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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
This book is centred about the Principle of virtual work and the related method for mechanical modelling. It aims at showing and enhancing the polyvalence and versatility of the virtual work approach in the mechanical modelling process. The virtual work statement is set as the principle at the root of a force modelling method that can be implemented on any geometrical description. After experimentally induced hypotheses have been made on the geometrical parameters that describe the concerned system and subsystems, the method provides a unifying framework for building up consistently associated force models where external and internal forces are introduced through their virtual rates of work. Systems described as three-dimensional, curvilinear or planar continua are considered: force models are established with the corresponding equations of motion; the validation process points out that enlarging the domain of relevance of the model for practical applications calls for an enrichment of the geometrical description that takes into account the underlying microstructure.
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Table of Contents
Cover
Dedication
Title
Copyright
Preface
1 The Emergence of the Principle of Virtual Velocities
1.1. In brief
1.2. Setting the principle as a cornerstone
1.3. The “simple machines”
1.4. Leonardo, Stevin, Galileo
1.5. Descartes and Bernoulli
1.6. Lagrange (1736–1813)
2 Dualization of Newton’s Laws
2.1. In brief
2.2. Newton’s statements
2.3. System of material points
2.4. Dualization and virtual work for a system of material points
2.5. Virtual work method for a system of material points
2.6. Practicing
3 Principle and Method of Virtual Work
3.1. Why and what for?
3.2. General presentation of the virtual work method
3.3. General results
3.4. Particular results
3.5. About equilibrium
4 Geometrical Modeling of the Three-dimensional Continuum
4.1. The concept of a continuum
4.2. System and subsystems
4.3. Continuity hypotheses
4.4. Validation of the model
4.5. Practicing
5 Kinematics of the Three-dimensional Continuum
5.1. Kinematics
5.2. Convective derivatives
5.3. Piecewise continuity and continuous differentiability
5.4. Comments
5.5. Explicit formulas in standard coordinate systems
5.6. Practicing
6 Classical Force Modeling for the Three-dimensional Continuum
6.1. Virtual motions
6.2. Virtual rates of work
6.3. Implementation of the principle of virtual work
6.4. Piecewise continuous and continuously differentiable fields
6.5. The stress vector approach
6.6. Local analysis
6.7. The hydrostatic pressure force modeling
6.8. Validation and implementation
6.9. Explicit formulas for the equation of motion in standard coordinate systems
6.10. Practicing
7 The Curvilinear One-dimensional Continuum
7.1. The problem of one-dimensional modeling
7.2. One-dimensional modeling without an oriented microstructure
7.3. One-dimensional model with an oriented microstructure
7.4. Relevance of the model
7.5. The Navier–Bernoulli condition
7.6. Analysis of systems
8 Two-dimensional Modeling of Plates and Thin Slabs
8.1. Modeling plates as two-dimensional continua
8.2. Virtual rates of work
8.3. Equations of motion
8.4. Physical interpretation and classical presentation
8.5. Piecewise continuous fields
8.6. Matching the model with the three-dimensional continuum
8.7. The Kirchhoff–Love condition
8.8. A illustrative example: circular plate under a distributed load
Appendix 1: Introduction to Tensor Calculus
Appendix 2: Differential Operators
Appendix 3: Distributors and Wrenches
Bibliography
Index
End User License Agreement
Guide
Cover
Table of Contents
Begin Reading
List of Figures
Chapter 1: The Emergence of the Principle of Virtual Velocities
Figure 1.1. Archimedes’ proof of the law of the lever
Figure 1.2. Covers of Stevin’s books
Figure 1.3. The inclined plane [GAL 38]
Figure 1.4. Illustrating Lagrange’s proof of the principle of virtual velocities
Chapter 2: Dualization of Newton’s Laws
Figure 2.1. Newton’s Principia
1
Figure 2.2. System of point masses: internal and external forces
Figure 2.3. Subsystem : S′ internal and external forces
Figure 2.4. System of point masses: law of mutual actions
Chapter 3: Principle and Method of Virtual Work
Figure 3.1. The virtuous circle of mechanical modeling
Figure 3.2. Principle of the virtual work modeling method
Figure 3.3. Schematic representation of the system S and a subsystem S′
Chapter 4: Geometrical Modeling of the Three-dimensional Continuum
Figure 4.1. The first step of the virtuous circle
Figure 4.2. Extrusion of a thick tube (document kindly supplied by M. Sauve, CEA)
Figure 4.3. Die stamping of an aluminum alloy specimen (document kindly supplied by A. Le Douaron)
Figure 4.4. Excerpt from Flow visualization [KLI 63]
Figure 4.5. Excerpt from Strain localization [DES 86]
Figure 4.6. Pathline of a particle
Figure 4.7. Punch indentation of a plasticine block (CEMEF, Sophia Antipolis)
Figure 4.8. Asymmetrical punch indentation test on a plasticine block (Application to tectonics in East Asia) [PEL 87]
Figure 4.9. “Slip line” pattern under a strip footing on a purely cohesive soil [HAB 84]
Figure 4.10. Foundation on a two-dimensional layer (reduced scale experiment): short exposure visualizing the velocity field [BON 72]
Figure 4.11. Flow around a flat plate (ONERA, France)
Chapter 5: Kinematics of the Three-dimensional Continuum
Figure 5.1. Material derivative of a vector
Figure 5.2. Distortion rate
Figure 5.3. Volume dilatation rate
Figure 5.4. Convective derivative of a volume integral
Figure 5.5. Convective derivative of a volume integral in the discontinuous case
Chapter 6: Classical Force Modeling for the Three-dimensional Continuum
Figure 6.1. Force modeling
Figure 6.2. External forces for the system
Figure 6.3. External forces for a subsystem
Figure 6.4. Contact forces at the boundary of a subsystem
Figure 6.5. Discontinuity of the virtual velocity field
Figure 6.6. Internal contact actions modeled by the stress vector
Figure 6.7. Small tetrahedron argument
Figure 6.8. Small parallelepiped argument for the symmetry proof
Figure 6.9. Small parallelepiped argument for the equation of motion
Figure 6.10. Discontinuity of the stress field
Figure 6.11. Stress vector acting on a facet
Figure 6.12. Components of the stress tensor in an orthonormal basis
Figure 6.13. Normal stress and tangential components
Figure 6.14. Stress vectors on facets perpendicular to the principal axes
Figure 6.15. Stress vectors in the case of an isotropic stress tensor
Figure 6.16. Equilibrium of a solid under gravity
Figure 6.17. The complete “virtuous circle”
Figure 6.18. Facet and Mohr’s plane
Figure 6.19. Facet and Mohr circles
Figure 6.20. Description of the Mohr circle
Figure 6.21. Rotating circular ring
Figure 6.22. Convex
Chapter 7: The Curvilinear One-dimensional Continuum
Figure 7.1. Marble Canyon (Arizona). Solferino footbridge (Paris) (© Jean Salençon)
Figure 7.2. One-dimensional geometrical modeling
Figure 7.3. “Squeezing” the three-dimensional system
Figure 7.4. Subsystem
Figure 7.5. Discontinuous virtual velocity field on
Figure 7.6. Frenet–Serret equations and local orthonormal basis
Figure 7.7. Concentrated external force acting on the system
Figure 7.8. a) The catenary problem. b) Funicular arch
Figure 7.9. The particle at the point P
Figure 7.10. External forces exerted on the system
Figure 7.11. Beam as the built-in support of a plate element
Figure 7.12. Beams connected by a rigid joint
Figure 7.13. External forces acting on the system
Figure 7.14. Internal force wrench at the field point
Figure 7.15. Fundamental law of dynamics applied to the infinitesimal line element
Figure 7.16. Matching the one-dimensional model with the three-dimensional one
Figure 7.17. Normal and shearing forces, twisting and bending moments
Figure 7.18. Virtual bending hinge
Figure 7.19. Assembly node
Figure 7.20. Ball and socket assembly joint
Figure 7.21. Geometrical instability of a planar truss made of pin-connected members
Chapter 8: Two-dimensional Modeling of Plates and Thin Slabs
Figure 8.1. Three-dimensional plate and director sheet
Figure 8.2. The particle at point P and associated microstructure
Figure 8.3. In-plane motion of a particle
Figure 8.4. Out-of-plane motion of a particle
Figure 8.5. Distributed forces acting on the system
Figure 8.6. Distributed boundary forces
Figure 8.7. Plate element supported by a beam
Figure 8.8. Boundary external forces on a subsystem
Figure 8.9. The infinitely small triangle argument applied to shear forces
Figure 8.10. Equilibrium of the resultant forces along
Figure 8.11. The infinitely small triangle arg ument applied to internal moments. Components of
Figure 8.12. Equilibrium of moments applied to an infinitely small parallelogram
Figure 8.13. Components of
Figure 8.14. Boundary conditions
Figure 8.15. Discontinuous virtual motion
Figure 8.16. Cylindrical volume (dS × A′ A)in the three-dimensional solid
Figure 8.17. Components of and
Figure 8.18. Hinge line in a discontinuous virtual motion
Figure 8.19. Circular plate under a uniformly distributed load
Figure 8.20. Virtual motions
Figure 8.21. Conical Kirchhoff–Love virtual motion
Figure 8.22. Bending of a reinforced plaster slab (M. Milicevic, Mons, Belgium)
Appendix 2: Differential Operators
Figure A2.1. Piecewise continuous and continuously differentiable vector field
Figure A2.2. Curl theorem
Figure A2.3. Divergence of the curl
Figure A2.4.
Figure A2.5.
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e1
In memory of Marcel Save (1930–2016)
Virtual Work Approach to Mechanical Modeling
Jean Salençon
First published 2018 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:
ISTE Ltd
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London SW19 4EU
UK
www.iste.co.uk
John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA
www.wiley.com
© ISTE Ltd 2018
The rights of Jean Salençon to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2017959464
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-295-3
Preface
This book was triggered by the suggestion from students at the City University of Hong Kong that I should publish a revised version of my textbook1 as a follow-up to the PhD courses they had been attending. As announced by its title, it comes out in a completely different spirit from the previous work, being centered on the principle of virtual work and the related method for mechanical modeling, with the purpose of showing and enhancing their polyvalence and versatility.
The concepts of virtual velocities and virtual work were progressively formulated through a 2000-year scientific process which started with the quest for a general principle to govern the leverage properties of “simple machines”, a process that can be considered as completed by the end of the 18th Century with Lagrange’s statement of the principle of virtual velocities, after the concepts of virtual velocities and energy had been specified by Johann Bernoulli. With the introduction of Lagrange’s multipliers, the principle led to the definition of internal and external binding forces as the dual variables associated with geometrical constraints, the very concept at the root of the virtual work mechanical modeling method.
The book starts with a brief overview of the genesis of the principle of virtual velocities in relation to the statics of simple machines.
Chapter 2 is then devoted to the complete statement of the principle of virtual work for the dynamics of a system of point masses. It involves external and internal forces and quantities of acceleration, introduces the concept of subsystems and underscores that, for them, external forces are only the result of a thought experiment. Through the dualization process performed on Newton’s Laws, linear forms, named virtual (rates of) work, are built up on the vector space of virtual motions for the system and similar linear forms for any subsystem on its own vector space of virtual motions. Virtual motions are defined by the rate of variation of the parameters (the positions of the point masses in that case) that describe the geometrical state of the system (or subsystem) without complying with any geometrical constraint. These rates of variation are the virtual velocities. Rigid body virtual motions are shown to play a specific role, as the linear forms expressing the virtual rate of work by internal forces, either for the system or any subsystem, are zero in such virtual motions. Conversely, as an introduction to the virtual work method in this particular case, it is shown how, given the geometrical model of point masses, the corresponding force model can be obtained by writing down the linear forms expressing the virtual rates of work in their most general form and then implementing the virtual work statements to specify them and derive the fundamental laws.
In Chapter 3, the principle of virtual work is generalized into a formulation suitable for any general geometrical description encompassing all the parameters which define the state of a system and its subsystems. Their non-constrained variations define the virtual motions generating vector spaces, and the linear forms expressing virtual rates of work are continuous. This leads to the presentation of the virtual work method, making it clear that the “virtuous circle” of mechanical modeling starts from experimentally-induced hypotheses on the geometrical parameters. The method then provides consistent processes to build up associated force models, which must be submitted to validation, the next step in the “circle”, in order to assess their domain of relevance. This chapter also takes advantage of the key role played by rigid body virtual motions to establish general results valid for any force modeling. They are expressed in terms of force wrenches.
The three-dimensional continuum model is chosen as a first example. Chapter 4 is devoted to defining the corresponding geometrical modeling from the physical intuition of a continuum gained through experimental observations. The concept is that of a system made of “diluted particles” whose geometrical state is characterized by their spatial position, under the constraint of piecewise continuity and continuous differentiability.
The kinematics of the model is analyzed in Chapter 5. Derived from the velocity field, tensor quantities are defined which measure the deformation rate and mean rigid body rotation rate of each “diluted” point particle. With the definition of convective (time) derivatives, the general theorems established in Chapter 3 are given more explicit expressions in the form of the Euler and Kinetic Energy theorems for the three-dimensional continuum.
In Chapter 6, the construction of the force model for the classical three-dimensional continuum, following the virtual work method, calls first for the expression of the virtual rate of work by external forces for the system, based upon physically-induced assumptions. The same assumptions are heuristically made for subsystems, together with the crucial hypothesis that constituent particles do not exert any action at a distance upon one another. As for the virtual rate of work by internal forces, it is assumed to be described by a volume density independent of the considered subsystem, a linear function of the virtual velocity field first gradient. In order to comply with the principle of virtual work, this expression is specified with a symmetric tensor field for the internal force model: the Cauchy stress tensor. The field and boundary equations of motion are then derived for the system and subsystems, showing that the initial assumptions about the virtual rates of work are consistent with each other and actually describe a physical model where particles only exert contact actions upon one another. The virtual work method thus plays a double role: first in organizing the whole modeling process, then as the final criterion for checking the consistency of the model obtained. The hydrostatic pressure model comes out as a particular case of the Cauchy stress tensor model. It can also be built up directly through the same method with a restricted expression of the virtual rate of work by internal forces on the same vector space of virtual motions; consequently, its relevance for practical applications is reduced.
Chapter 7 develops the same procedure for constructing the one-dimensional curvilinear continuum model. Taking the physical intuition of a slender three-dimensional medium being squeezed onto a director curve as a starting point, a first model is built up exactly in the same way as the three-dimensional continuum: the constituent particles are described as “diluted” material points on the director curve only characterized by their position vector from the geometrical viewpoint. The model obtained that way is physically relevant for wires and cables without stiffness, which turns out not to be sufficient since most practical applications are concerned with rods, beams, arches, etc. For a mechanical model better suited for practical issues, it is necessary to enrich the geometrical description of the constituent particle with parameters that better reflect the physical constitution of the original three-dimensional body. As it comes out from everyday practice that the cross-section of the original slender body plays a crucial role, the constitutive particle in this more sophisticated one-dimensional model is characterized on the director curve by its position vector and the orientation of an underlying microstructure to account for the transverse cross-section. Actual and virtual motions are defined by velocity distributor fields on the director curve. Consistently, external forces are modeled by wrenches and a line density of wrench, while internal forces are represented by a wrench field on the director curve. The field and boundary equations of motion are then derived for the system and subsystems in terms of wrenches, in the form of a conservation law as for the three-dimensional continuum. The internal force wrench field determines contact actions between adjacent particles. The connection between this model and the three-dimensional modeling of the original slender body is established through the principle of virtual work, which identifies the internal force wrench in the one-dimensional model with the wrench of the contact forces exerted on the transverse cross-section in the three-dimensional body.
Based upon a similar physical intuition, the two-dimensional modeling of plates and thin slabs through the virtual work method presented in Chapter 8 follows the same path as the one-dimensional modeling process in Chapter 7, with the director curve being substituted by a director sheet. Again, the constituent particles are diluted material points with an attached oriented microstructure which accounts for the role of the transverse material elements in the original plane and slender body. From the geometrical viewpoint, these particles are characterized by their position vector and the orientation of their attached microstructure with respect to the reference frame. Actual and virtual motions are defined by velocity distributor fields on the director sheet. External forces are modeled by surface and line densities of wrenches, while internal forces are represented by a tensorial wrench field defined on the director sheet. With the use of tensor quantities, the field and boundary equations can be written in a compact form easily identified as a conservation law. They can usually be split into in-plane and out-of-plane equations, which can be compared with their three-dimensional and one-dimensional counterparts, respectively. The internal force tensorial wrench field determines contact actions between adjacent particles in the two-dimensional model in the form of a wrench, in the same way as the Cauchy stress tensor determines the stress vector in the case of the three-dimensional classical continuum. The connection between this two-dimensional continuum modeling and the three-dimensional modeling of the original body through the principle of virtual work identifies the internal force tensorial wrench at the field point: for any given orientation, the wrench of contact actions in the two-dimensional continuum is equal to the wrench of the contact actions exerted on the corresponding element in the original body.
This selection of three typical mechanical continua, with five force models being constructed, first underscores the utmost and fundamental importance of the geometrical modeling that lies at the root of the force modeling process. Non-constrained variations of the chosen geometrical parameters define the virtual motions of the system and subsystems on which the virtual work method is implemented. In a non-axiomatic way, the method enhances the reference to physical intuition when it comes to the crucial step of writing down the linear forms dualizing the external and internal force rates of work and also, after the validation step, when it proves necessary to enrich the geometrical modeling in order to improve the physical and practical relevance of the force model.
From this latter point of view, the analogies and differences between the one-dimensional and three-dimensional continua are enlightening. In both cases, enlarging the practical relevance of the force model calls for a more sophisticated expression of the virtual rate of work by internal forces. For the one-dimensional curvilinear continuum, the inadequacy of the wires and supple cables model to account for beams and arches is obviously due to the fact that the underlying microstructure, namely the transverse cross-section, is not taken into account; thence, the geometrical modeling is enriched and virtual motions are defined by virtual velocity distributor fields instead of virtual velocity fields. Regarding the classical three-dimensional continuum, the hydrostatic pressure model only refers to the trace of the virtual velocity gradient, the virtual volume dilatation rate, in the expression of the virtual rate of work by internal forces. This amounts to taking just a mean vision of the virtual evolution of the constituent particle, thus neglecting its microstructure in some sense.
The analogy may then be pushed further. Considering that the wires and supple cables model is the exact one-dimensional counterpart of the classical three-dimensional continuum, we naturally imagine the three-dimensional counterpart of the beams and arches model, as a matter of fact similar to the two-dimensional model for plates and thin slabs, where the constituent diluted particles are characterized by their position vector and the orientation of an attached microstructure. The virtual work method is the most valuable tool for constructing this “micropolar” three-dimensional continuum in a consistent way.
Throughout these examples, it is observed that the equations of motion attached to the force models are not sufficient to determine the internal force field in a system, but for the particular case of a statically determinate one-dimensional element. In the general case, it is necessary to specify the mechanical constitutive equation of the constituent material. Given this physical data, such as the elastic, elastoplastic or viscoelastic constitutive laws, the principle of virtual work is the cornerstone of variational principles and numerical methods which are exposed extensively in numerous textbooks but do not fall within the scope of this book. Nevertheless, some examples related to the theory of yield design are briefly given as straightforward applications of the principle that yield relevant results in various practical circumstances.
Three appendices are devoted to the introduction and simple practice of mathematical concepts – Tensor Calculus, Differential Operators, Distributors and Wrenches – inasmuch as necessary for setting up the notations and making an autonomous reading possible.
Acknowledgments
I wish to express my gratitude to the many colleagues to whom I have been indebted, all along my teaching career, for helpful criticisms and suggestions.
Among them, I extend special thanks to Dr Jean-Michel Delbecq and Professors Michel Amestoy, Habibou Maitournam, Marcel Save and Pierre Suquet.
Finally, as a Senior Fellow, I wish to acknowledge the support of the Institute of Advanced Study at the City University of Hong Kong.
Jean SALENÇONDecember 2017
1
[SAL 01].
1The Emergence of the Principle of Virtual Velocities
1.1. In brief
The historical path to Lagrange’s statement of the principle of virtual velocities has been two-millennium long, a facet of what Benvenuto [BEN 91] calls “The Enigma of Force and the Foundations of Mechanics” and could “be regarded as vague meandering, impotent struggles, foolish attempts at reduction, and justified doubt about the nature of force” (Truesdell1). Many authors, after Lagrange himself [LAG 88a], have tracked the history and avatars of the concepts of virtual velocities and virtual work: a very comprehensive analysis appears in [DUH 05], [DUH 06] and we must also quote, among others, [DUG 50], [TRU 68], [BEN 81], [BEN 91] and [CAP 12]. The purpose of this chapter is just to present some milestones along this historical path, up to Lagrange’s contribution.
1.2. Setting the principle as a cornerstone
In the first edition of the Méchanique Analitique [LAG 88a], Lagrange had some very extolling words about the principle of virtual velocities:
“But this principle is not only very simple and very general in itself; as an invaluable and unique advantage it can also be expressed in a general formula which encompasses all the problems that can be proposed regarding equilibrium. We will expose this formula in all its extent; we will even try to present it in a more general way than done usually up to now, and present new applications”.
But leafing over his Complete Works as published in [LAG 88b], we find that in the subsequent editions, without dimming his enthusiasm for this fundamental principle of statics, he would be somewhat cautious about the possibility of laying it as a first stone (Méchanique Analitique, Section 1, Part 1, section 18):
“Regarding the nature of the principle of virtual velocities, it must be recognized that it is not self-evident enough to be settled as a primitive principle”. [Quant à la nature du principe des vitesses virtuelles, il faut convenir qu’il n’est pas assez évident par lui-même pour pouvoir être érigé en principe primitif.]
The name of Lagrange shines at the top of the list of the professors of Mechanics at the École polytechnique in Paris where he taught from 1794, when the school was founded, until 1799 (his first successor was Fourier). Some 200 years later, Germain, his 27th successor in charge of teaching Mechanics at the École polytechnique from 1973 to 1985, did take up the challenge of setting the “Principe des Puissances Virtuelles”2, as the cornerstone of his synthetic presentation of Mechanics [GER 86]. In the English version of our own textbook for the École polytechnique [SAL 01], where we followed the same track, we retained the wording Principle of Virtual Work for simplicity’s sake with the corresponding method for the modeling of forces being called the Virtual Work Method, thus dropping the reference to “virtual rates”, but we explicitly named the linear forms involved in the statements “virtual rates of work”.
1.3. The “simple machines”
When looking for the very roots of Mechanics, we inescapably encounter the study of the “simple machines” that provide mechanical advantage, or leverage, when applying a single active force to do work against a single load force, such as the weight of a body: “early theoretical thinking about statics and mechanics took as its references particular objects, things like the lever, used since ancient times as necessary tools”3. Aristotle’s (384–322 BC) Quaestionae Mechanicae (Mechanical Problems)4, as quoted by Benvenuto5, defines Mechanics as an art:
“Miraculously some facts occur in physics whose causes are unknown; that is, those artifices that appear to transgress Nature in favour of man…Thus, when it is necessary to do something that goes beyond Nature, the difficulties can be overcome with the assistance of art. Mechanics is the name of the art that helps us over these difficulties; as the poet Antiphon put it, “Art brings the victory that Nature impedes” ”.
Regarding the lever problem, he finds a marvelous explanation in the fact that the weight and “small force” describe their circular trajectories with different velocities:
“Among the problems included in this class are included those concerned with the lever. For it is strange that a great weight can be moved by a small force, and that, too, when a greater weight is involved. For the very same weight, which a man cannot move without a lever, he quickly moves by applying the weight of the lever. Now the original cause of all such phenomena is the circle; and this is natural, for it is in no way strange that something remarkable should result from something more remarkable, and the most remarkable fact is the combination of opposites with each other. The circle is made up of such opposites, for to begin with it is composed both of the moving and of the stationary, which are by nature opposite to each other … ‘Therefore, as has been said before, there is nothing strange in the circle being the first of all marvels’”.
“…Again, no two points on one line drawn as a radius from the centre travel at the same pace, but that which is further from the fixed centre travels more rapidly; it is due to this that many of the remarkable properties in the movement of circles arise”.
Aristotle’s Physicae Auscultationes (Physics) [ARI 09] is usually referred to for the introduction of the concept of (motive) “Power” ( or ) representing the product of the weight of the considered body by its velocity (the ratio of the displacement to the duration of the movement) in order to explain the principle of the rectilinear lever [DEG 08]. The equilibrium of the lever is just stated as the equality (equivalence) of the powers acting at each end, explaining the mechanical advantage by the comparison of the velocities of the active and load forces. The “rule of proportion” (Physics, vol. VII, Chapter V) clearly refers to motion, with the major ambiguity due to his reference to time that would be definitely ruled out by Descartes (section 1.5.1):
“Then, the movement A have moved B a distance G in a time D, then in the same time the same force A will move ½ B twice the distance G, and in ½ D it will move ½ B the whole distance for G: thus the rules of proportion will be observed”.
Archimedes’ (287–212 BC) approach to statics in De Planorum Æquilibriis (On the Equilibrium of Planes) [ARC] is completely different: “While Aristotle relates mechanics to a physical theory, aiming for a universal synthesis, Archimedes thinks of statics as a rational and autonomous science, founded on almost self-evident postulates and built upon rigorous mathematical demonstrations”6.
Figure 1.1.Archimedes’ proof of the law of the lever
A very illustrative example is given by his proof of the law of the lever (or steelyard) that can be sketched as follows (Figure 1.1). The initial accepted demand is that a lever with arms of equal length (p + q) over which the load – weight – is uniformly distributed is in equilibrium (a). Then, through a thought experiment (b), this lever is split into two parts of length 2p and 2q respectively (and the corresponding loads) which, anticipating the terminology we will use in the following chapters, we may call subsystems of the given physical system (the lever). Considering first the subsystem with length 2p , we can state, from the same initial demand, that it is in equilibrium about its midpoint Q where it exerts the load 2 p; in the same way, the subsystem with length 2q is in equilibrium about its midpoint P where it exerts the load 2q . These midpoints are respectively at a distance q and p from the midpoint O of the lever: thus, the equilibrium of the whole lever with equal arms is also the result of the equilibrium about O of the lever QP with unequal arms q and p , and loads 2p and 2q respectively (c).
This proof based upon a statical thought experiment does not refer to motion and does not call upon any general principle either. It has been discussed by many authors and various “improvements” were put forward that are listed and analyzed in [BEN 91] and won’t be discussed in this brief outline whose purpose is to introduce the two fundamental pathways that were to be followed all along the history of Mechanics. Schematically, we could say that Archimedes aimed at providing answers to given practical problems based upon a limited number of preliminary demands, while Aristotle would try to formulate a general principle, in the present case the equality of the powers of the active force and load force, to cope with any possible problem.
Making use of Descartes’ own words7 in his criticism of Galileo’s analyses of the steelyard and lever similar to Archimedes’ proof, Duhem8 commented on these approaches: Archimedes “plainly explains Quod ita sit but not Cur ita sit” (“What” but not “Why”) and, about Aristotle’s analysis (see section 1.5.1),
“This insight is, indeed, the seed from which will come out, through a twenty century development, the powerful ramifications of the Principle of virtual velocities”.
A similar comment had been made by Fourier in his Mémoire sur la Statique (A Memoir on Statics)9:
“One may add that his writings offer the first insights on the Principle of virtual velocities”.
1.4. Leonardo, Stevin, Galileo
It is clear that we are still very far from a general statement of the principle. The concepts must be extracted as essences through a long lasting trial-and-error process that cannot be extensively presented here following the historical timeline and quoting all contributions that are reported in the analyses of Duhem, Dugas, Benvenuto, Capecchi, etc. The story will be made short.
Among the many topics he covered in his manuscripts, which are stored and preserved in the Library of the Institut de France in Paris10, Leonardo da Vinci (1431–1519) detailed the properties of the simple machines (Ms. A, E, F, I and M) and tried to express them through a simple general law that turns out to be quite similar to Aristotle’s statement (Ms. F):
“If a power [puissance] moves a given body along a given length of space during a given time span, it will move half this body during the same time span along twice the given length of space. Or the same power [vertu] will move half this body along the same length of space in half the same time span”.
Simon Stevin (1548–1620) also referred to the lever problem: he discarded Aristotle’s argument about the velocities along the circular trajectories with the simple, hammer-like statement that [STE 05/08]:
“What is immobile does not describe circles, but two weights in equilibrium are immobile; thus two weights in equilibrium do not describe circles”11.
This actually underscores a true conceptual difficulty: why should the equilibrium of a system be studied by referring to motion?
Nevertheless, for the analysis of the inclined plane, Stevin derived the condition for the balance of forces using a diagram with a “wreath” or necklace containing evenly spaced round balls resting on a triangular wedge (Figure 1.2). He concluded that if the weights were not proportional to the lengths of the sides on which they rested, they would not be in equilibrium since the necklace would be in perpetual motion, which he considered obviously impossible. Incidentally, Stevin was so proud of his proof that the corresponding figure appears on the cover of his books De Beghinselen der Weeghconst [STE 86]12 and Hypomnemata Mathematica [STE 05/08] with his motto “Wonder en is gheen wonder” (“Magic is no magic” also translated by “Wonder, not miracle”) as a refutation of Aristotle’s “marvel”. But we may wonder whether this proof (although questionable) was not, as a matter of fact, a kind of kinematical thought experiment.
Figure 1.2.Covers of Stevin’s books
Also, in Hypomnemata Mathematica [STE 05/08], when dealing with pulleys and pulley blocks, Stevin wrote the following remark:
“Ut spatium agentis, ad spatium patientis, sic potentia patientis ad potentiam agentis”
that may be translated as
“As the space of the actor is to the space of the sufferer, so is the power of the sufferer to the power of the actor”13,
expressing that the displacement of the resistance is to the displacement of the power as the power to the resistance. We observe that there is no reference to time in this sentence that sounds like a rule of proportion. According to Benvenuto, Stevin would not give this statement the status of a principle, which he disliked, and considered it “as a criterion, not an explanation of equilibrium”14.
In Galileo’s (1564–1642) works [GAL 99, GAL 34, GAL 38], we encounter several occurrences of an implicit use of a concept close to what would be defined later on as virtual work. A famous example is related to the analysis of the inclined plane in Della Scienza Meccanica15:
“…Thence the weight F moves downwards, drawing the body E on the sloped plane, this body will cover a distance along AC equal to the one described by the weight F in its fall. But this should be observed: it is true that the body E will have covered all the line AC in the time the weight F falls down an equal length; but during this time, the body E will not have moved away from the common centre of weights more than the vertical length BC, while the weight F, falling down according to the vertical, has dropped a length equal to all the line AC. Recall that weights only resist an oblique motion inasmuch as they move away from the centre of the Earth… We can thus say rightly that the travel [viaggio] of the force [forza] F is to the travel [viaggio] of the force [forza] E in the same ratio as the length AC to the length CB”.
Figure 1.3.The inclined plane [GAL 38]
Despite the fact that this proof was only based upon the concomitant displacements or travels [viaggi] of the weight and the body with respect to the “common centre of weights”, Galileo, as a foreword, still referred to time and velocity in the Aristotelian spirit:
“Finally, let us not overlook the following consideration: as a principle, we said that necessarily, in any mechanical instrument, as much the force was increased via this instrument, as much, on the other hand, one would lose time or velocity”.
1.5. Descartes and Bernoulli
1.5.1. René Descartes (1596–1650)
The correspondence of Descartes (1596–1650)16, as published by Adam and Tannery shows, through many examples, that he had a much clearer vision of a virtual velocity principle than his predecessors or contemporaries, whom he would sometimes treat rather roughly as in a letter to Mersenne (November 15, 1638):
“Pour ce qu'a écrit Galilée touchant la balance et le levier, il explique fort bien quod ita sit, mais non pas cur ita sit, comme je fais par mon Principe. Et pour ceux qui disent que je devois considerer la vitesse, comme Galilée, plutot que l'espace, pour rendre raison des Machines, je croy, entre nous, que ce sont des gens qui n'en parlent que par fantaisie, sans entendre rien en cette matiere…”17
“Regarding what Galileo wrote about the steelyard and the lever, he plainly explains what happens but not why it happens, as I do it myself through my Principle. And as for those who pretend that I should consider velocity, as Galileo does, instead of space, I believe, between us, that they are just people who talk without any understanding of the matter at hand”.
He had stated his principle plainly, answering a letter from Constantijn Huygens (Christian’s father) on October 5, 1637 about the fundamental principle of the simple machines in its common form:
