Finding ##x_c(t=0)## for a system of coupled Masses & Springs

[Redwaves]

Homework Statement

Find $x_c(t=0)$ for which the frequency on the block a is $\frac{(3+\sqrt{4})}{2}$ and the amplitude is $\frac{(cos(3-\sqrt{7}t))}{2}$
The block B oscillate in pure sine.
All the blocks are at rest, thus $v_a = v_b = v_c = 0$

Relevant Equations

$x_a(t=0) = 6, x_b(t=0) =-6, x_c(t=0) = ?$

Hi,

First of all, I'm not sure at all how to start this question. I found the eigenvectors in a previous question, but I'm not sure if I need it to solve this one.

I think I need to use the expression for the position and velocity.
$a_n = C_n cos (\omega_n t + \alpha_n)$
$v_n = -\omega_n C_n sin (\omega_n t + \alpha_n)$

However, I don't how this can help me to find $x_c(t=0)$ for which the movement of the block A matches a frequency of $\frac{(3+\sqrt{4})}{2}$ and an amplitude $\frac{(cos(3-\sqrt{7}t))}{2}$

 

[BvU]

What are you talking about ? What block ?

$\$

 

[Redwaves]

What are you talking about ? What block ?

$\$

I drew the system. Sorry If it wasn't clear.

 

[haruspex]

I don't understand the expression for the amplitude. It should be a constant, not a function of time.

 

[Redwaves]

It is probably just me having hard time to understand the question. It says the amplitude is modulated by that function. However, I don't really know what that means. Once again I'm sorry.

 

[haruspex]

It says the amplitude is modulated by that function.

That suggests to me that A is supposed to move like $\sin(\frac{(3+\sqrt{7})}{2}t)\frac{(cos(3-\sqrt{7}t))}{2}$, but I would not have thought that consistent with B moving as pure sine. Though maybe C also having that sort of modulation makes that possible.

 

[Redwaves]

I think I found something with what you said.
In general if I have the eigenvectors and the eigenvalues and the system doesn't have any damping force, can I use $x_a(t) = C_1 X_{1a}cos(\omega_1 t + \alpha_1) + C_2 X_{2a}cos(\omega_2 t + \alpha_2) + C_3 X_{3a}cos(\omega_3 t + \alpha_3)$

What $X_{na}$ means. I mean is it some kind of displacement? I realized that I'm not even sure what $x_a(t) = C_1 X_{1a}cos(\omega_1 t + \alpha_1) + C_2 X_{2a}cos(\omega_2 t + \alpha_2) + C_3 X_{3a}cos(\omega_3 t + \alpha_3)$ really is.

I just realized that I have a graph for $x_a(t)$ which is a beats frequency graph.

I have my eigenvalues and eigenvectors which are $\omega_1^2 = 2k/m, \omega_2^2 = 7k/m, \omega_3^2 = 9k/m, <1| = (1,5,3), <2| = (3,0,-1), <3| = (1,-2,3)$

I know $x_c(t=0) = 8$

So I'm trying a lot of different thing just to find the correct answer. I really don't know what I'm doing.

 

[haruspex]

a beats frequency graph

Which is what you get from the expression in post #6.

 

[Redwaves]

I'm not sure to understand the $\sin$ If the amplitude of the beats frequencies is $2A \cos(\omega_1 - \omega_2)t/2$. Does it means that $x_a(t) = \cos(3+\sqrt{7})t / 2 (\cos (3-\sqrt{7})t/2)$.
Thus, $\omega_1 = 3$ and $\omega_? = \sqrt{7}$ ?

Using trig identity, I get $(\cos(a) + \cos(b))/2$

$x_a(t) = \frac{cos(3)}{2} + \frac{cos(\sqrt{7})}{2}$, so based of my eigenvalues. I can see that $x_a(t)$ oscillate in 2 modes only, the second and the third.

I have my vector for $x_n(t=0) = (6,-6,c)$ if I do a dot product with my eigenvector for the first mode I get $x_c(0) = 8$ which is the right answer! However, I don't know how it works. I mean why the dot product between my vector $x_n(0)$ and the first eigenvector gives me the correct answer.

 

[haruspex]

I'm not sure to understand the sin

I wrote "something like". The choice of sin rather than cos was arbitrary.

 

[Redwaves]

All right, I see.
Can you confirm that what I said is not totally wrong?

 

[haruspex]

why the dot product between my vector $x_n(0)$ and the first eigenvector gives me the correct answer.

I'd say it's because it does not oscillate in that mode.
For an eigenvector $e$, $\ddot x.e=\lambda x.e$. Setting $x(0).e=0$ gives $\ddot x.e=0$. Since $\dot x(0)=0$, that combo is constant.

 

[Redwaves]

I understand that at t=0, all the blocks doesn't moves, but does it means that the block A shouldn't oscillate in that mode? so, this is the only mode that A is at rest. Furthermore, I don't know why a dot product between an eigenvector and the position vector give me the position for C. What's the meaning of this eigenvector.

For me it's like if I did a dot product between my position vector and something for I whatever reason.

 

[haruspex]

I understand that at t=0, all the blocks doesn't moves, but does it means that the block A shouldn't oscillate in that mode? so, this is the only mode that A is at rest. Furthermore, I don't know why a dot product between an eigenvector and the position vector give me the position for C. What's the meaning of this eigenvector.

For me it's like if I did a dot product between my position vector and something for I whatever reason.

The eigenvector represents a linear combination of the displacements. Specifically, it is a combination which satisfies the equation I wrote in post #13:
$\ddot x.e=\lambda x.e$
where $\lambda$ is the corresponding eigenvalue.
You can check that works here.
As you can see, this means the scalar $x.e$ obeys 1D SHM.

You (somehow) determined that only modes 2 and 3 occur. We know that $\dot x.e$ is zero at t=0. If $x.e$ is also zero at t=0, the equation tells us that $\ddot x.e$ is zero at t=0. It follows that $x.e$ is always zero. This is precisely what we need for the mode not to occur (i.e. it is if and only if), so we can conclude $x.e$ is indeed zero at t=0 for mode 1.

 

[Redwaves]

I think I understand. That begin to make sense. I had no idea about the equation you wrote and if $\ddot{x}$ = 0 then the mode don't occur. Thanks !

 

[haruspex]

I think I understand. That begin to make sense. I had no idea about the equation you wrote and if $\ddot{x}$ = 0 then the mode don't occur. Thanks !

Well, it also requires $\dot x=0$.

Btw, this is not an area I ever studied, though I did encounter eigenvalues and eigenvectors in a couple of other contexts. I didn't know that equation either, but when you asked me about the physical significance of the eigenvector it made me give it some thought.

 

[Redwaves]

I think what you said make plenty of sense. It is exactly what I was looking for.