Solution of Equilibrium Equation in Dynamic Analysis. Mode Superposition. Dominik Hauswirth Method of Finite Elements II Page 1

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1 Soluton of Equlbrum Equaton n Dynamc Analyss Mode Superposton Domnk Hauswrth..7 Method of Fnte Elements II Page

2 Contents. Mode Superposton: Idea and Equatons. Example Modes 4. Include Dampng 5. Response contrbutons Method of Fnte Elements II Page

3 Mode Superposton Dynamc Equlbrum Equaton M U&& + CU& + KU = R Solvng methods: Central dfference Houbolt Wlson θ Newmark Method of Fnte Elements II Page 3

4 Mode Superposton Effectvty of ths methods Implcte methods (Houbolt, Wlson, Newmark).) Intal calculatons (LDL ) -> Number of operatons: o = n m k.) For each tme step (Mult.) -> Number of operatons o = n m k Where: n : matrx sze, m k : half bandwdth Number of operaton s growng wth bandwdth! Method of Fnte Elements II Page 4

5 Method of Fnte Elements II Page 5 Mode Superposton Reducton of bandwdth.) Optmze mesh topology -> Lmted effect.) ransformaton = k K = k K

6 Mode Superposton Modal generalzed dsplacements ransformaton wth unknown P, such that m k smaller U () t = P X () t Dynamc Equaton n modal dsplacements MU&& + CU& + KU = R MPX&& + CPX& + KPX = R P MPX&& + P CP X& P KP X ~ M ~ C ~ K = P{ R In theory many dfferent transformatons possble n practce only one transformaton matrx establshed! ~ R Method of Fnte Elements II Page 6

7 ransformaton matrx D-Elements; lnear Free-vbraton equlbrum solutons wth dampng neglected: MU&& + U KU = φ sn = ( ω ( t t ) Postulated soluton φ ω By nsertng U we obtan a egenproblem: Kφ Mω φ = wth n solutons for egenvector φ egenvalue ω Normalzaton: = = j φ Mφ j ω ω... ωn j Method of Fnte Elements II Page 7

8 ransformaton matrx D-Elements; lnear Defntons Φ = [ n φ, φ,..., φ ] Egenproblem for n equatons Ω ω = ω... ω n KΦ MΦΩ = Snce the egenvectors are M - orthonormal Φ MΦ = I Φ KΦ = Ω M and K dagonalzed wth RANSFORMAION MARIX Φ Φ = [ φ, φ,..., φn] U ( t) = Φ X ( t) Method of Fnte Elements II Page 8

9 D-Elements; lnear ransformed equaton, Dampng neglected Matrx equaton MU&& + KU = Φ MΦ X&& 3 + Φ3 KΦX I X&& + Ω X = Φ Ω R = Φ Sngle decoupled equaton && x r x () t + x() t = r ( t) () t = φ R() t ω =,,..., n ransformng the ntal condtons ( t = ) = φ M U R Method of Fnte Elements II Page 9

10 Method of Fnte Elements II Page D-Elements; lnear Soluton of decoupled equaton Duhamel ntegral () ( ) ( ) ( ) ( ) ( ) t t d t r t x t + + = ω β ω α τ τ ω τ ω cos sn sn ransformng to real dsplacement base () () = = n t x t U φ

11 Example Example 9.7; p. 789 Method of Fnte Elements II Page

12 Example Exact soluton wth mode superposton me hstory curve of exact soluton dsplacement 3 u_exakt u_exakt tme [s] Method of Fnte Elements II Page

13 Example Newmark method wo possbltes, leadng to the same result.) Integrate M U&& + KU = R straght forward -> SLOW.) ransform M U& + KU = R nto modal dsplacements X&& + Ω X = Φ R, ntegrate the decoupled equatons, transforme back -> FAS Method of Fnte Elements II Page 3

14 Example Newmark n MALAB t =.8 δ =.5 α =.5 Orgnally proposed as an uncondtonally stable scheme by Newmark Method of Fnte Elements II Page 4

15 Example Newmark method exact soluton vs. Newmark dsplacement 3 u_newmark u_newmark u_exakt u_exakt tme [s] Method of Fnte Elements II Page 5

16 Example Benchmark me hstory curve dsplacement.5 Newmark exakt tme [s] Method of Fnte Elements II Page 6

17 3 Modes Number of modes n calculaton For n lumped masses n a system, n modes were found - but for a good approxmaton often only a few are needed! Choosng the rght modes for calculaton In general the lowest frequences and modes are approxmated n the best way -> upper bounds for frequences are found How many and whch modes are taken for calculaton depends on the problem: Earthquake: In some cases only the lowest modes Shock: Many modes necessary, p > /3 n Method of Fnte Elements II Page 7

18 3 Modes Modes n Example 9.7 modes 4 3 dsplacement st mode u st mode u nd mode u nd mode u - - tme [s] Method of Fnte Elements II Page 8

19 3 Modes Only frst mode n Example 9.7 for dsplacement u u: frst mode - exact soluton 4 3 dsplacement exakt_u st mode u - - tme [s] Method of Fnte Elements II Page 9

20 3 Modes Error Measurement he accuracy of a soluton p < n can be measured ε p () t = R [ ] p p () t MU&& ( t) + KU ( t) R() t..and made better by the so called statc correcton R = R K U p = r ( Mφ ) () t = R() t Mode Superposton has more advantages then only the reducton of number of necessary operatons! Method of Fnte Elements II Page

21 4 Include dampng Include dampng Modal transformaton was derved wthout dampng ransformaton Matrx Φ dagonalzes M and K. but not a free chosen dampng Matrx C. In ths case the equatons stay coupled and mode Superposton sn t possble Raylegh dampng If C s a lnear combnaton of M and K decouplng s possble, ths s called Raylegh dampng C = α M + β K In ths case, there are only two free parameter for fttng the dampng rate Method of Fnte Elements II Page

22 4 Include dampng Decoupled equatons n case of Raylegh daampng && x ζ ( t) + ω ζ x& ( t) + ω x ( t) = α + βω ω If an accurate modellng of dampng s necessary: Drect ntegraton Caughey seres = r ( t) Method of Fnte Elements II Page

23 5 Response contrbutons Response contrbutons Solvng a dynamc loaded, damped system, two response contrbutons can be observed: ransent, damped out soluton part A permanent dynamc response, whch s the statc response multplcated by a dynamc load factor Method of Fnte Elements II Page 3