4. y" +gy 2 cos 36 Yn= m² +9=0 QUESTION 4 Determine the equation of motion for an undamped system at resonance governed

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4 Y Gy 2 Cos 36 Yn M 9 0 Question 4 Determine The Equation Of Motion For An Undamped System At Resonance Governed 1 (47.7 KiB) Viewed 12 times

4. y" +gy 2 cos 36 Yn= m² +9=0 QUESTION 4 Determine the equation of motion for an undamped system at resonance governed by d'y dt2 +9y2 cos 3t y(0) = 1, y(0) = 0. Sketch the solution. y (o): I y' (0) = 0 9h [Asin 36 + B cos 36 ] 9(0)=1 y'(o)= 0 M₁, M₂ = -0 ± 5-4 (1) (9) = ± 31 Up: Fox) (053€ Yps ² [Cc0336 + Dsin 36] T₂0 yp: (cos 3e + Dsin 36 X. fal Yp= C6 Cos 36+ De sin 3e y'p -3 (936 + ( cos 36 + 3 DE cos 3e + D sim 3 = [3D cas 36 -3c Sim 30 ] + [<cos 36+ Dsin 36] y'p= [-9D sin 36-gc cos 3e] € + [30 cos 3€ + 30 cos 36 - 3c sin36-30 5236] = [-19D sin 36-9cc0336 ] 6+ [ 6D cos 36 - 60 cos 36 -6 (Sin36] = 2c (0) (05 3 (0) + 2 0(0) Sin 3 (0) = (+0. (=O [-90 sin 3E-9ccos3€] + [60 cos 3E-6c sin3t ] + 9 [(cos 3£ + Dsin 3€] == 2005 36 Compare Cos 36 Sin 36 6D=2 -60=0 0= # C=0# :: y = A sin3t + Beos 3t + = + Sin 36 #
we have given y''+9y=2cos (3t),y(0)=1, y' (0)=0 the characteristic equation for homogeneous is r^2+9=0 implies r=31, -31 homogeneous solution is yh=Cicos (3t) +C2 sin3t for particular solution we guess yp-Atcos (3t) +Btsin (36) y'p=A[-3t+sin (3t) +cos (3t)] +B[3tcos (3t)+sin(3t)] = [A+3Bt] cos (3t) +[-3At+B) sin(3t) y' 'p=A[-3+ [3t+cos (3t) +sin (3t) 1-3sin (3t) 11+B [3* [-3tsin (3t) +cos (3t) 1+3 cos (3t) ] y' 'p=[-9At+6B] cos (3t) +[-6A-9Bt) sin (30) substitute above y, y'p, y''p into given differential equation y''+9y=2cos (30) [-9At+6B] cos (3t) + [-6A-9Bt] sin (3t) +9Atcos (3t) +9Btsin (3t) =2cos (3t) 6Bcos (3t) -6Asin (3t)=2cos (3t) plug the A=0, B-2/6=1/3 plug the A=0 and B=1/3 into yp yp=1/3-tsin (3t) = (tsin (3t))/3 y-yh+yp=C1cos (3t) +C2sin3t+ (tsin (3t))/3 y'=C1 (-3) sin (3t) +C23* cos3t+ (1/3)*(3tcos (3t)+sin (30)) plug the initial conditions into above equation y (0)=1 1-C1cos (3+0) +C2sin (3+0) + (tsin (3-0))/3 C1=1 y' (0)=0 0=C1 (-3) sin (0) +C23 cos (0) + (1/3) + (0 cos (0)+sin (0)) 3C2=0, C2-0 substitute C1=1, C2=0 into y y=cos (3t) + (tsin (3t))/3