Paper Summaries Any takers? Dynamics I Note on next lectures - PDF document

Paper Summaries • Any takers? Dynamics I • Note on next lecture’s papers: Linear Motion – Topic: Poupourri – NOT Angular Motion • Though a paper on Aerodynamic forces has been added Projects Plan for today • Proposals were due last Monday • Physics 101 for rigid body animation – Please submit if you haven’t already done so. • I will return proposals with feedback by Monday. Motivation Films Motivational Film • Two more Academy Award winning Pixar • Tin Toy (1988) films. – Technical challenge: life like baby. – First computer animated short to win an academy award.

Motivational Film Let’s get started • Geri’s Game (1997) • Physics for Rigid Body Dynamics – Technical challenge: Human & Clothing – Note: Change in original schedule Modeling • Today: Linear Motion • Monday: Rotational Motion – Geri makes a cameo appearance in Toy Story II. • Wednesday: Collisions • Monday, April 7: Numerical Integration / Constraints Using Physics in Animation The Source: Sir Isaac Newton • To achieve physically accurate motion, go • 1643-1727 to the source! • Discovered (amongst other things) – Keyframing accurate physical motion is – Calculus tedious. – Physics of Light • Use physics to calculate motion – Physics of Motion – Removes control from animator!! • Busy Man Laws of Motion Cartoon Laws of Motion • Law I • Law I – Every object in a state of uniform motion tends to – Any body suspended in space will remain in space until remain in that state of motion unless an external made aware of its situation. force is applied to it. (Inertia) • Law III • Law II: – Any body passing through solid matter will leave a – The acceleration of a body is proportional to the perforation conforming to its perimeter. resulting force acting on the body, and this • Law VII acceleration is in the same direction as the force. – Certain bodies can pass through solid walls painted to • Law III: resemble tunnel entrances; others cannot. – For every action there is an equal and opposite • Law IX reaction. – Everything falls faster than an anvil.

Terms Terms • For Linear Physical Motion • Mathematically defined – Mass = v ds / dt – Velocity • Measure of the amount of matter in a body – Acceleration = = 2 a dv / dt d s / dt • From Law II: Measure of the a body’s resistance to motion – Force – Velocity F = ma • Change of motion with respect to time – Acceleration • Change of velocity with respect to time – Force • In short, force is what makes objects accelerate Terms Physical units • Said another way: • You want physical motion, you need to use F ( t ) physical units = a ( t ) ∫∫ = 2 s ( t ) a ( t ) dt m ∫ = s ( t ) v ( t ) dt Quantity English SI (Metric) ∫ = v ( t ) a ( t ) dt Mass slug (s) kilogram (kg) Distance foot (ft) Meter (m) Velocity ft / sec m / sec • If we have force and mass, we can calculate Acceleration ft / sec 2 m / sec 2 motion. Force pound (lb) Newton s • ft / sec 2 kg • m / sec 2 Physical units Vectors • Is weight == mass? • Note that all quantities mentioned (except – This is actually still being debated for mass) are vector quantities – See Link on Web site – We are, after all, dealing in 3D motion • Our convention (used in Bourg book) – Mass = amount of matter in a body – Weight = Force resulting in acceleration due to gravity. • g = 9.8 m / sec 2 or 32 ft / sec 2 – Unambiguous in SI units – English units: slug = mass, pound = force

Center of Mass Initial value problems • In order to animate an object, we need to • Definition find p(t), for various values of t, given – the location where all of the mass of the system – Initial position of object could be considered to be located. – Initial velocity of object – For homogenous solid bodies that have a – Initial acceleration of object symmetrical shape, the center of mass is at the center of body's symmetry, its geometrical – Forces applied to object center. • All of the above are vector quantities Projectile motion Projectile motion • Shoot a projectile out of a cannon • analysis acceleration • Cannon is positioned at a given angle • No further acceleration except for gravity. -g Recall Solving differential equations • Means: F ( t ) – Analytically = a ( t ) ∫∫ = 2 s ( t ) a ( t ) dt m ∫ = s ( t ) v ( t ) dt – Numerically ∫ = v ( t ) a ( t ) dt

Analytic solution Analytic solution • For constant acceleration problem • We now have equations of motion: – Acceleration = = − = + φ a 0 a y g x ( t ) x ( v cos ) t x 0 0 – Velocity 1 v x = φ ∫ v cos = φ + = φ − v v sin a dt v sin gt = + φ − y ( t ) y ( v sin ) t gt 2 y 0 y 0 0 0 0 2 – Position 1 ∫ = + = + φ ∫ x x v dt x ( 0 v cos ) t y = y + v dt = y + ( v sin φ ) t − gt 2 0 x 0 0 y 0 0 2 Analytic solution Numerical Integration • Let’s see this in action • Remember, integration required – We won’t always be fortunate enough to be able to – Link perform the integration using calculus – Must code different equations for different motions. • In animation, numerical integration is usually used – Full discussion of Numerical Integration techniques next week • Quick and dirty hack – Euler Integration Leonard Euler Numerical Integration Object properties • 1707-1783 Calculate forces Position, orientation • Studied: Linear and angular velocity – Number theory Linear and angular momentum mass – Differential Equations – Newtonian Physics – Rotational Motion Update object properties Calculate accelerations Using mass, momenta

Euler Integration Euler Integration • Aka Method of Finite Differences F (t i+1 ) = F (t i ) + F ‘ (t i )* ∆ t • Approximates a curve with a series of straight lines corresponding to curve during F ’ (t i ) a given ∆ t • Uses tangent on the curve at a given point to guide the curve to the next point. F (t i ) Euler Integration Euler Integration • Equations • Euler Integration is fast, intuitive, and easy to code however, – x (t + ∆ t) = x(t) + (v(t) ∆ t) – v(t + ∆ t) = v(t) + (a(t) ∆ t) – It is also a very crude approximation to the integral – The smaller the ∆ t, the better the solution. – x(t + ∆ t) = x(t) + ((v(t) + v(t + ∆ t) /2) ∆ t • Also, the more work that is required. – x(t + ∆ t) = x(t) + v(t) ∆ t + ½ a(t) ∆ t 2 Euler Integration Rigid Body Simulation Object properties • Questions? Calculate forces Position, orientation Linear and angular velocity Linear and angular momentum mass • Break. Update object properties Calculate accelerations Using mass, momenta

Force Useful forces • F = ma • Gravity • a = F/m • Friction • From Force and mass you get acceleration • Impulse • Spring • Must consider the Sum of all forces • Wind (including rotational) • Add your own Gravity Gravity • Gravity is an attractive force between all pairs of massive objects • For objects interacting on this earth, the in the universe. acceleration due to gravity can be calculated • The gravitational force between two objects is given by a (fairly) simple mathematical equation. using the radius of the earth. m m – g = 9.8 m / sec 2 F = G 1 2 2 r – g = 32 ft / sec 2 • Where – m 1 and m 2 are the masses of the two objects – This acceleration is always towards the earth’s – r is the distance between the two objects surface. – G is the universal gravity constant = 6.67 x 10 -11 Nm 2 / kg 2 Friction Friction • Arises from interaction of surfaces in • Static Friction contact. – For objects not in motion – Fraction of the normal component of force • Always works against the direction of – Amount of force need to get object from rest moving relative motion of two objects. • Kinetic friction – For objects in motion – Fraction of the normal component of force – Amount of resistance due to friction

Static Friction Kinetic Friction Supporting object Supporting object F k v F s Resting contact Resting contact F F F N Normal force F N Normal force Static friction F s = u s * F N Kinetic friction F k = u k * F N Friction Impulse • Law III: Surfaces u s u k – For every action there is an equal and Dry glass on glass 0.94 0.4 opposite reaction. Dry iron on iron 1.1 0.15 Dry rubber on pavement 0.55 0.4 • Impulse is the equal and opposite reaction Dry steel on steel 0.78 0.42 after a collision Dry Teflon on Teflon 0.04 0.04 Dry wood on wood 0.38 0.2 • More when we talk about collisions Ice on Ice 0.1 0.03 Oiled steel on steel 0.1 0.08 Springs Springs • Force applied by stretching a • Hooke’s Law spring. – F = -kx • Given by Hooke’s Law – k = spring constant • Given in N/m – restoring force due to a spring is • Large k – stronger springs proportional to the length that the • Small k – looser springs spring is stretched – acts in the opposite direction. – F = -kx