Transverse movement of N masses

[mahler1]

Homework Statement .

Consider the system of $N$ masses shown in the figure (see picture 1).

a) Using the approximation for small angles, write the equation of transverse movement for the n-th particle.

b) Propose a solution of the form:
${ψ_n}^{p}(t)=A^{p}\cos(nk^{p}+α^{p})cos(w^{p}t+\phi^{p})$ ($p$ is in parenthesis in the original statement but I couldn't write it correctly here). Find the dispersion relation and decide whether it depends on the contour conditions.

The attempt at a solution.

For part a), I am making the assumption that $x_n-x_{n-1}=d$, $l_0$ and $k$ are the natural length and spring constant for all the springs and $m$ is the mass of all the particles. Taking these things into consideration, the equation for the $n$-th mass is (see picture 2)

(1)$\hat{y})$ $m\ddot{y_n}=-k(a_n-l_0)\sinθ_n+k(a_{n+1}-l_0)\sin_{n+1}$, with

$a_n=\sqrt{(y_n-y_{n-1})^2-d^2}$, $a_{n+1}=\sqrt{(y_{n+1}-y_n)^2+d^2}$. Now, for small oscillations, I can approximate $a_n$ and $a_{n+1}$ by $d$, i.e., $a_n≈d≈a_{n+1}$

so (1) reduces to

(2) $\hat{y})$ $\ddot{y_n}=-\dfrac{k}{m}(d-l_0)\dfrac{y_n-y_{n-1}}{d}+\dfrac{k}{m}(d-l_0)\dfrac{y_{n+1}-y_n}{d}$

if I call ${w_0}^2=\dfrac{k}{m}\dfrac{d-l_0}{d}$, then the equation (2) equals to

(3) $\ddot{y_n}={w_0}^2(y_{n+1}-2y_n+y_{n-1})$

I would like to know if I've done correctly part a). With regard to part b), I don't understand the notation of the proposed solution, to be more specific, I don't understand what $p$ is, is it an index? Could someone explain me what $\ddot{y_n}$ would be just to see what $p$ stands by?

Sorry if this isn't the adequate section for my thread, I didn't know whether to put it here or in another section.

 

[TSny]

Hello mahler1. Your equation of motion looks good to me.

The problem wants you to find solutions of your equation of motion in the particular form given in the statement of the problem. These types of solutions are called "normal modes". In a normal mode each particle oscillates with the same frequency $\omega^{(p)}$ but the individual particles will have amplitudes that differ from one particle to the next. Each normal mode is like a "standing wave". In solving the problem, you will see that more than one normal mode is possible. The index $(p)$ denotes a particular normal mode: $p = 1,2,...$. You will discover how many normal modes there are and what their frequencies are.

$\psi_n^{(p)}(t)$ is just another notation for your $y_n(t)$. That is, it represents the displacement of the nth particle as a function of time for the $p^{th}$ normal mode.

So, see if you can make the given form satisfy your equation of motion. You will need to think about the boundary conditions.

 

[mahler1]

Hello mahler1. Your equation of motion looks good to me.

The problem wants you to find solutions of your equation of motion in the particular form given in the statement of the problem. These types of solutions are called "normal modes". In a normal mode each particle oscillates with the same frequency $\omega^{(p)}$ but the individual particles will have amplitudes that differ from one particle to the next. Each normal mode is like a "standing wave". In solving the problem, you will see that more than one normal mode is possible. The index $(p)$ denotes a particular normal mode: $p = 1,2,...$. You will discover how many normal modes there are and what their frequencies are.

$\psi_n^{(p)}(t)$ is just another notation for your $y_n(t)$. That is, it represents the displacement of the nth particle as a function of time for the $p^{th}$ normal mode.

So, see if you can make the given form satisfy your equation of motion. You will need to think about the boundary conditions.

Very clear answer, thanks. I've supposed that the two extremes are fixed, i.e., $y_0(t)=0=y_{N+1}(t)$.

$y_{N+1}(t)=0$ if and only if $0=A^{(p)}\cos((N+1)k^{(p)}+α^{(p)})\cos(\phi^{(p)})$. From here, I could set $\phi^{(p)}=\dfrac{\pi}{2}$.

Now, using that $y_0(t)=0$ and that $\phi^{(p)}=\dfrac{\pi}{2}$, I get that $0=0=A^{(p)}\cos(α^{(p)})\cos(\dfrac{\pi}{2})$, so I would also conclude that $α^{(p)}=\dfrac{\pi}{2}$.

As you've said, I am going to have $N$ different frequencies corresponding to each normal mode. I am having trouble to find the form of each $\omega^{(p)}$, I think I must use equation (3) but I don't know how to. Could you help me with that?

 

[TSny]

I've supposed that the two extremes are fixed, i.e., $y_0(t)=0=y_{N+1}(t)$.

OK. Good. Note that these boundary conditions must be satisfied at all times t, not just t = 0. If you think about it, the phase constant $\phi^{(p)}$ determines the phase of the normal mode vibration at time $t = 0$. But there is no restriction on this initial phase. So, you will not be able to determine a value of $\phi^{(p)}$.

I suggest starting with the boundary condition at n = 0. Make sure this condition holds at all times t.

Then move on to the boundary condition at n = N+1.

Finally, invoke your equation (3).

 

[paranormal]

Dear student,

Thank you for your response. Your solution for part a) looks correct to me. You have correctly applied the small angle approximation and obtained an equation that describes the transverse movement of the n-th particle in terms of its neighboring particles.

In regards to part b), the notation here is a bit confusing. It seems that $p$ is just a general index, perhaps to indicate that there can be multiple solutions for the transverse movement of the particles. The notation for the proposed solution is also a bit unclear, but I believe that $\ddot{y_n}$ in this case would be the second derivative of the transverse position of the n-th particle with respect to time. This is because the general form of a cosine function is given by $A\cos(wt+\phi)$, where $A$ is the amplitude, $w$ is the angular frequency, and $\phi$ is the phase. By taking the second derivative of this function, you can obtain an equation for the acceleration of the particle, which is what is being described by $\ddot{y_n}$ in the proposed solution.

I hope this helps clarify things for you. Keep up the good work in your studies of transverse movement of N masses!