Techncal Report TR05 An Introducton to the Floatng Frame of Reference Formulaton for Small Deformaton n Flexble Multbody Dynamcs Antono Recuero and Dan Negrut May 11, 2016
Abstract Ths techncal report descrbes the bass of the nonlnear fnte element floatng frame of reference formulaton. Keywords: Floatng Frame of Reference Formulaton, Nonlnear Fnte Elements, Flexble Multbody Systems 1
Contents 1 Introducton 3 2 Floatng Frame of Reference 3 2.1 Knematcs of the FFR.............................. 3 2.2 Equatons of Moton............................... 4 3 Fnte Element FFR Formulaton 7 3.1 Local Deformaton and Reference Frames.................... 8 3.2 Descrpton of Local Deformaton........................ 9 3.3 Reference Condtons............................... 9 3.4 Knematcs of an FFR Fnte Element...................... 10 3.5 Model Order Reducton............................. 10 2
1 Introducton The dea of a floatng frame of reference (FFR) may have ts orgn n the analyss of astrodynamc systems. Such systems experence strong couplng of nerta forces, due to hgh acceleratons n ts reference (rgd body) moton, and flexblty due to the lght nature of the structures nvolved (see Ref. [2]). The dea of a body-wse co-rotated frame or floatng frame can be summarzed as the splttng of the overall moton of bodes that experence small deformaton nto a frame that captures the dynamcs of mean rgd body moton and a superposed flexble moton that defnes the deformaton state. Ths method requres one flexble body to have one sngle FFR to descrbe ts dynamcs, n contrast wth the co-rotatonal formulaton. In addton, the fact that the body s (structure s) elastcty can be defned va a constant stffness matrx opens up the possblty of usng general model order reducton technques to reduce the order of a lnear fnte element model. In what follows, the bass of the FFR formulaton s presented, whereas many dervatons are left to the reader as an exercse or gven as a reference. 2 Floatng Frame of Reference 2.1 Knematcs of the FFR From now on, we wll assume that Euler parameters wll be our choce of rotaton coordnates. Even though other choces can be made, t s, n any case, mportant to guarantee that sngulartes wll be avoded. The knematc state of the frame of reference of a body may be wrtten as [ qr = R T θ TT], (1) where q r s a set of Cartesan coordnates that locates the orgn of the FFR of body, O, wth respect to the global frame, θ s a set of Euler parameters that descrbe the orentaton of such a frame n the global coordnate system. A pont P n body can be descrbed, usng FFR knematcs, as r P = R + A ( θ ) ū P, (2) where A denotes the orentaton matrx of the FFR and ū P s the vector that defnes the poston of pont P wth respect to the FFR (superscrpt s dropped for smplcty). Ths vector can be splt nto ts reference poston and ts elastc dsplacement as follows ū P = ū P o + ū P f. (3) ū P o represents the undeformed poston of P, whereas ts elastc dsplacement, measured n the FFR, s denoted by the vector ū P f (see Fg. 1). The elastc dsplacement vector ūp f may be decomposed nto the product of a space-dependent matrx and a vector of tme-dependent flexble coordnates, as follows ū P f = S ( ) ū P o q f, (4) 3
where S (ū P o ) s a shape functon matrx referred to the reference (undeformed) confguraton and qf s the vector of flexble coordnates of body. By spellng out the knematc equatons n (3) and (4), the expresson of the deformed poston of the pont P n body may be wrtten as rp = R + A ( ) ū P o + S qf (5) Velocty expressons of a deformed pont P wthn the FFR formulaton are gven below: ū P f = S ( ū P o ) q f, (6) ṙ P = Ṙ + A ( ū P o + S q f) + A S q f. (7) Note that Eqs. (6)-(7) take nto account that ū P o and S do not depend on tme. Fgure 1: Basc knematcs of the FFR formulaton. The floatng frame axes are denoted by X Y Z, the ntal, unstressed confguraton s depcted wth sold lnes, whereas the current confguraton translated, rotated, and deformed s shown n dashed lnes. 2.2 Equatons of Moton Here, we present the dervaton equatons of moton of the system n abbrevated manner, omttng some dervatons. The nterested reader can consult the book by Shabana [8] to fnd out more on some detaled dervatons of the FFR formulaton equatons. Before movng on wth dervatons, we arrange, for convenence, the velocty vector n the followng form: Ṙ ṙp = [ I B A S ] θ = L q, (8) where matrx B s a lnear operator that acts on the rotaton coordnates, as follows A ū P = B θ, and B = A ū P Ḡ. The tlde denotes the skew symmetrc matrx 4 q f
operaton on a vector, and G s a lnear operator that transforms tme dervatves of rotaton parameters to the angular velocty vector of the reference moton of body. The form of G depends upon the selecton of rotatonal parameters. The equatons of moton of the FFR formulatons are to be derved usng Lagrange s equatons. For a flexble body descrbed usng a floatng frame, these equatons take the form d dt ( ) T T ( T q q ) T + C T q λ = Q e + Q a, (9) where T s the knetc energy, q and q are the generalzed coordnates and veloctes of body, respectvely, C q s the Jacoban of the constrant equatons, λ s the vector of Lagrange multplers, and Q e and Q a are the vectors of elastc and appled forces, respectvely. Note that, n general, the vectors of Lagrange multplers and appled forces may lnk the flexble body wth other (flexble) bodes and/or other nteractng systems. Mass Matrx The total knetc energy of body may be expressed as a volume ntegral: T = 1 ρ ṙ T 2 P ṙp dv (10) whch, n terms of the generalzed coordnates, may be wrtten as V T = 1 2 qt M q. (11) The mass matrx M s nonlnear n the rotaton coordnates and takes the followng form M = V ρ I B A S m RR m Rθ m Rf B T B B T A S dv = m θθ m θf, (12) sym. S T S sym. where submatrces have been named for convenence. Cross terms n the mass matrx, e.g. m Rf and m θf, ndcate that couplng between reference and flexble coordnates s captured. The Quadratc Velocty Vector The FFR formulaton ntroduces complex nerta terms whch represent Corols and centrfugal forces. Dervng the nerta terms of Lagrange equatons n Eq. (9), one obtans m ff ( ) d T T ( ) T T [ ( = M q + Ṁ q q T M q ) dt q q q } {{} Q v(quadratc velocty vector) ] T (13) 5
[ The fnal expresson for the quadratc velocty vector Q v = (Q v ) T R (Q v ) T θ terms of generalzed coordnates, reads ] T, (Q v ) T f n ( Q v ) ( Q v ) R = A ρ θ = ḠT ρ ( ) Q = v f ρ V V V S T [ ) ] ( ω 2ū + 2 ω S q f dv (14) [ ū T ) ] ( ω 2ū + 2 ū T ω S q f dv (15) [ ) ] ( ω 2ū + 2 ω S q f dv (16) and may be obtaned by dervng equaton (13) or va the Prncple of Vrtual Work. The Euler parameter dentty Ḡ θ = 0 s used above to smplfy the expressons. Checkng the correctness of (14) s left to the reader as an exercse. As an advanced topc, the ready may learn on the accurate calculaton of FFR velocty-dependent nerta terms n [9]. Appled Forces To nclude forces and moments n the system t s necessary to obtan ther generalzed counterparts by usng FFR knematcs. As an example, here we descrbe how to compute the generalzed force assocated wth a pont force. The vrtual work of a force descrbed n the global frame, F a, appled at P may be expressed as (17) δw a = F aδr P = Q aδq. (18) As a frst step, we wrte the varaton of the poston vector of a pont P n an FFR body as δrp = [I A ū ] δr Ḡ A S δθ. (19) Equaton (19) can be plugged nto (18) to yeld the followng generalzed force vector ( Q a ) R = F P a, ( Q a ) δq f θ = ḠT ū T A T F P a, ( Q a ) f = ST A T F P a, (20) where space-dependent vector and matrx, ū and S, respectvely, must be partcularzed at the poston of pont P. Note that the FFR formulaton does not mpose lmtatons as to the locaton or nature of the load. 6
Equatons of Moton In general form, the FFR equatons of moton of a body can be wrtten as follows M q + D q + K q + C T q λ = Q a + Q v, (21) whch, n turn, may be expressed n terms of submatrces as m RR m Rθ m Rf R 0 0 0 Ṙ 0 0 0 R m θθ m θf θ + 0 0 0 θ + 0 0 0 θ + sym. m ff q f 0 0 D ff q f 0 0 K ff q f C T R (Q a ) R (Q v ) R C θ λ = (Q a ) θ + (Q v ) θ. C T q f (Q a ) q f (Q v ) q f The new terms K ff and D ff refer to the stffness and dampng matrces of the structure. These two matrces may be obtaned by modelng the structure as a lnear system usng, for nstance, the fnte element method. 3 Fnte Element FFR Formulaton As shown n the prevous secton, the FFR formulaton leads to a local lnear problem that enables the use of classcal fnte element technology: Flexble coordnates can be consdered as fnte element degrees of freedom. Wthn the context of FFR, fnte elements contanng nfntesmal angles stll accurately descrbe rgd body moton. Ths fact, together wth the use of ntermedate element coordnate systems, allows to obtan exact modelng of rgd body nerta for flexble bodes. The ntroducton of ntermedate element frames, as descrbed by Shabana n [8], elmnates the ssue of ensurng that actual small strans are seen as small strans when descrbed n the body s floatng frame of reference. Complex geometrc shapes can be modeled by usng one sngle, co-rotated floatng frame per body due to the ntroducton of ntermedate frames. The co-rotatonal formulaton, whch uses element-wse coordnates to descrbe the state of fnte elements, also appeared due to need for modelng complex geometres [4]. The floatng frame of reference must be attach to the flexble body by prescrbng a set of condtons that wll ensure that no rgd body s degrees of freedom reman n the structure. That s, the FFR s degrees of freedom must descrbe reference moton, whereas the fnte elements coordnates must only descrbe dsplacements that lead to deformaton. Ths s accomplshed by mposng reference condtons, whch lnk the fnte element model to the FFR. The success of the FFR n practcal multbody applcatons owes much to the fact the t enables the use of lnear model order reducton technques, that s, numercal methods amng 7
at reducng the number of coordnates of a lnear system whle preservng statc/dynamc nformaton of the orgnal, large-scale model. Ths dea wll be elaborated on n a later subsecton. 3.1 Local Deformaton and Reference Frames We wll ntroduce a more complete notaton n ths subsecton to refer to fnte elements. The poston of a fnte element j n a body s expressed as follows: where e j 0 and e j f e j = e j 0 + e j f, (22) denote the rgd body and flexble coordnates of a fnte element j n a body. Several coordnate systems needed for a general problem are ntroduced: 1) X Y Z s a body coordnate system that represents the overall moton of the body but need not be rgdly attached to the flexble body. 2) An element-wse reference system X j Y j Z j s rgdly attached to the fnte element j. 3) Fnally, at the locaton of the body coordnate system, an ntermedate element coordnate system wth axes ntally parallel to X j Y j Z j s placed, whch s denoted by X j. Ths ntermedate frame s necessary to guarantee exact modelng of rgd body knematcs [8]. A way to nterpret the need for ths system s the followng: Snce the FFR formulaton assumes one sngle reference frame per body, there s a need to ensure that we account for the reference orentaton at the unstressed confguraton. Note that the ntermedate element coordnate system does not ntroduce addtonal coordnates n the formulaton snce t s defned by the geometry of the flexble body at the ntal confguraton. Y j Z j Fgure 2: The ntermedate frame of reference X j Y j Z j s shown n the fgure as a rotated coordnated system whch s parallel to the ntal fnte element frame, X j Y j Z j. The reference frame X j Y j Z j allows the transformaton of the fnte element coordnates n the element system, e j, nto the body reference frame, qj n. 8
3.2 Descrpton of Local Deformaton Connectvty condtons n the flexble body are establshed, as n any fnte element model, by dentfyng the nodes shared by elements. From now on, the connectvty problem wll be dsregarded and the dervatons wll be presented element-wse, wthout loss of generalty. To descrbe the knematcs of the deformaton of a fnte element j n a body, we need frst to defne the orentaton matrx descrbng the orentaton of the ntermedate element coordnate system wth respect to the body coordnate system or floatng frame, C j. The assumed dsplacement feld w j of an element can be expressed n the body frame as ū j = C j w j = C j S j e j, (23) where S j s the shape functon matrx of an element j n a body and e j element j s coordnates. Note that w j s the vector of s defned wth respect to the ntermedate element may be defned n the coordnate system (see Fg. 2). The vector of nodal coordnates e j body coordnate system as = C jt qn j, (24) n whch qn j s the vector of element j s nodal coordnates expressed n the body coordnate system. Makng use of Eqs. (23)-(24), one gets e j ū j = C j w j = C j S j C jt q j n. (25) Equaton (25) s key snce t expresses the poston coordnates of an arbtrary pont on the fnte element wth respect to the orgn of the body coordnate system [7]. 3.3 Reference Condtons Cartesan and rotatonal coordnates descrbe the locaton and atttude of the floatng frame of reference. Flexble coordnates, on the other hand, descrbe small deformaton of the body. However, so far, we have skpped descrbng how to elmnate the rgd body modes assocated wth the fnte element coordnates. In other words, the model descrbng the deformaton of the flexble body must be attached to the floatng frame by usng a set of reference condtons that must fully elmnate relatve rgd body moton. Reference condtons can mposed n dfferent ways; some of them are summarzed below: Free body. If the flexble body s not connected to other bodes, a sensble soluton to the problem of assgnng reference condtons to the flexble body s to keep the orgn of the coordnate system at the nstantaneous mass center and, smultaneously, ensure that ts axes are prncpal, that s, they defne the three products of nerta reman zero as the body deforms [1]. The reference condtons for mean axes may be mposed by selectng a proper set of algebrac constrants [6]. Clamped condtons. A smpler way of defnng reference condtons s by fully constranng the body coordnate system to be located and orented as one node at a fnte element. For nstance, the body frame may be fully constraned to the node lnkng a rotor blade to the rotor tself. 9
Smply supported. For beams, t s possble to attach the body coordnate frame to the flexble body n such a way that ts locaton s placed at one end of the beam and one of ts axes always ponts to the other end. Note that ths s the case of many robotc flexble manpulators. The choce of reference condtons must be consstent wth the problem at hand snce, n a practcal multbody system problem, the way flexble bodes are connected to other bodes may determne the way they deform. In other words, reference condtons must be selected such that they respect both the boundary condtons of the flexble body wthn the system and the separate model used to descrbe the lnear deformaton problem (e.g. the fnte element model). 3.4 Knematcs of an FFR Fnte Element For convenence, we rewrte Equaton (25) n the followng manner ū j = N j q j n, (26) where N j s a space-dependent matrx: N j = C j w j = C j S j C jt. Usng the new knematc descrpton that accounts for the use of fnte element technques, the poston vector r j of an arbtrary pont on element j of a body may be wrtten as r j = R + A ū j = R + A N j qn j = R + A N ( ) j qo + Brq j f, (27) where qo s the vector of nodal coordnates n the undeformed state,.e. they locate materal ponts wth respect to the body frame n the unformed state, qf s the vector of nodal deformatons, and Br s a matrx that may apply a set of reference condtons on the deformaton coordnates, e.g., n order to attach a node to the locaton of the reference frame. Equaton (27) may be dfferentated wth respect to tme to obtan ṙ j = Ṙ + A ( ω ū ) + A N j B r q j f. (28) Note that n Eq. (28) only matrx N j has spatal dependency. The knetc and elastc terms of the equatons of moton of the fnte element FFR formulaton can therefore be obtaned usng Eqs. (27)-(28). A set of nerta shape ntegrals nvolvng the fnte element shape functons and ther relatve orentaton wth respect to the body frame of reference may be calculated at a preprocessng stage as descrbed n Ref. [8]. Usng the knematc equatons presented n ths subsecton, Eqs. (12), (14), (20), and (21) can be rederved for the fnte element FFR formulaton. 3.5 Model Order Reducton The equatons of moton of the fnte element FFR formulaton may contan an excessve number of coordnates owng to the detaled fnte element models used to descrbe a flexble 10
body s dynamcs. The sngle frame of reference ntroduced to capture flexble body dynamcs allows for a separaton of rgd body and flexble coordnates. Snce flexble coordnates n the FFR formulaton descrbe lnear deformaton, any model order reducton technque may be appled. Tradtonally, methods such as Component Mode Synthess, Guyan condensaton, and purely egenmode analyss have been profusely employed to reduce the number of coordnates of the flexble body. In the past decade, there has been an nterest for the use of modern model order reducton technques n multbody system dynamcs, see [3, 5]. Proper use of model order reducton allows to analyze complex system at a hgh level of accuracy and reduced computatonal cost. References [1] OP Agrawal and AA Shabana. Applcaton of deformable-body mean axs to flexble multbody system dynamcs. Computer Methods n Appled Mechancs and Engneerng, 56(2):217 245, 1986. [2] B Fraejs De Veubeke. The dynamcs of flexble bodes. Internatonal Journal of Engneerng Scence, 14(10):895 913, 1976. [3] Jörg Fehr and Peter Eberhard. Smulaton process of flexble multbody systems wth non-modal model order reducton technques. Multbody System Dynamcs, 25(3):313 334, 2011. [4] CA Felppa and B Haugen. A unfed formulaton of small-stran corotatonal fnte elements: I. theory. Computer Methods n Appled Mechancs and Engneerng, 194(21):2285 2335, 2005. [5] P Koutsovasls and M Betelschmdt. Comparson of model reducton technques for large mechancal systems. Multbody System Dynamcs, 20(2):111 128, 2008. [6] Parvz E Nkravesh and Y-shh Ln. Use of prncpal axes as the floatng reference frame for a movng deformable body. Multbody System Dynamcs, 13(2):211 231, 2005. [7] AA Shabana. Dynamcs of large scale flexble mechancal systems. PhD thess, Ph. D. dssertaton, Unversty of Iowa, Iowa Cty, 1982. [8] Ahmed A Shabana. Dynamcs of multbody systems. Cambrdge unversty press, 2013. [9] Karm Sherf and Karn Nachbagauer. A detaled dervaton of the velocty-dependent nerta forces n the floatng frame of reference formulaton. Journal of Computatonal and Nonlnear Dynamcs, 9(4):044501, 2014. 11