Question

ONLY attempt to solve if you know what you are doing.

A mass of 1 kg is attached to a spring whose constant is 5 N/m. Initially, the mass is released 1 m below the equilibrium position with a downward velocity of 5 m/s, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocit y. a) Find the equation of motion if the mass is driven by an external force equal to f (t) 12 cos 2t 3 sin 2t. Graph the transient and steady-state solutions on the same coordinate axes. Graph the equation of motion. b) c)

Answer 1

Solution Assume that y (t) be the position of the mass at time t >0 The differential equation modelling the above situation is given by dv d,y dt In this case, m1.p 2, k 5, y (0)--1,y'(0)--5 f(t)-12 cos (2t)+3sin (2t) +2+5v-12 cos (2t+3 sin (2it The auxiliary equation of the homogeneous differential equation is The solution isye(r)=e"(c1 cos(2t)-e, sin (2t)) Let the particular solution be of the form y, (t)- acos(2r)+bsin (21) y' (t)-2asin (2t)2bcos (2t) and y*(t)--4acos (2t)- 4bsin (2t) Substitute these in the original di fferential equati on to get 4a cos(2t)-4bsin(21)+2-2asin (21)+2bcos(21)) +5 sacos(21)+bsin (21))-12 cos(2/)+3sin(21) -4acos (2t)-4bsin (2t)- 4a sin (2t)+4b cos(2t) +5acos(2t)+5bsin (2t)-12 cos(2)+3 sin (2t) (a+4b)cos(2t)+(b-4a)sin (2t)-12 cos(2t) +3 sin (2t)

Compare the coe ffi cients of the like terms to get a+4b-12 and b-4a 3 y,(t)-3sin(27) The general solution is y (t)-e"(ccos(2t)+c, sin (21))+3sin (2r) Apply y (0)--1 e (G cos (o)+c,sin (0))+3sin (o)--1 y(t)-e-cos(2f)+csin(2))+3 sin (2r) y(t)e(-cos(21)+c, sin(21))+e"(2sin(2r)+2c, cos(2)+6cos(21) Apply y' (0)--5, -(-cos(0)+c sin (0))+e (2 sin (0)+2c, cos(0))+ 6 cos(0)-5 (2c2)--12 t)e-cos (2t)- 2 sin (2t)) +3 sin (2t is the equation of motion Transient solution e "(-cos(21)-2sin (21)) Steady-state solution-3 sin(2r) Use an online graphing calculator to sketch transient and the steady-state solution of the given differential equation

Steady-state solution 10 -2 Transient solution -4 Use an online graphing calculator to sketch equati on of motion 15 2 y(,)-e-(-cos(2)-2sin(21))+3sin(2)