Coupled Oscillator Homework: Normal Modes & Frequencies

[Pqpolalk357]

Homework Statement

Two identical undamped oscillators, A and B, each of mass m and natural (angular) frequency $\omega_0$, are coupled in such a way that the coupling force exerted on A is $$\alpha m (\frac{d^2 x_A}{dt^2})$$, and the coupling force exerted on B is $$\alpha m (\frac{d^2 x_B}{dt^2})$$, where $$\alpha$$ is a coupling constant of magnitude less than 1. Describe the normal modes of the coupled system and find their frequencies.

I just need someone to explain to me what is the form of the differential equation with respect to each mass. The rest I can continue.

 

[Simon Bridge]

http://courses.washington.edu/phys2278/228wtr09/Phys_228_09_Lec_20_App_A.pdf
http://web.mit.edu/hyouk/www/mites2010/MITES_2010__Physics_III_-_Survey_of_Modern_Physics/MITES_2010__Physics_III_-_Survey_of_Modern_Physics/Entries/2010/6/28_Lecture_4___Classical_mechanics_-_Simple_harmonic_oscillator_%26_coupled_oscillators.html
... you have to use your knowledge of coupled oscillators and understanding of the term "coupling force" - along with your course notes - to work out the equations of motion.

 

[Pqpolalk357]

Could someone explain to me what is exactly is the "coupling force" ?

 

[Simon Bridge]

It is the force that each pendulum exerts on the other.
In a 2-mass, 3-spring system - it comes from the middle spring.

 

[paranormal]

I am happy to assist you with your homework on coupled oscillators. The differential equation for each mass in this system can be written as follows:

For mass A:
$m\frac{d^2x_A}{dt^2} + \omega_0^2x_A + \alpha m\frac{d^2x_B}{dt^2} = 0$

For mass B:
$m\frac{d^2x_B}{dt^2} + \omega_0^2x_B + \alpha m\frac{d^2x_A}{dt^2} = 0$

These equations represent the forces acting on each mass, taking into account the coupling force exerted by the other mass. To find the normal modes and frequencies of the coupled system, we can use the method of solving coupled differential equations. This involves substituting the equations for each mass into each other and solving for the normal modes and frequencies.

I hope this helps to clarify the form of the differential equations for you. If you need further assistance, please let me know. Good luck with your homework!