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The Rational Mechanics of Flexible or Elastic Bodies 1638–1788
Euler’s unification of the catenary and the elastica (1728)
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Author(s):
C. Truesdell
Publication date
(Print):
1960
Publisher:
Birkhäuser Basel
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Book Chapter
Publication date (Print):
1960
Pages
: 146-150
DOI:
10.1007/978-3-0348-5015-5_21
SO-VID:
6c74ef00-2981-4d43-87ac-572505fc0aff
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Book chapters
pp. 15
Remarks of the ancients on vibration and elasticity
pp. 18
Western researches before 1600
pp. 24
Beeckman on the suspension bridge (1614–1615), on vibration (1614–1618), and on elasticity (1620–1630)
pp. 28
Mersenne on vibration and rupture (1623–1636)
pp. 34
The vibrating string, the breaking of a beam, and the catenary in Galileo’s Discorsi (1638)
pp. 44
The unpublished work of Huygens on the suspension bridge (1646), the breaking of a beam (1662), the vibrating string (1673), and the vibrating rod (1688)
pp. 50
Pardies’ essays on the catenary and on elastic beams (1673)
pp. 53
Hooke’s law of spring (1675, 1678) and researches on the arch (1675), on ropes (1669), and on sound (1675–1681)
pp. 59
Mariotte and Leibniz on elastic beams (1684)
pp. 64
The contest to find the catenary (1690)
pp. 75
James Bernoulli’s researches on the general theory of flexible lines (1691–1704), and later work to 1717
pp. 88
James Bernoulli’s first researches on the elastica (1691–1694)
pp. 96
James Bernoulli’s attempts toward a theory of the neutral fibre (1695–1705)
pp. 109
Parent’s researches on the neutral fibre (1704–1713)
pp. 114
Researches on theoretical and experimental elasticity by James Riccati (1720–1723) and others
pp. 118
Experiments on the nodes of vibrating bodies by Noble and Pigot (1674), Sauveur (1696–1701), de la Hire (1709), and Zendrini (1715–1716)
pp. 129
Taylor’s analysis of the continuous vibrating string (1713)
pp. 132
John Bernoulli’s analysis of the loaded vibrating string (1727)
pp. 136
Summary: Euler’s heritage
pp. 142
Euler’s derivation of James Bernoulli’s law of bending from Hooke’s law of extension; introduction of the modulus of extension (1727)
pp. 146
Euler’s unification of the catenary and the elastica (1728)
pp. 150
Musschenbroek’s experimental discovery of the law of buckling in compression (1729)
pp. 154
Daniel Bernoulli’s discovery of the simple modes and proper frequencies of vibrating systems (1733), and related work of Euler
pp. 165
Daniel Bernoulli’s and Euler’s first calculations of simple modes and proper frequencies for the transverse vibrations of bars (1734–1735)
pp. 170
Further researches on the elastica; Euler’s general solution for linear vibration problems (1735–1739)
pp. 179
The first differential equations of motion: John Bernoulli’s and d’Alembert’s treatments of the hanging cord (1742–1743)
pp. 192
Daniel Bernoulli’s definitive work on the transverse vibrations of bars (1740–1742)
pp. 199
Euler’s treatise on elastic curves (1743). I. Static deflection
pp. 219
Euler’s treatise on elastic curves (1743). II. Vibrations
pp. 222
Euler’s equations of finite motion for linked systems and for the continuous string (1744)
pp. 229
Euler’s general solution for longitudinal vibration of the loaded elastic cord (1748)
pp. 234
Summary to 1748
pp. 237
D’Alembert’s first memoir (1746): the partial differential equation and its solution by an “equation”
pp. 244
Euler’s first memoir (1748): the solution by arbitrary functions
pp. 250
Euler’s “first principles of mechanics” (1750)
pp. 254
Daniel Bernoulli’s memoirs on the composition of simple modes (1753)
pp. 259
Euler’s second memoir (1754): the central importance of the partial differential equation
pp. 263
Lagrange’s first memoir (1759): the explicit solution for the loaded string, and a faulty passage to the limit
pp. 273
Miscellaneous polemics (1760–1767)
pp. 281
Euler’s researches on the propagation and reflection of waves (1764–1765)
pp. 286
Miscellaneous polemics to 1788
pp. 295
Summary of the theory of the vibrating string to 1788
pp. 301
The earliest researches on non-uniform strings: Simple modes for special cases, and Euler’s general inverse method (1752–1765)
pp. 311
Polemics, errors, and further special cases (1770–1788)
pp. 316
Plane vibrations of a heavy cord hung from one end
pp. 320
Euler’s faulty theories of the vibrations of curved rods (1760, 1774)
pp. 322
Euler’s definitive work on the six kinds of transverse vibrations of straight rods (1772–1774), and its experimental verification by Jordan Riccati (1782) and by Chladni (1787)
pp. 330
Euler’s theory of the vibrating membrane (1759)
pp. 335
Chladni’s experiments on the vibrations of free plates (1787)
pp. 341
Discrete models
pp. 345
Euler’s calculation of buckling loads for columns of non-uniform section (1757)
pp. 347
The distinction of different kinds of buckling by Daniel & John III Bernoulli (1766)
pp. 349
Lagrange’s two memoirs (1770, 1773)
pp. 356
Euler’s determination of the height at which a heavy column buckles (1776–1778).
pp. 371
Lagrange’s variational equations of motion for flexible lines (1761)
pp. 374
Euler’s theory of the skew elastic
pp. 383
Miscellaneous researches (1754–1768)
pp. 390
Euler’s introduction of shear force and derivation of the general equations of equilibrium for a deformable line (1771, 1774)
pp. 396
Coulomb’s introduction of interior shearing stress (1773)
pp. 401
Euler’s formal definition of the modulus of extension and final scaling laws for the frequencies and buckling loads of straight rods (1774, 1776)
pp. 405
Coulomb’s experiments on torsion
pp. 409
An evaluation from the eighteenth century: Lagrange’s Méchanique Analitique (1788)
pp. 412
Some later evaluations: Chladni (1802), T. Young (1807), Love (1892)
pp. 416
A modern evaluation
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