References - Springer978-3-642-12799-1/1 · References 1. Acary, ... Recent advances in liquid...
-
Upload
vuongthuan -
Category
Documents
-
view
218 -
download
0
Transcript of References - Springer978-3-642-12799-1/1 · References 1. Acary, ... Recent advances in liquid...
References
1. Acary, V., Brogliato, B.: Numerical methods for nonsmooth dynamical sys-tems: Applications in Mechanics and Electronics. Springer, Heidelberg (2008)
2. Andreaus, U., Casini, P., Vestroni, F.: Non-linear dynamics of a cracked can-tilever beam under harmonic excitation. International Journal of Non-LinearMechanics 42(3), 566–575 (2007)
3. Andrianov, I.V.: Asymptotic solutions for nonlinear systems with high degreesof nonlinearity. Prikl. Matem. Mekhanika (PMM) 57(5), 941–943 (1993)
4. Andrianov, I.V., Awrejcewicz, J.: Methods of small and large δ in the nonlineardynamics—a comparative analysis. Nonlinear Dynam. 23(1), 57–66 (2000)
5. Andrianov, I.V.: Asymptotics of nonlinear dynamical systems with a highdegree of nonlinearity. Doklady Mathematics 66(2), 270–273 (2002)
6. Andrianov, I.V., Awrejcewicz, J., Barantsev, R.G.: Asymptotic approachesin mechanics: New parameters and procedures. Applied Mechanics Re-views 56(1), 87–110 (2003)
7. Antonuccio, F.: Hyperbolic numbers and the Dirac spinor (1998),http://arxiv.org/abs/hep-th/9812036v1
8. Arnol′d, V.I.: Mathematical methods of classical mechanics. Springer, NewYork (1978)
9. Arnol′d, V.I.: Mathematical methods of classical mechanics. Springer, Heidel-berg (1978)
10. Ascher, U.M., Mattheij, R.M.M., Russell, R.D.: Numerical solution of bound-ary value problems for ordinary differential equations. Classics in AppliedMathematics, vol. 13. Society for Industrial and Applied Mathematics (SIAM),Philadelphia (1995); Corrected reprint of the 1988 original
11. Atkinson, C.P.: On the superposition method for determining frequencies ofnonlinear systems. In: ASME Proceedings of the 4th National Congress ofApplied Mechanics, pp. 57–62 (1962)
12. Auerbach, D., Cvitanovic, P., Eckmann, J.-P., Gunaratne, G., Procaccia, I.:Exploring chaotic motion through periodic orbits. Phys. Rev. Lett. 58(23),2387–2389 (1987)
13. Awrejcewicz, J., Bajaj, A.K., Lamarque, C.-H. (eds.): Nonlinearity, bifurca-tion and chaos: the doors to the future. Part II. World Scientific PublishingCo., Singapore (1999); Papers from the International Conference held in Do-bieszkow, September 16-18 (1996); Internat. J. Bifur. Chaos Appl. Sci. Engrg.9(3) (1999)
340 References
14. Awrejcewicz, J., Lamarque, C.-H.: Bifurcation and Chaos in Nonsmooth Me-chanical Systems. World Scientific, Singapore (2003)
15. Azeez, M.A.F., Vakakis, A.F., Manevitch, L.I.: Exact solutions of the problemof vibroimpact oscillations of a discrete system with two degrees of freedom.Prikl. Mat. Mekh. 63(4), 549–553 (1999)
16. Azeez, M.A.F., Vakakis, A.F., Manevich, L.I.: Exact solutions of the problemof vibro-impact oscillations of a discrete system with two degrees of freedom.Prikl. Mat. Mekh. 63(4), 549–553 (1999)
17. Babitsky, V.I.: Theory of Vibroimpact Systems and Applications. Springer,Berlin (1998)
18. Bahler, T.B.: Mathematica for Scientists and Engineers. Addison-Wesley, NewYork (1995)
19. Baker Jr., G.A., Graves-Morris, P.: Pade Approximants, 2nd edn. Encyclope-dia of Mathematics and Its Applications, vol. 59. Cambridge University Press,Cambridge (1987)
20. Balescu, R.: Statistical Dynamics, Matter out of Equilibrium. Imperial CollegePress, Singapore (1997)
21. Bateman, H., Erdelyi, A.: Higher Transcendental Functions. McGraw-Hill,New York (1955)
22. Belinfante, J.G.F., Kolman, B.: A survey of Lie groups and Lie algebras withapplications and computational methods. Society for Industrial and AppliedMathematics (SIAM), Philadelphia (1989); Reprint of the 1972 original
23. Bellman, R.: Introduction to Matrix Analysis. McGraw-Hill Company, NewYork (1960)
24. Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic analysis for peri-odic structures. North-Holland Publishing Co., Amsterdam (1978)
25. Blazejczyk-Okolewska, B., Czolczynski, K., Kapitaniak, T., Wojewoda, J.:Chaotic Mechanics in Systems with Impacts and Friction. World Scientific,Singapore (1999)
26. Boettcher, S., Bender, C.M.: Nonperturbative square-well approximation to aquantum theory. Journal of Mathematical Physics 31(11), 2579–2585 (1990)
27. Bogoliubov, N., Mitropollsky, Y.: Asymptotic Methods in the Theory of Non-linear Oscillations. Gordon and Breach, New York (1961)
28. Bogoljubow, N.N., Mitropolski, J.A.: Asymptotische Methoden in der Theorieder nichtlinearen Schwingungen. Akademie-Verlag, Berlin (1965)
29. Brogliato, B.: Nonsmooth Mechanics: Models, Dynamics and Control.Springer, Berlin (1999)
30. Brogliato, B.: Impacts in Mechanical Systems: Analysis and Modelling.Springer-Verlag, Berlin (2000)
31. Caughey, T.K., Vakakis, A.F.: A method for examining steady state solutionsof forced discrete systems with strong non-linearities. International Journal ofNon-Linear Mechanics 26(1), 89–103 (1966)
32. Chati, M., Rand, R., Mukherjee, S.: Modal analysis of a cracked beam. Journalof Sound and Vibration 207, 249–270 (1997)
33. Chen, S., Shaw, S.W.: Normal modes for piecewise linear vibratory systems.Nonlinear Dynamics 10, 135–163 (1996)
34. Cooper, K., Mickens, R.E.: Generalized harmonic balance/numerical methodfor determining analytical approximations to the periodic solutions of the x4/3
potential. Journal of Sound and Vibration 250, 951–954 (2002)
References 341
35. Coppola, V.T., Rand, R.H.: Computer algebra implementation of Lie trans-forms for hamiltonian systems: Application to the nonlinear stability of l4.ZAMM 69(9), 275–284 (1989)
36. Cveticanin, L.: Oscillator with strong quadratic damping force. Publicationsde L’institut Mathematique (Nouvelle serie) 85(99), 119–130 (2009)
37. Dankowicz, H., Paul, M.R.: Discontinuity-induced bifurcations in systems withhysteretic force interactions. Journal of Computational and Nonlinear Dynam-ics 4(Article 041009), 1–6 (2009)
38. Deprit, A.: Canonical transformations depending on a parameter. Celestialmechanics (1), 1–31 (1969)
39. Dimentberg, M.F.: Statistical Dynamics of Nonlinear and Time-Varying Sys-tems. John Wiley & Sons, New York (1988)
40. Dimentberg, M.F., Bratus, A.S.: Bounded parametric control of random vibra-tions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456(2002), 2351–2363(2000)
41. Dimentberg, M.F., Iourtchenko, D.V., Bratus’, A.S.: Transition from planarto whirling oscillations in a certain nonlinear system. Nonlinear Dynamics 23,165–174 (2000)
42. Feeny, B.A., Guran, A., Hinrichs, N., Popp, K.: A historical review on dryfriction and stick-slip phenomena. ASME Applied Mechanics Reviews 51, 321–341 (1998)
43. Ferrari, L., Boschi, C.D.E.: Nonautonomous and nonlinear effects in general-ized classical oscillators: A boundedness theorem. Physical Review E 62(3),R3039–R3042 (2000)
44. Fidlin, A.: Nonlinear Oscillations in Mechanical Engineering. Springer, Hei-delberg (2005)
45. Filippov, A.F.: Differential equations with discontinuous righthand sides.Kluwer Academic Publishers Group, Dordrecht (1988) (Translated from theRussian)
46. Fucik, S., Kufner, A.: Nonlinear differential equations. Elsevier, Amsterdam(1980); Studies in Applied Mechanics 2. Elsevier Scientific Publishing Com-pany, Amsterdam
47. Gendelman, O., Manevitch, L.I., Vakakis, A.F., M’Closkey, R.: Energy pump-ing in nonlinear mechanical oscillators. I. Dynamics of the underlying Hamil-tonian systems. Trans. ASME J. Appl. Mech. 68(1), 34–41 (2001)
48. Gendelman, O.V.: Modeling of inelastic impacts with the help of smooth func-tions. Chaos, Solitons and Fractals 28, 522–526 (2006)
49. Gendelman, O.V., Manevitch, L.I.: Discrete breathers in vibroimpact chains:Analytic solutions. Physical Review E 78(026609) (2008)
50. Giacaglia, G.E.O.: Perturbation methods in non-linear systems. Springer, NewYork (1972); Applied Mathematical Sciences, Vol. 8
51. Goldsmith, W.: Impact: The Theory and Physical Behaviour of Colliding.Courier Dover Publications, North Chelmsford (2001)
52. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 5thedn. Academic Press, Boston (1994)
53. Grebogi, C., Ott, E., Yorke, J.A.: Unstable periodic orbits and the dimensionsof multifractal chaotic attractorscontrolling chaos. Physical Review A 37(5),1711–1724 (1988)
54. Guckenheimer, J., Meloon, B.: Computing periodic orbits and their bifur-cations with automatic differentiation. SIAM J. Sci. Comput. 22(3), 951–985(electronic) (2000)
342 References
55. Guran, A., Pfeiffer, F., Popp, K.: Dynamics with Friction: Modeling, Analysisand Experiments. World Scientific, Singapore (2001)
56. Hahn, W.: Stability of motion. Springer Series in Nonlinear Dynamics.Springer, New York (1967)
57. Harvey, T.J.: Natural forcing functions in nonlinear systems. ASME Journalof Applied Mechanics 25, 352–356 (1958)
58. Hascoet, E., Herrmann, H.J., Loreto, V.: Shock propagation in a granularchain. Phys. Rev. E 59(3), 3202–3206 (1999)
59. Holm, D.D., Lynch, P.: Stepwise precession of the resonant swinging spring.SIAM J. Applied Dynamical Systems 1(1), 44–64 (2002)
60. Hong, J., Ji, J.-Y., Kim, H.: Power laws in nonlinear granular chain undergravity. Phys. Rev. Lett. 82(15), 3058–3061 (1999)
61. Hori, G.: Theory of general perturbations with unspecified canonical variables.Publ. Astron. Soc. Japan 18(4), 287–296 (1966)
62. Hori, G.: Mutual perturbations of 1: 1 commensurable small bodies with theuse of the canonical relative coordinates. I. In: Resonances in the motion ofplanets, satellites and asteroids, pp. 53–66. Univ. Sao Paulo, Sao Paulo (1985)
63. Hu, H., Xiong, Z.-G.: Oscillations in an x(2m+2)/(2n+1) potential. Journal ofSound and Vibration 259, 977–980 (2003)
64. Hunt, K.H., Crossley, F.R.E.: Coefficient of restitution interpreted as dampingin vibroimpact. ASME Journal of Applied Mechanics 97, 440–445 (1975)
65. Hutchins, C.M.: A history of violin research. J. Acoust. Soc. Am. 73(5), 1421–1440 (1983)
66. Ibrahim, R.A., Pilipchuk, V.N., Ikeda, T.: Recent advances in liquid sloshingdynamics. Applied Mechanics Reviews 54(2), 133–199 (2001)
67. Ibrahim, R.A.: Liquid Sloshing Dynamics. Cambridge University Press, NewYork (2005)
68. Ibrahim, R.A.: Vibro-Impact Dynamics: Modeling, Mapping and Applications.LNACM, vol. 43. Springer, Heidelberg (2009)
69. Ibrahim, R.A., Babitsky, V.I., Okuma, M. (eds.): Vibro-Impact Dynamics ofOcean Systems and Related Problems. Springer, Heidelberg (2009)
70. Ince, E.L.: Ordinary Differential Equations. Dover, New York (1956)71. Iomin, A., Fishman, S., Zaslavsky, G.M.: Quantum localization for a kicked
rotor with accelerator mode islands. Physical Review E 65(036215) (2002)72. Ivanov, A.P.: Dynamics of Systems with Mechanical Collisions. International
Program of Education, Moscow (1997) (in Russian)73. Ivanov, A.P.: Impact oscillations: linear theory of stability and bifurcations.
Journal of Sound and Vibration 178(3), 361–378 (1994)74. Jackson, L.B.: Signals, Systems, and Transforms. Addison-Wesley Publishing
Company, New York (1991)75. Jiang, D., Pierre, C., Shaw, S.W.: Large-amplitude non-linear normal modes of
piecewise linear systems. Journal of Sound and Vibration 272, 869–891 (2004)76. Kalamkarov, A.L., Andrianov, I.V., Danishevskyy, V.V.: Asymptotic ho-
mogenization of composite materials and structures. Applied Mechanics Re-views 62(030802), 1–20 (2009)
77. Kamenkov, G.V.: Izbrannye trudy v dvukh tomakh. Tom. I. Izdat, Nauka,Moscow (1971); Ustoichivost dvizheniya. Kolebaniya. Aerodinamika. [Stabilityof motion. Oscillations. Aerodynamics], With a biography of G. V. Kamenkov,a survey article on his works by V. G. Veretennikov, A. S. Galiullin, S. A.Gorbatenko and A. L. Kunicyn, and a bibliography, Edited by N. N. Krasovskiı
References 343
78. Kauderer, H.: Nichtlineare Mechanik. Springer, Berlin (1958)79. Kevorkian, J., Cole, J.D.: Multiple scale and singular perturbation methods.
Springer, New York (1996)80. Kinney, W.M., Rosenberg, R.M.: On steady state harmonic vibrations of non-
linear systems with many degrees of freedom. ASME Journal of Applied Me-chanics 33, 406–412 (1966)
81. Kobrinskii, A.E.: Dynamics of Mechanisms with Elastic Connections and Im-pact Systems. Iliffe Books, London (1969)
82. Koch, C.: Biophysics of Computation. Oxford University Press, Oxford (1999)Information processing in single neurons
83. Kollatz, L.: The eigen-value problems. Nauka, Moscow (1968)84. Kosevich, A.M., Kovalev, A.S.: Introduction to Nonlinear Physical Mechanics
(in Russian). Naukova Dumka, Kiev (1989)85. Kowalczyk, P., Di Bernardo, M., Champneys, A.R., Hogan, S.J., Homer,
M., Piiroinen, P.T., Kuznetsov, Y.A., Kuznetsov, Y.A., Nordmark, A.: Two-parameter discontinuity-induced bifurcations of limit cycles: Classification andopen problems. International Journal of Bifurcation and Chaos 16(3), 601–629(2006)
86. Kryloff, N., Bogoliuboff, N.: Introduction to Non-Linear Mechanics. PrincetonUniversity Press, Princeton (1943)
87. Krylov, N.M., Bogolyubov, N.N.: Vvedeniye v nelinejnuyu mekhaniku. ANUkrSSR, Kiev (1937)
88. Kutz, N.J.: Mode-locked soliton lasers. SIAM Review 48(4), 629–678 (2006)89. Landau, L.D., Lifschitz, E.M.: Lehrbuch der theoretischen Physik (Landau-
Lifschitz. Band I, 12th edn. Akademie-Verlag, Berlin (1987); Mechanik. [Me-chanics], Translated from the third Russian edition by Hardwin Jungclaussen,Edited and with a foreword by Paul Ziesche
90. Lavrent′ev, M.A., Shabat, B.V.: Problemy gidrodinamiki i ikh matematich-eskie modeli, 2nd edn., Izdat. “Nauka”, Moscow (1977)
91. Lee, Y.S., Nucera, F., Vakakis, A.F., McFarland, D.M., Bergman, L.A.: Pe-riodic orbits, damped transitions and targeted energy transfers in oscillatorswith vibro-impact attachments. Physica D 238(18), 1868–1896 (2009)
92. Lee, Y.S., Kerschen, G., Vakakis, A.F., Panagopoulos, P., Bergman, L., McFar-land, D.M.: Complicated dynamics of a linear oscillator with a light, essentiallynonlinear attachment. Physica D 204, 41–69 (2005)
93. Leine, R.I., Nijmeijer, H., Nijmeijer, H.: Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Springer, Heidelberg (2006)
94. Lewis, F.L., Dawson, D.M., Abdallah, C.T.: Robot Manipulator Control: The-ory and Practice. CRC Press, Boca Raton (2004)
95. Lichtenberg, A.J., Lieberman, M.A.: Regular and Stochastic Motion. Springer,New York (1983)
96. Lyapunov, A.M.: Investigation of a singular case of the problem of stability ofmotion. Mat. Sbornik 17, 252–333 (1893)
97. Malkin, I.G.: Some problems of the theory of nonlinear oscillations. Gosu-darstv. Izdat. Tehn.-Teor. Lit., Moscow (1956)
98. Manciu, M., Sen, S., Hurd, A.J.: Impulse propagation in dissipative and dis-ordered chains with power-low repulsive potentials. Physica D 157, 226–240(2001)
99. Manevich, A.I., Manevitch, L.I.: The Mechanics of Nonlinear Systems WithInternal Resonances. Imperial College Press, London (2005)
344 References
100. Manevich, L.I., Mikhlin, Y.V., Pilipchuk, V.N.: Metod normalnykh kolebaniidlya sushchestvenno nelineinykh sistem, Nauka, Moscow (1989)
101. Manevich, L.I.: New approach to beating phenomenon in coupled nonlinearoscillatory chains. Archive of Applied Mechanics 77, 301–312 (2007)
102. Manevitch, L.I.: The description of localized normal modes in a chain of non-linear coupled oscillators using complex variables. Nonlinear Dynamics 25,95–109 (2001)
103. Manevitch, L.I., Gendelman, O.V.: Oscillatory models of vibro-impact typefor essentially non-linear systems. Proceedings of the Institution of MechanicalEngineers, Part C: Journal of Mechanical Engineering Science 222(10), 2007–2043 (2008), doi: 10.1243/09544062JMES1057
104. Manevitch, L.I., Musienko, A.I.: Limiting phase trajectory and beating phe-nomena in systems of coupled nonlinear oscillators. In: 2nd International Con-ference on Nonlinear Normal Modes and Localization in Vibrating Systems,Samos, Greece, June 19-23, pp. 25–26 (2006)
105. Marsden, J.E.: Basic complex analysis. Freeman, San Francisco (1973)106. Maslov, V.P., Omel′janov, G.A.: Asymptotic soliton-like solutions of equations
with small dispersion. Uspekhi Mat. Nauk. 36(3), 63–126 (1981)107. Mickens, R.E.: Oscillations in an x4/3 potential. J. Sound Vibration 246, 375–
378 (2001)108. Mikhlin, Y.V., Reshetnikova, S.N.: Dynamical interaction of an elastic system
and a vibro-impact absorber. Mathematical Problems in Engineering (ArticleID 37980), 15 (2006)
109. Mikhlin, Y.V., Volok, A.M.: Solitary transversal waves and vibro-impact mo-tions in infinite chains and rods. International Journal of Solids and Struc-tures 37, 3403–3420 (2000)
110. Mikhlin, Y.V., Zhupiev, A.L.: An application of the Ince algebraization to thestability of the non-linear normal vibration modes. Internat. J. Non-LinearMech. 32(2), 393–409 (1997)
111. Minorsky, N.: Introduction to non-linear mechanics. J.W. Edwards, Ann Arbor(1947)
112. Mitropl’sky, Y.A., Senik, P.M.: Construction of asymptotic solution of an au-tonomouse system with strong nonlinearity. Doklady AN Ukr.SSR (UkrainianAcademy of Sciences Reports) 6, 839–844 (1961)
113. Moon, F.C.: Chaotic Vibrations. John Willey & Sons, New York (1987)114. Moser, J.: Recent developments in the theory of Hamiltonian systems. SIAM
Rev. 28(4), 459–485 (1986)115. Moser, J.K.: Lectures on Hamiltonian systems. In: Mem. Amer. Math. Soc.
No. 81, p. 60. Amer. Math. Soc., Providence (1968)116. Nayfeh, A.H.: Perturbation methods. John Wiley & Sons, New York (1973);
Pure and Applied Mathematics117. Nayfeh, A.H.: Perturbation methods in nonlinear dynamics. In: Nonlinear
dynamics aspects of particle accelerators (Santa Margherita di Pula, 1985),pp. 238–314. Springer, Berlin (1986)
118. Nayfeh, A.H.: Method of normal forms. John Wiley & Sons Inc., New York(1993); A Wiley-Interscience Publication
119. Nayfeh, A.H.: Nonlinear interactions: analytical computational, and exper-imental methods. John Wiley & Sons Inc., New York (2000); A Wiley-Interscience Publication
References 345
120. Nayfeh, A.H., Balachandran, B.: Applied nonlinear dynamics. John Wiley& Sons Inc, New York (1995) Analytical, computational, and experimentalmethods. A Wiley-Interscience Publication
121. Nesterenko, V.F.: Dynamics of Heterogeneous Materials. Springer, New York(2001)
122. Nesterov, S.V.: Examples of nonlinear Klein-Gordon equations, solvable interms of elementary functions. In: Proceedings of Moscow Institute of PowerEngineering, vol. 357, pp. 68–70 (1978)
123. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64(11),1196–1199 (1990)
124. Ozorio de Almeida, A.M.: Hamiltonian systems: chaos and quantization. Cam-bridge University Press, Cambridge (1988)
125. Parker, T.S., Chua, L.O.: Practical numerical algorithms for chaotic systems.Springer, New York (1989)
126. Peat, F.D.: Synchronicity: the bridge between matter and mind. BantamBooks, New York (1988)
127. Peterka, F.: Introduction to Oscillations of Mechanical Systems with InternalImpacts (in Czech). Academia, Prague (1981)
128. Pfeiffer, F.: Mechanical system dynamics. Springer, Heidelberg (2008)129. Pfeiffer, F., Glocker, C.: Multibody dynamics with unilateral contacts. Wiley,
New York (1996)130. Pfeiffer, F., Kunert, A.: Rattling models from deterministic to stochastic pro-
cesses. Nonlinear Dynamics 1(1), 63–74 (1990)131. Pierce, J.R.: Coupling of modes of propagation. Journal of Applied
Physics 25(2), 179–183 (1954)132. Pilipchuk, V.N.: The calculation of strongly nonlinear systems close to vi-
broimpact systems. Journal of Applied Mathematics and Mechanics 49(5),572–578 (1985)
133. Pilipchuk, V.N.: Transformation of oscillating systems by means of a pair ofnonsmooth periodic functions. Dokl. Akad. Nauk Ukrain. SSR Ser. A (4),37–40, 87 (1988)
134. Pilipchuk, V.N.: Transformation of the vibratory-systems by means of a pairof nonsmooth periodic-functions. Dopovidi Akademii Nauk Ukrainskoi RsrSeriya A-Fiziko-Matematichni Ta Technichni Nauki (in Ukrainian) 4, 36–38(1988)
135. Pilipchuk, V.N.: On the computation of periodic processes in mechanical sys-tems with the impulsive excitation. In: XXXI Sympozjon “Modelowanie wMechanice”, Zeszyty Naukowe Politechniki Slaskiej, Z.107, Gliwice (Poland),pp. 335–342. Politechnica Slaska (1992)
136. Pilipchuk, V.N.: On special trajectories in configuration space of non - linearvibrating systems. Mekhanika Tverdogo Tela (Mechanics of Solids) 3, 36–47(1995)
137. Pilipchuk, V.N.: Analytical study of vibrating systems with strong non-linearities by employing saw-tooth time transformations. J. Sound Vibra-tion 192(1), 43–64 (1996)
138. Pilipchuk, V.N.: On the computation of mechanical systems with impulseexcitation. Prikl. Mat. Mekh. 60(2), 223–232 (1996)
139. Pilipchuk, V.N.: Application of special nonsmooth temporal transformationsto linear and nonlinear systems under discontinuous and impulsive excitation.Nonlinear Dynam. 18(3), 203–234 (1999)
346 References
140. Pilipchuk, V.N.: Non-smooth spatio-temporal transformation for impulsivelyforced oscillators with rigid barriers. J. Sound Vibration 237(5), 915–919(2000)
141. Pilipchuk, V.N.: Principal trajectories of the forced vibration for discrete andcontinuous systems. Meccanica 35(6), 497–517 (2000)
142. Pilipchuk, V.N.: Non-smooth time decomposition for nonlinear models drivenby random pulses. Chaos Solitons Fractals 14(1), 129–143 (2002)
143. Pilipchuk, V.N.: Some remarks on nonsmooth transformations of space andtime for oscillatory systems with rigid barriers. Prikl. Mat. Mekh. 66(1), 33–40(2002)
144. Pilipchuk, V.N.: Temporal transformations and visualization diagrams fornonsmooth periodic motions. International Journal of Bifurcation andChaos 15(6), 1879–1899 (2005)
145. Pilipchuk, V.N.: A periodic version of Lie series for normal mode dynamics.Nonlinear Dynamics and System Theory 6(2), 187–190 (2006)
146. Pilipchuk, V.N., Ibrahim, R.A.: The dynamics of a non-linear system sim-ulating liquid sloshing impact in moving structures. Journal of Sound andVibration 205(5), 593–615 (1997)
147. Pilipchuk, V.N., Ibrahim, R.A.: Application of the Lie group transformationsto nonlinear dynamical systems. Trans. ASME J. Appl. Mech. 66(2), 439–447(1999)
148. Pilipchuk, V.N., Ibrahim, R.A.: Dynamics of a two-pendulum model withimpact interaction and an elastic support. Nonlinear Dynam. 21(3), 221–247(2000)
149. Pilipchuk, V.N., Starushenko, G.A.: On the representation of periodic solu-tions of differential equations by means of an oblique-angled saw-tooth trans-formation of the argument. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn.Tekh. Nauki (11), 25–28 (1997)
150. Pilipchuk, V.N., Starushenko, G.A.: A version of non-smooth transformationsfor one-dimensional elastic systems with a periodic structure. Journal of ap-plied mathematics and mechanics 61(2), 265–274 (1997)
151. Pilipchuk, V.N., Vakakis, A.F.: Nonlinear normal modes and wave transmis-sion in a class of periodic continuous systems. In: Dynamics and control ofdistributed systems, pp. 95–120. Cambridge Univ. Press, Cambridge (1998)
152. Pilipchuk, V.N., Vakakis, A.F.: Study of the oscillations of a nonlinearly sup-ported string using nonsmooth transformations. Journal of Vibration andAcoustics 120(2), 434–440 (1998)
153. Pilipchuk, V.N., Vakakis, A.F., Azeez, M.A.F.: Study of a class of subharmonicmotions using a nonsmooth temporal transformations (NSTT). Physica D 100,145–164 (1997)
154. Pilipchuk, V.N., Vakakis, A.F., Azeez, M.A.F.: Study of a class of subharmonicmotions using a non-smooth temporal transformation (NSTT). Phys. D 100(1-2), 145–164 (1997)
155. Pilipchuk, V.N.: Auto-localized modes in array of nonlinear coupled oscilla-tors. In: Manevich, A.I. (ed.) Problemy nelineinoi mekhaniki i fiziki materialov,Dnipropetrovsk, pp. 229–235 (1999) ISBN: 966-7476-10-3
156. Pilipchuk, V.N.: Impact modes in discrete vibrating systems with bilateralbarriers. International Journal of Nonlinear Mechanics 36(6), 999–1012 (2001)
157. Pilipchuk, V.N.: Transient mode localization in coupled strongly nonlinearexactly solvable oscillators. Nonlinear Dynamics 51(1-2), 245–258 (2008)
References 347
158. Pilipchuk, V.N.: Transitions from strongly to weakly-nonlinear dynamics in aclass of exactly solvable oscillators and nonlinear beat phenomena. NonlinearDynamics 52(4), 263–276 (2008)
159. Pilipchuk, V.N.: Transition from normal to local modes in an elastic beamsupported by nonlinear springs. Journal of Sound and Vibration 322, 554–563(2009)
160. Poincare, H.: Les methodes nouvelles de la mecanique celeste. Tome I. LibrairieScientifique et Technique Albert Blanchard, Paris, Solutions periodiques. Non-existence des integrales uniformes. Solutions asymptotiques. [Periodic solu-tions. Nonexistence of uniform integrals. Asymptotic solutions], Reprint ofthe, original, With a foreword by J. Kovalevsky, Bibliotheque Scientifique Al-bert Blanchard. [Albert Blanchard Scientific Library] (1987)
161. Poincare, H.: Science and method. Thoemmes Press, Bristol (1996); Trans-lated by Francis Maitland, With a preface by Bertrand Russell, Reprint of the1914 edition
162. Popp, K.: Non-smooth mechanical systems. Journal of Applied Mathematicsand Mechanics (PMM) 64(5), 765–772 (2000)
163. Qaisi, M.I.: Non-linear normal modes of a lumped parameter system. Journalof Sound and Vibration 205, 205–211 (1997)
164. Ramos, J.I.: Piecewise-linearized methods for oscillators with fractional-powernonlinearities. Journal of Sound and Vibration 300, 502–521 (2007)
165. Richtmyer, R.D.: Principles of advanced mathematical physics, vol. I.Springer, New York (1978); Texts and Monographs in Physics
166. Richtmyer, R.D.: Principles of Advanced Mathematical Physics. Springer,Berlin (1985)
167. Rosenberg, R.M.: The Ateb(h)-functions and their properties. Quart. Appl.Math. 21, 37–47 (1963)
168. Rosenberg, R.M.: Steady-state forced vibrations. Internat. J. Non-LinearMech. 1, 95–108 (1966)
169. Rowat, P.F., Selverston, A.I.: Oscillatory mechanisms in pairs of neuronsconnected with fast inhibitory synapses. Journal of Computational Neuro-science 4, 103–127 (1997)
170. Salenger, G., Vakakis, A.F., Gendelman, O., Manevitch, L., Andrianov, I.:Transitions from strongly to weakly nonlinear motions of damped nonlinearoscillators. Nonlinear Dynam. 20(2), 99–114 (1999)
171. Salenger, G.D., Vakakis, A.F.: Localized and periodic waves with discretenesseffects. Mech. Res. Comm. 25(1), 97–104 (1998)
172. Samoılenko, A.M., Boıchuk, A.A., Zhuravlev, V.F.: Weakly nonlinear bound-ary value problems for operator equations with impulse action. Ukraın. Mat.Zh., 49(2):272–288 (1997)
173. Scherz, P.: Practical Electronics for Inventors. McGraw-Hill, New York (2006)174. Scott, A.C., Lomdahl, P.S., Eilbeck, J.C.: Between the local-mode and normal-
mode limits. Chemical Physics Letters 113(1), 29–36 (1985)175. Sheng, G., Dukkipati, R., Pang, J.: Nonlinear dynamics of sub-10 nm flying
height air bearing slider in modern hard disk recording system. Mechanismand Machine Theory 41, 1230–1242 (2006)
176. Sobczyk, G.: The hyperbolic number plane. The College Mathematics Jour-nal 26(4), 268–280 (1995)
177. Sophianopoulos, D.S., Kounadis, A.N., Vakakis, A.F.: Complex dynamics ofperfect discrete systems under partial follower forces. Internat. J. Non-LinearMech. 37(6), 1121–1138 (2002)
348 References
178. Stakgold, I.: Green’s Functions and Boundary Value Problems. Wiley Inter-science, New York (1979)
179. Starushenko, G., Krulik, N., Tokarzewski, S.: Employment of non-symmetricalsaw-tooth argument transformation method in the elasticity theory for layeredcomposites. International Journal of Heat and Mass Transfer 45, 3055–3060(2002)
180. Stronge, W.J.: Impact Mechanics. Cambridge University Press, Cambridge(2000)
181. Thomsen, J.J., Fidlin, A.: Near-elastic vibro-impact analysis by discontinuoustransformations and averaging. Journal of Sound and Vibration 311, 386–407(2008)
182. Timoshenko, S.P., Yang, D.H., Wiver, U.: Kolebaniya v inzhenernom dele.Mashinostroeniye, Moscow (1985)
183. Tippetts, J.R.: Analysis of idealised oscillatory pipe flow. In: 2nd Interna-tional Symposium on Fluid - Control, Measurement, Mechanics - and FlowVisualisation, Sheffield, England, September 5-9 (1988)
184. Toda, M.: Nonlinear lattice and soliton theory. IEEE Transactions on Circuitsand Systems 30(8), 542–554 (1983)
185. Turner, J.D.: On the simulation of discontinuous functions. Journal of AppliedMechanics 68, 751–757 (2001)
186. Ueda, Y.: Randomly transitional phenomena in the system governed by Duff-ing’s equation. J. Statist. Phys. 20(2), 181–196 (1979)
187. Ulrych, S.: Relativistic quantum physics with hyperbolic numbers. PhysicsLetters B 625, 313 (2005)
188. Uzunov, I.M., Muschall, R., Golles, M., Kivshar, Y.S., Malomed, B.A., Led-erer, F.: Pulse switching in nonlinear fiber directional couplers. Phys. Rev.E 51, 2527–2537 (1995)
189. Vakakis, A.F., Atanackovic, T.M.: Buckling of an elastic ring forced by aperiodic array of compressive loads. ASME Journal of Applied Mechanics 66,361–367 (1999)
190. Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N., Zevin, A.A.:Normal modes and localization in nonlinear systems. John Wiley & Sons Inc.,New York (1996); A Wiley-Interscience Publication
191. Vedenova, E.G., Manevich, L.I., Pilipchuk, V.N.: Normal oscillations of astring with concentrated masses on nonlinearly elastic supports. Prikl. Mat.Mekh. 49(2), 203–211 (1985)
192. Vestroni, F., Luongo, A., Paolone, A.: A perturbation method for evaluatingnonlinear normal modes of a piecewise linear two-degrees-of-freedom system.Nonlinear Dynamics 54(4), 379–393 (2008)
193. Waluya, S.B., van Horssen, W.T.: On the periodic solutions of a generalizednon-linear Van-der-Pol oscillator. Journal of Sound and Vibration 268, 209–215 (2003)
194. Whitham, G.B.: Linear and nonlinear waves. John Wiley & Sons Inc., NewYork (1999); Reprint of the 1974 original. A Wiley-Interscience Publication
195. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cam-bridge University Press, Cambridge (1986)
196. Wiercigroch, M., de Kraker, B. (eds.): Applied Nonlinear Dynamics ans Chaosof Mechanical Systems with Discontinuities, vol. 28. World Scientific, Singa-pore (2000)
References 349
197. Zevin, A.A.: Localization of periodic oscillations in vibroimpact systems. In:XXXV Symposium Modeling in Mechanics, Gliwice (Poland), pp. 261–266.Politechnica Slaska (1996)
198. Zhupiev, A.L., Mikhlin, Y.V.: Stability and branching of normal oscillationsforms of nonlinear systems. Prikladnaya Matematica Mekhanika (PMM) 45,450–455 (1981)
199. Zhuravlev, V.F.: A method for analyzing vibration-impact systems by meansof special functions. Izvestiya AN SSSR Mekhanika Tverdogo Tela (Mechanicsof Solids) 11(2), 30–34 (1976)
200. Zhuravlev, V.F.: Equations of motion of mechanical systems with ideal one-sided links. Prikl. Mat. Mekh. 42(5), 781–788 (1978)
201. Zhuravlev, V.F.: The method of Lie series in the motion-separation problemin nonlinear mechanics. Prikl. Mat. Mekh. 47(4), 559–565 (1983)
202. Zhuravlev, V.F.: The application of monomial Lie groups to the problem ofasymptotically integrating equations of mechanics. Prikl. Mat. Mekh. 50(3),346–352 (1986)
203. Zhuravlev, V.F.: Particular directions in the configuration space of linear os-cillatory systems. Prikl. Mat. Mekh. 56(1), 16–23 (1992)
204. Zhuravlev, V.F., Klimov, D.M.: Prikladnye metody v teorii kolebanii, Nauka,Moscow (1988) Edited and with a foreword by A. Yu. Ishlinskiı
205. Zhuravlev, V.P., Klimov, D.M.: Applied methods in the theory of vibrations(in Russian), Nauka, Moscow (1988)
206. Zhusubaliyev, Z.T., Mosekilde, E.: Bifurcations and Chaos in Piecewise-smooth Dynamical Systems. World Scientific, Singapore (2003)
Appendix 351
APPENDIX 1: Mathematica�R� (Version 6) notebook for sawtooth power-series solutions for the oscillator �t,t x � xm � 0.
This module builds sawtooth power series solutions of the oscillator.The module is easy to modify on polynomial characteristics of the oscillator.
�� n � the number of iterations ���� m � the exponent ��STSR�n�, m�� :� Module��f, h, RHS, Ε, suc, x, X, H�,f�x�� :� xm;
h �i�0
n
Εi�Hi;
RHS � �Ε h f�i�0
n
Εi�Xi�Ξ�;
suc � Cancel�Coefficient�Normal�Series�RHS, �Ε, 0, n���,Table�Εi, �i, 1, n����;X0�Ξ�� :� A Ξ;Do�Xk�Τ�� � Integrate�suc��k�� ��Ξ Τ�, �Ξ, 0, Τ�,Assumptions � Τ � 0 && Re�m� � �1�, �k, 1, n��;H0 � H0 . Solve� Τ �X0�Τ� X1�Τ�� � 0 . Τ � 1, H0���1��;Do�Hk � Hk . Solve� Τ Xk1�Τ� � 0 . Τ � 1, Hk���1��, �k, 1, n��;�Table�Xk�Τ�, �k, 0, n��, Table�Hk, �k, 0, n � 1���
Successive approximations (execution):
n � 3;
AnalyticalSolution � STSR�n, Α�;Do�Xi�1 � AnalyticalSolution��1����i��, �i, 1, n 1��;Do�Hi�1 � AnalyticalSolution��2����i��, �i, 1, n��;Do�Print�"X", i, "�", Xi Simplify�, �i, 0, n��Do�Print�"H", i, "�", Hi Simplify�, �i, 0, n � 1��
X0�A Τ
X1��A Τ2�Α
2 � Α
X2��A�2 Α Α Τ �A Τ�1�Α �AΑ �3 � 2 Α� � �2 � Α� Τ �A Τ�Α�
2 �2 � Α�2 �3 � 2 Α�
X3��A1�2 Α Α Τ2�Α �A2 Α �1 � Α�2 �4 � 3 Α� � AΑ Α �8 � 10 Α � 3 Α2� Τ �A Τ�Α � ��2 � Α � 3 Α2 � Α3� Τ2 �A Τ�2 Α�
2 �2 � Α�3 �3 � 2 Α� �4 � 3 Α�
352 Appendix
H0�A1�Α �1 � Α�
H1�A1�Α Α �1 � Α�2 �2 � Α�
H2�A1�Α Α �1 � Α�3
2 �2 � Α�2 �3 � 2 Α�Successive approximation truncated series:
x �i�0
n
Xi; h �i�0
n�1
Hi; v �1
h
� Τ x;
Appendix 353
APPENDIX 2: Mathematica�R� (Version 6) notebook for sawtooth power-series expansion of periodic functions.
In[1]:= �� f � f�t� � periodic function of the period T ���� m � "length" of the series; see the example below ���� smoothness � True or False; use smoothness
� True to smooth the series ��TauSeries�f�, T�, m�, smoothness��:� Module��a,s1, s2, NLX, NLY,RE, IM, der, TauSpectrum�,a � T 4;s1 � �t �� a �; s2 � �t � 2�a � a �;
RE �1
2���f . s1� �f . s2��; IM �
1
2���f . s1� � �f . s2��;
der�F�� :� Apply�D, �F, Τ ��;NLX � NestList�der, RE, 2�m 1� . �Τ � 0�; NLY
� NestList�der, IM, 2�m 1� . �Τ � 0�;
Do��KX�2�i � 1� �k�1
i NLX��2�k��Factorial�2�k � 2�
, KX�2�i� �k�1
i NLX��2�k 1��Factorial�2�k � 1�
,
KY�2�i � 1� �k�0
i NLY��2�k��Factorial�2�k � 1�
,
KY�2�i� �k�0
i NLY��2�k 1��Factorial�2�k�
�, �i, �1, m�;
If�smoothness, �X � �RE . �Τ � 0�� i�1
2�m
KX�i��Τi
i�Τi2
i 2,
Y �i�0
2�m
KY�i���Τi � Τi2��, �X ��k�1
2�m NLX��k���Τk�1Factorial�k � 1�
,
Y ��k�1
2�m NLY��k���Τk�1Factorial�k � 1�
�
In[2]:= �� ������������� Example of usage ������������ ��
f�t�� :� Sin�Рt
23
T � 4;
In[4]:= �� � Expansions with no smoothing procedure �� ��fnosmooth�t�, m�� :� ��X Y e� . TauSeries�f�t�, T, m, False��
. �Τ � Τ�4�t T�, e � e�4�t T��
fnosmooth�t, 1�fnosmooth�t, 2�
354 Appendix
In[10]:= �� ��� Expansions with smoothing procedure ��� ��fsmooth�t�, m�� :� ��X Y e� . TauSeries�f�t�, T, m, True��
. �Τ � Τ�4�t T�, e � e�4�t T��fsmooth�t, 1�fsmooth�t, 2�fsmooth�t, 3�fsmooth�t, 4�
Out[11]= 0
Out[12]=3
8Π3
Τ�t�33
�Τ�t�55
Out[13]=3
8Π3
Τ�t�33
�Τ�t�55
�3 Π3
8�5 Π5
64
Τ�t�55
�Τ�t�77
Out[14]=3
8Π3
Τ�t�33
�Τ�t�55
�3 Π3
8�5 Π5
64
Τ�t�55
�Τ�t�77
�3 Π3
8�5 Π5
64�
91 Π7
15 360
Τ�t�77
�Τ�t�99
In[15]:= �� ������ Defining the basis functions ������ ��
��� :�2
Π�ArcSin� Sin�
Π
2�Ξ;
e�Ξ�� :� Sign� Cos�Π
2�Ξ;
Out[7]=1
8Π3 Τ�t�3 �
1
64Π5 Τ�t�5
Out[8]=1
8Π3 Τ�t�3 �
1
64Π5 Τ�t�5 �
13 Π7 Τ�t�715 360
Out[9]=1
8Π3 Τ�t�3 �
1
64Π5 Τ�t�5 �
13 Π7 Τ�t�715 360
�41 Π9 Τ�t�91 548288
Out[5]= 0
Out[6]=1
8Π3 Τ�t�3
fnosmooth�t, 3�fnosmooth�t, 4�fnosmooth�t, 5�
Appendix 355
In[18]:= �� ���������� Convergence with smoothing �������� ��
Plot�Evaluate��f�t�, fsmooth�t, 2�, fsmooth�t, 3�,fsmooth�t, 4��, �t, 0, 2 T��,
PlotStyle � ��Thickness�.005�, Color � Black�,�Thickness�.002�, Dashing��0.01, 0.01��, Color � Black�,�Thickness�.002�, Color � Black�,�Thickness�.003�, Color � Black��, AxesLabel � �"t", "x"�,
PlotRange � All�
Out[18]=
2 4 6 8t
�1.5
�1.0
�0.5
0.5
1.0
1.5
x
Out[17]=
2 4 6 8t
�4
�2
2
4
x
In[17]:= �� ��������� Convergence with no smoothing ������� ��
Plot�Evaluate��f�t�, fnosmooth�t, 2�, fnosmooth�t, 3�,fnosmooth�t, 4�, fnosmooth�t, 5��,�t, 0, 2 T��,PlotStyle � ��Thickness�.005�, Color � Black�,�Thickness�.002�, Dashing��0.01, 0.01��, Color � Black�,�Thickness�.002�,Color � Black�,�Thickness�.002�,Color � Black�,�Thickness�.003�, Color � Black��,
AxesLabel � �"t", "x"�, PlotRange � All�
Appendix 357
APPENDIX 3: Mathematica�R�(Version 4) notebook.
NSTT & Shooting method for periodic solutions of the oscillator
d2 �x
dt2� Ζ dx
dt� Ε x3 � B e� t
a�.
p�x�, v�, t�� :� Ζ v Ε x3;a �
Π
2 Ω;
�� Substitution x�t� � X��t a��Y��t a���e�t a�applied to the equation of motion: ��
eqX �1
a2� Τ,ΤX�Τ�
1
2� p�X�Τ� Y�Τ�,
1
a�� Τ X�Τ� Τ Y�Τ��, a Τ
p�X�Τ� � Y�Τ�,1
a��� Τ X�Τ� Τ Y�Τ��, 2 a � a Τ � 0 Simplify;
eqY �1
a2� Τ,ΤY�Τ�
1
2� p�X�Τ� Y�Τ�,
1
a�� Τ X�Τ� Τ Y�Τ��, a Τ
� p�X�Τ� � Y�Τ�,1
a��� Τ X�Τ� Τ Y�Τ��, 2 a � a Τ � B Simplify;
�� Parameters: ��Ω � 1.0; Ζ � 0.05; Ε � 1.0; B � 7.4;
Clear�g, h�; dg � 3; dh � 20;sol�g�, h�� :� NDSolve��eqX, eqY, X��1� � g, X'��1� � 0, Y��1� �� 0,Y'��1� � h�, �X, Y�, �Τ, �1, 1�, MaxSteps � Infinity�;Xn�g�, h�� :� X'�1� . sol�g, h���1��;Yn�g�, h�� :� Y�1� . sol�g, h���1��;
plx � ContourPlot�Xn�g, h�, �g, �dg, dg�, �h, �dh, dh�,Contours � �0�, ContourShading � False,FrameLabel � �"g", "h"�,PlotPoints � 100,RotateLabel � False,DisplayFunction � Identity�;
ply � ContourPlot�Yn�g, h�,�g,�dg, dg�,�h, �dh, dh�,Contours � �0�,ContourShading � False, FrameLabel � �"g", "h"�, PlotPoints � 100,
RotateLabel � False,
ContourStyle � �Dashing��0.01, 0.01���, DisplayFunction � Identity�;pxy � Show�plx, ply, DisplayFunction � $DisplayFunction�;
358 Appendix
-3 -2 -1 0 1 2 3g
-20
-15
-10
-5
0
5
10
15
h
�� Magnified portion of the diagram: ��
Show�pxy, PlotRange � ���0.3, 0�, ��5, �4���;
-0.25 -0.2 -0.15 -0.1 -0.05 0g
-4.8
-4.6
-4.4
-4.2
-4
h
Appendix 359
�� Find one of the roots: ��
�g, h� � �g, h� . FindRoot��Xn�g, h� � 0, Yn�g, h� � 0�, �g, �0.3, 0�,�h, �5, �4��
�0.113351, �4.51957�� Check precision: ��
sln � NDSolve��eqX, eqY, X��1� � g, X'��1� � 0, Y��1� �� 0, Y'��1� � h�,�X, Y�, �Τ, �1, 1��;
�Y�1�, X'�1�� . sln��1��
��2.09795�10�9, �1.47496�10�9��� Introduce the saw�tooth sine and the rectangular cosine: ��
�t�� :�2
Π�ArcSin� Sin�
Π
2�t;
e�t�� :� Sign� Cos�Π
2�t;
�� Graphic output compared to solution of the related Cauchy problem: ��
x�t�� :� �X��t a�� Y��t a���e�t a�� . sln��1��;
v�t�� :� 1
a��Y'��t a�� X'��t a���e�t a�� . sln��1��;
xxtt � Plot�Evaluate�x�t�, �t, 0, 4 a��, PlotRange � All,
AxesLabel � �"t", "x"�, PlotStyle � ��Thickness�.009���,TextStyle � �FontSize � 14�, DisplayFunction � Identity�;
xxvv � ParametricPlot�Evaluate��x�t�, v�t��, �t, 0, 4�a��, PlotRange � All,
AxesLabel � �"x", "v"�, Frame � True, PlotStyle � ��Thickness�.009���,TextStyle � �FontSize � 14�, DisplayFunction � Identity�;
Clear�y�;Tmax � 4�a; dirsol � NDSolve�� t,t y�t� � t y�t� Šy�t�3 � B e�t a�,
y�0� � x�0�, y'�0� � v�0��, y, �t, 0, Tmax�,MaxSteps � Infinity�; y � y . dirsol��1��;
yyvv � ParametricPlot�Evaluate��y�t�, y'�t��, �t, 0, Tmax��,PlotRange � All, AxesLabel � �"x", "v"�,PlotStyle � ��Thickness�.001���, TextStyle � �FontSize � 14�,PlotPoints � 500, Frame � True, DisplayFunction � Identity�;
Show�GraphicsArray���xxtt�, �xxvv�, �yyvv����;
360 Appendix
-3 -2 -1 0 1 2 3
-4
-2
0
2
4
x
v
-3 -2 -1 0 1 2 3
-4
-2
0
2
4
x
v
1 2 3 4 5 6t
-3-2-1
123
x
Appendix 361
APPENDIX 4: Mathematica�R� notebook.Conducts NSTT of the T-periodic differential equation of the form
�t,t x � f �x, �t x, t� � 0.
In[1]:= �� see examples below for the meaning of inputs ��
TauTransform�equation�, function�,argument�, period�, basis�� :�Module��sub, eqn, RE, IM�,sub � �function � X�� Y���e, argument function � �Y'�� X'���e� a, argument,argument function � �X''�� Y''���e� � a2�;
Q � Part�equation, 1� . sub;
eqn �1
2���Q . t �� a � �Q . t � 2�a � a ��
1
2���Q . t �� a � � �Q . t � 2�a � a �� e . a � period 4;
EQNX � Simplify�1
2���eqn . e � 1� �eqn . e � �1��;
EQNY � Simplify�1
2���eqn . e � 1� � �eqn . e � �1��;
subid � �X�� ��1
2��X�� X����, Y�� ��
1
2��X�� � X����,
X'�� ��1
2�� Τ X�Τ� Τ X��Τ��,Y'�Τ� ��
1
2�� Τ X�Τ� � Τ X��Τ��,
X''�� ��1
2� Τ,Τ�X�Τ� X��Τ��,Y''�Τ� ��
1
2� Τ,Τ�X�Τ�� X��Τ���;
IDP � Simplify��EQNX EQNY� . subid�;IDM � Simplify��EQNX � EQNY� . subid�;If�basis � 1, ��EQNX � 0, EQNY � 0�, �Y�1� � 0, Y��1� � 0, X'�1�� 0, X'��1� � 0��, ��IDP � 0, IDM � 0�,�X�1� � X��1� � 0,
X��1� � X���1� � 0, X'�1� X�'�1� � 0, X'��1� X�'��1� � 0���
EXAMPLES
In[2]:= �� If basis � 1, then the output is created in the standard
basis �1,e�, otherwise the output is in the idempotent basis
�e,e�����1e� 2,�1�e� 2� ��
In[3]:= TauTransform�x''�t� x�t�3 � 0, x�t�, t, 4�a, 1�
Out[3]= X��3 � 3 X�� Y��2 �X����a2
0, 3 X��2 Y�� � Y��3 �Y����a2
0�,
Y�1� 0, Y��1� 0, X��1� 0, X���1� 0�
362 Appendix
In[5]:= TauTransform�x''�t� x�t�3 � P Sin�t� � 0, x�t�, t, 2�Π, 0�
Out[5]= �P Sin�Π Τ
2� � X���3 �
4 X�����
Π2 0,�P Sin�Π Τ
2� � X���3 �
4 X�����
Π2 0�,
�X��1� � X��1� 0, �X���1� � X���1� 0, X���1� � X�
��1� 0, X�
���1� � X����1� 0�
In[6]:= TauTransform�x''�t� x�t�3 � P Sin�t� � 0, x�t�, t, 2�Π, 1�
Out[6]= �P Sin�Π Τ
2� � X�Τ�3 � 3 X�Τ� Y�Τ�2 �
4 X���Τ�Π2
0, 3 X�Τ�2 Y�Τ�
� Y��3 �4 Y����
Π2 0�,Y�1� 0, Y��1� 0, X��1� 0, X���1� 0�
In[7]:= �� If present, the function e�e�4t T� must be shown with
no argument; derivatives de dt are not allowed by this code,
but necessary generalizations are easy to implement ��
In[8]:= TauTransform�x''�t� �1 Α e ��x�t� Β x�t�3 � 0, x�t�, t, 4�a, 1�
Out[8]= X�Τ� � Β X�Τ�3 � Α Y�Τ� � 3 Β X�Τ� Y�Τ�2 �X���Τ�a2
0,
Α X�Τ� � Y�Τ� � 3 Β X�Τ�2 Y�Τ� � Β Y�Τ�3 �Y���Τ�a2
0�, Y�1� 0,
Y��1� 0, X��1� 0, X���1� 0�In[9]:= TauTransform�x''�t� �1 Α e ��x�t� Β x�t�3 � 0, x�t�, t, 4�a, 0�
Out[9]= �1 � Α� X��Τ� � Β X��Τ�3 �X�
����a2
0, ���1 � Α� X��Τ� � Β X��Τ�3
�X�
����a2
0�, �X��1� � X��1� 0, �X���1� � X���1� 0, X���1�
� X���1� 0, X�
���1� � X����1� 0�
PROJECT: Using the idempotent basis, find T-periodic solution of the linear oscillator under external and parametric rectangular wave periodic excitation
of the period T=4,
In[4]:= TauTransform�x''�t� x�t�3 � 0, x�t�, t, 4�a, 0�
Out[4]= X���3 �X�
����a2
0, X���3 �X�
����a2
0�, �X��1� � X��1� 0,
�X���1� � X���1� 0, X���1� � X�
��1� 0, X����1� � X�
���1� 0�
Appendix 363
In[12]:= �� Solves the boundary value problem analytically: ��
sol � DSolve�bvp, �X��, X����, � Simplify
Out[12]= X��� �
2 p 3 Cos� Ω
2� Sin� �3
2Ω� � Cos� �3
2Ω� � 4 Cos� �3
2Τ Ω� Sin� Ω
2�
3 Ω2 3 Cos� Ω
2� Sin� �3
2Ω� � Cos� �3
2Ω� Sin� Ω
2�
,
X��Τ� � � 2 p 3 3 Cos� Ω
2� Sin� 3
2Ω�� 4 3 Cos� Τ Ω
2� Sin� 3
2Ω�
� 3 Cos� 3
2Ω� Sin� Ω
2� � 3 Ω2 3 Cos� Ω
2� Sin� 3
2Ω�
� Cos� 3
2Ω� Sin� Ω
2� ��
In[13]:= �� Extracting two components of the solution: ��
X�� � X�� . sol��1��;X��� � X��� . sol��1��;
In[15]:= �� Back to the original x�t variables: ��
x � X���1
2��1 e� X����
1
2��1 � e� . �Τ � Τ�t�, e � e�t��;
v �1
T 4� Τ X�Τ��
1
2��1 e�� Τ X��Τ��
1
2��1 � e� .�Τ � Τ�t�,e � e�t��;
bvp � TauTransform�x''�t� Ω2� 1 1
2e �x�t� � p e � 0, x�t�, t, T, 0
Out[11]= �p �3
2Ω2 X��Τ� � X�
���� 0, p �1
2Ω2 X��Τ� � X�
���� 0�,�X��1� � X��1� 0, �X���1� � X���1� 0, X�
��1� � X���1� 0,
X����1� � X�
���1� 0�
�t,t x � �1� 12�e�t���Ω2�x � pe�t�.
In[10]:=�� Formulates boundary value problem in the idempotent basis: ��
T � 4;
In[17]:= �� Basis functions: ��
364 Appendix
In[19]:= �� Parameters and graphic output: ��٠� 20.0;
p � 1.0;
Plot�Evaluate�x, �t, 0, 8��, PlotRange � All,
AxesLabel � �"t", "x"�, PlotStyle � ��Thickness�.005����
Out[21]=2 4 6 8
t
�0.010
�0.005
0.005
0.010
x
In[22]:= �� Validating the code by durect numerical solution: ��x0 � x . t � 0;
v0 � v . t � 0;
dirsol � NDSolve��y''�t� Ω2� 1 1
2e �t� �y�t��p e�t��0,y�0��x0,
y'�0� � v0�, y, �t, 0, 50�, MaxSteps � Infinity;y � y . dirsol��1��;
In[26]:= Plot�Evaluate�y�t�, �t, 0, 8��, PlotRange � All,
AxesLabel � �"x", "v"�, PlotStyle � ��Thickness�.003����
Out[26]=2 4 6 8
x
�0.010
�0.005
0.005
0.010
v
�t�� :�2
Π�ArcSin� Sin�
Π
2�t;
e�t�� :� Sign� Cos�Π
2�t;