ModelingOfFriction

Description

Several elements of this library model Coulomb friction with the method proposed 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.

The friction equations are defined in base modelInterfaces.PartialFriction.In the following, there are given explanations.

Assume first the most simplest friction problem: A block sliding on a surface.The friction force f acts between the block surface and the environment surface andshall be a linear function of the relative velocity v between the two surfaces.When the relative velocity becomes zero, the two surfaces are stuck to each other andthe friction force is no longer a function of v. The element starts slidingagain if the friction force becomes larger than the maximum static friction forcef₀ (indicated by f0 in equations below).This element could be defined with a parameterized curve description leadingto the following equations:

forward  = s >  1;backward = s < -1;v = if forward  then s-1 elseif backward then s+1 else 0;f = if forward  then  f0 + f1*(s-1) elseif       backward then -f0 + f1*(s+1) else f0*s;

This model completely describes the simplified friction element ina declarative way. Unfortunately, currently it is not known how to transform suchan element description automatically in a form which can be simulated:

The block is described by the following equation:

m*der(v) = u - f

Note, that m is the mass of the block and u(t) is the given driving force.If the element is in its "forward sliding" mode, that is s ≥ 1,this model is described by:

m*der(v) = u - f       v = s - 1       f = f_0 + f_1*(s-1)

which can be easily transformed into state space form with v as the state.If the block becomes stuck, that is -1 ≤ s ≤ 1,the equation v = 0 becomesactive and therefore v can no longer be a state, that is an indexchange takes place. Besides the difficulty to handle the variable state change,there is a more serious problem:

Assume that the block is stuck and that s becomes greater than one.Before the event occurs, s ≤ 1 and v = 0;at the event instant s > 1 because this relation is the eventtriggering condition. The element switches into the forward sliding mode where vis a state which is initialized with its last value v = 0.Since v is a state, s is computed from vvia s := v + 1, resulting in s = 1,that is the relation s > 1 becomes false and the elementswitches back into the stuck mode. In other words, it is never possible to switch intothe forward sliding mode. Taking numerical errors into account, the situation is even worse.

The key to the solution is the observation that v = 0 in the stuckmode and when forward sliding starts, but der(v) > 0 when slidingstarts and der(v) = 0 in the stuck mode. Since the friction characteristicat zero velocity is no functional relationship, again a parameterized curve descriptionwith a new curve parameter s_a (denoted also sa below)has to be used leading to the following equations (note: at zero velocity):

startFor  = sa >  1;startBack = sa < -1;        a = der(v);        a = if startFor then sa-1 elseif startBack then sa+1 else 0;        f = if startFor then  f0  elseif startBack then  -f0 else f0*sa;

At zero velocity, these equations and the equation of the block form a mixed continuous/discreteset of equations which has to be solved at event instants (e.g. by a fix point iteration),When switching from sliding to stuck mode, the velocity is small or zero.Since the derivative of the constraint equation der(v) = 0 is fulfilledin the stuck mode, the velocity remains small even if v = 0 is not explicitlytaken into account. The approach to use the acceleration der(v) = 0 as"constraint" instead of v = 0, is often used in multi-body software.The benefit is that the velocity v remains a state in all switching configurations(there is a small, linear drift, but the friction element would have to stay stuck several daysbefore the drift becomes too large). Consequently, v is small but may have any sign whenswitching from stuck to sliding mode; if the friction element starts to slide, say in the forwarddirection, one has to wait until the velocity is really positive, before switching to forwardmode (note, that even for exact calculation without numerical errors a "waiting"phase is necessary, because v = 0 when sliding starts).Since der(v) > 0, this will occur after a small time period.This "waiting" procedure can be described by a state machine.Collecting all the pieces together, finally results in the following equationsof a simple friction element:

// part of mixed system of equationsstartFor  = pre(mode) == Stuck and sa >  1;startBack = pre(mode) == Stuck and sa < -1;        a = der(v);        a = if pre(mode) == Forward  or startFor  then  sa - 1    elseif               pre(mode) == Backward or startBack then  sa + 1    else 0;        f = if pre(mode) == Forward or startFor   then  f0 + f1*v elseif               pre(mode) == Backward or startBack then -f0 + f1*v else f0*sa;// state machine to determine configurationmode = if (pre(mode) == Forward  or startFor)  and v>0 then Forward  elseif          (pre(mode) == Backward or startBack) and v<0 then Backward else Stuck;

The above approach to model a simplified friction element is slightly generalized in modelInterfaces.PartialFriction:

• The sliding friction force has a nonlinear characteristic instead a linear one, by interpolation in a table of f(v) values.
• There may be a jump in the friction force when going from stuck to sliding mode (described with parameter peak).