MATHEMATICAL MODELSConstant velocity motionA function model: p(t) = V*t + P0 A 1st order differential equation model: p’(t) = V A 2nd order differential equation model: p’’(t) = 0 A falling objectNewton’s model: r’’(t)
= -G*M*MEarth/r(t)2 A simplified model: r’’(t) = -10 Motion of a rigid objectF=MA T=a’’/I Atmospheric drag for an objectAir density: D = 1.4*(1 – altitude/30,000) for altitude < 30,000 Force of atmospheric drag: Fdrag = Cdrag * D * v2 * cs / 2 Airplane lift and dragCL = 0.5 + 0.1*AoA Flift = CL*D** v2 * s CD = (5 - AoA )*1 + 0.7 for AoA <= 5 CD = (AoA - 5 )*1 + 0.7 for AoA > 5 Fdrag = CD*D** v2 * s Electrical componentsA resistor: vR(t) = i(t)R A capacitor: vC’(t) = i(t)/C An inductor vI(t) = i’(t)L Mechanical componentsA spring: Fs(t) = -K * p(t) A hydraulic damper: Fd(t) = -cp’(t) |
Sliding frictionCoulomb or sliding friction: FC(t) = -mNsign(p’(t)) Programmed modelsWe have programmed models for rocket thrusters, airplane thrust, aircraft elevator torque, electrical impulses, and rocky roads, by directly specifying their values at each subinterval in the computation. PHYSICAL LAWSLaws of physicsThe law of gravity: F = GM1M2/R2 The 2nd law of motion: F=MA The 2nd law of rotation: T=a’’/I Kirchoff’s Electrical Circuit LawsKirchoff’s voltage law: The sum of the voltages around a loop in a circuit is zero. Kirchoff’s current law: The sum of currents entering a node in a circuit is zero. |