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.