# Oscillation Question

## Homework Statement

What would be the mean value of oscillation x if f(t)=P. Where P is a constant force.

Not known.

## The Attempt at a Solution

Absolutely no idea where to start, happy to just be pointed in the right direction.

Thanks

## Answers and Replies

Defennder
Homework Helper
?? What is the setup and the corresponding assumptions? You left out a lot of details. You could start off by writing out the DE first then after that we could help you.

Thanks for replying. Thats it i'm afraid, that is all the question gives so I cant give anymore information. Ok, how would I do that, t is variable right?

dx
Homework Helper
Gold Member
There's no way that's the whole question. What is f?

I think it is. Here is a link to it, its question 1:

http://i423.photobucket.com/albums/pp315/skaboy607/Questions.png [Broken]

Does it make more sense if its read in conjunction with question 1?

Last edited by a moderator:
dx
Homework Helper
Gold Member
Yes, it does make more sense if read in conjunction with Q1.

ok, im still stuck what should be first steps to figuring out what it means and how to answer it?

dx
Homework Helper
Gold Member
You'll have to show some kind of attempt. Even if you don't know how to do it, give me some of your thoughts on the problem. What is the question asking you? How do you think you could solve it?

ok. Well ive done question 1 by drawing free body diagrams, and then writing the equations of motion for that mass. I could replace f(t) with P in that equation. Thinking about that system, I know that if a graph of data was collected after the force P was applied, then I think it would plot a sin wave. I dont know what the mean value of oscillation is but at a guess, I would say the height of the wave but then i'm not sure as that would be amplitude?

dx
Homework Helper
Gold Member
There's damping, so the oscillation will not be a sine wave.

Here's a huge hint: combine the constant force term C with kx like this:

kx - C = k(x - (C/k)).

dx
Homework Helper
Gold Member
The mean value is just the center of oscillation. Do the simpler case first: what would the center of oscillation be if the equation was

$$m\frac{d^2 x}{dt^2} + c\frac{dx}{dt} + kx = 0$$

?

Oh ok so damping-no sin wave, no damping-sin wave.

Right definately lost now. Is this what youve done.

mx(dbl dot)+c(dot)+kx=P
mx(dbl dot)+c(dot)+k(x-p/k)=0

hmmm I havent got a clue?

Thanks

dx
Homework Helper
Gold Member
Now, change variables to X = x - P/k. What do you get?

hmm

mx(dbl dot)+cx(dot)+kX=0

dx
Homework Helper
Gold Member
Right. Now what is the relationship between (d²x/dt²) and (d²X/dt²) and between (dx/dt) and (dX/dt)?

-(omega)^2*sin(omega)t and (omega)*cos(omega)t

Is that right.....?

dx
Homework Helper
Gold Member
What's that? Just differentiate X = x - P/k.

I was clutching at straws. What is the big X suppose to represent?

dx
Homework Helper
Gold Member
It's just a new variable that we're introducing so that the problem changes into another problem whose solution we already know.

The derivatives of X are clearly equal to those of x because they only differ by a constant. Is that clear? So the equation turns into the following equation:

m(d²X/dt²) + c(dX/dt) + kX = 0.

Now this is the equation without the driving force and you already know that the mean value of X is <X> = 0. But <X> = <x> - (C/k). Substitute <X> = 0 in this, and you will get <x> = C/k.

hmmmmmmmmm, sorry i dont really get it, know anywhere where I can read up about this?

I guess what i'm saying is if we know that the position of equilibrium is always 0 for a damped oscillator why is there an equation for it?

Thanks

dx
Homework Helper
Gold Member
The equation is the equation of motion. If you start off the oscillator by pulling it a little and giving it some initial velocity, the equation will determine how it moves after that.

Sure but ive read and you said in your post yesterday that force/k is equal to the mean value of oscillation. Which is always 0 for a damped oscillator right? So I guess I dont really see the point of the question that I was asked 'what is the mean value of oscillation?'

dx
Homework Helper
Gold Member
It is zero for a damped oscillator without the constant force term P. When there is a constant force term, the equilibrium position is P/k.

Ohh I see, so is the case of velocity and acceleration equal to 0 the same for a system with forced vibration?

Shameful as it may seem, I didnt undersand this that you said yesterday.

'The derivatives of X are clearly equal to those of x because they only differ by a constant. Is that clear? So the equation turns into the following equation:'