A detailed analysis of pendular motion is presented. Inertial effects, self-oscillation, and memory, together with non-constant moment of inertia, hysteresis, and negative damping are shown to be required for the comprehensive description of the free pendulum oscillatory regime. The effects of very high initial amplitudes, friction in the roller bearing axle, drag, and pendulum geometry are also analyzed and discussed. A model consisting of a fractional differential equation fits and explains high resolution and long-time experimental data gathered from standard action-camera videos.
REFERENCES
1.
Herman
Erlichson
, “
Galileo's pendulum
,” Phys. Teach.
37
, 478
–479
(1999
).2.
Christiaan
Huygens
, Horologium oscillatorium: sive de motu pendulorum ad horologia aptato demonstrationes geometricæ
(
F. Muguet
,
Paris
, 1966
).3.
Henry
Kater
and
Thomas
Young
, “
IV. An account of experiments for determining the length of the pendulum vibrating seconds in the latitude of London
,” Philos. Trans. R. Soc. London
108
, 33
–102
(1818
).4.
J. E.
Jackson
, “
The Cambridge pendulum apparatus
,” Geophys. J. Int.
4
, 375
–388
(1961
).5.
Iginio
Marson
and
Umberto
Riccardi
, “
A short walk along the gravimeters path
,” Int. J. Geophys.
2012
, 687813
(2012
).6.
D. G.
Blair
,
L.
Ju
, and
M.
Notcutt
, “
Ultrahigh Q pendulum suspensions for gravitational wave detectors
,” Rev. Sci. Instrum.
64
, 1899
–1904
(1993
).7.
V. P.
Mitrofanov
and
N. A.
Styazhkina
, “
Influence of surface adsorbed water on the pendulum damping in an external electric field
,” Phys. Lett. A
256
, 351
–355
(1999
).8.
T.
Uchiyama
,
T.
Tomaru
,
D.
Tatsumi
,
S.
Miyoki
,
M.
Ohashi
,
K.
Kuroda
,
T.
Suzuki
,
A.
Yamamoto
, and
T.
Shintomi
, “
Mechanical quality factor of a sapphire fiber at cryogenic temperatures
,” Phys. Lett. A
273
, 310
–315
(2000
).9.
G.
Cagnoli
,
L.
Gammaitoni
,
J.
Hough
,
J.
Kovalik
,
S.
McIntosh
,
M.
Punturo
, and
S.
Rowan
, “
Very high Q measurements on a fused silica monolithic pendulum for use in enhanced gravity wave detectors
,” Phys. Rev. Lett.
85
, 2442
–2445
(2000
).10.
F. M. S.
Lima
, “
Simple ‘log formulae’ for pendulum motion valid for any amplitude
,” Eur. J. Phys.
29
, 1091
–1098
(2008
).11.
Juan R.
Sanmartn
, “
O Botafumeiro: Parametric pumping in the middle ages
,” Am. J. Phys.
52
, 937
–945
(1984
).12.
Stephen
Wirkus
,
Richard
Rand
, and
Andy
Ruina
, “
How to pump a swing
,” Coll. Math. J.
29
, 266
–275
(1998
).13.
Denise S. D.
Stilling
and
Walerian
Szyszkowski
, “
Controlling angular oscillations through mass reconfiguration: A variable length pendulum case
,” Int. J. Non-Linear Mech.
37
, 89
–99
(2002
).14.
Auke A.
Post
,
Gert
de Groot
,
Andreas
Daffertshofer
, and
Peter J.
Beek
, “
Pumping a playground swing
,” Motor Control
11
, 136
–150
(2007
).15.
J. A.
Greenwood
,
K. L.
Johnson
,
S.-H.
Choi
, and
M. K.
Chaudhury
, “
Investigation of adhesion hysteresis between rubber and glass using a pendulum
,” J. Phys. D
42
, 035301
(2008
).16.
Martin
Obligado
,
Martin
Puy
, and
Mickael
Bourgoin
, “
Bi-stability of a pendular disk in laminar and turbulent flows
,” J. Fluid Mech.
728
, R2-1
–R2-11
(2013
).17.
R.
Cuerno
,
A. F.
Raada
, and
J. J.
Ruiz-Lorenzo
, “
Deterministic chaos in the elastic pendulum: A simple laboratory for nonlinear dynamics
,” Am. J. Phys.
60
, 73
–79
(1992
).18.
Robert
DeSerio
, “
Chaotic pendulum: The complete attractor
,” Am. J. Phys.
71
, 250
–257
(2003
).19.
M. Y.
Azbel
and
Per
Bak
, “
Analytical results on the periodically driven damped pendulum. Application to sliding charge-density waves and Josephson junctions
,” Phys. Rev. B
30
, 3722
–3727
(1984
).20.
Filipe J.
Romeiras
and
Edward
Ott
, “
Strange nonchaotic attractors of the damped pendulum with quasiperiodic forcing
,” Phys. Rev. A
35
, 4404
–4413
(1987
).21.
G. J.
Milburn
and
D. F.
Walls
, “
Quantum solutions of the damped harmonic oscillator
,” Am. J. Phys.
51
, 1134
–1136
(1983
).22.
Bernard
Yurke
, “
Quantizing the damped harmonic oscillator
,” Am. J. Phys.
54
, 1133
–1139
(1986
).23.
Danil
Doubochinski
and
Jonathan
Tennenbaum
, “
The macroscopic quantum effect in nonlinear oscillating systems: A possible bridge between classical and quantum physics
,” arXiv:0711.4892 (2007
).24.
A. I.
Shumaev
and
Z. A.
Maizelis
, “
Distribution functions of argumental oscillations of the Duboshinskiy pendulum
,” J. Appl. Phys.
121
, 154902
(2017
).25.
Marine
Pigneur
and
Jörg
Schmiedmayer
, “
Analytical pendulum model for a bosonic Josephson junction
,” Phys. Rev. A
98
, 063632
(2018
).26.
J.
Naudts
, “
Boltzmann entropy and the microcanonical ensemble
,” Europhys. Lett.
69
, 719
–724
(2005
).27.
Maarten
Baeten
and
Jan
Naudts
, “
On the thermodynamics of classical micro-canonical systems
,” Entropy
13
, 1186
–1199
(2011
).28.
A. A.
Lukichev
, “
Relaxation function for the non-Debye relaxation spectra description
,” Chem. Phys.
428
, 29
–33
(2014
).29.
A. A.
Lukichev
, “
Simple method for the dielectric relaxation function investigation
,” J. Non-Cryst. Solids
420
, 43
–47
(2015
).30.
A. A.
Lukichev
, “
Nonlinear relaxation functions. Physical meaning of the Jonscher's power law
,” J. Non-Cryst. Solids
442
, 17
–21
(2016
).31.
Alexander
Lukichev
, “
Physical meaning of the stretched exponential Kohlrausch function
,” Phys. Lett. A
383
, 2983
–2987
(2019
).32.
Michael R.
Matthews
, “
Pendulum motion: A case study in how history and philosophy can contribute to science education
,” in International Handbook of Research in History, Philosophy and Science Teaching
, edited by
Michael R.
Matthews
(
Springer Netherlands
,
Dordrecht
, 2014
), pp. 19
–56
.33.
Alexander B.
Rabinovich
, “
Seiches and harbor oscillations
,” Handbook of Coastal and Ocean Engineering
(
World Scientific
,
Singapore
, 2009
), pp. 193
–236
.34.
J. W.
Fee
, “
The apparent thixotropic properties of limb motion in totally relaxed subjects
,” in Proceedings of the IEEE 28th Annual Northeast Bioengineering Conference
(IEEE Cat. No.02CH37342) (2002
), pp. 33
–34
.35.
36.
João C.
Fernandes
,
Pedro J.
Sebastião
,
Luís N.
Gonçalves
, and
António
Ferraz
, “
Study of large-angle anharmonic oscillations of a physical pendulum using an acceleration sensor
,” Eur. J. Phys.
38
, 045004
(2017
).37.
Mobin
Kavyanpoor
and
Saeed
Shokrollahi
, “
Dynamic behaviors of a fractional order nonlinear oscillator
,” J. King Saud Univ.-Sci.
31
, 14
–20
(2019
).38.
Marco
Amabili
, “
Derivation of nonlinear damping from viscoelasticity in case of nonlinear vibrations
,” Nonlinear Dyn.
97
, 1785
–1797
(2019
).39.
Alejandro
Jenkins
, “
Self-oscillation
,” Phys. Rep.
525
, 167
–222
(2013
).40.
L. P.
Fulcher
and
B. F.
Davis
, “
Theoretical and experimental study of the motion of the simple pendulum
,” Am. J. Phys.
44
, 51
–55
(1976
).41.
Donald E.
Hall
and
Michael J.
Shea
, “
Large-amplitude pendulum experiment: Another approach
,” Am. J. Phys.
45
, 355
–357
(1977
).42.
S. C.
Zilio
, “
Measurement and analysis of large-angle pendulum motion
,” Am. J. Phys.
50
, 450
–452
(1982
).43.
Salvador
Gil
,
Andrs E.
Legarreta
, and
Daniel E.
Di Gregorio
, “
Measuring anharmonicity in a large amplitude pendulum
,” Am. J. Phys.
76
, 843
–847
(2008
).44.
Patrick T.
Squire
, “
Pendulum damping
,” Am. J. Phys.
54
, 984
–991
(1986
).45.
Lorenzo
Basano
and
Pasquale
Ottonello
, “
Digital pendulum damping: The single-oscillation approach
,” Am. J. Phys.
59
, 1018
–1023
(1991
).46.
L. F. C.
Zonetti
,
A. S. S.
Camargo
,
J.
Sartori
,
D. F.
de Sousa
, and
L. A. O.
Nunes
, “
A demonstration of dry and viscous damping of an oscillating pendulum
,” Eur. J. Phys.
20
, 85
–88
(1999
).47.
Xiao Jun
Wang
,
Chris
Schmitt
, and
Marvin
Payne
, “
Oscillations with three damping effects
,” Eur. J. Phys.
23
, 155
–164
(2002
).48.
M. E.
Bacon
and
Do
Dai Nguyen
, “
Real-world damping of a physical pendulum
,” Eur. J. Phys.
26
, 651
–655
(2005
).49.
John C.
Simbach
and
Joseph
Priest
, “
Another look at a damped physical pendulum
,” Am. J. Phys.
73
, 1079
–1080
(2005
).50.
B. R.
Smith
, “
The quadratically damped oscillator: A case study of a non-linear equation of motion
,” Am. J. Phys.
80
, 816
–824
(2012
).51.
Carl E.
Mungan
and
Trevor C.
Lipscombe
, “
Oscillations of a quadratically damped pendulum
,” Eur. J. Phys.
34
, 1243
–1253
(2013
).52.
R.
Hauko
and
R.
Repnik
, “
Damped harmonic oscillation: Linear or quadratic drag force?
,” Am. J. Phys.
87
, 910
–914
(2019
).53.
Varghese
Mathai
,
Laura A. W. M.
Loeffen
,
Timothy T. K.
Chan
, and
Sander
Wildeman
, “
Dynamics of heavy and buoyant underwater pendulums
,” J. Fluid Mech.
862
, 348
–363
(2019
).54.
Y. H.
Eng
,
W. S.
Lau
,
E.
Low
, and
G. G. L.
Seet
, “
Identification of the hydrodynamics coefficients of an underwater vehicle using free decay pendulum motion
,” in Proceedings of the International MultiConference of Engineers and Computer Scientists
(2008
), Vol.
2
, pp. 423
–430
.55.
Diogo
Bolster
,
Robert E.
Hershberger
, and
Russell J.
Donnelly
, “
Oscillating pendulum decay by emission of vortex rings
,” Phys. Rev. E
81
, 046317
(2010
).56.
S.
Whineray
, “
A cube-law air track oscillator
,” Eur. J. Phys.
12
, 90
–95
(1991
).57.
Peter F.
Hinrichsen
and
Chris I.
Larnder
, “
Combined viscous and dry friction damping of oscillatory motion
,” Am. J. Phys.
86
, 577
–584
(2018
).58.
Élise
Lorenceau
,
David
Quéré
,
Jean-Yves
Ollitrault
, and
Christophe
Clanet
, “
Gravitational oscillations of a liquid column in a pipe
,” Phys. Fluids
14
, 1985
–1992
(2002
).59.
Ryan P.
Smith
and
Eric H.
Matlis
, “
Gravity-driven fluid oscillations in a drinking straw
,” Am. J. Phys.
87
, 433
–435
(2019
).60.
L.
Basano
and
P.
Ottonello
, “
The air drag on an accelerating disk: A laboratory experiment
,” Am. J. Phys.
57
, 999
–1004
(1989
).61.
J.
Alho
,
H.
Silva
,
V.
Teodoro
, and
G.
Bonfait
, “
A simple pendulum studied with a low-cost wireless acquisition board
,” Phys. Educ.
54
, 015015
(2018
).62.
Christopher Isaac
Larnder
, “
Acceleration discontinuities in dry-friction oscillations
,” Am. J. Phys.
87
, 784
–784
(2019
).63.
B. P.
Mann
and
F. A.
Khasawneh
, “
An energy-balance approach for oscillator parameter identification
,” J. Sound Vib.
321
, 65
–78
(2009
).64.
Nikola
Jakšić
, “
Power law damping parameter identification
,” J. Sound Vib.
330
, 5878
–5893
(2011
).65.
The color pink was chosen because it produced the best contrast.
66.
67.
William H.
Press
and
Saul A.
Teukolsky
, “
Savitzky-Golay smoothing filters
,” Comput. Phys.
4
, 669
–672
(1990
).68.
E. V.
Ermanyuk
, “
The use of impulse response functions for evaluation of added mass and damping coefficient of a circular cylinder oscillating in linearly stratified fluid
,” Exp. Fluids
28
, 152
–159
(2000
).69.
Douglas
Neill
,
Dean
Livelybrooks
, and
Russell J.
Donnelly
, “
A pendulum experiment on added mass and the principle of equivalence
,” Am. J. Phys.
75
, 226
–229
(2007
).70.
J.
Messer
and
J.
Pantaleone
, “
The effective mass of a ball in the air
,” Phys. Teach.
48
, 52
–54
(2010
).71.
J.
Pantaleone
and
J.
Messer
, “
The added mass of a spherical projectile
,” Am. J. Phys.
79
, 1202
–1210
(2011
).72.
N.
Raza
,
I.
Mehmood
,
H.
Rafiuddin
, and
M.
Rafique
, “
Numerical simulation of added mass determination of standard ellipsoids
,” in Proceedings of 2012 9th International Bhurban Conference on Applied Sciences Technology (IBCAST)
(2012
), pp. 270
–273
.73.
Efstathios
Konstantinidis
, “
Added mass of a circular cylinder oscillating in a free stream
,” Proc. R. Soc. A
469
, 20130135
(2013
).74.
Randall
Peters
, “
Friction at the mesoscale
,” Contemp. Phys.
45
, 475
–490
(2004
).75.
Claude M.
Penchina
, “
Pseudowork-energy principle
,” Am. J. Phys.
46
, 295
–296
(1978
).76.
Jack
Copeland
, “
Work-energy theorem for variable mass systems
,” Am. J. Phys.
50
, 599
–601
(1982
).77.
Bruce Arne
Sherwood
, “
Pseudowork and real work
,” Am. J. Phys.
51
, 597
–602
(1983
).78.
Arnold B.
Arons
, “
Development of energy concepts in introductory physics courses
,” Am. J. Phys.
67
, 1063
–1067
(1999
).79.
J.
Gémez
and
M.
Fiolhais
, “
From mechanics to thermodynamics—analysis of selected examples
,” Eur. J. Phys.
34
, 345
–357
(2013
).80.
J.
Gémez
and
M.
Fiolhais
, “
Dissipation effects in mechanics and thermodynamics
,” Eur. J. Phys.
37
, 045101
(2016
).81.
J.
Gémez
and
M.
Fiolhais
, “
Principles of time evolution in classical physics
,” Eur. J. Phys.
39
, 045010
(2018
).82.
F. M. S.
Lima
, “
Analytical study of the critical behavior of the nonlinear pendulum
,” Am. J. Phys.
78
, 1146
–1151
(2010
).83.
Robert A.
Nelson
and
M. G.
Olsson
, “
The pendulum—Rich physics from a simple system
,” Am. J. Phys.
54
, 112
–121
(1986
).84.
Ken
Takahashi
and
D.
Thompson
, “
Measuring air resistance in a computerized laboratory
,” Am. J. Phys.
67
, 709
–711
(1999
).85.
Akhil
Arora
,
Rahul
Rawat
,
Sampreet
Kaur
, and
P.
Arun
, “
Study of the damped pendulum
,” arXiv:physics/0608071 (2006
).86.
Junke
Guo
, “
Motion of spheres falling through fluids
,” J. Hydraul. Res.
49
, 32
–41
(2011
).87.
Silvio R.
Dahmen
, “
The Mathematics and Physics of Diderot. I. On pendulums and air resistance
,” arXiv:1409.7446 (2014
).88.
Pascal
Klein
,
Andreas
Müller
,
Sebastian
Gröber
,
Alexander
Molz
, and
Jochen
Kuhn
, “
Rotational and frictional dynamics of the slamming of a door
,” Am. J. Phys.
85
, 30
–37
(2017
).89.
John R.
Graef
, “
On the generalized Liénard equation with negative damping
,” J. Differ. Equations
12
, 34
–62
(1972
).90.
Lewis P.
Fulcher
,
Ronald C.
Scherer
,
Artem
Melnykov
,
Vesela
Gateva
, and
Mark E.
Limes
, “
Negative Coulomb damping, limit cycles, and self-oscillation of the vocal folds
,” Am. J. Phys.
74
, 386
–393
(2006
).91.
R.
Stoop
,
A.
Kern
,
M. C.
Göpfert
,
D. A.
Smirnov
,
T. V.
Dikanev
, and
B. P.
Bezrucko
, “
A generalization of the van-der-Pol oscillator underlies active signal amplification in Drosophila hearing
,” Eur. Biophys. J.
35
, 511
–516
(2006
).92.
Yavor
Kostov
,
Ragib
Morshed
,
Barbara
Hling
,
Ray
Chen
, and
P. B.
Siegel
, “
Period-speed analysis of a pendulum
,” Am. J. Phys.
76
, 956
–962
(2008
).93.
Frank S.
Crawford
, “
Damping of a simple pendulum
,” Am. J. Phys.
43
, 276
–277
(1975
).94.
B.
Ravindra
and
A. K.
Mallik
, “
Performance of non-linear vibration isolators under harmonic excitation
,” J. Sound Vib.
170
, 325
–337
(1994
).95.
Jos P.
Baltans
,
Jos L.
Trueba
, and
Miguel A. F.
Sanjun
, “
Energy dissipation in a nonlinearly damped Duffing oscillator
,” Physica D
159
, 22
–34
(2001
).96.
Ronald
Mickens
, “
A combined equivalent linearization and averaging perturbation method for non-linear oscillator equations
,” J. Sound Vib.
264
, 1195
–1200
(2003
).97.
S. J.
Elliott
,
M.
Ghandchi Tehrani
, and
R. S.
Langley
, “
Nonlinear damping and quasi-linear modelling
,” Philos. Trans. R. Soc. A
373
, 20140402
(2015
).98.
A. R.
Plastino
,
R. S.
Wedemann
,
E. M. F.
Curado
,
F. D.
Nobre
, and
C.
Tsallis
, “
Nonlinear drag forces and the thermostatistics of overdamped motion
,” Phys. Rev. E
98
, 012129
(2018
).99.
P.
Flores
,
J.
Ambrósio
,
J. C.
Pimenta Claro
, and
Hamid M.
Lankarani
, “
Contact-impact force models for mechanical systems
,” Kinematics and Dynamics of Multibody Systems with Imperfect Joints: Models and Case Studies
(
Springer Berlin Heidelberg
,
Berlin, Heidelberg
, 2008
), pp. 47
–66
.100.
Onesmus
Muvengei
,
John
Kihiu
, and
Bernard
Ikua
, “
Computational implementation of lugre friction law in a revolute joint with clearance
,” in Proceedings of Sustainable Research and Innovation Conference
(2014
), pp. 99
–108
.101.
Pedro J.
Sebastião
, “
The art of model fitting to experimental results
,” Eur. J. Phys.
35
, 015017
(2013
).102.
Constantino
Grosse
, “
A program for the fitting of Debye, Cole–Cole, Cole–Davidson, and Havriliak–Negami dispersions to dielectric data
,” J. Colloid Interface Sci.
419
, 102
–106
(2014
).103.
N.
Minorsky
, “
Self-excited oscillations in dynamical systems possessing retarded action
,” J. Appl. Mech.
9
, 65
–71
(1942
).104.
Alfred Barnard
Basset
, A Treatise on Hydrodynamics: With Numerous Examples
(
Bell and Company
,
Deighton
, 1888
), Vol.
2
.105.
Wallis S.
Hamilton
, “
Fluid force on accelerating bodies
,” in Coastal Engineering
(1972
), pp. 1767
–1782
.106.
R. A.
Herringe
, “
On the motion of small spheres in oscillating liquids
,” Chem. Eng. J.
11
, 89
–99
(1976
).107.
P. J.
Thomas
, “
On the influence of the Basset history force on the motion of a particle through a fluid
,” Phys. Fluid A
4
, 2090
–2093
(1992
).108.
F.
Mainardi
,
P.
Pironi
, and
F.
Tampieri
, “
On a generalization of the basset problem via fractional calculus
,” Proc. CANCAM
95
, 836
–837
(1995
).109.
Tsang-Jung
Chang
and
Ben Chie
Yen
, “
Gravitational fall velocity of sphere in viscous fluid
,” J. Eng. Mech.
124
, 1193
–1199
(1998
).110.
F.
Candelier
,
J. R.
Angilella
, and
M.
Souhar
, “
On the effect of the Boussinesq-Basset force on the radial migration of a Stokes particle in a vortex
,” Phys. Fluids
16
, 1765
–1776
(2004
).111.
M. A. T.
van Hinsberg
,
J. H. M.
ten Thije Boonkkamp
, and
H. J. H.
Clercx
, “
An efficient, second order method for the approximation of the Basset history force
,” J. Comput. Phys.
230
, 1465
–1478
(2011
).112.
Dumitru
Baleanu
,
Roberto
Garra
, and
Ivo
Petras
, “
A fractional variational approach to the fractional basset-type equation
,” Rep. Math. Phys.
72
, 57
–64
(2013
).113.
Anton
Daitche
, “
On the role of the history force for inertial particles in turbulence
,” J. Fluid Mech.
782
, 567
–593
(2015
).114.
Subramanian
Annamalai
and
S.
Balachandar
, “
Faxén form of time-domain force on a sphere in unsteady spatially varying viscous compressible flows
,” J. Fluid Mech.
816
, 381
–411
(2017
).115.
Humphrey J.
Maris
, “
Note on the history effect in fluid mechanics
,” Am. J. Phys.
87
, 643
–645
(2019
).116.
F. B.
Tatom
, “
The basset term as a semiderivative
,” Appl. Sci. Res.
45
, 283
–285
(1988
).117.
F.
Mainardi
, “
Fractional calculus
,” in Fractals and Fractional Calculus in Continuum Mechanics
, edited by
A.
Carpinteri
and
F.
Mainardi
(
Springer Vienna
,
Vienna
, 1997
), pp. 291
–348
.118.
Andrea E.
Gonzáles
,
Fabián A.
Bombardelli
, and
Yarko I.
Niño
, “
Computation of the particle basset force with a fractional-derivative approach
,” J. Hydraul. Eng.
134
, 1513
–1520
(2008
).119.
Nikolay
Lukerchenko
, “
Discussion of ‘computation of the particle basset force with a fractional-derivative approach’ by FA Bombardelli, AE González, and YI Niño
,” J. Hydraul. Eng.
136
, 853
–854
(2010
).120.
Maolin
Du
,
Zaihua
Wang
, and
Haiyan
Hu
, “
Measuring memory with the order of fractional derivative
,” Sci. Rep.
3
, 1
–3
(2013
).121.
D. C.
Threlfall
, “
The inclusion of Coulomb friction in mechanisms programs with particular reference to DRAM au programme DRAM
,” Mech. Mach. Theory
13
, 475
–483
(1978
).122.
Herrmann
Richard
, Fractional Calculus: An Introduction for Physicists
(
World Scientific
,
Singapore
, 2014
).123.
Manuel D.
Ortigueira
and
J. A.
Tenreiro Machado
, “
What is a fractional derivative?
,” J. Comput. Phys.
293
, 4
–13
(2015
).124.
William H.
Press
,
Saul A.
Teukolsky
,
William T.
Vetterling
, and
Brian P.
Flannery
, Numerical Recipes 3rd Edition: The Art of Scientific Computing
, 3rd ed. (
Cambridge U. P
.,
New York, NY, USA
, 2007
).125.
Luis Nobre
Gonçalves
, “
Featpost and a review of 3D metapost packages
,” in TeX, XML, and Digital Typography
, edited by
Apostolos
Syropoulos
,
Karl
Berry
,
Yannis
Haralambous
,
Baden
Hughes
,
Steven
Peter
, and
John
Plaice
(
Springer Berlin Heidelberg
,
Berlin, Heidelberg
, 2004
), pp. 112
–124
.126.
L. B.
Magalas
, “
Snoek-Köster relaxation. New insights: New paradigms
,” J. Phys. IV
6
, 17
–19
(1996
).127.
See supplemental material at https://doi.org/10.1119/10.0001660 for (i) a detailed description of the OPA model; (ii) motivations to learn fractional calculus; (iii) a numerical method to solve a fractional differential equation; (iv) an additional bibliography; and (v) additional figures.
© 2020 American Association of Physics Teachers.
2020
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.