Theory for Gear Friction
The Friction node can optionally be added on gear pairs. In cases where friction is included on a gear pair, frictional forces in the plane perpendicular to the line of action are added on both gears.
By default, there is no friction on Gear Pair and Rack and Pinion. It is, however, included for Worm and Wheel. The worm and wheel pair has always neither a parallel nor an intersecting configuration, and in this configuration there is always a slip even under perfect conditions.
Friction can be applied to gear pairs only in time-dependent analysis.
Friction Model
The friction force is modeled using a continuous friction law, which is capable of modeling sliding-sticking phenomena. A strict application of Coulomb’s law involves discrete transition from sticking to sliding and vice versa, as dictated by a vanishing relative velocity. These discrete transitions cause numerical difficulties and to avoid them, the friction force is approximated with a continuous friction law:
where Ff is the friction force, μ is the frictional coefficient, N is the normal force, v is the slip velocity, and v0 is the characteristic slip velocity. The term
is called the regularization factor.
The regularization factor smooths the friction force discontinuity. The characteristic slip velocity should be made small in comparison to the characteristic relative velocities encountered during the simulation. The continuous friction law describes both sliding and sticking behavior; that is, it completely replaces Coulomb’s law. Sticking is replaced by creeping between the contacting bodies with a small relative velocity.
Normal Force
The normal force at the contact point can be taken as the computed contact force. Alternatively, it can also be entered explicitly.
Slip at Contact Point
The slip vector at the contact point of a gear pair is defined as:
A similar expression can be written for a worm and wheel pair. However, for a rack and pinion pair, the definition is slightly different and the slip vector is defined as:
In 2D, the slip vector expression reduces to:
In the case of a line contact model, the slip at the second contact point is defined as:
Friction Force
The friction force at the contact point and its virtual work contribution are defined as:
where Ff,r and Ff,max are the additional sliding resistance and the maximum friction force, respectively.
Similarly, in the case of a line contact model, the friction force at the second contact point and its virtual work contribution is defined as:
Energy Dissipation Rate
The energy dissipation rate due to friction at the contact point is defined as:
For a line contact model, the additional contribution in the energy dissipation rate is defined as: