Infinite-Extent 2D Mass-Spring System vibration and dispersion relation

[M_Abubakr]

Hi,
I am trying to find equation of motion and its solutions for a 2D infinite lumped mass spring system as depicted in figure. All the masses are identical, All the springs are identical, and even the horizontal and vertical periodicity is the same n=a.

I need to try find dispersive relation for such a system. Can anyone tell me how to solve this? I am fairly confident with mass-spring systems in one dimension with MDOF but I don't know how to solve this one and I can't find any similar problem solved over the internet.

 

[LightHero]

Thanks!The equation of motion for this system is given by:m*a*ddot{x_n} + k*(x_{n+1} + x_{n-1} - 2*x_n) = 0where m is the mass of each lumped mass, a is the periodicity of the system, k is the spring constant, and x_n is the displacement of the nth mass. Solving this equation is a bit involved, but it can be done using Fourier Analysis. The solution is:x_n = A*cos(w*n*a) + B*sin(w*n*a)where A and B are constants and w is the frequency of the system, which can be found from the dispersive relationw^2 = k/mHope this helps!