Electrical_Mechanical_Analogy

Electrical-Mechanical Analogy Dr Kevin Craig Professor of Mechanical Engineering Rensselaer Polytechnic Institute Electrical-Mechanical Analogy K Craig Electrical – Mechanical Analogies • A signal, element, or system which exhibits mathematical behavior identical to that of another, but physically different, signal, element, or system is called an analogous quantity or analog • Analogous quantities: force ⇔ voltage velocity ⇔ current displacement ⇔ damper ⇔ spring ⇔ mass ⇔ Electrical-Mechanical Analogy charge resistor capacitor inductor K Craig • Force causes velocity, just as voltage causes current • A damper dissipates mechanical energy into heat, just as a resistor dissipates electrical energy into heat • Springs and masses store energy in two different ways (potential energy and kinetic energy), just as capacitors and inductors store energy in two different ways (electric field and magnetic field) Spring Potential Energy Mass Kinetic Energy ( Kx ) f 2 q2 Kx = Ce = ⇔ = 2 K 2K 2C 1 Mv Li ⇔ 2 Capacitor Electric Field Energy Inductor Magnetic Field Energy • The product (f)(v) represents instantaneous mechanical power, just as (e)(i) represents instantaneous electrical power Electrical-Mechanical Analogy K Craig φ = ∫ ( e )dt q e= C e ∫ C R q ∫ φ e = iR L φ = Li i q = ∫ ( i )dt General Model Structure for Electrical Systems Electrical-Mechanical Analogy K Craig p = ∫ ( f )dt f = Kx f ∫ K B x ∫ p f = Bv M p = Mv v x = ∫ ( v )dt General Model Structure for Mechanical Systems Electrical-Mechanical Analogy K Craig force f ⇔ voltage e velocity v ⇔ current i damper B ⇔ resistor R spring K ⇔ capacitor 1/C mass M ⇔ inductor L Resistor e = Ri ⇔ di ⇔ Inductor e = L dt Capacitor e = ∫ i dt ⇔ C Electrical-Mechanical Analogy Electrical – Mechanical Analogies Damper f = Bv dv Mass f = M dt Spring f = K ∫ vdt K Craig RC Electrical System Spring-Damper Mechanical System fi − f B − f K = ein − e R − eC = ein − iR − eout = ⎛ deout ⎞ ein − ⎜ C ⎟ R − eout = ⎝ dt ⎠ de RC out + eout = ein dt eout = τ = RC ein RCD + Electrical-Mechanical Analogy f i − Bv − Kx = f i − Bv − f o = ⎛ fi ⎞ fi − B ⎜ o ⎟ − f o = ⎜K⎟ ⎝ ⎠ B i f o + f o = fi K fo B = τ= B fi K D +1 K K Craig LR Electrical System ein − e L − e R = di ein − L − eout = dt d ⎛ eout ⎞ ein − L ⎜ ⎟ − eout = dt ⎝ R ⎠ L deout + eout = ein R dt eout L = τ= L ein R D +1 R Electrical-Mechanical Analogy Mass-Damper Mechanical System fi − f B − f M = i f i − Bv − M v = ⎛ fi ⎞ fi − f o − M ⎜ o ⎟ = ⎜B⎟ ⎝ ⎠ M i fo + fo = fi B M fo τ= = B fi M D + B K Craig L R LRC Electrical System K B fo ein i C eout ein − e L − e R − eC = di ein − L − Ri − eout = dt d ⎛ de ⎞ ⎛ de ⎞ ein − L ⎜ C out ⎟ − R ⎜ C out ⎟ − eout = dt ⎝ dt ⎠ ⎝ dt ⎠ d eout de LC + RCdt out + eout = ein dt dt eout KS = = ein LCD + RCD + 1 D + 2ζ D + ω2n ωn ωn = R C ζ= KS = LC L Electrical-Mechanical Analogy M Mass-Spring-Damper Mechanical System fi − f K − f B − f M = +v fi i f i − Kx − Bv − M v = ⎛ fi ⎞ ⎛ fii ⎞ fi − fo − B ⎜ o ⎟ − M ⎜ o ⎟ = ⎜K⎟ ⎜K⎟ ⎝ ⎠ ⎝ ⎠ M ii B i fo + fo + fo = fi K K fo KS = = f i M D + B D + 1 D + 2ζ D + K K ω2n ωn ωn = K B KS = ζ= M KM K Craig • Inductor-Capacitor (LC) Oscillations Mass-Spring (MK) i=0 i=0 L C L eC eL M eC eL i L L eC v=0 x = +max C eL v=0 x = -max M i C eL K K C eC imax imax K K L C eC = eL = eL = L eC = i i L eL Electrical-Mechanical Analogy C C L eC e L M M v = max x=0 v = max x=0 C eC K Craig 10 Electrical-Mechanical Analogy K Craig 11 B K M +v F Resonance Electrical-Mechanical Analogy K Craig 12 Electrical-Mechanical Analogy K Craig 13