![“DynamicalSystems” — 2004/3/4 — page #4
4 L. Bevilacqua
initial difficulty of using the classical solid mechanics approach, in this case, lies
in the definition of the proper geometry. A strongly folded medium, except when
folded following very strict rules, doesn’t present a regular pattern. When we crush
a piece of paper the simple elements that compose the final complex configuration
are distributed at random and in different sizes. So, a preliminary problem to be
solved is gaining an understanding of both the local and the global geometry.
Let us think, for instance, of a paper or metal sheet densely folded to take the
shape of a ball. Classical structural analysis presents serious difficulties in deter-
mining the final configuration, for this involves a complex combination of buckling,
post-buckling, nonlinearities – both geometric and material – large displacements,
just to mention a few. If we are basically interested only in the geometry, is it
possible to establish a simple “global law” that would correlate some appropriate
variables leading to the characterization of the final shape? The aim of this paper
is to answer this question. A simple law is proposed as a kind of geometric consti-
tutive equation that is different in nature from the classical concept of constitutive
laws in mechanics. Some consequences are drawn from this basic law concerning
mass distribution, work and energy used in the packing process.
We believe that the results are plausible, that is, there are no violations of basic
principles, and there are no contradictions concerning the expected behavior of a
real material. But, to be recognized as scientifically valid, the theory need to be
tested against experimental results. Despite the fact that rigorous experiments are
missing, the development of a coherent theory is important, both from the viewpoint
of obtaining comments and suggestions on it, and from the viewpoint of developing
experimental methodologies.
In the next sections we will examine densely folded wires and densely folded
plates and shells. To the best of our knowledge, the current technical literature
does not include references on this subject. We have exposed the basic ideas of
this theory in [1] and [2]. This paper, however, is self-contained, it is a kind of
closure where the concepts are presented more clearly and precisely. Except for the
dynamic behavior, which is not included here, the other references are not necessary
to understand this paper. The dynamics of folded media still need further analysis.
The ideas advanced in [2] are at an exploratory stage and need several corrections.
2 Densely Folded Wires
Let us assume that a thin wire is pushed into a box with two predominant dimen-
sions, length (L) and width (h), while the depth is approximately equal to the wire
diameter, much smaller than L and h. It is difficult to make a prediction about the
geometry of the wire inside the box. The amount of wire packed in a box depends
on the wire diameter, the energy expended in the process and the mechanical prop-
erties of the wire material, particularly its ductility. We will explain later what is
understood as ductility in the present context.
How all those variables correlate with each other will be discussed in the sequel.
It is possible, however, to anticipate some dependence relationships by appealing
to common sense. It is expected, for instance, that by decreasing the wire diame-
ter, while keeping all other variables constant, the length of the wire packed inside
the box will increase. Also, by decreasing the ductility, that is, the capacity to](https://image.slidesharecdn.com/45534-250418204924-0a5f514a/85/Dynamical-systems-and-control-1st-Edition-Firdaus-E-Udwadia-21-320.jpg)
![“DynamicalSystems” — 2004/3/4 — page #5
A Geometric Approach to the Mechanics of Densely Folded Media 5
Figure 1 An ideal packing of a wire inside the box [L0 × H].
accumulate plastic deformation, while keeping all the other variables constant, it is
intuitively acceptable that the wire length in the box will increase. Other interpre-
tations are not so straightforward and will be discussed in the proper section.
Consider a thin box [L0 × h] and assume that a wire with diameter φ is pushed
into the box. In order to simplify the problem, it is assumed in the sequel that
plastic hinges will appear in the process, such that, after reaching the final stable
configuration, the wire geometry can be reduced to a sequence of straight segments
linked together, through plastic hinges, in the shape of a broken random line. It
will be assumed throughout this paper that we are dealing with a perfectly plastic
material.
Let us start with an ideal case, consisting of the configuration sequence following
the pattern shown in Figure 1.
It can be easily shown that for the n-th term in the sequence:
ln = L0
µ
1
(2n − 1)2
+ β2
¶1/2
(1a)
and
Ln =
M
X
m=1
l(m)
n = L0
¡
1 + (2n − 1)2
β2
¢1/2
, (2a)
where β = h/L0.
Clearly Ln is the total length of the wire inside the box corresponding to the
n-th term. For very thin wires relative to the box dimensions, i.e., φ ¿ L0 and
φ ¿ h, the number of folds is large, n À 1. The segment ln and the total length
Ln can then be approximated by:
lnL
∼
= L0β = h , (1b)
LnL
∼
= 2nLβL0 = 2nLh . (2b)](https://image.slidesharecdn.com/45534-250418204924-0a5f514a/85/Dynamical-systems-and-control-1st-Edition-Firdaus-E-Udwadia-22-320.jpg)
![“DynamicalSystems” — 2004/3/4 — page #6
6 L. Bevilacqua
Figure 2 Appropriate geometry of a wire densely packed in a box.
For this limit case, where n is very large we set n = nL. It follows from (1b) and
(2b) that, in this limit case, the wire occupies the total area of the box. Assuming
that the material is incompressible, the mass conservation principle requires:
Lnφ ≤ L0h , (3)
where the inequality sign holds when the confined wire fills up the box. Therefore
for n → nL we may write:
Ln
L0
∼
=
h
φ
∼
=
µ
φ
h
¶−1
. (4)
Combining (4) and (2b) the limit value nL can be estimated by:
nL
∼
=
1
2
L0
φ
. (5)
This is a limit value and clearly corresponds to the case where the wire fills up the
box. In general we might expect the wire geometry inside the box to be similar to
the line depicted in Figure 2, only partially covering the box in a random way.
It is not likely that the wire will be so densely packed as to fill up the box. For
the general case the expression (4) can then be written in the following form:
Ln
L0
∼
= e0
µ
h
φ
¶p
∼
= e0
µ
φ
h
¶−p
, (6)
where the exponent p is less than or equal to 1 (p ≤ 1). If p = 1 the line representing
the wire will cover the entire region [L0 × h], if it is less than 1 it will only partially
cover this same region. The constant e0 may be adjusted to fit experimental results.
Expression (6) can be put into a more convenient form for the current notation of
the fractionary geometry:
Γ = e0ρ1−D
(7a)
or
log Γ = log e0 + (1 − D) log ρ , (7b)
where Γ = L/L0, ρ = φ/h, and D is the fractal dimension. We have dropped the
subscript n in Ln for the sake of simplicity.](https://image.slidesharecdn.com/45534-250418204924-0a5f514a/85/Dynamical-systems-and-control-1st-Edition-Firdaus-E-Udwadia-23-320.jpg)
![“DynamicalSystems” — 2004/3/4 — page #11
A Geometric Approach to the Mechanics of Densely Folded Media 11
inside the box divided by the box length L0. This definition implies the homog-
enization of the specific mass, making it uniformly distributed along the box in
what, in general, is a good approximation. After some simple calculations it is then
possible to write:
µ =
µ0
M
P
m=1
l
(m)
n
L
n → nL , (15a)
Ω = e0ρ1−D
,
or
log Ω = log e0 + (1 − D) log ρ , (15b)
where Ω = µ/µ0.
Therefore the same law governing the packing capacity Γ also applies to the
packing density ρ. The representation in Figure 3 is equally valid for this case.
Analytically the parameter e0 in (15a,b) is the same as in (7a,b). Only experi-
mental evidence can confirm this result.
The limit cases have the same interpretation as for the packing capacity. Putting
e0 = 1, the limit case D = 1 is satisfied, for the virtual specific mass will coincide
with the wire specific mass. But there is no strong reason to abandon from the very
beginning the hypothesis of having e0 = e0(D). This might well be the case, and
at the present stage only experimental data can provide answers on this subject.
Now, consider the region MNPQ shown in Figure 2. It is plausible to assume
that the plastic energy stored in the wire inside the box is uniformly distributed
along the length L0. Therefore the energy stored in MNPQ Wj, is Lj/L0 times
the total energy necessary to pack the wire inside the box [L0 × h]. But also
the reference energy WR is proportional to L0 by definition, therefore the reference
energy corresponding to the box [Lj ×h] WRj is also Lj/L0 times WR corresponding
to the box [L0 ×h]. Since the plastic work density τ is the ratio Wj/WRj it is easily
seen that τ is invariant for any sub-region [Lj × h] of [L0 × h]. Combining (7a)
and (11) we get:
Γj = e0E1−D
j = e0E1−D
= Γ .
That is the packing capacity is the same for the box [L0 × h] and for any of its
sub-regions [Lj × h]. This result is coherent with the hypothesis of the uniform
distribution of the specific mass introduced before. It is also a confirmation of the
intrinsic self-similarity property required by the structure of the fractal geometry.
The above discussions lead to some conclusions that can be summarized as
follows:
Proposition 1 1. The geometry of densely folded wires packed in a two-dimen-
sional box – [L×h], L > h – has a random structure characterized by a fractal
dimension 1 < D < 2, provided that:
i. The slenderness ratio defined by ρ = φ/h, where φ is the wire diameter,
is sufficiently small. That is, ρ ¿ 1.
ii. The material is perfectly plastic.
iii. The final configuration is stable.](https://image.slidesharecdn.com/45534-250418204924-0a5f514a/85/Dynamical-systems-and-control-1st-Edition-Firdaus-E-Udwadia-28-320.jpg)
Dynamical systems and control 1st Edition Firdaus E. Udwadia
- 1. Dynamical systems and control 1st Edition Firdaus E. Udwadia pdf download https://ebookname.com/product/dynamical-systems-and-control-1st- edition-firdaus-e-udwadia/ Get Instant Ebook Downloads – Browse at https://ebookname.com
- 2. Instant digital products (PDF, ePub, MOBI) available Download now and explore formats that suit you... Dynamical systems Sternberg S. https://ebookname.com/product/dynamical-systems-sternberg-s/ Dynamical Systems and Group Actions Lewis Bowen https://ebookname.com/product/dynamical-systems-and-group- actions-lewis-bowen/ Adaptation in Dynamical Systems 1st Edition Ivan Tyukin https://ebookname.com/product/adaptation-in-dynamical- systems-1st-edition-ivan-tyukin/ CTOs at Work 1st Edition Scott Donaldson https://ebookname.com/product/ctos-at-work-1st-edition-scott- donaldson/
- 3. Handbook of Structural Equation Modeling 1st Edition Rick H. Hoyle https://ebookname.com/product/handbook-of-structural-equation- modeling-1st-edition-rick-h-hoyle/ Phenomenology Organizational Politics and IT Design The Social Study of Information Systems 1st Edition Gianluigi Viscusi https://ebookname.com/product/phenomenology-organizational- politics-and-it-design-the-social-study-of-information- systems-1st-edition-gianluigi-viscusi/ Dendrimers Towards Catalytic Material and Biomedical Uses 1st Edition Anne-Marie Caminade https://ebookname.com/product/dendrimers-towards-catalytic- material-and-biomedical-uses-1st-edition-anne-marie-caminade/ Blogging For Dummies 3rd Edition Susannah Gardner https://ebookname.com/product/blogging-for-dummies-3rd-edition- susannah-gardner/ Theories of counseling and psychotherapy systems strategies and skills 3rd ed., International ed. / Edition Seligman https://ebookname.com/product/theories-of-counseling-and- psychotherapy-systems-strategies-and-skills-3rd-ed-international- ed-edition-seligman/
- 4. Writing The Amish The Worlds Of John A Hostetler Pennsylvania German History Culture 1St Edition Edition David L. Weaver-Zercher https://ebookname.com/product/writing-the-amish-the-worlds-of- john-a-hostetler-pennsylvania-german-history-culture-1st-edition- edition-david-l-weaver-zercher/
- 7. Volume 1 Theory of Integro-Differential Equations V. Lakshmikantham and M. Rama Mohana Rao Volume 2 Stability Analysis: Nonlinear Mechanics Equations A.A. Martynyuk Volume 3 Stability of Motion of Nonautonomous Systems (Method of Limiting Equations) J. Kato, A.A. Martynyuk, and A.A. Shestakov Volume 4 Control Theory and its Applications E.O. Roxin Volume 5 Advances in Nonlinear Dynamics Edited by S. Sivasundaram and A.A. Martynyuk Volume 6 Solving Differential Problems by Multistep Initial and Boundary Value Methods L. Brugnano and D. Trigiante Volume 7 Dynamics of Machines with Variable Mass L. Cveticanin Volumes 8 Optimization of Linear Control Systems: Analytical Methods and Computational Algorithms F.A. Aliev and VB. Larin Volume 9 Dynamics and Control Edited by G. Leitmann, F.E. Udwadia and A.V Kryazhimskii Volume 10 Volterra Equations and Applications Edited by C. Corduneanu and I.W. Sandberg Volume 11 Nonlinear Problems in Aviation and Aerospace Edited by S. Sivasundaram Volume 12 Stabilization of Programmed Motion E. Ya. Smirnov Volume 13 Advances in Stability Theory at the End of the 20th Century A.A. Martynyuk Volume 14 Dichotomies and Stability in Nonautonomous Linear Systems Yu A. Mitropolskii, A.M. Samoilenko, and V. L Kulik Volume 15 Almost Periodic Solutions of Differential Equations in Banach Spaces Y. Hino, T. Naito, Nguyen Van Minh, and Jong Son Shin Volume 16 Functional Equations with Causal Operators C. Corduneanu Volume 17 Optimal Control of Growth of Wealth of Nations E.N. Chukwu Volume 18 Stability and Stabilization of Nonlinear Systems with Random Structure I. Ya Kats and A.A Martynyuk Volume 19 Lyapunov Method & Certain Differential Games V.I. Zhukovskii Volume 20 Stability of Differential Equations with Aftereffect N.V. Azbelev and P.M. Simonov Volume 21 Asymptotic Methods in Resonance Analytical Dynamics E.A. Grebenikov, Yu. A. Mitropolsky and Yu. A. Ryabov Volume 22 Dynamical Systems and Control Edited by Firdaus E. Udwadia, H.I. Weber, and George Leitmann Stability and Control: Theory, Methods and Applications A series of books and monographs on the theory of stability and control Edited by A.A. Martynyuk Institute of Mechanics, Kiev, Ukraine and V. Lakshmikantham Florida Institute of Technology, USA
- 8. Stability and Control: Theory, Methods and Applications Volume 22 EDITED BY Firdaus E. Udwadia University of Southern California USA H. I. Weber Pontifical Catholic University of Rio de Janeiro Brazil George Leitmann University of California, Berkeley USA CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London NewYork Washington, D.C. Dynamical Systems and Control
- 9. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specific clients, may be granted by CRC Press LLC, provided that $1.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 0-415-30997-2/04/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 2004 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-415-30997-2 Library of Congress Card Number 2004043591 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper Library of Congress Cataloging-in-Publication Data Dynamical systems and control / edited by F.E. Udwadia, G. Leitmann, and H.I. Weber p. cm. (Stability and control ; v.22) Includes bibliographical references and index. ISBN 0-415-30997-2 (alk. paper) 1. Dynamics. 2. Differentiable dynamical systems. 3. Control theory. I. Udwadia, F.E. II. Leitmann, George. III. Weber, H. (Hans) IV. International Workshop on Dynamics and Control (11th : 2000 : Rio de Janeiro, Brazil) V. Title. VI. Series. QA845.D93 2004 003‘.85—dc22 2004043591 TF1677-discl.fm Page 1 Monday, April 12, 2004 2:34 PM
- 10. “DynamicalSystems” — 2004/3/7 — page #v Contents List of contributors vii Preface xi Part I A geometric approach to the mechanics of densely folded media Luiz Bevilacqua 3 On a general principle of mechanics and its application to general non-ideal nonholonomic constraints Firdaus E. Udwadia 21 Mathematical analysis of vibrations of nonhomogeneous filament with one end load Marianna A. Shubov 33 Expanded point mapping analysis of periodic systems Henryk Flashner and Michael Golat 53 A preliminary analysis of the phase portrait’s structure of a nonlinear pendulum-mechanical system using the perturbed Hamiltonian formulation Débora Belato, Hans Ingo Weber and José Manoel Balthazar 77 A review of rigid-body collision models in the plane Edson Cataldo and Rubens Sampaio 91 Part II Optimal round-trip Earth–Mars trajectories for robotic flight and manned flight A. Miele, T. Wang and S. Mancuso 109 Aircraft take-off in windshear: a viability approach N. Seube, R. Moitie and G. Leitmann 127 Stability of torsional and vertical motion of suspension bridges subject to stochastic wind forces N.U. Ahmed 145
- 11. “DynamicalSystems” — 2004/3/7 — page #vi vi Contents Time delayed control of structural systems Firdaus E. Udwadia, Hubertus F. von Bremen, Ravi Kumar and Mohamed Hosseini 163 Robust real- and discrete-time control of a steer-by-wire system in cars Eduard Reithmeier 207 Optimal placement of piezoelectric sensor/actuators for smart structures vibration control Vicente Lopes, Jr., Valder Steffen, Jr. and Daniel J. Inman 221 A review of new vibration issues due to non-ideal energy sources J.M. Balthazar, R.M.L.R.F Brasil, H.I. Weber, A. Fenili, D. Belato, J.L.P. Felix and F.J. Garzelli 237 Identification of flexural stiffness parameters of beams José João de Espı́ndola and João Morais da Silva Neto 259 Active noise control caused by airflow through a rectangular duct Seyyed Said Dana, Naor Moraes Melo and Simplicio Arnaud da Silva 271 Dynamical features of an autonomous two-body floating system Helio Mitio Morishita and Jessé Rebello de Souza Junior 283 Dynamics and control of a flexible rotating arm through the movement of a sliding mass Agenor de Toledo Fleury and Frederico Ricardo Ferreira de Oliveira 299 Measuring chaos in gravitational waves Humberto Piccoli and Fernando Kokubun 319 Part III Estimation of the attractor for an uncertain epidemic model E. Crück, N. Seube and G. Leitmann 337 Liar paradox viewed by the fuzzy logic theory Ye-Hwa Chen 351 Pareto-improving cheating in an economic policy game Christophe Deissenberg and Francisco Alvarez Gonzalez 363 Dynamic investment behavior taking into account ageing of the capital goods Gustav Feichtinger, Richard F. Hartl, Peter Kort and Vladimir Veliov 379 A mathematical approach towards the issue of synchronization in neocortical neural networks R. Stoop and D. Blank 393 Optimal control of human posture using algorithms based on consistent approximations theory Luciano Luporini Menegaldo, Agenor de Toledo Fleury and Hans Ingo Weber 407 Subject Index 431
- 12. “DynamicalSystems” — 2004/3/7 — page #vii Contributors N.U. Ahmed, School of Information Technology and Engineering, Department of Mathematics, University of Ottawa, Ottawa, Ontario José Manoel Balthazar, Instituto de Geociências e Ciências Exatas – UNESP – Rio Claro, Caixa Postal 178, CEP 13500-230, Rio Claro, SP, Brasil Débora Belato, DPM – Faculdade de Engenharia Mecânica – UNICAMP, Caixa Postal 6122, CEP 13083-970, Campinas, SP, Brasil Luiz Bevilacqua, Laboratório Nacional de Computação Cientı́fica – LNCC, Av. Getúlio Vargas 333, Rio de Janeiro, RJ 25651-070, Brasil D. Blank, Institut für Neuroinformatik, ETHZ/UNIZH, Winterthurerstraße 190, CH-8057 Zürich R.M.L.R.F. Brasil, Dept. of Structural and Foundations Engineering, Polytech- nic School, University of São Paulo, P.O. Box 61548, 05424-930, SP, Brazil Edson Cataldo, Universidade Federal Fluminense (UFF), Departamento de Mate- mática Aplicada, PGMEC-Programa de Pós-Graduação em Engenharia Mecânica, Rua Mário Santos Braga, S/No-24020, Centro, Niterói, RJ, Brasil Ye-Hwa Chen, The George W. Woodruff School of Mechanical Engineering, Geor- gia Institute of Technology, Atlanta, Georgia 30332, USA E. Crück, Laboratoire de Recherches Balistiques et Aérodynamiques, BP 914, 27207 Vernon Cedex, France Seyyed Said Dana, Graduate Studies in Mechanical Engineering, Mechanical Engineering Department, Federal University of Paraiba, Campus I, 58059-900 Joao Pessoa, Paraiba, Brazil Christophe Deissenberg, CEFI, UMR CNRS 6126, Université de la Méditerranée (Aix-Marseille II), Château La Farge, Route des Milles, 13290 Les Milles, France José João de Espı́ndola, Department of Mechanical Engineering, Federal Uni- versity of Santa Catarina, Brazil Gustav Feichtinger, Institute for Econometrics, OR and Systems Theory, Uni- versity of Technology, Argentinierstrasse 8, A-1040 Vienna, Austria J.L.P. Felix, School of Mechanical Engineering, UNICAMP, P.O. Box 6122, 13800- 970, Campinas, SP, Brazil A. Fenili, School of Mechanical Engineering, UNICAMP, P.O. Box 6122, 13800- 970, Campinas, SP, Brazil
- 13. “DynamicalSystems” — 2004/3/7 — page #viii viii Contributors Henryk Flashner, Department of Aerospace and Mechanical Engineering, Uni- versity of Southern California, Los Angeles, CA 90089-1453 Agenor de Toledo Fleury, Control Systems Group/Mechanical & Electrical En- gineering Division, IPT/ São Paulo State Institute for Technological Research, P.O. Box 0141, 01064-970, São Paulo, SP, Brazil F.J. Garzelli, Dept. of Structural and Foundations Engineering, Polytechnic School, University of São Paulo, P.O. Box 61548, 05424-930, SP, Brazil Michael Golat, Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453 Francisco Alvarez Gonzalez, Dpto. Economia Cuantitativa, Universidad Com- plutense, Madrid, Spain Richard F. Hartl, Institute of Management, University of Vienna, Vienna, Austria Daniel J. Inman, Center for Intelligent Material Systems and Structures, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0261, USA Fernando Kokubun, Department of Physics, Federal University of Rio Grande, Rio Grande, RS, Brazil Peter Kort, Department of Econometrics and Operations Research and CentER, Tilburg University, Tilburg, The Netherlands G. Leitmann, College of Engineering, University of California, Berkeley CA 94720, USA Vicente Lopes, Jr., Department of Mechanical Engineering – UNESP-Ilha Solte- ira, 15385-000 Ilha Solteira, SP, Brazil S. Mancuso, Rice University, Houston, Texas, USA Naor Moraes Melo, Graduate Studies in Mechanical Engineering, Mechanical Engineering Department, Federal University of Paraiba, Campus I, 58059-900 Joao Pessoa, Paraiba, Brazil Luciano Luporini Menegaldo, São Paulo State Institute for Technological Re- search, Control System Group / Mechanical and Electrical Engineering Division, P.O. Box 0141, CEP 01604-970, São Paulo-SP, Brazil A. Miele, Rice University, Houston, Texas, USA Helio Mitio Morishita, University of São Paulo, Department of Naval Architec- ture and Ocean Engineering, Av. Prof. Mello Moraes, 2231, Cidade Universitária 05508-900, São Paulo, SP, Brazil Frederico Ricardo Ferreira de Oliveira, Mechanical Engineering Department/ Escola Politécnica, USP – University of São Paulo, P.O. Box 61548, 05508-900, São Paulo, SP, Brazil Humberto Piccoli, Department of Materials Science, Federal University of Rio Grande, Rio Grande, RS, Brazil Eduard Reithmeier, Institut für Meß- und Regelungstechnik, Universität Han- nover, 30167 Hannover, Germany Rubens Sampaio, Pontifı́cia Universidade Católica do Rio de Janeiro (PUC-Rio), Departamento de Engenharia Mecânica, Rua Marquês de São Vicente, 225, 22453- 900, Gávea, Rio de Janeiro, Brasil
- 14. “DynamicalSystems” — 2004/3/7 — page #ix Contributors ix N. Seube, Ecole Nationale Supérieure des Ingénieurs des Etudes et Techniques d’Armement, 29806 BREST Cedex, France Marianna A. Shubov, Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409, USA João Morais da Silva Neto, Department of Mechanical Engineering, Federal University of Santa Catarina, Brazil Simplicio Arnaud da Silva, Graduate Studies in Mechanical Engineering, Me- chanical Engineering Department, Federal University of Paraiba, Campus I, 58059- 900 Joao Pessoa, Paraiba, Brazil Jessé Rebello de Souza Junior, University of São Paulo, Department of Naval Architecture and Ocean Engineering, Av. Prof. Mello Moraes, 2231, Cidade Uni- versitária 05508-900, São Paulo, SP, Brazil Valder Steffen, Jr., School of Mechanical Engineering Federal University of Uber- lândia, 38400-902 Uberlândia, MG, Brazil R. Stoop, Institut für Neuroinformatik, ETHZ/UNIZH, Winterthurerstraße 190, CH-8057 Zürich F.E. Udwadia, Department of Aerospace and Mechanical Engineering, Civil En- gineering, Mathematics, and Operations and Information Management, 430K Olin Hall, University of Southern California, Los Angeles, CA 90089-1453 Vladimir Veliov, Institute for Econometrics, OR and Systems Theory, University of Technology, Argentinierstrasse 8, A-1040 Vienna, Austria T. Wang, Rice University, Houston, Texas, USA Hans Ingo Weber, DEM - Pontifı́cia Universidade Católica – PUC – RJ, CEP 22453-900, Rio de Janeiro, RJ, Brasil
- 15. “DynamicalSystems” — 2004/3/7 — page #x
- 16. “DynamicalSystems” — 2004/3/7 — page #xi Preface This book contains some of the papers that were presented at the 11th International Workshop on Dynamics and Control in Rio de Janeiro, October 9–11, 2000. The workshop brought together scientists and engineers in various diverse fields of dy- namics and control and offered a venue for the understanding of this core discipline to numerous areas of engineering and science, as well as economics and biology. It offered researchers the opportunity to gain advantage of specialized techniques and ideas that are well developed in areas different from their own fields of expertise. This cross-pollination among seemingly disparate fields was a major outcome of this workshop. The remarkable reach of the discipline of dynamics and control is clearly substan- tiated by the range and diversity of papers in this volume. And yet, all the papers share a strong central core and shed understanding on the multiplicity of physical, biological and economic phenomena through lines of reasoning that originate and grow from this discipline. I have separated the papers, for convenience, into three main groups, and the book is divided into three parts. The first group deals with fundamental advances in dynamics, dynamical systems, and control. These papers represent new ideas that could be applied to several areas of interest. The second deals with new and innovative techniques and their applications to a variety of interesting problems that range across a broad horizon: from the control of cars and robots, to the dynamics of ships and suspension bridges, to the determination of optimal spacecraft trajectories to Mars. The last group of papers relates to social, economic, and biological issues. These papers show the wealth of understanding that can be obtained through a dynamics and control approach when dealing with drug consumption, economic games, epidemics, neo-cortical synchronization, and human posture control. This workshop was funded in part by the US National Science Foundation and CPNq. The organizers are grateful for the support of these agencies. Firdaus E. Udwadia
- 17. “DynamicalSystems” — 2004/3/7 — page #xii
- 18. “DynamicalSystems” — 2004/3/4 — page #1 PART I
- 19. “DynamicalSystems” — 2004/3/4 — page #2
- 20. “DynamicalSystems” — 2004/3/4 — page #3 A Geometric Approach to the Mechanics of Densely Folded Media Luiz Bevilacqua Laboratório Nacional de Computação Cientı́fica – LNCC Av. Getúlio Vargas 333, Rio de Janeiro, RJ 25651-070, Brasil Tel: 024-233.6024, Fax: 024-233.6167, E-mail: bevi@lncc.br To date, the analysis of densely folded media has received little attention. The stress and strain analysis of these types of structures involves considerable dif- ficulties because of strong nonlinear effects. This paper presents a theory that could be classified as a geometric theory of folded media, in the sense that it ultimately leads to a kind of geometric constitutive law. In other words, a law that establishes the relationship between the geometry and other variables such as the stored energy, the apparent density and the mechanical properties of the material. More specifically, the theory presented here leads to funda- mental governing equations for the geometry of densely folded media, namely, wires, plates and shells, as functions of the respective slenderness ratios. With the help of these fundamental equations other relationships involving the ap- parent density and the energy are obtained. The structure of folded media according to the theory has a fractal representation and the fractal dimension is a function of the material ductility. Although at present we have no experi- ments to test the conjectures that arise from our analytical developments, the theory developed here is internally consistent and therefore provides a good basis for designing meaningful experiments. 1 Introduction Crush a sheet of paper till it becomes a small ball. This is an example of what we will call a folded medium. That is, we have in mind strongly folded media. The mechanical behavior of these kinds of structures could be analyzed as a very dense set of interconnected structural pieces in such a way as to form a continuum. The
- 21. “DynamicalSystems” — 2004/3/4 — page #4 4 L. Bevilacqua initial difficulty of using the classical solid mechanics approach, in this case, lies in the definition of the proper geometry. A strongly folded medium, except when folded following very strict rules, doesn’t present a regular pattern. When we crush a piece of paper the simple elements that compose the final complex configuration are distributed at random and in different sizes. So, a preliminary problem to be solved is gaining an understanding of both the local and the global geometry. Let us think, for instance, of a paper or metal sheet densely folded to take the shape of a ball. Classical structural analysis presents serious difficulties in deter- mining the final configuration, for this involves a complex combination of buckling, post-buckling, nonlinearities – both geometric and material – large displacements, just to mention a few. If we are basically interested only in the geometry, is it possible to establish a simple “global law” that would correlate some appropriate variables leading to the characterization of the final shape? The aim of this paper is to answer this question. A simple law is proposed as a kind of geometric consti- tutive equation that is different in nature from the classical concept of constitutive laws in mechanics. Some consequences are drawn from this basic law concerning mass distribution, work and energy used in the packing process. We believe that the results are plausible, that is, there are no violations of basic principles, and there are no contradictions concerning the expected behavior of a real material. But, to be recognized as scientifically valid, the theory need to be tested against experimental results. Despite the fact that rigorous experiments are missing, the development of a coherent theory is important, both from the viewpoint of obtaining comments and suggestions on it, and from the viewpoint of developing experimental methodologies. In the next sections we will examine densely folded wires and densely folded plates and shells. To the best of our knowledge, the current technical literature does not include references on this subject. We have exposed the basic ideas of this theory in [1] and [2]. This paper, however, is self-contained, it is a kind of closure where the concepts are presented more clearly and precisely. Except for the dynamic behavior, which is not included here, the other references are not necessary to understand this paper. The dynamics of folded media still need further analysis. The ideas advanced in [2] are at an exploratory stage and need several corrections. 2 Densely Folded Wires Let us assume that a thin wire is pushed into a box with two predominant dimen- sions, length (L) and width (h), while the depth is approximately equal to the wire diameter, much smaller than L and h. It is difficult to make a prediction about the geometry of the wire inside the box. The amount of wire packed in a box depends on the wire diameter, the energy expended in the process and the mechanical prop- erties of the wire material, particularly its ductility. We will explain later what is understood as ductility in the present context. How all those variables correlate with each other will be discussed in the sequel. It is possible, however, to anticipate some dependence relationships by appealing to common sense. It is expected, for instance, that by decreasing the wire diame- ter, while keeping all other variables constant, the length of the wire packed inside the box will increase. Also, by decreasing the ductility, that is, the capacity to
- 22. “DynamicalSystems” — 2004/3/4 — page #5 A Geometric Approach to the Mechanics of Densely Folded Media 5 Figure 1 An ideal packing of a wire inside the box [L0 × H]. accumulate plastic deformation, while keeping all the other variables constant, it is intuitively acceptable that the wire length in the box will increase. Other interpre- tations are not so straightforward and will be discussed in the proper section. Consider a thin box [L0 × h] and assume that a wire with diameter φ is pushed into the box. In order to simplify the problem, it is assumed in the sequel that plastic hinges will appear in the process, such that, after reaching the final stable configuration, the wire geometry can be reduced to a sequence of straight segments linked together, through plastic hinges, in the shape of a broken random line. It will be assumed throughout this paper that we are dealing with a perfectly plastic material. Let us start with an ideal case, consisting of the configuration sequence following the pattern shown in Figure 1. It can be easily shown that for the n-th term in the sequence: ln = L0 µ 1 (2n − 1)2 + β2 ¶1/2 (1a) and Ln = M X m=1 l(m) n = L0 ¡ 1 + (2n − 1)2 β2 ¢1/2 , (2a) where β = h/L0. Clearly Ln is the total length of the wire inside the box corresponding to the n-th term. For very thin wires relative to the box dimensions, i.e., φ ¿ L0 and φ ¿ h, the number of folds is large, n À 1. The segment ln and the total length Ln can then be approximated by: lnL ∼ = L0β = h , (1b) LnL ∼ = 2nLβL0 = 2nLh . (2b)
- 23. “DynamicalSystems” — 2004/3/4 — page #6 6 L. Bevilacqua Figure 2 Appropriate geometry of a wire densely packed in a box. For this limit case, where n is very large we set n = nL. It follows from (1b) and (2b) that, in this limit case, the wire occupies the total area of the box. Assuming that the material is incompressible, the mass conservation principle requires: Lnφ ≤ L0h , (3) where the inequality sign holds when the confined wire fills up the box. Therefore for n → nL we may write: Ln L0 ∼ = h φ ∼ = µ φ h ¶−1 . (4) Combining (4) and (2b) the limit value nL can be estimated by: nL ∼ = 1 2 L0 φ . (5) This is a limit value and clearly corresponds to the case where the wire fills up the box. In general we might expect the wire geometry inside the box to be similar to the line depicted in Figure 2, only partially covering the box in a random way. It is not likely that the wire will be so densely packed as to fill up the box. For the general case the expression (4) can then be written in the following form: Ln L0 ∼ = e0 µ h φ ¶p ∼ = e0 µ φ h ¶−p , (6) where the exponent p is less than or equal to 1 (p ≤ 1). If p = 1 the line representing the wire will cover the entire region [L0 × h], if it is less than 1 it will only partially cover this same region. The constant e0 may be adjusted to fit experimental results. Expression (6) can be put into a more convenient form for the current notation of the fractionary geometry: Γ = e0ρ1−D (7a) or log Γ = log e0 + (1 − D) log ρ , (7b) where Γ = L/L0, ρ = φ/h, and D is the fractal dimension. We have dropped the subscript n in Ln for the sake of simplicity.
- 24. “DynamicalSystems” — 2004/3/4 — page #7 A Geometric Approach to the Mechanics of Densely Folded Media 7 If the fractal dimension of the line representing the wire equals two (D = 2), that is, the line fills up the plane, expression (7a) reduces to (4), as it should be. For the other limit, D = 1, the one-dimensional Euclidean geometry is preserved, Γ = e0. In particular, for e0 = 1 we get Ln = L, that is, there is no folding at all. The extreme cases have no practical interest, but they provide a good assessment of the theory, showing that there is no contradiction, and the conjecture is plausible. The lower bound D = 1, corresponding to folding-free configurations, arises from two distinct origins. The first appears as a consequence of geometric constraints. Indeed, if h = φ, the wire fits the box perfectly. There is no room for bending, the stress distribution on the wire cross-section is uniform. We are in the presence of a pure axial force and the wire collapses under simple compression. The material properties do not play any particular role in this limit case. The second possibility has to do with the material properties and is independent of the geometry. Indeed, if the material is perfectly stiff, that is, does not admit any plastic strain, the collapse occurs without any permanent deformation. In other words, there is no stable folding configuration, which is contrary to one of the fundamental requirements of this theory. In real cases, however, we have an intermediate situation. Taking into account the discussion above, we may use the following criteria to establish the range of validity of the theory: 1. The ratio ρ = φ/h must remain inferior to 0.1: ρ < 0.1. 2. The contribution to the total dissipated plastic work (Wt) due to pure axial strain state (Wa) should be much smaller than the contribution due to bending (Wb): Wa ≤ 0.1Wt. The above conditions may be very strict, but only properly conducted experiments can give a conclusive answer. Figure 3 depicts the expected variation of log Γ against log ρ for different values of D. We have assumed that e0 is constant for all values of D, which could not be strictly true. More generally, put e0 = e0(D), in which case the lines corresponding to D1, D2 and D3 would not converge to the same point on the vertical axis. From Figure 3 it is clear that the packing capacity for a given value of ρ depends on the fractal dimension. Increasing values of D correspond to increasing packing capacities Γ. This means that D measures the propensity to incorporate plastic deformation. An experiment leading to points on the line with slope (1 − D1) in Figure 3 indicates geometric and material conditions much more favorable to incorporating permanent plastic deformation than an experiment that follows the line with slope (1 − D3). In order to define more precisely this behavior we will introduce the notion of apparent ductility. This notion will be better understood along with the determi- nation of the dissipated plastic work. As mentioned before, only perfect plastic materials are considered here and the final configuration is stable. This means that if the box is removed after the packing process, the final geometry will be preserved. The energy considered here is therefore the net energy necessary to introduce permanent plastic deformation. Let us start again with the ideal case. According to the fundamental assumptions plastic hinges will form in the vertices of the line representing the folded wire. The
- 25. “DynamicalSystems” — 2004/3/4 — page #8 8 L. Bevilacqua Figure 3 Expected variations of the packing capacity with the wire rigidity. Increasing fractal dimensions D1 > D2 > D3 correspond to decreasing rigidity. net work necessary to produce a rotation equal to θn in a typical plastic hinge as shown in Figure 1d is approximately equal to: τn = k̂σY φ3 θn , where k̂ is a constant, and σY is the yield stress of the perfect plastic material. Now using the notation shown in Figure 2, the rotation can be written as: θn = π − δn . But δn is of the order of L0 nh and for very large n, δn is very small. Therefore we may write: θn ∼ = g(U)π , where g(U) is a correction factor to take into account the material hardening, that is, the maximum rotation capacity of a typical hinge. The function g(U) expresses the material capacity to accumulate plastic deformation. If g(U) = 1 the material is extremely ductile and if g(U) = 0 there is no possible bending without failure; it is an extreme case of a brittle material. We are defining g(U) as a function of the material ductility U that will be discussed below. The total dissipated plastic work is given therefore by: Wn = nτn ∼ = nk̂σY g(U)πφ3 . (8) Introducing the value of n given by (2b) we obtain: Wn = π 2 k̂σY LnL h g(U)φ3 . Or with equation (7a): Wn = π 2 k̂e0σY L0h2 g(U)ρ4−D . (9)
- 26. “DynamicalSystems” — 2004/3/4 — page #9 A Geometric Approach to the Mechanics of Densely Folded Media 9 Now defining the reference plastic work as WR = π 2 k̂σY h3 G(U, β) , where G(U, β) = 1 β g(U) and β = h L0 , we obtain: Wn = WRρ4−D . (10) Call WR the reference dissipated plastic work. G(U, β) is the apparent ductility that involves both material properties g(u) and the geometry β. For the purpose of the present paper the ductility may be defined as: U = 1 − σY Eεu , where εu is the ultimate strain at fracture, and E is the Young modulus.1 U varies from zero, when εu = εY , and in this case the material is very stiff, and the failure is characterized by brittle fracture with no plastic deformation, to 1, for the ideal case of unlimited elongation at fracture εu → ∞. The packing capacity increases with the material ductility g(u) and with the ratio β = h/L0. The parameter β can be interpreted as the geometric ductility. We may assert therefore that, for a fixed value of h, the fractal dimension will be an increasing function of the apparent ductility G(U, β). Dropping the subscript n in (10) for the sake of simplicity, we finally obtain: τ = W WR = ρ4−D . (11) We will call τ the dissipated plastic work density. To illustrate the variation of dissipated plastic work with ratio ρ and packing capacity Γ consider the three points M, N and P shown in Figure 4. The points P and N correspond to the same value of ρ, and also to the same wire diameter, provided that h is fixed. We assume that the material properties are constant for all wires. Now clearly Γa < Γb and D2 < D1. Now from (11): WN WP = ρ−D1+D2 . (12) But since D2 < D1 and ρ < 1 the right hand term in (12) is greater than one. Then from (12) we may write: WN WP > 1 . 1For materials displaying a stress-strain curve that can be approximated by a bi-linear law, with σu as the ultimate stress at fracture, the ductility reads: U = 1 2 ³ 1 + σY σu ´³ 1 − σY Eσu ´ .
- 27. “DynamicalSystems” — 2004/3/4 — page #10 10 L. Bevilacqua Figure 4 Packing capacities for different combinations of the wire diameter, packing capacity and apparent ductility. Finally we may conclude that WP < WN . Consider now the points M and N. For these two points the packing capacity is the same and ρb < ρa. Therefore from (11) and (7a): WM WN = µ ρb ρa ¶3 . (13) But since ρb < ρa the right hand term in (13) is less than one, therefore WM WN < 1 . Then WM < WN . Finally consider the points P and M, on the line corresponding to the same fractal dimension. From (11), given that ρa > ρb and since the two points belong to a line with the same fractal dimension, that is, the same apparent ductility we have immediately: WM < WP . From equations (7b) and (11) it is possible to find an explicit expression for the fractal dimension as a function of the folding capacity Γ and the plastic work density τ: D = 1 − 3 log µ Γ e0 ¶ log τ − log µ Γ e0 ¶ . (14) Define now the packing density Ω, as the ratio between the apparent specific mass µ, per unit box length, and the specific mass of the wire µ0, per unit wire length. The apparent specific mass is defined as the total weight of the wire packed
- 28. “DynamicalSystems” — 2004/3/4 — page #11 A Geometric Approach to the Mechanics of Densely Folded Media 11 inside the box divided by the box length L0. This definition implies the homog- enization of the specific mass, making it uniformly distributed along the box in what, in general, is a good approximation. After some simple calculations it is then possible to write: µ = µ0 M P m=1 l (m) n L n → nL , (15a) Ω = e0ρ1−D , or log Ω = log e0 + (1 − D) log ρ , (15b) where Ω = µ/µ0. Therefore the same law governing the packing capacity Γ also applies to the packing density ρ. The representation in Figure 3 is equally valid for this case. Analytically the parameter e0 in (15a,b) is the same as in (7a,b). Only experi- mental evidence can confirm this result. The limit cases have the same interpretation as for the packing capacity. Putting e0 = 1, the limit case D = 1 is satisfied, for the virtual specific mass will coincide with the wire specific mass. But there is no strong reason to abandon from the very beginning the hypothesis of having e0 = e0(D). This might well be the case, and at the present stage only experimental data can provide answers on this subject. Now, consider the region MNPQ shown in Figure 2. It is plausible to assume that the plastic energy stored in the wire inside the box is uniformly distributed along the length L0. Therefore the energy stored in MNPQ Wj, is Lj/L0 times the total energy necessary to pack the wire inside the box [L0 × h]. But also the reference energy WR is proportional to L0 by definition, therefore the reference energy corresponding to the box [Lj ×h] WRj is also Lj/L0 times WR corresponding to the box [L0 ×h]. Since the plastic work density τ is the ratio Wj/WRj it is easily seen that τ is invariant for any sub-region [Lj × h] of [L0 × h]. Combining (7a) and (11) we get: Γj = e0E1−D j = e0E1−D = Γ . That is the packing capacity is the same for the box [L0 × h] and for any of its sub-regions [Lj × h]. This result is coherent with the hypothesis of the uniform distribution of the specific mass introduced before. It is also a confirmation of the intrinsic self-similarity property required by the structure of the fractal geometry. The above discussions lead to some conclusions that can be summarized as follows: Proposition 1 1. The geometry of densely folded wires packed in a two-dimen- sional box – [L×h], L > h – has a random structure characterized by a fractal dimension 1 < D < 2, provided that: i. The slenderness ratio defined by ρ = φ/h, where φ is the wire diameter, is sufficiently small. That is, ρ ¿ 1. ii. The material is perfectly plastic. iii. The final configuration is stable.
- 29. “DynamicalSystems” — 2004/3/4 — page #12 12 L. Bevilacqua 2. The packing capacity Γ = L/L0, and the packing density Ω = µ/µ0, vary with the slenderness ratio according to the power law: Γ = e0ρ1−D , and Ω = e0ρ1−D , where L is the wire length packed in the box; L0 is the box length; µ is the apparent specific mass; µ0 is the wire specific mass per unit length. The pa- rameter e0 is to be determined experimentally. 3. The fractal dimension D depends on the apparent ductility. Wires folding in a configuration with a high fractal dimension D will have a corresponding high apparent ductility. For a given value of the packing capacity Γ, the fractal dimension is a function of the dissipated plastic work density τ: D = 1 − 3 log µ Γ e0 ¶ log τ − log µ Γ e0 ¶ . 4. If the packing capacity is governed by a power law as given in item 2 above, then the geometry representing the wire final configuration is self-similar. Conversely, if the geometry is self-similar the packing configuration has a fractal structure as indicated in item 2 above. 3 Densely Folded Shells Let us move to a more complex case. Consider a uniform spherical thin shell with radius equal to R under uniform external pressure p as shown in Figure 5. Here, just as in the previous case, the shell is made of a perfect plastic material. When p reaches a critical value the shell collapses to form a complex surface com- posed of small tiles, in general of arbitrary shape, disposed around the rigid sphere of radius R0. The pressure continues to act till the entire shell is confined within the “spherical crust” bounded by two spheres, R0 and R0 + h. That is, the original shell is packed inside the “spherical crust” of thickness equal to h. Let us start with an ideal configuration as in the case of folded wires. Assume a regular folding such that the tiles have the shape of isosceles triangles and the fundamental element of 3-D geometry, that is, the surface generator consists of a pyramid whose basis is an equilateral triangle, and the height is equal to h as shown in Figure 6. A pineapple shell provides a good approximation to visualize this type of surface, which is a concave polyhedron. The edges of this surface, that is, the common lines of adjacent tiles are the rupture lines. The tiles rotate about these lines, developing a relatively complex mechanism and accumulating permanent plastic strain till the final configuration is reached. The final configuration is stable. The original shell remains folded within the “spherical crust” without any external or internal restrain.
- 30. Exploring the Variety of Random Documents with Different Content
- 31. For a moment she did not take the violets. But while she stood just inside, holding the door, a strange thing happened. . . . Again she saw the beautiful fall of the steps, the dark garden ringed with glittering ivy, the willows, the big bright sky. Again she felt the silence that was like a question. But this time she did not hesitate. She moved forward. Very softly and gently, as though fearful of making a ripple in that boundless pool of quiet she put her arms round her friend. “My dear,” murmured her happy friend, quite overcome by this gratitude. “They are really nothing. Just the simplest little thrippenny bunch.” But as she spoke she was enfolded—more tenderly, more beautifully embraced, held by such a sweet pressure and for so long that the poor dear’s mind positively reeled and she just had the strength to quaver: “Then you really don’t mind me too much?” “Good night, my friend,” whispered the other. “Come again soon.” “Oh, I will. I will.” This time she walked back to the studio slowly, and standing in the middle of the room with half-shut eyes she felt so light, so rested, as if she had woken up out of a childish sleep. Even the act of breathing was a joy. . . . The sommier was very untidy. All the cushions “like furious mountains” as she said; she put them in order before going over to the writing-table. “I have been thinking over our talk about the psychological novel,” she dashed off, “it really is intensely interesting.” . . . And so on and so on. At the end she wrote: “Good night, my friend. Come again soon.”
- 32. Pictures Eight o’clock in the morning. Miss Ada Moss lay in a black iron bedstead, staring up at the ceiling. Her room, a Bloomsbury top- floor back, smelled of soot and face powder and the paper of fried potatoes she brought in for supper the night before. “Oh, dear,” thought Miss Moss, “I am cold. I wonder why it is that I always wake up so cold in the mornings now. My knees and feet and my back—especially my back; it’s like a sheet of ice. And I always was such a one for being warm in the old days. It’s not as if I was skinny—I’m just the same full figure that I used to be. No, it’s because I don’t have a good hot dinner in the evenings.” A pageant of Good Hot Dinners passed across the ceiling, each of them accompanied by a bottle of Nourishing Stout. . . . “Even if I were to get up now,” she thought, “and have a sensible substantial breakfast . . .” A pageant of Sensible Substantial Breakfasts followed the dinners across the ceiling, shepherded by an enormous, white, uncut ham. Miss Moss shuddered and disappeared under the bedclothes. Suddenly, in bounced the landlady. “There’s a letter for you, Miss Moss.” “Oh,” said Miss Moss, far too friendly, “thank you very much, Mrs. Pine. It’s very good of you, I’m sure, to take the trouble.” “No trouble at all,” said the landlady. “I thought perhaps it was the letter you’d been expecting.” “Why,” said Miss Moss brightly, “yes, perhaps it is.” She put her head on one side and smiled vaguely at the letter. “I shouldn’t be surprised.”
- 33. The landlady’s eyes popped. “Well, I should, Miss Moss,” said she, “and that’s how it is. And I’ll trouble you to open it, if you please. Many is the lady in my place as would have done it for you and have been within her rights. For things can’t go on like this, Miss Moss, no indeed they can’t. What with week in week out and first you’ve got it and then you haven’t, and then it’s another letter lost in the post or another manager down at Brighton but will be back on Tuesday for certain—I’m fair sick and tired and I won’t stand it no more. Why should I, Miss Moss, I ask you, at a time like this, with prices flying up in the air and my poor dear lad in France? My sister Eliza was only saying to me yesterday—‘Minnie,’ she says, ‘you’re too soft- hearted. You could have let that room time and time again,’ says she, ‘and if people won’t look after themselves in times like these, nobody else will,’ she says. ‘She may have had a College eddication and sung in West End concerts,’ says she, ‘but if your Lizzie says what’s true,’ she says, ‘and she’s washing her own wovens and drying them on the towel rail, it’s easy to see where the finger’s pointing. And it’s high time you had done with it,’ says she.” Miss Moss gave no sign of having heard this. She sat up in bed, tore open her letter and read: “Dear Madam, Yours to hand. Am not producing at present, but have filed photo for future ref. Yours truly, BACKWASH FILM CO.” This letter seemed to afford her peculiar satisfaction; she read it through twice before replying to the landlady. “Well, Mrs. Pine, I think you’ll be sorry for what you said. This is from a manager, asking me to be there with evening dress at ten o’clock next Saturday morning.” But the landlady was too quick for her. She pounced, secured the letter.
- 34. “Oh, is it! Is it indeed!” she cried. “Give me back that letter. Give it back to me at once, you bad, wicked woman,” cried Miss Moss, who could not get out of bed because her nightdress was slit down the back. “Give me back my private letter.” The landlady began slowly backing out of the room, holding the letter to her buttoned bodice. “So it’s come to this, has it?” said she. “Well, Miss Moss, if I don’t get my rent at eight o’clock to-night, we’ll see who’s a bad, wicked woman—that’s all.” Here she nodded, mysteriously. “And I’ll keep this letter.” Here her voice rose. “It will be a pretty little bit of evidence!” And here it fell, sepulchral, “My lady.” The door banged and Miss Moss was alone. She flung off the bed clothes, and sitting by the side of the bed, furious and shivering, she stared at her fat white legs with their great knots of greeny-blue veins. “Cockroach! That’s what she is. She’s a cockroach!” said Miss Moss. “I could have her up for snatching my letter—I’m sure I could.” Still keeping on her nightdress she began to drag on her clothes. “Oh, if I could only pay that woman, I’d give her a piece of my mind that she wouldn’t forget. I’d tell her off proper.” She went over to the chest of drawers for a safety-pin, and seeing herself in the glass she gave a vague smile and shook her head. “Well, old girl,” she murmured, “you’re up against it this time, and no mistake.” But the person in the glass made an ugly face at her. “You silly thing,” scolded Miss Moss. “Now what’s the good of crying: you’ll only make your nose red. No, you get dressed and go out and try your luck—that’s what you’ve got to do.” She unhooked her vanity bag from the bedpost, rooted in it, shook it, turned it inside out. “I’ll have a nice cup of tea at an A B C to settle me before I go anywhere,” she decided. “I’ve got one and thrippence—yes, just one and three.”
- 35. Ten minutes later, a stout lady in blue serge, with a bunch of artificial “parmas” at her bosom, a black hat covered with purple pansies, white gloves, boots with white uppers, and a vanity bag containing one and three, sang in a low contralto voice: Sweet-heart, remember when days are forlorn It al-ways is dar-kest before the dawn. But the person in the glass made a face at her, and Miss Moss went out. There were grey crabs all the way down the street slopping water over grey stone steps. With his strange, hawking cry and the jangle of the cans the milk boy went his rounds. Outside Brittweiler’s Swiss House he made a splash, and an old brown cat without a tail appeared from nowhere, and began greedily and silently drinking up the spill. It gave Miss Moss a queer feeling to watch—a sinking—as you might say. But when she came to the A B C she found the door propped open; a man went in and out carrying trays of rolls, and there was nobody inside except a waitress doing her hair and the cashier unlocking cash-boxes. She stood in the middle of the floor but neither of them saw her. “My boy came home last night,” sang the waitress. “Oh, I say—how topping for you!” gurgled the cashier. “Yes, wasn’t it,” sang the waitress. “He brought me a sweet little brooch. Look, it’s got ‘Dieppe’ written on it.” The cashier ran across to look and put her arm round the waitress’ neck. “Oh, I say—how topping for you.” “Yes, isn’t it,” said the waitress. “O-oh, he is brahn. ‘Hullo,’ I said, ‘hullo, old mahogany.’” “Oh, I say,” gurgled the cashier, running back into her cage and nearly bumping into Miss Moss on the way. “You are a treat!” Then the man with the rolls came in again, swerving past her. “Can I have a cup of tea, Miss?” she asked.
- 36. But the waitress went on doing her hair. “Oh,” she sang, “we’re not open yet.” She turned round and waved her comb at the cashier. “Are we, dear?” “Oh, no,” said the cashier. Miss Moss went out. “I’ll go to Charing Cross. Yes, that’s what I’ll do,” she decided. “But I won’t have a cup of tea. No, I’ll have a coffee. There’s more of a tonic in coffee. . . . Cheeky, those girls are! Her boy came home last night; he brought her a brooch with ‘Dieppe’ written on it.” She began to cross the road. . . . “Look out, Fattie; don’t go to sleep!” yelled a taxi driver. She pretended not to hear. “No, I won’t go to Charing Cross,” she decided. “I’ll go straight to Kig and Kadgit. They’re open at nine. If I get there early Mr. Kadgit may have something by the morning’s post. . . . I’m very glad you turned up so early, Miss Moss. I’ve just heard from a manager who wants a lady to play. . . . I think you’ll just suit him. I’ll give you a card to go and see him. It’s three pounds a week and all found. If I were you I’d hop round as fast as I could. Lucky you turned up so early . . .” But there was nobody at Kig and Kadgit’s except the charwoman wiping over the “lino” in the passage. “Nobody here yet, Miss,” said the char. “Oh, isn’t Mr. Kadgit here?” said Miss Moss, trying to dodge the pail and brush. “Well, I’ll just wait a moment, if I may.” “You can’t wait in the waiting-room, Miss. I ’aven’t done it yet. Mr. Kadgit’s never ’ere before ’leven-thirty Saturdays. Sometimes ’e don’t come at all.” And the char began crawling towards her. “Dear me—how silly of me,” said Miss Moss. “I forgot it was Saturday.” “Mind your feet, please, Miss,” said the char. And Miss Moss was outside again. That was one thing about Beit and Bithems; it was lively. You walked into the waiting-room, into a great buzz of conversation, and
- 37. there was everybody; you knew almost everybody. The early ones sat on chairs and the later ones sat on the early ones’ laps, while the gentlemen leaned negligently against the walls or preened themselves in front of the admiring ladies. “Hello,” said Miss Moss, very gay. “Here we are again!” And young Mr. Clayton, playing the banjo on his walking-stick, sang: “Waiting for the Robert E. Lee.” “Mr. Bithem here yet?” asked Miss Moss, taking out an old dead powder puff and powdering her nose mauve. “Oh, yes, dear,” cried the chorus. “He’s been here for ages. We’ve all been waiting here for more than an hour.” “Dear me!” said Miss Moss. “Anything doing, do you think?” “Oh, a few jobs going for South Africa,” said young Mr. Clayton. “Hundred and fifty a week for two years, you know.” “Oh!” cried the chorus. “You are weird, Mr. Clayton. Isn’t he a cure? Isn’t he a scream, dear? Oh, Mr. Clayton, you do make me laugh. Isn’t he a comic?” A dark, mournful girl touched Miss Moss on the arm. “I just missed a lovely job yesterday,” she said. “Six weeks in the provinces and then the West End. The manager said I would have got it for certain if only I’d been robust enough. He said if my figure had been fuller, the part was made for me.” She stared at Miss Moss, and the dirty dark red rose under the brim of her hat looked, somehow, as though it shared the blow with her, and was crushed, too. “Oh, dear, that was hard lines,” said Miss Moss trying to appear indifferent. “What was it—if I may ask?” But the dark, mournful girl saw through her and a gleam of spite came into her heavy eyes. “Oh, no good to you, my dear,” said she. “He wanted someone young, you know—a dark Spanish type—my style, but more figure, that was all.”
- 38. The inner door opened and Mr. Bithem appeared in his shirt sleeves. He kept one hand on the door ready to whisk back again, and held up the other. “Look here, ladies——” and then he paused, grinned his famous grin before he said—“and bhoys.” The waiting-room laughed so loudly at this that he had to hold both hands up. “It’s no good waiting this morning. Come back Monday; I’m expecting several calls on Monday.” Miss Moss made a desperate rush forward. “Mr. Bithem, I wonder if you’ve heard from . . .” “Now let me see,” said Mr. Bithem slowly, staring; he had only seen Miss Moss four times a week for the past—how many weeks? “Now, who are you?” “Miss Ada Moss.” “Oh, yes, yes; of course, my dear. Not yet, my dear. Now I had a call for twenty-eight ladies to-day, but they had to be young and able to hop it a bit—see? And I had another call for sixteen—but they had to know something about sand-dancing. Look here, my dear, I’m up to the eyebrows this morning. Come back on Monday week; it’s no good coming before that.” He gave her a whole grin to herself and patted her fat back. “Hearts of oak, dear lady,” said Mr. Bithem, “hearts of oak!” At the North-East Film Company the crowd was all the way up the stairs. Miss Moss found herself next to a fair little baby thing about thirty in a white lace hat with cherries round it. “What a crowd!” said she. “Anything special on?” “Didn’t you know, dear?” said the baby, opening her immense pale eyes. “There was a call at nine-thirty for attractive girls. We’ve all been waiting for hours. Have you played for this company before?” Miss Moss put her head on one side. “No, I don’t think I have.” “They’re a lovely company to play for,” said the baby. “A friend of mine has a friend who gets thirty pounds a day. . . . Have you arcted much for the fil-lums?”
- 39. “Well, I’m not an actress by profession,” confessed Miss Moss. “I’m a contralto singer. But things have been so bad lately that I’ve been doing a little.” “It’s like that, isn’t it, dear?” said the baby. “I had a splendid education at the College of Music,” said Miss Moss, “and I got my silver medal for singing. I’ve often sung at West End concerts. But I thought, for a change, I’d try my luck . . .” “Yes, it’s like that, isn’t it, dear?” said the baby. At that moment a beautiful typist appeared at the top of the stairs. “Are you all waiting for the North-East call?” “Yes!” cried the chorus. “Well, it’s off. I’ve just had a phone through.” “But look here! What about our expenses?” shouted a voice. The typist looked down at them, and she couldn’t help laughing. “Oh, you weren’t to have been paid. The North-East never pay their crowds.” There was only a little round window at the Bitter Orange Company. No waiting-room—nobody at all except a girl, who came to the window when Miss Moss knocked, and said: “Well?” “Can I see the producer, please?” said Miss Moss pleasantly. The girl leaned on the window-bar, half shut her eyes and seemed to go to sleep for a moment. Miss Moss smiled at her. The girl not only frowned; she seemed to smell something vaguely unpleasant; she sniffed. Suddenly she moved away, came back with a paper and thrust it at Miss Moss. “Fill up the form!” said she. And banged the window down. “Can you aviate—high-dive—drive a car—buck-jump—shoot?” read Miss Moss. She walked along the street asking herself those questions. There was a high, cold wind blowing; it tugged at her, slapped her face, jeered; it knew she could not answer them. In the Square Gardens she found a little wire basket to drop the form into. And then she sat down on one of the benches to powder her nose.
- 40. But the person in the pocket mirror made a hideous face at her, and that was too much for Miss Moss; she had a good cry. It cheered her wonderfully. “Well, that’s over,” she sighed. “It’s one comfort to be off my feet. And my nose will soon get cool in the air. . . . It’s very nice in here. Look at the sparrows. Cheep. Cheep. How close they come. I expect somebody feeds them. No, I’ve nothing for you, you cheeky little things. . . .” She looked away from them. What was the big building opposite—the Café de Madrid? My goodness, what a smack that little child came down! Poor little mite! Never mind—up again. . . . By eight o’clock to-night . . . Café de Madrid. “I could just go in and sit there and have a coffee, that’s all,” thought Miss Moss. “It’s such a place for artists too. I might just have a stroke of luck. . . . A dark handsome gentleman in a fur coat comes in with a friend, and sits at my table, perhaps. ‘No, old chap, I’ve searched London for a contralto and I can’t find a soul. You see, the music is difficult; have a look at it.’” And Miss Moss heard herself saying: “Excuse me, I happen to be a contralto, and I have sung that part many times. . . . Extraordinary! ‘Come back to my studio and I’ll try your voice now.’ . . . Ten pounds a week. . . . Why should I feel nervous? It’s not nervousness. Why shouldn’t I go to the Café de Madrid? I’m a respectable woman—I’m a contralto singer. And I’m only trembling because I’ve had nothing to eat to-day. . . . ‘A nice little piece of evidence, my lady.’ . . . Very well, Mrs. Pine. Café de Madrid. They have concerts there in the evenings. . . . ‘Why don’t they begin?’ The contralto has not arrived. . . . ‘Excuse me, I happen to be a contralto; I have sung that music many times.’” It was almost dark in the café. Men, palms, red plush seats, white marble tables, waiters in aprons, Miss Moss walked through them all. Hardly had she sat down when a very stout gentleman wearing a very small hat that floated on the top of his head like a little yacht flopped into the chair opposite hers. “Good evening!” said he. Miss Moss said, in her cheerful way: “Good evening!”
- 41. “Fine evening,” said the stout gentleman. “Yes, very fine. Quite a treat, isn’t it?” said she. He crooked a sausage finger at the waiter—“Bring me a large whisky”—and turned to Miss Moss. “What’s yours?” “Well, I think I’ll take a brandy if it’s all the same.” Five minutes later the stout gentleman leaned across the table and blew a puff of cigar smoke full in her face. “That’s a tempting bit o’ ribbon!” said he. Miss Moss blushed until a pulse at the top of her head that she never had felt before pounded away. “I always was one for pink,” said she. The stout gentleman considered her, drumming with her fingers on the table. “I like ’em firm and well covered,” said he. Miss Moss, to her surprise, gave a loud snigger. Five minutes later the stout gentleman heaved himself up. “Well, am I goin’ your way, or are you comin’ mine?” he asked. “I’ll come with you, if it’s all the same,” said Miss Moss. And she sailed after the little yacht out of the café.
- 42. The Man without a Temperament He stood at the hall door turning the ring, turning the heavy signet ring upon his little finger while his glance travelled coolly, deliberately, over the round tables and basket chairs scattered about the glassed-in verandah. He pursed his lips—he might have been going to whistle—but he did not whistle—only turned the ring— turned the ring on his pink, freshly washed hands. Over in the corner sat The Two Topknots, drinking a decoction they always drank at this hour—something whitish, greyish, in glasses, with little husks floating on the top—and rooting in a tin full of paper shavings for pieces of speckled biscuit, which they broke, dropped into the glasses and fished for with spoons. Their two coils of knitting, like two snakes, slumbered beside the tray. The American Woman sat where she always sat against the glass wall, in the shadow of a great creeping thing with wide open purple eyes that pressed—that flattened itself against the glass, hungrily watching her. And she knoo it was there—she knoo it was looking at her just that way. She played up to it; she gave herself little airs. Sometimes she even pointed at it, crying: “Isn’t that the most terrible thing you’ve ever seen! Isn’t that ghoulish!” It was on the other side of the verandah, after all . . . and besides it couldn’t touch her, could it, Klaymongso? She was an American Woman, wasn’t she Klaymongso, and she’d just go right away to her Consul. Klaymongso, curled in her lap, with her torn antique brocade bag, a grubby handkerchief, and a pile of letters from home on top of him, sneezed for reply. The other tables were empty. A glance passed between the American and the Topknots. She gave a foreign little shrug; they
- 43. waved an understanding biscuit. But he saw nothing. Now he was still, now from his eyes you saw he listened. “Hoo-e-zip-zoo-oo!” sounded the lift. The iron cage clanged open. Light dragging steps sounded across the hall, coming towards him. A hand, like a leaf, fell on his shoulder. A soft voice said: “Let’s go and sit over there— where we can see the drive. The trees are so lovely.” And he moved forward with the hand still on his shoulder, and the light, dragging steps beside his. He pulled out a chair and she sank into it, slowly, leaning her head against the back, her arms falling along the sides. “Won’t you bring the other up closer? It’s such miles away.” But he did not move. “Where’s your shawl?” he asked. “Oh!” She gave a little groan of dismay. “How silly I am, I’ve left it upstairs on the bed. Never mind. Please don’t go for it. I shan’t want it, I know I shan’t.” “You’d better have it.” And he turned and swiftly crossed the verandah into the dim hall with its scarlet plush and gilt furniture— conjuror’s furniture—its Notice of Services at the English Church, its green baize board with the unclaimed letters climbing the black lattice, huge “Presentation” clock that struck the hours at the half- hours, bundles of sticks and umbrellas and sunshades in the clasp of a brown wooden bear, past the two crippled palms, two ancient beggars at the foot of the staircase, up the marble stairs three at a time, past the life-size group on the landing of two stout peasant children with their marble pinnies full of marble grapes, and along the corridor, with its piled-up wreckage of old tin boxes, leather trunks, canvas hold-alls, to their room. The servant girl was in their room, singing loudly while she emptied soapy water into a pail. The windows were open wide, the shutters put back, and the light glared in. She had thrown the carpets and the big white pillows over the balcony rails; the nets were looped up from the beds; on the writing table there stood a pan of fluff and match-ends. When she saw him her small impudent eyes snapped and her singing changed to humming. But he gave no
- 44. sign. His eyes searched the glaring room. Where the devil was the shawl! “Vous desirez, Monsieur?” mocked the servant girl. No answer. He had seen it. He strode across the room, grabbed the grey cobweb and went out, banging the door. The servant girl’s voice at its loudest and shrillest followed him along the corridor. “Oh, there you are. What happened? What kept you? The tea’s here, you see. I’ve just sent Antonio off for the hot water. Isn’t it extraordinary? I must have told him about it sixty times at least, and still he doesn’t bring it. Thank you. That’s very nice. One does just feel the air when one bends forward.” “Thanks.” He took his tea and sat down in the other chair. “No, nothing to eat.” “Oh do! Just one, you had so little at lunch and it’s hours before dinner.” Her shawl dropped off as she bent forward to hand him the biscuits. He took one and put it in his saucer. “Oh, those trees along the drive,” she cried, “I could look at them for ever. They are like the most exquisite huge ferns. And you see that one with the grey-silver bark and the clusters of cream coloured flowers, I pulled down a head of them yesterday to smell and the scent”—she shut her eyes at the memory and her voice thinned away, faint, airy—“was like freshly ground nutmegs.” A little pause. She turned to him and smiled. “You do know what nutmegs smell like—do you, Robert?” And he smiled back at her. “Now how am I going to prove to you that I do?” Back came Antonio with not only the hot water—with letters on a salver and three rolls of paper. “Oh, the post! Oh, how lovely! Oh, Robert, they mustn’t be all for you! Have they just come, Antonio?” Her thin hands flew up and hovered over the letters that Antonio offered her, bending forward.
- 45. “Just this moment, Signora,” grinned Antonio. “I took-a them from the postman myself. I made-a the postman give them for me.” “Noble Antonio!” laughed she. “There—those are mine, Robert; the rest are yours.” Antonio wheeled sharply, stiffened, the grin went out of his face. His striped linen jacket and his flat gleaming fringe made him look like a wooden doll. Mr. Salesby put the letters into his pocket; the papers lay on the table. He turned the ring, turned the signet ring on his little finger and stared in front of him, blinking, vacant. But she—with her teacup in one hand, the sheets of thin paper in the other, her head tilted back, her lips open, a brush of bright colour on her cheek-bones, sipped, sipped, drank . . . drank. . . . “From Lottie,” came her soft murmur. “Poor dear . . . such trouble . . . left foot. She thought . . . neuritis . . . Doctor Blyth . . . flat foot . . . massage. So many robins this year . . . maid most satisfactory . . . Indian Colonel . . . every grain of rice separate . . . very heavy fall of snow.” And her wide lighted eyes looked up from the letter. “Snow, Robert! Think of it!” And she touched the little dark violets pinned on her thin bosom and went back to the letter. . . . Snow. Snow in London. Millie with the early morning cup of tea. “There’s been a terrible fall of snow in the night, Sir.” “Oh, has there, Millie?” The curtains ring apart, letting in the pale, reluctant light. He raises himself in the bed; he catches a glimpse of the solid houses opposite framed in white, of their window boxes full of great sprays of white coral. . . . In the bathroom—overlooking the back garden. Snow—heavy snow over everything. The lawn is covered with a wavy pattern of cat’s paws; there is a thick, thick icing on the garden table; the withered pods of the laburnum tree are white tassels; only here and there in the ivy is a dark leaf showing. . . . Warming his back at the dining-room fire, the paper drying over a chair. Millie with the bacon. “Oh, if you please, Sir, there’s two little boys come as will do the steps and front for a shilling, shall I let
- 46. them?” . . . And then flying lightly, lightly down the stairs—Jinnie. “Oh, Robert, isn’t it wonderful! Oh, what a pity it has to melt. Where’s the pussy-wee?” “I’ll get him from Millie” . . . “Millie, you might just hand me up the kitten if you’ve got him down there.” “Very good, Sir.” He feels the little beating heart under his hand. “Come on, old chap, your Missus wants you.” “Oh, Robert, do show him the snow—his first snow. Shall I open the window and give him a little piece on his paw to hold? . . .” “Well, that’s very satisfactory on the whole—very. Poor Lottie! Darling Anne! How I only wish I could send them something of this,” she cried, waving her letters at the brilliant, dazzling garden. “More tea, Robert? Robert dear, more tea?” “No, thanks, no. It was very good,” he drawled. “Well mine wasn’t. Mine was just like chopped hay. Oh, here comes the Honeymoon Couple.” Half striding, half running, carrying a basket between them and rods and lines, they came up the drive, up the shallow steps. “My! have you been out fishing?” cried the American Woman. They were out of breath, they panted: “Yes, yes, we have been out in a little boat all day. We have caught seven. Four are good to eat. But three we shall give away. To the children.” Mrs. Salesby turned her chair to look; the Topknots laid the snakes down. They were a very dark young couple—black hair, olive skin, brilliant eyes and teeth. He was dressed “English fashion” in a flannel jacket, white trousers and shoes. Round his neck he wore a silk scarf; his head, with his hair brushed back, was bare. And he kept mopping his forehead, rubbing his hands with a brilliant handkerchief. Her white skirt had a patch of wet; her neck and throat were stained a deep pink. When she lifted her arms big half- hoops of perspiration showed under her arm-pits; her hair clung in wet curls to her cheeks. She looked as though her young husband had been dipping her in the sea, and fishing her out again to dry in the sun and then—in with her again—all day.
- 47. “Would Klaymongso like a fish?” they cried. Their laughing voices charged with excitement beat against the glassed-in verandah like birds, and a strange saltish smell came from the basket. “You will sleep well to-night,” said a Topknot, picking her ear with a knitting needle while the other Topknot smiled and nodded. The Honeymoon Couple looked at each other. A great wave seemed to go over them. They gasped, gulped, staggered a little and then came up laughing—laughing. “We cannot go upstairs, we are too tired. We must have tea just as we are. Here—coffee. No—tea. No—coffee. Tea—coffee, Antonio!” Mrs. Salesby turned. “Robert! Robert!” Where was he? He wasn’t there. Oh, there he was at the other end of the verandah, with his back turned, smoking a cigarette. “Robert, shall we go for our little turn?” “Right.” He stumped the cigarette into an ash-tray and sauntered over, his eyes on the ground. “Will you be warm enough?” “Oh, quite.” “Sure?” “Well,” she put her hand on his arm, “perhaps”—and gave his arm the faintest pressure—“it’s not upstairs, it’s only in the hall—perhaps you’d get me my cape. Hanging up.” He came back with it and she bent her small head while he dropped it on her shoulders. Then, very stiff, he offered her his arm. She bowed sweetly to the people on the verandah while he just covered a yawn, and they went down the steps together. “Vous avez voo ça!” said the American Woman. “He is not a man,” said the Two Topknots, “he is an ox. I say to my sister in the morning and at night when we are in bed, I tell her —No man is he, but an ox!” Wheeling, tumbling, swooping, the laughter of the Honeymoon Couple dashed against the glass of the verandah. The sun was still high. Every leaf, every flower in the garden lay open, motionless, as if exhausted, and a sweet, rich, rank smell filled
- 48. the quivering air. Out of the thick, fleshy leaves of a cactus there rose an aloe stem loaded with pale flowers that looked as though they had been cut out of butter; light flashed upon the lifted spears of the palms; over a bed of scarlet waxen flowers some big black insects “zoom-zoomed”; a great, gaudy creeper, orange splashed with jet, sprawled against a wall. “I don’t need my cape after all,” said she. “It’s really too warm.” So he took it off and carried it over his arm. “Let us go down this path here. I feel so well to-day—marvellously better. Good heavens —look at those children! And to think it’s November!” In a corner of the garden there were two brimming tubs of water. Three little girls, having thoughtfully taken off their drawers and hung them on a bush, their skirts clasped to their waists, were standing in the tubs and tramping up and down. They screamed, their hair fell over their faces, they splashed one another. But suddenly, the smallest, who had a tub to herself, glanced up and saw who was looking. For a moment she seemed overcome with terror, then clumsily she struggled and strained out of her tub, and still holding her clothes above her waist. “The Englishman! The Englishman!” she shrieked and fled away to hide. Shrieking and screaming, the other two followed her. In a moment they were gone; in a moment there was nothing but the two brimming tubs and their little drawers on the bush. “How—very—extraordinary!” said she. “What made them so frightened? Surely they were much too young to . . .” She looked up at him. She thought he looked pale—but wonderfully handsome with that great tropical tree behind him with its long, spiked thorns. For a moment he did not answer. Then he met her glance, and smiling his slow smile, “Très rum!” said he. Très rum! Oh, she felt quite faint. Oh, why should she love him so much just because he said a thing like that. Très rum! That was Robert all over. Nobody else but Robert could ever say such a thing. To be so wonderful, so brilliant, so learned, and then to say in that queer, boyish voice. . . . She could have wept.
- 49. “You know you’re very absurd, sometimes,” said she. “I am,” he answered. And they walked on. But she was tired. She had had enough. She did not want to walk any more. “Leave me here and go for a little constitutional, won’t you? I’ll be in one of these long chairs. What a good thing you’ve got my cape; you won’t have to go upstairs for a rug. Thank you, Robert, I shall look at that delicious heliotrope. . . . You won’t be gone long?” “No—no. You don’t mind being left?” “Silly! I want you to go. I can’t expect you to drag after your invalid wife every minute. . . . How long will you be?” He took out his watch. “It’s just after half-past four. I’ll be back at a quarter past five.” “Back at a quarter past five,” she repeated, and she lay still in the long chair and folded her hands. He turned away. Suddenly he was back again. “Look here, would you like my watch?” And he dangled it before her. “Oh!” She caught her breath. “Very, very much.” And she clasped the watch, the warm watch, the darling watch in her fingers. “Now go quickly.” The gates of the Pension Villa Excelsior were open wide, jammed open against some bold geraniums. Stooping a little, staring straight ahead, walking swiftly, he passed through them and began climbing the hill that wound behind the town like a great rope looping the villas together. The dust lay thick. A carriage came bowling along driving towards the Excelsior. In it sat the General and the Countess; they had been for his daily airing. Mr. Salesby stepped to one side but the dust beat up, thick, white, stifling like wool. The Countess just had time to nudge the General. “There he goes,” she said spitefully. But the General gave a loud caw and refused to look. “It is the Englishman,” said the driver, turning round and smiling. And the Countess threw up her hands and nodded so amiably that
- 50. he spat with satisfaction and gave the stumbling horse a cut. On—on—past the finest villas in the town, magnificent palaces, palaces worth coming any distance to see, past the public gardens with the carved grottoes and statues and stone animals drinking at the fountain, into a poorer quarter. Here the road ran narrow and foul between high lean houses, the ground floors of which were scooped and hollowed into stables and carpenters’ shops. At a fountain ahead of him two old hags were beating linen. As he passed them they squatted back on their haunches, stared, and then their “A-hak-kak-kak!” with the slap, slap, of the stone on the linen sounded after him. He reached the top of the hill; he turned a corner and the town was hidden. Down he looked into a deep valley with a dried up river bed at the bottom. This side and that was covered with small dilapidated houses that had broken stone verandahs where the fruit lay drying, tomato lanes in the garden, and from the gates to the doors a trellis of vines. The late sunlight, deep, golden, lay in the cup of the valley; there was a smell of charcoal in the air. In the gardens the men were cutting grapes. He watched a man standing in the greenish shade, raising up, holding a black cluster in one hand, taking the knife from his belt, cutting, laying the bunch in a flat boat-shaped basket. The man worked leisurely, silently, taking hundreds of years over the job. On the hedges on the other side of the road there were grapes small as berries, growing wild, growing among the stones. He leaned against a wall, filled his pipe, put a match to it. . . . Leaned across a gate, turned up the collar of his mackintosh. It was going to rain. It didn’t matter, he was prepared for it. You didn’t expect anything else in November. He looked over the bare field. From the corner by the gate there came the smell of swedes, a great stack of them, wet, rank coloured. Two men passed walking towards the straggling village. “Good day!” “Good day!” By Jove! he had to hurry if he was going to catch that train home. Over the gate, across
- 51. a field, over the stile, into the lane, swinging along in the drifting rain and dusk. . . . Just home in time for a bath and a change before supper. . . . In the drawing-room; Jinnie is sitting pretty nearly in the fire. “Oh, Robert, I didn’t hear you come in. Did you have a good time? How nice you smell! A present?” “Some bits of blackberry I picked for you. Pretty colour.” “Oh, lovely, Robert! Dennis and Beaty are coming to supper.” Supper—cold beef, potatoes in their jackets, claret, household bread. They are gay—everybody’s laughing. “Oh, we all know Robert,” says Dennis, breathing on his eyeglasses and polishing them. “By the way, Dennis, I picked up a very jolly little edition of . . .” A clock struck. He wheeled sharply. What time was it. Five? A quarter past? Back, back the way he came. As he passed through the gates he saw her on the look-out. She got up, waved and slowly she came to meet him, dragging the heavy cape. In her hand she carried a spray of heliotrope. “You’re late,” she cried gaily. “You’re three minutes late. Here’s your watch, it’s been very good while you were away. Did you have a nice time? Was it lovely? Tell me. Where did you go?” “I say—put this on,” he said, taking the cape from her. “Yes, I will. Yes, it’s getting chilly. Shall we go up to our room?” When they reached the lift she was coughing. He frowned. “It’s nothing. I haven’t been out too late. Don’t be cross.” She sat down on one of the red plush chairs while he rang and rang, and then, getting no answer, kept his finger on the bell. “Oh, Robert, do you think you ought to?” “Ought to what?” The door of the salon opened. “What is that? Who is making that noise?” sounded from within. Klaymongso began to yelp. “Caw! Caw! Caw!” came from the General. A Topknot darted out with one hand to her ear, opened the staff door, “Mr. Queet! Mr. Queet!” she bawled. That brought the manager up at a run.
- 52. “Is that you ringing the bell, Mr. Salesby? Do you want the lift? Very good, Sir. I’ll take you up myself. Antonio wouldn’t have been a minute, he was just taking off his apron——” And having ushered them in, the oily manager went to the door of the salon. “Very sorry you should have been troubled, ladies and gentlemen.” Salesby stood in the cage, sucking in his cheeks, staring at the ceiling and turning the ring, turning the signet ring on his little finger. . . . Arrived in their room he went swiftly over to the washstand, shook the bottle, poured her out a dose and brought it across. “Sit down. Drink it. And don’t talk.” And he stood over her while she obeyed. Then he took the glass, rinsed it and put it back in its case. “Would you like a cushion?” “No, I’m quite all right. Come over here. Sit down by me just a minute, will you, Robert? Ah, that’s very nice.” She turned and thrust the piece of heliotrope in the lapel of his coat. “That,” she said, “is most becoming.” And then she leaned her head against his shoulder, and he put his arm round her. “Robert——” her voice like a sigh—like a breath. “Yes——” They sat there for a long while. The sky flamed, paled; the two white beds were like two ships. . . . At last he heard the servant girl running along the corridor with the hot water cans, and gently he released her and turned on the light. “Oh, what time is it? Oh, what a heavenly evening. Oh, Robert, I was thinking while you were away this afternoon . . .” They were the last couple to enter the dining-room. The Countess was there with her lorgnette and her fan, the General was there with his special chair and the air cushion and the small rug over his knees. The American Woman was there showing Klaymongso a copy of the Saturday Evening Post. . . . “We’re having a feast of reason and a flow of soul.” The Two Topknots were there feeling over the peaches and the pears in their dish of fruit, and putting aside all they considered unripe or overripe to show to the manager, and the
- 53. Welcome to our website – the ideal destination for book lovers and knowledge seekers. With a mission to inspire endlessly, we offer a vast collection of books, ranging from classic literary works to specialized publications, self-development books, and children's literature. Each book is a new journey of discovery, expanding knowledge and enriching the soul of the reade Our website is not just a platform for buying books, but a bridge connecting readers to the timeless values of culture and wisdom. With an elegant, user-friendly interface and an intelligent search system, we are committed to providing a quick and convenient shopping experience. Additionally, our special promotions and home delivery services ensure that you save time and fully enjoy the joy of reading. Let us accompany you on the journey of exploring knowledge and personal growth! ebookname.com