Mathematical Models of Beams and Cables E-Book

Angelo Luongo

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 462

Veröffentlichungsjahr: 2013

Beschreibung

Nonlinear models of elastic and visco-elastic onedimensional continuous structures (beams and cables) are formulated by the authors of this title. Several models of increasing complexity are presented: straight/curved, planar/non-planar, extensible/inextensible, shearable/unshearable, warpingunsensitive/ sensitive, prestressed/unprestressed beams, both in statics and dynamics. Typical engineering problems are solved via perturbation and/or numerical approaches, such as bifurcation and stability under potential and/or tangential loads, parametric excitation, nonlinear dynamics and aeroelasticity. Contents 1. A One-Dimensional Beam Metamodel. 2. Straight Beams. 3. Curved Beams. 4. Internally Constrained Beams. 5. Flexible Cables. 6. Stiff Cables. 7. Locally-Deformable Thin-Walled Beams. 8. Distortion-Constrained Thin-Walled Beams.

Leseprobe

Table of Contents

Preface

Introduction

I.1 Derived one-dimensional models

I.2 Direct one-dimensional models

I.3 Cables and strings

I.4 Locally deformable beams

I.5 An overview of literature

I.6 An overview of the book

I.7 Notation

List of Main Symbols

Chapter 1: A One-Dimensional Beam Metamodel

1.1 Models and metamodel

1.2 Internally unconstrained beams

1.3 Internally constrained beams

1.4 Internally unconstrained prestressed beams

1.5 Internally constrained prestressed beams

1.6 The variational formulation

1.7 Example: the linear Timoshenko beam

1.8. Summary

Chapter 2: Straight Beams

2.1 Kinematics

2.2 Dynamics

2.3 Constitutive law

2.4 The Fundamental Problem

2.5 The planar beam

2.6 Summary

Chapter 3: Curved Beams

3.1 The reference configuration and the initial curvature

3.2 The beam model in the 3D-space

3.3 The planar curved beam

3.4 Summary

Chapter 4: Internally Constrained Beams

4.1 Stiff beams and internal constraints

4.2 The general approach

4.3 The unshearable straight beam in 3D

4.4 The unshearable straight planar beam

4.5 The inextensible and unshearable straight beam in 3D

4.6 The inextensible and unshearable straight planar beam

4.7 The inextensible, unshearable and untwistable straight beam

4.8 The foil-beam

4.9 The shear–shear

4.10 The planar unshearable and inextensible curved beam

4.11 Summary

Chapter 5: Flexible Cables

5.1 Flexible cables as a limit of slender beams

5.2 Unprestressed cables

5.3 Prestressed cables

5.4 Shallow cables

5.5 Inextensible cables

5.6 Summary

Chapter 6: Stiff Cables

6.1 Motivations

6.2 Unprestressed stiff cables

6.3 Prestressed stiff cables

6.4 Reduced models

6.5 Inextensible stiff cables

6.6 Summary

Chapter 7: Locally-Deformable Thin-Walled Beams

7.1 Motivations

7.2 A one-dimensional direct model for double-symmetric TWB

7.3 A one-dimensional direct model for non-symmetric TWB

7.4 Identification strategy from 3D-models of TWB

7.5 A fiber-model of TWB

7.6 Warpable, cross-undeformable TWB

7.7 Unwarpable, cross-deformable, planar TWB

7.8 Summary

Chapter 8: Distortion-Constrained Thin-Walled Beams

8.1 Introduction

8.2 Internal constraints

8.3 The non-uniform torsion problem for bi-symmetric cross-sections

8.4 The general problem for warpable TWB

8.5 Cross-deformable planar TWB

8.6 Summary

Bibliography

Index

To Fiorella (A.L.)

To my parents Elio & Marisa (D.Z.)

First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2013

The rights of Angelo Luongo and Daniele Zulli to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2013948060

British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library

Preface

It is customary, in the engineering community, to distinguish linear from nonlinear theories. As a matter of fact, “Dynamics” and “Nonlinear Dynamics”, as well as “Buckling” and “Post-Buckling” are considered to be basically different disciplines. Usually, the adjective “linear” is omitted so that theories, “by default”, are understood to be linear, and nonlinearities, when accounted for, must be explicitly mentioned. This habit is probably due to the fact that, usually, linear models are believed sufficient to solve most of the technical problems encountered in practice. Linear theories, therefore, are well rooted in the knowledge of any engineer, whereas nonlinear theories are considered to be of the competence of few specialists.

The opposite, instead, occurs in the field of Applied Mathematics. For example, “Continuum Mechanics” and “Dynamical System Theory” are understood as intrinsically nonlinear, with no need for further specifications. If nonlinearities are instead ignored, mathematicians stress that the problem has been linearized. This practice is probably due to the fact that mathematicians are not very interested in solving technical problems and, therefore, they are not stimulated by “simplifying” models, but rather by formulating theories and proving theorems of the widest generality.

The theory of beams, which is the subject of this book, is not an exception to this rule. The relevant linear theory is well-known to mechanical and civil engineers, but only some researchers and few PhD students operating in the field are confident with nonlinear theories. Thus, if an engineer tries to enter this new world, he or she must face the reading of books whose mathematics is often beyond his or her knowledge; the focus is on the formulation in itself, and often little (or even no) attention is paid to algorithms and examples. In other words, the engineer has to face an environment where mathematics “is a end, but not a means”. On the other hand, there exist some good books specialized in one of the subjects discussed above. However, very often, they are entirely devoted to illustrating algorithms and results, while modeling is overlooked or required as a reader’s prerequisite.

All previous considerations seemed to be good reasons for the authors to write a new book, which is aimed (a) at introducing the reader to geometrically exact nonlinear modeling of one-dimensional structures, by using elementary mathematics only; (b) at consistently deriving approximate models in order to render the relevant equations amenable to semi-analytical solutions; and (c) at giving a comprehensive overview of different engineering problems, in which the application of nonlinear theories is mandatory. However, after having written a few chapters, the project soon appeared too ambitious to be fully realized in a single volume. Therefore, it was decided to devote a first book to models and a forthcoming book (Nonlinear Beam and Cable Mechanics in Engineering Applications) to algorithms and phenomena, with the aim of guiding the reader throughout the whole process of the engineering design. The two books, therefore, should be intended as sequential and closely related, but, at the same time, independent, so that only one of the two aspects, theoretical or algorithmic-phenomenological, can be addressed by the readers, consistently with their interest.

In this book, several models of elastic and viscoelastic beams, both in statics and dynamics, are analyzed. They are discussed in order of increasing complexity by including straight/curved, planar/non-planar, extensible/inextensible, shearable/unshearable, warpable/unwarpable, cross-deformable/undeformable and prestressed/unprestressed beams. String and cables, straight or curved, perfectly flexible or endowed with flexural–torsional stiffnesses, are also addressed. Modeling is developed via a direct approach, based on one-dimensional polar or Cauchy continua.

In summary, this book is an attempt (a) to make it easy to learn the nonlinear theory of beams and cables and (b) to formulate consistent approximate models, leading to reasonably simple mathematical problems.

The book is mainly devoted to researchers and PhD students in Civil and Mechanical Engineering, as well as in Applied Mathematics. It is also hoped to be useful for professional engineers. It requires only the basic knowledge of Mathematical Analysis, Linear Algebra and Continuum Mechanics, generally covered in engineering courses.

Angelo LuongoDaniele ZulliSeptember 2013

Introduction

Here we summarize the main concepts to be discussed later and illustrate the guiding factor of this book. Firstly, the modeling problem for a beam is addressed by comparing two different philosophies: derivation from a three-dimensional (3D) Cauchy continuum, or direct formulation as a one-dimensional (1D) polar continuum. Secondly, string and cables are successively considered as degenerate models of perfectly flexible beams, and the circumstances in which flexural and torsional stiffnesses have to be considered are discussed. Thirdly, more sophisticated models of beams with deformable cross-sections are addressed. Finally, a quick overview of the literature and of this book is given.

I.1 Derived one-dimensional models

As is well-known, the Fundamental Problem of Continuum Mechanics, formulated in the context of the Lagrangian description, consists of evaluating stresses, strains and displacements in a body, when this is loaded by assigned volume and surface forces, and, moreover, displacements are prescribed on a portion of the boundary. When this problem is addressed for a beam, a 3D Cauchy continuum model could be applied, which would lead to a system of partial differential equations, in which each (scalar, vector or tensor) magnitude is a function of three coordinates (time understood), which span the volume occupied by the body in the reference configuration. Such an approach, however, although “exact” in the context of continuum mechanics, is almost unpractical for the difficulty of solving the governing equations, so that it is advisable to resort to “approximate” models that exploit the geometric peculiarity of the body, namely its slenderness. The main object of the analysis consists of formulating a 1D (rather than a 3D) model in which all the magnitudes involved depend on only one coordinate, e.g. a curvilinear abscissa s running along the (unstretched) curve S.

I.2 Direct one-dimensional models

In the former approach, the 1D model was derived from a 3D one, by constraining, in some sense, the kinematics. However, we could wonder if it is possible to formulate a “direct” 1D model of a beam, by avoiding “derivation” from a more complex system. This, indeed, is possible, by referring, however, to a continuum richer than Cauchy’s, namely to a polar continuum (also known as Cosserat’s continuum or a structured continuum). A polar continuum is made of material particles that are endowed with orientation; they, therefore, can translate, as the particles of the Cauchy continuum, but, in addition, can rotate. (This rotation is often called the micro-rotation, to distinguish it from the macro-rotation that one observes in a Cauchy continuum, when one looks not at a particle, but at a small neighborhood of it). With this idea in mind, we can consider the beam as a 1D object, geometrically described by its axis S, which, however, is made up of orientable material particles, in other words, as a 1D polar continuum. Each point of this continuum possesses six degrees of freedom, as the 3D beam made up of rigid sections. If we compare the two points of view (3D and 1D objects), we observe that the cross-sections disappear in the latter model; however, they are replaced by body-points, able to describe, via their orientation, the cross-section attitude, thus regaining the information lost. (Of course, if the cross-section was also able to deform itself, in addition to rotating, more information should be borne by the structured continuum, as we will discuss later).

How can we account for the orientation of body-points? The best way is to consider that a triad of orthonormal unit vectors ai(s), called directors, is attached to each of them. The rotation of the triad, described by a rotation tensor R(s), describes the rotation of the body-point. Although not strictly necessary, one can think that two of the directors lie, in any configuration, in the plane of the (now disappeared) cross-section, whereas the third one, orthogonal to former ones, is initially aligned with the beam axis, but, after the deformation, loses this parallelism.

It is interesting to compare derived and direct models not only from a kinematic point of view, but also from a dynamical perspective. In a Cauchy 3D continuum, in-contact points only exchange forces per unit area; in a 1D polar continuum, instead, they exchange forces and couples. When we consider a beam as a 3D object, we have to integrate the stresses on the cross-section, to evaluate their resultants, namely the axial and shear forces, and the bending and torsional moments. When, instead, we use the 1D polar continuum, the internal forces already represent these resultants. Therefore, the absence of an “internal arm”, which is responsible for the moment in the 3D model, is regained, in the 1D polar continuum, by the dynamic property of the body-points to exchange couples.

Using a direct model, instead of deriving it from a 3D continuum, offers some advantages. Indeed, the direct model is capable to describe beam-like structures, as trussed or framed beams (Vierendel-like), or, moreover, spiral structures (as helicoidal springs), when one is not interested in the local behavior (e.g. of the single truss), but rather in the global behavior of the structure as a whole. These systems put in evidence a new question: how can we endow the model with a constitutive law? The problem, however, is not specific to beam-like structures, but also concerns massive beams and represents, in some sense, the drawback of direct models. If, indeed, in the derived models, the constitutive law is a straightforward consequence of that of the Cauchy continuum, in contrast, in the direct model, the law has to be independently enforced. The problem can be tackled by considering a representative volume of the refined model (e.g. one period length of a periodic structure) and establishing an energy equivalence between this and an equally long segment of the rough model.

In this book, we follow the direct approach and will show how to formulate constitutive laws for massive and beam-like structures.

I.3 Cables and strings

When the beam possesses a very high slenderness ratio (e.g. of the order of 103, or more), it becomes extremely flexible. For example, if it is disposed (without stretch) on a horizontal line, hinged at the ends, and then made subject to its own weight, it undergoes large, prevalently transverse, displacements, thus resulting as axially stressed rather than subject to a bending moment. Such an object, of course, would be of no utility in structural engineering, where beams are designed to carry loads. However, if the beam is put into a state of tensile prestress, it can assume a significant geometric stiffness, which makes the structure work. As is well-known from the elementary theory of beams, the geometric stiffness is produced by the perturbation of the pre-existent state of equilibrium, which brings unbalanced forces up, able to balance the incremental loads. Such a mechanical system is usually not recognized as a beam, but rather as a string or cable, the first denomination being preferred when the axis is almost rectilinear (i.e. when the prestress is mainly due to tensile end forces), and the second when the axis is significantly curved (or piecewise linear, i.e. when the prestress is due to carried weights).

The simplest (and nearly universally used) model for a string completely neglects the flexural (and torsional) stiffness. Therefore, a string is viewed as a perfectly flexible, idealized beam, whose cross-sectional area is lumped at the centroid, and, consequently, has zero moment of inertia. More precisely, we can also say that a string is a 1D Cauchy continuum (instead of polar), embedded in a 3D environment. As a result of this idealization, the string does not possess its own specific shape, since it can assume infinite natural configurations in which stress and strain simultaneously vanish. In the linear field, therefore, the string is a kinematically undetermined and statically impossible continuum system; prestress, however, makes the equilibrium possible in a configuration very close to the prestressed one (often called adjacent configuration), as occurs, for example, for a mathematical pendulum prestressed by the gravity force, when it is disturbed by a transverse force. The prestressed configuration is usually taken as reference configuration in a Lagrangian approach.

There exist, however, some problems in which the perfectly flexible model is not adequate to give an answer. We cite two of these circumstances. (a) If the string is clamped at the ends, a boundary layer manifests itself in a narrow zone close to the clamps, in which the bending effects cannot be neglected. The same occurs close to transverse loads applied to the string, as those transmitted by a pulley moving along the cable. Without going into detail, which is out of the scope of this book, it can be checked that the flexural stiffness appears as a small term affecting the highest derivatives in the equations of motions. This term can be neglected almost everywhere (to obtain the so-called outer solution), but not close to singular points (where an inner solution must be sought). Here, in order to satisfy the boundary conditions, the solution becomes fast varying, thus rendering the (elsewhere) small terms comparable to the leading terms. If, therefore, one uses the perfectly flexible model, one has to renounce satisfying some of the boundary conditions and, consequently, investigating the mechanical state in the boundary layers. (b) As a second example, let us consider the effect of wind on iced strings. Ice accretion modifies the (usually) original circular shape of the section and consequently its aerodynamic properties. Therefore, wind loads depend on the attitude of the cross-section, and change during motion, as a consequence of the twisting of the string. To correctly analyze this interaction phenomenon, torsional effects need to be introduced into the structural model.

One could, of course, use a complete model of a prestressed beam to analyze all problems involving strings and cables, without introducing the drastic simplification concerning their flexibility. This is usually done in purely numerical investigations, where all the terms of the beam model are retained in the analysis, and used together with prestress. However, such models are complex and suffer from ill-conditioning, since the relevant stiffness matrices are nearly singular, for the presence of small terms. An approximated simple model, accounting for the essential terms, is discussed later in this book.

I.4 Locally deformable beams

As we observed before, there are problems in which deformations of the cross-section cannot be ignored. The question assumes great importance when the beam is thin-walled, open or closed. As a first example, it is known from the Vlasov theory [VLA 61] that warping of open thin-walled beams, when caused by non-uniform torsion, induces stresses normal to the cross-section (equivalent to a bi-moment), variable along the beam-axis; these, in turn, trigger tangential stresses equivalent to a (so-called secondary) torsional moment which cannot be ignored. Consequently, the torsional stiffness of the beam turns out to be much higher than the de Saint-Venant stiffness. As a second example, it is known from the Brazier theory [BRA 27] that when a tubular beam is bent, the original circular middle line undergoes ovalization, with flattening and consequent reduction of the cross-section inertia moment.

A proper modeling of a thin-walled beam therefore calls for accounting for in-plane and/or out-of-plane deformability of the cross-sections. We will refer to these beams as locally deformable (and to the former as locally undeformable). The task can, again, be accomplished via derivation from a 3D model or via a direct approach, as here briefly outlined.

The modern generalized beam theory (GBT, see e.g. [GON 07, BEB 08, BAS 09, SIL 10, CAM 10, GON 10] for a wide overview) derives a 1D model from the 3D Cauchy continuum. It is based on the semi-variational (or Kantorovitch) method, according to which the displacements u(r, s) are expressed as a linear combination of known shape functions ψ(r) and unknown amplitude functions α(s). A variation principle leads to a set of ordinary differential equations in the amplitudes. Of course, if the shape functions only describe rigid motions of the cross-section, the GBT furnishes the locally undeformable model; for this reason, it is called generalized, since it includes the standard model. However, it is much more powerful, because it is capable of accounting for changes of shape of the cross-section, including warping.

If a direct model is desired, the classical Cosserat continuum must be endowed with additional kinematic descriptors α(s), here called distortional variables, whose meaning, at least initially, is not necessary to be specified. These descriptors entail that the strains of the beam increase in number, with respect to the standard model. In the context of a first-gradient theory, strains consist of the descriptors themselves, α(s) := α(s) and of their first derivatives β(s) := α′(s). The use of a variational principle leads to balance equations in the displacements and distortional variables. The main difficulty of this approach consists of assigning the constitutive law, which links the generalized strains to their dual generalized stresses (i.e. distortional and bi-distortional stresses). This task can be accomplished by an identification procedure from a 3D model, based on an equivalence in energy. The operation leads to attributing a geometrical meaning to the distortional variables and a mechanical meaning to the dual stress quantities.

In this book, we will follow the direct approach and use a 3D fiber-model to identify the constitutive law.

I.5 An overview of literature

The literature on beams and cables is extensive and continuously produced over the years. A huge quantity of books and scientific papers are issued daily on the topics, and it would be in any case impossible to try to overview most of them. Therefore, we limit ourselves, here, to cite only some classical and recent books, considering that the literature on beams and cables should be necessarily embedded in the wider context of continuum mechanics, rational mechanics, linear and nonlinear dynamics and stability theory.

From this point of view, the main reference is given to classic benchmarks on continuum mechanics [GUR 82, GUR 72, GUR 83, GUR 00, TRU 77, TRU 66, TRU 04, CIA 88, VIL 77, GRE 92, LAN 70, LEI 74, MAL 69, MAR 93, OGD 07, POD 00, SOU 73, TIM 51, FUN 01, HOL 00, DEN 87, WAS 82, ODE 82, RED 02] as well as to more recent contributions [GUR 10, OGD 97, CHA 13, CON 07, WEG 09, DIM 11, DYM 13, ROM 06, BER 09, BAR 10, BRI 13, ERE 13, ESL 13, RED 10, RED 13].

Fundamental concepts about the mathematical framework, general mechanics, stability theory and nonlinear dynamics can be found in [COU 53, CHO 01, CLO 03, FIN 08, GAL 07, GAN 13, GOL 80, GRE 10, KOK 06, KUI 99, LAN 97, LEI 87, LOV 89, MAR 12, MEI 70, MEI 97, MEI 01, MEI 80, NAY 79, NAY 73, WEI 74, PRE 13].

Many books move freely from the general continuum mechanics to more specific theories of beams, plates or shells. Among them, reference is made to [ANT 05, LOV 44, NAY 04, BIG 12, HOW 09, LAC 13, RUB 00]. Sometimes the subject is approached in the finite element context, as in [BAT 82, WRI 08, ZIE 05, IBR 09].

On the other hand, books specifically devoted to cables, beams and/or shells are [ANT 72, CAP 89, VIL 97, TIM 65, TIM 59, VLA 61, FER 06, LIB 06, IRV 81, MAG 12, MUR 86, OBO 13, ROS 11, VIN 89, VOR 99, WAN 00, HOD 06, HOD 11] and those devoted to their stability are [BOL 64, BOL 63, ATA 97, BAŽ 03, ELI 01, LEI 87, SIM 06, AMA 08, TIM 63].

As a specific choice of the authors, only a few journal papers are reported here, the literature overview being more turned towards books. Nevertheless, we find it important to cite here some fundamental contributions reported in papers: on general framework and continuum mechanics [GER 73, DIC 96], on cables [IRV 74, REG 04a, REG 04b, IBR 04, PER 87, LU 94, LEE 92, BUR 88, TJA 98, TRI 84, GAT 02, GOY 07, GOY 05], on beams [SIM 85, SIM 86, SIM 88, SIM 91, CRE 91, CRE 78a, CRE 78b, ZAR 94, CRE 88a, CRE 88b], on thin-walled beams [BRA 27, RIZ 96, DIC 99, REI 59, REI 83b, REI 84, REI 87, RUT 06], and on beam-like structures [DIC 90] and GBT [SIL 03, GON 07, BAS 09, BEB 08, CAM 06, SIL 10, CAM 10, GON 10].

I.6 An overview of the book

All the concepts discussed above are detailed in this book. Here, a short overview is presented. The book is ideally divided into four parts: (a) metamodel (Chapter 1), (b) locally undeformable beams (Chapters 2–4), (c) cables and strings (Chapters 5 and 6), and (d) locally deformable thin-walled beams (Chapters 7 and 8).

In Chapter 1 a metamodel is introduced, which works as a progenitor for specific models to be dealt with later. Here, unprestressed, prestressed and internally constrained beams, with or without prestress, are studied. The virtual power principle is used to derive the balance equations [GER 73]. The variational formulation is also illustrated. Exact equations are derived in operator form and then linearized around the reference configuration.

Planar straight beams, locally undeformable, are addressed in Chapter 2. Exact kinematics and balance equations are derived. Homogenization procedures are outlined to derive the constitutive law. The model is developed in a 3D environment. The planar beam is drawn as a particular case.

The previous analysis is extended in Chapter 3 to curved beams (arches), both in-space and in-plane.

Chapter 4 deals with internally constrained beams for which two basically different approaches (mixed and displacement formulations) are followed, respectively, including or not the reactive stresses produced by the internal constraints. Several cases of internally constrained beams are considered; a few among them are unshearable, inextensible, untwistable, and shear–shear–torsional beams.

In Chapter 5 flexible cables and strings are analyzed as 1D bodies not endowed with flexural and/or torsional stiffnesses. Both unprestressed and prestressed cables are considered and linearized equations are derived for the latter group. Approximated equations for shallow cables, horizontal or inclined, are obtained. Finally, inextensible cables are addressed.

In Chapter 6 stiff cables, equipped with flexural and torsional stiffnesses, are considered. A simplified model, based on the hypothesis of small curvature and large elongation, is developed.

A 1D model of a thin-walled beam undergoing in-plane and out-of-plane distortions of the cross-section is formulated in Chapter 7. Nonlinear hyper-elastic laws are obtained by the homogenization process of a 3D fiber-model. Due to their cumbersome expressions, governing equations are explicitly given only for simple cases, although the illustrated procedure is general.

The theory is specialized in Chapter 8 to locally deformable thin-walled beams with internal constraints, of the Vlasov, Bredt and Brazier type, able to supply nonlinear equations which generalize the underlying linear theories, respectively.

I.7 Notation

Throughout the book, a scalar quantity is denoted by a Roman or Greek italic letter (e.g. u or ω). A bold Roman or Greek letter denotes a vector or a tensor: the former (mostly, but not exclusively) lowercase (e.g. u or ω) and the latter (exclusively) uppercase (e.g. R). A bold italic letter refers to a column-matrix or matrix (e.g. u, ω, R) which, typically, are the scalar representation of the homonymous vector quantities in a specified basis, as discussed below.

Vectors and tensors attached to bases

Scalar representation of vectors and tensors in different bases

The aforementioned transformations describe the change, under a rotation, of absolute geometric entities into new geometric enties. As is well-known to the reader, they should not be confused with the transformations undergone by the components of a (sole) geometric entity when the basis is rotated. To relate the components of a given vector w or tensor T in different bases, or , we first note that the matrix of the change of basis is (where square brackets denote component evaluation in the basis indicated as an index); therefore:

[1]

Components of attached vectors and tensors in their “natural bases”

To stress the independence of the two concepts previously discussed, both vectors r, , as well as both the tensors L, , could be represented either in or , and denoted by respectively. Of course, it appears “more natural” to express , in and r, L in since, in those bases, they are more meaningful. When we apply the component transformations to the attached vectors, we have:

[2]

and, for tensors:

[3]

In conclusion, as expected, we find that the attached vectors and tensors have the same components in their respective “natural bases”.

Notation adopted

In this book, with a few exceptions to be clearly stated later, an overbar affixed on a vector or tensor denotes that that entity is attached to the basis ; the same symbol without a bar denotes that the vector or tensor has been transformed by a rotation which led to . When, instead, an overbar appears on column matrices, matrices, or their components, it denotes that a vector or tensor (regardless if it is attached to the basis or not) has been represented in , and, without a bar, in . As an example, and , which are related by ; similarly, and , related by .

List of Main Symbols