Abstract
In this study, we investigate the importance of the fourth-order time derivative that appears in the equations derived by Jacques Antoine Charles Bresse in 1859, as well as in equations that were derived by Stephen Prokofievich Timoshenko and Paul Ehrenfest during years 1912 and 1913 and reported by Timoshenko in the 1916 book on the theory of elasticity in the Russian language and then in two papers dated 1920 and 1921, in English. The first part of the study demonstrates that Timoshenko and Ehrenfest did not overestimate the importance of the fourth-order derivative term in their equations. The second part deals with the debate on the so-called second spectrum attendant in the original set of equations. It is shown that in the truncated Timoshenko—Ehrenfest equations—which is asymptotically consistent with elasticity theory—“the second spectrum” issue does not arise. Thus, the two parts of this study are intricately interrelated with each other.
References
1.
Todhunter
, I.
, and Pearson
, K.
1886 and 1893
, A History of the Theory of Elasticity and of the Strength of Materials
, Vols. I and II
, Cambridge University Press
, Cambridge
, UK (see also New York: Dover, 1960).2.
Bresse
, J. A. C.
, 1859
, Cours de Mécanique Appliquée—Résistance des Matériaux et Stabilité des Constructions
, Gauthier-Villars
, Paris
(in French).3.
Rayleigh
, J. W.
, and Strutt
, B.
, 1877
, The Theory of Sound
, Vol. 1, Macmillan and Co.
, London
.4.
Timoshenko
, S. P.
, 1972
, A Course of Elasticity Theory. Part 2: Rods and Plates
, 2nd ed., A.E. Collins Publishers
, St. Petersburg
, 1916 (in Russian), Kiev: “Naukova Dumka” Publishers, 341, pp. 337
–338
.5.
Timoshenko
, S. P.
, 1920
, “On the Differential Equation for the Flexural Vibrations of Prismatic Rods
,” Glas. Hrvatskoga Prirod. Drustva
, 32
, pp. 55
–57
.6.
Timoshenko
, S. P.
, 1921
, “On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bar
,” Philos. Mag. Ser.
, 6
(245
), pp. 744
–746
. 7.
Timoshenko
S. P.
, 1922
, On the Transverse of Bars of Uniform Cross Sections
, Philosophical Magazine, Series 6, Vol. 43
, 125
–131
, The Collected Papers, 288–290, McGraw-Hill, New York, 1953).8.
Timoshenko
, S. P.
, 1928
, Vibration Problems in Engineering
, Constable and Company
, London
, p. 231
.9.
Timoshenko
, S. P.
, 1937
, Vibration Problems in Engineering
, 2nd ed., Van Nostrand Reinhold Company
, New York
.10.
Timoshenko
, S. P.
, and Young
, D. H.
, 1955
, Vibration Problems in Engineering
, 3rd ed., Van Nostrand Reinhold Company
, New York
.11.
Timoshenko
, S. P.
, Young
, D. H.
, and Weaver
, W.
, Jr., 1974
, Vibration Problems in Engineering
, 4th ed., Wiley
, New York
.12.
Weaver
, W.
, Jr., Timoshenko
, S. P.
, and Young
, D. M.
, 1990
, Vibration Problems in Engineering
, Wiley
, New York
.13.
Wang
, C. M.
, Reddy
, J. N.
, and Lee
, K. H.
, 2000
, Shear Deformable Beams and Plates: Relationship With Classical Solutions
, Elsevier
, Oxford, UK
.14.
Elishakoff
, I.
, 2020
, Handbook on Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories
, World Scientific
, Singapore
.15.
Shi
, G.
, and Guo
, Q.
, 2021
, “On the Appropriate Rotary Inertia in Timoshenko Beam Theory
,” Int. J. Appl. Mech. Eng.
, 13
(04
), p. 2150055
. 16.
Fung
, Y. C.
, and Tong
, P.
, 2001
, Classical and Computational Solid Mechanics
, World Scientific
, River Ridge, NJ
.17.
Stephen
, N. G.
, 1982
, “The Second Frequency Spectrum of Timoshenko Beams
,” J. Sound Vib.
, 80
(4
), pp. 578
–582
. 18.
Stephen
, N. G.
, 2006
, “The Second Spectrum of Timoshenko Beam Theory—Further Assessment
,” J. Sound Vib.
, 292
(1–2
), pp. 372
–389
. 19.
Stephen
, N. G.
, and Puchegger
, S.
, 2006
, “On the Valid Frequency Range of Timoshenko Beam Theory
,” J. Sound Vib.
, 297
(3–5
), pp. 1082
–1087
. 20.
Elishakoff
, I.
, Kaplunov
, J.
, and Nolde
, E.
, 2015
, “Celebrating the Centenary of Timoshenko’s Study of Effects of Shear Deformation and Rotary Inertia
,” Appl. Mech. Rev.
, 67
(6
), p. 060802
. 21.
Díaz-de-Anda
, A.
, Flores
, J.
, Gutiérrez
, L.
, Méndez-Sánchez
, R. A.
, Monsivais
, G.
, and Morales
, A.
, 2012
, “Experimental Study of the Timoshenko Beam Theory Predictions
,” J. Sound Vib.
, 331
(26
), pp. 5732
–5744
. 22.
Patra
, A. K.
, Gopalakrishnan
, S.
, and Ganguli
, R.
, 2020
, “Existence of Second Spectrums of Timoshenko Beam and Mindlin–Herrmann Rod Theories on the Basis of Atomistic Studies
,” Acta Mech.
, 231
, pp. 1159
–1171
. 23.
De Rosa
, M. A.
, Lippiello
, M.
, and Elishakoff
, I.
, 2022
, “Variational Derivation of Truncated Timoshenko-Ehrenfest Beam Theory
,” J. Appl. Comput. Mech.
, 8
(3
), pp. 996
–1004
.24.
Challamel
, N.
, private communication, 20 April 2022.25.
Elishakoff
, I.
, and Lubliner
, E.
, 1985
, “Random Vibration of a Structure via Classical and Nonclassical Theories,” Probabilistic Methods in the Mechanics of Solids and Structures
, S.
Eggwertz
, and N. C.
Lind
, eds., Springer
, Berlin
, pp. 455
–467
.26.
Cazzani
, A.
, Stochino
, F.
, and Turco
, E.
, 2016
, “On the Whole Spectrum of Timoshenko Beams, Part 1: A Theoretical Revisitation
,” Z. Angew. Math. Phys.
, 67
(2
), p. 24
. 27.
Goens
, E.
, 1931
, “Über die Bestimmung des Elastizitätsmodulus von Stäben mit Hifle von Biegungsschwingungen
,” Ann. Phys.
, 11
(6
), pp. 649
–678
. (in German). 28.
Trail-Nash
, R. W.
, and Collar
, A. R.
, 1953
, “The Effect of Shear Flexibility and Rotary Inertia on the Bending Vibrations of Beams
,” Q. J. Mech. Appl. Math.
, 6
(2
), pp. 186
–222
. 29.
Abbas
, B. A. H.
, and Thomas
, J.
, 1977
, “The Second Frequency Spectrum of Timoshenko Beams
,” J. Sound Vib.
, 51
(1
), pp. 123
–137
. 30.
Bhashyam
, G. R.
, and Prathap
, G.
, 1981
, “The Second Frequency Spectrum of Timoshenko Beams
,” J. Sound Vib.
, 76
(3
), pp. 407
–420
. 31.
Hathout
, I. A.
, Leipholz
, H.
, and Singha
, F. K.
, 1980
, “Sensitivity of the Frequencies of Damped Timoshenko Beam
,” J. Eng. Sci. Univ. Riyadh
, 6
(2
), pp. 113
–121
.32.
Levinson
, M.
, and Cooke
, D. W.
, 1981
, “On the Two Frequency Spectra of Timoshenko Beams
,” J. Sound Vib.
, 84
(3
), pp. 319
–326
. 33.
Bhaskar
, A.
, 2009
, “Elastic Waves in Timoshenko Beams: The ‘Lost and Found’ of an Eigenmode
,” Proc. R. Soc. London, Ser. A
, 465
(2101
), pp. 239
–255
. 34.
Nesterenko
, V. V.
, 1993
, “A Theory for Transverse Vibrations of the Timoshenko Beam
,” PMM. J. Appl. Math. Mech.
, 57
(4
), pp. 669
–677
. 35.
Chervyakov
, A. M.
, and Nesterenko
, V. V.
, 1993
, “Is It Possible to Assign Physical Meaning to Field Theory With Higher Derivatives?
,” Phys, Rev. D
, 48
(12
), pp. 5811
–5817
. 36.
Postnov
, V. A.
, Kalinin
, V. S.
, and Rostovtsev
, D. M.
, 1983
, Ship Vibrations
, Sudostroenie Publishing House
, Leningrad
(in Russian).37.
Postnov
, V. A.
, and Fedorov
, A. S.
, 1963
, “On the Second Frequency Spectrum in the General Vibration of Ship’s Hull
,” Coll. Pap. Sudprom Sci. Tech. Soc.
, 54
, pp. 60
–77
. (in Russian).38.
Kumai
, T.
, 1959
, “On a Second Spectrum of the Natural Frequency of the Flexural Vibration of Ship’s Hull
,” Eur. Shipbuild.
, (7
).39.
Manevich
, A. I.
, 2015
, “Dynamics of Timoshenko Beam on Linear and Nonlinear Foundation: Phase Relations, Significance of the Second Spectrum, Stability
,” J. Sound Vib.
, 344
, pp. 209
–220
. 40.
van Rensburg
, N. F. J.
, and van der Merve
, A. J.
, 2006
, “Natural Frequencies and Modes of a Timoshenko Beam
,” Wave Motion
, 44
(1
), pp. 58
–69
. 41.
Pilkey
, W.
, 2005
, Formulas for Stress, Strain and Structural
, Wiley
, New York
.42.
Gopalakrishnan
, S.
, 2016
, Wave Propagation in Materials and Structures
, CRC Press
, Boca Raton, FL
.43.
Miklowitz
, J.
, 1978
, The Theory of Elastic Waves and Waveguides
, North Holland Publishing Company
, Amsterdam
.44.
Almeida Júnior
, D. S.
, and Ramos
, A.
, 2017
, “On the Nature of Dissipative Timoshenko Systems at Light of the Second Spectrum of Frequency
,” Z. Angew. Math. Phys.
, 68
(6
), pp. 1
–31
. 45.
Barr
, A. D. S.
, 1956
, “Some Notes on the Resonance of Timoshenko Beams and the Effects of Lateral Inertia on Flexural Vibration
,” Proceedings of the 9th International Congress of Applied Mechanics
, Brussels, Belgium
, Sept. 5–13
, Vol. 7, pp. 448
–458
.46.
Geist
, B.
, and McLaughlin
, J. R.
, 1997
, “Double Eigenvalues for the Uniform Timoshenko Beam
,” Appl. Math. Lett.
, 10
(3
), pp. 129
–134
. 47.
O’Reilly
, O. M.
, and Turcotte
, J. C.
, 1996
, “Another Mode of Vibration in a Timoshenko Beam
,” J. Sound Vib.
, 198
(4
), pp. 517
–521
. 48.
Chan
, K. T.
, Wang
, X. Q.
, So
, R. M.
, and Reid
, S. R.
, 2002
, “Superposed Standing Waves on a Timoshenko Beam
,” Proc. R. Soc. London, Ser. A
, 458
, pp. 83
–108
. 49.
Monsivais
, G.
, Díaz-de-Anda
, A.
, Flores
, J.
, Gutiérrez
, L.
, and Morales
, A.
, 2016
, “Experimental Study of the Timoshenko Beam Theory Predictions: Further Results
,” J. Sound Vib.
, 375
, pp. 187
–199
. 50.
da Costa Azevêdo
, A. S.
, Vasconcelos
, A. C.
, and dos Santos Hoefel
, S.
, 2016
, “Dynamic Analysis of Elastically Supported Timoshenko Beam
,” Revista Interdisciplinar de Pesquisa em Engenharia-RIPE
, 2
(34
), pp. 71
–85
.51.
da Silva Bezerra
, W. K.
, Soares
, L. S.
, and dos Santos Hoefel
, S.
, 2017
, Second Spectrum of Timoshenko Beam on Pasternak Foundation
, CILAMCE, Florianopolis, Santa Catarina, Brazil.52.
Gul
, U.
, and Aydogdu
, M.
, 2018
, “Wave Propagation Analysis in Beams Using Shear Deformable Beam Theories Considering Second Spectrum
,” J. Mech.
, 34
(3
), pp. 279
–289
. 53.
Gul
, U.
, Aydogdu
, M.
, and Karacam
, F.
, 2019
, “Dynamics of a Functionally Graded Timoshenko Beam Considering New Spectrums
,” Compos. Struct.
, 207
, pp. 273
–291
. 54.
Anderson
, R. A.
, 1953
, “Flexural Vibrations in Uniform Beams According to the Timoshenko Theory
,” ASME J. Appl. Mech.
, 20
(4
), pp. 504
–510
. 55.
Dolph
, C. L.
, 1954
, “On the Timoshenko Theory of Transverse Beam Vibrations
,” Q. Appl. Math.
, 12
(2
), pp. 175
–187
. 56.
Huang
, T. C.
, 1961
, “The Effect of Rotatory Inertia and of Shear Deformations on the Frequency and Normal Mode Equation Beams With Simple End Conditions
,” ASME J. Appl. Mech.
, 28
(4
), pp. 579
–584
. 57.
Young
, D. H.
, 1962
, “Continuous Systems,” Handbook of Engineering Mechanics
, 4th ed., W.
Flügge
, ed., McGraw Hill
, New York
, pp. 61.14
–61.18
.58.
Downs
, B.
, 1976
, “Transverse Vibration of a Uniform, Simply Supported Timoshenko Beam Without Transverse Deflection
,” ASME J. Appl. Mech.
, 43
(4
), pp. 671
–674
. 59.
Amato
, M.
, Elishakoff
, I.
, and Reddy
, J. N.
, 2021
, “Flutter of a Multicomponent Beam in a Supersonic Flow
,” AIAA J.
, 59
(11
), pp. 4342
–4353
. 60.
Ramos
, A. J.
, Aouadi
, M.
, Almeida Júnior
, D. S.
, Freitas
, M. M.
, and Araújo
, M. L.
, 2021
, “A New Stabilization Scenario for Timoshenko Systems With Thermo-diffusion Effects in Second Spectrum Perspective
,” Arch. Math.
, 116
(2
), pp. 203
–219
. 61.
Kaplunov
, J.
, 2022
, private communication, April 7.62.
Almeida Júnior
, D. S.
, Ramos
, A. J. A.
, Santos
, M. L.
, and Gutemberg
, L. R. M.
, 2018
, “Asymptotic Behavior of Weakly Dissipative Bresse-Timoshenko System on Influence of the Second Spectrum of Frequency
,” Z. fur Angew. Math. Mech.
, 98
(8
), pp. 1320
–1333
. 63.
Almeida Júnior
, D. S.
, Ramos
, A. J. A.
, Soufyane
, A.
, Freitas
, M. M.
, and Santos
, M. L.
, 2022
, “Impact of the Damaging Consequences of the Second Spectrum on the Stabilization of Nonlinear Timoshenko Systems
,” Acta Appl. Math.
, 180
(1
), pp. 1
–23
. 64.
Bhat
, K. S.
, Sarkar
, K.
, Ganguli
, R.
, and Elishakoff
, I.
, 2018
, “Slope-Inertia Model of Non-uniform and Inhomogeneous Bresse-Timoshenko Beams
,” AIAA J.
, 56
(10
), pp. 4158
–4168
. 65.
Khasawneh
, F. A.
, and Segalman
, D.
, 2019
, “Exact and Numerically Stable Expressions for Euler-Bernoulli and Timoshenko Beam Modes
,” Appl. Acoust.
, 151
, pp. 215
–228
. 66.
Xia
, G.
, Shu
, W.
, and Stanciulescu
, I.
, 2020
, “Analytical and Numerical Studies on the Slope Inertia-Based Timoshenko Beam
,” J. Sound Vib.
, 473
, p. 115227
. 67.
Lei
, T.
, Zheng
, Y.
, Yu
, R.
, Yan
, Y.
, and Xu
, B.
, 2022
, “Dynamic Response of Slope Inertia-Based Timoshenko Beam Under a Moving Load
,” Appl. Sci.
, 12
(6
), p. 3045
. 68.
Elishakoff
, I.
, and Lubliner
, E.
, 1988
, “Random Vibrations of Structures With Complicating Effects,” Stochastic Structural Dynamics–Progress: Theory and Applications
, S. T.
Ariaratnam
, G. I.
Schuëller
, and I.
Elishakoff
, eds., Elsevier, London
, London
, pp. 47
–64
.69.
Almeida Júnior
, D. S.
, Elishakoff
, I.
, Ramos
, A. J. A.
, and Rosário Miranda
, L. G.
, 2019
, “The Hypothesis of Equal Wave Speeds for Stabilization of Timoshenko Beam is Not Necessary Anymore: The Time Delay Cases
,” IMA J. Appl. Math.
, 84
(4
), pp. 763
–796
. 70.
Carrera
, E.
, Elishakoff
, I.
, and Petrolo
, M.
, 2021
, “Who Needs Refined Structural Theories?
,” Compos. Struct.
, 264
, p. 113671
. 71.
Challamel
, N.
, and Elishakoff
, I.
, 2019
, “A Brief History of First-Order Shear-Deformable Beam and Plate Models
,” Mech. Res. Commun.
, 102
, p. 103389
. 72.
Elishakoff
, I.
, Hache
, F.
, and Challamel
, N.
, 2018
, “Variational Derivation of Governing Differential Equations for Truncated Version of Bresse-Timoshenko Beams
,” J. Sound Vib.
, 435
, pp. 409
–430
. 73.
Elishakoff
, I.
, Li
, Y.
, Challamel
, N.
, and Reddy
, J. N.
, 2022
, “Simplified Timoshenko–Ehrenfest Beam Equation to Analyze Metamaterials
,” J. Appl. Phys.
, 131
(10
), p. 104902
. 74.
Erofeev
, V. I.
, Kazhaev
, V. V.
, Lisenkova Ye
, Y.
, and Semerikova
, N. P.
, 2009
, “Comparative Analysis of Dynamical Behavior of Beams in Bernoulli-Euler, Rayleigh, and Timoshenko Models Lying on Elastic Foundation
,” Vestnik Nauchno-Tekhnicheskogo Razvitiya, 8
(24
), pp. 18
–26
. (in Russian).75.
Faghidian
, S. A.
, Żur
, K. K.
, Reddy
, J. N.
, and Ferreira
, A. J. M.
, 2021
, “On the Wave Dispersion in Functionally Graded Porous Timoshenko-Ehrenfest Nanobeams Based on the Higher-Order Nonlocal Gradient Elasticity
,” Compos. Struct.
, 279
, p. 114819
.76.
Han
, S. M.
, Benaroya
, H.
, and Wei
, T.
, 1999
, “Dynamics of Transversely Vibrating Beams Using Four Engineering Theories
,” J. Sound Vib.
, 225
(5
), pp. 935
–988
. 77.
Méndez-Sánchez
, R. A.
, Morales
, A.
, and Flores
, J.
, 2005
, “Experimental Check in the Accuracy of Timoshenko’s Beam Theory
,” J. Sound Vib.
, 279
(1–2
), pp. 508
–512
. 78.
Nesterenko
, V. V.
, and Chervyakov
, A. M.
, 1994
, “Parabolic Approximation to the Theory of Transverse Vibrations of Rods and Beams
,” J. Appl. Mech. Tech. Phys.
, 35
(2
), pp. 151
–154
. (in Russian). 79.
Shi
, G.
, and Voyiadjis
, G. Z.
, 2011
, “A Sixth-Order Theory of Shear Deformable Beams With Variational Consistent Boundary Conditions
,” Appl. Acoust.
, 78
(2
), p. 021019
.80.
Timoshenko
, S. P.
, 1953
, Collected Papers
, McGraw-Hill
, New York
.81.
Timoshenko
, S. P.
, 1963
, Remembrances
, Boulogne-sur-Seine
, Paris
(in Russian).82.
Timoshenko
, S. P.
, 1968
, As I Remember: The Autobiography of Stephen P. Timoshenko
, Van Nostrand
, Princeton
.83.
Timoshenko
, S. P.
, 1972
, A Course in Theory of Elasticity
, 2nd ed., Naukova Dumka Publishers
, Kiev
(in Russian).84.
Timoshenko
, S. P.
, 1993
, Vospominaniya (Remembrances)
, Naukova Dumka Publishers
, Kiev
(in Russian).85.
Timoshenko
, S. P.
2006
, Errinerungen—eine Autobiographie
, Ernst und Sohn, Berlin
(in German).86.
Timoshenko
, S. P.
, 2014
, Vospominaniya (Remembrances)
, Vyzovskaya Kniga Publishers
, Moscow
(in Russian).Copyright © 2022 by ASME
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