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 matwml@nus.edu.sghttp://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 !