USSC3002 Oscillations and Waves Lecture 6 Forced Oscillations

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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_at_nus.edu.sg http//www.math.nus/matwm
l Tel [phone]
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FORCED OSCILLATIONS
1 degree of freedom (DOF) systems

Mechanics
Electronics
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 gt 1 DOF systems ?
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ENERGY

The mechanical equation
can be rewritten as
where
so energy can increase, decrease depending on F.
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GENERAL SOLUTION
Defining
and
gives
hence
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GREENs FUNCTION AND STEADY STATE
Therefore, the solution u satisfies
where the Greens Function G is defined by
For F bounded on
the steady state solution is
Question 1. What is the form of G ?
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GREENs FUNCTION
Since
and
etc. implies
hence
Diracs Delta function.
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GREENs FUNCTION FOR DISTINCT EIGENVALUES
Since
and
let
Complex Roots
Real Roots
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GREENs FUNCTION FOR CRITICAL DAMPENING
Then
hence
and
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SINUSOIDAL RESPONSE
define
We choose
and observe that
solves
iff
(gt.5 iff u. c. d.)
Define
is maximized by choosing
to be the
resonant frequency
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TRANSIENT RESPONSE
If
solves
then so does
where
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
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INHOMOGENEOUS WAVE EQUATION
The inhomogeneous initial value problem for u
admits the solution
Proof. Pages 23-24 in Coulson and Jeffrey.
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TUTORIAL 6

• Show that the Greens function G(t)?0 for large t.

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 may need to use Greens Theorem.
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