Finding Coefficients for Coupled Harmonic Oscillator

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Discussion Overview

The discussion revolves around finding coefficients for a coupled harmonic oscillator, specifically focusing on determining these coefficients based on initial conditions such as position and velocity. The conversation includes both theoretical and practical aspects, with references to normal modes and methods for solving the equations of motion.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant seeks the best method to determine coefficients for a coupled harmonic oscillator given initial conditions, mentioning the need for a solution applicable to multiple masses.
  • Another participant provides equations for position and velocity of the masses, suggesting that the initial conditions can be used to set up a system of equations to solve for the coefficients.
  • A third participant proposes introducing sine and cosine variables to the equations, resulting in a larger system of equations that includes trigonometric identities, which could simplify the solution process.
  • One participant expresses a need for an algorithm to find the coefficients, especially for scaling the solution to n equal masses.
  • Another participant suggests using Lagrangian mechanics as a faster method for solving equations of motion, although this is not the preferred approach for the original poster.
  • The original poster clarifies that they intend to use Newtonian mechanics for their explanation and mentions having written a complex solution in terms of real and imaginary parts to facilitate solving it.

Areas of Agreement / Disagreement

Participants present various methods and approaches to solving the problem, with no consensus on a single method. Some propose using Lagrangian mechanics while others focus on Newtonian mechanics, indicating a divergence in preferred methodologies.

Contextual Notes

The discussion includes assumptions about the applicability of methods to systems with multiple degrees of freedom and the need for clarity in explaining concepts to different audiences. There are also unresolved aspects regarding the choice of methods and the complexity of the equations involved.

mrandersdk
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I've been looking at a coupled harmonic oscillator, and normal modes of this:

http://en.wikipedia.org/wiki/Normal_mode#Example_.E2.80.94_normal_modes_of_coupled_oscillators

At the bottom of this example it says:

This corresponds to the masses moving in the opposite directions, while the center of mass remains stationary. The general solution is a superposition of the normal modes where c1, c2, φ1, and φ2, are determined by the initial conditions of the problem.

What is the best way to determine these coefficients, given some initial conditions (start position and velocity)?

This is only for two masses, but the method should also work for n equal masses. It is fine if I need to program me out of it, but how should i do this?
 
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According to that website, the equatios are:

x1(t) = c1*cos(w1*t+phi1)+c2*cos(w2*t+phi2)
x2(t) = c1*cos(w1*t+phi1)-c2*cos(w2*t+phi2)

Taking the derivative, you get 2 more equations

v1(t) = -c1*w1*sin(w1*t+phi1)-c2*w2*sin(w2*t+phi2)
v2(t) = -c1*w1*sin(w1*t+phi1)+c2*w2*sin(w2*t+phi2)

If your initial conditions are:

x0_1 = x1(t=0)
x0_2 = x2(t=0)
v0_1 = v1(t=0)
v0_2 = v2(t=0)

then your equations become:

x0_1 = c1*cos(phi1)+c2*cos(phi2)
x0_2 = c1*cos(phi1)-c2*cos(phi2)
v0_1 = -c1*w1*sin(phi1)-c2*w2*sin(phi2)
v0_2 = -c1*w1*sin(phi1)+c2*w2*sin(phi2)

Now you have 4 equations and 4 unknowns and can solve the coefficients.
 
Last edited:
Actually, to solve this you need more variables and more equations. Instead of phi1 and phi2, use these:

sp1 = sin(phi1)
cp1 = cos(phi1)
sp2 = sin(phi2)
cp2 = cos(phi2)

Adding 2 new equations, you now have these 6 equations:

x0_1 = c1*cp1+c2*cp2
x0_2 = c1*cp1-c2*cp2
v0_1 = -c1*w1*sp1-c2*w2*sp2
v0_2 = -c1*w1*sp1+c2*w2*sp2
sp1^2+cp1^2 = 1
sp2^2+cp2^2 = 1

Once you've solved sp1,cp1,sp2, and cp2:

phi1 = atan2(sp1,cp1)
phi2 = atan2(sp2,cp2)
 
maybe I was a bit unclear, i know how to set up the equations, but what algorithm should i use to find the coefficients, especially when i need to scale it to n equal masses.
 
I would use http://en.wikipedia.org/wiki/Lagrangian_mechanics" . It's basically an energy method approach to solving equations of motion for systems with multiple degrees of freedom much more quickly than using Newtonian mechanics. There are examples at the link above.
 
Last edited by a moderator:
ok thanks. I know lagrangian mechanics, but need to solve it with Newtonian mechanics, because i need to explain it to someone who doesn't know it.

I've wrote the complex solution as the real and imaginary part and then i could solve it. Thanks for all your sugestions.
 

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