Damped and Undamped Resonance - Gordon State College
Dr. ZABDAWI
Damped and Undamped Resonance
Section 5.3 Forced Motion.
[pic] - [pic] ; [pic]
m[pic] [pic]
[pic]
[pic]
This is a 2[pic] order linear, non-homogeneous ODE with constant coefficients.
Use either: a) Method of undetermined coefficients.
or b) Method of variation of parameters.
Example 1.
Interpret & solve the initial value problem.
[pic] [pic] [pic]
(1) [pic] , [pic]
=[pic]
The associated homogeneous equation:
[pic] r = [pic]
= [pic]
[pic]
Use the method of undetermined coefficients:
[pic]
[pic]
[pic]
Substitute in (1)
[pic]
[pic] [pic] (2)
[pic] (3)
[pic] -4(2) + (3) [pic]
(3) [pic] A = [pic]
= [pic]
[pic][pic]
[pic]
[pic] [pic] [pic]
[pic] = [pic] = [pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic][pic]
Example 2 : Transient and Steady State Solutions
Solve: [pic] [pic], [pic]
The associated homogeneous equation is: [pic][pic]
[pic]
[pic]
[pic]
Let’s use the method of undetermined coefficients:
[pic]
[pic]
[pic]
[pic]
[pic]
[pic] (1)
[pic] (2)
2(1) + (2) 5B = 10 [pic] , A = 0
[pic]
X(0) = 0 [pic] [pic]
[pic]
[pic]
[pic] [pic]
[pic]
Or [pic]
Goes [pic] as t[pic]
Example 3 : Forced Undamped Motion
Solve the IVP
[pic], x(0) = 0, [pic]
[pic] [pic] [pic]
[pic]
[pic]
[pic]
[pic]
Substitute back into the D.E.
[pic]
[pic] [pic]
[pic] [pic]
[pic][pic], [pic] [pic] [pic]
[pic] [pic] [pic]
[pic] [pic]
[pic] [pic]
[pic]
[pic] [pic] [pic]
[pic]
[pic]
[pic] [pic] [pic]
[pic]
[pic] [pic], [pic]
[pic] , [pic]
What Happens when [pic], Forcing Frequency = Natural Frequency
Answer: Resonance Occurs because [pic]
[pic]
[pic]=[pic]
= [pic]
= [pic]
X(t) = [pic]
[pic]
PURE RESONANCE (above figure)
Alternatively; I would like to solve the problem again when [pic]
[pic], [pic], [pic]
[pic] [pic] [pic]
[pic]
[pic]
[pic]
[pic][pic]
Substitute back into the ODE
[pic]+[pic] = [pic]
[pic] [pic]
[pic]
[pic] [pic] [pic]
[pic] [pic]
[pic][pic]
[pic] [pic] [pic]
[pic]
[pic]
[pic] [pic] [pic] [pic] [pic]
[pic] [pic]
Which is the same exact answer as [pic]
[pic]
Resonance Curve: In the Case of Underdamped Vibrations
The ODE is; [pic] (1)
The associated homogeneous equation is:
[pic]
[pic]
[pic] ; but [pic]
[pic] [pic]
[pic]
[pic]
[pic]
[pic]
Substitute in (1)
[-[pic]] = [pic]
[pic] ([pic]
[pic] ([pic] = 0
[pic]
A = [pic] = [pic]
B = [pic] = [pic]
[pic]
Use the reduction formula
[pic] = [pic]
[pic] = [pic]
[pic] = [pic]
[pic] [pic]
Where C = [pic]
[pic] , [pic]
[pic][pic]
[pic]
[pic]
[pic]
And I hereby verified the equations in the book on page 204.
@ Steady State [pic] = 0
[pic] [pic]
Where again [pic]
[pic]
[pic][pic] [pic] [pic]
The maximum value of [pic]
Occurs when [pic]
[pic] [pic]
[pic] [pic]
[pic] ; [pic]
[pic] 2[pic]
2[pic]
[pic][pic]
So the maximum oscillations occur when the external force has a period = [pic]
a frequency = [pic]
When this happens, the system is said to be in Resonance.
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