DuffingType Oscillators with AmplitudeIndependent Period


 Arthur Atkins
 10 months ago
 Views:
Transcription
1 DuffingType Oscillators with AmplitudeIndependent Period Ivana N. Kovacic and Richard H. Rand Abstract Nonlinear oscillators with hardening and softening cubic Duffing nonlinearity are considered. Such classical conservative oscillators are known to have an amplitudedependent period. In this work, we design oscillators with the Duffingtype restoring force but an amplitudeindependent period. We present their Lagrangians, equations of motion, conservation laws, as well as solutions for motion. 1 Introduction Classical Duffing oscillators are governed by Rx C x x 3 D : (1) Their restoring force F D x x 3 includes a linear geometric term as well as a cubic geometric term: a positive sign in front of the cubic term corresponds to a hardening Duffing oscillator (HDO) and the negative one to a softening Duffing oscillator (SDO) [4, 6]. Unlike the majority of nonlinear oscillators, both of them have a closedform exact solution, which is expressed in terms of Jacobi cn or sn elliptic functions. These solutions corresponding to the following initial conditions x./ D A; Px./ D ; () I.N. Kovacic ( ) Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, 115 Novi Sad, Serbia R.H. Rand Department of Mathematics, Cornell University, Ithaca, NY 14853, USA Department of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA Springer International Publishing Switzerland 14 J. Awrejcewicz (ed.), Applied NonLinear Dynamical Systems, Springer Proceedings in Mathematics & Statistics 93, DOI 1.17/
2 I.N. Kovacic and R.H. Rand Table 1 Solutions for motion of the HDO and SDO ( in case of the SDO one requires jaj <1 for the closed orbits around the stable origin) Solution for motion Frequency Elliptic modulus HDO: x HDO D Acn.! HDO t;k HDO /! HDO D 1 C A khdo D A D! HDO SDO : x SDO D Asn.! SDO t;k SDO /! SDO A D 1 k SDO D A D! SDO A.1CA / A 1 A 1 T 8 SDO p HDO A Fig. 1 Period of the HDO (blue solid line)andsdo(red dashed line) as a function of amplitude A are given in Table 1 together with their frequencies! and elliptic moduli k (see, for example, [4, 6] for more detail). Given the fact that the period of cn and sn is T D 4K.k/ =!, where K.k/ stands for the complete elliptic integral of the first kind, the following expressions are obtained for the period of the HDO and SDO: T HDO D r A 4K p 1 C A.1CA / ; T SDO D 4K r A 1 A q 1 A! : (3) As seen from these expressions and Fig. 1, the period of the HDO and SDO depends on the amplitude A. This leads us to the question of designing an oscillator having the Duffingtype restoring force F D x x 3, but a constant, amplitudeindependent period, corresponding to the socalled isochronous oscillators [1, ]. In what follows we present such Duffingtype oscillators, both hardening and softening, modelled by Rx C G.x; Px/ C x x 3 D ; (4) finding also the corresponding Lagrangians, conservation laws, as well as solutions for motion.
3 DuffingType Oscillators with AmplitudeIndependent Period 3 Derivation The derivation of the mathematical model (4) is based on the transformation approach [5], in which the kinetic energy E k and potential energy E p of nonlinear oscillators are made equal to the one of the simple harmonic oscillator (SHO) E ksho D 1 P X ; E psho D 1 X ; (5) which is known to have a constant, amplitudeindependent period (note that its generalized coordinate is labelled here by X)..1 Case I To find the first mathematical model of the form (4), we assume that E p D E p.x/ and let E p E psho D X =, obtaining X D p E p : (6) Then, we also make the kinetic energy E k of nonlinear oscillators equal to the one of the SHO and use Eq. (6) to derive E k D X P E D p Px ; (7) 4E p where Ep D de p=dx. The differential equation of motion stemming from the Lagrangian L D E k E p has a general form Rx C! E p E p Px C E p Ep E p Ep D : (8) The last term on the lefthand side E p =Ep is required to correspond to the Duffing restoring force F =x x 3, which gives the potential energy E p D Equation (7) now yields the kinetic energy E k D x.1 x / : (9) 1.1 x / 3 Px ; (1)
4 4 I.N. Kovacic and R.H. Rand so that Eq. (8) becomes Rx 3x 1 x Px C x x 3 D : (11) Based on X = C P X = D const:, the system exhibits the first integral 1.1 x / 3 Px C x 1 x D A 1 A : (1) Taking the solution for motion of the SHO in the form X D a cos.t C /, where a and are constants, and using Eqs. (6), (9) and (), the solution for motion of Eq. (11) is obtained A p cos t D x p : (13) 1 A 1 x Numerical verifications of the analytical results for the motion (13) and phase trajectories (1) are shown in Figs. and 3 for the HDO and SDO, respectively. These figures confirm that the analytical results coincide with the solutions obtained by solving the equation of motion (11) numerically. In addition, they illustrate the fact that the period stays constant despite the fact that the amplitude A changes, i.e., that the systems perform isochronous oscillations. a b ẋ.5 x.  p 4p 6p t.5 x Fig. Isochronous oscillations of the HDO, Eq. (11) for A D :5;.5;.75:(a) time histories obtained numerically from Eq. (11) (black dots) and from Eq. (13) (blue solid line); (b) phase trajectories obtained numerically from Eq. (11) (black dots) and from Eq. (1) (blue solid line) (upper signs are used in all these equations)
5 DuffingType Oscillators with AmplitudeIndependent Period 5 a.5 b.5 ẋ x. x p 4p 6p t  Fig. 3 Isochronous oscillations of the SDO, Eq. (11) for A D :5;.5;.75: (a) time histories obtained numerically from Eq. (11) (black dots) and from Eq. (13) (red dashed line); (b) phase trajectories obtained numerically from Eq. (11) (black dots) and from Eq. (1) (red dashed line) (lower signs are used in all these equations). Case II In this case we consider the system whose potential and kinetic energies are E p D 1 X D 1.xf/ ; E k D 1 X P D 1 Pxf C x f ; (14) where f f.i /, I D R t x.t/ dt, and f D df =d I. The corresponding Lagrange s equation is Rx C 3x Px f f C x C f f x3 D : (15) This system has two independent first integrals. The first one is the energy conservation law stemming from XP = C X = D const: Pxf C x f C.x f/ D h 1 ; h 1 D const: (16) The other first integral is related to the principle of conservation of momentum for the SHO XP C R t Xdt D X P./ D const.byusingx and XP from (14) and knowing that di=dt D x, we obtain Z Px f C x f C f.i / di D h ; h D const: (17)
6 6 I.N. Kovacic and R.H. Rand In addition, as the solution for motion for the SHO can be written down as X D a sin.t C /, the following should be satisfied: a sin.t C / D xf; acos.t C / DPxf C x f : (18) For the hardeningtype nonlinearity in Eq. (15), one requires f =f D 1, which leads to Z t f HDO D exp x.t/ dt ; (19) and the equation of motion takes the form Rx C 3x Px C x C x 3 D : () Two first integrals (16) and (17) are h x3 exp.x 1 / C x i C x D h 1 ; (1) and where x 1 D Z t with initial conditions being [see Eq. ()] exp.x 1 / x 3 C x C 1 D h ; () x.t/ dt; x DPx 1 D x; x 3 DPx DPx; (3) x 1./ D ; x./ DPx 1./ D A; x 3./ DRx 1./ D : (4) Equation (1) is plotted in Fig. 4aforh 1 D 1. To analyze phase trajectories in more detail, Eq. () is squared and divided by Eq. (1) to obtain x3 C x C 1 x3 C x D B; B D const: (5) C x This expression agrees with the first integral obtained and studied in [3] and is plotted in Fig. 4b, where periodic solutions correspond to the case B > 1. Note that for the initial conditions (4), one has B D 1 C 1=A, which implies that B is always higher than unity.
7 DuffingType Oscillators with AmplitudeIndependent Period 7 a x 3 x 1 x b 1 5 B=1.5 B=1.1 x 3 B= B=55 B=.1 B= x Fig. 4 (a) 3D plot of Eq.(1); (b) phase trajectories obtained from Eq. (5) for different values of B By using (18) and (19) one can derive Px sin.t C / C x sin.t C / x cos.t C / D : (6) Its solution satisfying Eq. () is sin t C arc tan 1 A x D q 1 C 1 cos : (7) t C arc tan 1 A A
8 8 I.N. Kovacic and R.H. Rand.5 x.  p 4p 6p t Fig. 5 Time response of the HDO, Eq. () for A D :5;.5;.75: numerically obtained solution from Eq. (8) (black dots) and from Eq. (7) (blue solid line) To compare the analytical solution for motion (7) with a numerically obtained solution of the equation of motion (), the latter is written down in the form «x 1 C 3 Px 1 Rx 1 CPx 1 CPx 1 3 D ; (8) and numerically integrated by using the initial conditions (4). This comparison, plotted in Fig. 5, shows that these two types of solution are in full agreement as well as that the period is amplitudeindependent. The equation of motion (15) corresponds to the SDO if f =f D 1, which is satisfied for Z t f SDO D cos x.t/ dt : (9) This equation of motion is now given by Z t Rx 3x Px tan x.t/ dt C x x 3 D : (3) By using the notation given in Eq. (3), the equation of motion (3) transforms to «x 1 3 Px 1 Rx 1 tan x 1 CPx 1 Px 1 3 D ; (31) with the initial conditions given in Eq. (4). Two first integrals (16) and (17) become x3 cos x 1 x sin x 1 C x cos x 1 D h 1 ; (3)
9 DuffingType Oscillators with AmplitudeIndependent Period 9 and x 3 cos x 1 x sin x 1 C sin x 1 D h : (33) These two integrals can be manipulated to exclude x 1 and to derive.x 1/.h 1 C. 1 h 1 C h /x C x4 / C.1 h 1 h C. 1 C h 1 h /x /.h 1.x 1/ x.h C x 1//x 3 C.. 1 C h 1 /.1 C h 1 /h C h4 /x4 3 D : (34) For the initial conditions (4) one has h 1 D A and h D. Introducing these values into Eq. (34) and solving it with respect to x 3, the following explicit solution for phase trajectories is obtained: s x 3 D.x 1/ A x 1 A : (35) Combining equations in (18) and using a D A and D =, we derive Its solution satisfying Eq. () is Px Acos t x p x A cos t C xasin t D : (36) x D A cos t p 1 A sin t : (37) This solution is plotted in Fig. 6 together with the numerical solution of Eq. (31) with Eq. (4) for different values of A. These solutions coincide and confirm isochronicity..5 x.  p 4p 6p t Fig. 6 Time response of the SDO, Eq. (3) with A D :5;.5;.75: numerically obtained solution from Eq. (31) (black dots) and from Eq. (37) (red dashed line)
10 1 I.N. Kovacic and R.H. Rand 3 Conclusions In this work we have considered nonlinear oscillators with a hardening and softening Duffing restoring force. Unlike classical conservative Duffing oscillators, which have an amplitudedependent period, the designed Duffingtype oscillators have the period that does not change with their amplitude and are, thus, isochronous. Two separate cases are considered with respect to the form of their potential and kinetic energy, which are made equal to the corresponding energies of the SHO, which is known to be isochronous. Corresponding equations of motions are derived, as well as their solutions for motion. Numerical verifications of these isochronous solutions are provided. In addition, two independent first integrals are presented: the energyconservation law and the principle of conservation of momentum. Acknowledgements Ivana Kovacic acknowledges support received from the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina (Project No ). References 1. Calogero, F.: Isochronous Systems. Oxford University Press, Oxford (8). Calogero, F.: Isochronous dynamical systems. Phil. Trans. R. Soc. A 369, (11) 3. Iacono,R., Russo, F.: Class of solvable nonlinear oscillators with isochronous orbits. Phys. Rev. E 83, 761 (4 pages) (11) 4. Kovacic, I., Brennan, M.J.: The Duffing Equation: Nonlinear Oscillators and their Behaviour. Wiley, Chichester (11) 5. Kovacic, I., Rand, R.: About a class of nonlinear oscillators with amplitudeindependent frequency. Nonlinear Dyn. 74, (13) 6. Rand, R.H.: Lecture notes on nonlinear vibrations (version 53). handle/1813/8989
Dynamics of a massspringpendulum system with vastly different frequencies
Dynamics of a massspringpendulum system with vastly different frequencies Hiba Sheheitli, hs497@cornell.edu Richard H. Rand, rhr2@cornell.edu Cornell University, Ithaca, NY, USA Abstract. We investigate
More informationThree ways of treating a linear delay differential equation
Proceedings of the 5th International Conference on Nonlinear Dynamics NDKhPI2016 September 2730, 2016, Kharkov, Ukraine Three ways of treating a linear delay differential equation Si Mohamed Sah 1 *,
More informationPhysics 115/242 Comparison of methods for integrating the simple harmonic oscillator.
Physics 115/4 Comparison of methods for integrating the simple harmonic oscillator. Peter Young I. THE SIMPLE HARMONIC OSCILLATOR The energy (sometimes called the Hamiltonian ) of the simple harmonic oscillator
More information18.12 FORCEDDAMPED VIBRATIONS
8. ORCEDDAMPED VIBRATIONS Vibrations A mass m is attached to a helical spring and is suspended from a fixed support as before. Damping is also provided in the system ith a dashpot (ig. 8.). Before the
More informationLecture 5. Outline: Limit Cycles. Definition and examples How to rule out limit cycles. PoincareBendixson theorem Hopf bifurcations Poincare maps
Lecture 5 Outline: Limit Cycles Definition and examples How to rule out limit cycles Gradient systems Liapunov functions Dulacs criterion PoincareBendixson theorem Hopf bifurcations Poincare maps Limit
More informationLogistic Map, Euler & RungeKutta Method and LotkaVolterra Equations
Logistic Map, Euler & RungeKutta Method and LotkaVolterra Equations S. Y. Ha and J. Park Department of Mathematical Sciences Seoul National University Sep 23, 2013 Contents 1 Logistic Map 2 Euler and
More informationElectric and Magnetic Forces in Lagrangian and Hamiltonian Formalism
Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More information2:2:1 Resonance in the Quasiperiodic Mathieu Equation
Nonlinear Dynamics 31: 367 374, 003. 003 Kluwer Academic Publishers. Printed in the Netherlands. ::1 Resonance in the Quasiperiodic Mathieu Equation RICHARD RAND Department of Theoretical and Applied Mechanics,
More information7 Pendulum. Part II: More complicated situations
MATH 35, by T. Lakoba, University of Vermont 60 7 Pendulum. Part II: More complicated situations In this Lecture, we will pursue two main goals. First, we will take a glimpse at a method of Classical Mechanics
More informationIllustrating Dynamical Symmetries in Classical Mechanics: The LaplaceRungeLenz Vector Revisited
Illustrating Dynamical Symmetries in Classical Mechanics: The LaplaceRungeLenz Vector Revisited Ross C. O Connell and Kannan Jagannathan Physics Department, Amherst College Amherst, MA 010025000 Abstract
More informationReport EProject Henriette Laabsch Toni Luhdo Steffen Mitzscherling Jens Paasche Thomas Pache
Potsdam, August 006 Report EProject Henriette Laabsch 7685 Toni Luhdo 7589 Steffen Mitzscherling 7540 Jens Paasche 7575 Thomas Pache 754 Introduction From 7 th February till 3 rd March, we had our laboratory
More informationA Classical Approach to the StarkEffect. Mridul Mehta Advisor: Prof. Enrique J. Galvez Physics Dept., Colgate University
A Classical Approach to the StarkEffect Mridul Mehta Advisor: Prof. Enrique J. Galvez Physics Dept., Colgate University Abstract The state of an atom in the presence of an external electric field is known
More informationTwoBody Problem. Central Potential. 1D Motion
TwoBody Problem. Central Potential. D Motion The simplest nontrivial dynamical problem is the problem of two particles. The equations of motion read. m r = F 2, () We already know that the center of
More informationPoincaré Map, Floquet Theory, and Stability of Periodic Orbits
Poincaré Map, Floquet Theory, and Stability of Periodic Orbits CDS140A Lecturer: W.S. Koon Fall, 2006 1 Poincaré Maps Definition (Poincaré Map): Consider ẋ = f(x) with periodic solution x(t). Construct
More informationIntroduction to Seismology Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 12.510 Introduction
More informationImproving convergence of incremental harmonic balance method using homotopy analysis method
Acta Mech Sin (2009) 25:707 712 DOI 10.1007/s1040900902564 RESEARCH PAPER Improving convergence of incremental harmonic balance method using homotopy analysis method Yanmao Chen Jike Liu Received: 10
More informationTHEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)
4 Vector fields Last updated: November 26, 2009. (Under construction.) 4.1 Tangent vectors as derivations After we have introduced topological notions, we can come back to analysis on manifolds. Let M
More informationRelative Stability of Mathieu Equation in Third Zone
Relative tability of Mathieu Equation in Third Zone N. Mahmoudian, Ph.D. tudent, Nina.Mahmoudian@ndsu.nodak.edu Tel: 701.231.8303, Fax: 701.2318913 G. Nakhaie Jazar, Assistant Professor, Reza.N.Jazar@ndsu.nodak.edu
More informationNo. 5 Discrete variational principle the first integrals of the In view of the face that only the momentum integrals can be obtained by the abo
Vol 14 No 5, May 005 cfl 005 Chin. Phys. Soc. 10091963/005/14(05)/88805 Chinese Physics IOP Publishing Ltd Discrete variational principle the first integrals of the conservative holonomic systems in
More information2. Determine whether the following pair of functions are linearly dependent, or linearly independent:
Topics to be covered on the exam include: Recognizing, and verifying solutions to homogeneous secondorder linear differential equations, and their corresponding Initial Value Problems Recognizing and
More informationHarmonic balance approach to periodic solutions of nonlinear
Harmonic balance approach to periodic solutions of nonlinear jerk equations Author Gottlieb, Hans Published 004 Journal Title Journal of Sound and Vibration DOI https://doi.org/10.1016/s00460x(03)0099
More informationLecture Notes for PHY 405 Classical Mechanics
Lecture Notes for PHY 405 Classical Mechanics From Thorton & Marion s Classical Mechanics Prepared by Dr. Joseph M. Hahn Saint Mary s University Department of Astronomy & Physics September 1, 2005 Chapter
More informationM345 A16 Notes for Geometric Mechanics: Oct Nov 2011
M345 A16 Notes for Geometric Mechanics: Oct Nov 2011 Text for the course: Professor Darryl D Holm 25 October 2011 Imperial College London d.holm@ic.ac.uk http://www.ma.ic.ac.uk/~dholm/ Geometric Mechanics
More information1 f. result from periodic disturbance same period (frequency) as source Longitudinal or Transverse Waves Characterized by
result from periodic disturbance same period (frequency) as source Longitudinal or Transverse Waves Characterized by amplitude (how far do the bits move from their equilibrium positions? Amplitude of MEDIUM)
More informationDynamics of multiple pendula without gravity
June 13, 2013 Institute of Physics University of Zielona Góra Chaos 2013 Istambul, 1115 June Dynamics of multiple pendula without gravity Wojciech Szumiński http://midoriyama.pl For Anna Finding new
More informationVibrations and Waves Physics Year 1. Handout 1: Course Details
Vibrations and Waves JanFeb 2011 Handout 1: Course Details Office Hours Vibrations and Waves Physics Year 1 Handout 1: Course Details Dr Carl Paterson (Blackett 621, carl.paterson@imperial.ac.uk Office
More informationHyperbolicType Orbits in the Schwarzschild Metric
HyperbolicType Orbits in the Schwarzschild Metric F.T. Hioe* and David Kuebel Department of Physics, St. John Fisher College, Rochester, NY 468 and Department of Physics & Astronomy, University of Rochester,
More informationAutonomous system = system without inputs
Autonomous system = system without inputs State space representation B(A,C) = {y there is x, such that σx = Ax, y = Cx } x is the state, n := dim(x) is the state dimension, y is the output Polynomial representation
More informationA SIMPLE, DIRECT PROOF OF UNIQUENESS FOR SOLUTIONS OF THE HAMILTONJACOBI EQUATIONS OF EIKONAL TYPE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 100, Number 2, June 1987 A SIMPLE, DIRECT PROOF OF UNIQUENESS FOR SOLUTIONS OF THE HAMILTONJACOBI EQUATIONS OF EIKONAL TYPE HITOSHI ISHII ABSTRACT.
More informationChapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = kx
Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = kx When the mass is released, the spring will pull
More informationLecture 6. Angular Momentum and Roataion Invariance of L. x = xcosφ + ysinφ ' y = ycosφ xsinφ for φ << 1, sinφ φ, cosφ 1 x ' x + yφ y ' = y xφ,
Lecture 6 Conservation of Angular Momentum (talk about LL p. 9.3) In the previous lecture we saw that time invariance of the L leads to Energy Conservation, and translation invariance of L leads to Momentum
More informationPath integrals in quantum mechanics
Path integrals in quantum mechanics Phys V3500/G8099 handout #1 References: there s a nice discussion of this material in the first chapter of L.S. Schulman, Techniques and applications of path integration.
More information1. < 0: the eigenvalues are real and have opposite signs; the fixed point is a saddle point
Solving a Linear System τ = trace(a) = a + d = λ 1 + λ 2 λ 1,2 = τ± = det(a) = ad bc = λ 1 λ 2 Classification of Fixed Points τ 2 4 1. < 0: the eigenvalues are real and have opposite signs; the fixed point
More informationThe logistic difference equation and the route to chaotic behaviour
The logistic difference equation and the route to chaotic behaviour Level 1 module in Modelling course in population and evolutionary biology (701141800) Module author: Sebastian Bonhoeffer Course director:
More informationThe development of algebraic methods to compute
Ion Energy in Quadrupole Mass Spectrometry Vladimir Baranov MDS SCIEX, Concord, Ontario, Canada Application of an analytical solution of the Mathieu equation in conjunction with algebraic presentation
More informationDerivation of the analytical solution for torque free motion of a rigid body
Derivation of the analytical solution for torque free motion of a rigid body Dimitar N. Dimitrov Abstract In this report the rotational motion of a single rigid body in the absence of external forces and
More information2DVolterraLotka Modeling For 2 Species
Majalat AlUlum AlInsaniya wat  Tatbiqiya 2DVolterraLotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose
More informationELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky
ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky Equilibrium points and linearization Eigenvalue decomposition and modal form State transition matrix and matrix exponential Stability ELEC 3035 (Part
More information14. Matrix treatment of polarization
14. Matri treatment of polarization This lecture Polarized Light : linear, circular, elliptical Jones Vectors for Polarized Light Jones Matrices for Polarizers, Phase Retarders, Rotators (Linear) Polarization
More informationNonchaotic random behaviour in the second order autonomous system
Vol 16 No 8, August 2007 c 2007 Chin. Phys. Soc. 10091963/2007/1608)/228506 Chinese Physics and IOP Publishing Ltd Nonchaotic random behaviour in the second order autonomous system Xu Yun ) a), Zhang
More informationDefinition 5.1. A vector field v on a manifold M is map M T M such that for all x M, v(x) T x M.
5 Vector fields Last updated: March 12, 2012. 5.1 Definition and general properties We first need to define what a vector field is. Definition 5.1. A vector field v on a manifold M is map M T M such that
More informationLecture 1 : p q dq = n q h (1)
Lecture 1 : The WilsonSommerfeld Quantization Rule The success of the Bohr model, as measured by its agreement with experiment, was certainly very striking, but it only accentuated the mysterious nature
More informationOSCILLATORS WITH DELAY COUPLING
Proceedings of DETC'99 1999 ASME Design Engineering Technical Conferences September 115, 1999, Las Vegas, Nevada, USA D E T C 9 9 / V IB 8 3 1 8 BIFURCATIONS IN THE DYNAMICS OF TWO COUPLED VAN DER POL
More informationPhysics 121, April 3, Equilibrium and Simple Harmonic Motion. Physics 121. April 3, Physics 121. April 3, Course Information
Physics 121, April 3, 2008. Equilibrium and Simple Harmonic Motion. Physics 121. April 3, 2008. Course Information Topics to be discussed today: Requirements for Equilibrium (a brief review) Stress and
More informationVibrations: Second Order Systems with One Degree of Freedom, Free Response
Single Degree of Freedom System 1.003J/1.053J Dynamics and Control I, Spring 007 Professor Thomas Peacock 5//007 Lecture 0 Vibrations: Second Order Systems with One Degree of Freedom, Free Response Single
More informationModal Decomposition and the TimeDomain Response of Linear Systems 1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING.151 Advanced System Dynamics and Control Modal Decomposition and the TimeDomain Response of Linear Systems 1 In a previous handout
More informationGeneralized projective synchronization between two chaotic gyros with nonlinear damping
Generalized projective synchronization between two chaotic gyros with nonlinear damping Min FuHong( ) Department of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, China
More informationOscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is
Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 18. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring
More informationSolution for the nonlinear relativistic harmonic oscillator via LaplaceAdomian decomposition method
arxiv:1606.03336v1 [math.ca] 27 May 2016 Solution for the nonlinear relativistic harmonic oscillator via LaplaceAdomian decomposition method O. GonzálezGaxiola a, J. A. Santiago a, J. Ruiz de Chávez
More informationMath 216 First Midterm 19 October, 2017
Math 6 First Midterm 9 October, 7 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationNonlinear System Analysis
Nonlinear System Analysis Lyapunov Based Approach Lecture 4 Module 1 Dr. Laxmidhar Behera Department of Electrical Engineering, Indian Institute of Technology, Kanpur. January 4, 2003 Intelligent Control
More informationC. Show your answer in part B agrees with your answer in part A in the limit that the constant c 0.
Problem #1 A. A projectile of mass m is shot vertically in the gravitational field. Its initial velocity is v o. Assuming there is no air resistance, how high does m go? B. Now assume the projectile is
More informationCALCULATION OF NONLINEAR VIBRATIONS OF PIECEWISELINEAR SYSTEMS USING THE SHOOTING METHOD
Vietnam Journal of Mechanics, VAST, Vol. 34, No. 3 (2012), pp. 157 167 CALCULATION OF NONLINEAR VIBRATIONS OF PIECEWISELINEAR SYSTEMS USING THE SHOOTING METHOD Nguyen Van Khang, Hoang Manh Cuong, Nguyen
More informationPhysics 106b: Lecture 7 25 January, 2018
Physics 106b: Lecture 7 25 January, 2018 Hamiltonian Chaos: Introduction Integrable Systems We start with systems that do not exhibit chaos, but instead have simple periodic motion (like the SHO) with
More informationOrbital Motion in Schwarzschild Geometry
Physics 4 Lecture 29 Orbital Motion in Schwarzschild Geometry Lecture 29 Physics 4 Classical Mechanics II November 9th, 2007 We have seen, through the study of the weak field solutions of Einstein s equation
More informationThe Two Body Central Force Problem
The Two Body Central Force Problem Physics W3003 March 6, 2015 1 The setup 1.1 Formulation of problem The twobody central potential problem is defined by the (conserved) total energy E = 1 2 m 1Ṙ2 1
More informationLinear Spring Oscillator with Two Different Types of Damping
WDS'5 Proceedings of Contributed Papers, Part III, 649 654, 25. ISBN 886732592 MATFYZPRESS Linear Spring Oscillator with Two Different Types of Damping J. Bartoš Masaryk University, Faculty of Science,
More informationPhysics 2101 S c e t c i cti n o 3 n 3 March 31st Announcements: Quiz today about Ch. 14 Class Website:
Physics 2101 Section 3 March 31 st Announcements: Quiz today about Ch. 14 Class Website: http://www.phys.lsu.edu/classes/spring2010/phys2101 3/ http://www.phys.lsu.edu/~jzhang/teaching.html Simple Harmonic
More informationWave Equation in One Dimension: Vibrating Strings and Pressure Waves
BENG 1: Mathematical Methods in Bioengineering Lecture 19 Wave Equation in One Dimension: Vibrating Strings and Pressure Waves References Haberman APDE, Ch. 4 and Ch. 1. http://en.wikipedia.org/wiki/wave_equation
More informationControl Systems. Time response. L. Lanari
Control Systems Time response L. Lanari outline zerostate solution matrix exponential total response (sum of zerostate and zeroinput responses) Dirac impulse impulse response change of coordinates (state)
More informationTaylor series. Chapter Introduction From geometric series to Taylor polynomials
Chapter 2 Taylor series 2. Introduction The topic of this chapter is find approximations of functions in terms of power series, also called Taylor series. Such series can be described informally as infinite
More informationeff (r) which contains the influence of angular momentum. On the left is
1 Fig. 13.1. The radial eigenfunctions R nl (r) of bound states in a squarewell potential for three angularmomentum values, l = 0, 1, 2, are shown as continuous lines in the left column. The form V (r)
More informationThe Principle of Least Action
The Principle of Least Action In their neverending search for general principles, from which various laws of Physics could be derived, physicists, and most notably theoretical physicists, have often made
More informationTravelling waves. Chapter 8. 1 Introduction
Chapter 8 Travelling waves 1 Introduction One of the cornerstones in the study of both linear and nonlinear PDEs is the wave propagation. A wave is a recognizable signal which is transferred from one part
More informationSOLVED PROBLEMS ON TAYLOR AND MACLAURIN SERIES
SOLVED PROBLEMS ON TAYLOR AND MACLAURIN SERIES TAYLOR AND MACLAURIN SERIES Taylor Series of a function f at x = a is ( f k )( a) ( x a) k k! It is a Power Series centered at a. Maclaurin Series of a function
More informationDerivation of the harmonic oscillator propagator using the Feynman path integral and
Home Search Collections Journals About Contact us My IOPscience Derivation of the harmonic oscillator propagator using the Feynman path integral and recursive relations This content has been downloaded
More information1 Lagrangian for a continuous system
Lagrangian for a continuous system Let s start with an example from mechanics to get the big idea. The physical system of interest is a string of length and mass per unit length fixed at both ends, and
More informationLECTURE 16 GAUSS QUADRATURE In general for NewtonCotes (equispaced interpolation points/ data points/ integration points/ nodes).
CE 025  Lecture 6 LECTURE 6 GAUSS QUADRATURE In general for ewtoncotes (equispaced interpolation points/ data points/ integration points/ nodes). x E x S fx dx hw' o f o + w' f + + w' f + E 84 f 0 f
More information21.55 Worksheet 7  preparation problems  question 1:
Dynamics 76. Worksheet 7  preparation problems  question : A coupled oscillator with two masses m and positions x (t) and x (t) is described by the following equations of motion: ẍ x + 8x ẍ x +x A. Write
More informationChaotic motion. Phys 750 Lecture 9
Chaotic motion Phys 750 Lecture 9 Finitedifference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to
More informationNonlinear Oscillations Analysis of the Elevator Cable in a Drum Drive Elevator System
From the SelectedWorks of Hassan Askari Fall September, 214 Nonlinear Oscillations Analysis of the Elevator Cable in a Drum Drive Elevator System Hassan Askari Available at: https://works.bepress.com/hassan_askari/3/
More informationExtended Harmonic Map Equations and the Chaotic Soliton Solutions. Gang Ren 1,* and YiShi Duan 2,*
Extended Harmonic Map Equations and the Chaotic Soliton Solutions Gang Ren,* and YiShi Duan,* The Molecular Foundry, Lawrence erkeley National Laboratory, erkeley C 9470, US The Department of Physics,
More information2. The Schrödinger equation for oneparticle problems. 5. Atoms and the periodic table of chemical elements
1 Historical introduction The Schrödinger equation for oneparticle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical
More information2:1 Resonance in the delayed nonlinear Mathieu equation
Nonlinear Dyn 007) 50:31 35 DOI 10.1007/s110710069165 ORIGINAL ARTICLE :1 Resonance in the delayed nonlinear Mathieu equation Tina M. Morrison Richard H. Rand Received: 9 March 006 / Accepted: 19 September
More informationTimedelay feedback control in a delayed dynamical chaos system and its applications
Timedelay feedback control in a delayed dynamical chaos system and its applications Ye ZhiYong( ), Yang Guang( ), and Deng CunBing( ) School of Mathematics and Physics, Chongqing University of Technology,
More informationBifurcation control and chaos in a linear impulsive system
Vol 8 No 2, December 2009 c 2009 Chin. Phys. Soc. 674056/2009/82)/523507 Chinese Physics B and IOP Publishing Ltd Bifurcation control and chaos in a linear impulsive system Jiang GuiRong 蒋贵荣 ) a)b),
More informationIntroduction to the Calculus of Variations
236861 Numerical Geometry of Images Tutorial 1 Introduction to the Calculus of Variations Alex Bronstein c 2005 1 Calculus Calculus of variations 1. Function Functional f : R n R Example: f(x, y) =x 2
More informationSection 5.8. Taylor Series
Difference Equations to Differential Equations Section 5.8 Taylor Series In this section we will put together much of the work of Sections 5.5.7 in the context of a discussion of Taylor series. We begin
More informationA Mathematical Trivium
A Mathematical Trivium V.I. Arnold 1991 1. Sketch the graph of the derivative and the graph of the integral of a function given by a freehand graph. 2. Find the limit lim x 0 sin tan x tan sin x arcsin
More informationASTATISM IN NONLINEAR CONTROL SYSTEMS WITH APPLICATION TO ROBOTICS
dx dt DIFFERENTIAL EQUATIONS AND CONTROL PROCESSES N 1, 1997 Electronic Journal, reg. N P23275 at 07.03.97 http://www.neva.ru/journal email: diff@osipenko.stu.neva.ru Control problems in nonlinear systems
More informationApproximation for the largeangle simple pendulum period
Approximation for the largeangle simple pendulum period A. Beléndez, J. J. Rodes, T. Beléndez and A. Hernández Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal. Universidad de Alicante.
More informationNonlinear Control Lecture 2:Phase Plane Analysis
Nonlinear Control Lecture 2:Phase Plane Analysis Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 r. Farzaneh Abdollahi Nonlinear Control Lecture 2 1/53
More informationOn the Lebesgue constant of barycentric rational interpolation at equidistant nodes
On the Lebesgue constant of barycentric rational interpolation at equidistant nodes by Len Bos, Stefano De Marchi, Kai Hormann and Georges Klein Report No. 0 May 0 Université de Fribourg (Suisse Département
More informationCarefully tear this page off the rest of your exam. UNIVERSITY OF TORONTO SCARBOROUGH MATA37H3 : Calculus for Mathematical Sciences II
Carefully tear this page off the rest of your exam. UNIVERSITY OF TORONTO SCARBOROUGH MATA37H3 : Calculus for Mathematical Sciences II REFERENCE SHEET The (natural logarithm and exponential functions (
More informationTopics in Stochastic Geometry. Lecture 4 The Boolean model
Institut für Stochastik Karlsruher Institut für Technologie Topics in Stochastic Geometry Lecture 4 The Boolean model Lectures presented at the Department of Mathematical Sciences University of Bath May
More informationMonte Carlo simulations of harmonic and anharmonic oscillators in discrete Euclidean time
Monte Carlo simulations of harmonic and anharmonic oscillators in discrete Euclidean time DESY Summer Student Programme, 214 Ronnie Rodgers University of Oxford, United Kingdom Laura Raes University of
More informationComputational Physics (6810): Session 8
Computational Physics (6810): Session 8 Dick Furnstahl Nuclear Theory Group OSU Physics Department February 24, 2014 Differential equation solving Session 7 Preview Session 8 Stuff Solving differential
More informationResearch Article Parametric Evaluations of the RogersRamanujan Continued Fraction
International Mathematics and Mathematical Sciences Volume 011, Article ID 940839, 11 pages doi:10.1155/011/940839 Research Article Parametric Evaluations of the RogersRamanujan Continued Fraction Nikos
More informationKink, singular soliton and periodic solutions to class of nonlinear equations
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1939466 Vol. 10 Issue 1 (June 015 pp. 1  Applications and Applied Mathematics: An International Journal (AAM Kink singular soliton and periodic
More informationStable OneDimensional Dissipative Solitons in Complex CubicQuintic Ginzburg Landau Equation
Vol. 112 (2007) ACTA PHYSICA POLONICA A No. 5 Proceedings of the International School and Conference on Optics and Optical Materials, ISCOM07, Belgrade, Serbia, September 3 7, 2007 Stable OneDimensional
More informationWHITE NOISE APPROACH TO FEYNMAN INTEGRALS. Takeyuki Hida
J. Korean Math. Soc. 38 (21), No. 2, pp. 275 281 WHITE NOISE APPROACH TO FEYNMAN INTEGRALS Takeyuki Hida Abstract. The trajectory of a classical dynamics is detrmined by the least action principle. As
More informationLongitudinal Waves. waves in which the particle or oscillator motion is in the same direction as the wave propagation
Longitudinal Waves waves in which the particle or oscillator motion is in the same direction as the wave propagation Longitudinal waves propagate as sound waves in all phases of matter, plasmas, gases,
More informationHarmonic balance approach for a degenerate torus of a nonlinear jerk equation
Harmonic balance approach for a degenerate torus of a nonlinear jerk equation Author Gottlieb, Hans Published 2009 Journal Title Journal of Sound and Vibration DOI https://doi.org/10.1016/j.jsv.2008.11.035
More informationLectures on Periodic Orbits
Lectures on Periodic Orbits 11 February 2009 Most of the contents of these notes can found in any typical text on dynamical systems, most notably Strogatz [1994], Perko [2001] and Verhulst [1996]. Complete
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS AP PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS AP PHYSICS LSN 111: SIMPLE HARMONIC MOTION LSN 11: ENERGY IN THE SIMPLE HARMONIC OSCILLATOR LSN 113: PERIOD AND THE SINUSOIDAL NATURE OF SHM Introductory Video:
More informationMAT137 Calculus! Lecture 45
official website http://uoft.me/mat137 MAT137 Calculus! Lecture 45 Today: Taylor Polynomials Taylor Series Next: Taylor Series Power Series Definition (Power Series) A power series is a series of the form
More information1 + z 1 x (2x y)e x2 xy. xe x2 xy. x x3 e x, lim x log(x). (3 + i) 2 17i + 1. = 1 2e + e 2 = cosh(1) 1 + i, 2 + 3i, 13 exp i arctan
Complex Analysis I MT333P Problems/Homework Recommended Reading: Bak Newman: Complex Analysis Springer Conway: Functions of One Complex Variable Springer Ahlfors: Complex Analysis McGrawHill Jaenich:
More informationSchrödinger equation for central potentials
Chapter 2 Schrödinger equation for central potentials In this chapter we will extend the concepts and methods introduced in the previous chapter for a onedimensional problem to a specific and very important
More informationNUMERICAL STUDY OF INTRINSIC FEATURES OF ISOLAS IN A 2DOF NONLINEAR SYSTEM
NUMERICAL STUDY OF INTRINSIC FEATURES OF ISOLAS IN A 2DOF NONLINEAR SYSTEM T. Detroux 1, J.P. Noël 1, L. Masset 1, G. Kerschen 1, L.N. Virgin 2 1 Space Structures and Systems Lab., Department of Aerospace
More information