| This model can be found in the 20-sim demo
directory.
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Using Backlash
Description
Although it should be avoided at all times, backlash is still very
common in drivetrains. The use of backlash in models is not trivial.
In 20-sim backlash is modelled by a combination of springs. Inside the
play (i.e. the region where both side of the backlash model are free
to move) both sides are connected by a spring-damper combination with
very small stiffness and damping. Outside the play (i.e. both sides
of the backlash model are now rigidly connected) both sides are connected
by a spring-damper combination with high stiffness and damping.
If we show a plot of the static force F
versus the backlash position x
(not that we have chosen x = 0 in the middle of the play) we get plot
like:
In simulations the sudden change
of stiffness leads to stiff models that are hard to simulate. A useful
backlash model therefore has a more gradual change of stiffness. In
20-sim this gradual change is denoted by the relative round-off parameter
ep.
For large values of ep
(> 0.01) the change in stiffness is too gradual and the model does
not represent real backlash behavior anymore. For very small values
of ep
(< 1e-4) there is a very steep change in stiffness and the model
represents a near perfect backlash behavior, but is hard to simulate.
How to set the backlash parameters
The simulation looks like:
The plot shows the backlash
position x. Due to the sinusoidal force, the backlash bounces from and
to each side. In this example the backlash model has default parameter
values:
|
s = 1 [mm]
k1 = 0.1 [N/m]
k2 = 1e6 [N/m]
d1 = 0.01 [N.s/m]
d2 = 1e5 [N.s/m]
ep = 0.01
|
width of the play [m]
Stiffness in the play [N/m]
Stiffness outside the play [N/m]
Damping inside the play [Ns/m]
Damping outside the play [Ns/m]
Relative round off
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These are good values to start
with. As in the simulation plot, always inspect the backlash position
x. The various parameters can be tuned in the order shown below, until
a good result is found, before paying attention to the rest of the model.
The rest of the tuning can be found in the 20-sim model itself.
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Note
The use of backlash models
leads to stiff simulation models. I.e. models with high and low
resonance frequencies. These models lead to very small stepsizes
in most integration methods. The Backward Differentiation Formula
(used in the example) and the Vode Adams method are best suited
for stiff models.
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