and Elliptical. Their effects on total deformation are also considered.

Introduction

Automotive transmissions are designed to use in transmission power systems between the engine and the wheels with an ability to control the torque and rotational speed output.

Pitting occurs when the cyclic stresses in a gear tooth cause small cracks at or below the surface of the tooth face.

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These cracks will propagate into pits as the gear material fractures and are removed from the tooth. The size of the pit then grows as the removed material presents further stress concentrations on the tooth face. While the parameters dictating tooth bending fatigue are limited to geometry, material and loading levels, a large number of parameters interact to define contact fatigue conditions of gears [1]. Due to high service load, harsh operating conditions or fatigue, faults may develop in gears. Gear faults are responsible for gearbox failures.

Most of these come from damage to the gear teeth such as pitting, cracking, and spalling. Crack is no lubrication related failure mode while pitting is a lubrication-related failure mode [3] Contact fatigue of gears is greatly influenced by tooth geometry as well as surface finish and lubrication parameters.

These variables combined with gear material, operating speed, and temperature define the conditions for cracks to nucleate and grow into sizable pits. In this study, pits shapes effect on mesh stiffness is studied. The pit shapes considered are circular, elliptical and rectangular [4]. According to the American Society for Metals, 'pitting occurs when fatigue crack is initiated on the tooth surface or just below the surface'. Pits are the surface crack caused by metal to metal contact of severities due to low lubricant film thickness.

High-speed gears with a smooth surface and good film thickness may experience pitting due to subsurface cracks [5]. Fakher Chaari et al [3] presented the analytical modeling of tooth cracks and quantification of the gear mesh stiffness reduction due to the tooth crack. The validation of analytical formulation is done with the finite element model. Pitting growth is shown in figure 1. Ten et al [4] – Experimentally measured pitting growth under different levels of load. To create surface pitting in a relatively short time frame, lubricant oil without anti-wear properties was employed. Prolonged operation time resulted in pitting spreading to other gear teeth. The pitted area progression from 6.3% to 41.7% of the gear tooth surface under the test condition. Chaari et al.[3] analytically studied the effect of a single pit on the time-varying mesh stiffness of gears.

The pit was modeled as a rectangular shape as shown in Figure 4. Xihui Liang et al [9] Pitting occurs when fatigue cracks are initiated on the tooth surface or just below the surface. Pits are the result of surface cracks caused by metal-to-metal contact of asperities or defects due to low lubricant film thickness. Ashish Agrawal et al [10] – analyze spur gear mesh stiffness by the mathematical and numerical method using MATLAB and MS Excel spreadsheets. Rincon et al. [11] evaluated the pitting effect on the time-varying mesh stiffness of gears using the finite element method. A single pit was modeled in an elliptical shape as shown in Figure 5. They developed a model to study and predicted the mesh stiffness from the model including the bending stiffness, axial stiffness and shear stress mesh stiffness of gears in the mesh using the MATLAB software package based on the numerical method.

Mesh Stiffness Calculation for Gears with Different Shapes of Pitting

In this study, tooth pits are modeled in a circular shape, rectangular shape and elliptical shape as did in [5]. The MATLAB software is used to find the mesh stiffness of spur gear. In this case, it is assumed to be without friction, manufacturing error and gear body is treated as rigid [7]. Our objective is to compare the effect on the mesh stiffness of different shapes of pits (Circular, Elliptical, and Rectangular) as a function of the gear rotation angle. For single tooth pair, gear and pinion teeth are considered in series combination. The single tooth contact pair mesh stiffness in a different shapes (circular, rectangular and elliptical) pitting are written below .

Where, Ktc, Ktr, and Kte are total mesh stiffness of the circular shape, rectangular shape and elliptical shape of pitting respectably and kb1, ka1, and ks1 are the series combination of bending stiffness, axial stiffness and shear stiffness of pinion respectably and kb2, ka2 and ks2 are the series combination of bending stiffness, axial stiffness and shear stiffness of gear respectably and kh is the Hertzian contact stiffness. [3,5,6]

Table 1: Gear Parameters Used in the Study

Circular Pitting Calculation

The series combination of bending stiffness (Kb), axial stiffness (Ka), shear stiffness (Ks) and Hertzian contact stiffness as calculated by [3] is given as

Where, Rb, Rr and L represent the base circle radius, root circle radius and tooth width of the gear respectively h is the distance between the gear contact point and the tooth central line,d is the distance between the gear contact point and the gear root, x is the distance to tooth root, Ax and Ix indicate the area and the area moment of inertia of the tooth section, respectively. α2 is the half tooth angle on the base circle, α3 is the describes the approximated half tooth angle on the root circle and α is the angular displacement. α2 and α3. Where Z1 and Z2 are the numbers of teeth of the gear and pinion and ∝_0 is the pressure angle.

For the gear tooth with pitting, the expressions of h, hx, Ix, and Ax are different from the ones given above for a perfect gear tooth. The tooth contact width is not constant L. ∆L_x,∆A_x and ∆I_x are used to represent the reduction of tooth contact width, area and area moment of inertia of the tooth respectively.

Where the distance contact point is x. the expression of ∆L_x,∆A_x and ∆I_x are given below [4]

∆L_x = {(2√(r^2 - (u -x)^2 ), x∈[u-r, u+r] @0 , others)┤

∆A_x = {(∆L_x δ, x∈[u-r, u+r] @0 , others)┤ A_x ∆A_x (h_x -δ/2)

∆I_x={(1/12 ∆L_x δ^3+ ((A_x ∆A_x (h_x -δ/2)))^2/(A_x - ∆A_x ), x∈[u-r, u+r] @0 , others)┤

Mesh Stiffness Derivation for Gear With Single Tooth Pit:

Chen, Y.et al [4 ]derived mesh stiffness equations for gears with a single circular tooth. The circular pits do not overlap with each other and all the circles are within tooth surface area, the Bending stiffness, shear stiffness, and axial stiffness can be obtained through equations (16), (17) and (18), respectively[6].

1/k_b =([1-((Z-2.5) cos⁡(α_1 ) cos⁡(α_3 ))/(Z cos⁡(α_0 ) )]^3-(1-cos⁡(α_1 ) cos⁡(α_2 ) )^3)/(2EL cos⁡(α_1 ) sin^3⁡(α_2 ) )+ ∫_((-α)_1)^(α_2) ((3{1+cosα_1 [(α_2-α)sinα-cosα] })^2 (α_2-α)cosα)/E(2L[sinα+(α_2-α)cosα]^3-3 (∇L)_x/(R_b^3 )) da 1/k_s =(1.2 (1+v) cos^2⁡(α_1 ) (cosα_2 - (Z-2.5)/(Zcosα_0 ) cosα_3 ))/(EL sin⁡(α_2 ) )+ ∫_((-α)_1)^(α_2)((1.2 (1+v) (α_2-α)cosα cos^2⁡(α_1 ))/E(L[sinα+(α_2-α)cosα]-0.5 (∇A)_x/R_b ) da)

1/k_a =( sin^2⁡(α_1 )(cosα_2 - (Z-2.5)/(Zcosα_0 ) cosα_3 ))/(2EL sin⁡(α_2 ) )+ ∫_((-α)_1)^(α_2)(( (α_2-α)cosα sin^2⁡(α_1 ))/E(2L[sinα+(α_2-α)cosα]- (∇A)_x/R_b ) da)

For a pair of gear with tooth pitting, the tooth contact width is (L -∇L_x) rather than L. correspondingly, the Hertzian contact stiffness for gear pairs with a circular tooth pit can be expressed as:

K_h=(πE(L-∇L_x))/(4(1-μ^2))

Where E, L, μ denotes young modulus, tooth width, and Poisson's ratio respectively.

Mesh Stiffness Calculation for Rectangular Pitting

1/k_a =∫_(-a1)^a2(((a2-a)*cosa*(sina1)^2)/(2*E*L*(sina+(a2-a)*cosa)) da- ∫_(-a1)^a2((∆L*(a2-a)*cosa*(sina1)^2)/(2*E*L^2*(sina+(a2-a)*cosa) ) da + ∫_(-a1)^a2((∆L*(a2-a)*cosa*(sina1)^2)/(2*E*L^2*(sina+(a2-a)*cosa-t1/2R1) )+ ∫_0^(x1-x2)((sina1)^2/(2*E*L*h_x ) dx))))

1/k_s =∫_(-a1)^a2((3(1+cosa1((a2-a)sina-cosa)))^2 (a2-a)cosa)/(2EL(sina+(a2-a)cosa))^3 da- ∫_(-a1)^a2(((3∆L(1+cosa1((a2-a)sina-cosa)))^2 (a2-a)cosa)/(2EL^2 ((sina+(a2-a)cosa))^3 ) da+ ∫_(-a1)^a2(((3∆L(1+cosa1((a2-a)sina-cosa)))^2 (a2-a)cosa)/(2EL^2 (sina+(a2-a)cosa-t1/2R1)^3 ) da)+ ∫_0^(x1-x2)((3(R1-R1cosa1*cosa2+xcosa1))^2/(2*E*L*h_x^3 ) dx))

Analytical Calculation for finding Deflection due to bending [6]:

δ_b=F (cos)^2 α∑_(i=1)^n(e_i {(d_i-e_i d_i+1/3 (e^2)_i)/(E^' I ̅_i )+1/(s_h GA ̅_i )+((tan)^2 α_m)/(A ̅_i E^' )} )

Bending stiffness of the tooth can be given by

(k)_b=F/δ_b

Other parameters like modulus of elasticity, a moment of inertia, and area can be given as

E^I=(E(1-v))/((1+v)(1-2v))

1/A ̅_i =(1/A_i +1/A_(i+1) )/2

Analytical Calculation for finding Deflection due to bending [6]:

δ_f=(F〖cos〗^2 α_m)/WE {L^* ((u_f)^2/(s^2)_f )+M^* (u_f/s_f )+P^* (1+Q^* t_(g^2 ) α_m)}

The coefficient L^*, M^*, P^*,Q^* can be calculated by polynomial functions

X_i (h_(f_i ),θ_f )=A_i/(θ^2)_f +(B)_i (h^2)_(f_i )+(C_i h_(f_i ))/θ_f +D_i/θ_f +E_i h_fi+F_(i )

Fillet foundation stiffness can be given as

k_f=F/δ_f

Local deformation then expressed by

δ_h=F/k_h

A program is created for calculating the mesh stiffness using MATLAB programming code. The input parameters for the program are pressure angle, module, addendum coefficient, number of teeth on the pinion, gear ratio, modulus of elasticity, Poisson’s ratio and gear root diameter. By using these basic parameters, the other parameters such as base circle diameter, pitch circle diameter, addendum circle diameter, base pitch, contact ratio, etc. are calculated. The mesh stiffness is calculated to know the meshing of spur gears during transmission processes. The locations and sizes of the contact zone are calculated using the contact ratio and base pitch.

A large number of points are imagined along the total contact zone for one mesh cycle. These points are taken to know spur gear mesh stiffness behavior along the complete mesh cycle. The pinion roll angle is used as a quantity for selecting these points for which mesh stiffness is to be calculated. The formulas for the first three stiffnesses for circular pitting [Eq. (16) – Eq. (18)], for rectangular pitting, [Eq. (20) – Eq. (22) and for elliptical pitting [Eq. (23) – Eq. (30)] depend on tooth geometry. These values are used to solve the mesh stiffness in the MATLAB command window and find out the equivalent mesh stiffness of different shapes of pitting [Eq. (1), Eq. (2) and Eq. (3)]. The Hertzian contact stiffness is constant all shape (HG, CPG, RPG, and EPG) of pitting for gear pairs with a written as Eq. (19).

Finally, the three stiffness’s are calculated by using Eq.1, Eq.2and Eq. 3 and these stiffnesses are as follows:

• Equivalent mesh stiffness of circular pitting
• Equivalent mesh stiffness of rectangular pitting
• Equivalent mesh stiffness of elliptical pitting

Result

The single tooth mesh stiffness within one mesh cycle is plotted for healthy gear in figure 8 circular shape pitting in Figure 9, rectangular shape pitting in Figure 10, elliptical shape pitting in Figure 11 and comparative mesh stiffness of gear in Figure 12 represents the evolution of gear mesh stiffness along the pinion roll angle corresponding to module = 3.2mm and pressure angle = 20 deg. It has been observed that stiffness of the driving gear tooth decreases, whereas the stiffness of the driven gear tooth increases along the path of contact.

Conclusion

The results of this study sketched and evaluated in the previous chapter and the above section allow for some useful conclusions to be made. With the gear manufacturing and finishing processes all being equal in this study, the influence on gear material has been isolated as the sole variable influencing pitting fatigue life. The following conclusions can be made - For all the considered cases in MATLAB, tooth effective mesh Stiffness of elliptical shape pitting of Spur gear is lower than the same Spur gear but the different shapes of (circular and rectangular) pitting as shown in Fig. 12(Elliptical shape pitting is 82.46%, circular shape pitting is 83.7%, rectangular shape pitting is 91.36% and healthy gear)

References

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