Lecture 10: Multi-body simulation 1 (MBS 1) Steinbachstraße 53 B Tel.: (80)

Transcription

1 Leture : Mut-bod uaton (MBS ) eron-n-harge: Dp.-Ing. Mare Fe Stenbahtraße 5 B e.: (8) 744 E-a: M.Fe@wz.rwth-aahen.de

2 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-

3 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Struture Fundaenta of the ut-bod-uaton Defnton and uaton-opton Kneat: euaton of veot- and aeeraton / Eape of a der-rank ehan Invere dna: auaton of reatve fore / Eape of a der-rank ehan Dna: dervaton of the euaton of oton / Eape of a der-rank ehan Mut-bod-uaton of ahne too Modeng of ahne eeent (Gudng te, bearng,...) Eape of a -a ng-ahne Avaabe Software-te, ope of erve erpetve Appaton n vehe engneerng Appaton n aeropae engneerng Appaton n ehana engneerng Appaton n ra vehe engneerng

4 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Struture he eture dvded nto 4 part. Frt, the fundaenta of utbod ethod w be handed. After defnng the ut-bod ethod and the ubeuent pobe opton for ana, the euaton for kneat, nvere dna and dna w be derved b ean of an eape of a pe der rank ehan. he odeng and uaton of a ahne too preented a utbod ode aordng to thee fundaenta. he freuen repone funton of a ng-ahne ode wth rgd bode and febe jont (gude, bearng, drve) deterned. he preent avaabe ut-bod uaton te and ther effen are preented n another eton. An overvew of the an reovabe probe w be provded b eape of nuerou uaton fro vehe engneerng, forua one, aeropae engneerng, ehana engneerng and ra vehe engneerng. Nuerou and detaed febe ut-bod ode of ahne too are dued n the ubeuent eture Mut-bod uaton.

5 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-4 What ut-bod-uaton? ehana te M F An aount of rgd bode that have a and nerta properte but an not defor. Contrant defne how the part are attahed and how the are aowed to ove reatve to eah other. he oton of a part an be dtated a a funton of te or a a funton of the reatve poton of evera part. he oton of the ehana te derbed b agebra, kneat euaton or b dfferenta euaton and eterna fore whereb the oton derbed b a pha aw., J he kneat and dna behavour of thee ehana te are haratered b oton wth arge aptude whh reut n geoetra non-nearte. hee non-nrarte affet the agebra euaton and the dfferenta euaton of the ehana te.

6 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-5 What ut-bod uaton?. A nuber of bode that are nked together. due to the kneat ontrant the bode ove ontent to eah another.. oton an be derbed b: varaton n te of the poton of a bod varaton n te of the reatve poton of ndvdua bode to eah another. 4 kneat ontrant an be arked b jont or dret agebra kneat ontrant Jont: Revoute jont wthdraw 5 degree of freedo ranatona jont wthdraw 5 degree of freedo Cndra jont wthdraw 4 degree of freedo Cardan jont wthdraw 4 degree of freedo Ba-and-oket jont wthdraw degree of freedo (phera jont). Dna effet between bode: Sprng-daper-atuator-eeent Contat eeent ower vetor 5. ead to oton dfferenta euaton: dfferenta agebra euaton (DAE) are derved Dfferene to FEM MBS an auate arge dpaeent and rotaton (geoetr nonnearte) FEM jut near auaton -> oton uaton jut wth MBS-progra

7 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-6 Ana opton n ut-bod uaton te Kneat-ana Dna-ana Invere dna Stat-ana he oton of the ehana te ndependent of eterna fore. he varaton n te of the aboute or reatve poton of eah part prerbed. he varaton n te of the poton, veot and aeeraton of the other part auated b the outon of the non-near te of euaton for the poton and the near te of euaton for the veot and aeeraton. Moton of the te aued b eterna fore. A pea ae the auaton of the eubru under the effet of eterna fore whh are nvarant over te. he oton of the ehana te under the effet of eterna fore ut be ontent wth the euaton of oton whh predeterned b jont. he dna-euaton are dfferenta euaton or a obnaton of dfferenta euaton and agebra euaton. Hbrd for of the dna- and kneat-ana. he varaton of te of one or evera bode of the ehana te dtated and eadng to a opete deternaton of the poton, the veot and the aeeraton of the te. Afterward the agebra euaton of oton are oved b the ue of the known poton, veote and aeeraton to deterne the fore whh aue thee oton. he tat-ana deterne the poton of the ehana te where the aeeraton of the part whh are burdened b eterna oad and gravt na. he reutng poton the poton of eubru.

8 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-7 Ana opton n ut-bod uaton te aeb ana -MBS-progra tre to aebe the ehan n the odeed onfguraton -So the underng non-near euaton te ha to be oved -h tep arred out before eah uaton over error n the odeng (redundant euaton te) kneat uaton. poton of a bode anaed dependng on the te. otona ntroduton defned b pefng a aw of oton for: aboute oordnate of a bod reatve poton of an bode to one another degree of freedo ut be denta ( prnpe of drve ut orrepond to the nuber of odeed degree of freedo) dna ana there are kneat ontrant: euaton te: obnaton of agebra euaton and dfferenta euaton nvere dna. oton pattern of bode pefed. drve fore and the nterna fore of the jont are auated. tat auaton tate of eubru oton of a bode where the aeeraton na poton of bode deterned a a reut of fore kneat ontrant are repaed b febe eeent (prngdaper-eeent, ). If oveent on b the effet of eterna fore and no kneat ontrant: euaton te: nora dfferenta euaton te. f

9 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-8 Kneat Kneat fundaenta euaton of a der-rank ehan fundaenta euaton of a der-rank ehan K Φ n o n n o o n o ) ( [ ] Generazed oordnate t t D ω Φ ), ( t t D K Φ Φ Φ ), ( ) ( ), ( 9 euaton for 9 varabe Drve-euaton Kneat ontrant euaton ϕ ' ' ' ' ϕ ' ' rank oupe der Kneat hee

10 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-9 Kneat fundaenta euaton for a der-rank ehan A der-rank ehan erve a anat tud of the anae derbed above. he oton euaton of a der-rank ehan houd be derved n the foowng (tranferabe to an ope ehan) poton and oaton of a bode foruaton arred out wth the o-aed generazed oordnate. Eape ontant anguar veot ω of the rank defned -> euaton te of 9 unknown and 9 euaton an be oved. Contran Nora autoata generated b the oftware Jont : hnge of the rank -> jut rotaton aowed Jont : kneat ontrant for the jont between rank and oupe -> hnge pont ut onde on both bode Jont 4: kneat ontrant for the de wth the prat jont -> orentaton and -oordnate annot be atered reatve to the frae A are obned n a ontrant vetor that houd be zero. Euaton te 9 generazed oordnate wth 8 kneat ontrant -> ehan ha one degree of freedo o deterne the oordnate n dependene on the te, a drveeuaton ut be foruated.

11 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Anaton and reut of the der-rank ehan

12 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Anaton and reut for the der-rank ehan he der-rank ehan wa odeed n a ut-bod uaton te. he pture how the ode a we a the -poton, veot and aeeraton of the der. Bede the reut preented here, the poton, veot and aeeraton of a generazed oordnate of the ehan are auated b the MBS-progra.

13 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Newton-Raphon ethod for the outon of the poton Souton of the non-near te of euaton K Φ ( ) Φ(, t) D Φ (, t) Known varabe n non-near euaton Φ () n unknown varabe vektor of the outon [ * * ] * L * n aor epanon at an approaton () for an better approaton () under dregard of ter of hgher order: Φ( ( ) ) Φ ( ( ) ) [ ( ) ( ) ] If the Jaoban Φ () ( ) n t nguar the euaton ant t be reoved nto () : ( ) ( ) ( ) ( ) Φ ( ) Φ( )

14 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Newton-Raphon Method for ovng the poton euaton Non near euaton te Mot the Newton Raphon ethod apped a outon. the frt vaue of the generazed oordnate at the te zero (t) known.. aor deveopent to approate () (negetng ter of hgher order) t neear to deterne the Jaoban Matr Φ () parta dervaton of the te of euaton aordng to the generazed oordnate te (ut not be nguar) 4 euaton error toerane for the euaton te Φ ( ) < εe ( ) ( ) ( ) 5 a we a a outon error toerane < ε outon vetor pefed. for the toerane erve a nterrupton rteron for the teraton.

15 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-4 Newton-Raphon ethod for the outon of the poton () (t) ponoa (t) t Δt Φ(, t) [ ( ) ( ] Φ ( ( ) ) Φ ( ( ) ) ) ε ε e 4

16 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-5 Newton Raphon outon agorth. Make an nta etaton euaton () of the outon for the te of. In teraton.,... deterne Φ ( ) and Φ ( ). If the aboute vaue of a error and hange n the approated outon fuf the foowng ondton: () () ( ) k ( Φ ) < ε e ( ) k ( ) k < ε k,... n the outon an be topped () ( ) If Φ ( ) nguar and Φ ( ) go bak to tep and tart the proe wth a new etaton. Otherwe, go to tep. ( ) ( ) ( ). Sove Φ ( ) Δ Φ( ). Set bak to tep wth ntead of. ( ) ( ) ( ) Δ and go

17 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-6 Dervaton of the veot and aeeraton euaton Euaton of the poton Φ Φ(, t) Euaton of the veot Φ (, t) Φ Φ Φ t d dt Φ Φ t Φ t If the Jaoban Matr n t nguar! Euaton of the aeeraton Φ Φ (, t) ( Φ ) Φ Φ t t Φ Φ Φ ( Φ ) Φ Φ ) t t tt ( Φ ) ( Φ ) tt t t t If the Jaoban Matr n t nguar!

18 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-7 Dervaton of the veot and aeeraton euaton deterne the veot: tepora dervaton of a generazed oordnate aordng to the te neear to of eah ndvdua bod. You have to ue the han rue. he atr Φ that appear n the veot and aeeraton Jaoban Matr.

19 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-8 Deternaton of the veot for the der-rank ehan Deternaton of the veot for the der-rank ehan [ ] t t t dt d t Φ Φ Φ Φ Φ Φ Φ ), ( t Φ ω ϕ ϕ ϕ Φ o n o o n n o n Lnear te of euaton! K Φ n o n n o o n o ) (

20 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-9 Deternaton of the veot for the der-rank ehan he euaton how the reaton of the Jaoban Matr for ovng the veot euaton.

21 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Cauaton of the veot varaton n Matab

22 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Cauaton of the veot varaton n Matab DEGRAD p/8; RADDEG /DEGRAD; g 986; [;6;5]; Jp [.e4;.e5; 6e4];./; 56.8/; phstart 5.; phstart_r DEGRAD*phStart; phstart_r an(-/*n(phstart_r)); phstart RADDEG*phStart_r; oega *p; ttart ; tend ; npont ; t ttart; [ *o(phstart_r) ;*n(phstart_r); phstart_r;... **o(phstart_r)*o(phstart_r); **n(phstart_r)*n(phstart_r); phstart_r;... **o(phstart_r)**o(phstart_r); ; ]; Re poen(); dp(['reduu for Start poton ', nutr(re)]); t ttart; dt (tend-ttart)/npont; opton optet('dpa','off','largesae','off'); for :npont fove('poen',,opton); ; d jaoban()\rh_ve(); end p(:,) ; dp(:,) d; tp() t; t t dt; funton en poen() goba t oega phstart_r % Contrant and drver euaton for the der rank ehan o(()); n(()); o((6)); n((6)); a ()-*; en() a; en() () - *; en() () * - (4) *; en(4) () * - (5) *; en(5) (4) * - (7); en(6) (5) * - (8); en(7) (8); en(8) (9); en(9) () - phstart_r - oega*t; funton J jaoban() % Jaoban of the ontrant euaton n poen. goba ph (); ph (6); n(ph); o(ph); n(ph); o(ph); J(,:) [ * ]; J(,:) [ -* ]; J(,:) [ -* - -* ]; J(4,:) [ * - * ]; J(5,:) [ -* - ]; J(6,:) [ * - ]; J(7,:) [ ]; J(8,:) [ ]; J(9,:) [ ];

23 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Deternaton of the aeeraton for the der-rank ehan Deternaton of the aeeraton for the der-rank ehan tt t tt t Φ Φ Φ Φ ϕ ϕ ϕ Φ, n o n n o o n o ) ( t), ( tt t Φ Φ Φ Φ Φ

24 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Deternaton of the aeeraton for the der-rank ehan he euaton how the deternaton of the neear ter for the aeeraton euaton of the der-rank ehan.

25 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-4 Cauaton of the aeeraton varaton n Matab

26 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-5 Cauaton of the aeeraton varaton n Matab DEGRAD p/8; RADDEG /DEGRAD; g 986; [;6;5]; Jp [.e4;.e5; 6e4];./; 56.8/; phstart 5.; phstart_r DEGRAD*phStart; phstart_r an(-/*n(phstart_r)); phstart RADDEG*phStart_r; oega *p; ttart ; tend ; npont ; t ttart; [ *o(phstart_r) ;*n(phstart_r); phstart_r;... **o(phstart_r)*o(phstart_r); **n(phstart_r)*n(phstart_r); phstart_r;... **o(phstart_r)**o(phstart_r); ; ]; Re poen(); dp(['reduu for Start poton ', nutr(re)]); t ttart; dt (tend-ttart)/npont; opton optet('dpa','off','largesae','off'); for :npont fove('poen',,opton); ; d jaoban()\rh_ve(); dd jaoban()\rh_a(,d); end p(:,) ; dp(:,) d; ddp(:,) dd; tp() t; t t dt; goba ph (); ph (6); n(ph); o(ph); n(ph); o(ph); J(,:) [ * ]; J(,:) [ -* ]; J(,:) [ -* - -* ]; J(4,:) [ * - * ]; J(5,:) [ -* - ]; J(6,:) [ * - ]; J(7,:) [ ]; J(8,:) [ ]; J(9,:) [ ]; funton J jaoban_() % dervatve of the Jaoban of the ontrant euaton n poen. goba ph (); ph (6); n(ph); o(ph); n(ph); o(ph); J(,:) [ * ]; J(,:) [ * ]; J(,:) [ -* -* ]; J(4,:) [ -* -* ]; J(5,:) [ -* ]; J(6,:) [ -* ]; J(7,:) [ ]; J(8,:) [ ]; J(9,:) [ ]; funton J jaoban() % Jaoban of the ontrant euaton n poen.

27 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-6 Euaton of oton for a two-denona, rgd bod () r r d(p) r r δr r A ' δ r r d( p) δ r F( ) d( ) δr δ B ' r B ' t B n o o n r r B ' A ' δr δr r d( ) [ δr ( B A) δ r B] ' d( ) δ ' B [ B A] 4 5 F( ) d( ) δ ' B F( ) d( ) ' d( )

28 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-7 Euaton of oton for a two-denona, rgd bod () Fore that at on the dfferenta a eeent d() n pont opre the eterna fore F() per unt a and nterna fore f(,r) per unt a n the pont and R. he Newto aw of oton for the dfferenta a d() gven b: r d( ) F( ) d( ) Varaton foruaton: δr r d( ) δr F( ) d( ) (rnpe of vrtua work) If the vetor r r A ' r repaed b the foowng epreon and th ter ondered for a dervaton and vrtua dpaeent, then one get the opete euaton gven n the above utraton. h aed Varaton for of the euaton of oton for two-denona te. he ndvdua nubered part of the euaton are epaned n deta n the utraton beow.

29 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-8 Euaton of oton for a two-denona, rgd bod () δr r d( ) δr r [ δr ( B A) δ r B] ' d( ) da ' d( δ ' B [ B A] δ δ δ J ' ' ' B ' B ' d( ) ' ) d( ) d( ) δ ' B A ' d( ) If the entre of gravt n the orgn of the oordnate te B B E, B A J : a oent of nerta about the aw a 4 δr F( ) d( ) F 5 δ ' B F( ) [ r F] δ [ J ' ] δ r

30 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-9 Euaton of oton for a two-denona, rgd bod (). Varaton for of the aeeratve fore of the bod. he ntegra ter beoe zero when the bod oordnate te onde wth the entre of gravt. h pfe the euaton.. Utate, on one ntegra ter derved through the atr pfaton that ndate the poar a oent of nerta of the pane bod. he ter ndate the varaton for of the anguar aeeratve torue of the bod. 4. h ter ndate the varaton for of the reutng eterna fore atng on the bod. h opre the apped power and e.g. the ontat fore. 5. h ter ndate the varaton for of the reutng eterna oent atng on the bod. he beow rght euaton repreent the varaton for of the euaton of oton of a rgd bod n pane wth a bod oordnate te whh fed n the entre of gravt.

31 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Euaton of oton n ter of generazed oordnate Q [ ] [ F F F F F F ] z z z [ M Q] δ Φ δ For a vrtua dpaeent whh are ontent wth the ontrant euaton Φ,t. ( ) h ter ut be oped utaneou. [ ] [ ] M Q δ Φ δ M Φ Q M Φ Q δ M Φ Φ Q ( Φ ) Φ Φ t tt Dfferenta euaton Agebra euaton (aeeraton)

32 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Euaton of oton n ter of generazed oordnate () If the eterna fore and oent are obned n a generazed fore Q, ar to the generazed oordnate n a vetor, euaton [ r F δ[ J' ] ] δ r an be reforuated n the foowng anner: [ M Q] δ. h euaton ut be ontent for a vrtua dpaeent δ wth the atve kneat ontrant n the te. At the ae te, the generaed euaton ut ao fuf the foowng euaton: Φ (, t) A the vrtua dpaeent are etated at a partuar te, the ondton for kneat aowabe vrtua dpaeent δ an be deterned fro the dervaton of the above euaton at the partuar te: Φ δ. he Jaoban auated for the oordnate, whh op wth the euaton Φ (, t). hu, the euaton of oton for te wth kneat ontrant are gven b δ [ M Q], whh ut be et for a vrtua dpaeent δ that op wth the euaton Φ δ. h produe a ed euaton te wth dfferenta and agebra euaton. o ove thee te, the Lagrange utper theore ued: A vetor b gven a n vetor of ontant, the vetor a n vetor of the varabe and the atr A a ontant n atr. If b fuf for a, the euaton A, then there a vetor of the Lagrange utper, o that: b A. B ovng to the etng probe, Lagrange utper for of the oton euaton produe: M Φ ogether wth the aeeraton euaton Φ Φ Φ Φ Q ( ) γ the ed te of dfferenta-agebra euaton (DAE) produed. M Φ Φ Q γ t tt

33 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Invere dna of kneata deterned te F k j M Φ Φ Q ( Φ ) Φ Φ t tt k F k j k j j j j δ δr δr M A A C C δr δr k '' '' δ B Φ ' δ δ r r δ Φ k r δr k A δ B ' Q ' A δ δr '' C A Φ k r k k k k k '' ''k ''k ( δr Φ r δ Φ ) δr F δ k ' k k '' ''k '' k ( Φ B Φ ) δr F δ δ r F '' k k k '' k ' k k k C A Φr ( B Φr Φ )

34 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V- Invere dna of kneata deterned te he varaton for of the euaton of oton an be foruated for bod n the foowng anner: δ M δ k Φ δ Φ k k δ A Q If the kneat ontrant k pt up and repaed b nterna fore, the ae oton houd be the reut. h produe the foowng ondton for the nterna fore: k k k k k k ( δr Φ δ Φ ) δr F δ r Wth the foowng reatonhp r r A A oeffent oparon dever: k r C A Φ F k k k k B Φr Φ k k δ r δr B δ ranforaton fro the ''-oordnate te δ r A C δr δr δr B δ A C δr B δ uttng n the euaton for the nterna fore δr δr C F k A δ Φ k r k k δ k k ( Φ B Φ ) r k

35 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-4 Invere kneat-ana for the der-rank ehan () Invere kneat-ana for the der-rank ehan () ( ) Φ Φ Φ Φ Φ Φ M Q Q M tt t ( ) ( ) ( ) J g J g J g o n o n o n o n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) g J g J J g J g g J g J g J g tan tan tan tan tan tan tan o

36 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-5 Invere kneat-ana for the der-rank ehan () Step : Deternng the Lagrange utper

37 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-6 Invere kneat-ana for the der-rank ehan () Invere kneat-ana for the der-rank ehan () ( ) r r B A C F Φ Φ Φ ' '' '' C A o n n o B n o o n Φ r [ ] Φ o n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) g J g J g t t J g F tan o n n o o n o o n n o o n n o o n n o '' '' ' ' F F z

38 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-7 Invere kneat-ana for the der-rank ehan () Step : Deternng the nterna fore F, on the rank bearng

39 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-8 Invere kneat-ana for the der-rank ehan () F '' '' o n 9 n o F ' ' ' F z F '' '' o n n o 4 F F z z ' F F F '' '' '' '' o n o n n o n o F ' ' F ' ' z F '' 4 '' 4 o n n o 7 8 F ' F z ' F

40 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-9 Invere kneat-ana for the der-rank ehan () Step : Deternng a nterna fore.

41 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-4 Cauaton of the reatve fore n Matab 8 Lagerkräfte 6 4 Kraft [N] - -4 F rank-ground Kurbe-Gete F rank-ground Kurbe Gete F rank-oupe Kurbe Koppe -6 F rank-oupe Kurbe Koppe F oupe-der Koppe Sheber F oupe-der Koppe Sheber F der-ground Sheber Gete Zet []

42 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-4 Cauaton of the reatve fore n Matab goba oega t Jp g phstart_r DEGRAD p/8; RADDEG /DEGRAD; g 986; [;6;5]; Jp [.e4;.e5; 6e4];./; 56.8/; phstart 5.; phstart_r DEGRAD*phStart; phstart_r an(-/*n(phstart_r)); phstart RADDEG*phStart_r; oega *p; ttart ; tend ; npont ; t ttart; [ *o(phstart_r) ;*n(phstart_r); phstart_r;... **o(phstart_r)*o(phstart_r); **n(phstart_r)*n(phstart_r); phstart_r;... **o(phstart_r)**o(phstart_r); ; ]; funton F nv_dna(,d,dd) % nvere dna of the der-rank ehan goba Jp g M dag([() () Jp() () () Jp() () () Jp()]); Q [ ; -()*g; ; ; -()*g; ; ; -()*g; ]; QNeu Q - M*dd; abda jaoban()'\qneu; F(,) -abda(); F(,) -abda(); F(,) -abda(9); F(4,) -abda(); F(5,) -abda(4); F(6,) ; F(7,) -abda(5); F(8,) -abda(6); F(9,) ; F(,) -abda(7); F(,) -abda(8); F(,) ; Re poen(); dp(['reduu for Start poton ', nutr(re)]); t ttart; dt (tend-ttart)/npont; opton optet('dpa','off','largesae','off'); for :npont fove('poen',,opton); ; d jaoban()\rh_ve(); dd jaoban()\rh_a(,d); F nv_dna(,d,dd); end p(:,) ; dp(:,) d; ddp(:,) dd; Fp(:,) F./e; tp() t; t t dt;

43 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-4 Dna-ana of the der-rank ehan wth febe bearng ' ' ' he reatve fore and are repaed b eterna fore whh at on the rank. Q A A Q ( k, k,,,,, ϕ,,, ϕ ) ' ϕ ' ' ϕ k Φ( ) o n o n ωt o n Φ Φ u ( Φ u Φ v ) v γ Φ t Φ tt γ u: Vetor of the dependent, generazed oordnate v: Vetor of the ndependent, generazed oordnate

44 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-4 Dna-ana of the der rank ehan wth febe rankhaft bearng If the nuber of kneat ontrant e than the nuber of the generaed oordnate, the generaed oordnate an be dvded nto dependent and ndependent oordnate. he dependent oordnate are dtnguhed b the fat that the are nked wth other oordnate through kneat ontrant and thu are no onger ndependent.

45 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-44 Dependent and ndependent Dependent and ndependent generazed oordnate generazed oordnate () () ω ω Φ Φ Φ v u t v u

46 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-45 Dependent and ndependent generazed oordnate () Converon of the euaton of veot for the der rank ehan.

47 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-46 Dependent and ndependent Dependent and ndependent generazed oordnate generazed oordnate () () ( ) ω ω ω ω ω Φ Φ Φ v t u v u

48 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-47 Dependent and ndependent generazed oordnate () Foruaton of the veot of the dependent oordnate a funton of the ndependent oordnate.

49 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-48 Dependent and ndependent Dependent and ndependent generazed oordnate generazed oordnate () () γ Φ Φ v u v u

50 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-49 Dependent and ndependent generazed oordnate () Euaton of aeeraton for the der rank ehan dvded nto dependent and ndependent euaton.

51 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-5 Dervaton of the euaton of aeeraton for dependent Dervaton of the euaton of aeeraton for dependent oordnate oordnate [ ] γ Φ Φ v u v u

52 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-5 Dervaton of the aeeraton euaton for the dependent oordnate Souton of the aeeraton of the dependent oordnate a funton of the ndependent oordnate.

53 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-5 Euaton of oton n ter of Euaton of oton n ter of dep dep. and. and ndep ndep. generazed oordnate generazed oordnate ( ) ( ) [ ] g g k k k g k J J J Φ Φ Av Au v u vv vu uv uu Q Q M M M M

54 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-5 Euaton of oton n ter of dependent and ndependent generazed oordnate Foruaton of the euaton of oton n dependent and ndependent oordnate.

55 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-54 Reduton on a ordnar dfferenta euaton (ODE) M Φ M M Φ uu Q γ vu u uv M v Au Au uu uv Φ Q Φ [ Q M u M v ] u M M Φ vu vv u Φ A v Φ u Φ u u v v v vv Au uv uu [ γ Φ v] M v Φ Φ [ Q M v M u] v Q γ Dfferenta euaton Agebra euaton (aeeraton) Av v u Φ u u u [ γ Φ v ] v Q Av DAE dfferentaagebra euaton Mˆ v v ( v, v, t) v Qˆ ( v, v, t) vv vu uv uu ( M M Φ u Φ v Φ v Φ u M Φ v Φ u M Φ u Φ v ) Av vu Au uu Q M Φ γ Φ Φ [ Q M Φ γ] u v u u v ODE ordnar dfferenta euaton

56 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-55 Reduton on a ordnar dfferenta euaton (ODE) A ter n the dfferenta-agebra euaton that depend on the vetor u, an be nterpreted a funton of v. If one onvert the euaton te aordng, utate a reguar dfferenta euaton derved n the ndependent, generaed oordnate v that an be oved b the ue of tandard ethod.

57 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-56 v u uu u v uv u v v u vu vv v M M M M Mˆ Φ Φ Φ Φ Φ Φ Φ Φ [ ] ( ) ( ) Φ Φ γ Φ Φ Φ γ Φ Φ Φ Φ γ Φ Φ A A A A A A u v u uu Au u v Av u uu u v Au u v u vu Av v g Q Q g Q g J g Q Q g Q g M Q Q M Q M Q Qˆ ( ) ( ) t v v Q v t v v M v v,, ˆ,, ˆ

58 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-57 Euaton of oton for the der rank ehan Conrete euaton foraton for the der rank ehan.

59 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-58 Suaton wth a febe rank-bearng

60 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-59 Suaton wth a febe rank-bearng he uaton how the fore progreon n -dreton of the bearng for two dfferent dapng paraeter. he der rank ehan drven b the rank wth a ontant anguar veot. he uker dng out at hgher dapng vaue an be ear dentfed.

61 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-6 Appaton of the ut-bod-uaton n ahne too engneerng Gudng-te Spnde Modeng of frae oponent a rgd bode or febe bode. F Modeng of gudng-te, ba-rewpnde and bearng a febe onnetor, uh a prng-daper and buhng. he pha properte of thee eeent an be near or non-near. δ Modeng of the feed drve b the ue of a oed ontro oop. Bearng Stat-ana: Deforaton at the too entre pont aued b tat fore (e.g. uttng proe) Dna-ana Deforaton at the too entre pont aued b dna fore (e.g. uttng proe) Defeton at the too entre pont b proeedng aong a prograed path. (dnaa path varaton)

62 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-6 Appaton of the ut-bod-uaton n ahne too engneerng Wth MBS an probe an be oved (ahne too degn) Stat: Mode: frae oponent -> rgd and a-affted bod, jont-> prng-daper-eeent outon: febt too ow beaue frae oponent are rgd -oupng oaton have a bg nfuene on tota febt -> o effet of gudewa arrangeent an be heked - gude oad an be deterned Dna ana the repone of the ahne a a reut of an etaton an be auated (e.g. through the uttng proe). Moton uaton. atua ahne oton an be defned through drve-euaton ut-bod ode wth febe jont-> dna trak devaton an be uated -> optaton through: Ma arrangeent of bode hange n jont properte ouped uaton ntegrate ontro oop the feed drve n the ode

63 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-6 Modeng of gudng-te and ba-rew-drve Modeng of gudng-te and ba-rew-drve Stffne Dapng de N N E E k k k k z z e/,,, /,, Stffne Dapng pnde houng N N E E k k k k z z e/,,, /,, Stffne Dapng oun N N E E k k k k z z e/,,, /,, Subttute te of the ba-rew-drve

64 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-6 Modeng of gudng-te and ba-rew-drve gude. dentfed b:. tranatona tffne:. n tenon/preure dreton 4. de dreton 5. rotatona tffne around a rotatona ae (ro, aw, pth). 6. Moreover, a dapng. In the ode: B the ue of eeent that an app prng and daper fore between oupng oaton ubjeted to dpaeent and reatve veot. approprate vaue an be ondered for eah fro eaurng tet of the ahne eeent. Eape runner bok of the gudng te 5 tffne vaue for tranaton degree of freedo for tranatona dpaeent 6 dapng haratert vaue Eape ba rew drve depted b a prng-daper-eeent n feed drve dreton.

65 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-64 Natura freuene and near ode of a -a ng-ahne f 4,8 Hz f 47,7 Hz f 9, Hz Longtudna oton of the de n dreton of the drve Longtudna oton of the oun n dreton of the drve Longtudna oton of the pnde houng n dreton of the drve

66 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-65 Natura freuene and near ode of a -a ng-ahne 4.8 HZ, the de oved. au de a and the ted tffne-> frt natura freuen our n the feed drve of the de. k 5N / μ f 47, 6 π π kg Hz natura freuen ower Due to the effetve dapng and the other febte f 47.7 Hz, ongtudna oton of the oun n -dreton atera bukng of the oun and the pnde houng about the -a. f 9. Hz, ongtudna oton of the pnde houng n dreton of the feed drve (n z-dreton). a atera bukng of the oun about the -a. Conuon atera bukng of upper oun and the de oud be rta durng operaton n and z-dreton. Optzaton a reduton of the oun, nreae of the gude tffne

67 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-66 Freuen repone funton n -dreton of the ng-ahne - Nahgebgketfreuenzgang G Euaton to evauate the freuen repone funton f 4,8 Hz G( jω) ( t) e F( t) e jωt jωt dt dt E E F ( jω) ( jω) therefore: F (t) Input gna (t) Output gna E F ( jω) Cope energ petru of the nput gna E ( jω) Cope energ petru of the output gna Nahgebgket [µ/n] - - f 94, Hz Freuenz [Hz]

68 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-67 Freuen repone funton n -dreton of the ngahne On etaton of the ahne at the too entre pont (C), a dpaeent of the eted too our dependng on the freuen. he rato fro dpaeent at the C to etng fore hown n the above utraton a freuen repone funton n -dreton. he etng power pont n -dreton. Fro the dpaeent, on the -part were eaured. Fro the freuen repone funton one an ao deterne tat febt (febt at the freuen of Hz) and the dna febte n reonane oaton. In the preent eape, a tat febt of.9 µ/n eaured n -dreton, whh orrepond to a tffne of N/µ. Wth a odeed tffne of 5 N/µ n the -feed drve, th kewe produe bg dpaeent porton fro the febe gudng te. Wth f 4.8 Hz, there a dna febt of appro..5 µ/n. h reut n an reonane tep-up of.7 n oparon to the tat febt.

69 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-68 Avaabe Software-te Aeb Kneat Stat Dna vn. Dn. Contro Opt. FEM-Int. CAD-Ip. Hdrau Engne Ra Whee ADAMS DADS Spak Vua Natran Sa ro Mehana NEWEUL Moton Work Meano Moton Doa Autoev Aaka Mehana Dna LMS Inte GbH MSC Adept ehnoog Rand ehnooge rof. Shehen Sod Dna Sateh DnaS OnLne Dna IM

70 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-69 Mut-bod uaton too For odeng and uaton of ut-bod te, there are an oftware te. A dfferene of the oftware te an be aheved partuar aordng to the nuber of ana opton and tehna dpne. he dfferent opton for ana nude: Aeb ana Kneat ana Dna ana Invere dna ana Stat ana h funtonat upported b ot oftware too n partuar b thoe ted here. Another dfferene e n the pea dpne: Hdrau Contro tehnue Ra ontat Whee ontat Here, approprate etended eeent brare are provded b the oftware too to ntegrate ontro oop and hdrau eeent a we a to ode pea whee and ra ontat (vehe engneerng, ra vehe engneerng). o be abe to ue produt data dret fro the CAD ode, varou nterfae to CAD te are offered. he auaton too are part ntegrated n the CAD te dret. On the other hand, t pobe to eorze geoetr and nerta data through dret or tandard nterfae. he ot popuar tandard nterfae ued nude: IGES SE araod DXF / DWG SL o etend the auaton opton, bede rgd bode, one an even onder febe bode. For th purpoe, a range of oftware pakage offer port opton for bode wth tat and dna febt properte. In addton to the anua, eperene-upported optaton, t ao pobe to autoata opte the ehana te aordng to defned target funton.

71 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-7 Appaton n vehe engneerng

72 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-7 Overvew: Appaton n vehe engneerng op eft: Febe tre ode op entre: Curve drve of a fork ft truk Conderng tre ontat a we a ontat between ar bod and ground n ae of turn over op rght: Curve drve of a paenger ar Conderng tre ontat and the ha Botto eft: Stre ana of a front upenon onderng febe bode Botto rght: et benh for a port ar ha

73 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-7 Appaton n aeropae engneerng

74 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-7 Overvew: Appaton n aeropae engneerng op eft: ouh down of a andng gear op rght: Landng proedure of an arraft Conderng the arraft a febe truture Botto eft: aoad farng eparaton of ARIANE 4 Conderng febe truture Botto rght: Run of an arraft over ground wave on an arfed Landng gear tunng

75 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-74 Appaton n ehana engneerng

76 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-75 Overvew: Appaton n ehana engneerng Left: et of the hdrau deve of an eavator O preure dependng on the pton poton Rght: Ipat drng ahne Contat probe

77 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-76 Appaton n ra vehe engneerng

78 Vrtua ahne too Modeng and Suaton Leture: Mut-bod uaton V-77 Overvew: Appaton n ra vehe engneerng Left: Boge of a ra vehe a febe truture Natura freuen ana Centre: Boge uaton of a ra vehe on a tet trak Rght: ran run over a febe brdge