WEEK 1 SUMMARY Energy approach to equation of motion: i.e. find trajectory x(t) if U(x) is known dx • dt = 2 ± E ! U(x)] [ m • Special case of U(x)=(½)kx2 found period T (indep of A), found x(t) = A cos(ωt+φ) • Found 3 other forms of A cos(ωt+φ) • Learned to apply initial conditions to determine A, φ, and also the arbitrary parameters in other 3 forms • Harder case of U(θ)= MgLcm(1-cos θ) found -> HO for small theta found T numerically measured T for pendulum • Learned to argue qualitatively about time to move certain distances, comparing T for diff U • Did NOT learn equation of motion θ(t) Energy Energy approach to equation of motion: position Complex numbers: • rectangular and polar form and Argand diag. • complex conjugate • Euler relation exp(i" ) = cos " + isin " • solving one complex equation is actually solving 2 simultaneous equations ! WEEK 2 SUMMARY Free, undamped oscillators k m L No friction k I C q m x m!! x = !kx 1 q!! = ! q LC ! ! r; r = L θ Common notation for all T m mg g !! !"# ! L 2 !! ! + " 0! = 0 Force approach to equation of motion of FREE, UNDAMPED HARMONIC OSCILLATOR: i.e. find trajectory θ(t) if F(θ) is known • Special case of F(θ)=-sin(θ) -> small angle approx: F(θ)=-θ =>2nd order DE, Found sinusoidal motion ! (t) = Cei" 0t + C *e#i" 0t ! (t) = A cos (" 0t + $ ) Applied initial conditions as before. E = K + U = 12 I!! 2 + 12 mgL! 2 Free, damped oscillators k friction m x m!! x = !kx ! bx! 1 ! LI + q + RI = 0 C R 1 q!! + q! + q=0 L LC ! r = Lcm θ Common notation for all T m mg g !! ! " # ! # b'!! L !!! + 2 "!! + # 02! = 0 Force approach to equation of motion of FREE, DAMPED OSCILLATOR • Add damping force to eqn of motion • Found decaying sinusoid x(t) = Ce =e ! " t+i# 1t ! "t x(t) = Ae $%Ce " #t * ! " t!i# 1t +C e +i# 1t * !i# 1t +C e [cos($1t + % )] &' FREE, DAMPED OSCILLATOR • Damping time τ=1/β • measures number of oscillations in Q = ! decay time • apply initial conditions, energy decay x(t) = Ae " #t " #0 = T 2$ [cos($1t + % )] WEEK 3 SUMMARY DRIVEN, DAMPED OSCILLATOR L I C Vocosωt R q V0 e " Lq!! " " Rq! = 0 C V0 i! t 2 q!! + 2 # q! + ! 0 q = e L i! t q ( t ) = Re #$ q0 e e %& i!q i" t "2 #! q0 = ; tan (q = 2 1/2 2 2 !0 " ! $(! 2 " ! 2 ) + 4 # 2! 2 & 12 % 0 ' V0 L CHARGE "Resonance" Charge Amplitude |q0| q0 = ω0 Charge Phase φq 0 $(! 2 " ! % 0 V0 L ) 2 2 + 4 # 2! 2 & ' Driving Frequency ω------> % "2 #$ ( !q = arctan ' 2 2 * &$ 0 " $ ) -π/2 -π ω0 1/2 CURRENT dq ( t ) I (t ) = = i! q ( t ) dt Resonance Current Amplitude |I0| I0 = ω0 Current π/2 Phase φI 0 ! V0 L $(! " ! % 2 0 ) 2 2 1/2 + 4# ! & ' 2 2 Driving Frequency ω------> & #2 $% ) " ! I = + arctan ( 2 2 + 2 '% 0 # % * -π/2 ω0 ADMITTANCE I Y (! ) = Vapp NOT time dependent, but IS freq dependent. Resonance Admittance Amplitude |Y0| Y = ω0 Addmittance π/2 Phase φI 0 ! L $(! 2 " ! % 0 ) 2 2 1/2 + 4 # 2! 2 & ' Driving Frequency ω------> & #2 $% ) " ! I = + arctan ( 2 2 + 2 '% 0 # % * -π/2 ω0 DRIVEN, DAMPED OSCILLATOR • can also rewrite diff eq in terms of I and solve directly (same result of course) L I Vocosωt C R q V0 e " Lq!! " " Rq! = 0 C V0 i! t 2 q!! + 2 # q! + ! 0 q = e L V0 i! t 2 $ !!! q + 2 # q!! + ! 0 q! = i! e L I V 0 i! t !! ! I + 2# I + = i! e LC L i! t FOURIER SERIES – periodic functions are sums of sines and cosines of integer multiples of a fundamental frequency. These “basis functions are orthonormal f ( t ) = " an cos n! t + bn sin n! t f �t�,g�t� A n 0 �A f �t��g�t� A 0 T�4 T�2 3T�4 T Time, t �A f �t��g�t� A 0 �A f �t�,g�t� A T�4 T�2 3T�4 T T�4 T�2 3T�4 T Time, t 0 T�4 T�2 3T�4 T Time, t �A T 2 sin ( p! t ) sin ( q! t ) dt = # pq " T 0 T 2 cos ( p! t ) cos ( q! t ) dt = # pq " T 0 Time, t ODD functions f(t) = -f(-t). Their Fourier representation must also be in terms of odd functions, namely sines. Suppose we have an odd periodic function f(t) like our sawtooth wave and you have to find its Fourier series " bn sin(n!t ) n=1,2... Then the unknown coefficients can be evaluated this way Integrate over the period of the fundamental Here s the coefficient of the sin(ωnt) term! Plot it on your spectrum! 2 bn = T normalize properly T " f (t)sin ( n! t ) dt 0 the function the harmonic f �t� A 0 Time domain �A T�4 T�2 3T�4 ( ) f ( t ) = # 2 sin n" f t n! n Frequency domain Time, t T 2A f (t ) = A ! t 0 < t < Tf Tf B frequency Fundamantal freq = 2π/T coefficient of the sin(ωnt) term! Integrate over the period of the fundamental 2 bn = T normalize properly T " f (t)sin ( n! t ) dt 0 the function the harmonic mod Y phase�Y� DRIVING AN OCILLATOR WITH A PERIODIC FORCING FUNCTION THAT IS NOT A PURE SINE Ω0 Ω0 2Ω0 2Ω0 3Ω0 3Ω0 Ω �rad�s� Ω �rad�s� B Oscillator response Important to know where the fundamental freq of the forcing funcOon lies in relaOon to the oscillator max response freq! Forcing funcOon frequency DRIVING AN OCILLATOR WITH A PERIODIC FORCING FUNCTION THAT IS NOT A PURE SINE Vapp = V0 e i! f t + 2V0 e i 2! f t (given) i! t i 2! t q!! + 2 " q! + ! 02 q = V0 e f + 2V0 e f (Kirchoff) # q = q! + q2! (linear diff eq - superposition) # I = I! f + I 2! f I = q! mod Y I = Y! f Vapp,! f + Y2! f Vapp,2! I= + !f L ( % !2 $!2 f '& 0 ) 2 1/2 + 4 " 2! 2f (* ) ( 2! ) f ( % ! 2 $ ( 2! )2 f '& 0 ) L e ,+ % $2 "! f (/ i. +arctan ' 2 2 *1 .- 2 '& ! 0 $! f *)10 1/2 e 2 2 + 4 " 2 ( 2! ) f ( *) V0 e i! f t phase�Y� Ω0 Ω0 , % $2 " 2 ! (/ + f *1 i. +arctan ' 2 ' ! 0 $ ( 2 ! )2 * 1 .2 f )0 & 2V0 e ( ) i 2! f t 2Ω0 2Ω0 3Ω0 3Ω0 Ω �rad�s� Ω �rad�s� FT - you know this time Black box Z(ω) frequency Observe what (LRC) black box does to an impulse function This was harmonic response expt - you know what black box (LRC) does to a single freq Black box Z(ω) mod Y ? Are these connected by FT?? They d better be you find out! phase�Y� Ω0 2Ω0 3Ω0 Ω �rad�s� Ω0 2Ω0 3Ω0 Ω �rad�s�