System Modeling Concepts

Transcription

1 1. E+2 1. E+1 1. E+4 1. E L i n e a r F r e q u e n y ( H z ) L i n e a r F r e q u e n y ( H z ) Strutural Dynami odeling ehniques & odal nalysis ethods ] ] n ] n a ] a 2 ω ] E a ] System odeling Conets m k 2 2 m k m m k k 1 1 Peter vitabile ehanial Engineering Deartment niversity of assahusetts Lowell System odeling Conets 1 Dr. Peter vitabile

2 System odeling Conets System models are generated from omonent models for a variety of aliations odal ethods Comonent ode Synthesis Frequeny Based Substruturing System odeling Conets 2 Dr. Peter vitabile

3 odal to odal with ie atri For omonent (modal omonent): 2 ω ] ] For omonent (modal omonent): 2 ω ] ] System odeling Conets Comonent 2] ω ] 2 ] ω ] Comonent 3 Dr. Peter vitabile

4 4 Dr. Peter vitabile System odeling Conets odal to odal with ie atri nouled Systems Comonent Comonent ] ] ] ] Ω Ω && && ] ] ω 2 ] ] ω 2

5 odal to odal with ie atri he two systems are onneted with tie matries ] ] he tie matri an be rojeted from hysial to modal sae using ] ] ] ] ] ] ] ] with ] ] ] Comonent 2] ω ] ] ] 2 ] ω ] Comonent System odeling Conets 5 Dr. Peter vitabile

6 6 Dr. Peter vitabile System odeling Conets dding this to the unouled equations gives he eigensolution of this gives the new system frequenies and mode shaes Comonent Comonent ] ] ] ] ] ] Ω Ω && && odal to odal with ie atri

7 odal to Physial with Constraint Comonent Comonent (hysial) an be artitioned into onnetion DOF () and other or interior DOF (i) i ] ] Comonent (modal) only onsiders onnetion DOF ] i ] i ] ii ] ] i ] ] ] i ii 2 ω ] ] Comonent System odeling Conets 7 Dr. Peter vitabile

8 8 Dr. Peter vitabile System odeling Conets he onstraint equation is given by and the unouled equations are given as and the relationshi of onstraint is Comonent Comonent odal to Physial with Constraint ] ] ] ] ] ] ] ] ] ] + i ii i i i ii i i && && && ] ] ] ] i i i

9 odal to Physial with Constraint Comonent Substituting and utting into normal form gives * && i * + && or ] i ] ] ] ] ] ] ] i] ] ] i ii ] + ] ] && & i ] i] ] ] i ii ] Comonent i ] System odeling Conets 9 Dr. Peter vitabile

10 odal to odal with Physial System dded Comonent and (modal omonents): Comonent ] 2 2 ] ω ω ] ] Physial ],] Comonent γ (hysial omonent) only onnetion DOF: i Comonent ] ] i ] ] ] i ii ] ] i ] ] ] i ii System odeling Conets 1 Dr. Peter vitabile

11 11 Dr. Peter vitabile System odeling Conets odal to odal with Physial System dded nouled system Constraint relation is Comonent Comonent Physial ],] ] ] ] ] ] ] + && && && ] t

12 12 Dr. Peter vitabile System odeling Conets odal to odal with Physial System dded For all DOF Comonent Comonent Physial ],] ] ] ] ] ] ] ] ] ] ] * t t z

13 13 Dr. Peter vitabile System odeling Conets odal to odal with Physial System dded his leads to where Comonent Comonent Physial ],] ] ] ] ] ] ] * * + && && ] ] ] ] ] ] ] ] ] ] ] ] ]] ] ] ]] ] ] ] ] ] ] ]] ] ]] ] ]] ] ] ]] t t t t t t t t t t t t t t t t * ** * **

14 Physial System with odal Comonents dded Comonent and (modal omonents): Comonent ] 2 2 ] ω ω ] ] Physial ],] Comonent γ (hysial omonent) only onnetion DOF: i Comonent ] ] i ] ] ] i ii ] ] i ] ] ] i ii System odeling Conets 14 Dr. Peter vitabile

15 Physial System with odal Comonents dded nouled system ] i] ] ] i ii ] ] i] ] ] + i ii ] ] Constraint relation is && && && && i i ] ] ] ] Comonent Physial ],] Comonent System odeling Conets 15 Dr. Peter vitabile

16 16 Dr. Peter vitabile System odeling Conets For all DOF Comonent Comonent Physial ],] Physial System with odal Comonents dded ] ] ] ] ] i i

17 17 Dr. Peter vitabile System odeling Conets his then beomes where Comonent Comonent Physial ],] Physial System odal Comonents dded ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] + + i i i i ii * ** ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] + + i i i i ii * ** ] ] ] ] ] ] i * i * + && && &&

18 Comonent ode Synthesis wo omonents: and For eah omonent, the equation of motion an be written in artitioned form, with junture oordinates i interior oordinates i i ii && && i + i i ii i f f i System odeling Conets 18 Dr. Peter vitabile

19 Comonent ode Synthesis We an reresent the hysial oordinates in terms of omonent generalized oordinates : Ψ where Ψ a matri of omonent modes. For the Craig-Bamton method, onstraint modes and fied-interfae normal modes are used as omonent modes. System odeling Conets 19 Dr. Peter vitabile

20 Comonent ode Synthesis Fied-nterfae normal modes m (where m refers to ket modes) ll junture oordinates are onstrained Obtain the normal modes by solving the eigenvalue roblem: 2 k ω m ( ) t is assumed that the modes are saled to unit modal mass. System odeling Conets 2 Dr. Peter vitabile

21 Comonent ode Synthesis Constraint modes Partition hysial oordinates into two sets:, the onstrained oordinates the oordinates relative to whih the onstraint modes will be defined; and i, the remaining (interior) oordinates Statially imose a unit dislaement on one onstrained oordinate, and maintain a zero dislaement on the other onstrained oordinates. he remaining i oordinates are free to deform. System odeling Conets 21 Dr. Peter vitabile

22 Comonent ode Synthesis Constraint modes he onstraint mode matri is therefore C 1 i ii i Where onstraint mode transformation System odeling Conets 22 Dr. Peter vitabile

23 Comonent ode Synthesis lying the Craig Bamton method: Let + C for eah omonent, where m m m ket fied-interfae normal modes, and C onstraint modes. System odeling Conets 23 Dr. Peter vitabile

24 Comonent ode Synthesis Sine the normal modes are fied-interfae modes, and, this an be written in artitioned form as where i Ci im m identity assoiated with unit dislaement at onnetion DOF (for onstraint modes) 1 C i Resulting onstraint modes, equal to im Normal modes of system with onstraints alied (fied-interfae normal modes) Note: n i n all oints (as usual) onstrained oints i interior oints ii i System odeling Conets 24 Dr. Peter vitabile

25 Comonent ode Synthesis n general, for eah omonent, the mass and stiffness matries are µ Ψ κ Ψ Ψ Ψ where Ψ is the matri of omonent modes. System odeling Conets 25 Dr. Peter vitabile

26 Comonent ode Synthesis sing the Craig Bamton method, the mass is given by µ µ m µ µ m µ mm where µ µ µ mm m C µ mm m i im ( C + m ) ii i ( miici + mi ) + mici + m ij System odeling Conets 26 Dr. Peter vitabile

27 Comonent ode Synthesis sing the Craig Bamton method, the stiffness is given by κ κm κ κm κmm Where κ κ κ mm m Λ κ m mm i Ω 2 1 ii i Note that κ is the Guyan redued stiffness matri System odeling Conets 27 Dr. Peter vitabile

28 Comonent ode Synthesis o assemble system matries, first write the equation of interfae dislaement omatibility (with being the deendent oordinate): Written in the form D this is ] m m therefore D D D DD D ] ] D deendent linearly indeendent System odeling Conets 28 Dr. Peter vitabile

29 29 Dr. Peter vitabile System odeling Conets Comonent ode Synthesis hen define S so, where original generalized oordinates, and q new generalized oordinates. n this ase Sq m m m m

30 3 Dr. Peter vitabile System odeling Conets Comonent ode Synthesis o get synthesized system and matries: S S S S κ µ µ µ µ µ µ µ µ µ mm m m mm m m

31 System odeling Conets hen m m m mm m mm where mm mm m m µ mm mm ( ) ( ) m m + µ µ µ m m System odeling Conets 31 Dr. Peter vitabile

32 System odeling Conets nd Where mm Λ mm mm Λ mm κ + κ mm diagonal matri, modal stiffness of diagonal matri, modal stiffness of kk full matri of stiffness terms for redued and System odeling Conets 32 Dr. Peter vitabile

33 System odeling Conets hen solve the equation of motion for the assembled system: & q + q ransform from q to oordinates using Sq and then from to u (hysial oordinates) using i C i im m System odeling Conets 33 Dr. Peter vitabile

34 medane odeling ehniques Consider a antilever beam. t is desired to estimate the FRF between oint and b when the ti of the beam is inned to ground. n artiular, the FRF h b when a b a System odeling Conets 34 Dr. Peter vitabile

35 medane odeling ehniques he resonse at "a" is related to the fore at "a" and "b" through b a h f + a ab b h aa f a where a is the vertial translation at the ti of the beam. With the onstraint a, the fore at oint "a" beomes f h a -1 aa h ab f b System odeling Conets 35 Dr. Peter vitabile

36 medane odeling ehniques he resonse at "" due to an eitation at a and "b" is h f + h f a a b b b a n order to inlude the effets of the onstraint at a, the fore at oint "a" with the onstraint a, hanges this equation to ~ b h f b h b - h whih are obtained from the unonstrained system a h -1 aa h ab System odeling Conets 36 Dr. Peter vitabile

37 Summary of medane odeling Frequeny Resonse Funtions an also be used to investigate strutural modifiations. he FRF an be written as H ij (jω) m k 1 q k u ik u (jω jk k ) + q k u * ik u (jω * jk * k ) sing fore balane and omatibility equations, the effets of a modifiation an be written in terms of the unmodified system as H F + H a F a ab H b 1 aa H ab H F + H a a aa F b F b F a b b a H ~ b F b H b H a H 1 aa H ab System odeling Conets 37 Dr. Peter vitabile

38 Frequeny Based System odeling ehniques Consider ombining two systems together SYSE (S) COPONEN () COPONEN (B) a-dofs -DOFs -DOFs b-dofs System odeling Conets 38 Dr. Peter vitabile

39 Frequeny Based System odeling ehniques he equation of motion for eah omonent is { X } n H] nn { F} n SYSE (S) COPONEN () a-dofs -DOFs COPONEN (B) -DOFs b-dofs where n a + for omonent () n b + for omonent (B). Note that the number of "" oordinates are the same on omonent () and (B) System odeling Conets 39 Dr. Peter vitabile

40 FBS odeling ehniques COPONEN () SYSE (S) COPONEN (B) a-dofs -DOFs -DOFs b-dofs Comonent an be artitioned as { X } { X } a n ] H ] aa H ] H H ] a a nn { F } a { F } n (4-8) (4-8) Comonent B an be artitioned as { B X } { B X } b n B] B] H H B] B H H ] b b bb nn { B F } { B F } b n (4-9) System odeling Conets 4 Dr. Peter vitabile

41 FBS odeling ehniques COPONEN () SYSE (S) COPONEN (B) When rigidly onneting Comonent to Comonent B, omatibility imlies that { } { B} { S X } X X and equilibrium at the "" DOFs requires that { } { B} { S F } + F F a-dofs -DOFs -DOFs b-dofs (4-1) (4-11) where "S" suersrit is used to reresent system omrised of Comonent rigidly ouled to Comonent B at the onnetion DOFs "" System odeling Conets 41 Dr. Peter vitabile

42 FBS odeling ehniques COPONEN () SYSE (S) COPONEN (B) a-dofs -DOFs -DOFs b-dofs he FRFs of the unouled system an be defined as { S X } { S X } { S X } a b n S] S] S H ] aa H a H S H ] a H H S] S] S H H H ] ba { S F } a { S} F { S F } b n From the artitioned equations for Comonent and Comonent B, the onnetion DOF are b ab b bb nn { } ] { } ] { X } H a F a + H F (4-12) (4-13) { B} B] { B} ] B] { B X } H F + H b F b (4-14) System odeling Conets 42 Dr. Peter vitabile

43 FBS odeling ehniques COPONEN () SYSE (S) COPONEN (B) a-dofs -DOFs -DOFs b-dofs hese two equations an be equated and used to solve for the onnetion fore as (4-15) { ~ } ] B] ] 1 B] { B} ] { } B] { S F H + H H F H F + H F } ] b From these equations derived above, the ouled system FRFs an be determined in terms of the unouled FRFs of the individual omonents. Equations (4-8), (4-9), (4-12) and (4-15) are used in the develoment of the ouled system. b a a System odeling Conets 43 Dr. Peter vitabile

44 FBS odeling ehniques COPONEN () SYSE (S) COPONEN (B) s an eamle, will be derived S] H aa he first equation of (4-12) of the ouled system is { S} S] { S} S] { S} S] { S X } a H aa F a + H a F + H ab F b and the first equation of (4-8) of the unouled system is a-dofs -DOFs -DOFs b-dofs (4-16) { } ] { } ] { X } a H aa F a + H a F (4-17) System odeling Conets 44 Dr. Peter vitabile

45 FBS odeling ehniques COPONEN () SYSE (S) COPONEN (B) When the systems are ouled, the fore on "" from "B" is ~ given by F from (4-15) { } and the orresonding resonse assoiated with { S} a that ouling fore is X whih then beomes { S} ] { } ] { X } a H aa F a + H a F then (4-16) and (4-18) ombine to give ~ a-dofs -DOFs -DOFs b-dofs (4-18) S] { S} S] { S} S] { S H } aa F a + H a F + H ab F ] { } ] { ~ H } aa F a + H a F b (4-19) System odeling Conets 45 Dr. Peter vitabile

46 FBS odeling ehniques COPONEN () SYSE (S) COPONEN (B) a-dofs -DOFs -DOFs b-dofs S] aa he FRF, is develoed H realizing that { S S B } F are zero F { F } { } b b Substituting (4-15) into (4-19) and simlifying, allows for the alulation of H in terms of the unouled FRF matries as S] aa S] ] ] ] B] ] 1 H ] aa H aa H a H + H H a (4-2) System odeling Conets 46 Dr. Peter vitabile

47 FBS odeling ehniques COPONEN EVLON Shematially this is shown as h S ij h ij ] B] ] 1 { H H } + H H j i FRFs desribing outut resonse oints FRFs desribing onnetion oints FRFs desribing inut fore oints i j COPONEN CONNECON PONS COPONEN B System odeling Conets 47 Dr. Peter vitabile

48 FBS odeling ehniques COPONEN EVLON dditional relations are dervied as S] ] ] B] ] 1 B H ] a H a H + H H S] ] ] B] ] 1 H ] ab H a H + H H b S] ] ] B] ] 1 B H ] H H + H H S] ] ] B] ] 1 B H ] b H H + H H b S] B] B] ] B] ] 1 B H ] bb H bb H b H + H H b (4-21) (4-22) (4-23) (4-24) (4-25) System odeling Conets 48 Dr. Peter vitabile