Study of General Dynamic Modeling Using the Craig-Bampton Method

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1 Study of General Dynamic Modeling Using the Craig-Bampton Method Shinji Mitani (JAXA) Abstract: This paper proposes a more general dynamic model of the constrained mode using the Craig-Bampton method under appropriate assumptions. The proposed dynamic model can treat structural objects with flexible appendages and multiple contact points, which cannot be handled using the traditional Likins model. In addition, the relationship between the Craig-Bampton submatrix and Likins coupling matrix is clarified and it is proved that the proposed model is identical to the Likins model for appendages with a single contact point. Some numerical calculations verify the accuracy and efficacy of the proposed model. 部分構造法を用いた汎用ダイナミックモデル構築手法の研究摘要 : 適切な仮定の下で Craig-Bampton 法を用いることにより, 一般的な拘束モードモデルを提案する. 提案する拘束モードモデルは, 従来の Likins モデルでは取り扱えない複数の接触点を有する柔軟付属構造物を扱える. また, Craig-Bampton 部分行列と Likins カップリング行列との関係を明らかにする. 提案モデルは, 単一の接触点を有する付属の場合,Likins モデルと同一であることが証明される. 数値計算により, 提案モデルの精度と有効性を検証する. Nomenclature [aa] : Vector cross-product matrix [aa]bb aa bb HH : Craig-Bampton transformation matrix KK kk : Stiffness matrix of substructure kk MM kk : Mass matrix of substructure kk : Modal DOFs SS kk : System coupling matrix uu : External force and torque vector [ff ττ ] xx : Translation and rotation vector [δδ ββ ] ΦΦ : Fixed base mode shapes ΦΦ : Rigid body mode matrix ΨΨ : Rigid body vector. Introduction Accurate analytical dynamic representations, usually in the form of a finite element model (FEM) are used to analyze structural loads and control system design [, ]. Dynamic models associated with Craig-Bampton (C-B) substructure representations have been extensively studied [3-7]. This paper discusses the relationship between the constrained mode model derived by the C-B method under appropriate assumptions and the traditional Likins model [8]. Substructure (body) Substructure (appendage) Contact point between substructures SAP, Antenna, attached with single point Mission payload, attached with multiple points Fig. Traditional model (Left) and future model (Right) The proposed dynamic model can treat structural objects with flexible appendages and multiple contact points (Fig. right), which cannot be handled by the traditional Likins model (Fig. left).. Dynamic Model Construction Method.. Craig-Bampton Method Boundary point variation xx BB = [xx BB xx BBNN ] Internal point variation Substructure Fig. Division to substructures and boundary points The C-B method is used for analyzing each by dividing the object into certain partial structures. Combining the results of each analysis reveals the overall result. The C-B method is often used to analyze the vibration of satellites and rockets. As shown in Fig., the boundary points of the FEM, BB,, BB NN, divide the entire structure into two substructures (left part) and (right part). The Eigenvalue analysis for each partial structure was conducted by keeping the boundary DOFs fixed to obtain the mode shapes Φ. Subsequently, by separating between internal points DOFs (use the subscript of the letter II) and boundary points (use the subscript of the letter BB), the static equilibrium equation is KK BBBB KK IIII Suppose uu II =, then KK BBBB xx II Substructure xx BB KK IIII xxii uu BB uu () II

2 xx II = KK IIII KK IIII xx BB ΨΨxx BB () where ΨΨ is the matrix of shapes giving the static I-set displacement of the substructure for a unit displacement of the B-set. Therefore the displacement xx II can be represented as xx II = ΨΨxx BB + ΦΦ (3) which is the sum of forced displacement xx II added to xx BB and free response displacement xx II when fixed with xx BB. The boundary DOFs xx BB and modal DOFs are unknown state variables. The Craig-Bampton transformation is defined as xx = xx BB xx BB II xxii ΨΨxx BB + ΦΦΦΦ OO ΨΨ ΦΦ xx BB HH xx BB (4) where ΨΨ rr kk 6NN is the rigid body vector and ΦΦ rr kk rr kk is fixed base mode shapes. HH 6(NN+EE) (6NN+rr kk) is known as the Craig-Bampton transformation matrix. The merit of the C-B method is the fact that it allows the problem size to be reduced. Pre-multiplying by HH MM = HH MMMM, KK = HH KKKK (5) Define the C-B mass and stiffness matrices as MM BBBB MM MM BBBB, KK KK BBBB OO 6NN rrkk (6) MM MM OO rrkk 6NN KK MM BBBB = MM BBBB + ΨΨ MM IIII ΨΨ + MM BBBB ΨΨ + ΨΨ MM IIII MM BBBB = MM = ΨΨ MM IIII ΦΦ + MM BBBB ΦΦ (7) MM = ΦΦ MM IIII ΦΦ KK BBBB = KK BBBB KK BBBB KK IIII KK IIII = KK BBBB + KK BBBB ΨΨ KK = ΦΦ (8) KK IIII ΦΦ The important properties of the C-B mass and stiffness matrices are as follows. The submatrix MM BBBB is the boundary mass matrix - total mass properties translated to the boundary points and KK BBBB is the interface stiffness matrix - stiffness associated with displacing one boundary DOF while the others are kept fixed. If the boundary point is a single grid (i.e. non-redundant) then KK BBBB = OO. KK BBBB is known as a Guyan reduction. The C-B method combines the motion of boundary points with modes of a structure assuming the boundary points remain fixed. Applying the C-B method to substructures and, superposing these MM, KK xx BB xx BB uu BB MM + KK = LL {uu II } (9) {uu II } MM BBBB + MM BBBB MM BBBB MM BBBB MM = MM MM MM MM KK = KK BBBB + KK BBBB KK KK () () II 6 6 [ΨΨ] [ΨΨ] LL = [ΦΦ] () [ΦΦ] The derived dynamic equation (9) would be equivalent to the original full FEM one if all elastic modal DOFs remained. The bracket [ ] kk represents C-B substructure matrices (7) and (8) of the substructure, while {uu II } kk is the external force and torque vector acting on the internal points of substructure kk... The Constrained mode model representation similar to the Craig-Bampton method First, the rigid body transformation of Eq. (9) ΦΦ HH (6NN+rr +rr ) 6 is defined as where ΦΦ BB 6NN 6 an ΦΦ BBjj ΦΦ HH = (ΦΦ BB ) OO 6 rr OO 6 rr (3) ΦΦ BB = (ΦΦ BB ) (ΦΦ BBNN ) (4) 6 6 represents the rigid body modes of each boundary grid point (the center of rotation is C.G. of the overall structure.) To simplify the discussion, assume that the system coordinates are parallel to the principal axes of the overall inertia. Then ΦΦ BBjj is ΦΦ BBjj = II 3 3 [ rr ] BBjj, jj =,, NN (5) OO 3 3 II 3 3 [ rr ] BB (zz BBjj zz GG ) xx BBjj xx GG (6) zz BBjj zz GG (yy BBjj yy GG ) jj yy BBjj yy GG (xx BBjj xx GG ) where [xx BBjj yy BBjj zz BBjj ] represents the coordinates of boundary point BB jj, and [xx GG yy GG zz GG] is the C.G. of the overall structure. Pre-multiplying ΦΦ HH Eq. (3) to MM Eq. (), (ΦΦ HH ) MM ΦΦ HH = (ΦΦ BB ) MM BBBB ΦΦ BB = MM ΛΛ (7) where MM ΛΛ 6 6 is the rigid body modal mass matrix MM ΛΛ mm II 3 3 II xxxx IIyyyy II zzzz having a total mass mm and the principal moment of inertia [II xxxx II yyyy II zzzz]. Similarly, (8) (ΦΦ HH ) KKΦΦ HH = (ΦΦ BB ) KK BBBB ΦΦ BB = OO 6 6 (9) since Eigen frequencies of the rigid body mode are all zeros. Secondly, define {xx xx NN } using ΦΦ BBjj ( jj =,, NN ) Eq. (5) as xx BBjj = ΦΦ BBjj xx jj, jj =,, NN () In addition, define {xx δδxx δδxx NN } as NN xx = NN xx jj jj=, δδxx jj = NN xx jj xx ()

3 This transformation matrix WW can be represented as xx xx δδxx = WW xx NN δδxx NN II 6 6 II 6 6 II 6 6 xx II 6 6 (NN )II 6 6 δδxx II 6 6 () II 6 6 II 6 6 (NN )II 6 6 δδxx NN MM ΛΛ MM SS SS OO 6 6 MM MM δδδδ δδδδ δδδδ δδxx SS δδδδ MM KK δδδδ + KK SS δδδδ MM If the boundary point is a single grid ( NN = ), then ΦΦ BB = ΦΦ BB, WW = II 6 6. Using Eqs. () and (), define the similarity transformation HH as DDDD HH II rr rr II rr rr (3) where DD diag(φφ BBjj ). Premultiplying Eq. (9) by HH, HH MM HH xx HH + HH KKHH xx HH = HH LL {uu II } (4) {uu II } uu BB where xx HH {xx δδxx δδxx NN } and HH MM HH = WW DD MM BBBB DDDD WW DD MM BBBB WW DD MM BBBB MM BBBB DDDD MM MM BBBB DDDD MM Define the diagonal block matrix VV jj as VV jj diagww jj ww jj II 6 6 (5) ww NNNN II 6 6 (6) where ww jj {ww jj II 6 6 ww NNNNII 6 6 } is the jj -th block column in the transformation matrix WW Eq. (). It should be noted that VV = II 6NN 6NN from Eq. (). Using the representation of Eq. (6), WW DD MM BBBB DDDD (ΦΦ BB ) VV MM BBBB VV ΦΦ BB (ΦΦ BB ) VV MM BBBB VV NN ΦΦ BB = (ΦΦ BB ) VV NN MM BBBB VV ΦΦ BB (ΦΦ BB ) VV NN MM BBBB VV NN ΦΦ BB In addition, using VV = II 6NN 6NN and Eq. (7), (7) (ΦΦ BB ) VV MM BBBB VV ΦΦ BB = (ΦΦ BB ) MM BBBB ΦΦ BB = MM ΛΛ (8) In the similar way, then (ΦΦ BB ) VV WW DD = (ΦΦ BB ) (9) (ΦΦ BB ) VV GG NN xx where WW DD MM BBBB kk = (ΦΦ BB ) MM BBBB kk GGMM BBBB kk, kk =, (3) SS kk (ΦΦ BB ) MM BBBB kk, kk =, (3) is the important value obtained in this paper. Likewise, HH KKHH and WW DD uu BB can be simplified. Consequently, Eq. (9) can be transformed as the following constrained mode model system representation xx (ΦΦ BB ) (ΦΦ BB ) [ΨΨ] (ΦΦ BB ) [ΨΨ] uu δδxx BB GG GG[ΨΨ] = GG[ΨΨ] KK [ΦΦ] {uu II } [ΦΦ] {uu II } (3) where MM = [(ΦΦ BB ) VV MM BBBB VV ΦΦ BB (ΦΦ BB ) VV MM BBBB VV NN ΦΦ BB ] MM δδδδ (33) (ΦΦ BB ) VV MM BBBB VV ΦΦ BB (ΦΦ BB ) VV MM BBBB VV NN ΦΦ BB = (34) (ΦΦ BB ) VV NN MM BBBB VV ΦΦ BB (ΦΦ BB ) VV NN MM BBBB VV NN ΦΦ BB KK δδδδ (ΦΦ BB ) VV KK BBBB VV ΦΦ BB (ΦΦ BB ) VV KK BBBB VV NN ΦΦ BB = (35) (ΦΦ BB ) VV NN KK BBBB VV ΦΦ BB (ΦΦ BB ) VV NN KK BBBB VV NN ΦΦ BB (ΦΦ BB ) VV MM BBBB kk δδδδ kk = GGMM BBBB kk = (36) (ΦΦ BB ) VV NN MM BBBB kk Noting [ rr ] =, the external force and torque vectors ff BBjj (ΦΦ BBjj ) uu BBjj = II 3 3 OO 3 3 [ rr ] ff BB jj II 3 3 ττ (37) BBjj BBjj ff BBjj + ττ BBjj Conversely, (ΦΦ BB ) [ΨΨ] kk {uu II } kk is the force and torque vector in the overall C.G acting on the internal point of the substructure kk. Therefore, the total force and torque vector in the overall C.G is NN UU (ΦΦ BBjj ) uu BBjj jj= + (ΦΦ BB ) [ΨΨ] kk {uu II } kk kk= (38).3. Assumed Constrained Mode Model Representation Now suppose the following two assumptions: The rigid body mode transformation is only considered in substructure, including boundary points xx BB. The variation of the overall C.G, depending on the elastic transformation of substructure, is negligible and not considered. These are the same assumptions as those when deriving the well-known Likins constrained mode model. It should be

4 noted that Eq. (3) lacks the similarity characteristics to Eq. (9). Since the boundary points xx BB are transformed rigidlly in the overall C.G, the boundary DOF degenerates to 6 from 6NN. Therefore, xx = xx jj and δxx jj =. In addition, the elastic DOF of the substructure is all zero. Then Eq. (3) simply reduces to MM ΛΛ SS xx SS MM + OO 6 6 UU xx KK {ΦΦ uu II } (39) In the representation of Eq. (39), MM ΛΛ is the overall rigid body modal mass matrix Eq. (8). The matrixes MM and KK are modal mass matrix Eq. (7) and modal stiffness matrix Eq. (8) respectively when the boundary DOFs are all fixed. Therefore the representation Eq. (39) is equivalent to that of Likins constrained mode model. It should be noted that SS kk Eq. (3) is a generalized system coupling matrix, which is applicable when there are multiple boundary points..4. The Compatibility with Likins Model and the Mass Conservation Law It will be shown that if the boundary point is a single grid BB, SS kk Eq. (3) is equal to the Likins system coupling matrix. The subscript kk of the substructure is not shown in the following section. When the constrained mass matrix is used, MM BBBB = OO. Then Eq. (7) leads to MM BBBB = ΨΨ MM IIII ΦΦ + MM BBBB ΦΦ = (ΦΦ II ) MM IIII ΦΦ (4) where (ΦΦ II ) MM IIII ΦΦ is the transposition of the modal participation factor matrix [9]. The matrix (ΦΦ II ) MM IIII ΦΦ is equal to the matrix which aligned the tandem with Likins zero- and first-order coupling matrix ΔΔ 3 6EE,ΔΔ 3 6EE. The proof is shown in Appendix A: MM BBBB = (ΦΦ II ) MM IIII ΦΦ = ΔΔ ΔΔ (4) where ΔΔ,ΔΔ are defined as ΔΔ = [δδ, δδ,6ee ] ΔΔ = [δδ, δδ,6ee ] (4) and the i-th modal Likins zero- and first-order coupling vector δδ,ii 3, δδ,ii 3 are defined respectively as EE δδ,ii mm kk ii kk kk= EE δδ,ii II kk ii kk + mm kk rr kk ii kk kk= (43) (44) Substituting Eq. (4) to Eq. (3), the system coupling matrix SS is newly obtained as SS = (ΦΦ BB ) ΔΔ (45) ΔΔ ΔΔ + ΔΔ ΔΔ which is the same form as the well-known Likins system coupling matrix. In addition, it is known that in terms of the Likins zero- and first-order coupling matrix ΔΔ,ΔΔ, the following conservation law holds []: ΔΔ ΔΔ = mm EE II 3 3 (46) ΔΔ ΔΔ = II EE where mm EE is the mass of the appendage and II EE is the moment of inertia centered in the attachment point. It should be noted that modal masses of ΦΦ are normalized as when making ΔΔ and ΔΔ. The conversation law corresponding to Eq. (46) is MM BBBB MM BBBB = ΔΔ ΔΔ ΔΔ ΔΔ ΔΔ ΔΔ ΔΔ ΔΔ mm EEII 3 3 (47) II EE The proof of Eq. (46) is shown in Appendix B. 3. Numerical Examples 3.. Example A: The Appendage with a Single Point Z (m) Y (m) Figure 3. The structure in example A The example structure to be handled in this section is shown in Fig. 3. The satellite body is replaced by V-shaped high stiffness heavy beams as a satellite body and one paddle with a single point (No. 3 point in pink) is attached to the one side. Because this appendage is connected to the main body at the single grid point, it can be handled as a traditional Likins constrained mode model. Each partial structure is a beam element with common isotropic material properties. The material properties of the element are shown in Table, where the bending stiffness EEEE, the torsional stiffness GGGG, the beam cross-section area AA, the density ρρ and the moment of inertia per unit length ρρρρ. Table. The beam element properties Sub- EEEE EEEE ρρρρ GGGG ρρρρ structure [N] [Nm ] [kg/m] [Nm ] [kgm] 5.E5.E6.5E.E6.5E4.E3 4.E3.5 4.E3 3. Since there are points of FEM nodes in total, the number of DOFs is 7. All 7 vector modes and natural frequencies can be obtained by analyzing the whole FEM. The natural frequencies obtained by both the traditional Likins model and the obtained model Eq. (39) are shown in Table when the

5 fifth-order elastic modes are taken (rr = 5). The results of the two models are totally equivalent. Table. The comparison with FEM and the models (Ex. A) Mode No. FEM All Constrained mode model (rr = 5) Likins method C-B method Change [%] Secondly, the both results of calculating the system coupling matrix SS are the same as In addition, both results of confirming the conversation laws Eqs. (46) and (47) are also the same as ΔΔ ΔΔ = ΔΔ ΔΔ = ΔΔ ΔΔ = The mass properties of appendage are mm EE = [kg] and II EE = diag( ) [kgm ]. Therefore, in the case of single point connection, it is confirmed numerically that both models are equivalent. 3.. Example B: Appendage with Multiple Points Y (m) - - X (m) Z (m) Figure 4. The structure in example B The example structure to be handled in this section is shown in Fig. 4. The satellite body has been replaced by four beams and flexible appendages are attached at 4 points (Nos., 3, 4 and 5 points in pink). In this case, therefore, it cannot be modeled using the conventional Likins approach. It is assumed that the material properties of each substructure are the same as those in Table. Eigen frequencies in All FEM, C-B Method Eq. (3) and Assumed C-B Method Eq. (39) are shown respectively in Table 3. The result of Eq. (3) is the most accurate, but the result of Eq. (39) is sufficiently precise for practical use. Figure 5 shows the plot of the variation in RMS error of elastic mode frequencies up to the th order, taking the horizontal axis as the adoption order of appendage rr. Using Eq. (3), combining the all-elastic mode of substructure (rr = 6) and taking a total of 96 modes, the result matches the all FEM except for the calculation error. Furthermore, Eq. (39) has converged to sufficient accuracy if taken up to rr =. Table 3. The comparison with FEM and the models (Ex. B) Constrained Mode Model (rr = ) Mode No. FEM All (Hz) C-B Method (rr = ) Assumed C-B Method Change [%] Root Mean Square Error C-B Method (r=6) C-B Method (r=) Assumed C-B Method Degrees of Freedom Figure 5. RMS error of the elastic mode frequencies. 4. Conclusions This paper shows that the similarity transformation of the model obtained from the Craig-Bampton method leads to a more general constrained mode model. In addition, assuming an approximation like that of Likins derives an approximate model representation to be handled as a multi-point connection structure of flexible structures. Some numerical calculations confirmed the approximate validity of the proposed model. It can be analytically shown that in the case of a single point

6 connection, the proposed model is equivalent to the traditional constrained mode model representation by Likins. Appendix A: Proof of Eq. (4) The transposition of the ii -th modal participant factor (ΦΦ II ) MM IIII φφ ii can be expanded to the sum of II 3 3 rr kk mmkk kk II 3 3 II 3 3 II kk ii mm where φφ ii kk [( ii kk ) ii kk kk kk ii II kk ii kk + mm kk rr kk ii kk ( ii kk ) ] is the ii-th mode vector of the element kk. From Eqs. (4) and (43), the following is obtained: EE (ΦΦ II ) mm kk kk MM IIII φφ ii = ii II kk kk ii + mm kk rr kk kk ii kk= = δδ,ii δδ,ii Therefore, the collocated matrix side-by-side with all modes of (ΦΦ II ) MM IIII φφ ii leads to Eq. (4). (Q.E.D.) Flexibility on the Control of a Spacecraft Full-coupled Model, Advances in the Astronautical Sciences, pp. 97-5,. 7) P. Gasbarri, et. al.: Control-oriented Modelization of a Satellite with Large Flexible Appendages and Use of Worst-case Analysis to Verify Robustness to Model Uncertainties of Attitude Control, Acta Astronautica, pp. 4-6,. 8) P. W. Likins: Dynamics and Control of Flexible Space Vehicles, JPL Technical Report 3-39, 97. 9) K. Komatsu: Mechanical Structure Vibrations, Morikita pp. 5-8, pp. 6-66, 9 in Japanese.. ) P. C. Hughes: Modal Identities for Elastic Bodies, Journal of Applied Mechanics, Vol. 47, No. march, pp , 98. Appendix B: Proof of Eq. (46) then Using ΦΦ, the modal mass of which is normalized as, MM BBBB MM BBBB ΦΦ MM IIII ΦΦ = II 6 6 ΦΦ = ΦΦ MM IIII = (ΦΦ II ) MM IIII ΦΦΦΦ MM IIII ΦΦ II = (ΦΦ II ) MM IIII ΦΦ II Since the center of the rotation in the rigid body mode ΦΦ II is the attachment of the appendage, MM BBBB MM BBBB = mm EEII 3 3 II EE The off-diagonal elements are non-zero. (Q.E.D.) Acknowledgements The author would like to thank Prof. Keiji Komatsu at ISAS / JAXA for providing technical suggestions and advice. References ) Attitude control study team: The handbook of spacecraft dynamics and control, Baifukan, pp. 95-4, 7 in Japanese. ) J. L. Junkins and Y. Kim: Introduction to Dynamics and Control of Flexible Structures, AIAA Education Series, pp. 85-8, ) R. R. Craig, Jr. and M. C. C. Bampton: Coupling of Substructures for Dynamic Analyses, AIAA Journal, Vol. 6, No. 7, pp , ) S. Gordon: FMECI The Book Craig-Bampton Method URL 5) D. C. Kammer and M. J. Triller: Selection of Component Modes for Craig-Bampton Substructure Representations, Transactions of the ASME, 8 pp. 64-7, ) P. Gasbarri, et. al.: Second Order Effects of the

On Attitude Control of Microsatellite Using Shape Variable Elements 形状可変機能を用いた超小型衛星の姿勢制御について

The 4th Workshop on JAXA: Astrodynamics and Flight Mechanics, Sagamihara, July 015. On Attitude Control of Microsatellite Using Shape Variable Elements By Kyosuke Tawara 1) and Saburo Matunaga ) 1) Department