Question

Homework7: Problem 24 Problem List Next Problem Previous Problem (1 point) Consider the initial value problem my" +cy'+ky F(t) , (0 ) 0, y y'(0)-0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m 2 klograms, c8 kilograms per second, k 80 Newtons per meter, and the applied force in Newtons is ifosts/2 30 F(0) if t> /2. 0 remain contiuous when the applied force is discontinuous. a. Solve the initial value problem, using that the displacement y(t) and velocity y '(t) For 0sts/2, v(t)- help (formulas) For tT/2, y(t) help (formulas) b. Determine the long-term behavior of the system. Is lim y(t) 0? If it is , enter zero. if not, enter a function that approximates y(t) for very large positive values of t. help (formulas) For very large positive values of t, y(t) Submit Answers Preview My Answers You have attempted this problem 0 times You have unlimited attempts remaining.

Answer 1

F (t)30xu -0)-30 xut- 2 8y80y 30xu(t-0)-30xut-1 2y" 2 21+81(y)+80(y)=30L(u(-0))-30. 2(s?Y ()-(0)0) +8(s (s)-y(0))+80Y (s) = 30 _ S S 30-플 30 +8s +80)Y (s) (203 S S 30 30 Y (s) s(232 +8s+80) s2 +8s 80 31 31 s+4 s4 Y (s) 8 s 8 s4s40 8 s 8 s 4s 40 31 3 3 1 s +4 s4 Y (s) 8 s4s4+36 8 s4s40 8 s 8 s 31 3 s 2 3 Y (s) 8 s 2 8 (s+2) +6 4 (s +2) - s 2 31 3 3 8 (s+2)+6 4 (s+2 +6 8 s 3 3 3 1 Y(s) 8 -21 -2t cos 6 sin 6t 4 6 1-5s6 3 3 31 Sin 6 2 4 6 2 2 co