State-Space model of a mechanical system in MATLAB/Simulink

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1 Avalale onlne a Proceda ngneerng 48 (0 ) MMaMS 0 Sae-Sace odel of a echancal syse n MATLAB/Suln Peer Svá a * arna Hroncová a a Techncal Unversy of Košce Faculy of Mechancal ngneerng Lená Košce Slova reulc Asrac Ths aer descres soluon of he equaons of oon of he echancal syse y usng Sae-Sace locs n MATLAB/Suln. I deals wh he echancal syse wh wo degrees of freedo. Sae-Sace loc soluon s coared wh soluon ade y an alernave aroach usng so called Transfer Fcn loc. quaons are also derved y Newon's second law Lagrange s equaons and he Halon s equaons. 0 The Auhors. Pulshed y lsever Ld. 0 Pulshed y lsever Ld.Selecon and/or eer-revew under resonsly of he Branch Offce of Slova Meallurgcal Socey a Faculy Selecon of and/or Meallurgy eer-revew and Faculy under of Mechancal resonsly ngneerng of he Branch Techncal Offce Unversy of Slova of Košce Meallurgcal Oen access Socey under a CC Faculy BY-NC-N of Meallurgy lcense. and Faculy of Mechancal ngneerng Techncal Unversy of Košce. Keywords: Sae Sace Transfer Funcon nec energy oenal energy Mala Suln. Noenclaure H y u A B C asses sffness of he srngs coeffcens of vscous dang Halon s funcon nec energy oenal energy Raylegh s dssave funcon oenu of he -h eer sae vecor ouu vecor nu vecor sae ar nu ar ouu ar feedforward ar. Inroducon In he frs ar we derved he equaons of oon of he echancal syse wh wo degrees of freedo usng he * Corresondng auhor. Tel.: al address: eer.sva@ue.s Pulshed y lsever Ld.Selecon and/or eer-revew under resonsly of he Branch Offce of Slova Meallurgcal Socey a Faculy of Meallurgy and Faculy of Mechancal ngneerng Techncal Unversy of Košce Oen access under CC BY-NC-N lcense. do:0.06/j.roeng

2 630 Peer Svá and arna Hroncová / Proceda ngneerng 48 ( 0 ) second Newon's law. Ne he hese equaons are derved y Lagrange's equaons of he second nd. Fnally we deerne he sae equaons y usng Halon's equaons []. The a s o descre he use of Sae-Sace locs and Transfer Fcn of he dynac syse n Mala/Suln. The oaned resuls are coared wh drec soluon n Mala. The resuls are resened n grahcal for.. Newon s quaons In hs secon we descre he colaon of equaons of oon of he echancal syse wh wo degrees of freedo wh Lagrange equaons of he second nd and he Halon equaons. These equaons are hen solved n Mala wh Runge-Kua ehod. The resuls rovde nforaon aou dslaceen velocy and acceleraon of ndvdual eers of he echancal syse as a funcon of e. A haroncally varale force F() s used for he odel ecaon (Fg. ). Fg.. aed ass-srng syse wh wo degrees of freedo. We consder a echancal syse wh wo degrees of freedo of oveen (Fg. ) whch consss of odes wh asses and conneced wh srngs wh sffnesses and and daers wh lnear dang coeffcens and conneced o a rgd frae [5-0]. The syse erfors lnear oon n drecon of srngs and daers aes. The weghs of he srngs are no consdered. A haroncally varale force F() F 0 sn() s used for he odel ecaon. The resecve eers erfor lnear forced oscllang oon. quaons of he oon: reacon forces n he srngs dang forces Fr Fd Fr Fd F () F r F () F F r d d. Fr. ( ) (3). Fd. ( ). (4) The a s o deerne he resonse of he oscllang syse - dslaceens () (). Ths however requres o solve a syse of dfferenal equaons. For a echancal syse wh wo degrees of freedo s a nonhoogeneous syse of nd order lnear dfferenal equaons wh consan coeffcens. 3. Lagrange s equaons Ne we deerne he equaons of oon of he aove enoned syse usng Lagrange s equaons of he second nd. For hs we need o deerne nec and oenal energy of he syse and he Raylegh dssave funcon. If we defne he generalzed coordnaes as q q and he generalzed veloces as q q he nec energy of he syse s: (5) oenal energy ( ) (6) Raylegh s dssave funcon ( ) (7) generalzed forces F( ) 0. (8)

3 63 Peer Svá and arna Hroncová / Proceda ngneerng 48 ( 0 ) Susung he generalzed coordnaes he oenal energy of he syse: ( ) (9) Raylegh s dssave funcon ( ). (0) Lagrange s equaons of he second nd for he echancal syse (Fg. ) are: d () d. () Afer susuon of he aral dervaons we oan equaons of oon n he for: ( ) ( ) () F (3) ( ) ( ) (4) where are veloces and are acceleraons of he asses and. These equaons are lnear dfferenal equaons of second order wh consan coeffcens. 4. Halon s equaons The Halon s funcon H H (q ) whch s he su of nec and oenal energes can e used o deerne he equaons of oon of he echancal syse wh wo degrees of freedo. I has a for: H (5). H (6) Then Halon's equaons: H (7) H. (8) Sae varales of he echancal syse fro Fg. are he dslaceens and oenu. The nec energy eressed y he oenu s: (9) oenal energy

4 63 Peer Svá and arna Hroncová / Proceda ngneerng 48 ( 0 ) Raylegh s dssave funcon (0). () Generalzed lnear force for vscous dang s eressed y Raylegh's dssave funcon relaonsh: generalzed forces () 0. (3) F quaons of he echancal syse are oaned n he for: d d d d ( ) d d ( ) H (4) H (5) H (6) In ar for: sae vecor s d d d d d d H. (7) () () F() A. B. () (8) 0 [ ] T X. (9) quaons of he echancal syse n ar for for ().v () are: d d dv dv A. v v F B. 0 (30)

5 Peer Svá and arna Hroncová / Proceda ngneerng 48 ( 0 ) and sae vecor [ v v ] T X. (3) We have deerned he equaons of sae of a echancal syse wh wo degrees of freedo for he sae varales and v v of resecve ojecs wh asses. 5. Sae Sace equaon n MATLAB/Suln Soluon of he nonhoogenous syse of dfferenal equaons of a echancal syse wh wo degrees of freedo s frs done n Mala/Suln usng Sae-Sace and Transfer Fcn locs [7] []. In conrol engneerng a sae sace reresenaon s a aheacal odel of a hyscal syse as a se of nu ouu and sae varales relaed y frs-order dfferenal equaons. To asrac fro he nuer of nus ouus and saes he varales are eressed as vecors. Addonally f he dynacal syse s lnear and e nvaran he dfferenal and algerac equaons ay e wren n ar for. The sae sace reresenaon (also nown as he "e-doan aroach") rovdes a convenen and coac way o odel and analyze syses wh ulle nus and ouus. Unle he frequency doan aroach he use of he sae sace reresenaon s no led o syses wh lnear coonens and zero nal condons. "Sae sace" refers o he sace whose aes are he sae varales. The sae of he syse can e reresened as a vecor whn ha sace. The os general sae-sace reresenaon of a lnear syse wh u nus y ouus and n sae varales s wren n he followng for (Fg. ): Fg.. Bloc dagra reresenaon of he sae sace equaons. For Sae-Sace equaons n for [7]: We oan: v v A. B u(). (3) C. u() y.. (33) A. v v F B. 0 (34) F y C... (35) v 0 v A B C are he resecve arces of he echancal syse defned as follows:

6 634 Peer Svá and arna Hroncová / Proceda ngneerng 48 ( 0 ) ( ) ( ) A B 0 C (36) 0 0 The nu u() s reresened y he ecaon forces: And he ouu y() s reresened y he dslaceens () and (): F u. (37) 0 () () y.. (38) v v Paraeers used n he rocess of soluon (Fg. 3) of he echancal syse were: 70 g 40 g 500 N/ 50 N/ 0 N/(.s - ) 50 N/(.s - ) F 0 00 N rad.s - nal condons (0) 0 v(0) 0 /s [-4]. Fg. 3 The loc dagra n Suln Ouu n Scoe loc - dslaceens and veloces v v are shown n Fg. 4. [] [] slaceen a of he eer and () () v [/s] v [/s] Velocy v and v of he eer a vv() vv() (a) [s] () [s] Fg. 4. Sulaon resus n Suln for (a) dslaceen () () and () velocy v () v (). 6. Transfer Fcn odel The oaned Sae-Sace reresenaon of he echancal syse can e n he Mala easly ransfored o an equvalen Transfer Fcn odel (Fg. 5) y ssf funcon. The syna of hs coand n Mala s:

7 Peer Svá and arna Hroncová / Proceda ngneerng 48 ( 0 ) >> [nu den]ssf(abc) Fg. 5 The loc dagra Transfer Fcn Solvng he echancal syse y usng Sae-Sace equaons and he Transfer Funcon (Fg. 6) gave he sae resuls as we ancaed slaceen a wh Transfer Fcn () () [] [] [s] Fg. 6 Sulaon resuls n Suln wh Transfer Fcn 7. Concluson The aer descres he colaon of he equaons of oon of a echancal syse wh wo degrees of freedo n Mala/Suln y usng Sae - Sace and Transfer funcon. Resuls are shown grahcally. The calculaon s done for a odel wh force ecaon n Mala and Suln o llusrae he ehodology. The conruon of hs wor s rarly educaonal esecally n he feld of Aled Mechancs and Mecharoncs. The aove rocedure resens he ossly of raccal leenaon of hs soluon o sle equaons of oon of a echancal syse n Mala/Suln. Acnowledgeen Ths conruon s he resul of he rojec leenaon: Cener for research of conrol of echncal envronenal and huan rss for eranen develoen of roducon and roducs n echancal engneerng (ITMS: ) suored y he Research & eveloen Oeraonal Progra funded y he RF. Ths wor was also suored n ar y he Mnsry of ducaon of he Slovaa Foundaon under Gran VGA /05/ and Gran VGA /089/. References [] Vavrníová V. Hroncová Modelovane ana v rosredí Suln. Aca Mechanca Slovaca Košce Slovaa. s ISSN [] Gero A Mecharoncs. lena Košce Slovaa. [3] Karan P Výoy a sulace v rograech Mala a Suln. Couer Press Brno eso. ISBN [4] Kozá Š. Kajan S Mala Suln I. STU Braslava Braslava Slovenso. [5] Ogaha K Syse ynacs. Prence Hall Inc. nglewood Clffs New Jersey. [6] Srado J. Mchalíe M. Mudrí J. Slavovsý J. 99. ynaa srojov. Alfa Braslava. Slovaa. ISBN [7] Horáe P Syses and odels. Naladaelsví VUT Praha eso. [8] Sega Š. Sega J. 0. Modellng and Ozaon of Vehcle susenson wh Magneorheologcal aers In: 7h Inernaonal Conference ynacs of Rgd and eforale Bodes 0 Úsí nad Lae 0. ISBN [9] Hroncová. 0. Sudes of coale sofware ools for asrac and concree desgn of echaronc syses and lnng he for he urose of logcal and hyscal odelng sseraon hess Košce. [0] Vole J. Sega Š. Souu J Analycý výoe verálních osuv rolejusu Šoda Tr ejezdu sousavy eáže dle SN ve sanovených odech. Výzuná zráva. 05/07 FVTM UJP v Úsí nad Lae. eso.