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Prove that the motion of a mass $m$ on a linear spring with constant $k$, has the form $$y (t) = A \sin(wt+f),$$ where $t$ is the time and $A, w, f$ are constants. We know that for $t = 0, y(0)=y_{0}$ and $y'(0)=v_{0}$. If, in addition, the mass is subject to external force $F (t) = F_{0} \sin (w_{0}t)$, where $F_{0}$ the amplitude and $w_{0}$ the cyclic frequency, then calculate the amplitude of the motion.

When the mass is subject to external force $F (t) = F_{0} \sin (w_{0}t)$,we get this differential equation: $$y''+w^{2}y=\frac{F_{0}}{m} \sin(w_{0}t),$$ which has the solution: $$y(t)=c_{1} \cos(wt)+c_{2} \sin(wt)+\frac{F_{0}}{m(w-w_{0}^{2})} \sin(w_{0}t),$$ where $c_{1}=y_{0} $ and $ c_{2}=\frac{v_{0}}{w}-\frac{F_{0}w_{0}}{mw(w-w_{0}^{2})}$. Right? But how can I find the amplitude of the motion?

jinawee
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Mary Star
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2 Answers2

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Ampliyude $A$ will be determined by
$[\dfrac{v_0}{\omega ^2} + x_0^2]= A^2 $

To see how this equation is derived read this answer given by me:
i've posted here:
Using $\sin()$ or $\cos()$ for computing SHM?.

Community
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user31782
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Amplitude is the maximum displacement at steady state. That is okay for a working definition. So, all you need to do is find out is the maximum value of $|y(t)|$ at steady state.

There some small algebraic errors in your solution. That said,

$y(t) = y_{cf}(t) + y_p(t)$

To illustrate my point better, let me use the following form for $y_{cf}(t)$

$ y_{cf}(t) = A e^{-i\lambda_1t} + Be^{-i\lambda_2t} $ with $ \lambda_1 , \lambda_2 > 0$

There is no difference between this and the form with sines and cosines apart from the values of the constants.

At steady state, that is, $\lim{t\to\infty}$ , $y_{cf}(t) = 0$

This means that the amplitude is simply, $|y_p(t)|= |\frac{F_0}{m(\omega^2 - \omega_0^2)}\sin (\omega_0 t)| = \frac{F_0}{|m(\omega^2 - \omega_0^2)|} $

sundeep
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