The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems

Transcription

1 Swss Federal Insue of Page 1 The Fne Elemen Mehod for he Analyss of Non-Lnear and Dynamc Sysems Prof. Dr. Mchael Havbro Faber Dr. Nebojsa Mojslovc Swss Federal Insue of ETH Zurch, Swzerland Mehod of Fne Elemens II

2 Swss Federal Insue of Conens of Today's Lecure Page Soluon of Equlbrum Equaons n Dynamc Analyss Mode Superposon Modal Generalzed Dsplacemens Analyss wh Dampng Negleced Analyss wh Dampng Included Mehod of Fne Elemens II

3 Swss Federal Insue of Mode Superposon Page 3 Modal Generalzed Dsplacemens The drec negraon mehods necessae ha he fne elemen equaons are evaluaed for each me sep The bandwdh of he marxes M, C and K depend on he numberng of he fne elemen nodal pons In prncple we could ry o rearrange he nodal pon numberng bu hs approach s cumbersome and has lmaons Insead we ransform he equaons no a form whch h n erms of numercal effor s less expensve - by a change of bass Mehod of Fne Elemens II

4 Swss Federal Insue of Mode Superposon Page 4 Change of Bass o Modal Coordnaes The followng ransformaon s nroduced: U() = PX() P: n x n square marx X(): me dependen vecor of order n MX () + CX () + KX () = R () T T T T M = P MP, C = P CP, K = P KP, R ( ) = P R Mehod of Fne Elemens II

5 Swss Federal Insue of Mode Superposon MX () + CX () + KX () = R () M = P T MP, C = P T CP, K = P T KP, R ( ) = P T R Page 5 Change of Bass o Modal Coordnaes The queson s how o choose P? A good choce s o ake bass n he free vbraon soluon neglecng g dampng,.e.: MU + KU = 0 U = Φsn ω( ) whch has a soluon of he form 0 KΦ = ω MΦ ( ω Φ ),( ω Φ ),...,( ω n Φ n ) 1 1 Mehod of Fne Elemens II Mode shape vecors

6 Swss Federal Insue of Mode Superposon Page 6 Change of Bass o Modal Coordnaes Any of he soluons sasfy MU + KU = 0 ( ),( ),...,(,( ) ω 1Φ 1 ω Φ ω nφ n The n soluons may be wren as: KΦ MΦΩ, Φ T T = KΦ= Ω ; Φ MΦ = I ω 1 ω Φ= Φ 1, Φ,..., Φ n ; Ω = ω n wh: [ ] Mehod of Fne Elemens II

7 Swss Federal Insue of Mode Superposon Page 7 Change of Bass o Modal Coordnaes Now usng U () = ΦX () n MX () + CX () + KX () = R () M = P T MP, C = P T CP, K = P T KP, R ( ) = P T R we ge X Φ CΦX Ω X Φ R T T () + () + () = () wh X = Φ M U ; X = Φ M U 0 T 0 0 T 0 Mehod of Fne Elemens II

8 Swss Federal Insue of Mode Superposon Page 8 Analyss wh dampng negleced Here we sar wh: X Ω X Φ R () + () = T ().e.: x () + x () = r () ω T r () = Φ R() Can be solved usng he drec negraon schemes or he Duhamel negral 1 x () = r( τ )sn ω ( τ) dτ + α snω+ β cosω ω 0 and we conver o he dsplacemens hrough: h n U() = Φ x () = 1 Mehod of Fne Elemens II

9 Swss Federal Insue of Mode Superposon Page 9 Analyss wh dampng negleced Comparng mode superposon wh drec negraon we have so far only changed he bass before negrang The soluons mus hus be he same! Wheher o use mode superposon or drec negraon s hus only a maer of effcency! However, dependng on he dsrbuon and frequency conens of he loadng he mode superposon mehod can be much more effcen ha drec negraon Mehod of Fne Elemens II

10 Swss Federal Insue of Mode Superposon Page 10 Analyss wh dampng negleced Consderng agan x x r r () () (), () Φ T + ω = = R() and seng r ( ) = 0, = 1,,..., n and eher 0 0 U or U are a mulple of only Φ j hen only xj () 0 Mehod of Fne Elemens II

11 Swss Federal Insue of Mode Superposon Page 11 Analyss wh dampng negleced f nsead we se 0 U = 0 U = 0 and R() = MΦ j f() hen only x () 0 j Example 9.8 shows ha for a one degree of freedom sysem x() + ω x () = Rsnω here s: ˆ 3 R / ω 1 R ˆ ω / ω x() = sn ˆ ω+ ( )snω 1 ˆ ω / ω ω 1 ˆ ω / ω = Dx + x sa rans Mehod of Fne Elemens II

12 Swss Federal Insue of Mode Superposon ˆ Page 1 x() + ω x () = Rsnω 3 R / ω 1 R ˆ ω / ω x() = sn ˆ ω+ ( )snω 1 ˆ ω / ω ω 1 ˆ ω / ω = Dx + x sa rans Analyss wh dampng negleced T Φ CΦ = ωζδ j ζ = modal dampng parameers δ = Kronecker dela δ = 1, for = j else δ = 1 j j j Proporonal dampng Mehod of Fne Elemens II

13 Swss Federal Insue of Mode Superposon Page 13 Analyss wh dampng negleced Consderng agan x () + ω x() = r() T r () = Φ R() 1 x() = r( τ )sn ω ( τ) dτ + αsnω+ βcosω ω n U() = Φx() = 1 0 we realze ha an approxmaon may be nroduced by only consderng some of he mode shapes Mehod of Fne Elemens II

14 Swss Federal Insue of Mode Superposon Page 14 Analyss wh dampng negleced Consderng agan x () + ω x() = r() T r () = Φ R() 1 x() = r( τ )sn ω ( τ) dτ + αsnω+ βcosω ω U p p 0 () = Φx() = 1 we realze ha an approxmaon may be nroduced by only consderng some (p) of he mode shapes Mehod of Fne Elemens II

15 Swss Federal Insue of Mode Superposon Page 15 Analyss wh dampng negleced The error made may hen be esmaed as: p ε () = p p R() MU () + KU () R () Ths s a measure of he degree o whch nodal pon loads are balanced by nera and elasc nodal pon forces. Mehod of Fne Elemens II

16 Swss Federal Insue of Mode Superposon Page 16 Analyss wh dampng negleced The unbalance n nodal forces s: Δ R = R r( MΦ) p = 1 If he problem s modeled appropraely he unbalance should a mos amoun o he sac response Therefore we can calculae a sac correcon K Δ U () = Δ R () for mes of specal neres (e.g. wh max response) Mehod of Fne Elemens II

17 Swss Federal Insue of Mode Superposon Page 17 Analyss wh dampng ncluded d In case of proporonal dampng e.g.: T Φ CΦ = ωζ δ j ζ = modal dampng parameers δ = Kronecker dela δ = 1, for = j else δ = 1 j j j The soluon may be found as: x () () () (), () Φ T + ωζ x + ω x = r r = R() 1 ζω ( τ) ζω x () = r( τ ) e sn ω ( τ) dτ + e ( α snω+ β cos ω) ω 0 ω = ω 1 ζ Mehod of Fne Elemens II

18 Swss Federal Insue of Mode Superposon Page 18 Analyss wh dampng ncluded d Assume ha we know he dampng raos for a number of modes.e.: ζ, = 1,,.. r If r = Raylegh dampng can be used n he form: C= α M + β K Examples llusrae how o use hs approach. Mehod of Fne Elemens II