Dynamic Physics for Simulation and Game Programming

Transcription

1 Dynamic Physics for Simulation and Game Programming Mike Bailey Think Geek physics-dynamic.pptx

2 Discrete Dynamics 2. Force equals mass times acceleration (F=ma) a F m v t a vat v v at x t v x vt x xvt These can be thought of as summing areas under curves

3 Integrating the Physics Equations if Acceleration is Constant a t t vat area

4 What if Acceleration is not Constant? a t t vat area

5 What is v(t+ t)? a Accumulated v so far The velocity is off by this much! Acceleration at the start of the time step vat t t Time Step vt ( t) vt () vvt () at This is close, but is clearly not exactly right! t

6 a What is v(t+ t)?????? vat Acceleration at the start of the time step vt ( t) vt () vvt () at t { t t Time Step The problem is that we are treating all of the quantities as if they always have the value that they had at the start of the Time Step, even though they don t. This is known as a First Order solution.

7 What does a First Order Solution look like in a Program? You need a way to hold the entire state of the system. This will be the input to our numerical integrator. You also need a way to return the derivatives once you determine them. struct state { float time; float x; float vx; } ; dx x dt v dv dt x a x struct derivatives { float vx; float ax; } ; struct state State; struct derivatives Derivatives; void GetDerivs( State, Derivatives ) {... } The outputs are the derivatives of the state variables. The inputs are the state, which consists of all variables necessary to completely describe the state of the physical system. The outputs are the derivatives of each state variable.

8 What does a First Order Solution look like in a Program? void AdvanceOneTimeStep( ) { GetDerivs( State, Derivatives ); // get derivatives } State.x = State.x + Derivatives.vx * t; // use derivatives State.vx = State.vx + Derivatives.ax * t; // use derivatives State.t = State.t + t ; The application, then, consists of: Initialize( ); AdvanceOneTimeStep( ); Finish( );

9 What is v(t+ t)? A Second Order solution is obtained by doing the First Order solution, determining all quantities at time t t then averaging them with the quantities at time t and then treating them as constant throughout the interval. 1 2 a Ft () Ft ( t) at () m at ( t) m 3 4 at () at ( t) vat aavgt t 2. vt ( t) vt ( ) v vt () t{ t

10 What does a Second Order Solution look like in a Program? void AdvanceOneTimeStep( ) { GetDerivs( State, Derivatives1); State2.t = State.t + t; State2.x = State.x + Derivatives1.vx * t; State2.vx = State.vx + Derivatives1.ax * t; } GetDerivs( State2, Derivatives2 ); aavg = ( Derivatives1.ax + Derivatives2.ax) / 2.; vavg = ( Derivatives1.vx + Derivatives2.vx) / 2.; State.x = State.x + vavg * t; State.vx = State.vx + aavg * t; State.t = State.t + t ; The application, then, consists of: Initialize( ); AdvanceOneTimeStep( ); Finish( );

11 The Runge-Kutta Fourth Order Solution v a 1 1 GetDerivs(, t x, v) v2 t t t GetDerivs( t, x v 1, v a 1 ) a v3 t t t GetDerivs( t, x v 2, v a 2 ) a v a4 GetDerivs( t t, x v t, v a t) xt ( t) x t v v v v ( ) vt ( t) v 6 a1 a2 a3 a4 Adapted from:

12 The Runge-Kutta Fourth Order Solution void AdvanceOneTimeStep( ) { GetDerivs( State, Derivatives1 ); State2.t = State.t + t/2.; State2.x = State.x + Derivatives1.vx * ( t/2.); State2.vx = State.vx + Derivatives1.ax * ( t/2.); GetDerivs( State2, Derivatives2 ); State3.t = State.t + t/2.; State3.x = State.x + Derivatives2.vx * ( t/2); State3.vx = State.vx + Derivatives2.ax * ( t/2.); GetDerivs( State3, Derivatives3 ); State4.t = State.t + t; State4.x = State.x + Derivatives3.vx * t; State4.vx = State.vx + Derivatives3.ax * t; GetDerivs( State4, Derivatives4 ); } State.x = State.x + ( t/6.) * ( Derivatives1.vx + 2.*Derivatives2.vx + 2.*Derivatives3.vx + Derivatives4.vx ); State.vx = State.vx + ( t/6.) * (Derivatives1.ax + 2.*Derivatives2.ax + 2.*Derivatives3.ax + Derivatives4.ax ); Adapted from:

13 Solving Motion where there is a Spring Fspring ky This is known as Hooke s law +y, +F F W ky v t t m m void GetDerivs( State, Derivatives ) { Derivatives.vy = State.vy; Derivatives.ay = ( W K*State.y ) / MASS; } Weight

14 Air Resistance Force F 1 2 v AC 2 drag y d Fluid density +y Y Velocity Drag Coefficient Cross-sectional area Body Falling Air Resistance always acts in a direction opposite to the velocity of the object W

15 C d Some Drag Coefficients Item 2.1 a smooth brick 0.9 a typical bicycle plus cyclist 0.4 rough sphere 0.1 smooth sphere laminar flat plate turbulent flat plate bullet person (upright position) 1.28 flat plate perpendicular to flow skier wires and cables Empire State Building Eiffel Tower

16 Solving Motion where there is Air Resistance 1 2 F W Sign( vy) vyacd v t 2 t m m air kg 3 m The Sign( ) function returns +1. or -1., depending on the sign of argument void GetDerivs( State, Derivatives ) { Derivatives.vy = State.vy; Derivatives.ay = ( W.5*Sign(State.vy)*DENSITY* State.vy * State.vy *AREA*DRAG ) / MASS; }

17 Terminal Velocity v W 1 2 vyac 2 m d t When a body is in free fall, it is being accelerated by the force of gravity. However, as it accelerates, it is encountering more and more air resistance force. At some velocity, these two forces balance each other out and the velocity becomes constant, that is, v=0. This is known as the terminal velocity. The velocity becomes constant when v = 0 : W 1 2 vyac 2 d 0 v t 2W AC d

18 Human Terminal Velocity Assume: Weight = 200 pounds = 890 Newtons C d = 1.28 A = 6 ft 2 = m 2 air kg m v t 2W m mph AC sec d

19 How about a Cliff Jumper on a Bungee Cord? Fspring ky 1 2 Fdrag Sign( vy ) vc y d A 2 Body Falling 1 2 F W kysign( vy) vyacd v t 2 t m m W void GetDerivs( State, Derivatives ) { Derivatives.vy = State.vy; Derivatives.ay = ( W K*State.y -.5*Sign(State.vy)*DENSITY*State.vy*State.vy*AREA*DRAG ) / MASS; }

20 Lift Another Good Force to Know About F lift vacl Coefficient of Lift, for a given angle of attack Air density Airspeed Platform area

21 Coefficient of Lift vs. Angle of Attack Angle of Attack

22 Lift and Drag Dramatically Working Together Flight of a Frisbee

23 Friction Force Another Good Force to Know About Ffriction N Normal force (i.e., amount of force that is perpendicular to the surface) Coefficient of Friction N N W

24 Some Coefficients of Friction Materials Aluminum Steel 0.61 Copper Steel 0.53 Brass Steel 0.51 Cast iron Copper 1.05 Cast iron Zinc 0.85 Concrete (wet) Rubber 0.30 Concrete (dry) Rubber 1.0 Concrete Wood 0.62 Copper Glass 0.68 Glass Glass 0.94 Dry & clean μ Lubricated Metal Wood (wet) Polythene Steel Steel Steel Steel Teflon Teflon Teflon Wood Wood (wet)

25 Buoyancy Another Good Force to Know About Archimedes Principle says that the buoyancy force on an object in a fluid is the weight of the fluid that is being displaced by the object. air 4.66x10 pounds / in 5 3 helium 0.65x10 pounds / in 5 3 } Densities So, for a helium balloon that is one foot in diameter (i.e., radius=6 inches), it has its weight pulling it down and a buoyancy force pushing it up. The net force pushing it up because of the gas inside the balloon is: F V V V ( ) buoyancy balloon helium balloon air balloon helium air 4 Vballoon r in Fbuoyancy in (4.01x10 pounds / in ) pounds Note that this must still counterbalance the weight of the balloon material, or the balloon will not fly.

26 Spinning Motion: Dynamics T I Moment of Inertia an angular mass (newton-meters-sec 2 =kg-meters 3 ) T t I T t t I Torque an angular force (newton-meters)

27 Spinning Motion: What does this look like in a Program? struct state { float t; float x; float vx; float theta; float omega; } ; struct derivatives { float vx; float ax; float omega; float alpha; } ; The state and derivative vectors now include angular components void GetDerivs( State, Derivatives ) { Derivatives.vx = State.vx; Derivatives.x = SomeOfAllForces / MASS; Derivatives.omega = State.omega; Derivatives.alpha = SomeOfAllTorques / INERTIA }