Answer Happy

. A dynamical system is composed of two bodies of masses m₁ and m2, which are placed on a horizontal non-smooth surface, and three springs with Hooke's constants ka, k and ke, respectively, as shown in the figure. The motion of the two bodies is damped by friction with the surface, which is proportional to their velocities, with coefficients ₁ and 2 respectively. The displacements of the two bodies with respect to equilibrium are denoted by r(t) and r2(t) re- spectively (these displacements are small enough that the springs remain in the linear regime, consistently with Hooke's law, throughout the motion). m Ka kb kc H₂ H₂ (a) Determine the total force acting on each of the two bodies. (b) Write down the equation of motion of the system for the vector r(t) = (26))in in a matrix form. (c) Assume from now on that m₁ = m₂ = m, and ka = k = k = k and also #₁=₂ μ. Show that the equation of motion can be brought to the form: 2 d [(a)² + 2) +²U] z(t) = 0 where U is a matrix and determine A, w and U in terms of p, k and m. (d) Define new coordinates y(t) = (366)) with r(t) = Ry(t) where R is a (time independent) matrix such that the equation of motion for y(t) is diagonal. Determine R and the new equation of motion. (e) Find the general solution for y(t) assuming that the parameters are such that one mode is over-damped and the other is under-damped. What is the range of values of w/A allowing this? (f) Determine the general solution for xi(t) and x₂(t).
So, we have the two differential as (@) + (-+- что у му би+ Сентен и кат Non, Considering, them are get and y₂ = die 1st onder dife (+)2-12 + (+₂)2-K₂X=0 m₂ y +(₂+1) + K+K-K₂=0 are the Ra's. (2) Now, it the metier-fre 10 00 1 00 00 000 Me D 0. + D P 0 hu 9-2 A-- ] O 2 -K (444)-11 (1)-(4) 9 +B=0 (0) Fim my My on, begeg m² + MX+ DX-D O +24x + 5xx-xx-0 *+*+ ** = 0 4 x+3x2+²x o Du . PX- V₁= K₂ K₂ K² lyhy +225 +2²] x = 0 A