Simulation of generally shaped 3D elastic body dynamics with large motion using transfer matrix method incorporating model order reduction

Abstract

The multibody system transfer matrix method (MSTMM) is an efficient formalism for systems involving rigid bodies and simple elastic elements like beams. The paper tackles the challenge of setting up transfer matrices as building blocks for generally shaped, three-dimensional elastic bodies, and establishes a novel connection between the transfer matrix method and a large family of model order reduction (MOR) methods. An articulated manipulator arm serves as a demonstration example throughout the paper to first explain the basic concepts of MSTMM. Then, a classical floating frame of reference formulation (FFRF) of flexible bodies is adapted to MSTMM to express their equations of motion in a global inertial frame. The favorable ideas from FFRF are transferred into the typical MSTMM formulation yielding the transfer equation of a general elastic body. These ideas include: (i) splitting the overall motion into a large rigid body motion and a small elastic deformation; (ii) discretization of the deformation field by using the linear finite element (FE) method; (iii) calculating the standard input data from commercial FE data for generally shaped bodies; and (iv) MOR, to find the most suitable shape functions and speed up the simulation. Next, the procedure for solving elastic multibody system dynamics using MSTMM is illustrated. In a final numerical simulation, the proposed elastic MSTMM concept for a coupled rigid-flexible multibody system shows good agreement with a classical Newton-Euler formulation. The algorithm is based on the classical FFRF without additional assumptions and thus can be used with any MOR method available in MatMorembs.

Appendix: FFRF expressed in the inertial coordinate system

Herein, we first discuss the kinematics and then the EoM of a free elastic body using FFRF expressed in the global inertial frame $K_{I}$.

In the current configuration of the elastic body shown in Fig. 3, the global coordinates of an arbitrary material point $P$ on the body can be described as

(31)

by the coordinates of $P$ in $K_{R}$ defined in the reference configuration (not drawn), and displacement of $P$ in the current configuration w.r.t. its reference configuration defined in $K_{R}$. The first- and second-time derivatives of Eq. (31) read as

(32)

(33)

respectively, where is the absolute angular velocity of $K_{R}$ defined in $K_{I}$ and defines the time differentiation of in the reference coordinate system $K_{R}$.

The orientation of a coordinate system $K_{P}$ rigidly attached at $P$ is given by

(34)

where $\boldsymbol{A}_{RP}$ and $\boldsymbol{A}_{RP_{0}}$ are the direction cosine matrices defining the orientation of $K_{P}$ w.r.t. $K_{R}$ in the current and reference configuration, respectively; $\boldsymbol{A}_{P_{0}P}$ corresponds to the infinitesimal angular displacement at point $P$, whose coordinates in $K_{R}$ and $K_{P_{0}}$ (not drawn in Fig. 3) are and , respectively. The last equality holds because

The first- and second-time derivatives of Eq. (34) lead to angular velocity and acceleration of $K_{P}$ that can be expressed as

(35)

(36)

where and are the angular velocities of $K_{P}$ w.r.t. $K_{R}$ defined in $K_{I}$ and $K_{R}$, respectively.

With the generalized velocity $\boldsymbol{\eta}$ defined in Eq. (5), equations (32) and (33) can be rewritten as Eqs. (6) and (7); equations (35) and (36) can be rewritten as Eqs. (9) and (10). In the following, the EoM is briefly derived in $K_{I}$ using Jourdain’s principle.

For a free elastic body shown in Fig. 3, Jourdain’s principle reads as

$$ \delta P_{i}+\delta P_{e}+\delta P_{v}+\delta P_{s} = 0, $$

(37)

where $\delta P_{i}$, $\delta P_{e}$, $\delta P_{v}$ and $\delta P_{s}$ represent the virtual powers of inertial forces, internal elastic forces, external volume forces, and external surface forces, respectively. With Eqs. (6) and (7), the virtual power of inertial forces can be expressed as

(38)

where $\delta \boldsymbol{\eta}$ is the virtual velocity of the elastic body and the integration is over the volume $V_{0}$ in the reference configuration. By substituting Eqs. (5) and (8), after tedious deduction, one may arrive at $\boldsymbol{M}(^{R}\boldsymbol{q})$ and as defined in Eq. (12) with

where definitions of submatrices are [25]:

(39)

where $k=1(1) n_{q}$ and $\boldsymbol{\varPhi}_{P_{(:,k)}}$ is the $k$th column of $\boldsymbol{\varPhi}_{P}$. These unhandy volume integrals are either constant or depend on $^{R}\boldsymbol{q}$ and uniformly denoted as . To avoid updating in each time integration step, these integrals are approximated as using a Taylor series, where $\boldsymbol{X}_{0}$, $\boldsymbol{X}_{1q_{i}}$, and $\boldsymbol{X}_{1 \dot{q}_{i}}$ contain constant shape integrals and are called SID [12], to be computed only once before time integration as a pre-processing step. These SID can be further expressed using six constant elementary volume integrals $\boldsymbol{C}1$—$\boldsymbol{C}6$ [11] which depend on the FE shape functions and can be extracted from the mass matrix $\boldsymbol{M}_{F}$ of the generally shaped free FE system using three projectors: translation projector $\boldsymbol{S}_{t}$, rotation projector $\boldsymbol{S}_{r}$, and elastic projector $\boldsymbol{S}_{e}$, see Ref. [11, 13] for details. It should be noted that in Eq. (12) the above shape integrals are never computed explicitly, but only in combination with projection matrices $\boldsymbol{V}$ and $\boldsymbol{W}$ according to Eq. (16), which results in shape integrals with much lower dimensions. This is why MOR is always implemented before the calculation of shape integrals.

Since the deformation is supposed to be small, the strain and stress can be measured in $K_{R}$ using linear elastic theory. This yields the virtual power of internal elastic forces as [25, 30]

$$ \delta P_{e} = - \int _{V_{0}} \delta \dot{\hat{\boldsymbol{G}}}^{\mathrm{T}} \hat{\boldsymbol{P}} \mathrm{d} V = - \int _{V_{0}} \delta \left ( \boldsymbol{L}_{L} {}^{R} \dot{\boldsymbol{u}} \right ) ^{\mathrm{T}} \hat{\boldsymbol{C}} \left ( \boldsymbol{L}_{L} {}^{R} \dot{\boldsymbol{u}} \right ) \mathrm{d} V =: -\delta \boldsymbol{\eta}^{\mathrm{T}} \boldsymbol{h}_{e}, $$

(40)

where $\dot{\hat{\boldsymbol{G}}} = \boldsymbol{L}_{L} {}^{R} \dot{\boldsymbol{u}}$ denotes the time derivative of the strain vector and $\hat{\boldsymbol{P}} = \hat{\boldsymbol{C}} \hat{\boldsymbol{G}}$ is the corresponding stress vector following Hooke’s law with Young’s modulus $E$ and Poisson’s ratio $\nu$ of the material. Further,

$$ \boldsymbol{L}_{L} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} \partial _{1} & 0 & 0 \\ 0 & \partial _{2} & 0 \\ 0 & 0 & \partial _{3} \\ \partial _{2} & \partial _{1} & 0 \\ 0 & \partial _{3} & \partial _{2} \\ \partial _{3} & 0 & \partial _{1} \end{array}\displaystyle \right ], $$

$$ \hat{\boldsymbol{C}} = \frac{E}{(1+\nu )(1-2\nu )} \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1-\nu & \nu & \nu & 0 & 0 & 0 \\ \nu & 1-\nu & \nu & 0 & 0 & 0 \\ \nu & \nu & 1-\nu & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1-2\nu}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1-2\nu}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1-2\nu}{2} \end{array}\displaystyle \right ], $$

$\partial _{i} := \partial / \partial R_{i}$. Substituting Eqs. (4) and (5) into Eq. (40) yields

where $\underset{n_{q} \times n_{q}}{\boldsymbol{K}_{e}} = \int _{V_{0}} \boldsymbol{B}_{L}^{\mathrm{T}} \hat{\boldsymbol{C}} \boldsymbol{B}_{ \mathrm{L}} \mathrm{d} V$ and $\underset{6 \times n_{q}}{\boldsymbol{B}_{L}} = \boldsymbol{L}_{L} \boldsymbol{\varPhi _{P}}$. If damping matrix $\boldsymbol{D}_{e}$ is considered, then

The virtual power of external volume forces is

(41)

where ${}^{I} \boldsymbol{g}$ is the acceleration of gravity defined in $K_{I}$. Substituting Eqs. (5), (6) and abbreviations (39) into Eq. (41) yields

Finally, the virtual power of external surface forces (discrete case only) can be obtained as

(42)

where ${}^{I} \boldsymbol{f}_{P_{k}}$ and ${}^{I} \boldsymbol{l}_{P_{k}}$ denote the concentrated forces and torques acting on points $P_{k}$, respectively. Substituting Eqs. (6) and (9) into Eq. (42) leads to

$$ \boldsymbol{h}_{s}(^{R}\boldsymbol{q}) = \sum _{k} \left ( \boldsymbol{L}_{TP_{k}}^{ \mathrm{T}}{ }^{I} \boldsymbol{f}_{P_{k}} + \boldsymbol{L}_{RP_{k}}^{ \mathrm{T}} {}^{I}\boldsymbol{l}_{P_{k}} \right ). $$

(43)

Then, substitute Eqs. (38), (40), (41) and (42) into Eq. (37). For a free elastic body, with independent variation $\delta{\boldsymbol{\eta}}^{\mathrm {T}}$, Eq. (37) eventually becomes Eq. (12).