Example: Spring Mass System Classic example: spring mass system consider simple case first d ki= 1 M-1K0 M= diag(mi) k1+k2+k5-k5 k 2 k5 k3+k4+hs -k3 k k2+k3 Eigenvalues and eigenvectors of the undamped system 入1=±0.77i入2=±185iA3=2.00i U 1.00 1.00 1.00 1.41 1.41 0.00 ±0.77i 士1.85i ±2.00i ±0.77i ±1.85i 千2.00i 士1.08i 干2.61i� � Example: Spring Mass System • Classic example: spring mass system consider simple case first: mi = 1, and ki = 1 M1 k1 k2 k3 k4 k5 M3 M2 Z1 Z3 Z2 x = z1 z2 z3 z˙1 z˙2 z˙3 ⎡ ⎤ ⎢ 0 I ⎣ ⎥ A = ⎦ M = diag (mi) −M−1K 0 k1 + k2 + k5 ⎡ ⎤ ⎢ −k5 −k2 ⎢ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ K = −k5 k3 + k4 + k5 −k3 −k2 −k3 k2 + k3 • Eigenvalues and eigenvectors of the undamped system λ1 = ±0.77i λ2 = ±1.85i λ3 = ±2.00i v1 v2 v3 1.00 1.00 1.00 1.00 1.00 −1.00 1.41 −1.41 0.00 ±0.77i ±1.85i ±2.00i ±0.77i ±1.85i �2.00i ±1.08i �2.61i 0.00 1