Influence of initial shift on undamped frictionless forced oscillations

[Tymofei]

I have general equation for undamped forced oscillations (no friction) which is:

x(t) = v0/ωn⋅sin(ωn⋅t)+(x0 − f0/(ωn^2−ω^2)⋅cos(ωn⋅t)+f0/(ωn^2−ω^2)⋅cos(ω⋅t)

I just wonder about,what type of motion should occur when initial conditions are both 0 (i.e v0=0 and x0=0). My intuitive expectation is that as there is no 'natural' oscillations at beginning,vibration has to be depending only on last term.

But in formulation we can see that it will not depend only on applied force and its frequency since it will still have second term as:

−f0/(ωn^2−ω^2)⋅cos(ωn⋅t)
which will be summed with oscillation coming from last term yielding complicating oscillation with two different frequencies.
Why is that happening? Is that force during acting produces also that 'natural oscillation' which after joins together with him aka last term?

And interestingly, to produce clear vibration caused just by last term without any other 'natural' vibrations (first two terms) we need to start movement with 0 velocity and SHIFTED position [x0=f0/(ωn^2−ω^2)⋅cos(ωn⋅t)] its so counter intuitive.

 

[rude man]

This is a straightforward problem involving 2nd order linear ordinary differential equations. You can solve it either classically or (my preference) Laplace transform.

The resujlt is a summation of sine terms in both natural and forced frequencies. As I think you pointed out, the amplitude of the ensuing motion increases as the forced freqency approaches the natural frequency, going to infinity as the two frequencies equate.

If however there is even a slight amount of damping then the mathematics becomes extremely complicated and would include separate cases of underdamping, critical damping or overdamping. But in the steady-state there would be motion at the forced frequency only.

And interestingly, to produce clear vibration caused just by last term without any other 'natural' vibrations (first two terms) we need to start movement with 0 velocity and SHIFTED position [x0=f0/(ωn^2−ω^2)⋅cos(ωn⋅t)]

This is not so because there would exist two natural frequency oscillations, 90 degrees out of phase with one another.

 

[rude man]

I would add that using one's intuition becomes tricky when dealing with such concepts as a fully undamped system since no such system exists.

 

[Tymofei]

This is a straightforward problem involving 2nd order linear ordinary differential equations. You can solve it either classically or (my preference) Laplace transform.

The resujlt is a summation of sine terms in both natural and forced frequencies. As I think you pointed out, the amplitude of the ensuing motion increases as the forced freqency approaches the natural frequency, going to infinity as the two frequencies equate.

If however there is even a slight amount of damping then the mathematics becomes extremely complicated and would include separate cases of underdamping, critical damping or overdamping. But in the steady-state there would be motion at the forced frequency only. This is not so because there would exist two natural frequency oscillations, 90 degrees out of phase with one another.

I already solved all differantial equations also for dumped ones what I am looking for is just more intuitive understanding of mechanics behind aka 'whats really going on there" .As you pointed out,its really hard when you living and observing world where damping and friction is everywhere.
By the way can you explain what did you meant by

"This is not so because there would exist two natural frequency oscillations, 90 degrees out of phase with one another. "
From my point of view,as there is no friction and damping,there will no any steady state.Im not going to do here math but we can express first two terms with natural freq. as Ccos(ω0t−γ) .So there will be two different oscillations superpositioning forever without steady state.Only thing that really surprised me is that needed initial shift for clear vibration caused by last term(as we canceling first terms by giving those initial conditions).And secondly I am still confused about that natural oscillation occurring with 0 0 inital conditions beside forced oscillation term(last).In paper everything fits ok.But in my model in head its confusing.My last explenation is maybe that force when it touches at t=0 object on spring,that impuls also creates additional natural oscillation which doesn't diseppears later and as there is no friction due to conservation of energy that initial natural oscillation doesn't do any work its just adding and subtracting kinetic energy stored in spring which in will give zero.

 

[rude man]

By the way can you explain what did you meant by

"This is not so because there would exist two natural frequency oscillations, 90 degrees out of phase with one another. "

From my point of view,as there is no friction and damping,there will no any steady state.

I don't know what you mean by this. Without damping you get a steady state sinusoidal solution without an exponential decay term.

Given applied force $F sin(\omega_ft)$, no dampig, natural frequency $\omega_n$ we get two terms, one in $sin(\omega_nt)$ and one in $sin(\omega_ft)$.

When an initial displacement $x_0$ is added, we get a third term $x_0 cos(\omega_nt)$. Thus the two natural frequency terms do not cancel; as I said there is a 90 degree phase shift between them.

You say you have solved the complete motion including the finite initial displacement. Give me your complete x(t) and we may be able to resolve the problem.

 

[mpresic3]

I notice your last term varies as a cosine. The last term at t = 0, is nonzero, and your initial conditions involved in your other terms is zero. It might be instructive to solve the problem with a sine in the source term. If you do that notice the force at t = 0 is zero just as the initial conditions are. I think that initially all the motion will involve the forcing frequency initially but as time goes on, you will pick up the natural frequency, but that is just a guess. Actually without damping, it seems like the natural frequency term (the transient) doesn't develop, so I will need to think of this again.

 

[rude man]

I notice your last term varies as a cosine. The last term at t = 0, is nonzero, and your initial conditions involved in your other terms is zero. It might be instructive to solve the problem with a sine in the source term. If you do that notice the force at t = 0 is zero just as the initial conditions are. I think that initially all the motion will involve the forcing frequency initially but as time goes on, you will pick up the natural frequency, but that is just a guess. Actually without damping, it seems like the natural frequency term (the transient) doesn't develop, so I will need to think of this again.

What you need to do is solve the differential equation with your initial displacement included. Then you will know what really happens and you can ponder how it fits in with your ituition.

 

[mpresic3]

Sure, and I did that with a sinusoidal forcing term ( a force that started = 0 at t = 0). My forcing term was:
F0 sin (w * t), (not F0 cos (w * t)). I solved the problem with damping. I can go to my notes (when I can find them) and let the damping go to zero, to refresh my recollection, but right now, I'm just suggesting you might try it. Maybe it is just as simple as adding a phase of pi/2 to the forcing term with what you have already calculated.

 

[rude man]

Maybe it is just as simple as adding a phase of pi/2 to the forcing term with what you have already calculated.

The phase of the forcing function is immaterial. You will always get two terms in $\omega_n$ separated by 90 degrees.

 

[mpresic3]

In response to Tymofei. I reread your initial post. Now I see what you are getting at. You have written the solution for the equations of motion in x(t). I assume the calculation was done correctly. It looks correct.
I agree if you put v0 = 0 and start x0 at the shifted position, you will get ONLY one frequency, (the forcing frequency). and this is a bit counter-intuitive or perhaps not. It depends on how much experience you have with springs and masses or electrical circuits.

Part of the question you may be addressing is why is there a phase shift in the problem. In my electromagnetics course we solved a analogous problem with capacitors, inductors, and resistors. The physical process my professor advanced concerning how the B field had to build up in the inductor and the phase lags etc. seemed to make sense to me. I have not examined these arguments for many years but I am just writing this is food for thought.

I have not made a calculation at this point. but if you take a derivative with respect to time and look at the velocity as a function of time. If you want a velocity as a function of the forcing frequency only, (I suspect) you can achieve this by a "shifted" velocity (with an initial zero (not shifted) position). Basically what may be occurring is in order to evolve the system with only one frequency for future time, you need to account for the phase shift and prescribe your forcing function accordingly.

 

[rude man]

The $\omega_n$ term can indeed be eliminated, but with an initial velocity, not displacement.

The required initial velocity is $\frac {F\omega_e} {m(\omega_n^2 - \omega_e^2)}$ using my previous nomenclature.