![“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/45534-250516191703-65d00a66/85/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/45534-250516191703-65d00a66/85/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/45534-250516191703-65d00a66/85/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/45534-250516191703-65d00a66/85/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 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. Discovering Diverse Content Through Random Scribd Documents
- 31. A Lily of the Valley. UST a breath of fragrance On the breeze—alas! A lily of the valley Dying in the grass. Just a recollection Followed with a sigh; Just a teardrop dripping Down the cheek, and why? May 16, 1887.
- 33. Lines to the Old Year. AREWELL, Old Year, the shades are growing deep, Thou art dethroned and vanishes your power; I sit alone with folded hands and weep, While close the minutes chase our parting hour. Your lips are dumb, and with a feeble hand You turn the pages of the year’s great book, While my wet cheeks are with an odor fanned, Like that the summer breeze from violets shook. I gaze into the volume. Undiscerned Some scenes advance, like phantoms hurry by, And thoughts look from the leaves now swifter turned As meaningless as would a stranger’s eye. I meet familiar names in Death’s long list, I pass new graves where tears have thawed the snows, I search my heart lest something I have missed, But in its garden find no dying rose. Thou hast been kind to me; no marble urn Chills the warm pulses of my heart to night, And from the thought my pen doth gladly turn To offer homage ere you take your flight. Bright recollections thou hast left instead, That twinkle in the firmament of thought, And lover-like I sit and gaze o’erhead Upon the starry gems thy hand has wrought. Far down the by-path of a summer dream,
- 34. Glad voices call and fingers beckon me— An oar dips music from a moonlit stream, Where in thy prime I sailed, Old Year, with thee And now, e’en in the shadow of thy hearse, Ungarland save with fated mistletoe, While midnight fiends the hours call like a curse, You clasp my hand and smiling on me—go. Farewell! A friend thou’st been to me, and I Shall wander through the burial ground of years, And often with an introspective eye Search out thy grave and water it with tears.
- 36. Why I Smile. smile because the world is fair; Because the sky is blue. Because I find, no matter where I go, a friend that’s true. I smile because the earth is green, The sun so near and bright, Because the days that o’er us lean Are full of warmth and light. I smile as past the yards I go, Though strange and new the place, The violets seem my step to know, And look up in my face. I smile to hear the robin’s note. He comes so newly dressed, A love song throbbing in his throat, A rose pinned on his breast. And so the truth I’ll not disown, Because the spring is nigh; My heart has somewhat better grown, And I forget to sigh. Mt. Vernon, Ill.
- 39. My Phantom Ships. heard the plunging of the sea Like a wild steed pursuing me, And dark and frothy was the main; But suddenly a checking rein Seemed drawn, and panting on the shore, I heard the billows’ frightful roar. My dream betook a different hue, Caught from the ocean’s changeful blue. A door was opened in my heart, From which I saw each fear depart, And there from some far, happy isle, The sea breeze came as would a smile Oh! it was sweet to wander there, The sky o’erhanging still and bare. A cloud, in some soft raiment dressed, Leaned like a bride upon the west; The sea-gulls floated on the breeze Like blossoms blown from April trees. The wind just kissed by summer’s mouth Walked like a lover from the South; And jewels from a sunbeam’s hand Were sprinkled on the snowy sand; The breakers ran along the beach, And scattered shells within my reach. I stooped and held one to my ear, And listened as to voices dear; And then methought far, far away,
- 40. Where purple mists made dim the day, I saw the motion of a ship That from the heavens seemed to slip. On, on it came with fluttering sail, Strong blew the steady ocean gale. The waves were running thick and high, And kept the ship close to the sky; It seemed a picture on the sea, “A picture,” thought I, “can it be?” But from the waves the wind withdrew And brought the sailors close to view. The pilot pointed to the shore, And then to gems and shining ore Piled up against the good ship’s side That leaned so brave upon the tide. Oh! there were silks of colors soft, And plumes that proudly waved aloft; And there were jewels, bags of gold, From caves o’er which the water rolled, And coral crowns—gifts of the sea— And all of this for whom? For me. With open arms to meet the ship I ran, and proudly curled my lip. No one should know from whence it came, And none should share my wealth and fame. My gowns of silk with me should roam, My gold I’d closet at my home. Ah, me! I knew not what I thought. The ship was by a whirlwind caught. It staggered out upon the sea— I heard the sailors cursing me;
- 41. A flash fell from the lowering night, And down the brave ship sank from sight. * * * * * I walk again upon the sands With aching heart and empty hands. Sometimes a piece of broken mast Upon the tide goes sailing past; And, where the sun so friendly shone, A shadow on the sand has grown. A strange and half-distracted dream Comes just behind the sea-gull’s scream. The sinking ship again I see, The sailors hurl their oaths at me, And like an echo from the grave Is the sad song of wind and wave. But somewhere, under bluer skies, Another ship in harbor lies. Its flags are flying free and fast, The sails are white, and strong the mast. ’Tis loaded, too, with precious freight, And for the same I stand and wait. When it comes home I’ll happy be, And all share my joy with me. My wines at other feasts I’ll pour, The sorrowful shall smile—yea, more, The poor shall not be turned away, And one and all shall bless the day. Pablo Beach, Fla., January, 1887.
- 44. The Weight of a Word. AVE you ever thought of the weight of a word That falls in the heart like the song of a bird, That gladdens the springtime of memory and youth And garlands with cedar the banner of Truth, That moistens the harvesting spot of the brain Like dew-drops that fall on the meadow of grain Or that shrivels the germ and destroys the fruit And lies like a worm at the lifeless root? I saw a farmer at break of day Hoeing his corn in a careful way; An enemy came with a drouth in his eye, Discouraged the worker and hurried by. The keen-edged blade of the faithful hoe Dulled on the earth in the long corn row; The weeds sprung up and their feathers tossed Over the field and the crop was—lost. A sailor launched on an angry bay When the heavens entombed the face of day The wind arose like a beast in pain, And shook on the billows his yellow name, The storm beat down as if cursed the cloud, And the waves held up a dripping shroud— But, hark! o’er the waters that wildly raved Came a word of cheer and he was—saved. A poet passed with a song of God Hid in his heart like a gem in a clod. His lips were framed to pronounce the thought, And the music of rhythm its magic wrought;
- 45. Feeble at first was the happy trill, Low was the echo that answered the hill, But a jealous friend spoke near his side, And on his lips the sweet song—died. A woman paused where a chandelier Threw in the darkness its poisoned spear; Weary and footsore from journeying long, She had strayed unawares from the right to the wrong. Angels were beck’ning her back from the den, Hell and its demons were beck’ning her in; The tone of an urchin, like one who forgives, Drew her back and in heaven that sweet word—lives. Words! Words! They are little, yet mighty and brave; They rescue a nation, an empire save; They close up the gaps in a fresh bleeding heart That sickness and sorrow have severed apart, They fall on the path, like a ray of the sun, Where the shadows of death lay so heavy upon; They lighten the earth over our blessed dead, A word that will comfort, oh! leave not unsaid.
- 48. An Apology. TO J. D. N. Y pen is mournful—you ask why When all the time my face is glad, And though contentment lights my eye, You say my verse is strangely sad; So serious that e’en the strain You can detect, as on the pane You know the patter in the night, Although the cloud is hid from sight. You asked me once to change my tone, “To trim my pen for gayer verse,” And, laughing, said ’twas like a moan That followed close behind a hearse. My muse was saddened at the stroke, And in my heart new chords awoke, Chords that vibrate like the bell That tolled one day a funeral knell. I would not have them otherwise; I claim my caged bird’s song more sweet Because ’tis sad, than one which tries The echo merrier to repeat. How quickly I would turn aside, And soon forget a boist’rous tide, To hear the brooklet, sad and low, Sing in a minor key I know. I’ll not attempt Hood’s humorous style, I do not crave John Gilpin’s ride.
- 49. It was my custom, when a child, To linger at my mother’s side When she would sing “The Old Church Yard,” That told how soft and green its sward. “The angels that watched ’round the tomb” Crept, as she sang, into our room. ’Tis said the clown will never jest When folded is the showman’s tent; That she who pathos renders best Has loudest laugh in merriment. Thus, vice versa is the theme, Or, “all things are not what they seem.” Sadness to Joy is as a twin, One rules without, one rules within. My life is full of love and joy, My heart-strings, though, with sadness tuned. Then do not ask me to destroy The mournful measures; it would wound My Muse—the playmate of my youth— Who taught me early many a truth From others’ woes, and bid me think While she supplied the pen and ink.
- 51. Speak Kindly. PEAK kindly in the morning, When you are leaving home, And give the day a lighter heart Into the week to roam. Leave kind words as mementoes To be handled and caressed, And watch the noon-time hour arrive In gold and tinsel dressed. Speak kindly in the evening! When on the walk is heard A tired footstep that you know, Speak one refreshing word, And see the glad light springing From the heart into the eye, As sometimes from behind a cloud A star leaps to the sky. Speak kindly to the children That crowd around your chair, The tender lips that lean on yours Kiss, smooth the flaxen hair; Some day a room that’s lonesome The little ones may own, And home be empty as the nest From which the birds have flown. Speak kindly to the stranger Who passes through the town, A loving word is light of weight— Not so would prove a frown.
- 52. One is a precious jewel The heart would grasp in sleep, The other like a demon’s gift The memory loathes to keep. Speak kindly to the sorrowful Who stand beside the dead, The heart can lean against a word Though thorny seems the bed. And oh, to those discouraged Who faint upon the way, Stop, stop—if just a moment— And something kindly say. Speak kindly to the fallen ones, Your voice may help them rise; A word right-spoken oft unclasps The gate beyond the skies. Speak kindly, and the future You’ll find God looking through! Speak of another as you’d have Him always speak of you.
- 54. Those Willing Hands IN MEMORY OF MISS FANNIE STEVENS. HOSE willing hands—they’re still to-night— The life has from them fled; They’re folded from the longing sight, So cold and pale and dead. The busy veins have idle grown, Like a long famished rill, That once in such an eager tone Called soft from hill to hill. Dear hands, I’ve felt their pressure oft, In a sad time gone by; They moved about the years as soft As clouds move through the sky. They screened the rainstorm from my heart, And let the moonlight in, And showed, while shadows fell athwart, Tracks where the sun had been. They were such willing, willing hands, They stilled the mournful tear, Unwound the pattern of God’s plans, And made his problems clear. They did not reach to high-grown bowers, Where rarest blossoms bloom; But culled the blessed, purer flowers, And bore them to the tomb. Poor hands—they are so still and white, The rose that shared their rest
- 55. Is shrinking from the long, dark night, And falling on her breast. The wreath is wilted on the mound Where long the sunshine stands, But angels have the sleeper found, And clasped those willing hands.
- 57. Look Into the Past. OOK into the past—there are pictures Detaining the sunshine of May, All aquiver with light they turn to the sight, Like a flower that faces the day. How restful the hillsides and shady! The brook like a song passeth by, And the trespassing moon floats about through noon, Like a bubble blown up in the sky. Look into the past! It is happy; Its voices are voices of youth; There is no idle jest to disturb the heart’s rest, And its banners wear mottoes of truth; Look back at the glad, happy faces That walk with our childhood abreast, And show me to-day, though it be miles away, A spot that can offer such rest. Say not that the years long escaping, Show graves of a cankering joy. Because we have found that new pleasures abound, Must we cast off our first childish toy? Because some old love has disturbed us, And filled a lost hour full of gloom, Are we never to go, when the sun lieth low, And stand by the neglected tomb?
- 60. A Little Face. TO “C.” little face to look at, A little face to kiss; Is there anything, I wonder, That’s half so sweet as this? A little cheek to dimple When smiles begin to grow A little mouth betraying Which way the kisses go. A slender little ringlet, A rosy little ear; A little chin to quiver When falls the little tear. A little face to look at, A little face to kiss; Is there anything, I wonder, That’s half so sweet as this? A little hand so fragile All through the night to hold Two little feet so tender To tuck in from the cold. Two eyes to watch the sunbeam That with the shadow plays— A darling little baby To kiss and love always.
- 62. 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