ASPECTS CONCERNING A DYNAMIC MODEL FOR A SYSTEM WITH TWO DEGREES OF FREEDOM
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1 Buetin of the Trnsivni University of Brşov Series I: Engineering Sciences Vo. 4 (53 No. - ASPECTS CONCERNING A DYNAMIC MODEL FOR A SYSTEM WITH TWO DEGREES OF FREEDOM M. BOTIŞ C. HARBIC Abstrct: In this rtice is resente ynic oe for syste with two egrees of freeo n the controer use for contro in osition. The ynic oe uses Lgrngen eutions n for rejection of erturbtions in osition of orienttion tht er wi be esigne one controer of tye PD in Siuin. After the nysis of the syste hving two egrees of freeo PD tye controer ws nyze for the contro of the syste in isceent for ge cinetic egree of freeo. Key wors: ynic oe controer PD syste of orienttion.. Introuction The syste of orienttion is one for coun on which is ut through two cinetic joints tfor with she ie is where sor nes (Figure re fixe. The is on which the sor nes re fixe hs two egrees of freeo (Figure.. Mteri n Methos For ction whoe syste on two egrees of freeo re use hyruic grou which ctions two hyruic otors one rotry n the other iner. The in urose of the two egrees of freeo is is to obtin the best energetic conversion of sor energy into eectric energy. Becuse the syste of orienttion wors in n environent where there re ny vritions ie win see seisic os teerture vritions it is necessry to esign coenstor for osition of orienttion ie PD controers. Syste of contro isoses of ny igit n nogue inut n outut rt of this inuts n oututs re use for ifferent roceures. Becuse the see of win is very rge owing to environent where the structure is ce there is roceure to reuce the surfce exose on irection of win tht bring the structure in contro osition. There re so roceures tht ow recoring infortion bout ccity of conversion of sor energy in eectric energy n reibe woring of hyruic syste. The in oveent is e in stes consiering tht the sor hour hs 5. In orer to hve n incience nge nor to the sor nes n higher erio of tie the iy rottion tes ce by oting isceent w which uring on 5 cross the stes cceertion - stey stte - eceertion. Therefore the in Det. of Civi Engineering Trnsivni University of Brşov.
2 48 Buetin of the Trnsivni University of Brşov Series I Vo. 4 (53 No. - oveent tes ce consiering the isceent w on erios of 5. After finishing stge between 8 n 8 the orienttion syste of sor nes is ten in the strt osition by rotting counter cocwise with one rottion of 6. The seson rottion oveent which eens on the seson tes ce between n 45 n it is erenicur on the in rottion oveent. Becuse this oveent is e between big erios of tie n is not necessry high ccurcy of ositioning we ecie tht for this egree of freeo to not erfor the isceent contro of orienttion syste of sor nes. During the execution of seson rottion the cceertions tht er re s ue to the fct tht e erios of cceertion n eceertion re one in ong erio of tie. Cery tht the rnge of vrition of inerti t the rottion oveent roun the in xis is very big becuse for ech osition between n 45 in the seconry rottion coue the isc on which re fixe the sor nes execute the in rottion oveent between 8 n 8. One of the robes tht occur t the construction of this in of structure is the trnsitory resonse of structure t cceertion n eceertion of syste ue to the fct tht for one coete oveent re necessry on verge 8 cceertions n eceertions of riving syste uring one y in the in coue of rottion [3]... Dynic oe for syste with two egrees of freeo n controer PID For the ynic oe in this er the uthor consiers tht concentrte ss n eeent of the syste re rigi eeents. The boies tht coose the syste cn be consiere rigi boies becuse when the coun n is for syste of orienttion sor ne ws esigne coex o nysis for coonents of structure ws Fig.. The structure hving two cinetic egrees of freeo ositione on the University hi
3 Botiş M. et.: Asects Concerning Dynic Moe for Syste with Two Degrees of Freeo 49 erfore. Perio for coonents of structure is T <. s in this wy coonents cn be consiere s rigi boies [] (Figure. Fig.. Dynic oe - for syste with two egrees of freeo with concentrte sses n rigi eeents Cinetic reters-osition ngur see n ngur cceertion for ss n re: - osition: r r ( - ngur see: ω ω ( - ngur cceertion: ε ε (3 - iner see:. v v (4 Kinetic energy for boy with ss is:. ( z E (5 Kinetic energy for boy with ss is:. ( ( E x z y Grvittion energy for boy with ss is: U g. (6
4 5 Buetin of the Trnsivni University of Brşov Series I Vo. 4 (53 No. - Grvittion energy for boy with ss is: U g(. Generize force tht ctions in the joints of the structure re obtine through Lgrnge eutions: t L L Q. (7 where: L - ngrngen for syste of two boies - generize coorinte - generize veocity Q - generize force tht ctions in the joints of structure. Generize force tht ctions in the first joint: Q z ( g( z ( y y z ( z g Generize force tht ctions in secon joint: Q x ( ( y z z g. The schee for the PD controer use to contro the two egrees of freeo syste is resente in Figure 3. The PD controer [] ws esigne bse on the w of oentu ccuus. The. oentu vue generte by the PD controer esigne in Siuin. In orer to eterine K n K v constnts the foowing schee ws reize (Figure 3. Moveent ws in isceent veocity n cceertion were iorte fro MATLAB rogr (Figure 4: τ M ( ( u ig( τ M ( ( ig( u N( e ig( ig( e e N( (8 where τ the vector of generize oents in the coue M( is the syste s trix of inerti N ( is the noniner ters vector is the esigne cceertions vector ig( the igon trix fore by the controer erivtive tune reters ig( the igon trix fore by the controer roortion tune reters e is the errors vector e is the veocity errors vector. The error vrition w in the cse of the PD controer is reresente in Figure 4b: τ M ( ( ig( τ M ( N( τ e ig( ig( e N( e ig( e M ( τ (9 where is the vector of esurbe cceertions in the coue τ is the vector of isturbing oents. The tune reters of the PD controer re eterine for the conition s the syste to be criticy e: V ς ςω V ω ω. (
5 Botiş M. et.: Asects Concerning Dynic Moe for Syste with Two Degrees of Freeo 5 Fig. 3. PD contro for one egree of freeo for the orientting sor nes structure resonnce henoen t ω ω s / where ω s is the own freuency of the structure. The biggest irecision fro the oveent w re in the cceerting n eceerting erios of the syste ue to inerti forces. 3. Concusions Fig. 4. Acceertion veocity isceent-controer PD Fig. 4b. Positioning error-controer PD Increg the ω freuency es to ecreg the erturbtion M (τ which hs to be rejecte. The vue of ω ust be suerior iite in orer to voi the Configurtion of the stiffness structure for the yon s for the tfor on which the sor nes re fixe ws one in orer to iniize the retive isceents. Consiering s isceents it yies tht the eeents of the structure cn be consiere rigi in this wy the ynic egrees of freeo re becoing iortnt. In orer to consier the orientting structure s being coose of rigi boies the own erio of the structure s eeents ws uner. s. Becuse on the first ynic egree of freeo the oveent is rre n the oveent w in cceertion veocity n isceent oes not iy significnt vrition of the ction force on the secon egree of freeo in orer to irove the contro reters ynic oe with ge egree of freeo ws consiere the one tht the oveent uring the y tes ce.
6 5 Buetin of the Trnsivni University of Brşov Series I Vo. 4 (53 No. - PD contro in osition for the tfor owe increg the ositioning recision in orer to xiize the over energetic efficiency. For the cse in which win cts erenicur on the tfor roceure ws crete to reuce the exose surfce by orientting the tfor re to the groun. In orer to consier so in this cse the two egree of freeo syste s ge cinetic egree of freeo syste the oveents re successive one on ech cinetic egree of freeo. By nyzing the obtine resuts it yies tht the sest ositioning error is obtine in the cse of PD controer hving otiize tune reters. In orer to vnce the ositioning recision one cn consier increg the orientting structure s eeents stiffness by roer configurtion. References. Botiş M.: Meto eeentuui finit (The Finite Eeent Metho. Cuj Noc. Noc Str Pubishing House.. Ogt K.: Moern Contro Engineering. 5 th Eition. New ersey. Pretince-H Sorensen A..: A Survey of Dynic Positioning Contro Systes. In: Annu Reviews in Contro 35 ( Issue
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