# Oscillations: mass in the center of an octahedron -- eigenvalues?

• leialee
In summary: So the task is to find normal modes and frequencies...In summary, the problem is to find the normal modes and frequencies of a system where a mass is connected to 6 different springs of equal length and no pre-stress. The equilibrium position is in the center and gravity is not taken into account. The relevant equations involve using Hook's law to calculate the forces exerted by each spring and then solving for the equations of motion using a matrix. However, there is a discrepancy in the equations as they contain mixed terms involving the coordinates x, y, and z. The assumption of equal relaxed lengths for the springs may not hold and could potentially lead to
leialee
You have an infinitesimally small mass in the center of octahedron. Mass is connected with 6 different springs (k_1, k_2, ... k_6) to corners of octahedron.
Equilibrium position is in the center, you don't take into account gravity, only springs.
Find normal modes and frequencies.

Relevant equations
##m\ddot R=\sum_{n=1}^{6} \vec F_i##
##R=(x,y,z) ##is a vector of location of the mass and ##A## is matrix.
My idea is to transform differential equations that I will get for each coordinate into this form:
##\ddot R+AR=0 ##

https://www.physicsforums.com/attachments/105010

## The Attempt at a Solution

When you move the mass away from equilibrium position, every spring works on it with its own force, problem is 3D so I used vectors.
##\vec F_i=\frac{\vec r_i}{|\vec r_i|}u_i k_i##, where I used Hook's law: ##F=-kx##, and in using this law I keep in mind that ##x,y,z## are small.
##u_i=b-|\vec r_i|##, where ##\vec r_i = \vec R - \vec R_i##.
##R_i## is a vector from center to corner for each spring (picture)
https://www.physicsforums.com/attachments/105009
For every sping i defined its vector ##R_i##
##\vec R_1=b(1,0,0)\\
\vec R_2=b(0,1,0)\\
\vec R_3=b(-1,0,0)\\
\vec R_4=b(0,-1,0)\\
\vec R_5=b(0,0,1)\\
\vec R_6=b(0,0,-1)##
Where ##b## is a length of un-expanded spring, ##b=a\frac {\sqrt{2} } {2}## and ##a## is a length of a side of octahedron. I marked ##r^2=x^2+y^2+z^2##.
So i get:
##\vec r_1= (x,y,z)-(b,0,0)=(x-b,y,z)\\
\vec r_2=(x,y-b,z)\\
\vec r_3=(x+b,y,z)\\
\vec r_4=(x,y+b,z)\\
\vec r_5=(x,y,z-b)\\
\vec r_6=(x,y,z+b)##

and for the lenghts of thse vectors:
##
|\vec r_1|= \sqrt{(x-b)^2 + y^2 + z^2}=\sqrt{r^2+b^2-2xb}\\
|\vec r_2|=\sqrt{x^2 + (y-b)^2 + z^2}=\sqrt{r^2+b^2-2yb}\\
|\vec r_3|=\sqrt{(x+b)^2 + y^2 + z^2}=\sqrt{r^2+b^2+2xb}\\
|\vec r_4|=\sqrt{x^2 + (y+b)^2 + z^2}=\sqrt{r^2+b^2+2yb}\\
|\vec r_5|=\sqrt{x^2 + y^2 + (z-b)^2}=\sqrt{r^2+b^2-2zb}\\
|\vec r_6|=\sqrt{x^2 + y^2 + (z+b)^2}=\sqrt{r^2+b^2+2zb}##

My equation is then:
##m\ddot R=\sum_{n=1}^{6} \vec F_i=\sum_{n=1}^{6} k_i u_i \frac{\vec r_i}{|\vec r_i|}##, and i mark ##u_i \frac{\vec r_i}{|\vec r_i|}=\vec r_i (\frac{b}{|\vec r_i|}-1)##

I started solving for each coordinate, and equation for x is:
$$m\ddot x=k_1(\frac {b}{\sqrt{r^2+b^2-2xb}}-1)(x-b)+k_2(\frac {b}{\sqrt{r^2+b^2-2yb}}-1)x+k_3(\frac {b}{\sqrt{r^2+b^2+2xb}}-1)x+k_4(\frac {b}{\sqrt{r^2+b^2+2yb}}-1)x+k_5(\frac {b}{\sqrt{r^2+b^2-2zb}}-1)x+k_6(\frac {b}{\sqrt{r^2+b^2+2zb}}-1)x=\\ -x(\sum_{n=1}^{6} k_i)+ k_1 b - k_3 b+ b^2(\frac {k_3}{\sqrt{r^2+b^2+2xb}}-\frac {k_1}{\sqrt{r^2+b^2-2xb}})+ \\bx(\frac {k_1}{\sqrt{r^2+b^2-2xb}}+\frac {k_2}{\sqrt{r^2+b^2+2xb}}+\frac {k_3}{\sqrt{r^2+b^2-2yb}}+\frac {k_4}{\sqrt{r^2+b^2+2yb}}+\frac {k_5}{\sqrt{r^2+b^2-2zb}}+\frac {k_6}{\sqrt{r^2+b^2+2zb}})$$ (1)

And then similar equation for ##y## and ##z##.
And here is where i have a problem. As far as i know i should get linear equatios so i can put together matrix A but no matter how i try to approximate, i get mixed parts of equations.

I went like this: because ##x,y,z##are small i neglected all ##r^2## and then developed roots with Taylor approximation to linear order.
##\frac{1}{\sqrt{1\pm x}}=1\mp \frac{1}{2}x##
Which i thought would go nicely and everything odd will fall out, even constant ##k_1 b - k_3 b## went away.
But I get such equations:
##m\ddot x= -x(k_3+k_1)+\frac{x}{b}(k_1 x + k_2 y-k_3 x -k_4 y +k_5 z -k_6 z)## and similar for y, z.
This all translates into a matrix which isn't constant but has ##x,y,z## dependencies.

So my question would be, how to develop equation 1 differently so i only get linear dependencies, meaning:
##\ddot x= a_1x+a_2y+a_3z\\
\ddot y=b_1x+b_2y+b_3z\\
\ddot z=c_1x+c_2y+c_3z## so I can then get this system: ##\ddot R+AR=0 ##

...or is my whole approach wrong? In that case...any suggestions, advice?
We are also allowed to use numerical approaches, but given I'm looking for normal modes i think matrix is a way to go.
Anything would be helpful at this point, because I'm on the verge of just neglecting mixed parts and getting diagonal matrix from the equations alone.

leialee said:
Where b is a length of unexpanded spring
Not sure you can assume this is the same for each spring. If they have different constants, the same relaxed length, yet equilibrium is in the centre than that says there is a relationship between the constants. So maybe they have different relaxed lengths, and the central position is where each spring exerts the same force.

haruspex said:
Not sure you can assume this is the same for each spring. If they have different constants, the same relaxed length, yet equilibrium is in the centre than that says there is a relationship between the constants. So maybe they have different relaxed lengths, and the central position is where each spring exerts the same force.
True, that is one of my problems - as you don't take into account gravity, are springs pre-stressed or not? if they are relaxed and same lenghts equilibrium is still in the center, but if they are pre-stressed then equlibrium (which the task says is in the center) gives you relationship between ##k_1## and ##k_3##, ##k_2## and ##k_4##, ##k_5## and ##k_6##.
So, arm? Not sure anymore?

leialee said:
True, that is one of my problems - as you don't take into account gravity, are springs pre-stressed or not? if they are relaxed and same lenghts equilibrium is still in the center, but if they are pre-stressed then equlibrium (which the task says is in the center) gives you relationship between ##k_1## and ##k_3##, ##k_2## and ##k_4##, ##k_5## and ##k_6##.
So, arm? Not sure anymore?
Actually the task's instruction is scpecific: all springs have the same length and are not pre-stressed.

leialee said:
Actually the task's instruction is scpecific: all springs have the same length and are not pre-stressed.
Ok. I realized later I was wrong to say it implied a relationship between the constants.

Since the forces equations look messy, I tried looking at potential energy instead.
I considered a small perturbation length s at angle θ to the X axis. I represent the corners of the octagon as being at distance R from the origin and at angles θI to the X axis.
Approximating as far as the s2 terms, when the fog clears I get ##\frac{s^2}2\Sigma k_i\cos^2(\theta_i-\theta)##

leialee said:
But I get such equations:
##m\ddot x= -x(k_3+k_1)+\frac{x}{b}(k_1 x + k_2 y-k_3 x -k_4 y +k_5 z -k_6 z)## and similar for y, z.
...
i'm on the verge of just neglecting mixed parts and getting diagonal matrix from the equations alone.
Yes, I think you are right to drop the nonlinear terms. So, you get a fairly trivial solution.

Thank you all, i solved this later on - i did drop nonlinear terms and that was it.

## 1. What is an oscillation?

An oscillation is a repetitive back-and-forth movement between two points. It can occur in various systems, such as a pendulum or a spring, and is characterized by a certain frequency and amplitude.

## 2. What is the "mass in the center of an octahedron" in relation to oscillations?

The "mass in the center of an octahedron" refers to a physical system in which a mass is suspended at the center of an octahedron structure. This system can exhibit oscillatory behavior, with the mass moving back and forth between the corners of the octahedron.

## 3. What are eigenvalues in the context of oscillations?

Eigenvalues are a mathematical concept used to describe the behavior of a system. In the context of oscillations, eigenvalues refer to the values that determine the frequency and amplitude of the oscillation. They can be calculated using mathematical equations and can provide insight into the behavior of the system.

## 4. How is the oscillation of a "mass in the center of an octahedron" system measured?

The oscillation of a "mass in the center of an octahedron" system can be measured in several ways. One method is to use a timer to measure the frequency of the oscillation, which is the number of complete cycles per unit time. Another method is to measure the amplitude, which is the maximum displacement of the mass from its equilibrium position.

## 5. What are some real-life applications of oscillations of a "mass in the center of an octahedron" system?

Oscillations of a "mass in the center of an octahedron" system can be found in many real-life situations. One example is the oscillation of a pendulum, which can be approximated as a "mass in the center of an octahedron" system. Other examples include the movement of a swing, the motion of a spring, and the vibrations of a guitar string. Understanding the behavior of these systems can have practical applications in fields such as engineering, physics, and music.

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