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

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

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

Gravity

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:

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

Spring Mesh Edges => springs Internal springs to stabilize shape

Damping Calm down spring oscillations

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?

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

Examples 1, 4, 16 time samples per frame

Examples Multiple connectivity; fixing a node adding gravity and wind

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

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

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

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

Constrain Forces (soft constraints)