Physically Based Animation - Computer Animation: Algorithms and Techniques - page 201

spring

mass point

mass point

damper

FIGURE 7.1
Spring-damper configuration.

t ¼ Ia (7.11)

More physics will be introduced as various topics are covered, but this should give the reader a
starting point if physics is only a vague recollection.

7.1.1 Spring-damper pair
One of the most useful tools to incorporate some use of forces into an animation is the use of a spring-
damper pair as in Figure 7.1 . Consider a mass point, p 1 , connected by a spring to a fixed point, p 2 , with
rest length L r and current Length L c . the resulting force on p 1 would be computed as in Equation 7.12 .
As mentioned before, the spring represents a force to maintain a relationship between two points or
insert a control force into the system. The damper is used to restrict the motion and keep the system
from reacting too violently.

p 2

p 2 p 1 p 1

p 2 f ¼ k s ð L c L r Þk d p 1 (7.12)

k p 2 p 1 k k p 2 p 1 k

In physical simulations, the challenge in using a spring-damper system is in selecting an appropriate
time step, setting the spring and damper constants, and setting the mass of the object so that the result-
ing motion is desirable. Some combination of too small a time step, a large mass, a high damping con-
stant, and a low spring constant might result in a system that is not very lively, where the object slowly
moves toward the fixed point going slower and slower. On the other hand, a larger time step, a smaller
mass, a lower damping constant, and a higher spring constant might result in a system that moves too
much, resulting in a mass that oscillates around the fixed point, getting further and further away from it
each time step. Balancing these parameters to get the desired effect can be tricky.

7.2 Spring animation examples
7.2.1 Flexible objects
In Chapter 4 , kinematic approaches to animating flexible bodies are discussed in which the animator is
responsible for defining the source and target shapes as well as implying how the shapes are to be inter-
polated. Various physically based approaches have been proposed that model elastic and inelastic
behavior, viscoelasticity, plasticity, and fracture (e.g., [ 11 ] [ 13 ] [ 17 ] [ 23 ][ 24 ][ 25 ]). Here, the simplest
approach is presented. Flexibility is modeled by a mass-spring-damper system simulating the reaction
of the body to external forces.

Mass-spring-damper modeling of flexible objects
Probably the most common approach to modeling flexible shapes is the mass-spring-damper model.
The most straightforward strategy is to model each vertex of an object as a point mass and each edge
of the object as a spring. Each spring's rest length is set equal to the original length of the edge. A mass