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.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download