Strongly Nonlinear Oscillators: Analytical Solutions

241 Pages·2014·6.41 MB·English

by  Livija Cveticanin (auth.)

Undergraduate Lecture Notes in Physics
Livija Cveticanin
Strongly
Nonlinear
Oscillators
Analytical Solutions
Undergraduate Lecture Notes in Physics
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Livija Cveticanin
Strongly Nonlinear
Oscillators
Analytical Solutions
123
LivijaCveticanin
Faculty ofTechnical Sciences
Universityof NoviSad
NoviSad
Serbia
ISSN 2192-4791 ISSN 2192-4805 (electronic)
ISBN 978-3-319-05271-7 ISBN 978-3-319-05272-4 (eBook)
DOI 10.1007/978-3-319-05272-4
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Preface
This book is the result of my long-time investigations and interest in the field of
nonlinearvibration.Theintentionofthistextistogivetheapproximateanalytical
solution procedures for strong nonlinear oscillators and to explain some of the
phenomena that occur in such systems. The book considers the free and forced
vibrations, takes the positiveand negativedamping ofVan der Poltype,analyzes
the criteria for deterministic chaos, and investigates the parametrically excited
vibration in one-degree-of-freedom oscillators. Special attention is given to
vibration properties of a two and one mass systems with two-degrees-of-freedom
where the oscillation of a rotor, as a practical device, is discussed. The ideal and
nonideal nonlinear mechanical systems are also treated where the jump phenom-
ena, the Sommerfeld effect, and the control of the system are included. The basic
partforallconsiderationsisapurenonlinearoscillatorwhoseorderofnonlinearity
is any positive rational number (integer or non-integer). This type of nonlinearity
is the generalization for the previously discussed linear or pure cubic oscillators
and oscillators with small nonlinearity. All the suggested solution procedures are
based on the exact or approximate solution of the strong nonlinear differential
equation which is the mathematical model of the corresponding oscillator.
I hope that the book will be suitable as a textbook for students in nonlinear
vibrations, but also for those who are researching the nonlinear phenomena in
oscillatorysystemsinmechanics,mechanicaldevices,electromechanicalsystems,
electric circuits, physics, chemistry, etc.
The book has an intention to give some practical information to engineers and
technicians dealing with the problem of vibration and its elimination. The results
of investigation show that independently of the amplitude and frequency of
excitation force by proper treatment of the strong nonlinear system, the vibration
level may be kept at a small level. Namely, in mechanical systems like cutting
machines with periodical motion of cutting tools, presses, supports for machines,
seats in vehicles, etc., but also in electronics (electromechanical devices like
microactuators and micro oscillators) the requirement of small oscillations but
withoutintroducingofdampers,whichcauseenergydissipationanddecreasingof
the efficiency of machines, can be achieved by proper use of the nonlinear prop-
erties of the system.
Theresultspublishedinthisbookareapplicableforimprovementindesigning,
forexample,ofmusicinstrumentsandtheirpartssuchasthehammersinapiano.
v
vi Preface
Attheotherside,theinvestigationisalsoofpotentialinterestformodelinghuman
voice production in cases where the vocal cords and voice producing fold are
damaged.
Finally, I have to thank to my family for the support to write and publish this
book.
Novi Sad, Serbia Livija Cveticanin
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Nonlinear Oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Physical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Mathematical Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Pure Nonlinear Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 Qualitative Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Exact Period of Vibration . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Exact Periodical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.2 Cubic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Adopted Lindstedt-Poincaré Method . . . . . . . . . . . . . . . . . . . . 26
3.4 Modified Lindstedt-Poincaré Method. . . . . . . . . . . . . . . . . . . . 30
3.4.1 Comparison of the LP and MLP Methods . . . . . . . . . . . 31
3.4.2 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Exact Amplitude, Period and Velocity Method . . . . . . . . . . . . . 33
3.6 Solution in the Form of Jacobi Elliptic Function. . . . . . . . . . . . 34
3.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7 Solution in the Form of a Trigonometric Function. . . . . . . . . . . 38
3.7.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.7.2 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.8 Pure Nonlinear Oscillator with Linear Damping . . . . . . . . . . . . 41
3.8.1 Parameter Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.8.2 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Free Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Homotopy-Perturbation Technique. . . . . . . . . . . . . . . . . . . . . . 51
4.1.1 Duffing Oscillator with a Quadratic Term . . . . . . . . . . . 54
4.1.2 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Averaging Solution Procedure. . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Solution in the Form of an Ateb Function . . . . . . . . . . . . . . . . 58
vii
viii Contents
4.3.1 Small Nonlinear Deflection Functions. . . . . . . . . . . . . . 59
4.3.2 Differential Equation with a Linear Dominant Term. . . . 62
4.4 Solution in the Form of the Jacobi Elliptic Function . . . . . . . . . 65
4.4.1 Oscillator with Nonlinear Elastic Force. . . . . . . . . . . . . 67
4.5 Solution in the Form of a Trigonometric Function. . . . . . . . . . . 71
4.5.1 Oscillator with Small Linear Damping. . . . . . . . . . . . . . 73
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.7 Oscillator with Linear Damping . . . . . . . . . . . . . . . . . . . . . . . 77
4.7.1 Van der Pol Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . 79
4.7.2 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 Oscillators with Time Variable Parameters. . . . . . . . . . . . . . . . . . 87
5.1 Oscillators with Slow Time Variable Parameters. . . . . . . . . . . . 88
5.2 Solution in the Form of the Ateb Function. . . . . . . . . . . . . . . . 88
5.2.1 Oscillator with Linear Time Variable Parameter. . . . . . . 91
5.3 Solution in the Form of a Trigonometric Function. . . . . . . . . . . 93
5.3.1 Linear Oscillator with Time Variable Parameters . . . . . . 95
5.3.2 Non-integer Order Nonlinear Oscillator. . . . . . . . . . . . . 96
5.3.3 Levi-Civita Oscillator with a Small Damping. . . . . . . . . 96
5.3.4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4 Solution in the Form of a Jacobi Elliptic Function. . . . . . . . . . . 101
5.4.1 Van der Pol Oscillator with Time Variable Mass . . . . . . 103
5.4.2 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.5 Parametrically Excited Strong Nonlinear Oscillator. . . . . . . . . . 111
5.5.1 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.5.2 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.5.3 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6 Forced Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.1 Oscillator with Constant Excitation Force. . . . . . . . . . . . . . . . . 128
6.1.1 Solution of the Odd-Integer Order Oscillator . . . . . . . . . 131
6.1.2 The Oscillator with Additional Small Nonlinearity . . . . . 134
6.1.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.1.4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.2 Harmonically Excited Pure Nonlinear Oscillator . . . . . . . . . . . . 142
6.2.1 Pure Odd-Order Nonlinear Oscillator. . . . . . . . . . . . . . . 142
6.2.2 Bifurcation in the Oscillator. . . . . . . . . . . . . . . . . . . . . 145
6.2.3 Harmonically Forced Pure Cubic Oscillator . . . . . . . . . . 147
6.2.4 Numerical Simulation and Discussion . . . . . . . . . . . . . . 153
6.2.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Contents ix
7 Two-Degree-of-Freedom Oscillator. . . . . . . . . . . . . . . . . . . . . . . . 161
7.1 Two-Mass System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.1.1 Two-Degree-of-Freedom Van der Pol Oscillator. . . . . . . 164
7.1.2 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.2 Complex-Valued Differential Equation. . . . . . . . . . . . . . . . . . . 173
7.2.1 Adopted Krylov-Bogolubov Method . . . . . . . . . . . . . . . 174
7.2.2 Method Based on the First Integrals . . . . . . . . . . . . . . . 176
7.2.3 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8 Chaos in Oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.1 Chaos in Ideal Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.1.1 Homoclinic Orbits in the Unperturbed System . . . . . . . . 193
8.1.2 Melnikov’s Criteria for Chaos . . . . . . . . . . . . . . . . . . . 195
8.1.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.1.4 Lyapunov Exponents and Bifurcation Diagrams . . . . . . . 202
8.1.5 Control of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.1.6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.2 Chaos in Non-ideal Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . 206
8.2.1 Modeling of the System. . . . . . . . . . . . . . . . . . . . . . . . 206
8.2.2 Asymptotic Solving Method. . . . . . . . . . . . . . . . . . . . . 208
8.2.3 Stability and Sommerfeld Effect. . . . . . . . . . . . . . . . . . 209
8.2.4 Numerical Simulation and Chaotic Behavior . . . . . . . . . 214
8.2.5 Control of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.2.6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Appendix A: Periodical Ateb Functions . . . . . . . . . . . . . . . . . . . . . . . 223
Appendix B: Averaging of Ateb Functions. . . . . . . . . . . . . . . . . . . . . 227
Appendix C: Jacobi Elliptic Functions. . . . . . . . . . . . . . . . . . . . . . . . 231
Appendix D: Euler’s Integrals of the First and Second Kind. . . . . . . . 233
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237