Day 2: Dynamics. Part I: Multiple-rigid-body Dynamics

Transcription

1 PhD Coure o Advaced Mechaic of Mechaical Sytem Day : Dyamic Shaopig Bai hb@m-teach.aau.dk Dept. Mechaical ad Maufacturig Eg., AAU Sept 8-, Part I: Multiple-rigid-body Dyamic Newto-Euler Equatio Lagrage Equatio Numerical implemetatio Formulatio for improved performace Other coideratio Sytem

2 Example of dyamic modelig Dyamic modelig of Hydrama 9D dump truck Dyamic modelig of HMF 4-K5 Kuckle Boom Loader Sytem 3 Fudametal: Newto-Euler equatio Tralatio equatio for a body with (.) Rotatioal equatio (Euler equatio) (.) Remark: The liear acceleratio by Newto d law i give for ceter of ma. The liear acceleratio chage withi the rigid body. The liear acceleratio i NOT depedet to liear velocity The agular acceleratio IS depedet to the agular velocity The momet i Euler equatio i take about ceter of ma. NOT other poit i the fixed frame. I ome cae, Euler equatio ca alo be decribed i term of local compoet, i.e., I thi way, J i a cotat. (.3) Sytem 4

3 5 Newto-Euler equatio For a igle body i a multibody ytem, it EOM i: I matrix form or Note:. ad f are i the ame directio, but ad are geerally ot. All local coordiate are etablihed at their ceter of ma Motio equatio of multiple ucotraied bodie (free body) r i X Y Z G i i. r i, f i.. i i i C.M. i i i i g b v M Sytem (.4) Newto-Eular equatio Aembly together the EOM for all bodie I hort form The equatio ca be ued i ivere dyamic, i.e., to fid force eeded for a deired motio. Sytem 6 (.5) ; r r v g b v M with f f J J r r M M M

4 Dyamic aalyi of a ytem: Lagrage multiplier approach Eq. (.3) i etablihed o the bai of ucotraied bodie. I aother word, we have to develop freebody diagram to fid out the reactio force at each joit. Alteratively, we ca make ue of Lagrage multiplier to iclude the cotrait force (iteral) i the equatio. M v b M g v b g M v b g ad Aume there are m kiematic cotrait equatio, let The D Lagrage multiplier Dyamic equatio with Lagrage multiplier: T Mv b D g Sytem g c i T i ext I eq. (.5), the iteral force are ot icluded explicitly. (.5) 7 Newto-Euler equatio Eq. (.5) tad for (+m) ukow: the acceleratio ad m Lagrage multiplier. I order to have ufficiet umber of equatio, we have to upply m more equatio. The acceleratio equatio ca thu be ued. Aembly with acceleratio equatio: We have I matrix form T Mv b D g Dv ; ext Dv M D D T v g ext b (.6) Now the equatio ca be readily olved umerically. Sytem 8

5 Example A multibody ytem with four pherical joit Sytem 9 Example (cot d) Velocity ad acceleratio equatio, takig the firt cotrait a a example Sytem Matrix D of the ytem Make ue of A A A A A A r r r r 4D 4C 3C 3B D B A A I I I I I I I I

6 Lagrage Equatio (with idepedet coordiate) Let the vector q repreet a et of ukow idepedet coordiate Let T (q, q ) be the kietic eergy of the ytem, V(q) the potetial eergy ad The Lagragia: L= T-V Q ex (q) the vector of geeralized exteral force (o coervative force) actig alog the depedet coordiate q of a cotraied mechaical ytem. The Lagrage equatio of the firt kid i d dt L L ex Q q q (.7) Sytem Lagrage Equatio (with depedet coordiate) Let the vector q repreet a et of ukow depedet coordiate, m i the total umber of idepedet cotrait equatio (geometric ad kiematic), ad f=-m i the umber of dyamic degree of freedom (dof). The cotrait coditio are writte i the followig geeral form: ( q, t) Let T (q, q ) be the kietic eergy of the ytem, V(q) the potetial eergy ad The Lagrage equatio of the ecod kid: (.8) The matrix q i the Jacobia matrix of the cotrait equatio (.8). The vector i (.8) defie the Lagrage multiplier. Sytem

7 Example : A -dof pedulum Formulatio the equatio of motoi New-Euler equatio Sytem 3 Lagrage equatio. Kietic eegry. Potetial eergy 3. Lagragia fucito 4. Lagrage equatio Sytem 4

8 Numerical olutio While there are may way availalbe, a coveiet method i to covert them ito DAE of firt order. Thi implie that, give the firt order derivative of depedet variable at time t, we ca ue a itegrator to fid their ew value for time t+t. Algorithm. tart at a time t=t i which the poitio ad velocity are kow.. Ue eq. (.6) to olve the acceleratio at time t=t. Thi tep i referred a fuctio evaluatio T 3. Let y { v, q} t, which ca be ued a iput to a itegrator of firt order differetial equatio to get T ẏ tt 4. Update time t ad go to tep Sytem 5 Algorithm (fuctio evaluatio) (vel. traformatio ) Refer to appedix A (Nikraveh book) for the traformatio betwee agular rate ad agular velocitie. For Euler agle of ZXZ covetio, the traformatio i (.9) Sytem 6

9 Example 3. Dyamic imulatio A fallig-cro (matlab code i available at the coure webite) Aumptio: Four metal ball coectted rigidly by rod The impact of the cro o the groud i characteried with a prig cotat ad a dampig coefficiet t = ekuder.5 z.5 y y x x.5 Sytem 7 New formulatio for improved tability With the formulatio give i eq. (.6), the problem of umerical tability raie. The reao lie i the itegratio of diffetial equatio of the cotrait. The geeral olutio of thi differetial equatio i ubouded, thu leadig to utable olutio. Look at the ecod order of differetial equatio of the cotrait (.) It geeral olutio i i a form of where a ad a deped o the iitial coditio. (.) Solutio i the above form i ubouded, makig the ytem utable. A remedy to fix thi problem i to mix the ytem of differetial equatio with algebraic equatio (cotrait) Sytem 8

10 Baumgarte Stabilizatio Thi i a cotrol theory baed method. The idea i to eforce/impoe the cotrait at the acceleratio level by addig the cotait equatio i the differetial equatio a where ad are cotat (.) The olutio i ow i a form of (.3) with (.4) If ad are egative, the geeral olutio i bouded, thu the tability i guarateed. Sytem 9 A ew formulatio With the Baumgarte Stabilizatio method, the dyamic equatio of a multibody ytem ca be rewritte a (.5) where (.6) or Remark: g (.7) The cotat ad, depedig o the problem, are uually et to equal value betwee to. The ew formulatio i imple---modifyig imply the origial differetial equatio, ad computatioal efficiet. It ha limitatio, for example, i dealig with ear-igularity cofiguratio. Further readig: Chapter 5, J. Jalo ad E. Bayo, Kiematic ad dyamic imulatio of multibody ytem. Sytem

11 Example 4: Numerical experimet.we ue the ame example of our exercie (ee lide ext page) The error i defied a the rm value of reidual of all cotrait equatio. 7 x - error plot (baic equatio) x -3 error plot (alpha=beta=) x -3 error plot (alpha=beta=5) x -4 error plot (alpha=beta=) x -4 error plot (alpha=beta=5) Sytem Example Sytem

12 Sigularity-free dyamic equatio olvig I the previou formulatio, the ytem may likely reach a igular cofiguratio, whe the leadig matrix become igular. The coequece of a igle poitio i kiematically a udde chage i the umber of degree of freedom, either lot or icreae oe or more degree. I the umerical olvig, the acceleratio caot be computed, the dyamic imulatio may either crah or lead to a large error. The ear-igular cofiguratio will lead to amplified roud-off error ad reultig erroeou olutio. A imple way to deal with the igularity i to detect the ill-coditioig of the Jacobia ad let the itegrator tep over it. New formulatio are available for robut olutio. Oe i the pealty-augmeted Lagragia formulatio (ch..6, Jalo book): (.8) Sytem 3 Coulomb frictio model Frictio force (ch.., Jalo book) To icooperate with Coulomb frictio, a additioal item ca be added to the equatio of motio a With Where i frictioal coefficiet, u(q) i a matrix decribig joit type ad geometry (.9) (.) Sytem 4

13 Impact ad colliio I certai cae, the dyamic modellig may get ivolved with impact ad colliio. It i ormally aumed that the duratio of impact i very hort ad impact force i large, all other remaiig force ca thu be eglected. Aother aumptio i the poitio of the ytem doe ot chage durig the impact, oly velocity chage. The equatio of motio, drig the impact, i: (.) Where P i i the itegral effect of the impact force Q i, i.e. (.) Sytem 5 Backlah (ch..3) Other coideratio of dyamic modelig I dyamic modelig, the complexity of the model deped o the problem. Alway tart modelig from the baic model ad validate. Check error of computatio The bet validatio i experimet, whe applicable. Further readig: P. E. Nikraveh, AN OVERVIEW OF SEVERAL FORMULATIONS FOR MULTIBODY DYNAMICS, D. Talab ad T. Roche (ed.), Product Egieerig, 89 6 Sytem 6

14 Part II. Gear Dyamic Gear tramiio are commoly ued mechaical ytem. A a traditioal ubject of reearch ad developmet, gear dyamic ha bee exteively tudied over ceturie. However, the demadig for high peed, large-cale tramiio i applicatio like highpeed machiig ad wid turbie, poe ew challege for the deig ad aalyi of gear tramiio. Thi part of the lecture i aimed at providig a overview of theorie for gear dyamic modelig. The followig cotet are icluded: Gear baic Kiematic of gear tramiio Rigid body gear dyamic Flexible body gear dyamic Sytem 7. Gear baic Gear are the mot commo mea ued for power tramiio They ca be applied betwee two haft which are Parallel Colliear Perpedicular ad iterectig Perpedicular ad oiterectig Iclied at ay arbitrary agle. Spur gear Mot commo ued form for parallel haft Suitable for low to medium peed applicatio Relatively high ratio ca be achieved (< 7) Steel, bra, broze, cat iro, ad platic. Helical gear Teeth are at a agle---helix agle Ued for parallel haft Teeth egage gradually reducig hock Sytem 8

15 . Gear baic 3. Bevel gear They have coical hape Ued to chage directio Two axe are coplaar 4. Worm gear For large peed reductio: oe tur of worm v oe teeth of gear Tramiio betwee two perpedicular ad o-iterectig haft Oe-way tramiio Two axe are ot coplaar 5. Rack ad piio A rack i a gear whoe pitch diameter i ifiite Ued to covert rotary motio to traight lie motio Sytem 9 Gear geometry Root diameter Diameter of the gear, meaured at the bae of the tooth. Addedum circle (outer circle) the circle that coicide with the top of the teeth of a gear Bae circle: a circle of referece of gear teeth profile geeratio pitch circle A imagiary circle that i cetered at gear ceter ad paig the pitch poit Addedum ad dededum Sytem 3

16 Gear geometry Pitch poit, p Poit where the lie of actio croe a lie joiig the two gear axe. Pitch circle, pitch lie Circle cetered o ad perpedicular to the axi, ad paig through the pitch poit. Pitch diameter, d=m.n Diameter of a pitch circle. Equal to twice the perpedicular ditace from the axi to the pitch poit (R). The omial gear ize i uually the pitch diameter. Module, m The pitch diameter divided by the umber of teeth. Preure agle, arcco( / r), r b r co r b r b R b Sytem 3 Gear geometry Lie of actio A lie or curve alog which two tooth urface are taget to each other. A lie where teeth egagemet happe Cotact ratio Ratio of the arc of actio to the circular pitch It tad for the average umber of gear tooth pair i cotact o a pair of mehig gear. Backlah: The differece betwee the circle thicke of oe gear ad the tooth pace of the matig gear Sytem 3

17 . Gear Kiematic o the relatiohip betwee gear dimeio ad motio Gear ratio (GR): the ratio of umber of teeth o drive gear to the umber of teeth o driver gear Ndrive N GR m N N (.) Velocity ratio (VR): the ratio of the agular peed of the driver gear (gear ) to the agular peed of the drive gear (gear ) driver VR (.) drive Pitch lie velocity: the velocity of the pitch poit v t driver r r More expreio of velocity ratio (.3) VR driver drive r d N r d N (.4) Sytem 33 Gear trai kiematic. i a erie of matig gearet. Trai ratio (TR) i the product of all velocity ratio TR i out ( 3 VR )( VR )( VR )... Pay attetio to the directio of gear rotatio Plaet gear trai To fid the TR, we firt fix the arm, the (.5) TR ( VR )( VR )( VR3 )... where relative agular velocitie are, Thu gear gear arm gear gearn gearn gearn (.6) arm ( VR )( VR )( VR3 )... gear gearn arm arm (.7) Plaet gear trai ad equivalet dof mechaim Sytem 34

18 Example: Aalyzig of plaetary gear trai Problem: determie the trai ratio betwee the u gear ad the arm. Give data: The u ha 4 teeth, the plaet teeth, ad the rig gear 8 teeth. The arm i the iput ad the u i the output. The rig gear i held tatioary. Solutio: We look at the gear trai from the u to the rig That i or Hece ( N N u plaet )( ) N plaet Nrig ( 4 )( u rig u arm ) 8.5 u / arm arm arm arm TR u / 3 arm Sytem 35 For a gear tramiio how i the figure power i T poweri power out Rigid body gear dyamic ; powerout T T T (.8) T N T N Reflected torque (.9) T R T (.) Reflected momet of iertia: whe viewed from the iput gear J R J Equatio of motio of gear trai T T ( J J R ) (.) (.) Sytem 36

19 Example Equatio of motio of gear trai T m N,J T l J l T m T lr ( J J J ) lr m br J m N, J Where T m = maximum available motor peak torque J m = motor rotor iertia J br = reflected momet of iertia of gearbox T lr = reflected load torque, icludig frictioal torque, for example, T lr = T gr +M fr The tartig torque (acceleratio torque plu teady torque) T m The acceleratio T J m lr Tm TlR J J lr Steady torque ( J J J ) lr m br br Tm T lr 37 Sytem 3. Gear dyamic with flexible bodie Baic oliear gear dyamic model Goverig eq. m u k e m u (3.) Where: equivalet ma II I me R I R I (3.) A baic model Tramiio error I dyamic modelig, the followig problem eed to be ovled. Periodic gear mehig force. Determiatio of mehig tiffe 3. Ifluece of variace uch a backlah u R R (3.3) Sytem 38

20 Other type of model A compreheive model. z phae agle. The compoite error e i the um of tooth error of the piio ad gear. Typical model for a gear trai Mathematical model ued i gear dyamic A review, H. Nevzat Özgüve ad D.R. Houe, Joural of Soud ad Vibratio, Vol, Iue 3, March 988, Page OPTIMIZATION METHODS FOR SPUR GEAR DYNAMICS, Marco Barbieri, et al., proc. ENOC 8, Sait Peterburg, Ruia, Jue, 3 July, 4 8 Sytem 39 Gear modelig I dyamic modelig, the followig problem eed to be ovled. Dicotiuou gear mehig force. Determiatio of peroidic mehig tiffe 3. Ifluece of variace uch a backlah The mehig force calculatio ha to coider the egagemet ituatio:. poitive workig coditio, a ituatio i which drivig gear drive drive gear.. egative workig coditio. I uch a ituatio, a drive gear i i cotact with the backide of the drivig gear. 3. o egagemet. Sytem 4

21 Model of mehig force with backlah The mehig force i alo ubject to backlah(clearace) Sytem 4 Gear mehig tiffe, ref. [4] The tiffe at a poit j i (3.4) where y B i the deformatio due to beam effect ad y L i that due to urface cotact, which ca be calculated by With where I i i the area momet of iertia of the i-th elemet ad S i i the cro-ectio area of the teeth at elemet i, ad W i tooth width Sytem 4

22 Gear mehig tiffe (cot d) The effective modulu of elaticity E e i defied to take ito accout of the ifluece of width of the teeth profile, which i where E i the material Youg modulu ad v i the Poio' ratio. The tiffe of the gear mehig i the ivere of combied compliace of two gear at the egagig poitio (3.5) Alteratively, the mehig tiffe ca be determied through FEA Sytem 43 Cae Study: Model with flexible gear teeth ad haft [5] Aumptio: coitig of oe pair of pur gear iclude geometric variatio like backlah where Sytem 44

23 . Cae tudy (cot d) By itroducig the dyamic tramiio error, ad relative rotatio, we ca elimiate oe degree of freedom (3.6) Sytem 45 Frequecy repoe Cae Study (cot d) Simulaito: Noliearity aalyi Nolliearity i weakeed with the decreaig of haft torioal tiffe Sytem 46

24 Cae tudy (cot d) Load-harig imulatio (3.7) Time hitory with idetical iput torque The load-harig i ot eitive to differece betwee the iput torque Time hitory with lightly differet iput torque Sytem 47 Additioal Referece. Advaced Deig of Mechaical Sytem: From Aalyi to Optimizatio. Ambroio, Jorge A. C.; Eberhard, Peter (Ed.), Spriger, 9. Mechaim ad Dyamic of Machiery (third editio), by Hamilto H. Mabie ad Fred W. Ocvirk, Wiley, Joh & So 3. Mechaical Vibratio (4 th iteratioal editio), by S. S. Rao, Pretice Hall, NJ, 4. R.W. Corell. Compliace ad tre eitivity of pur gear teeth. ASME J. Mechaical Deig, 3(): , S. Bai, M. Haatrup, P. Rigal. Dyamic Modelig of a Wid Turbie Yawig Mechaim for Load-harig Aalyi I Proc. d Joit Iteratioal Coferece o Multibody Sytem Dyamic., Stuttgart, Germay Sytem 48