USSC3002 Oscillations and Waves Lecture 6 Forced Oscillations
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected]://www.math.nus/~matwml
Tel (65) 6874-2749
1
FORCED OSCILLATIONS
2
1 degree of freedom (DOF) systems
Mechanics )(2
2
tFkudt
dup
dt
udm
Question 1. What do F and E model ? Question 2. What happens to energy ?Question 3. Is u determined by F, E ?Question 4. Can they describe > 1 DOF systems ?
)(12
2
tEuCdt
duR
dt
udL Electronics
ENERGY
3
The mechanical equation )(
2
2
tFkudt
dup
dt
udm
so energy can increase, decrease depending on F.
2212
21),( kuumuuE
where
can be rewritten as
)()(2 tutFupdt
dE
GENERAL SOLUTION
4
Defining
u
uw
and
gives
)(
01)(tF
Atm
At etwedt
d
mp
mkA
10
dsetu
tuetw
sF
t
t
stAm
ttA
)(
0
0
0 )(1
0
0)(
)(
)()(
hence
GREEN’s FUNCTION AND STEADY STATE
5
Therefore, the solution u satisfies
where
),[,)()(),()( 01
0
0
ttdsstFsGttctu
tt
m
.0,0;0,)(
ss
sGeAs
Question 1. What is the form of G ?
For F bounded on ],( t the steady state solution is
RtdsstFsGtu m
,)()()( 1
),( 0ttc
and where the Green’s Function G is defined byis a solution of the homogeneous eqn
GREEN’s FUNCTION
6
Since
dsstFkds
dp
ds
dmsGtuk
dt
dp
dt
dmtF m
)()()()(
2
21
2
2
)()(2
2
2
2
stFkds
dp
ds
dmstFk
dt
dp
dt
dm
and
0)(lim
sGs
( from
.)()()()( 12
2
dsstFsdssGkds
dp
ds
dmstF m
and )()(12
2
ssGkds
dp
ds
dm m
Dirac’s Delta ‘function’.
1)(lim;0)(lim;0,0)(002
2
sGsGssGk
ds
dp
ds
dm
ss
ss ececsG )( )
Therefore
GREEN’s FUNCTION
7
Theorem If then
dssdssGkm
k
ds
d
m
p
ds
d tt
)()(
2
2
Proof
)()(2
2
ssGkm
k
ds
d
m
p
ds
d
Compute the indefinite integral twice
1)(lim;0)(lim00
sds
dGsG
ss
0for
0for
1
0)()()(
t
tdssG
m
ktG
m
pt
ds
dG t
so G is continuous at
0for
0for 0)()()(
u
u
ududttG
m
kdttG
m
puG
u tu
0 and the result follows since
0for 0)( ssG
GREEN’s FUNCTION FOR DISTINCT EIGENVALUES
8
Since and
ccsGs
)(lim0
ccsGs
)(lim0
0),()()(1
011 1
seesGc
c ss
0,sin)( 21 ssesG smp
m
k
m
p
m
p
2
22
Complex Roots
Real Roots
letm
k
m
p
2
2
0,sinh)( 21 ssesG smp
GREEN’s FUNCTION FOR CRITICAL DAMPENING
9
Then
hence 00
)(lim csGs
100
)(lim ccsGs
0,)(1
0
1
01
1
0
ssesG
c
c s
0,)( 2 sessG smp
m
p
2
0,)( 10 ssececsG ss
and
SINUSOIDAL RESPONSE
10
and observe that
solves
12 )()( ipmkc
RtetF ti ,)( define
)(2
2
tFkudt
dup
dt
udm
,0We choose
Rtectu ti ,)()(
iff
)(c
202
2
2
11,0max
2,0max
Qm
p
m
k
is maximized by choosing
pmk
mk Q ,0
resonant frequency
to be the
Define
(>.5 iff u. c. d.)
TRANSIENT RESPONSE
11
If
then so does
RttFkudt
dup
dt
udm ),(
2
2
wherehs uuu hu is any solution
of the homogeneous equation. Therefore, we can obtain the solution for any nonhomogeneous initial value problem for sinusoidal F as the sum of a sinusoidal solution and a transient solution of the homogeneous initial value problem given by
su solves
)0()0(
)0()0(
)(
)(
s
sAt
h
h
uu
uue
tu
tu
INHOMOGENEOUS WAVE EQUATION
12
The inhomogeneous initial value problem for u
admits the solution
),(2
22
2
2
txfx
uc
t
u
Proof. Pages 23-24 in Coulson and Jeffrey.
ctx
ctxc dssk
ctxhctxhtxu )(
2
)()(),( 2
1
t tcx
tcxc ddyyf
0
)(
)(21 ),(
)()0,( xhxu )()0,( xkxu
TUTORIAL 61. Show that the Green’s function for any damped harmonic oscillator (that we derived in these notes) satisfies G(t)0 for large t
13
2. Show the equivalence on the bottom of page 6.
3. Use the Greens functions to derive the steady state solution to the nonhomogeneous oscillation equation with complex sinusoidal forcing term. 4. Outline an approach to solve the general multidim. nonhomogeneous oscillation equation.
5. Verify that the solution on page 11 satisfies the initial values and the nonhomogeneous equation. You do not need to use Green’s Theorem !
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