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Modeling Dynamic Systems

Modeling Dynamic Systems. Basic Quantities From Earthquake Records Fourier Transform, Frequency Domain Single Degree of Freedom Systems (SDOF) Elastic Response Spectra Multi-Degree of Freedom Systems, (MDOF) Modal Analysis Dynamic Analysis by Modal Methods Method of Complex Response.

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Modeling Dynamic Systems

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  1. Modeling Dynamic Systems • Basic Quantities From Earthquake Records • Fourier Transform, Frequency Domain • Single Degree of Freedom Systems (SDOF) Elastic Response Spectra • Multi-Degree of Freedom Systems, (MDOF) Modal Analysis • Dynamic Analysis by Modal Methods • Method of Complex Response

  2. Earthquake Records

  3. Numerical Concept

  4. Acceleration vs. Time

  5. Acceleration vs Time t=16 to 20 sec

  6. Harmonic Motion

  7. Fourier Transform

  8. Fourier Transform; El Centro

  9. xt x m m x c c k/2 k/2 k/2 k/2 xg (a) (b) Earthquake Elastic Response Spectra

  10. Duhamel's Integral t p(τ)

  11. Elastic Response Spectrum

  12. x3 m3 c3 k3/2 k3 /2 x2 m2 c2 k2/2 k2/2 x1 m1 y3 y2 c1 y4 k1/2 k1/2 y1 y5 θ3 θ2 θ5 θ4 θ1 (a) (b) Multi-Degree of Freedom

  13. Modal Analysis

  14. Modal Damping

  15. FEM Frequency Domain

  16. Finite Elements u1 u7 G1,ρ1,ν1 u2 u8

  17. Method of Complex Response • Given earthquake acceleration vs. time, ü(t) • FFT => ω1, ω 2 , ω 3...ωn ; {p}1 ,{p}2 ,{p}3,{p}n • Recall that • Solve • FFT-1 => ü (t)

  18. 212,428 nodes, 189,078 brick elements and 1500 shell elements Circular boundary to reduce reflections

  19. Finite Element Model of Three-Bent Bridge

  20. Zoom 1 Zoom 2

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