DYNAMIC ANALYSIS OF A CONSTRAINED ROTATING FLEXIBLE EXTENSIBLE LINK WITH SEMI-PERIODIC IMPACT

Transcription

1 11 th Internatonal Conference on Vbraton Problems Z. Dmtrovová et al. (eds.) Lsbon, Portugal, 9-12 September 2013 DYAMIC AALYSIS OF A COSTRAIED ROTATIG FLEXIBLE EXTESIBLE LIK WITH SEMI-PERIODIC IMPACT Mha Dupac* 1, Samak orooz 2 1 Bournemouth Unversty School of Desgn, Engneerng and Computng Talbot Campus, Fern Barrow, Poole, Dorset, BH12 5BB, UK mdupac@bournamouth.ac.uk 2 Bournemouth Unversty School of Desgn, Engneerng and Computng Talbot Campus, Fern Barrow, Dorset, BH12 5BB, UK snorooz@bournemouth.ac.uk Keywords: Extensble Mechansm, Perodc Impact, Flexblty, Clearance. Abstract. In ths paper the modelng of an extensble mechansm wth a rgd crank and a flexble lnk subject to sem-perodc mpact s consdered and ts dynamcal behavor analyzed. The flexble extensble lnk havng one end constraned to a predefned trajectory, rotate wth a constant angular velocty n a horzontal plane. Sem-perodc mpact occurs between the non-constraned end of the flexble extensble lnk and a rgd support. The dynamc evoluton of the system s nvestgated and the flexural response of the flexble lnk analyzed under the combned effect of flexblty, sem-perodc mpacts and clearance.

2 1 ITRODUCTIO The modellng and smulaton of rotatng mechancal and robotc systems have receved attenton for several years. The study of ther dynamcs was consdered mportant both from a desgn perspectve, ncludng applcatons such as robot manpulators and helcopter rotors, as well as for ther dynamc stablty. Snce the majorty of those mechancal models contan flexble or rgd parts connected by jonts, ther dynamcal response s affected by the parts deformaton, clearance or mpact between the components, whch results n a medum to low ablty to perform precson manpulaton, ncreased components and assembly vbraton and subsequently, hgh levels of nose. Due to the dynamcal stress caused by the moton and the mpact of the parts whch affect the vbraton characterstcs of the mechancal system the mechancal system may fal or perform at a very low capacty. Interestng revews about mpact dynamcs and control of flexble-jont and dual-arm robots and extensble members can be found n the papers gven by [1-4]. The mpact control of a flexble lnk and the combned vbraton control of a flexble lnkage mechansm for dampng ts vbratons and mpact effects have been dscussed n [5, 6]. The lnear control a slender flexble beam - attached to a rgd lnk of varable length - undergong rotatons and perodc mpacts was studed n [7]. Rotatng extensble flexble beams have also been studed as part of the modellng, smulaton and control of robotc systems. The dynamc behavour of a translatng flexble beam wth a prsmatc jont and rotatonal moton was studed n [8] and an expermental verfcaton was consdered n [9]. A dynamc fnte element modellng of a translatng and rotatng flexble lnk has been studed n [10], and a study regardng the free vbraton of an extensble rotatng beam was consdered n [11]. Classcal examples of flexble mechancal systems, ther forward, nverse and mpulsve dynamcs, and ther stablty have been presented n [12-14]. The study presented n [15] presents a dynamc analyss of a planar mechansm wth slder jonts and clearance. The smulaton of the non-smooth translatonal jonts wth clearance has receved attenton n the work of Zhuang and Wang [16]. A dynamcal analyss of some classcal mechancal systems wth lumped masses and mpact was performed n [17, 18]. Mechancal systems wth mpact, clearance and dfferent types of contact force models has consdered n [19-23]. In ths paper the modellng and smulaton of an extensble mechansm wth a rgd crank and a constraned flexble lnk subject to sem-perodc mpacts was consdered. Smulatons have been performed a double ellptc-crcular/ellptc constraned trajectory n order to explore mechansm behavour under sem-perodc mpacts. A clearance vs. a non-clearance model of the extensble mechansm wth a flexble lnk was consdered n order to analyse the effect of clearance on the dynamcal behavour of the system. 2 SYSTEM MODEL A flexble extensble lnk composed of a rgd gude of length l CD and a flexble lnk denoted by PS, rotate about ts fxed end C as shown n Fg. 1. A Cartesan reference frame Oxy havng the orgn at O and the unt vectors and j s consdered. The flexble lnk, modelled as n [24] usng n successve equal rods P P 1(where P 1 =P and P n =S) connected wth torsonal sprngs, can slde nsde the gude CD. Each one of the rgd rods of the flexble lnk has the mass m m and moment of nerta J, and each one of the sprngs used to model lnk flexblty has the stffness k EJ lps computed as n [24, 25], where E s the Young modulus and 2

3 l PS n 1 l PP 1 s the length of the flexble lnk. The length l CP represents the dstance between the end C of the gude and the constraned trajectory of the system. Fgure 1: Constraned Flexble Extensble Lnk wth Rgd Support and sem-perodc Impacts Model. An ellptc-crcular/ellptc double trajectory s consdered. The constraned trajectory s a double ellptc-crcular/ellptc trajectory composed of two trajectores one ellptcal and one half ellptcal half crcular as shown n Fg. 1. The ellptcal trajectory has the centre at the orgn O of the Cartesan frame and has the transverse dameter of A A of length d A A, and the conjugate dameter F F of length d F F. The crcular/ellptc trajectory has the rght sde a half ellpse and the left sde a half crcle havng the dameter equal wth the conjugate dameter F F of the ellptcal trajectory. The rgd rod P P s 1 2 lnked to the constraned trajectory wth a slot-jont at P 1. The angles between the lnks P 1 and P 1P 2 denoted by are named relatve angles (for any 1, n 1), whle the angles between the lnks P P 1 and the horzontal drecton denoted by are named absolute angles. The angle between the gude and the horzontal drecton s denoted by, and n the case of a flexble lnk wth no clearance s always equal wth the angle 1. A motor torque M 1 acts on end C of the rgd gude of the mechancal system. The mpacted rgd lnk BE of length l BE can rotate around the fxed pont B as shown n n Fg. 1. The rgd lnk s connected to the ground through a sprng havng the stffness k BE, such that, after the mpact between the flexble lnk and the rgd fxed (mpacted) lnk, the lnk oscllatons damped rapdly to zero before a new mpact wll occur. 3 DYAMIC MODEL OF THE FLEXIBLE AD IMPACTIG LIK The dynamc model for the constraned flexble lnk as well as for the rgd mpactng lnk can be expressed based on the poston, velocty and acceleraton vector of the centre of the mass C of each rgd rod P P 1, 1, n of the lnk and rgd mpactng lnk respectvely. The poston centre of lnk PP 1s gven byrc xc yc j, where and j are the unt vectors of P 3

4 the assocated Cartesan reference frame Oxy. The velocty and acceleraton vector of the center of the mass C of each rgd rods PP 1of the constraned flexble extensble lnk s the dervatve and respectvely the double dervatve wth respect to tme of the poston vector and s gven by v C r x y j and respectvelya r x y j. C 3.1 Flexble Lnk Dynamc Model wth no Clearance C C For the no clearance model, when the flexble lnk translates parallel to ts support, the dynamcs of the flexble extensble lnk can be expressed usng the Lagrange dfferental equaton of moton C C C C r C d T T Q dt q (1) q where Q are the generalzed forces, T s the total knetc energy of the system expressed as T n 1 T n 2 2 m v C IC T s the knetc energy of each th lnk,, q are the generalzed coordnates, the subscrpt represents the number of the generalzed forces/coordnates. The model descrbed n [17] was used to descrbe the generalzed forces actng on each lnk. 3.2 Flexble Lnk Dynamc Model wth Clearance The same clearance model consdered n [24], where the flexble lnk of the extensble can translate and rotate about ts support, was used for ths study. Due to the clearance model, the flexble lnk may mpact the rgd support as shown n Fg. 2. Fgure 2: Constraned Flexble Extensble Lnk wth Rgd Support and Clearance Model, (a) o Impact, (b) Impact on one Pont, (c) Impact on two Ponts 4

5 The mpact model between the flexble lnk and the gude s shown n Fg. 5. The man mpact cases are a) no mpact between the flexble lnk and the gude (lnk support), b) mpact of the flexble lnk on a sngle pont of the gude, and, c) mpact of the flexble lnk wth two ponts of the gude. For ths study, multple mpacts have not consdered but only presented snce mpacts at the same tme nstant can be statstcally excluded. The dscontnuous model, used n ths paper to evaluate mpacts, assumes that the mpact occurs nstantaneously and that no change of the system confguraton occurs durng contact. The ntegraton of equatons of moton s halted at the tme of mpact and a momentum balance s performed to calculate the post mpact veloctes of the system components. The resttuton coeffcent s employed to quantfy the dsspaton energy n the process. Dfferent types of mpact and contact force models have been dscussed n [7, 17] and [19-23]. To derve the equaton of moton wth mpact a smlar approach as n [7, 17] was consdered. The mpact dfferental equaton of moton can be wrtten as T T P, (2) k u k u k where T s the total knetc energy of the system, ts T u k ta are the generalzed momenta, P k are the generalzed mpulses assocated wth the coordnate u k, t a and t s represents the moments of tme of approach and separaton, respectvely before and after the mpact. Consderng t Rdt Ry j the force exerted durng the mpact (only a vertcal component s consdered n t s a ths study) by the flexble lng at contact, the generalzed mpulses can be expressed by ts v Pk Rdt where the velocty of the mpact pont v can be expressed q k ta as v vm ωr. Usng ewton s coeffcent of resttuton e, the velocty of approach and separaton for the mpact ponts (on the flexble lnk) and M (on the mpacted lnk) can be expressed as v v v, v v v, (3) where v t a and v, M t a a v and t s t M a t a v M t s s t M s t s are the flexble lnk velocty and the mpacted lnk velocty at tme t a and t s before and after the mpact. Usng Eq. (3) and the defnton of the coeffcent of resttuton e (usng ewton s formulaton), one can wrte ev v. (4) 4 SIMULATIOS AD RESULTS a The results from the computer smulatons are presented n ths secton usng the followng numercal values: the length of the gude l CD = 0.09 m, the length of the flexble lnk l PS = 0.07 m, and the clearance model, and the clearance between the gude and the extensble lnk (one sde) of m, chosen to accentuate the mpact effect between the gude and the flexble lnk. The next materal propertes are consdered for the gude and for the movng parts: densty (kgm 3 ) 7850, Young modulus Pa and Posson Rato 0.3. The rotatng end C of the gude s located on the Ox axs at 0.02 m from the orgn O of the Cartesan reference frame, where the centre of the ellptc-crcular/ellptc constraned trajectory s located. The prncpal axes of the ellptc constraned trajectory have the next length: the transverse dame- s 5

6 ter of 0.08 m, the conjugate dameter of 0.06 m, and the sem-crcle radus of 0.03 m. The mpactng rgd lnk s located at m from the orgn O of the Cartesan reference frame. The dynamcal behavour of the extensble mechansm wth a flexble lnk, no clearance and sem-perodc mpacts due to ts constraned ellptc-crcular/ellptc trajectory s shown n Fg. 3.a, that s, the Ox and Oy trajectores of the end S of the constraned extensble lnk plotted vs. the crank angle. From Fg. 3.a one can observe the moment of mpact and the mmedate perturbaton the system exhbt due to the mpact, as well as the tme frame n whch the system damped the perturbaton. Snce perturbaton of the trajectory s an mportant factor whch affects system dynamcs, the behavour of the extensble flexble lnk consdered for the ellptc-crcular/ellptc constraned trajectory (shown n Fg. 1) and subject to semperodc mpact was consdered for a clearance vs. a non-clearance model. (a) (b) Fgure 3: Dynamcal behavour of the extensble flexble lnk wth sem-perodc mpacts. The Ox (green and blue colour) and Oy (red colour) trajectores of the end S of the constraned extensble flexble lnk for: (a) no clearance, (b) clearance The dynamcal behavour of the extensble flexble lnk wth clearance and sem-perodc mpacts s shown n Fg. 3.b, that s, the Ox and Oy trajectores of the end S of the constraned extensble lnk plotted vs. the crank angle. It was observed that the clearance model add more exctaton to the system as one can see on both the exctaton tme as well as the ampltude of the exctatons whch can be clearly observed on Fg. 3.a and Fg. 3.b. (a) (b) Fgure 4: Dvergence d of the trajectores of the moton (a) no clearance of the flexble lnk, (b) clearance of the flexble lnk The dynamcal behavour of the mpacted rgd lnk s shown n Fg. 4.a and Fg. 4.a for the clearance vs. a non-clearance model respectvely. One can observe that the magntude of 6

7 oscllatons decrease n tme because of the dampng as well as because the elastc part s retractng nsde the rgd gude. It was also observed that for the clearance model, the ntal ampltude of the oscllatons s a lttle bt hgher when compared wth the no-clearance model. Fgure 5: Dynamcal behavour of the extensble flexble lnk wth sem-perodc mpacts plotted vs. the crank angle n a polar coordnate system. Trajectory of the end of the constraned extensble flexble lnk wth: (a) no clearance, (b) clearance The dynamcal behavour of the extensble mechansm wth a flexble lnk, no clearance and sem-perodc mpacts are plotted vs. the crank angle n a polar coordnate system as shown n Fg. 5. From Fg. 5.a and Fg. 5.b one can observe that the trajectores of the extensble mechansm wth a flexble lnk and wth clearance as well the one wthout clearance dverges n tme, that s, a separaton of nearby trajectores and a clear sgn of nonlnear moton. It was observed also that the clearance model add more exctaton to the system, both the ampltude of the exctatons and trajectores separatons are clearly vsble n ths case. One can conclude that the tme evoluton of the system s manly affected by the semperodc mpacts as well as by lnk flexblty, clearance and velocty, that s, the combned effect of lnk flexblty and clearance accentuate trajectory dvergence and affect system stablty. 5 COCLUSIOS In ths paper the modellng of an extensble mechansm wth a double constraned flexble lnk and a rgd crank subject to sem-perodc mpacts s presented and ts behavour analysed. Accurate smulatons for an ellptc-crcular/ellptc constraned trajectory are performed. A dynamcal analyss s carred out n order to compare the dynamcal response of the flexble lnk wth clearance vs. no clearance under the combned effect of the flexblty and semperodc mpacts. Trajectores dvergence has been observed for the constraned extensble mechansm wth and wthout clearance and sem-perodc mpacts, however, t was observed that the system behave more unstable n the clearance case. It was concluded that parameters such as clearance, rotatonal velocty, mpact and flexblty have a great nfluence on the dynamcal stablty of mechancal system performance and stablty. Expermental tests wll be performed n order to valdate and generalze the smulatons reported n ths study. 7

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