Three-dimensional Finite Rotations Research Articles

Geometrically exact beam theory with embedded strong discontinuities for the modeling of failure in structures. Part I: Formulation and finite element implementation

We introduce a three-dimensional geometrically nonlinear Reissner beam theory with embedded strong discontinuities for the modeling of failure in structures and discuss its finite element implementation. Existing embedded beam theories are geometrically linear or two-dimensional, motivating the need for the present work. We propose a geometrically nonlinear beam theory that accounts for cracks through displacement discontinuities in 3D. To represent the three modes of fracture inside an element, we enrich each displacement field with an incompatible mode parameter in the form of a jump, an additional degree of freedom. We then eliminate these nodeless degrees of freedom through static condensation and evaluate them within the framework of inelasticity by utilizing an operator split solution procedure. Seeing that the coupled strain-softening problem is non-convex, we present an alternating minimization (staggered) algorithm, thus retaining a positive definite stiffness matrix. Finally, the four-parameter representation by quaternions describes a three-dimensional finite rotation. We demonstrate a very satisfying and robust performance of these new finite elements in several numerical examples, including the fracture of random lattice structures with application to fibrous materials. We show that accounting for geometrical nonlinearity in the beam formulation is necessary for direct numerical simulations of fiber networks regardless of the density.

Dynamic Modeling of an Insect Wing Subject to Three-Dimensional Rotation

A dynamic model of an insect wing is developed treating the wing as a deformable body subject to three-dimensional finite rotation about a fixed point at the base of the wing. Discretization of a stationary wing is first conducted via finite element analysis to determine the natural frequencies and mode shapes of the wing. By formulating and discretizing the kinetic and potential energy, we derive the equation of motion governing the modal response of a flapping wing using Lagrange's equation. The equation of motion indicates Coriolis, Euler, and centrifugal forces resulting from the finite rotation are responsible for the wings elastic deformation. Numerical integration reveals a beat phenomenon that arises from the Coriolis excitation in the first vibration mode. The beat phenomenon is insensitive to yaw amplitudes and nonzero initial conditions but diminishes in the presence of damping. The beat phenomenon can potentially be used to estimate gyroscopic forces.

New kinematic parameters of finite rotation of a solid

A new family of kinematic (global and local) orientation parameters of a solid is presented and described. All kinematic parameters are obtained by the method of mapping the variables onto the corresponding oriented subspace (hyperplane). In particular, the method of stereographic projection of a point of a five-dimensional sphere S6 ⊂ R6 onto the oriented hyperplane R5 is presented for the classical direction cosines of the angles determining the orientation of two coordinate systems. A family of global kinematic parameters obtained by the method of mapping of five-dimensional kinematic Hopf parameters given in the space R5 onto the four-dimensional oriented subspace R4 is described. The theorem of the homeomorphism of two topological spaces (the four-dimensional sphere S4 ⊂R5 with one deleted point and the oriented hyperplane R4) is used to establish the correspondence between five- and four-dimensional kinematic parameters defined in the corresponding spaces. It is also shown what global four-dimensional orientation parameters, quaternions defined in the subspace R4 are associated with the classical local parameters, i.e., the Rodrigues and Gibbs three-dimensional finite rotation vectors. The projection method is used to obtain the kinematic differential equations (KDE) of rotation corresponding to the five- and four-dimensional orientation parameters. All above-introduced kinematic orientation parameters of a solid permit efficiently solving the classical Darboux problem by using the corresponding KDE, i.e., determining the body current angular position in the space R3 from the known (measured) angular velocity of the object rotation and its initial position in space.

Three-dimensional finite rotations treatment based on a minimal set parameterization and vector space operations in beam elements

A minimal set vectorial parameterization involving vector space operations is proposed for finite 3D rotations in structural analysis. In this approach, based on the updated Lagrangian description, complex manipulations required to obtain conservative descriptions and well-posed transformation matrices are avoided. In particular, slopes are used instead of rotation parameters to compute the nonlinear representations of the strain measures in the inertial frame of reference. This approach is applied to a geometrically nonlinear formulation for 3D beam elements in the hypotheses of large rotations and small strains. Numerical tests have been carried out to validate the developed technique in the frame structures context.

Macroscopic behavior of non-polar tetrahedratic nematic liquid crystals

We discuss the symmetry properties and the macroscopic behavior of a nematic liquid crystal phase with D(2d) symmetry. Such a phase is a prime candidate for nematic phases made from banana-shaped molecules where the usual quadrupolar order coexists with octupolar (tetrahedratic) order. The resulting nematic phase is nonpolar. While this phase could resemble the classic D (infinityh) nematic in the polarizing microscope, it has many static as well as reversible and irreversible properties unknown to nonpolar nematics without octupolar order. In particular, there is a linear gradient term in the free energy that selects parity leading to ambidextrously helical ground states when the molecules are achiral. In addition, there are static and irreversible coupling terms of a type only met otherwise in macroscopically chiral liquid crystals, e.g. the ambidextrous analogues of Lehmann-type effects known from cholesteric liquid crystals. We also discuss the role of hydrodynamic rotations about the nematic director. For example, we show how strong external fields could alter the D(2d) symmetry, and describe the non-hydrodynamic aspects of the dynamics, if the two order structures, the nematic and the tetrahedratic one, rotate relative to each other. Finally, we discuss certain nonlinear aspects of the dynamics related to the non-commutativity of three-dimensional finite rotations as well as other structural nonlinear hydrodynamic effects.

A History of the Description of the Three-Dimensional Finite Rotation

A history of the description of a three-dimensional finite rotation is given starting with Cardano in the middle of the sixteenth century and ending with Bryan in the beginning of the past century. Description means both a textual description and/or a mathematical representation. To appreciate the historical context of the milestones reached over the centuries, the background and personality of the main players in this history are given. At the end, a short critical discussion is added, reviewing the present names of rotation parameters in use related to the scientists which have been considered here.

Finite rotations in dynamics of beams and implicit time-stepping schemes

We examine theoretical and computational aspects of three-dimensional finite rotations pertinent to the dynamics of beams. The model problem chosen for consideration is the Reissner beam theory capable of modelling finite strains and finite rotations in geometrically exact manner. Special emphasis is placed on clarifying the geometry aspects, finite rotation updates and the associated linearization procedure pertaining to different choices of rotation parameters. The latter is shown to play an important role in constructing the optimal implementation of a time-stepping scheme. © 1998 John Wiley & Sons, Ltd.

弾性立体ラーメンの大たわみ挙動の解析とその検証実験

An efficient computational method has been developed by the authors for predicting the large deflection behavior of spatial rigid frames. The method is based on the beam-column finite element method with the incremental perturbation technique. The three-dimensional finite rotations are exactly incorporated in the method. This paper shows that the author's method accurately predicts the experimental results on the large deflection behavior of model rigid frames. The model frames are made of styrene and acrylic plates which keep elastic up to large deflection range. It is also shown that the finite element model with warping function is able to predict accurately the theoretical lateral buckling load of a simply supported H beam.

On the choice of finite rotation parameters

In this work we discuss some aspects of the three-dimensional finite rotations pertinent to the formulation and computational treatment of the geometrically exact structural theories. Among various possibilities to parameterize the finite rotations, special attention is dedicated to a choice featuring an incremental rotation vector. Some computational aspects pertinent to the implementation of the Newton iterative scheme and the Newmark time-stepping algorithm applied to solving these problems are examined. Representative numerical simulations are presented in order to illustrate the performance of the proposed formulation.

Stress Resultant Geometrically Exact Shell Theory for Finite Rotations and Its Finite Element Implementation

This article reviews the significant progress on shell problem theoretical foundation and numerical implementation attained over a period of the last several years. First, a careful consideration of the three-dimensional finite rotations is given including the choice of optimal parameters, their admissible variations and the much revealing relationship between different parameters. A non-conventional derivation of the stress resultant shell theory is presented, which makes use of the virtual work principle and local Cartesian frames. The presented derivation introduces no simplifying hypotheses regarding the shell balance equations, hence the resulting shell theory is referred to as being geometrically exact. The strain measures energy-conjugate to the chosen stress resultants are identified and the nature of the stress resultants with respect to the three-dimensional stress tensor is explained along with the resulting constitutive restrictions. Comments are made regarding a rather useful extension of the shell theory which accounts for the rotational degree of freedom about the director, the so-called drilling rotation. A linear shell theory is obtained as a very useful by-product of the present work, by linearizing the present nonlinear shell theory about the reference configuration. It is shown that this non-conventional approach not only clarifies an often confusing derivation of the linear shell theory, but also leads to a novel linear shell theory capable of delivering significantly improved results and essentially exact solutions to the standard linear benchmark problems. Another important aspect of the nonlinear shell problem solution, the finite element approximation of the shell theory, is also discussed. The model problem of assumed shear strain interpolation is used to illustrate that numerical implementation which preserves the salient features of the theoretical formulation often brings an improved final result. For the selected rotation parameterization and the finite element interpolation, the issues of the consistent linearization of the nonlinear shell problem are addressed. In a number of numerical simulations, the latter is proved to play a crucial role not only in ensuring the robust performance of the Newton solution procedure, but also in linear and nonlinear buckling problems of shells. Several directions for future research are pointed out and some contemporary works of special interest are listed. There are 127 references at the end of the article.

Théorie géométriquement exacte des coques en rotations finies et son implantation éléments finis

ABSTRACT In this article, we review the significant progress on shell problem theoretical foundation and numerical implementation attained over a period of the last several years. First, a careful consideration of the three-dimensional finite rotations is given including the choice of optimal parameters, their admissible variations and the much revealing relationship between different parameters. A non-conventional derivation of the stress resultant shell theory is presented, which makes use of thevirtual work principle and local Cartesian frames. The presented derivation introduces no simplifying hypotheses regarding the shell balance equations, hence the resulting shell theory is referred to being the geometrically exact. The strain measures energy-conjugate to the chosen stress resultants are identified and the nature of the stress resultants with respect to the three-dimensional stress tensor is explained along with the resulting constitutive restrictions. Comments are made regarding a rather useful e...

Rotation about an inclined axis: Three dimensional matrices for reconstructing paleomagnetic and structural data

We derive matrices for calculating the three-dimensional finite rotation that restores a rigid block to its pre-displacement attitude. To use this technique an initial horizontal bedding is assumed, and the following orientations must be known: the expected field orientation of magnetization, the situ orientation of the paleomagnetic vector and the bedding normal. The restoring rotation is decomposed into two convenient steps, represented by rotation matrices, such that they restore the in situ paleomagnetic vector together with the tilted bedding normal to their original orientation. These auxiliary rotations combine algebraically to a single finite rotation from which an inclined axis and amount of rotation arc obtained. We demonstrate this method by reconstructing a distinct section of the Mount Sedom salt diapir (Dead Sea Rift Valley, Israel). Here, the conventional tilt correction that brings the bedding to horizontal fails to position the paleomagnetic vector in its expected orientation. Only a rotation of 125 ° about an axis which plunges 50 ° southward can restore both bedding and magnetization to their expected orientations.

Nonlinear dynamic analysis of frame structures

A geometrically nonlinear dynamic analysis method is presented for frames which may be subjected to finite rotations in three-dimensional space. The proposed method is based on the static geometrically nonlinear analysis method reported by Yoshida et al., in which the governing incremental equilibrium equation is represented by the coordinates after the deformation themselves rather than conventional displacements. The governing dynamic equilibrium equation for each element is obtained from the static equation by adding the inertia term. In the solution procedure, a modified Steffensen's iteration process is introduced and combined with the two-step approximation and iterative correction solution procedure developed for static analysis. A numerical example of a curved cantilever beam under lateral loads indicates the effectiveness of the proposed method in cases with three-dimensional finite rotations. Forced vibration analyses of a two-hinged shallow arch are conducted under centrally concentrated loading with several loading amplitudes. The resulting dynamic buckling load is compared with that given by Gregory and Plaut in 1982, who used Galerkin method, and shows good agreement.

On the Parametrization of the Three-Dimensional Rotation Group

Parameterization of group of rotations of Euclidean three-dimensional space applied to integration of rigid body motion equations

Effect of Finite Rotations on Gyroscopic Sensing Devices

Abstract The well-known noncommutativity of three-dimensional finite rotations has long been a curiosity in mechanics since, in actual solution of dynamical problems, the angular velocity, which is conveniently representable as a vector, plays a more natural role. In modern inertial guidance systems, however, the orientation of a body in space, i.e., a rotation, is of primary engineering interest. In this paper a simple method of determining orientation from the time history of three body components of angular velocity is developed by means of a new theorem in kinematics. As a special case of this theorem it is shown that a gyro subjected to a regime of rotations which returns it to the original space orientation will, in general, produce a residual signal. It will have experienced a nonzero and easily calculated mean angular velocity about its input axis. Some implications of the theorem for the design of inertial guidance systems and for the testing of gyros are discussed.