What is the Differential Equation for a Forced Driven Oscillator with Damping?

In summary, the conversation discusses a problem involving a spring with a spring constant of k, an object with mass M1, and damping friction b. The top end of the spring is wiggled with an amplitude A and frequency w. The problem aims to find the differential equation for this scenario. After some discussion, it is decided that the force of the spring should be modified to include the wiggling motion, resulting in a consistent differential equation. To solve the problem, it is suggested to assume a sinusoidal response and find the conditions for this solution to hold. It is also mentioned that to find the transient solution, one can simply add 0 to both sides of the equation.
  • #1
drszdrsz
5
0

Homework Statement



http://fatcat.ftj.agh.edu.pl/~i7zebrow/rysunek.jpg
tring constant is k,
object mass is [tex]M_{1}[/tex]
Damping friction is b
and we wiggle the top end of spring in the above diagram with amount Asin(wt)
(Where A is a amplitude and w is a frequency).

Homework Equations


Spring Force:
[tex]Fs=-kx[/tex]
Damping force is:
[tex]Fb=-b\frac{dx}{dt}[/tex]

The Attempt at a Solution


I don't need full solution.I just looking for differential equation for this problem.My attempt is:
[tex]M_{1}\frac{d^{2}x}{dt^{2}}+b\frac{dx}{dt}+kx=Asin(\omega t)[/tex]
I am sure about left side of the eguation but the right side is propably wrong I hope that someone can tell me where I made an mistake.
 
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  • #2
Look at the dimensions, you are comparing force with distance. That is dimensionally inconsistent!

Consider the extension of the spring as a function of time, how does the oscillation of the end-point affect it? How does this translate into the force the spring exerts on the mass?
 
  • #3
Do you mean that the force of spring should looks like this:

[tex]Fs=-k(x-Asin(wt))[/tex]
and now the differential equation will be:
[tex]
M_{1}\frac{d^{2}x}{dt^{2}}+b\frac{dx}{dt}+kx=k*Asin( \omega t)
[/tex]
Now the differential is dimensionally consistent.But, is that correct??
 
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  • #4
That's how I would approach the problem, you might want to wait for confirmation from someone else that that's the way to go.

As for actually solving the differential equation, assume that the response, [tex]x(t)[/tex] is also a sinusoid. (Think about it, you're driving it at a certain frequency, why would it do anything but oscillate at that frequency?)

That is to say, assume a general solution of the form [tex]x(t)=B_0\sin{(\omega t)}[/tex] and then find [tex]B_0[/tex] and the conditions under which that is indeed a solution.
 
  • #5
Ok thanks a lot.

ps.
I'm not sure about solution. there mus be transitory solution too(otherwise the stationary solution that is sinusoid).
 
  • #6
drszdrsz said:
Ok thanks a lot.

ps.
I'm not sure about solution. there mus be transitory solution too(otherwise the stationary solution that is sinusoid).

To find the transient solution, all you need to do is add 0 to both sides! Remember that the sum of solutions is also a solution for such linear differential equations!
Do you remember doing this to find transients?

If not, this is how it's done (Very long and interesting lecture on the subject!):
http://ocw.mit.edu/OcwWeb/Physics/8-03Fall-2004/VideoLectures/detail/embed04.htm [Broken]
 
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  • #7
Thank you for the link:) it's very helpful.
 

1. What is a forced driven oscillator?

A forced driven oscillator is a physical system that oscillates under the influence of an external force or driving force.

2. How does a forced driven oscillator differ from a simple harmonic oscillator?

A simple harmonic oscillator is a system that oscillates with a constant amplitude and frequency, while a forced driven oscillator experiences changes in amplitude and frequency due to the external force.

3. What factors affect the behavior of a forced driven oscillator?

The behavior of a forced driven oscillator is affected by the amplitude, frequency, and phase of the driving force, as well as the natural frequency and damping coefficient of the system.

4. How is the motion of a forced driven oscillator described mathematically?

The motion of a forced driven oscillator can be described by a second-order differential equation, typically in the form of mx'' + kx + cx' = F0sin(ωt + φ), where m is the mass, x is the displacement, k is the spring constant, c is the damping coefficient, F0 is the amplitude of the driving force, ω is the angular frequency, and φ is the phase angle.

5. What real-life applications involve forced driven oscillators?

Forced driven oscillators can be found in many systems such as pendulums, musical instruments, and electronic circuits. They are also used in mechanical systems to reduce vibrations and in optics to control the frequency of laser light.

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