Question

Question. Systems of ODEs of higher order can be solved by the Laplace transform method. As an important application, typical of many similar mechanical systems, consider coupled vibrating masses on springs. Wrovov The mechanical system in the Figure consists of two bodies of mass 1 on three springs of the same spring constant k and of negligibly small masses of the springs. Also damping is assumed to be practically zero. Then the model of the physical system is the system of Ordinary Differential Equations given by: dr2 = -kyı + k(yz- y of these -kyz – k(yz – ys) Here y, and y, are the displacements of the bodies from their positions of static equilibrium. These ODES follow from Newton's second law, Mass x Acceleration = Force, for a single body. Regard downward forces as positive and upward as negative. On the upper body, -ky, is the force of the upper spring and ky2 = y.) that of the middle spring (Y2 - yı) being the net change in spring length. On the lower body, -k(y, - y.) is the force of the middle spring and -ky, that of the lower spring. Initial conditions are y(0) = 1, y,0) = 1, y(0) = v3k ,y(0) = -V3k Use Laplace transforms to solve for yı(t) and yz(t) which satisfy above given conditions. (Hint: Use simultaneous equations to so solve Lly) and L{y, ) once apply Laplace transforms to the given equations

Answer 1

301 y = -ky, tkc4,-9,) = -aky, try - A LI dt? dth dyz : -ky, kly, y) - 2kytky, Y, (O) = 8 y 10) = JBK taplaw of A Y 10) = 1 / 2 (0) = - 534 Selo uptoledoaryd = es Phyl-as Concept: - L (d?/d+²) = f(s) - sf (o) -fo) & L (dflat) - SFCS) - f10 Laplace of s² Lly) - syco) - y lo = -2K Lly) + K LLY) •=5 lly,) - S[!] - [ J3k 1 = -2k Lcy,) tk Llys Let's cenote Lly) by Fies) & Let's denote Lly) = Fz (s) = = s² F (S) -s- Jak = - 2K F, (s) + K {(5) - (C) - Laplace of eq s² Lly) -sy 10 - 10) = -2K (ly) tk Lly) .. - s² (FCS) -s t53k = -2K £ 10 + KF (S) - (D

IP-2 eq 0 + eq s? ( ficut ) - 25 = -K firs) - KF, (5) 8² Ch co + f (6) -25= -k( flot hy(s)} fotfis & 25 25 stk ea © - s² CF (S) - fz (S)) - Jok = - 3k f1 (s) + 3K fyls) 3² (f 10 -26)) - 50k = -3k( F (S) - fls)) (5² +34) [f, cw-fucs] = Sök & fic soff2C) = Jok t o 5 +34 aal & ea + Jok 2f, 15 = 25 2 , 2K S3k

= cos Jkt & S2 sinJak t 1 . sin 52 Formulas P {als on a small a la fecond ma . Cosat Now frome & G 18)= 25 - fils) stk = - 23 - Is & Sok stk 2(3²+36) Rewrite is (si con se si e Aus 18 - Cos(Tv 4) – sa smlok t)