Taliyah Spencer

2022-04-23

Pendulum with Dirac Comb excitation

$l\ddot{\theta}+b\dot{\theta}+g\theta =G,\sum _{-\mathrm{\infty}}^{\mathrm{\infty}}\delta (t-nT)$

where l, b, g, G are constants and$T=\frac{2\pi}{\omega}$ .

Show that the resulting motion is given by

$\theta \left(t\right)=\frac{G}{Tl{\omega}^{2}}+\frac{2G\mathrm{cos}(\omega t-\frac{\pi}{2})}{Tb\omega}+$ [terms with frequencies $\ge 2\omega$ ]

and explain why the higher frequency terms are supressed.

My first take was to rearrange to

$\ddot{\theta}+\frac{b}{l}\dot{\theta}+{\omega}^{2}\theta =\frac{G}{l},\sum _{-\mathrm{\infty}}^{\mathrm{\infty}}\delta (t-nT)$

where$\omega =\sqrt{\frac{g}{l}}$

Then, taking the Laplace transform of both sides I got

$\mathrm{\Theta}\left(s\right)=\frac{G}{l,\sqrt{{\omega}^{2}-{\left(\frac{b}{2l}\right)}^{2}}},\frac{\sqrt{{\omega}^{2}-{\left(\frac{b}{2l}\right)}^{2}}}{{(s+\frac{b}{2l})}^{2}-({\left(\frac{b}{2l}\right)}^{2}-{\omega}^{2})},\frac{1}{1-{e}^{-sT}}$

which, as far as I'm concerned transforms to

$\theta \left(t\right)=\frac{G}{l,\sqrt{{\omega}^{2}-{\left(\frac{b}{2l}\right)}^{2}}}\sum _{n=0}^{\mathrm{\infty}},H(t-nT),{e}^{-\frac{b}{2l}(t-nT)}\mathrm{sin}\left(\sqrt{{\omega}^{2}-{\left(\frac{b}{2l}\right)}^{2}}(t-nT)\right)$

And, assuming that this is a correct form of the solution, I can't see how that is equivalent with the function given in the question. I reckon it has something to do with using Fourier series/transform instead? If so, I'm not sure how to do that. Or, is there a way to convert my solution into the given one?

I've been struggling with this for a good few days now, so any help would be much appreciated.

where l, b, g, G are constants and

Show that the resulting motion is given by

and explain why the higher frequency terms are supressed.

My first take was to rearrange to

where

Then, taking the Laplace transform of both sides I got

which, as far as I'm concerned transforms to

And, assuming that this is a correct form of the solution, I can't see how that is equivalent with the function given in the question. I reckon it has something to do with using Fourier series/transform instead? If so, I'm not sure how to do that. Or, is there a way to convert my solution into the given one?

I've been struggling with this for a good few days now, so any help would be much appreciated.

Ezakhenitne

Beginner2022-04-24Added 10 answers

Step 1

The distinction between apparent solutions via Laplace transform and via Fourier transform depends meaningfully on the function-space that the "target" function is in, and the desired solution is in.

Suppressing some of the constants to reduce clutter, consider the equation $\theta {}^{\u2033}+a{\theta}^{\prime}+\theta =\sum _{n\in \mathbb{Z}}{\delta}_{n}$

Under various external context assumptions, we might suppose that the solution θ is at worst a tempered distribution (the right-hand side is such). This excludes exponentially-growing (either forward or backward) solutions $\theta$. In particular, unless the constants a,b on the left-hand side are such that solutions to the characteristic equation ${\lambda}^{2}+a\lambda +b=0$, the homogeneous equation has no tempered-distributions solutions (except 0).

Step 2

The Poisson summation formula is the assertion that $\sum _{n}{\delta}_{n}$ is its own Fourier transform. Thus, again up to essentially irrelevant constants,

$-{x}^{2}\hat{\theta}+bix\hat{\theta}+c\hat{\theta}=\sum _{n}{\delta}_{n}$

which suggest that

$\hat{\theta}=\sum _{n}\frac{{\delta}_{n}}{-{x}^{2}+bix+c}=\sum _{n}\frac{{\delta}_{n}}{-{n}^{2}+bin+c}$

Transforming back, again suppressing constants, $\theta \left(t\right)=\sum _{n}\frac{{e}^{2\pi int}}{-{n}^{2}+bin+c}$.

What is the derivative of the work function?

How to use implicit differentiation to find $\frac{dy}{dx}$ given $3{x}^{2}+3{y}^{2}=2$?

How to differentiate $y=\mathrm{log}{x}^{2}$?

The solution of a differential equation y′′+3y′+2y=0 is of the form

A) ${c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

B) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{3x}$

C) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{-2x}$

D) ${c}_{1}{e}^{-2x}+{c}_{2}{2}^{-x}$How to find instantaneous velocity from a position vs. time graph?

How to implicitly differentiate $\sqrt{xy}=x-2y$?

What is 2xy differentiated implicitly?

How to find the sum of the infinite geometric series given $1+\frac{2}{3}+\frac{4}{9}+...$?

Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?

A. 4.2

B. 4.4

C. 4.7

D. 5.1What is the derivative of $\frac{x+1}{y}$?

How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

How to find the volume of a cone using an integral?

What is the surface area of the solid created by revolving $f\left(x\right)={e}^{2-x},x\in [1,2]$ around the x axis?

How to differentiate ${x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}=4$?

The differential coefficient of $\mathrm{sec}\left({\mathrm{tan}}^{-1}\left(x\right)\right)$.