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

• Redwaves
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.

#### 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}## Last edited:
• Delta2
What are you talking about ? What block ?

##\ ##

BvU said:
What are you talking about ? What block ?

##\ ##

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

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

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.

Redwaves said:
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.

Last edited:
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.

Last edited:
Redwaves said:
a beats frequency graph
Which is what you get from the expression in post #6.

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.

Last edited:
Redwaves said:
I'm not sure to understand the sin
I wrote "something like". The choice of sin rather than cos was arbitrary.

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

Redwaves said:
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.

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.

Redwaves said:
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 !

Redwaves said:
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.

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