Question

Discuss the effect of the frequency of the driving force on the amplitude and phase of the oscillator, by deriving the solution of the relevant differential equation.

A spring stretches by 1.96 m when a 2 kg mass is attached. The system is then submerged in liquid which imparts a damping force numerically equal to 4 times the velocity of the mass. Find the value of the steady state solution after T/2 second if an external force f(t)= 2sin 2t(kg.m/sec2) is applied to the system.

Answer 1

wrp - Bosd and tong motion wr-pr if f 2 f feia -19 maze e + Foeipt/k= 2 m de 2 a natural frequency dt 2 w x - 2K ax Ee'pt Beid - t -ig let us consider the applied force it is to eipt and akp = B sing p is the frequency of the applied force. So w pr) + (2 kp) - By (sin No + cosp) or We apple this force to the damped harmonic so that the effect of damping can be B = Jor pugų akupy 2 kp Hemoved. mis the mass of the oscillators, kis the Now Asif 9 = tant (261 -- 1} (ww_pr Beosotissing constant d is the dauping constant spring Blosptising) then the equation of motion becomes - 4x-ddy B at ww-pry + 4u8v wa in a ~ 224 is we ignore complimentary function then the of the oscillators salection is It - F f d2x ei ipt +2kdx ther = feipt Amplitude of the f de ² It driving force Scar por jo 4 xypr We have to solve this by using api & Particular Integral, a complimentary fonction. Now let x = Ágipt (w pr dx = ipß cipt that means p=2 & f = 2 de preipt ta I takkipeipt + Ácipt Kan 43 416 a A L-pr+2kip twr] =f f (Wx-p") tiake x = I (T-7) √32 = eft-q) Perunit mass Jiw ) t que per p Bitte deriven force frequency + complitude of enter that Vw pr9 79kor Alf f(t) = 2 sinat h d= 4, K = 2 isthe (20p dez m=2 ta 180 LI m se, – ph Aript fert 8 ton 19 (0 - 9 f(1) =23inT -0 = ax wao PA) a, azo - 0x1.96.