Inner Contact Force Calculation

This model calculates the force between two teeth on an outer spur gear and an inner spur gear. It also determines when this force should be applied, dependent on the isFront parameter and toothIndexShift. 

At initialization, a tooth pair will be selected with different indices, one tooth on the outer gear in frame_z1 and one tooth on the outer gear in frame_z2. Normally, six of these components are put in the component GearForceCalculationInner, which results in six tooth pairs. To visualize where these tooth pairs are located, it is possible to turn on a parameter called "debugPoints", which will result in the following visualization. 

Figure 1: Debug points.

Two spheres with the same color should be paired up together. This way, you can see if the forces have been placed at appropriate positions.

Contact Ratio

The contact ratio is calculated continuously and can be found by plotting the variable epsilon. Epsilon is calculated through the following equations:

The contact ratio should never be below 1 for a stable simulation result. If it is below 1, it might mean that the gears are too far apart, and the conditions for ContactForcePoints to switch teeth might not be triggered correctly.

Forces

The forces calculated depend on the distance between two coordinates in a tooth pair. The coordinates will always be on the line of contact (LoC). The force used is based on Hertz contact theory between two cylinders. 

It is assumed that the teeth have contact in the whole gearwheel width, i.e. no crowning. Equation for the Hertz contact width b:

and displacement of cylinders, delta:

where F is the force, nu is the Poisson ratio, Emod is Young's modulus, L is the length of the cylinder, and R is the radius of the cylinder. To simplify the equations, Ered is defined as:

With the preceding equations, delta as a function of contact force can be solved. However, the simulation time can be improved if the following equations are rewritten as shown. The error in deformation delta is normally less than 1% and in contact stress even better.

The relation between contact force and indentation depth can be written as:

where with good accuracy, k₁ can be set to:

and

Then, the load indentation depth can be written as:

With the notation used in the source code:

This describes the force dependent on indentation depth and geometry of the cylinders. The geometry of the cylinders in contact will change when moving on the line of contact, i.e. the curvature radii rk1 and rk2 will either start with a small radius and increase when moving on the LoC or start with a big radius and decrease when moving on the LoC. They are defined through the following equation:

where rb1 and rb2 are the base circle radii of wheel 1 and wheel 2, respectively, and ToothBaseAngle describes the angle from the start of when the base of a tooth intersects with the LoC to its current angle. LoCTangent is a vector describing the LoC tangent line.

The forces are visualized by a blue and red arrow for each tooth pair, representing the force from wheel one and wheel two, respectively.

Contact Stress

The Hertz contact stress for a cylinder contact is:

ISO Standard Calculation

To be able to compare the forces from simulation to theoretical textbook values, there is a secondary implementation of the force. What can be compared is the contact pressure sigmaH and the indentation depth delta. The theoretical versions of these variables are called sigmaHTheoretical and deltaApprox. The theoretical values are calculated according to the formulas below:

where Kl is the load factor and Kv is the dynamic load factor (default value for both is 1).

where zH is the form factor, zM is the material factor, and zE is the gear ratio factor.

References

[1]  Dahl, M., H. Wettergren, and H. Tidefelt. "Modelica Spur Gears with Hertzian Contact Forces." Proceedings of the 12th International Modelica Conference, Prague, Czech Republic, May 15-17, 2017. Linköping Electronic Conference Proceedings 132, no. 82 (2017): 755-763. http://dx.doi.org/10.3384/ecp17132755.