Question

PART A

(1 point) A mass m = 4 kg is attached to both a spring with spring constant k = 197 N/m and a dash-pot with damping constant c=4N s/m. The mass is started in motion with initial position to 3 m and initial velocity vo = 6 m/s. Determine the position function r(t) in meters. x(1) Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form x(t) = Cie pl cos(wit - Qi). Determine C1,wi , and p. C wi Q1 = (assume 0 <«i < 2) p= Graph the function (t) together with the "amplitude envelope" curves r = -Ce-jit and I=Cje-pit Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected ( so c= 0). Solve the resulting differential equation to find the position function u(t). In this case the position function u(t) can be written as u(t) = Cocos(wot - 20). Determine Co, wo and co. Co = Wo = 00= (assume 0 < ao < 27 )

PART B

PART C

PART D

Answer 1

small c, Given mass 4ng A K. 197 N/m 4 Ns/m 3 m 6 m/s Equation of motion mät cà + kr = 0 for X(+) Cosw't de-rt $1 8- i'm 4 2x4 0.5 2m. S / به к m 4m2 = w 197 4 42 4. 42 w'? 197 4 0. 25 7 rad/s

-0.5 XL) = 3.186 e A e oost cosfat- do) Ае Xt xt=0) A Casa, à (t) t - 0.5A é Cos(-2) - FAE - Ost .0.5t Sin(tt-d) à(to) = -0.5 A Cosa, + FA sind, = ilto) = Vo = 6 = -A casa, + 7A sind,

and ② S from 2. 3 A cos d, ang Nos 6 6 = - A Cosd, + FA Sind, eanation / Cosa, t 7 Sind, z. -) 을 14 Costa -0.5 22 7 tana, + =) 7 tom dia 2.5 =) di 2 tam! (4) 1 2) X, 19.65° = 0.343 radian = from 3= A cosai A 11 3 cosa, 3.186.

4 (-0.5x) 3.186.e + Red plot cos(7.8 -0.343) - -3.186.e(-0.5x). cos(7.8 – 0.343) Blue Plot C 12- -21