26 4. MASS-SPRING MODELS
f
i
f
j
i j
Figure 4.3: A single spring connecting particle i and j applies equal and opposite forces to the
particles along the direction connecting the particles.
k
1
k
1
> k
2
> k
3
||x
i
- x
j
||- L
k
2
k
3
Force
Compression Elongation
Figure 4.4: e spring force will be linear in the amount of stretching or compression. A larger
spring constant will result in a bigger force response for a certain elongation or compression.
e horizontal axis shows the deviation from the rest length.
4.3.3 SPRING DAMPING FORCE
Obtaining stable simulation results is critically dependent on having damping forces in the sys-
tem. We hinted at this when discussing the stability of the oscillatory equation and integration
using explicit integration in Section 3.3.2 of the previous chapter. e most simple way to model
a damping force for a particle is to add a force that opposes the motion. For a particle i connected
to particle j we have the damping force acting on particle i as
d
i
.x/ D k
d
v
i
v
j
D d
j
.x/
(4.7)
with k
d
the damping coefficient. is mimics the real-world behavior of energy dissipation.
Note that this damping model is easy but far from perfect. It prevents bending of the cloth
and it penalizes rigid rotations of the spring. Adding a small amount of damping will result in
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