General Periodic Response

General Periodic Response - CE 573: Structural Dynamics...

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CE 573: Structural Dynamics General Periodic Response Theory ( ) () () 11 cos sin NN on n nn mu cu ku F t a a n t b n t ϖ == ++= ≈+ + ∑∑ ±± ± Complementary Solution: [ ] () cos s in . t cD D ut e A t B t ξω ωω =+ Particular Solution: 1 22 2 2 2 For constant term: ( ) . / 2 For sin : ( ) sin , tan . 1 12 // For cos ( ) sin cos . o p n np n n n n a ut k bk nr bn t u t n t nr n r ak an t u t n t n t n r n r π ξ ϖϖ θ ϖθ ξξ = ⎛⎞ =− = ⎜⎟ ⎝⎠ −+ Since equation of motion is linear, principle of superposition can be used. ( ) ( ) 1 2 2 1 cos sin , tan . 1 General solution is ( ) ( ) ( ). N o p n n cp t t a nr kk n r u t = ∴= + = ⇒= + Note: can get resonances when “frequency ratio” = 1; i.e., 1. n n ω =⇔ = Example lb in 0 lb 0 2 sec 15 kips; 2136 ; 0.05; ( ) ( 5) ( ). 200 lb 2 5 sec t W k Ft t ≤≤ == = = + = From last example: rad rad sec sec 1 0 0 1 4 2 1 4 1.2566 ; ( ) 120 lb + lb - sin cos cos 1 sin 55 5 N n t n n ππ = 2 5 5 t ⎧⎫ + ⎨⎬ ⎩⎭ >> weight = 15000; k = 2136; xi = 0.05; wbar = 2*pi/5 wbar = 1.2566 >> w = sqrt(k*386.4/weight) w = 7.4178
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>> r = wbar/w r = 0.1694 >> syms n t >> thetan = atan(2*xi*n*r/(1-n^2*r^2)); >> vpa(thetan,5) ans = atan(.16941e-1*n/(1.-.28699e-1*n^2)) >> a0=120;an=(200/pi)*(-1/n)*sin(4*pi*n/5);bn=(200/pi)*(1/n)*(cos(4*pi*n/5)-1);
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General Periodic Response - CE 573: Structural Dynamics...

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