Question

A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c). The mass is set in motion with initial position X, and initial velocity vo Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t) =C, e-pt cos (0,t-a). Also, find the undamped position function u(t) = Cocos (0,0+ - ) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (SO C= 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of (t) and u(t). m=1, c=6, k= 9, Xo = 5, 10 = -8

Answer 1

Answer: Givers that Consider the mass m attached to the spring with initial position to and initial velocity vo WMA mal C= 6 K: X05 Vo: -8 The free damped motion is with damping but there is no the external force. The differential equation takes the form 2" +2px' + woxo The value of p is givers by p. clam PS: 3 is 2(1) WÓ = km - 9 = 3: No, so the motion under damped The pseudo frequency is W, - vwo-pe 19 -9 20 Substitute the values of p and wo x" + 6x + 9x = 0 The characteristic equation <"+6x' +92=4*4

7 +6 Y+0 + 3y + 3x + 9 = 0 Y(7+3) + 3(x+3): 0 (v + 3) = 0 -) = -3, -3 Hence the roots are 8 = - 82 = -3 The solution is given by xct): cest+Gzēst The initial condition xo = 5 gives 200) = qē360) + Cé310) 5 = c + C2 The initial condition Vo = -8 gives x'(t) = -36, e3t 3 ca ezt x'(o) --342360).3 gē360) -8 = -34-342 -8:-3 (9+(2) Therefore 4= 5 6 7/3 for undanped motion the differential equation mx" +Kx=0 » 2"+9x=0 1 x" +92 - 0 x (t) = 5est - 1/3

7 + 9 + 7 -9 8 = + 3 8 +3 82 = 3 given by The undamped function is xct), A COS Y t + B sin ot Hence the position function is act) : Acos (30) + B sin(-30) At no 5 gives 5 - Acos (30) + B Sin(-30) ► A=5 At vo=-8 gives x'lt): -A (3) sin(3 t) + (-3) B COS (-30) alo) -3A Sinco) - 38 Cos (0) -8 = -3B Undamped 37 21t) ult) = B = 8/3 function ult)- 5 cos 3t + 8/3 Sin (-34) The - a +