Solving Oscillating Mass-Spring System w/ Non-Resonant Force

[Yann]

1. Homework Statement and 2. Homework Equations

Find position/velocity of a mass m attached to a spring of constant k when subjected to an oscillatinf roce

$$F(t) = F sin(Bt)$$

With $$B\not = \sqrt{k/m}$$

The Attempt at a Solution

Model;

$$mx'' + kx = F \sin(Bt)$$

I have no idea if/how it can be solved (without a computer, of course). Because;

$$mr^2 + k = 0$$

Gives

$$r = ±\sqrt{-k/m}$$

As $$B\not = \sqrt{k/m}$$ it can't be an answer.

 

[Astronuc]

The natural frequency would be
$$\omega = \sqrt{k/m}$$

so the problem is asking for the response when sinusoidal forcing function has frequency different from the natural frequency.

See - http://en.wikipedia.org/wiki/Harmonic_oscillator#Driven_harmonic_oscillator

 

[Yann]

I'm not sure I understand your point, it's more a problem of math than a problem of physics, I must solve;

$$mx'' + kx = F \sin(Bt)$$

But I don't know how

 

[learningphysics]

I'm not sure I understand your point, it's more a problem of math than a problem of physics, I must solve;

$$mx'' + kx = F \sin(Bt)$$

But I don't know how

First solve the homogenous equation first:

mx'' + kx = 0

Then you need a particular solution for
$$mx'' + kx = F \sin(Bt)$$

just plug in x = Asin(Bt) into the differential equation and solve for A...

Then your general solution is the solution for the homogenous equation + the particular solution Asin(Bt)...

And finally you need to deal with initial conditions...

 

[Yann]

Thx for the help, I solved the diff. equation. But will the solution to the differential equation give me the position or the velocity at time t ? And there's no initial condition, only $$B\not = \sqrt{k/m}$$, I don't know what to do with it.

 

[Mindscrape]

You actually need two boundary conditions, but since you don't have them you can probably just leave the two constants unsolved for.