Introduction to Compact Dynamical Modeling. I.2.a Conservations and Constitute Laws. Luca Daniel Massachusetts Institute of Technology

Transcription

1 NSF & NIH Introducton to ompact Dnamcal Modelng I.2.a onservatons and onsttute Laws Luca Danel Massachusetts Insttute of Technolog L- Four representatve examples Power dstrbuton networks (e.g. for chp or whole countr) Network of struts and jonts n space frame mechancal structures, or buldngs or brdges lamp Ol, natural gas or blood transport networks T Heat transport networks (e.g. heat conductng bars) T end L2-2

2 ourse Outlne Quck Sneak Prevew I. ssemblng Models from Phscal Problems I.. Sample problems I.2. ssemblng models from networks of dnamcal sstems I.2.a. onservaton & onsttute Laws I.2.b. Node-branch & nodal analss I.3. ssemblng models from PDE Solvers II. Smulatng Models III. Model Order Reducton for Lnear Sstems IV. Model Order Reducton for Non-Lnear Sstems V. Parameterzed Model Order Reducton L2-3 L-3 Modelng Electrcal Power Dstrbuton Networks v ache PU Decoder Power suppl provde current at a certan voltage. Functonal blocks draw current. The wre resstance generates losses. L2-4

3 Modelng Electrcal Power Dstrbuton Networks The Power Suppl becomes: a Voltage Source Power suppl current + Voltage I V Vs V Vs onsttutve Equaton Phscal Smbol rcut Element L2-5 Modelng Electrcal Power Dstrbuton Networks The Functonal locks become: urrent Sources PU I V + - I s I Is Phscal Smbol rcut Element onsttutve Equaton L2-6

4 Modelng Electrcal Power Dstrbuton Networks The Metal Lnes become: Resstors I V + - Phscal Smbol rcut model onsttutve Equaton (Ohm s Law) Length R resstvt rea Desgn Parameters I Materal Propert R V L2-7 Modelng Electrcal Power Dstrbuton Networks Puttng t all together + - I ILU ID ache LU Decoder Power Suppl voltage source Functonal locks current sources Wres become resstors Result s a schematc L2-8

5 Formulatng Equatons from Schematcs Step : Identfng Unknowns R R 2 s s s R D R 4 3 ssgn each node a voltage, wth one node as (reference) L2-9 Formulatng Equatons from Schematcs Step : Identfng Unknowns E R R E R 2 s s s R D D 4 R ssgn each element a current 3 L2-

6 Formulatng Equatons from Schematcs Step 2: onservaton Laws E R R E R s + s E 2 s + s s s R D D 4 D s s R 3 s Sum of currents (current conservaton law) L2- Formulatng Equatons from Schematcs Step 2: onsttutve Equatons E R E R V R D R s E D V E R E V R R s 2 s D R D V D 4 Use onsttutve Equatons to relate branch currents to node voltages (urrents flow from plus node to mnus node) R R V 3 L2-2

7 Motvatons Modelng of Networks onservaton Laws and onsttutve equatons Electrcal power dstrbuton network example Ol/gas/blood transport network example Heat transport network example Structural network example Node-branch Nodal analss Outlne Stead State nalss of Networks Dnamcal nalss of Networks Model Order Reducton L2-3 U.S. Natural Gas Ppelne Network (29) L2-4

8 Isolate Prmar Flow Pathwas L2-5 Modelng the cardovascular sstem E Pcture thanks to JungHoon Lee, (Merck) Is the bran gettng enough blood flow? L2-6

9 onservaton Laws and onsttutve Equatons lood Flow onsttutve Relaton For now let s smplf: no wall deformatons, no convectve term (lnear), no nerta of the flud (no turbulence), bascall just the pressure term: P lood Flow through one short arter secton Δx P + q + +, bloodflow κ P P Δx P q +, q +, P + R Δx κ L2-7 onservaton Laws and onsttutve Equatons lood Flow onservaton Law P Net Flow out of ontrol Volume q, + q, + q+ 2, q, P q q, + + 2, P + control volume P +2 L2-8

10 onservaton Laws and onsttutve Equatons lood Flow onservaton Law Pressure analogous to Voltage lood Flow analogous to urrent P R, P R, +2 R, + P + P +2 L2-9 Motvatons Modelng of Networks onservaton Laws and onsttutve equatons Electrcal power dstrbuton network example Ol/gas/blood transport network example Heat transport network example Structural network example Node-branch Nodal analss Outlne Stead State nalss of Networks Dnamcal nalss of Networks Model Order Reducton L2-2

11 Demonstraton Example Heat onductng ar lamp power Lamp ( ) u t T T end L2-2 onservaton Laws and onsttutve Equatons Heat Flow: -D Example Incomng Heat T () Near End Temperature Unt Length Rod T () Far End Temperature Queston: What s the temperature dstrbuton along the bar T T () T () x L2-22

12 onservaton Laws and onsttutve Equatons Heat Flow: Dscrete Representaton ) ut the bar nto short sectons 2) ssgn each cut a temperature T () T () T T2 TN TN L2-23 onservaton Laws and onsttutve Equatons Heat Flow: onsttutve Relaton T Heat Flow through one secton Δx T+ T T + h+, heat flow κ Δx T h +, T + Δx R thermal κ h +, L2-24

13 onservaton Laws and onsttutve Equatons Heat Flow: onservaton Law Net Heat Flow out of ontrol Volume h ~ h Δx, h +, s control volume ~ Incomng Heat ( ) h s Heat out from left Heat n from rght Incomng heat per unt length T h, T h +, T + Δx L2-25 onservaton Laws and onsttutve Equatons Heat Flow: rcut nalog Temperature analogous to Electrcal Voltage / lood Pressure Heat Flow analogous to Electrcal urrent / lood Flow T R κ Δ x TN + - vs + ~ - T() h Δx v T() s s s L2-26

14 Motvatons Modelng of Networks onservaton Laws and onsttutve equatons Electrcal power dstrbuton network example Ol/gas/blood transport network example Heat transport network example Structural network example Node-branch Nodal analss Outlne Stead State nalss of Networks Dnamcal nalss of Networks Model Order Reducton L2-27 pplcaton Problems Oscllatons n a Space Frame What s the oscllaton ampltude? L2-28

15 pplcaton Problems Oscllatons n a Space Frame: Smplfed Structure Struts olts Ground Load Example Smplfed for Illustraton L2-29 Modelng the Frame The eams become: Struts eam Phscal Smbol Strut f f r ( x, ) r ( x, ) f ( f x, f ) L2-3

16 Modelng the Frame The beams become: struts (,) ê κ, L x f k eˆ k ( L L ) x onsttutve Equaton (Hooke s Law) E L c L Unstretched Length ross-sectonal rea c E Young's Modulus Desgn Parameters Materal Propert L2-3 Modelng the Frame The eams become: Struts f f r ( x, ) Strut lˆ r ( x, ) f ( f x, f ) r r r L r ( L ) l ˆ k L r r l ˆ k ( L L) L2-32

17 Formulatng Equatons from Schematcs Struts Example - Step : Identfng Unknowns ( ) x, ( ) x 2, 2 D, Y X, hnged ssgn each jont an X,Y poston, wth one jont as zero (reference). L2-33 Formulatng Equatons from Schematcs Struts Example - Step : Identfng Unknowns ( f f ), x,, ( f f ), x,, ( f f ), x,, ( f D f ), x, D, D f load ssgn each strut an X and Y force component. L2-34

18 f f, x, + f + f, x, Formulatng Equatons from Schematcs Struts Example - Step 2: onservaton Laws + f + f, x, ( f f ) ( f, x, f, ), x,, ( f f ), x,, ( f D f ), x, D, f f, x, + f + f D, x D, + f + f loadx, load, D f load,, Force Equlbrum Sum of X-drected forces at a jont Sum of Y-drected forces at a jont L2-35 f f, x, Formulatng Equatons from Schematcs Struts Example - Step 3: onsttutve Equatons x k L k L ( L, L ) ( x, ) ( L L ), f f, x, x x L 2 k 2 k L x f k L L ( ), x, L f k L L ( ),, L ( L L ), ( L L ),, D ( ) x 2, 2 2 f load x f k L L ( ) 2 D, x D, D LD f k L L ( ) 2 D, D, D LD Use onsttutve Equatons to relate strut forces to jont postons. L2-36

19 omparng onservaton Laws V R V V R + s s q q, pump q +, q q q, +, pump ~ Incom ng H eat ( ) h s h ~ h Δx, h +, s T h T h T, +, + Δ x f f f L f f + f L L2-37 Summar of ke ponts Pcked Four Representatve Examples Electrcal power, ol/gas/blood, Struts and Jonts, Heat transport Two Tpes of Equatons onsttutve Equaton rcut: current-voltage relatonshp rteres: pressure drop - blood flow relatonshp Heat ar: temperature drop - heat flow relatonshp Struts: force-dsplacement relatonshp onservaton/alance Laws rcut: Sum of urrents at each node rteres: Sum of blood flows out of control volume Heat ar: Sum of heat flows out of control volume Struts: Sum of Forces at each jont L2-38

20 Group ctvt 2 Form groups of 2 or 3: mxed background! Each student wll choose one case stud: Identf the nodal quanttes and edge quanttes (alread done n actvt ) Identf conservaton laws (or condtons that specf how nodes are connected) Identf consttutve equatons (or condtons that specf the behavor of the edges) Each student wll then present hs conclusons about hs/her case stud, whle the others wll help hm/her b askng questons and provdng suggestons or constructve crtcsm L2-39