Coupled oscillators and normal modes question

Click For Summary
SUMMARY

The discussion centers on the analysis of coupled oscillators involving two equal masses connected by three equal springs. When one mass is fixed, the oscillation period is established as three seconds, leading to angular frequency calculations. The derived periods of oscillation for both normal modes are confirmed as t=sqrt(6) and t=3sqrt(2). The relationship between angular frequency and oscillation period is clearly defined through the equation t=2π/w, and the discussion emphasizes the importance of understanding the dynamics of coupled systems.

PREREQUISITES
  • Understanding of coupled oscillators
  • Knowledge of normal modes in mechanical systems
  • Familiarity with angular frequency and oscillation period calculations
  • Basic proficiency in matrix setup for dynamic systems
NEXT STEPS
  • Study the derivation of normal modes in coupled oscillators
  • Learn about the mathematical modeling of oscillatory systems using matrices
  • Explore the implications of spring constants on oscillation periods
  • Investigate the effects of damping on coupled oscillators
USEFUL FOR

This discussion is beneficial for physics students, mechanical engineers, and anyone interested in the dynamics of oscillatory systems and normal mode analysis.

warfreak131
Messages
186
Reaction score
0

Homework Statement



Two equal masses are held on a frictionless track by 3 equal springs, attached to two rigid posts. If either of the masses is clamped, the period (t=2pi/w) of one oscillation is three seconds.

If both masses are free, what is the periods of oscillation of both normal modes?

Homework Equations





The Attempt at a Solution



We learned in class that one fixed mass resulted in w=sqrt(2k/m), both traveling in the same direction with equal distance between them results in w=sqrt(k/m), and both traveling in opposite directions results in w=sqrt(3k/m).

So I know that 3=2pi/w, but I don't know where that gets me.

I know that the answer to this question is t=sqrt(6) and t=3sqrt(2), but I can't imagine how these numbers and formulas can go together to make them.
 
Physics news on Phys.org
I'd start by setting up a matrix to describe the acceleration of each mass in terms of the position of each mass
 

Similar threads

Coupled oscillators -- period of normal modes
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Error tolerant normal mode frequency
  • · Replies 7 ·
Replies
7
Views
2K
Normal modes of four coupled oscillating masses (Kleppner and Kolenkow)
  • · Replies 4 ·
Replies
4
Views
3K
Understanding solution to equations of motion of n coupled oscillators
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Normal modes of vibration from the total energy
  • · Replies 3 ·
Replies
3
Views
1K
Normal modes and degrees of freedom in coupled oscillators
  • · Replies 5 ·
Replies
5
Views
5K
Undergrad What is “normal” about normal frequencies and normal modes?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Pair of interacting systems, driven coupled harmonic oscillators
  • · Replies 7 ·
Replies
7
Views
2K
Graduate Equipartition theorem and Coupled harmonic oscillator system
  • · Replies 3 ·
Replies
3
Views
2K
Normal Modes and Normal Frequencies
  • · Replies 6 ·
Replies
6
Views
2K