If the mass m is displaced by -Δy, then lower side springs are compressed and upper side springs are stretched.
Compressed length Δy2 = -(1/2)Δy and Streched length Δy1 = -(1/√2) Δy ( see the figure )
Restoring forces acting on mass m is shown in figure .
If we resolve each forces into horizontal and vertical components, horizontal components cancel with each other as can be seen from figure.
Vertical componenets of resolved forces give net resultant force as given below
F = 2F1 sin45 + 2 F2 sin30 = - 2 k [ (1/√2) Δy ] (1/√2) - 2 (2k) ( Δy/2 )( 1/2 ) = -2k Δy
acceleration = (-2k/m) Δy
angular speed , ω = 2π/T = ( 2k/m )1/2
where T is period of oscillation,