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Transcript
Lecture 1 - Background from 1A Revision of key concepts with application to driven oscillators: Aims: Review of complex numbers: Addition; Multiplication. Revision of oscillator dynamics: Free oscillator - damping regimes; Driven oscillator - resonance. Concept of impedance. Superposed vibrations. 1 Waves1 Complex representation Complex nos. and the Argand diagram: Use complex number A, where the real part represents the physical quantity. A Aoei a1 ia2 Amplitude Phase Amplitude follows from: Phase follows from: Ao2 a12 a22 tan a2 a1 Harmonic oscillation: z Aeit Aoei t Displacement z Aeit x z Velocity z i Aeit x z Ao sin t Acceleration z i 2 Aeit x z 2 Ao cost 2 Ao cost Waves1 Manipulation of Complex Nos. I Addition A Aoeio A1ei1 A2ei2 Ao2 A12 A22 2 A1 A2 cos 2 1 A1 sin 1 A2 sin 2 tan o A1 cos1 A2 cos 2 A Ao coso A1 cos1 A2 cos2 The real part of the sum is the sum of the real parts. 3 Waves1 Manipulation of Complex Nos. II Multiplication WARNING: z1z2 z1 z2 One cannot simply multiply the two complex numbers. Example (i): To calculate (velocity)2 . Take velocity v = Voeit with Vo real. 2 2 2 Instantaneous value: v Vo cos t Mean value: v 2 1 Vo2 1 vv 1 v 2 2 2 2 Example (ii). Power, (Force . Velocity). Take f = Foei(t+) with Fo real. Instantaneous value: f v FoVo cost cost Mean value: 1 FoVo cos2t cos 2 1 1 FoVo cos fv 2 2 4 Waves1 The damped oscillator Equation of motion mx sx bx Restoring force 1.1 Dissipation (damping) Rearranging gives x 2x o2 x 0 2 b m o2 s m 1.2 Natural resonant frequency Two independent solutions of the form x=Aept. Substitution gives the two values of p, (i.e. p1, p2), from roots of quadratic: p 2 2p o2 0 p1 / 2 2 o2 General solution to [1.2] x e t A1e qt A2e qt q 2 2 o2 5 Waves1 Damping régimes Heavy damping q 2 0 or o q t A2e q t x A1e Sum of decaying exponentials. 2 Critical damping q 0 or o t x e A1 A2t Swiftest return to equilibrium. Light damping q i1, where 12 o2 , t A1ei1t A2ei1t xe Damped vibration. x0 0, x 0 1, o 1 Heavy : 6 Critical : 1 Light : 6 0.02 Waves1 Driven Oscillator Oscillatory applied force (frequency ): Force: f Fe it Equation of motion: F x 2x o2 x eit m Use complex variable, z, to describe displacement: i.e. x z F z 2z o2 z eit m 1.3 Steady state solution MUST be an oscillation at it frequency . So z Ae A gives the magnitude and phase of the “displacement response”. Substitute z into [1.3] F to get 2 2 A i 2 o A F m o2 2 i 2 The “velocity response” is z iAeit m . F m e i t 2 o2 2 i 7 Waves1 Impedance Mechanical impedance Note, the velocity response is proportional to the driving force, i.e. Force =constant(complex) x velocity Mechanical impedance Z = force applied / velocity response 2 o2 Z m 2 i In general it is complex and, evidently, frequency dependent. Electrical impedance Z=applied voltage/current response Example, series electrical circuit: Z R iL 1 iC We can write the mechanical impedance in a similar form: Z 2m im imo2 b im s i 8 Waves1