Convolution Integral (s.d.o.f. system)

[DaNud]

< Mentor Note -- Poster has been reminded that they need to show their work on schoolwork questions >

Does anybody know how to solve this exercise?

Derive the response of an undamped single-degree-offreedom system to force f(t)=F_0*cos(w_n*t)*u(t) with null initial conditions, w_n=(k/m)^1/2 and u(t) being the unit step function by convolution integral. Compare the solution with the result obtained with an alternative approach.

I don't even know how to start.

 

[Khashishi]

What have you learned about transfer functions and the convolution theorem?

 

[DaNud]

I learned that can be applied only for linear system because I am using the superposition principle. I am summing up all the random forces of a generic F(t) at time F(delta). I don't know how can be solved in terms of mathematics.

 

[Khashishi]

Is your undamped single-degree-of-freedom system a linear system? If yes, write down the transfer function.

 

[DaNud]

Yes the transfer function is:

a + w_n*x = F₀*cos(w_n*t)*u(t)

where a is the acceleration.

 

[Khashishi]

No, that's not a transfer function. That's an equation of motion. The transfer function is the Laplace transform of the LHS.

 

[DaNud]

I don't know how to do it. Could you please explain me?
However the equation of motion is a + w_n*x = F0/m *cos(w_n*t)*u(t)

 

[BvU]

[Mentor's note: Post merged from another thread.]

Hello Nud, a belated

According to the PF rules "don't know where to start" isn't good enough. So tell us what you've got in terms of subject know-how !

At least a convolution integral should be in the "relevant equations"

Then: continuing/repeating an existing thread in another subforum is considered spamming and frowned upon ! (general PF rules)

On a positive note, some guidance:
If you don't know how to get started, perhaps you can explore the alternative approach (to the convolution approach) which is hard work...
There is a solution to the homogeneous equation $\ddot x + \omega_n^2 x = 0$. You must know that already, right ?
Next you need a particular solution, which you may well also know about ?

You have no damping, so the solution to the homogeneous equation will be there for all T > 0

--

More help -- but credit is now low -- : from the original thread I gather you have not been paying much attention when the transfer function concept was treated. My advice: do catch up ! It's very useful and makes this exercise a breeze .

--

One final lifebuoy: Read up on Laplace Transforms ! - the word convolution appears there !

--

 

[rude man]

Don't start with the Laplace transfer function. That can come later when you're asked for an "alternative approach".
You first need to find the unit impulse response to your system. The system is defined by your 2nd order undamped diff. equation which you have correctly given in your post 7 except "a" needs to be expressed in terms of x and w_n should be w_n^2.

One way is to transform the system equation with the Fourier integral, find the unit impulse response h(t) from that, then convolve h(t) with the input time function using the convolution integral. That however is not sticking to the time domain; in fact it is close to what is asked for later as the "alternative approach". It's kind of cheating, but I think it's what they want you to do.

So, can you use a time domain method to solve for the impulse response? The answer is Yes. It can be done in 2 steps:
1. assume input to the quiescent x(0) = x'(0) = 0 system is f(t)/m = cU(t) for a time T. The impulse is here represented as a pulse of width T and height c, with c → ∞, T → 0 but with cT = 1. Solve the diff. eq. and find the new initial conditions x(T) and x'(T), then
2. re-solve the diff. eq. with the new initial conditions x(T) and x'(T), with zero f(t).
When you take the limit as c → ∞ and T → 0, holding cT = 1, you will get the unit impulse response to the system x(t) = h(t).

Either way of getting h(t) you now need to use the convolution integral to convolve the input f(t) with h(t).
The "alternative approach" can be the Laplace transform mentioned in post 6.

 

[DaNud]

Thanks to everybody.
I am trying to solve with the method suggested by rude man.
I obtained

h(t)=(1/(w_n*m))*(sin(w_n*t))

Now I have to calculate my convolution integral

x(t)=∫(F(j)*h(t-j)dj

calculated b/w 0 and t
where j is a dummy variable.
My slides talk about the shifting procedure (t-j) but I don't understand how can be done practically. Could you please give me one more hint?

 

[rude man]

h(t)=(1/(w_n*m))*(sin(w_n*t))

Up to here eveything is OK. Well done!

Now I have to calculate my convolution integral
x(t)=∫(F(j)*h(t-j)dj
calculated b/w 0 and t
where j is a dummy variable.

Correct also.

My slides talk about the shifting procedure (t-j) but I don't understand how can be done practically. Could you please give me one more hint?

The shifting procedure is better done with discrete convolutions. I suggest you carry out this integral in closed form.

Then, you're ready for the "alternative approach" which has been suggested in previous posts.