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An Introduction to the Floating Frame of Reference Formulation

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An Introduction to the Floating Frame of Reference Formulation Technical Report TR05 An Introduction to the Floating Frame of Reference Formulation for Small Deformation in Flexible Multibody Dynamics Antonio Recuero and Dan Negrut May 11, 2016 Abstract This technical report describes the basis of the nonlinear finite element floating frame of reference formulation. Keywords: Floating Frame of Reference Formulation, Nonlinear Finite Elements, Flexible Multibody Systems 1 Contents 1 Introduction 3 2 Floating Frame of Reference 3 2.1 Kinematics of the FFR . .3 2.2 Equations of Motion . .4 3 Finite Element FFR Formulation 7 3.1 Local Deformation and Reference Frames . .8 3.2 Description of Local Deformation . .9 3.3 Reference Conditions . .9 3.4 Kinematics of an FFR Finite Element . 10 3.5 Model Order Reduction . 10 2 1 Introduction The idea of a floating frame of reference (FFR) may have its origin in the analysis of astrodynamic systems. Such systems experience strong coupling of inertia forces, due to high accelerations in its reference (rigid body) motion, and flexibility due to the light nature of the structures involved (see Ref. [2]). The idea of a body-wise co-rotated frame or floating frame can be summarized as the splitting of the overall motion of bodies that experience small deformation into a frame that captures the dynamics of mean rigid body motion and a superposed flexible motion that defines the deformation state. This method requires one flexible body to have one single FFR to describe its dynamics, in contrast with the co-rotational formulation. In addition, the fact that the body's (structure's) elasticity can be defined via a constant stiffness matrix opens up the possibility of using general model order reduction techniques to reduce the order of a linear finite element model. In what follows, the basis of the FFR formulation is presented, whereas many derivations are left to the reader as an exercise or given as a reference. 2 Floating Frame of Reference 2.1 Kinematics of the FFR From now on, we will assume that Euler parameters will be our choice of rotation coordinates. Even though other choices can be made, it is, in any case, important to guarantee that singularities will be avoided. The kinematic state of the frame of reference of a body i may be written as h i i iT iTT qr = R θ ; (1) i i where qr is a set of Cartesian coordinates that locates the origin of the FFR of body i, O , with respect to the global frame, θi is a set of Euler parameters that describe the orientation of such a frame in the global coordinate system. A point P in body i can be described, using FFR kinematics, as i i i i P rP = R + A θ u¯ ; (2) where Ai denotes the orientation matrix of the FFR and u¯P is the vector that defines the position of point P with respect to the FFR (superscript i is dropped for simplicity). This vector can be split into its reference position and its elastic displacement as follows P P P u¯ = u¯o + u¯f : (3) P u¯o represents the undeformed position of P , whereas its elastic displacement, measured in P P the FFR, is denoted by the vector u¯f (see Fig. 1). The elastic displacement vector u¯f may be decomposed into the product of a space-dependent matrix and a vector of time-dependent flexible coordinates, as follows P i P i u¯f = S u¯o qf ; (4) 3 i P where S (u¯o ) is a shape function matrix referred to the reference (undeformed) configuration i and qf is the vector of flexible coordinates of body i. By spelling out the kinematic equations in (3) and (4), the expression of the deformed position of the point P in body i may be written as i i i P i i rP = R + A u¯o + S qf (5) Velocity expressions of a deformed point P within the FFR formulation are given below: P i P i u¯_ f = S u¯o q_f ; (6) i _ i _ i P i i i i i r_P = R + A u¯o + S qf + A S q_f : (7) P i Note that Eqs. (6)-(7) take into account that u¯o and S do not depend on time. Figure 1: Basic kinematics of the FFR formulation. The floating frame axes are denoted by hXiY iZii, the initial, unstressed configuration is depicted with solid lines, whereas the current configuration {translated, rotated, and deformed{ is shown in dashed lines. 2.2 Equations of Motion Here, we present the derivation equations of motion of the system in abbreviated manner, omitting some derivations. The interested reader can consult the book by Shabana [8] to find out more on some detailed derivations of the FFR formulation equations. Before moving on with derivations, we arrange, for convenience, the velocity vector in the following form: 2R_ i3 i i i i _i i i r_P = IB A S 4 θ 5 = L q_ ; (8) i q_f where matrix Bi is a linear operator that acts on the rotation coordinates, as follows Aiu¯P = Biθ_i, and Bi = −Aiu~¯P G¯i. The tilde ?~ denotes the skew symmetric matrix 4 operation on a vector ?, and Gi is a linear operator that transforms time derivatives of rotation parameters to the angular velocity vector of the reference motion of body i. The form of Gi depends upon the selection of rotational parameters. The equations of motion of the FFR formulations are to be derived using Lagrange's equations. For a flexible body i described using a floating frame, these equations take the form i T i T d @T @T T i i − + C i λ = Q + Q ; (9) dt @q_i @qi q e a where T is the kinetic energy, qi and q_i are the generalized coordinates and velocities of body i, respectively, Cqi is the Jacobian of the constraint equations, λ is the vector of Lagrange i i multipliers, and Qe and Qa are the vectors of elastic and applied forces, respectively. Note that, in general, the vectors of Lagrange multipliers and applied forces may link the flexible body i with other (flexible) bodies and/or other interacting systems. Mass Matrix The total kinetic energy of body i may be expressed as a volume integral: 1 Z T i = ρir_iTr_i dV i (10) 2 P P V i which, in terms of the generalized coordinates, may be written as 1 T i = qiTM iqi: (11) 2 The mass matrix M i is nonlinear in the rotation coordinates and takes the following form 2 i i i 3 2 3i Z IB A S mRR mRθ mRf i i iT i iT i i i M = ρ 4 B B B A S 5dV = 4 mθθ mθf 5 ; (12) iT i V i sym: S S sym: mff where submatrices have been named for convenience. Cross terms in the mass matrix, e.g. mRf and mθf , indicate that coupling between reference and flexible coordinates is captured. The Quadratic Velocity Vector The FFR formulation introduces complex inertia terms which represent Coriolis and centrifugal forces. Deriving the inertia terms of Lagrange equations in Eq. (9), one obtains d @T i T @T i T @ T − = M iq¨i + M_ iq_i − q_iTM iq_i (13) dt @q_i @qi @qi | i {z } Qv(Quadratic velocity vector) 5 T i h iT iT iTi The final expression for the quadratic velocity vector Qv = (Qv)R (Qv)θ (Qv)f , in terms of generalized coordinates, reads Z h i i i i ~ 2 i ~ i i i Qv R = −A ρ !¯ i u¯ + 2!¯ iS q_f dV (14) V i Z h i i ¯iT i ~iT~ 2 i ~iT ~ i i i Qv θ = G ρ u¯ !¯ i u¯ + 2u¯ !¯ iS q_f dV (15) V i Z h i i i iT ~ 2 i ~ i i i Qv f = −ρ S !¯ i u¯ + 2!¯ iS q_f dV (16) V i (17) and may be obtained by deriving equation (13) or via the Principle of Virtual Work. The i Euler parameter identity G¯_ θ_i = 0 is used above to simplify the expressions. Checking the correctness of (14) is left to the reader as an exercise. As an advanced topic, the ready may learn on the accurate calculation of FFR velocity-dependent inertia terms in [9]. Applied Forces To include forces and moments in the system it is necessary to obtain their generalized counterparts by using FFR kinematics. As an example, here we describe how to compute the generalized force associated with a point force. The virtual work of a force described in the i global frame, Fa, applied at P may be expressed as i i i i i δWa = FaδrP = Qaδq : (18) As a first step, we write the variation of the position vector of a point P in an FFR body i as 2δRi3 h i i i i ¯i i i i δrP = I −A u~¯ G A S 4 δθ 5 : (19) i δqf Equation (19) can be plugged into (18) to yield the following generalized force vector i P i ¯iT ~iT iT P i iT iT P Qa R = Fa ; Qa θ = −G u¯ A Fa ; Qa f = S A Fa ; (20) where space-dependent vector and matrix, u¯i and Si, respectively, must be particularized at the position of point P . Note that the FFR formulation does not impose limitations as to the location or nature of the load. 6 Equations of Motion In general form, the FFR equations of motion of a body can be written as follows i i i i i i T M q¨ + D q_ + K q + Cqi λ = Qa + Qv; (21) which, in turn, may be expressed in terms of submatrices as 2 3i2 3i 2 3i2 3i 2 3i2 3i mRR mRθ mRf R¨ 0 0 0 R_ 0 0 0 R 4 mθθ mθf 5 4 θ¨5 + 40 0 0 5 4 θ_ 5 + 40 0 0 5 4 θ 5 + sym: mff q¨f 0 0 Dff q_f 0 0 Kff qf 2 T 3 2 3 2 3 CRi (Qa)Ri (Qv)Ri T 4Cθi 5 λ = 4(Qa)θi 5 + 4(Qv)θi 5 : T C i (Qa) i (Qv) i qf qf qf The new terms Kff and Dff refer to the stiffness and damping matrices of the structure.
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