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

AI Thread Summary
The discussion focuses on solving a mass-spring system subjected to a non-resonant sinusoidal force, represented by the equation mx'' + kx = F sin(Bt), where B is not equal to the natural frequency √(k/m). The initial approach involves solving the homogeneous equation mx'' + kx = 0 before finding a particular solution by substituting x = Asin(Bt) into the differential equation. The general solution combines the homogeneous and particular solutions, but the lack of initial conditions complicates the final solution. Participants note that while the differential equation can be solved, the absence of boundary conditions means constants may remain unsolved. Ultimately, the discussion emphasizes the mathematical challenges of the problem rather than the physical concepts involved.
Yann
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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.
 
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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
 
Yann said:
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...
 
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.
 
You actually need two boundary conditions, but since you don't have them you can probably just leave the two constants unsolved for.
 
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