Spur Gear Stress Analysis

SPUR GEAR STRESS ANALYSIS

Objective: A pair of spur gears is simulated for a static structural test scenario. The material of the gears is varied and the test is simulated to determine the stress, stress intensity and deformation in each case.

Geometry: The test sample is a pair of spur gears of equal diameter of 18.5mm and 13 teeth each. The pre-prepared geometry is loaded and the simulation set-up is carried out inn Ansys Mechanical.

Meshing: The geometry is meshed in Ansys Mechanical with an element size of 1mm to adhere to the node/element number limitation of the student edition. Meshing method has been left to Automatic and the smoothening effect has been kept low.

A mesh metric that identifies elements according to their quality is used to plot a graph of the element quality. It can be seen that most of the elements lie in the high quality region of 0.8 -1.0 and very few elements in the lower quality region.

Contact Definition: A frictional contact is defined in between the two gears. The mating surfaces of the gears i.e. the face and flank regions of the gear teeth are defined as the contact surfaces. The right gear is defined as the contact body and the left gear as the target body.

The contact formulation is set for “Augmented Lagrange”. The penetration force (F_normal) governing equation for this type of formulation is:

F_normal = k_normal * X_normal + λ (i)

Where k_normal is the stiffness of the mesh, X_normal is the depth of penetration and λ is the augmentation constant that reduces the dependency of penetration force on the stiffness of the system. This formulation provides the best convergence and can be applied to any kind of contact.

Joint Definition: Body-ground revolute joints are defined for both the gears. The inner surface of the gears are used to define the joints. This constraints the gears to only one degree of freedom i.e. rotation about the Z-axis (R_z).

Analysis Settings: The “step controls” option is edited to incorporate 6 “number of steps” for the analysis. At Auto Time-stepping, the Initial Time Step is set to 0.2s and the maximum and minimum time-steps are set to 0.5s and 5e-02s respectively. The simulation is set for large deflection.

Moment/Rotation Definition: For the right gear, a rotation of 30 degrees is provided for each step. Hence the gear turns 180 degrees every 6s.

The left gear is provided with a constant moment of -10Nm at every step. The negative sign ensures that the rotation of the gear is opposite to that of the mating gear.

Material Definition: The analysis is run for three different materials in three separate simulations.

Case 1: In this case, Cast Iron (ductile) material is assigned to the gears. The material exhibits the following properties:

Sl. No	Particulars	Magnitude
1.	Density	7.13e-06 kg/mm3
2.	Young’s Modulus	1.71e+05MPa
3.	Thermal Conductivity	0.0359 W/mm ͘C
4.	Specific Heat	4.77e+05 mJ/kg ͘C
5.	Yield Strength	344MPa
6.	Ultimate Strength	548MPa

Case 2: The geometry is assigned the material of Cast Steel, with the following properties:

Sl. No	Particulars	Magnitude
1.	Density	7.82e-06 kg/mm3
2.	Young’s Modulus	2.03e+05MPa
3.	Thermal Conductivity	0.0604 W/mm ͘C
4.	Specific Heat	4.96e+05 mJ/kg ͘C
5.	Yield Strength	349MPa
6.	Ultimate Strength	505MPa

Case 3: The material assigned in this case is Cast Bronze which exhibits the following properties:

Sl. No	Particulars	Magnitude
1.	Density	8.81e-06 kg/mm3
2.	Young’s Modulus	0.8e+05MPa
3.	Thermal Conductivity	0.0604 W/mm ͘C
4.	Specific Heat	3.84e+05 mJ/kg ͘C
5.	Yield Strength	144MPa
6.	Ultimate Strength	267MPa

The simulation is run for the three cases are the results are tabulated and plotted.

RESULTS:

The total deformation, Equivalent (Von-Mises) stress, Equivalent Elastic Strain and the Stress Intensity for the three cases have been tabulated below.

Particulars (max value)	Case 1	Case 2	Case 3
Total Deformation (mm)	30	30	30
Equivalent Stress (MPa)	322.09	322.01	321.52
Stress Intensity (MPa)	329.91	331.32	333.3
Equivalent Strain	1.8977e-03	1.5982e-03	4.0497e-03

1. Total Deformation: The maximum and minimum deformations in all the cases are 30mm and 18.5mm respectively. Maximum deformation takes place at the Top Land of the gear teeth and the minimum deformation at the surface of the inner diameter.

The Deformation vs Time graph shows the maximum, minimum and average deformations as green, red and blue lines respectively.

2. Equivalent Stress: Maximum Equivalent stress is developed in case 3 where Cast Bronze has been assigned as material.

The red, blue and green lines indicate the maximum, average and minimum equivalent stresses computed for the geometry respectively.

Case 1

Case 2

Case 3

3. Stress Intensity: The maximum stress intensity is also developed in the bronze gear on account of its low strength and Young’s Modulus value.

Case 1

Case 2

Case 3

CONCLUSION:

The simulations terminated successfully. A mesh size of 1mm provided the most accurate results without any errors. Any further reduction in mesh size would lead to severe distortion of elements under load and demand changes in the time step size.

The deformation values obtained coincide with the inner and outer diameters of the gear (30mm and 18.5mm respectively). As the load is rotational in nature, the deformation measured is basically displacement of the any point on the gear (in this case the points on the teeth and inner diameter) on a 180 degree rotation of the gears. Thus the maximum and minimum displacement positions on rotation are the diametrically (outer and inner respectively) opposite points on the gear regardless of the material assigned.

Analysis of the stresses reveal that the highest stress intensities are concentrated at the base of the gear teeth, where a fillet radius has been provided between the gear teeth and the Dedendum circle.

Most probable region of fracture

Since the stress intensity in this region is the highest in all the cases, this region has the highest probability of fracturing under load.

This analysis reveals that under the given load and boundary conditions, the lowest equivalent stress and strain is developed in Case 2 i.e. the spur gear that was assigned Cast steel material. Hence from an analysis point of view, the Cast steel spur gear will be preferred for production over the other two gear materials.