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

<br /> F(t) = F sin(Bt)<br />

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

The Attempt at a Solution

Model;

<br /> mx&#039;&#039; + kx = F \sin(Bt)<br />

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

<br /> mr^2 + k = 0<br />

Gives

<br /> r = ±\sqrt{-k/m}<br />

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;

<br /> mx&#039;&#039; + kx = F \sin(Bt)<br />

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;

<br /> mx&#039;&#039; + kx = F \sin(Bt)<br />

But I don't know how

First solve the homogenous equation first:

mx'' + kx = 0

Then you need a particular solution for
<br /> mx&#039;&#039; + kx = F \sin(Bt)<br />

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.