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

2. Basic Physics Accumulate forces, calculate acceleration
Implicit assumption is constant acceleration over time slice

3. Integration
For arbitrary function, f
We know the derivative (acceleration). We need to update the function based on derivative information
Much mathematics about how to update f based on its derivative
Runga-Kutta integration
Implicit Euler integration
Etc.
Accuracy v. time slice

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

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

6. Damping
Calm down spring oscillations

7. 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?

8. 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

9. 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

10. Angular Springs Use dot product of normals (cosine)
Take inverse cosine and use angle
Place a linear spring between ends of triangles

11. Constrain Forces (soft constraints)
Create temporary restoring forces (springs) when constraint violated F F F
Fix to surface
Non-penetration

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

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

14. Gravity

15. 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: