1. Forces - basic physics Gravity Friction - static and kinetic Viscosity
Springs (Hooke’s law)
Damping

2. Friction Supporting object Fs Resting contact F FN Normal force
Static friction Fs
Resting contact F
Normal force FN

3. Friction Supporting object Kinetic friction Fk v Resting contact F FN
Normal force
Static friction

4. Gravity

5. Viscosity kv - depends on shape of object
n - depends on properties of liquid
For spherical object:
Terminal velocity - viscosity and gravity balance
E.g., for sphere:

6. Springs (Hooke’s law) Spring’s rest length: exerts zero force x xrest

7. Spring Mesh
Edges => springs
Internal springs to stabilize shape

8. Damping
Calm down spring oscillations

9. Mass-Spring-Damper System
Define point masses
postion
velocity
mass
force
fixed?
Define springs
point 1
point 2
rest length
kspring
kdamper
Multiple time samples per frame?

10. Mass-Spring-Damper System
For each point
Initialize force with wind
For each spring
Calculate spring-damper force
spring.point1.force += force
spring.point2.force -= force
For each point
acc = gravity
acc += mass/force
newVel = velocity + acc*dt
position += dt*(velocity+newVelocity)/2
Velocity = newVelocity

11. Examples
1, 4, 16 time samples per frame

12. Examples
Multiple connectivity; fixing a node adding gravity and wind

13. Examples Adding damping; making a flag
Use controlled random numbers to add interest

14. Randomize Controlled randomness adds more interest
To initial values (positions, velocities)
To force fields (wind direction, wind speed)
To spring constants, masses
To joint angles
Proximal joints: lower amplitude
Distal joints: higher amplitude
Coordinate frequence and phase

15. Angular Springs Use angles (cosine) between normals
Place a linear spring between ends of triangles

16. Constrain Forces (soft constraints)
Fix to surface
Non-penetration

17. Constrain Forces (soft constraints)