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Multibody systems simulation software 109
the I marker from the J marker resolved parallel to the axes of the I marker
or DZ(I, J, I). Note that the third marker in brackets has been added to the
system variable DZ to indicate that the component of displacement is
resolved using this frame rather than by default the ground reference frame.
A first attempt at modelling this force could be achieved with the following
SFORCE statement. In this example 0206 is the I marker, 0307 is the J
marker, the length of the bump stop is 50 mm and a linear stiffness for the
bump stop of 300 N/mm is used.
SFORCE/0607,I0206,J0307,TRANS,
FUNCTIONIF(DZ(0206,0307,0206)-50:300*(50-DZ
(0206,0307,0206)),0,0)
This example has introduced an arithmetic IF that allows us to conditionally
program the value of the FUNCTION and hence the force returned by
this statement. The format used here is
IF (expression 1: expression 2, expression 3, expression 4)
Expression 1 is evaluated and the value obtained is used to determine which
expression is used to evaluate the FUNCTION as follows:
IF expression 1 0 then the FUNCTION expression 2
IF expression 1 0 then the FUNCTION expression 3
IF expression 1 0 then the FUNCTION expression 4
In this case expression 1 is DZ(0206,0307,0206)-50. Clearly when this is
greater than zero the gap is open and so the calculated force from expression
4 is zero. When expression 1 is equal to zero the I marker is just making
contact with the face of the bump stop but no deformation has taken place
and so the force is still zero. When expression 1 is less than zero then the
contact force is generated using 300*(50-DZ(0206,0307,0206)). Note that
this has been programmed to ensure that a positive value is generated for
the bump stop as it compresses.
Although the method introduced here might work it is possible that it will
cause problems during the numerical integration of the solution. This is
because the arithmetic IF causes the force to switch on and off instantaneously
about the point of contact. Such modelling is usually undesirable
and some method is needed to improve the formulation by ‘smoothing’ the
transition as contact occurs. This could be achieved by introducing another
feature known as a STEP FUNCTION as follows:
SFORCE/0607,I0206,J0307,TRANS,
FUNCTIONSTEP(VARVAL(1),0,0,0.5,1)*(300*(50-DZ
(0206,0307,0206)))
VARIABLE/1,FUNCTION50-DZ(0206,0307,0206)
Although referred to here as a step users of similar multibody systems software
packages may also think of it as a ramp since the change in value
returned by the function is not an instantaneous step change. The STEP
function used here uses a cubic polynomial to smooth the transition from
one state to another as shown in Figure 3.32.
This example also introduces the use of VARIABLES that can be used
to program equations and substitute the returned value, VARVAL(id), into