Question

2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a vertical plane under the influence of gravity can be modelled with the second order non-homogeneous ODE 0"(t) + 270'(t) +w20(t) = f(t), (2) where wa = g/l. The second term in the equation represents the damping force (e.g. air resistance) for the given constant 7 > 0. The model can be used to approximate the motion of a magnetic pendulum bob being driven by the force, f(t), due to an electrical field (e.g. robot arm). (a) Write down the characteristic equation of the associated homogeneous equation, and determine an expression for the roots in terms of and w. (b) Write down the general solution to the associated homogeneous equation for 7<w. (c) Write down the general solution to the associated homogeneous equation for 7=w. (d) Write down the general solution to the associated homogeneous equation for 7 >w. (e) Determine the behaviour of the associated homogeneous equation as to for all three of your answers to (b), (c) and (d). (f) Use the method of undetermined coefficients to find a particular solution to the non-homogeneous equation (2) for the given driving force f(t) = E cos(wt), where E >O is a given constant. (g) Determine the maximum and minimum values of the angular displacement, e(t), as t →

Answer 1

tws o=0 ODE Lecond order non-homogeneous Pantan sito 01 (4) + 9V O'CH + WA 0 (t) = f(t) w2a corruponding homogenous differential equation w20= Oll(t) + 20 o'lt) + d20 dth tando dt dt dt2 da tavolo + wa) =0 characteristic equation of the expoulated homogenors equection. ma +90 mt whao ma-21 I Vaujany x 1 xw? 21 -20 t dava_4W? ma

-28 + Vucra_wa) 2 ma -20 t ar 02-12 marv z dve-w? mir-y + y 2-wa لاسا - In g=-V-Vya_w Pair fafhen VLw v22 w2 12- W2Lo m=-it Vra-w2 ma - V IV-a from a there a=v2-wl m = -0th oi treneral soluhun. Vt OH) = f Cox24 lainrahe OH) = (4 col dve - Waft la fin Vra-wed et

20 y2zwe 3 10lt)=4 ett at et qng and Vidana ri Ceneral solulation of addouated homegenus equatin Paste DE va-wazo on from a marit Vra-we ma-v. word are -V, V treneral solution of the addouaked homagenery equerra -V+ Parrel khen Dyw from a mard & ſ2-w2 Maard + Nyawa -VINViwa Octave toe o(t) a (6 tlg tje Or уяг u? 22-war O -ỷ yếu