Question

Consider the mass-spring-damper system depicted in the figure below, where the input of the system is the applied force F(t) and the output of the system is xít) that is the displacement of the mass according to the coordinate system defined in that figure. Assume that force F(t) is applied for t> 0 and the system is in static equilibrium before t=0 and z(t) is measured from the static equilibrium. b m F Also, the mass of the block, the spring constant and viscous friction coefficient are denoted as m, k and b, respectively. Take g=10 m/sec2 A) Derive a mathematical model between 2(t) and F(t) as a differential equation in terms of m, k, b, and by drawing free-body-diagram (s) and using appropriate physical law(s). B) By taking m=1, b = 4, k = 4, determine the transfer function of this system. C) By taking m = 1, b = 4, k = 4, determine the impulse response of this system D) By taking m 1,5 = 4, k 4, and by assuming of all zero initial-conditions, determine x(t) for F(t) = 4u(t) where uſt)is the unit-step function.

Answer 1

ka ↑ a bi m Tam = Fret I 21) FIO F(t) - ka-bà CAL From Newtons second law, ma > märką tbi = f(t) Taking Laplace trenstorm on both sides, with zero initial conditions, (8) (ms*+bs+k) *Cs) - $0) x(s) T(3) +6) mstbstk f(s) st 45 +4 C) For impulse response, f(s) = 1. = X(s) Susty (3+2) 26) i-l teat, (5+234 For muit input, f(s) Ś rol) x(5) S(ST2) 2 t - at téct. alth = t -zt -26 dl- tie -2 lo t mat -26 - te 0 -26 26 -te [ c 4 -zt Lite -2t e 2 4 4 CS Scanned with CamScanner