Can different masses be used in normal modes?

[miscellanyous]

Homework Statement

A particle of mass m1 is attached to a wall by a spring of constant k. A second particle of mass m2 is attached to a different wall by another sping of constant k. The two masses are attached to each other by a third spring of constant k. Let $$x_1$$ and $$x_2$$ be the displacement of the respective particles from their equilibrium positions. Find the normal coordinates for the system.

This problem either with spings or pendula appears in many classical mechanics textbooks (I am using the book by Thornton & Marion). The wrinkle is that the two particles have different masses in this case.

Homework Equations

for the first mass:
$$m_{1} \ddot{x_1} = -k x_1 - k \left(x_1 - x_2 \right)$$
for the second mass:
$$m_{2} \ddot{x_2} = -k x_2 - k \left(x_2 - x_1 \right)$$

the eigenfrequencies arise from taking $$\texttt{det} \left( A_{j k} - \omega^2 m_{j k} \right) = 0$$
where $$m_{j k} = ( (m_1, 0), (0, m_2) )$$ and $$A_{j k} = ( (2 k, -k), (-k, 2 k) )$$ are both 2 x 2 matrices.

These are used to find conditions on the eigenvectors $$a_{j r}$$ which satisfy
$$\left( A_{j k} - \omega_r^2 m_{j k} \right) a_{j r} = 0$$
where $$\omega_r^2$$ is the rth eigenfrequency.

The eigenvectors transform the original coordinates $$x_j$$ into the normal coordinates $$\eta_j$$ via
$$x_j = \sum_r a_{j r} \eta_r$$.

The Attempt at a Solution

The eigenfrequencies are:
$$\omega^2_{\pm} = \frac{k}{m_1 m_2} \left( m_1 + m_2 \pm \sqrt{m_1^2 - m_1 m_2 + m_2^2} \right)$$

The (unnormalized) eigenvectors are
$$a_{j \pm} = \left(1, \frac{1}{m_2} \left( - m_1 + m_2 \pm \sqrt{m_1^2 - m_1 m_2 + m_2^2} \right) \right)$$

Then the new coordinates are found by
$$x_1 = \eta_1 + \eta_2$$
$$x_2 = \left( m_1 + m_2 + \sqrt{m_1^2 - m_1 m_2 + m_2^2} \right) \eta_2 + \left( m_1 + m_2 - \sqrt{m_1^2 - m_1 m_2 + m_2^2} \right) \eta_2$$

Note that in the limit $$m_1 = m_2$$ we obtain the textbook result
$$x_1 = \eta_1 + \eta_2$$
$$x_2 = \eta_2 - \eta_1$$

So it seems like I have solved the problem - but I haven't. If I check the solution by substituting the new coordinates into the equations of motion I find that the equations have not been decoupled, so these cannot be the normal coordinates.

I wish anyone who attempts this problem good luck. I have attached a Mathematica program which shows this result.

Chris

 

[dwintz02]

Well, this is a pretty long problem, so let's start towards the beginning. When you take the $$\texttt{det} \left( A_{j k} - \omega^2 m_{j k} \right) = 0$$ what is the quadratic equation you get for omega squared? I have done the problem up to this point but want to make sure we're on the same page before I proceed with the ridiculous amount of algebra/simplifying. Yes, sorry for the laziness, haha.

 

[miscellanyous]

When you take the $$\texttt{det} \left( A_{j k} - \omega^2 m_{j k} \right) = 0$$ what is the quadratic equation you get for omega squared?

The eigenvalue equation is
$$\left( 2 k - \omega^2 m_1 \right) \left( 2 k - \omega^2 m_2 \right) - k^2 = 0$$
which has two roots in the variable $$\omega^2$$:
$$\omega^2_{\pm} = \frac{k}{m_1 m_2} \left( m_1 + m_2 \pm \sqrt{m_1^2 - m_1 m_2 + m_2^2} \right)$$

You are right, there is a rediculous amount of algebra which is why I opted for a computer algebra package (see the file attached to my original post). I believe my program is correct and it does give the correct normal coordinates for $$m_1 = m_2$$.

 

[dwintz02]

Ok, I got the same characteristic frequencies that you did.

Given the formulas you've posted, your normalized coordinates cannot be right. When I plug in m1=m2 into x2, I get something like 4*m*n2 which cannot be right. I think one of your etas needs to switch an index.

When I computed the eigenfrequencies, I multiplied that matrix out and just set the first component equal to 1, and then solved for the 2nd component, is this what you did?

 

[miscellanyous]

When I computed the eigenfrequencies, I multiplied that matrix out and just set the first component equal to 1, and then solved for the 2nd component, is this what you did?

Yes, this sounds similar to what I did. I made an error in my original post regarding the eigenvectors . The correct eigenvectors are:
$$a_{j (1)} = \left(1, \frac{1}{m_2} \left( - m_1 + m_2 + \sqrt{m_1^2 - m_1 m_2 + m_2^2} \right) \right)$$
$$a_{j (2)} = \left(1, \frac{1}{m_2} \left( - m_1 + m_2 - \sqrt{m_1^2 - m_1 m_2 + m_2^2} \right) \right)$$

Then
$$x_1 = a_{1 (1)} \eta_1 + a_{1 (2)} \eta_2 = \eta_1 + \eta_2$$

$$x_2 = a_{2 (1)} \eta_1 + a_{2 (2)} \eta_2$$
$$= \frac{1}{m_2} \left( - m_1 + m_2 + \sqrt{m_1^2 - m_1 m_2 + m_2^2} \right) \eta_1 + \frac{1}{m_2} \left( - m_1 + m_2 - \sqrt{m_1^2 - m_1 m_2 + m_2^2} \right) \eta_2$$

When $$m_1 = m_2$$ this gives the right result. Sorry for the error. Thanks for picking it up.

Chris

 

[dwintz02]

So, wait. Did this allow you to solve the problem or was that a typo?

 

[miscellanyous]

So, wait. Did this allow you to solve the problem or was that a typo?

It was a typo. If you sub the "normal coordinates" just found into the equations of motion you will find they do not separate as they should. Or I don't see how they separate.

 

[dwintz02]

These are used to find conditions on the eigenvectors $$a_{j r}$$ which satisfy
$$\left( A_{j k} - \omega_r^2 m_{j k} \right) a_{j r} = 0$$
where $$\omega_r^2$$ is the rth eigenfrequency.

The (unnormalized) eigenvectors are
$$a_{j \pm} = \left(1, \frac{1}{m_2} \left( - m_1 + m_2 \pm \sqrt{m_1^2 - m_1 m_2 + m_2^2} \right) \right)$$

I'm getting different eigenvectors than you because I don't know how you went about solving for yours, so I just want to check if yours are ok. Here's my method: multiplying out the your first equation and letting the components of my eigenvectors be 'a' and 'b' I get:

$$a(-m_1\omega^2+2k)-bk=0$$
$$-ak+b(-m_2\omega^2+2k)=0$$

Letting a=1, and solving for b from the first equation

$$b=2-\frac{m_1\omega^2}{k}$$

Plugging in for $$\omega^2$$, I get eigenvectors:

$$(1,2-\frac{1}{m_1}(m_1+m_2\pm\sqrt{m_1^2-m_1m_2+m_2^2}))$$
which also checks with the limit
$$m_1=m_2$$

If your method matches mine for the eigenvectors, I suggest tinkering with the final equation when you sub the normal coordinates back into the original differential equation and just keep playing with it until you get the right answer or proof that you messed up somewhere.

 

[miscellanyous]

I'm getting different eigenvectors than you because I don't know how you went about solving for yours, so I just want to check if yours are ok. Here's my method...

It seems like we arrive at the same eigenvectors which is reassuring. Thanks for all your hard work on this problem.

I am becoming skeptical that normal modes exist for cases when the masses are different or the sping constant connecting m1 to the wall is different than the spring constant connecting m2 to the wall. In these cases the two masses will naturally oscillate at different frequencies. Consequently it is impossible for their motion to be completely in phase or completely out of phase as is the case in the textbook treatment of normal modes (i.e. m1 = m2).