Brake

Description

This component models a brake, i.e., a component where a frictionalforce is acting between the housing and a flange and a controlled normalforce presses the flange to the housing in order to increase friction.The normal force fn has to be provided as input signal f_normalized in a normalized form(0 ≤ f_normalized ≤ 1),fn = fn_max*f_normalized, where fn_max has to be provided as parameter.Friction in the brake is modelled in the following way:

When the absolute velocity "v" is not zero, the friction forceis a function of the velocity dependent friction coefficient mue(v) , ofthe normal force "fn", and of a geometry constant "cgeo" which takes intoaccount the geometry of the device and the assumptions on the frictiondistributions:

        frictional_force = cgeo * mue(v) * fn

Typical values of coefficients of friction:

      dry operation   :  mue = 0.2 .. 0.4      operating in oil:  mue = 0.05 .. 0.1

The positive part of the friction characteristic mue(v), v >= 0, is defined via table mue_pos (first column = v, second column = mue). Currently, only linear interpolation in the table is supported.

When the absolute velocity becomes zero, the elements connected by the friction element become stuck, i.e., the absolute position remains constant. In this phase the friction force is calculated from a force balance due to the requirement, that the absolute acceleration shall be zero. The elements begin to slide when the friction force exceeds a threshold value, called the maximum static friction force, computed via:

       frictional_force = peak * cgeo * mue(w=0) * fn   (peak >= 1)

This procedure is implemented in a "clean" way by state events andleads to continuous/discrete systems of equations if friction elementsare dynamically coupled. The method is described in:

Otter M., Elmqvist H., and Mattsson S.E. (1999):

Hybrid Modeling in Modelica based on the Synchronous Data Flow Principle. CACSD'99, Aug. 22.-26, Hawaii.

More precise friction models take into account the elasticity of thematerial when the two elements are "stuck", as well as other effects,like hysteresis. This has the advantage that the friction element canbe completely described by a differential equation without events. Thedrawback is that the system becomes stiff (about 10-20 times slowersimulation) and that more material constants have to be supplied whichrequires more sophisticated identification. For more details, see thefollowing references, especially (Armstrong and Canudas de Wit 1996):

Armstrong B. (1991):

Control of Machines with Friction. Kluwer Academic Press, Boston MA.

Armstrong B., and Canudas de Wit C. (1996):

Friction Modeling and Compensation. The Control Handbook, edited by W.S.Levine, CRC Press, pp. 1369-1382.

Canudas de Wit C., Olsson H., Åström K.J., and Lischinsky P. (1995):

A new model for control of systems with friction. IEEE Transactions on Automatic Control, Vol. 40, No. 3, pp. 419-425.