advertisement

Name________________________ Student I.D.___________________ Math 2250-4 Quiz 12 Solutions April 20, 2012 1) Find the general solution x1 t , x2 t T to the homogeneous system of second order differential equations, which could result from two masses coupled together with springs: x1 ## t =K3 x1 C x2 x2 ## t = 2 x1 K 2 x2 . x1 ## t x2 ## t K3 K l 2 1 K2 K l = K3 1 2 K2 (8 points) x1 t x2 t 2 = lC 3 lC 2 K 2 = l C 5 lC 4 = lC 4 lC 1 . l =K1 w = Kl = 1 the homogeneous system is K2 1 0 K2 1 0 1 / 0v= 2 K1 0 0 0 0 2 . l =K4 w = 2 the homogeneous system is 1 1 0 1 1 0 1 / 0v= 2 2 0 0 0 0 K1 . So, x1 t x2 t = c1cos t C c2sin t 1 C c3cos 2 t C c4sin 2 t 2 1 K1 . 2) Use Newton's second law to derive the system of differential equations above, from the configuration below. x1 t , x2 t are the displacements from equilibrium of two masses. The mass values and spring constants are as shown. (2 points) 2 x1 ## t =K4 x1 C 2 x2 K x1 =K6 x1 C 2 x2 1 x1 ## t =K2 x2 K x1 = 2 x1 K 2 x2 . This is equivalent to the system in problem 1 (divide the first DE by 2).