Question

(4) Consider the 2nd order equation for a mass-spring-damper system, mx'' + bx' + kx = f(t)

a) Assuming f(t) is a step function, find the Laplacian transform, X(s) (include terms for the initial conditions xo, vo).

b) Assume m = 1, b = 5, and k = 6, and x(0) = 3, x’(0) = 0. Find the time-domain solution (take the inverse transform).

(5)Find the Laplace transform of y(t) from the differential equation, assuming u(t) is a step function.

α4(d^4y /dt^4)+ α3(d^3y/ dt^3)+ α2(d^2y/ dt^2)+ α1(dy /dt)+ α0y = β(du/ dt) + u (t)

+ u(t) (6) At t = 0, the voltage across the capacitor is Vo (= 1/C ꭍ idt]to), then the switch is closed and current flows through the circuit. Find Vc(t)

Answer 1

Taking Lapluc Taanstonm on both sidra, yn [스2 2.С.)-дж(o) _x'(03] + b[bx(4) ← x(о)]+kx(n) = f x(1) = f-CA) =- 3ぶ+150+1 Laplace Tnansboam, we 七 S -2t 3

(5) Guven, du d t dt dt dt Now cinc wt·Ja closed, cunnent toouds