Question

Consider the mass-spring system for mass 0.7 kg, spring constant N/m, and an oscillating force 2.6 cos(0) Newtons. That is, ost) What positive angular frequency a leads to resonan help (numbers) What is the resonance part of the solution (without the complementary solution); help (formulas)

Answer 1

0.7 The given diff. equa is U 0.7x" +5158 9 216 C5 (67) ---0) The reduced homogeneous diff. equin of ) i 0.72"+5-52' =0 6) The charecteristic equit of the diff. equn (6) is 0.7* +5.5. 20 ERA - 2-5555 5 i stá 55 The solen of the equr (6) is given by B x = + = 5 + 5 + 8 g (5 + R AM 8. If we take a are arbitary constant .. wt = 555 + 7 w = 1 & 2.8031 (correct up to I four decimal . In this case we can take particular pregral of the (i) can be written as I where cfd are 20.= ct cos (601) + Dt Sin (wt) constante ter Xp ce Cos(wt) etwson (wt) + D Sin (007) + Duteswt) e sc GWDED Sön (wt) + tak csin (wt) +D 6 (wt) ::M02-wcsin(wt) +D w Cos (et) twec Sirwt) +D G(wt). " Auf ew los (WI) - DW sin (WA)) determine

:: Xp=2w6csin(we) + D C66t))+6w"-c Gog(@t) - D Sin(wt)) Since xp is a solution of the diff. equin (8), we get 0.74% +5,5% = 26 Cos (wt) 25,14W Ecser (web) +D G (W140710f-cos(wsl)-D ein (wt)) +5:57 Cecos (w&D + Dein (wt)) = 216 coz (WI) -- tii) 355 ; - Putting t-o in the equ" (15) de get 104WD = 216 5) De zon = V V 168 Putting t = 7 w in the equa (is) we get 13 14W (-9) +0.769) _D)+5+5 () (D)=0 0,- lol Weep-0.7 (9)w (*) +5.5(W) 643)-0 00,-1.4W C-7. 44+. 4 -0. ar, -1.4WC - 188 + tertentu so OV, -1.46C- 13% + 132 so a, - 14W C =0 = @=0. X XP = 16 t sin (WA). . 7p v hornet fin (JF +)