Solving Coupled Oscillators: Find Spring Constants

[Unicorn.]

Hi,
I didn't understand the exercise so I didn't do it and my teacher doesn't give the details of the solutions. If somewone can help me and explain me the steps.. it'd be great !

Three block of mass m=0.13kg are connected with three springs of constant k1 k2 k3 and

Two of the normal modes of the system, expressed in terms of displacement are :
|1>=( 1 0 -3) |2>= ( 27 -10 9)

Knowing that the lower frequency that isn't necessarly the first mode frequency is 20 rad/s. Find the spring constants.

All i did is that i gave the equations of motion so i got few points for that, and found the |3> as they're perpendicular between them.

Thank you, I have an exam tomorrow and need to understand this !

 

[Nate810]

To answer this question, you need to use the equation of motion for a system of connected masses and springs. The equation of motion can be written as: ma + k1x1 + k2(x2-x1) + k3(x3-x2) = 0 where m is the mass of each block, k1, k2, and k3 are the spring constants, and x1, x2, and x3 are the displacements of the blocks. Since you have the two normal modes of the system, you can write the equation of motion in terms of the normal mode amplitudes. The equation of motion can then be written as: (m/2)(|1> + |2>) + (k1+k2)|1> + (k2+k3)|2> = 0 From this equation, you can solve for the spring constants using the given normal modes and frequency. First, you need to find the angular frequency of the system, which is given by: ω = √(k1+k2+k3)/m Then, you can substitute this expression into the equation of motion to get: (m/2)(|1> + |2>) + (ω2/2)(|1> + |2>) = 0 Solving for the spring constants yields: k1 = (ω2 - 4mω2/2)/2 k2 = (ω2 - 4mω2/2)/2 k3 = (ω2 - 4mω2/2)/2 Finally, substituting in the given frequency of 20 rad/s yields the spring constants: k1 = 400 N/m k2 = 400 N/m k3 = 400 N/m