Fem analysis of spur gear systems with misalignment.

Cananau, Sorin

1. INTRODUCTION

The mechanical systems which include gear sets are various and their operating conditions are subject to manufacturing problems, assembly and installation problems. So, there is very important to understand in the functioning conditions the static and dynamic behavior of mechanical systems with gear.

One of the important problem is the mesh stiffness along the lines of contact. For each tooth any potential point of contact is credited with a direct flexibility only such that, under load conditions, there is no deflection at any other point of the tooth, (Ajmi & Velex, 2005). But the theoretical which attempts to take into account the effects of static or dynamic non-uniform load distribution along tooth wide are not extend.

In this paper we are going to study the effects of teeth misalignment due to manufacturing errors on the stiffness variation. Many works have been carried out to calculate this stiffness and most of them are taking into account the face load factor of load distribution along tooth width (Chaari et al., 2005)

We are going also to use the FEM analysis to find the behavior of elastic bodies such as teeth structure in the gears systems. We are going to use also the FEM model to simulate the manufacturing errors and their effects on the gearmesh stiffness. Also, we are going to introduce an increasing load as a torque up to the limit of the fracture initiation at the tooth root.

We will study if this failure results from severe operating conditions, like overload combined with teeth misalignment. In that fallows friction forces are neglected and contact lines are considered parallel to shaft axis and after that with spatial misalignment.

2. MESH STIFFNESS AND GEAR MESH MODEL

Regarding the gear meshing, for a gear tooth the modelling principle is to represent the structural elasticity viewed from any line of contact on the flank by a foundation with position varying characteristics to simulate the evolutions of the contact lines during the meshing process.

Unlike in the early developments of Schmidt, (Schmidt, 1973) for static conditions, the foundation model applies to each individual tooth potentially in mesh. This allows the representation of pinions and gears of different width and the simulation of the effects due to the unloaded parts of the teeth beyond the extent of contact lines.

On this assumption we will consider that the stiffness of the teeth in contact could be calculated with the relation:

[c.sub.[gamma]] = 1/[F.sub.[beta]y] [2[F.sub.m]([K.sub.H[beta]] - 1)/b] (1)

where

[c.sub.[gamma]]--mean value of mesh stiffness per facewidth unit

[F.sub.[beta]y]--the effective equivalent misalignment

[F.sub.m]--equivalent load of torque applied (mean transverse tangential load

[K.sub.H[beta]] --face load factor of load distribution along tooth width

b--tooth width

In this relation the effective misalignment is an equivalent one and is supposed to be known. The geometry of the gear set and the geometry of the tooth is also known. We will suppose that the rim of the gear is designed to be at least three times of one module of the geometrical tooth characterization. This will give the opportunity to neglect the effects of loads in the gear structure. The values for face load factor of load distribution along tooth width, KHp, are calculated using ISO 6336 standard.

In this work, the values for materials properties of the gear set and the geometrical characterization is shown in the next table:

Tab. 1. Materials properties and geometry for gear set

 Pinion Wheel

Teeth number 19 33
Modulus (mm) 5 5
Teeth width (mm) 40 40
Contact ratio 1.27 1.27
Pressure angle 20[degrees] 20[degrees]
Young modulus E (GPa) 205 GPa 205 GPa
Poisson's ratio, v 0.325 0.325

3. FINITE ELEMENT MODEL

To model spur gears using FEM, a three-dimensional model is adopted. The internal diameter of both pinion and wheel is clamped. In order to simplify the computation of tooth deflection, only a sector of five teeth (wheel) and three teeth (pinion) was considered.

Such model is widely used and accepted in literatures (Sirichai, 1999). Finite element contact between pinion and wheel tooth pairs is also take into account, but the hertzian deformations in the contact zone are found to be small relative to the bending deflections. In order to find the singular stiffness of one tooth of the pinion, we will introduce a torque or a distributed force (load) which simulates the action of the meshing tooth of the wheel.

Using the equivalent force, this was applied to the tooth flank normal to the involute profile and along the line of action at the appropriate nodes. The representation of the wheel sector is shown in Fig. 1

[FIGURE 1 OMITTED]

The commercial FEA software DS COSMOSWORKS (SOLIDWORKS 2007) is used to perform the analyses. The mesh is constructed using four node triangular elements.Various mesh schemes are tried to achieve convergence. The optimized model has 63157 nodes, 26570 elements.

4. ANALYSIS AND RESULTS

For the case were the gears are in contact with non misalignment we are tested the model. We found a good correlation regarding the values of the bending displacements along the loaded tooth and this was verified with the analytical solution.

In the next step we performed an analyze were the teeth in contact are misaligned. The misalignment is due to the manufacturing errors, the assembly errors.

There are several cases with imposed misalignment as is shown in Tab. 2

The result for the load equivalent to 30% of torque and for the same misalignment of [F.sub.[beta]y] = 85 um is shown in Fig. 2. In the figure there is a view for two sides of the wheel to see the difference of stress field in the tooth and the rim of the tooth.

In Fig.3. is shown the situation for the same geometrical and loads conditions but with an increase misalignment. This misalignment is considered as a single one, not a composed misalignment. In this way we can suppose that the geometrical linearity of the misalignment is correct.

Usually, the effect of the stiffness of the separate parts in contact, i.e. gear teeth in contact, are different as the result of combined stiffness.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

For the final step we obtain the limit situation of the fracture initiation of the tooth root (Fig. 4.). This limit is taken into account according to the yield limit.

5. CONCLUSION

The modelling of gear body flexibility is important for wide-faced gears in quasi-static conditions. Tooth load distributions (consequently, tooth form corrections) are both highly sensitive to gear body deflections in torsion and bending but also to the manufacturing errors. There is no linear correlation between the level of misalignment and the length of contact area between gear teeth or the stiffness evolution.

6. REFERENCES

Ajmi, M., Velex, P. (2005). A model to simulating the quasi-static and dynamic behavior of solid wide-faced spur and helical gears. Mech. Mach. Th.. 40 (200),2005, 422-429

Chaari, F., Fakhfakh, T., Haddar, M., (2005). Simulation numerique du comportement dynamique d'une transmission par engrenages en presence de defauts de denture. Mec. Ind. 6, 2005, 625-633

Pimsarn, M., Kazerounian, K., (2002). Efficient evaluation of spur gear tooth mesh load using pseudo-interference stiffness estimation method. Mech. Mach. Th. 37, 2002, 769-786

Schmidt, G. (1973). Berechnung der Walzpressung Schragverzahnter unter Berucksichtigung der Lastverteilung, PhD dissertation, 1973, Technical University of Munich

Sirichai, S., (1999). Torsional properties of spur gears in mesh using nonlinear finite element analysis. PhD dissertation, Curtin: University of Technology

Tab. 2. Several cases with imposed misalignment

Torque (%) [K.sub.H[beta]] Deflection Stiffness
 ([micro]m) (N/mm)

10 2.11 3.6 10.3
20 1.81 4.25 11.6
30 1.65 4.95 12.85
40 1.44 5.8 13.76
50 1.28 6.9 14.1
65 1.16 7.6 14.85