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�