Question

For a mass-spring oscillator, Newton's second law implies that the position y(t) of the mass is governed by the second-order differential equation my'' (t) + by' (t) + ky(t) = 0. (a) Find the equation of motion for the vibrating spring with damping if m= 10 kg, b = 100 kg/sec, k = 260 kg/sec?. y(0) = 0.3 m, and y'(0) = -0.4 m/sec. (b) After how many seconds will the mass in part (a) first cross the equilibrium point? (c) Find the frequency of oscillation for the spring system of part (a). (d) The corresponding undamped system has a frequency of oscillation of approximately 0.812 cycles per second. What effect does the damping have on the frequency of oscillation? What other effects does it have on the solution? (a) y(t)=

Answer 1

Gioen dato, bloolis k=260 tgls 2 a m=lokg o.1m yol y'(o) = - 0iq mps tho eguation my" (+) + bylt) tky (E)=0 loy" t? t looy' (t) + 260 y(t) =D find th characters HC eguolion JOD? + Yoon +260 = 0 D² tiod t 26 11 -10 + 9 102 - 4X1 X26 2x 0 je t zi 2 - 5 + The solution of the differential equation y(+) 5t [c, Gast tąsint 11

condition He boundary aprey t=0 y = 0.3 m from eg e 500 l caso t ca sino ] 0] = 1X (C+0) 0.3= Cizon second Apply the boundary condition ý= & mis t=0 differentiating eg - 5t G'CHI - Ci sint + C2 Cast] + [ ci Cestit Casint (-5051) q I! [-exo +C2 ] + [4 + 6x0](-501) o ¢ C2 - 501 -0.4 = C2-5x0.3 ( 2 = 1

so for vibo motion vibroling equation I spring is fo ū (f) 0.3 Cast + lil sind] 55 e equilibrium an G dy dt from es " die Oh [-on sint +116014] + [ 0.9 Cost + 1.1 Sint l'l sine ] (-5ēt) o -0.3 sint til ost -1.5 Cast - 5.5 sint 5.0 Sint -ooq Cest - 5.8 sint = o q est 0.06 8 9 tant - tant ton [ro t tov(-0.0607] ta 3.945º -T-3.9458 I t = 3.18 0.0688 (t=3.07275

Compare ego Goort standard equation Ý ² + 2q wnyt only =o on 2 2 E Wn= 100 260 To Wn=5:09 radis 2 Y EX5. og lo are for 5.09 In = 0.01 cycle/ 6 981 = natural frequeng find the osallation frequency al own Iraz Wd - 5,09 | 1.9812 Wd - 0.987 rad 18 = 25 fd I fd = 157 cycle is or fa= 0.157 Hz af frequency The S und amped fo=o. 812 Cycle damping fequency fd = fon J1-62 fo 1.9812 o. 1575 cyclols- fon 7 fd. the damping will frequency of oscillation 0,812 reduce the