《结构工程中的数学方法》课程教学课件（讲稿）intro_eigenvalue

2AlgebraicproblemsinmatrixformEigenvalueProblemsSystemstructure-introductoryexampleeguilibriumofadeformedframedeformationstructureequilibriumIP1IP1[P,α,oarXαraPP2P2KoBxWWSB点AKATByAaaIOAOB72.3v=2.PB=B·α+0.0aa2P-=Px=k.oo+0.=Ba75aMichaelBeer,EngineeringMathematics

Michael Beer, Engineering Mathematics equilibrium of a deformed frame System structure − introductory example 2 Algebraic problems in matrix form ● Eigenvalue Problems 75a B B x y = ⋅α A k x AA = ⋅δ δA δB P1 P2 α δA α,δ « a deformation P1 a a A B a a kA kB P2 structure x y equilibrium P1 P2 α Ax Bx Ay By B k x BB = ⋅δ A P y 1 = B 2P y 1 = ⋅ A BP x x2 = + ( A B ) 1 a α= ⋅ δ +δ ( ) 1 B B A B 2 P k a ⋅ ⋅δ = ⋅ δ +δ ( ) 1 AA 2 A B 2 P k P a ⋅ ⋅δ = + ⋅ δ +δ

2AlgebraicproblemsinmatrixformEigenvalueProblemsSystem structure-introductory example (cont'd)systeminmatrixform2.P2·P2.P2·P=0=Paaaacase study:influence of P2.P2 ·PPi [kN] P2 [kN]Xi [mm] X2 [mm]aa500.000.0000.00002 ·P2.P5013.680.526aa2507.371.05120010.010.0》specificparameters:220020.020.0X1=8AX2=8B,a=4.0m,0.123816.223.8ka = 300 kN/m, kg = 200 kN/m0.223832.447.62400.00infinitely manyPP30022180-1200= X, =1.5.XX6800-120=×,=1.5xPP20022det(...) = 075bMichaelBeer,EngineeringMathematics

System structure − introductory example (cont'd) 2 Algebraic problems in matrix form ● system in matrix form 1 1 A A 2 1 1 B B 2P 2P k a a P 2P 2P 0 k a a     ⋅ ⋅   − −       δ     ⋅ =     ⋅ ⋅     δ     − −       Eigenvalue Problems 1 1 B BA 2P 2P k 0 a a   ⋅ ⋅ − ⋅δ − ⋅δ =     1 1 A A B 2 2P 2P k P a a   ⋅ ⋅ − ⋅δ − ⋅δ =     » specific parameters: x1 = δA, x2 = δB, a = 4.0 m, kA = 300 kN/m, kB = 200 kN/m 1 1 1 2 1 1 2 P P 300 2 2 x P P P x 0 200 2 2       − −         ⋅ =           − −       » case study: influence of P1 P1 [kN] P2 [kN] x1 [mm] x2 [mm] 50 0.00 0.000 0.000 50 1 3.68 0.526 50 2 7.37 1.05 200 1 10.0 10.0 200 2 20.0 20.0 1 2 180 120 0 x 120 80 x 0  −     ⋅ =      −     ⇒= ⋅ x 1.5 x 2 1  ⇒= ⋅ x 1.5 x 2 1 det(.) = 0 238 0.1 16.2 23.8 238 0.2 32.4 47.6 240 0.00 infinitely many Michael Beer, Engineering Mathematics 75b