MECH370: Modelling, Simulation and Analysis of Physical Systems
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1 MECH370: Modelling, Siultion nd Anlysis o Physicl Systes Chpter Modeling o Trnsltionl Mechnicl Syste Eleents nd eleent lws o trnsltionl echnicl systes Free body digr (FD Interconnection lws Obtining the syste odel Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes, Youin Zhng (CU
2 Modelling (Ch.,3,4,5,6,9,0,, Siultion (4 Anlysis (7,8 Course Outline. Deinition nd clssiiction o dynic systes (chpter. Trnsltionl echnicl systes (chpter 3. Stndrd ors or syste odels (chpter 3 4. loc digrs nd coputer siultion with Mtlb/Siulin (chpter 4 5. Rottionl echnicl systes (chpter 5 6. Electricl systes (chpter 6 7. Anlysis nd solution techniques or liner systes (chpters 7 nd 8 8. Developing liner odel (chpter 9 9. Electroechnicl systes (chpter 0 0.Therl nd luid systes (chpters, Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes
3 Why Mtheticl Models re Needed? Review Anlogous Systes Cn hve the se theticl odel or dierent types o physicl systes Coon nlysis ethods nd tools cn be used Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 3
4 Procedure o Syste Modeling Review Divide the syste into idelized coponents Apply physicl lws to the eleents Apply interconnection lws between eleents Cobine the equtions to obtin the odel Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 4
5 Overview o Eleent Models in Physicl Systes Mechnicl Trnsltionl Models M orce/velocity orce/position Mss M dv/dt M d /dt Dper (Viscous riction Spring (Stiness v v dt d/dt Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 5
6 Overview o Eleent Models in Physicl Systes Mechnicl Rottionl Models J torque/velocity torque/position T, θ Inerti T J dω/dt T J dθ /dt T, θ Viscous riction T ω T dθ/dt s Stiness T s ω dt T s θ T, θ Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 6
7 Overview o Eleent Models in Physicl Systes Electricl Coponent Models i + v _ i + v _ voltge/current voltge/chrge Inductnce v L di/dt v L dq /dt Resistnce v R i v R dq/dt i + v _ Cpcitnce v /C i dt v /C q Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 7
8 Overview o Eleent Models in Physicl Systes Trnsortion Models i i v N N v, Trnsorer v N i N v N i N, L L Lever L L L L T, θ N Gers T N θ N T N θ N T, θ N Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 8
9 Mtheticl Modelling o Mechnicl Systes Eleentry prts A ens or storing inetic energy (ss or inerti A ens or storing potentil energy (spring or elsticity A ens by which energy is grdully dissipted (dper Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 9
10 Mtheticl Modelling o Mechnicl Systes Motion in echnicl systes cn be Trnsltionl Rottionl, or Cobintion o bove Mechnicl systes cn be o two types Trnsltionl systes Rottionl systes Vribles tht describe otion Displceent, Velocity, v Accelertion, Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 0
11 Modeling o trnsltionl echnicl systes : displceent ( v: velocity (/sec Vribles : ccelertion (/sec : orce (N P: power (N/sec, Wtt W: wor (energy (N, J v & Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes d dt dv d d ( dt dt dt P v & W( t & dw dt All these vribles re unctions o tie, t. Power is deined to be the rte t which energy is supplied or dissipted. W( t 0 + t t 0 P( t dt M
12 Modeling o trnsltionl echnicl systes Vribles (cont d The energy supplied between tie t o nd t is t t 0 P( t dt The totl energy supplied ro tie t 0 up to ny lter tie is W( t W( t 0 + t t 0 P( t dt N (Newton-Meter J (Joule Wtt Joule per Second Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes
13 Modeling o trnsltionl echnicl systes Eleents Three priry eleents o interest Mss (inerti (or M Stiness (spring Friction - Dissiption (dper Usully we del with equivlent,, Distributed ss -> luped ss Luped preters Mss intins otion Stiness restores otion Dping eliintes otion Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 3
14 Eleents Mss Newton s second lw: su o orces cting on body tie rte o chnge o oentu v d d dv ( v, i 0 dt dt dt Assuptions:. otions deined with respect to inertil reerence re. sclr quntities ( degree o reedo g Energy v Kinetic EK Potentil E P gh Initil conditions v 0, h 0 Grvity Norl orce N Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 4
15 Eleents (cont d Stiness (N/ Stiness is the resistnce o n elstic body to delection or deortion by n pplied orce Most coon: idel spring d 0 + d 0 elongtion cused by d ( t length o spring when no orce is pplied d 0 + ( t ( t : stiness constnt d ( t Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 5 d 0
16 Eleents (cont d The liner spring is n pproition o soething lie Liner Non-liner Multiple pplied orces: ( Δ Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 6
17 Eleents (cont d Dping (N-s/ Also nown s viscous riction or liner riction. Friction is the orce tht opposes the reltive otion or tendency o such otion o two surces in contct Δv, where Δv v -v nd viscosity constnt/coeicient v v v Above let: is proportionl to contct re nd viscosity o oil. is inversely proportionl to the thicness o il. Above right: is sll enough to be neglected (this is lwys n pproition. Dping is used to odel dshpot (dper, e.g. shoc bsorbers on crs. Dping is used to odel dshpot (dper e.g. shoc bsorbers on crs. Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 7 v
18 Eleent Lws Mss: Newton s nd Lw Stiness (Spring: d dt ( dv dt i i Friction (Dping: ( v v Δv : viscosity constnt, unit: N-s/ Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 8 v v v v v No riction Idelized shoc bsorber (dshpot
19 Eleent Lws (cont d Viscous riction: Δv Δv Coulob (dry riction: Asign( Δv A -A Δv Drg: CΔ Δv Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 9
20 Interconnection Lws Deterine how to connect eleents D Alebert s lw Just re-stteent o Newton s nd lw, suing eternlly pplied orces to ss ( or ( 0 i et i I you thin o s n dditionl orce I (the inertil orce, or D Alebert s orce, you cn then consider it long with ll other orces nd write i i 0 Lw o rection orces (Newton s 3 rd lw All orces occur in equl nd opposite pirs (ction/rection et i Force eerted by n eleent is equl nd opposite to the orce on the eleent. Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 0
21 Lw o Displceents I the ends o two eleents re connected, these ends re orced to ove with the se displceent, velocity, nd ccelertion. v v v v v v Newton s nd lw t point: The su o the orces t connection between eleents equls zero. v dv n 0 P / P i i 0 dt + 0 no ss Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes y
22 Modeling Steps Understnd syste unction, deine proble, nd identiy input/output vribles. Drw sipliied schetics using bsic (idelized eleents. Develop theticl odel (dierentil equtions Identiy reerence point nd positive direction Write eleentl equtions s well s interconnecting equtions by pplying physicl lws. Drw Free-ody-Digr (FD or ech bsic eleent. Cobine equtions by eliinting interedite vribles. Vlidte odel by copring siultion results with physicl esureents. Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes
23 Obtining the Syste Model Eple: ss-spring-dper syste, v Free ody Digr (FD: Force eerted by the spring Newton s nd Lw: dv ( t ( t v dt dv d + v + ( t, v dt dt d d + + ( t dt dt && + & + ( t Force eerted by the dshpot (t I (t the pplied orce the initil orce y Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 3
24 Obtining the Syste Model (cont d D Alebert s Lw (use the ide o n inertil orce, I (t i 0 ( t && + & + ( t & & 0 I ( t y I Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 4
25 Obtining the Syste Model (cont d Energy distribution: EOM o the bove siple ss-spring-dper syste && Contibutin o Inerti + & o Contribtin the Dper + Contribuion o the Spring Totl Applied Force We now wnt to loo t the energy distribution o the syste. How should we do it? Multiply the bove eqution by the velocity v: (since P is deined s Pv & & + & & + & ( t 443 & Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 5 Power Integrte the nd eqution w.r.t. (with respect to tie: t && dt & + t ΔE chnge o enetic energy t & dt & t t & t dt 0 0 Energy dissipted by dper t + dt & t ΔE P Chnge o potentil energy t ( t & ( t dt t W Totl wor done by the pplied orce (t ro tie t to 0 t
26 Obtining the Syste Model (cont d Eple: (t,v (t Drw the FDs nd write the equtions o the syste in ters o nd. Free body digr (FD: ( & & ( - ( & & ( - (t Newton s nd Lw gives: && && ( & & + ( t ( & ( & ( or && ( & & ( 0 && + ( & & + ( ( t Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 6
27 Obtining the Syste Model (cont d How bout deriving the equtions using D Alebert s Lw? Free body digr (FD: ( & & ( - ( & & ( - (t ( & & ( & & (t & ( - ( - & ( & & + ( t ( & ( & Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 7 ( && 0 & 0 or && ( & & ( 0 && + ( & & + ( ( t
28 Obtining the Syste Model (cont d Eple: Displceent ecittion (t Drw the FD nd write the eqution o the syste. Free body digr (FD: Newton s nd Lw gives: & ( & ( t & ( (t- && & & & ( ( t + ( ( t Do not need n eqution or since it is deined unction (t! Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 8
29 Obtining the Syste Model (cont d Eple: Reltive displceent z Free body digr (FD: Newton s nd Lw gives: && & z && z& (&& + && z ( Use bsolute ccelertion ( +z Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 9 & z & && + z z& & && + z + z& + && z
30 Assuption or idel pulley: Idel Pulley Eleent T T No ss, no riction, no slippge between cble nd cylinder. Cble is lwys in tension. Cble cnnot stretch. I the pulley is not idel, its ss nd ny rictionl eects ust be considered. Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 30
31 Idel Pulley Eleent (cont d Eple: Find the eqution describing the syste with pulley. Free body digr (FD: Equtions: (t ( - & M: && && M : && && ( t + ( + & + ( ( + & & (t + ( - g + soething new here! g + & ( t & ( g T Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 3
32 Idel Pulley Eleent (cont d Copre the bove syste including n idel pulley with ollowing syste: Horizontl y Free body digr (FD: (t ( - & ( - & Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 3
33 Lecture Notes on Lecture Notes on MECH 370 MECH 370 Modelling Modelling, Siultion nd Anlysis o Physicl Systes, Siultion nd Anlysis o Physicl Systes Chpter Chpter 33 Idel Pulley Eleent (cont Idel Pulley Eleent (cont d d ( ( ( t ( v v ( t ( ( ( t M : or M: & && & && & & && & && p.3 (8, ( ( t ( v v v v & && & & Equtions or the syste: Finlly, (t (t ( t && ( t ( t && && y ( t ( + + : M & && ( g : M & &&
34 Prllel Cobintions: Prllel Cobintions Newton s nd lw: && where, Generl cse: FD: eq + or, K & -( eq + - equivlent spring stiness eq y eq + Siilrly, eq + Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 34
35 Prllel Cobintions (cont d For spring in prllel, or ( ( + ( 3 K eq + ( For dpers in prllel, or ( & & ( + ( 3 eq + & & ( & & Eleents re in prllel i the irst end o ech is connected to the se body nd the reining ends re connected to coon body. Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 35
36 Eple: Equivlent syste: Equtions: Prllel Cobintions (cont d && eq eq where, then, && + ( t ( ( t & eq eq 3 eq Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 36 FD: & + + eq + eq & + + 3, ( t 3 & & ( & & + 3 eq + y & (t & 3
37 Series Cobintions Eple: (t A Drw the FDs nd write n eqution only in ters o. FD: (t ( - ( - (0- y & A Newton's nd lw or Newton's nd lw or : && ( & A: ( + 0 ( t ( ( Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 37
38 Series Cobintions (cont d Substitute ( into ( to get: && && ( where, eq + eq ( & + ( t & ( t Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 38
39 Sury Understnd the syste nd identiy the eleents nd vribles. Divide syste into idelized eleents: ss, stiness, riction, pulley. Find the eleent lws: Use the interconnection lws: && i, Δ, Δv Newton s 3 rd lw Lw o displceent Newton s nd lw i i 0 Apply Newton s nd lw to sses, nodes with unnown v nd use eleent, interconnection lws to deterine orces in ters o, v. Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 39
40 Reding Reding nd Eercise Chpter Assignent # E:.,.,.6,., 3.3, 3.8 (to be hnded in or ring Due: Fri., 6/7/07 t lecture E:.0,.7, 3., 3.5, 3.8 (or your prctice, no need to hnd in Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 40
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