Nonlinear Physical Systems E-Book

Oleg N. Kirillov

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 649

Veröffentlichungsjahr: 2013

Beschreibung

Bringing together 18 chapters written by leading experts in dynamical systems, operator theory, partial differential equations, and solid and fluid mechanics, this book presents state-of-the-art approaches to a wide spectrum of new and challenging stability problems. Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations focuses on problems of spectral analysis, stability and bifurcations arising in the nonlinear partial differential equations of modern physics. Bifurcations and stability of solitary waves, geometrical optics stability analysis in hydro- and magnetohydrodynamics, and dissipation-induced instabilities are treated with the use of the theory of Krein and Pontryagin space, index theory, the theory of multi-parameter eigenvalue problems and modern asymptotic and perturbative approaches. Each chapter contains mechanical and physical examples, and the combination of advanced material and more tutorial elements makes this book attractive for both experts and non-specialists keen to expand their knowledge on modern methods and trends in stability theory. Contents 1. Surprising Instabilities of Simple Elastic Structures, Davide Bigoni, Diego Misseroni, Giovanni Noselli and Daniele Zaccaria. 2. WKB Solutions Near an Unstable Equilibrium and Applications, Jean-François Bony, Setsuro Fujiié, Thierry Ramond and Maher Zerzeri, partially supported by French ANR project NOSEVOL. 3. The Sign Exchange Bifurcation in a Family of Linear Hamiltonian Systems, Richard Cushman, Johnathan Robbins and Dimitrii Sadovskii. 4. Dissipation Effect on Local and Global Fluid-Elastic Instabilities, Olivier Doaré. 5. Tunneling, Librations and Normal Forms in a Quantum Double Well with a Magnetic Field, Sergey Yu. Dobrokhotov and Anatoly Yu. Anikin. 6. Stability of Dipole Gap Solitons in Two-Dimensional Lattice Potentials, Nir Dror and Boris A. Malomed. 7. Representation of Wave Energy of a Rotating Flow in Terms of the Dispersion Relation, Yasuhide Fukumoto, Makoto Hirota and Youichi Mie. 8. Determining the Stability Domain of Perturbed Four-Dimensional Systems in 1:1 Resonance, Igor Hoveijn and Oleg N. Kirillov. 9. Index Theorems for Polynomial Pencils, Richard Kollár and Radomír Bosák. 10. Investigating Stability and Finding New Solutions in Conservative Fluid Flows Through Bifurcation Approaches, Paolo Luzzatto-Fegiz and Charles H.K. Williamson. 11. Evolution Equations for Finite Amplitude Waves in Parallel Shear Flows, Sherwin A. Maslowe. 12. Continuum Hamiltonian Hopf Bifurcation I, Philip J. Morrison and George I. Hagstrom. 13. Continuum Hamiltonian Hopf Bifurcation II, George I. Hagstrom and Philip J. Morrison. 14. Energy Stability Analysis for a Hybrid Fluid-Kinetic Plasma Model, Philip J. Morrison, Emanuele Tassi and Cesare Tronci. 15. Accurate Estimates for the Exponential Decay of Semigroups with Non-Self-Adjoint Generators, Francis Nier. 16. Stability Optimization for Polynomials and Matrices, Michael L. Overton. 17. Spectral Stability of Nonlinear Waves in KdV-Type Evolution Equations, Dmitry E. Pelinovsky. 18. Unfreezing Casimir Invariants: Singular Perturbations Giving Rise to Forbidden Instabilities, Zensho Yoshida and Philip J. Morrison. About the Authors Oleg N. Kirillov has been a Research Fellow at the Magneto-Hydrodynamics Division of the Helmholtz-Zentrum Dresden-Rossendorf in Germany since 2011. His research interests include non-conservative stability problems of structural mechanics and physics, perturbation theory of non-self-adjoint boundary eigenvalue problems, magnetohydrodynamics, friction-induced oscillations, dissipation-induced instabilities and non-Hermitian problems of optics and microwave physics. Since 2013 he has served as an Associate Editor for the journal Frontiers in Mathematical Physics. Dmitry E. Pelinovsky has been Professor at McMaster University in Canada since 2000. His research profile includes work with nonlinear partial differential equations, discrete dynamical systems, spectral theory, integrable systems, and numerical analysis. He served as the guest editor of the special issue of the journals Chaos in 2005 and Applicable Analysis in 2010. He is an Associate Editor of the journal Communications in Nonlinear Science and Numerical Simulations. This book is devoted to the problems of spectral analysis, stability and bifurcations arising from the nonlinear partial differential equations of modern physics. Leading experts in dynamical systems, operator theory, partial differential equations, and solid and fluid mechanics present state-of-the-art approaches to a wide spectrum of new challenging stability problems. Bifurcations and stability of solitary waves, geometrical optics stability analysis in hydro- and magnetohydrodynamics and dissipation-induced instabilities will be treated with the use of the theory of Krein and Pontryagin space, index theory, the theory of multi-parameter eigenvalue problems and modern asymptotic and perturbative approaches. All chapters contain mechanical and physical examples and combine both tutorial and advanced sections, making them attractive both to experts in the field and non-specialists interested in knowing more about modern methods and trends in stability theory.

Leseprobe

Table of Contents

Preface

Chapter 1: Surprising Instabilities of Simple Elastic Structures

1.1. Introduction

1.2. Buckling in tension

1.3. The effect of constraint’s curvature

1.4. The Ziegler pendulum made unstable by Coulomb friction

1.5. Conclusions

1.6. Acknowledgments

1.7. Bibliography

Chapter 2: WKB Solutions Near an Unstable Equilibrium and Applications

2.1. Introduction

2.2. Connection of microlocal solutions near a hyperbolic fixed point

2.3. Applications to semi–classical resonances

2.4. Acknowledgment

2.5. Bibliography

Chapter 3: The Sign Exchange Bifurcation in a Family of Linear Hamiltonian Systems

3.1. Statement of problem

3.2. Bifurcation values of γ

3.3. Versal normal forms near the bifurcation values

3.4. Infinitesimally symplectic normal form

3.5. Global issues

3.6. Bibliography

Chapter 4: Dissipation Effect on Local and Global Fluid–Elastic Instabilities

4.1. Introduction

4.2. Local and global stability analyses

4.3. The fluid–conveying pipe: a model problem

4.4. Effect of damping on the local and global stability of the fluid–conveying pipe

4.5. Application to energy harvesting

4.6. Conclusion

4.7. Bibliography

Chapter 5: Tunneling, Librations and Normal Forms in a Quantum Double Well with a Magnetic Field

5.1. Introduction

5.2. 1D Landau–Lifshitz splitting formula and its analog for the ground states

5.3. The splitting formula in multi–dimensional case

5.4. Normal forms and complex Lagrangian manifolds

5.5. Constructing the asymptotics for the eigenfunctions in tunnel problems

5.6. Splitting of the eigenvalues in the presence of magnetic field

5.7. Proof of main theorem (a sketch)

5.8. Conclusion

5.9. Acknowledgments

5.10. Bibliography

Chapter 6: Stability of Dipole Gap Solitons in Two–Dimensional Lattice Potentials

6.1. Introduction

6.2. The model

6.3. Solitons in the first bandgap: the SF nonlinearity

6.4. Stability GSs in the second bandgap

6.5. Conclusions

6.6. Bibliography

Chapter 7: Representation of Wave Energy of a Rotating Flow in Terms of the Dispersion Relation

7.1. Introduction

7.2. Lagrangian approach to wave energy

7.3. Kelvin waves

7.4. Wave energy in terms of the dispersion relation

7.5. Conclusion

7.6. Bibliography

Chapter 8: Determining the Stability Domain of Perturbed Four–Dimensional Systems in 1:1 Resonance

8.1. Introduction

8.2. Methods

8.3. Examples

8.4. Conclusions

8.5. Bibliography

Chapter 9: Index Theorems for Polynomial Pencils

9.1. Introduction

9.2. Krein signature

9.3. Index theorems for linear pencils and linearized Hamiltonians

9.4. Graphical interpretation of index theorems

9.5. Conclusions

9.6. Acknowledgments

9.7. Bibliography

Chapter 10: Investigating Stability and Finding New Solutions in Conservative Fluid Flows Through Bifurcation Approaches

10.1. Introduction

10.2. Counting positive–energy modes from IVI diagrams

10.3. An approximate prediction for the onset of resonance in 2D vortices

10.4. An example: three corotating vortices

10.5. Comparison with exact eigenvalues and discussion

10.6. Conclusions

10.7. Bibliography

Chapter 11: Evolution Equations for Finite Amplitude Waves in Parallel Shear Flows

11.1. Introduction

11.2. Wave packets

11.3. Critical layer theory

11.4. Nonlinear instabilities governed by integro–differential equations

11.5. Concluding remarks

11.6. Bibliography

Chapter 12: Continuum Hamiltonian Hopf Bifurcation I

12.1. Introduction

12.2. Discrete Hamiltonian bifurcations

12.3. Continuum Hamiltonian bifurcations

12.4. Summary and conclusions

12.5. Acknowledgments

12.6. Bibliography

Chapter 13: Continuum Hamiltonian Hopf Bifurcation II

13.1. Introduction

13.2. Mathematical aspects of the continuum Hamiltonian Hopf bifurcation

13.3. Application to Vlasov–Poisson

13.4. Canonical infinite–dimensional case

13.5. Commentary: degeneracy and nonlinearity

13.6. Summary and conclusions

13.7. Acknowledgments

13.8. Bibliography

Chapter 14: Energy Stability Analysis for a Hybrid Fluid–Kinetic Plasma Model

14.1. Introduction

14.2. Stability and the energy–Casimir method

14.3. Planar Hamiltonian hybrid model

14.4. Energy–Casimir stability analysis

14.5. Conclusions

14.6. Acknowledgments

14.7. Appendix A: derivation of hybrid Hamiltonian structure

14.8. Appendix B: Casimir verification

14.9. Bibliography

Chapter 15: Accurate Estimates for the Exponential Decay of Semigroups with Non–Self–Adjoint Generators

15.1. Introduction

15.2. Relevant quantities for sectorial operators

15.3. Natural examples

15.4. Artificial examples

15.5. Conclusion

15.6. Bibliography

Chapter 16: Stability Optimization for Polynomials and Matrices

16.1. Optimization of roots of polynomials

16.2. Optimization of eigenvalues of matrices

16.3. Concluding remarks

16.4. Acknowledgments

16.5. Bibliography

Chapter 17: Spectral Stability of Nonlinear Waves in KdV–Type Evolution Equations

17.1. Introduction

17.2. Historical remarks and examples

17.3. Proof of theorem 17.1

17.4. Generalization of theorem 17.1 for a periodic nonlinear wave

17.5. Conclusion

17.6. Bibliography

Chapter 18: Unfreezing Casimir Invariants: Singular Perturbations Giving Rise to Forbidden Instabilities

18.1. Introduction

18.2. Preliminaries: noncanonical Hamiltonian systems and Casimir invariants

18.3. Foliation by adiabatic invariants

18.4. Canonization atop Casimir leaves

18.5. Application to tearing–mode theory

18.6. Conclusion

18.7. Acknowledgments

18.8. Bibliography

List of Authors

Index

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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Preface

The BIRS Workshop on Spectral Analysis, Stability and Bifurcations in Modern Nonlinear Physical Systems1 brought together a unique combination of experts in modern dynamical systems, mathematical physics, partial differential equations (PDEs), numerical analysis, operator theory and applications.

One of the immediate outcomes of the meeting is this post-conference volume of papers from the participants of the workshops making its materials available to a wider audience. This book presents unique viewpoints of the participants on the history, current state of the art and prospects of research in their fields contributing to the progress of stability theory. In this book, we have compiled a collection of essays – mathematical, physical and mechanical. The contributions show connections between different approaches, applications and ideas. We believe that such a book could set the benchmarks and goals for the next generation of researchers and be a true event in modern stability theory. The other outcomes will be seen over a long period of time, when the ideas formulated and discussed during the workshop, as well as new collaborations made, will lead to new scientific publications and new research discoveries.

This book covers the problems of spectral analysis, stability and bifurcations arising from the nonlinear PDEs of modern physics. Bifurcations and stability of solitary waves, stability analysis in hydro- and magnetohydrodynamics and dissipation-induced instabilities will be treated with the use of the theory of Krein and Pontryagin space, index theory, the theory of multiparameter eigenvalue problems and modern asymptotic and perturbative approaches. All chapters contain mechanical and physical examples and combine both tutorial and advanced sections, making them attractive both to professionals working in the field and non-specialists interested in knowing more about modern methods and trends in stability theory.

Chapter 1, written by Davide Bigoni and his colleagues, opens the book and presents the reader with sophisticated experiments with simple mechanical structures demonstrating buckling under tensile dead loading (without elements subject to compression at all) and flutter or oscillatory instability of a two-link pendulum that is caused by Coulomb friction. This new look at the classical mechanics is directly motivated by the successes of modern materials science.

The semi-classical n-dimensional quantum tunneling effect, through a hyperbolic fixed point, is treated by Jean-François Bony et al. in Chapter 2. The transfer operator which solves this microlocal Cauchy problem appears to be a Fourier integral operator which gives outgoing waves in terms of incoming waves. As an application, the longtime behavior of the Schrödinger group at barrier top is described in term of resonances with explicit generalized spectral projections. Another application is to obtain resonances free regions for homoclinic trapped sets.

A semi-classical limit of a quantum problem on angular momenta interacting in a magnetic field has led Richard Cushman and his colleagues to a curious one-parameter family of Hamiltonian systems in Chapter 3. Their system exhibits an S1-equivariant sign exchange bifurcation in its linearization about an equilibrium point. The stability of this bifurcation under small S1-invariant perturbations by linear Hamiltonian vector fields is shown in an instructive manner involving the method of versal deformations.

In Chapter 4, Olivier Doaré discusses the counter-intuitive destabilizing effect of damping in the problems of fluid–structure interaction. A model problem considered is a fluid–conveying pipe where the viscous damping is shown to destabilize the negative energy waves. The fluid-conveying pipe is a model problem for many fluid-elastic systems where a compliant structure interacts with a flow, such as flags, plates, shells, walls or wings. The model is of particular interest in the modern energy-harvesting applications.

Sergey Dobrokhotov and Anatoly Anikin discuss in Chapter 5 the splitting of the lowest eigenvalues of the multidimensional Schrödinger operator with the double-well potential. As a rule, the splitting formula is based on the instanton, which is a singular trajectory of the Newtonian system with inverted potential. However, a physically relevant form of the formula should involve, as the authors demonstrate, not the instanton but an appropriate unstable periodic trajectory (libration).

Periodic potentials and solitons are the subject of Chapter 6, written by Nir Dror and Boris Malomed. To stabilize the solitons in a two-dimensional Bose-Einstein condensate, a linear periodic potential is induced by means of the optical lattices, which are the interference patterns created by laser beams shone through the condensate. Such periodic potentials give rise to bandgaps in the corresponding linear spectrum, which, in combination with the self-focusing or self-defocusing nonlinearity, support various types of localized mode. The authors demonstrate that bound complexes built of the dipole solitons, in the form of bi-dipoles and four-dipole non-topological states, vortices and quadrupoles, are all stable if the underlying dipole is stable.

A steady Euler flow of an inviscid incompressible fluid is characterized as an extremum of the total kinetic energy with respect to perturbations constrained to an isovortical sheet. Yasuhide Fukumoto et al. analyze in Chapter 7 the criticality in the Hamiltonian to calculate the energy of three-dimensional waves on a steady vortical flow and to calculate the mean flow induced by nonlinear interaction of waves with themselves. The energy of waves on a rotating flow is expressible in terms of a derivative of the dispersion relation with respect to the frequency.

Pure imaginary eigenvalues in 1:1 semi-simple resonance (diabolical points in the physics language) typically occur in rotationally symmetrical non-dissipative models of physics and engineering. Its unfolding caused by symmetry-breaking and non-conservative perturbation is a reason for many instabilities such as the rotating polygon instability of swirling free surface flow. In Chapter 8, Igor Hoveijn and Oleg Kirillov map all possible singularities on the boundary of the stability domain of perturbed four-dimensional systems in 1:1 resonance and apply the result to the study of the enhancement of the modulation instability with dissipation.

Since the time of the celebrated Kelvin–Tait–Chetaev theorem, counts of unstable point spectra and other related counts that are referred to as index theorems have appeared across various distinct and unrelated fields due to their simple structure and importance for stability applications. Richard Kollár and Radomír Bosák give in Chapter 9 a unique and comprehensive survey of the index theorems motivated by very different physical, algebraic and control theory applications and also present a graphical Krein signature theory. The latter makes the proofs of index theorems for linearized Hamiltonians extremely elegant in the finite dimensional setting: a general result implying Vakhitov–Kolokolov criterion (or Grillakis–Shatah–Strauss criterion) as a corollary generalized to problems with arbitrary kernels, and a count of real eigenvalues for linearized Hamiltonian systems in canonical form.

Chapter 10 provides an example of counting unstable eigenvalues in the problems of vortex dynamics presented by Paolo Luzzatto-Fegiz and Charles H.K. Williamson. They demonstrate that the turning points in impulse of the vortex array correspond to a change in the number of unstable modes. Furthermore, whether the isovortical rearrangements involve the introduction or removal of an unstable mode can be inferred from the shape of a fold in the phase velocity–impulse plot.

In Chapter 11, the fluid dynamical theme is continued by Sherwin Maslowe who provides a general and comprehensive survey of the finite amplitude theory and discusses in detail the critical layer analyses that indicate, in particular, important resolution requirements for computational schemes.

A main motivation for studying Hamiltonian systems is their universality. In Chapter 12, Philip Morrison and George Hagstrom show how infinite-dimensional noncanonical Hamiltonian systems enlarge this universality class. Any specific system within the classes of systems considered may possess steady-state bifurcations, positive and negative energy modes and Krein’s theorem for the Hamiltonian Hopf bifurcations. An analogous situation transpires for the continuous steady-state and Hamiltonian Hopf bifurcations. However, continuous spectra are difficult to deal with mathematically and functional analysis is essential. For example, we can interpret the continuous Hamiltonian Hopf bifurcation as the Hamiltonian Hopf bifurcation with the second mode coming from the continuous spectrum. Chapter 12 sets the stage for the explicit treatment of bifurcations with the continuous spectrum that is considered in Chapter 13.

A hybrid fluid-kinetic model of plasma physics considered by Philip Morrison and his coauthors in Chapter 14 combines a magnetohydrodynamics (MHD) part for a description of bulk fluid components and a Vlasov kinetic theory part that describes an energetic plasma component. In the considered model, a Hamiltonian structure is found that allows the authors to implement the energy-Casimir method for an explicit derivation of sufficient stability conditions.

Semigroups (or dynamical systems) of contractions in Hilbert space with non-self-adjoint generators considered by Francis Nier in Chapter 15 are motivated by the linearization of incompressible 2D-Navier-Stokes equation in the vortex formulation around Oseen vortices and by the Feller semigroup associated with the Langevin dynamics, which solves the Kramers–Fokker–Planck equation. The accurate estimates for the exponential decay of such semigroups with parameter-dependent non-self-adjoint generators obtained by the author substantially involve the theory of pseudo-spectrum.

The theory of pseudo-spectrum reappears in Chapter 16 where Michael Overton gives a broad survey of recent achievements in stability optimization for polynomials and matrices. The optimization problems discussed in this chapter typically lead to optimizers that are polynomials with multiple roots or matrices with non-derogatory multiple eigenvalues. The higher their multiplicity, the more these multiple roots or eigenvalues are sensitive to small perturbations; furthermore, computing these minimizers numerically is difficult. Instead of optimizing eigenvalues it is proposed to consider optimization of the pseudo-spectral radius and pseudo-spectral abscissa, which is computationally less difficult than for the spectral radius and spectral abscissa.

In Chapter 17, Dmitry Pelinovsky returns to the index theory and proves the index theorem in a rather general setting motivated by the problems of stability of nonlinear waves in KdV-type evolution equations. The directions leading to further extensions of this result are pointed out.

In the final Chapter 18, Zensho Yoshida and Philip Morrison describe several facets of noncanonical Hamiltonian systems, namely, the Poisson operator (field tensor) of a noncanonical Hamiltonian system has a non-trivial kernel (and thus, a cokernel) that foliates the phase space (Poisson manifold), imposing topological constraints on the dynamics. When we can “integrate” the kernel of the Poisson operator to construct Casimir elements, the Casimir leafs foliate the Poisson manifold and, then, the effective energy is the energy-Casimir functional. The theory is applied to the tearing-mode instability, where a tearing mode is regarded as an equilibrium point on a helical-flux Casimir leaf. As long as the helical-flux is constrained, the tearing mode cannot grow. However, it is shown that a singular perturbation that allows the system to change the helical flux can cause a tearing mode to grow if it has an excess energy with respect to a fiducial energy of the Beltrami equilibrium at the bifurcation point.

Oleg N. Kirillov

Dmitry E. Pelinovsky

October 2013

1  Took place at the Banff International Research Station for Mathematical Innovation and Discovery, Banff, Canada on 4–9 November 2012. For more information see http://www.birs.ca/events/2012/5-day-workshops/12w5073.

Chapter 1

Surprising Instabilities of Simple Elastic Structures

In this chapter, examples of structures buckling in tension are presented, where no compressed elements are present, slightly different from those previously proposed by the authors. These simple structures exhibit interesting postcritical behaviors; for instance, multiple configurations of vanishing external force are evidenced in one case. Flutter instability as induced by dry friction is also considered in the Ziegler pendulum, with the same arrangement presented by Bigoni and Noselli [BIG 11], but now considering the dynamical effects due to the mass of the wheel, which was previously neglected. It is shown that, for the values of rotational inertia pertinent to our experimental setup, this effect does not change the overall behavior, so that previous results remain fully confirmed.

1.1. Introduction

The first example of an elastic structure buckling for a tensile dead load, without elements subject to compression, has been provided by Zaccaria et al. [ZAC 11]. This finding opens new possibilities in the design of compliant structures. In this chapter, we present a single-degree-of-freedom structure (different from – and slightly generalizing – that found by [ZAC 11]), an example that shows that the previously investigated systems are elements of a broad set of structures behaving in a, perhaps, “unexpected way”. Moreover, we present a simple generalization of a single-degree-of-freedom system, further revealing the effects of the constraint’s curvature analyzed by Bigoni et al. [BIG 12b]. The presence of an additional spring has an important effect on the post-critical behavior, so that two configurations (in addition to the trivial one) corresponding to a null external force are found.

Finally, we reconsider the frictional instability setup analyzed by Bigoni and Noselli [BIG 11], where a follower tangential load is transmitted by friction at a freely rotating wheel mounted at the end of a Ziegler pendulum [ZIE 77]. The application of a follower tangential load to a structure was a problem previously unsolved [ELI 05, KOI 96], but important from both a theoretical (see, for instance, [KIR 10]) and applicative point of view (for instance, to energy harvesting [DOA 11]). Within the same setting considered by Bigoni and Noselli [BIG 11], we now analyze the effects on dynamics of the inertia of the wheel and we show that, for the values of inertia pertinent to the experimental setting used, these effects are negligible, so that previous results are now fully confirmed.

1.2. Buckling in tension

Structures buckling under tensile dead loading (without elements subject to compression) were discovered by Zaccaria et al. [ZAC 11], who pointed out the simple example of the single-degree-of-freedom system as shown in Figure 1.1.

Figure 1.1.A single-degree-of-freedom structural model showing bifurcation under tensile dead loading, where two rigid rods are connected through a slider [ZAC 11]

They also developed the concept by replacing the rigid rods with deformable elements. Though the finding by Zaccaria et al. [ZAC 11] might seem an isolated case, we state, on the contrary, that a broad class of structures buckling in tension can be invented. To substantiate this statement, we provide, as an example, the new single-degree-of-freedom system as shown in Figure 1.2, where two rigid rods are connected through a roller constrained to slide orthogonally to the left rod.

For this structure, bifurcation load and equilibrium paths can be calculated by considering the bifurcation mode illustrated in Figure 1.2 and defined by the rotation angle ϕ. The elongation of the system and the total potential energy are, respectively,

[1.1]

so that the force at equilibrium satisfies

[1.2]

[1.3]

1.3. The effect of constraint’s curvature

The strong effects related to the curvature of the profile on which a structure end is constrained to slide have been highlighted by Bigoni et al. [BIG 12b], who showed how to exploit a constraint to induce two critical loads (one in tension and one in compression) in a single-degree-of-freedom elastic structure. This structure, as shown in Figure 1.3, can be easily generalized by including an additional elastic spring on the hinge sliding along the profile, as shown in Figure 1.4.

Figure 1.3.Post-critical behavior in tension of a single-degree-of-freedom structure. The structure has two critical loads, one in tension and one in compression [BIG 12b]

Figure 1.4.A single-degree-of-freedom structure with a linear-elastic hinge constrained to slide along a generic profile at the right end and a rotational linear-elastic spring at the left end

Bifurcation loads can be calculated by considering a deformed mode defined by the rotation angle ϕ, assumed to be positive when clockwise. The potential energy of the system is

[1.4]

so that the axial force at equilibrium becomes

[1.5]

When the profile of the constraint is circular, with radius Rc and dimensionless signed curvature as shown in the inset of Figures 1.5 and 1.6, the axial load at equilibrium satisfies

[1.6]

[1.7]

For an imperfect system, characterized by an initial inclination of the rod ϕ0, the potential energy becomes

[1.8]

so that the axial force at equilibrium is

[1.9]

which for a circular profile becomes

[1.10]

1.4. The Ziegler pendulum made unstable by Coulomb friction

The first experimental evidence of flutter and divergence instability related to dry friction has recently been provided by Bigoni and Noselli [BIG 11]. In their experimental study, essentially based on the Ziegler’s double pendulum [ZIE 77], Coulomb friction was exploited in order to provide the system with a tangential follower force of frictional origin. This goal was achieved by endowing the double pendulum with a freely rotating wheel, constrained to slide with friction on a horizontal plate (see Figure 1.7 for the experimental setting and Figure 1.8 for a sequence of images revealing flutter instability).

Figure 1.7.The experimental setting used by Bigoni and Noselli [BIG 11] to show the connection between Coulomb friction and dynamic instabilities such as flutter and divergence. A Ziegler double pendulum is endowed at its tip with a freely rotating wheel, constrained to slide on a horizontal plate and providing the system with a follower force of frictional origin

Figure 1.8.A sequence of images (taken from a movie recorded with a Sony handycam at 25 frames per second) of the structure shown in Figure 1.7 and exhibiting flutter instability. The whole sequence of images was recorded in 0.40 s and the time interval between two images was 0.08 s

Note that, to generate a force of the frictional type, a transversal reaction between plate and wheel is needed, which during the experiments was created by hanging a dead weight W on the left of the structure, used as a lever.

In their experimental study, Bigoni and Noselli [BIG 11] analyzed the stability of the double pendulum using the five different wheels, as shown in Figure 1.9; however, in their numerical analyses, the wheel was assumed to be massless, so the aim of this section is to show the effects on the system’s dynamic of a heavy wheel.

Figure 1.9.The five different wheels used in the experimental tests by Bigoni and Noselli [BIG 11]. (1) Aluminum wheel with V-shaped cross-section, external diameter 15 mm, thickness 5 mm, weight 3 g; (2) cylindrical steel wheel, external diameter 25 mm, thickness 5 mm, weight 18 g; (3) cylindrical steel wheel, external diameter 25 mm, thickness 6 mm weight 22 g; (4) steel wheel with V-shaped cross-section, external diameter 25 mm, thickness 6 mm, weight 17 g; (5) cylindrical steel wheel, external diameter 25 mm, thickness 10 mm, weight 36 g

[1.11]

Figure 1.10.A three-degree-of-freedom system subject to a tangential follower forcePand orthogonal follower forceTprovided by a freely rotating wheel sliding with friction on a plate, which moves with velocity of modulus υp. The two rods, of linear mass density ρ are rigid and connected through two rotational springs of stiffness k1andk2and viscosity β1and β2. The wheel has mass mw, radius rw and thickness hw

where the two scalar quantities P and T have been introduced. Note that P and T are positive quantities when the forces acting on the wheel are directed as in Figure 1.10, and, in general, their absolute values equal to |P| and |T|, respectively.

The assumption of Coulomb friction at the contact point between the wheel and the plate allows us to write

[1.12]

where R is the vertical reaction applied at the wheel and orthogonal to the moving plane, μs and μd are the static and dynamic friction coefficients, respectively, and and are the radial and the tangential components of the velocity of the wheel with respect to the plate, which can be expressed in the forms

[1.13]

The system is characterized by three-degrees-of-freedom, denoted by α1, α2 and α3, and the latter representing the rotation of the wheel about its axis (see Figure 1.10). Moreover, mw, rw and hw are the mass, the radius and the thickness of the wheel.

The principle of virtual works, denoting the scalar product with “⋅”, is written as

[1.14]

holding for every virtual displacement δC, δG1 and δG2, functions of the virtual rotations δα1, δα2 and δα3.

In equation [1.14], m1, m2 and mw are, respectively, the mass of the rod of length l1, the mass of the rod of length l2 and the mass of the wheel, whereas I13, I23, Iw3 and Iwr are, respectively, the principal moment of inertia of the two rods about the vertical axis and the principal moment of inertia of the wheel about the vertical axis and its rotation axis.

Now imposing condition [1.14] and invoking the arbitrariness of δα1, δα2 and δα3 we arrive at the system of three nonlinear differential equations, governing the dynamics of the system

[1.15]

We note from equations [1.12]–[1.15] that α1, α2, α3, P and T are the five unknowns, function of time. Moreover, in the case in which sliding between the wheel and the plate is active, a situation corresponding to , one additional condition has to be imposed in order to find the solution, namely, that the force applied to the wheel, P + T, must be directed parallel, but opposite to the relative plate/wheel velocity, , a condition yielding

[1.16]

The nonlinear system of equations has been numerically solved, and for this purpose the function “NDSolve” of Mathematica 6.0 has been used, together with a viscous smooth approximation of the friction law [1.12] (see [ODE 85, BIG 11]).

In Figure 1.11, a comparison is found (in terms of α1 and α2) between the numerical results for the case of a massless (solid curves) and a heavy (dashed curve) wheel. These results have been obtained for a dead weight W corresponding to the onset of flutter instability and assuming wheel number 3 as shown in Figure 1.9. From the results shown in Figure 1.11, we can conclude that the inertia of the wheel only slightly contributes to the motion of the system and can therefore be neglected.

1.5. Conclusions

Instability in tension, effects of a constraint’s curvature and follower loads induced by dry Coulomb friction are new phenomena that open an important perspective in the design of structures that can become unstable at prescribed loads.

New examples of structures exhibiting buckling under tensile dead loading have been given, slightly generalizing previous findings by the authors and showing that a broad set of systems behaving in a counterintuitive and innovative way can be invented and practically realized.

The effects of a constraint’s curvature have been further investigated: we have shown that the introduction on a curved constraint profile of an elastic, torsional spring strongly affects the post-critical behavior of the system and may lead to multiple equilibrium configurations, corresponding to an external force of zero magnitude.

Finally, we have presented also a detailed analysis of flutter instability as induced by dry friction in the Ziegler double pendulum. In this system, dynamical effects related to a heavy frictional constraint have been determined. The results show that these are negligible for the values of a constraint’s inertia pertinent to our experimental setting, but may become interesting in other situations.

The structures considered in our study can be combined to design flexible systems and artificial materials, which may find broad applications, even at the micro- and nanoscale.

1.6. Acknowledgments

Financial support from the European FP7 – Intercer2 project (PIAP-GA-2011-286110-INTERCER2) is gratefully acknowledged.

1.7. Bibliography

[BIG 11] Bigoni D., Noselli G., “Experimental evidence of flutter and divergence instabilities induced by dry friction”, Journal of the Mechanics and Physics of Solids, vol. 59, no. 10, pp. 2208–2226, 2011.

[BIG 12a] Bigoni D., Nonlinear Solid Mechanics. Bifurcation Theory and Material Instability, Cambridge University Press, 2012.

[BIG 12b] Bigoni D., Misseroni D., Noselli G., et al., “Effects of the constraint’s curvature on structural instability: tensile buckling and multiple bifurcations”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 468, no. 2144, pp. 2191–2209, 2012.

[DOA 11] Doare O., Michelin S., “Piezoelectric coupling in energy-harvesting fluttering flexible plates: linear stability analysis and conversion efficiency”, Journal of Fluids and Structures, vol. 27, no. 8, pp. 1357–1375, 2011.

[ELI 05] Elishakoff I., “Controversy associated with the so-called ‘follower force’: critical overview”, Applied Mechanics Reviews, vol. 58, no. 2, pp. 117–142, 2005.

[KIR 10] Kirillov O.N., Verhulst F., “Paradoxes of dissipation-induced destabilization or who opened Whitney’s umbrella?”, Zeitschrift für Angewandete Mathematik und Mechanik, vol. 90, no. 6, pp. 462–488, 2010.

[KOI 96] Koiter W.T., “Unrealistic follower forces”, Journal of Sound and Vibration, vol. 194, no. 4, pp. 636–638, 1996.

[ODE 85] Oden J.T., Martins J.A.C., “Models and computational methods for dynamic friction phenomena”, Computer Methods in Applied Mechanics and Engineering, vol. 52, no. 1–3, pp. 527–634, 1985.

[SUG 95] Sugiyama Y., Katayama K., Kinoi S., “Flutter of a cantilevered column under rocket thrust”, Journal of Aerospace Engineering, vol. 8, no. 1, pp. 9–15, 1995.

[ZAC 11] Zaccaria D., Bigoni D., Noselli G., et al., “Structures buckling under tensile dead load”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 467, no. 2130, pp. 1686–1700, 2011.

[ZIE 77] Ziegler H., Principles of Structural Stability, Birkhäuser Verlag, Basel, Stuttgart, 1977.

Chapter written by Davide Bigoni, Diego Misseroni, Giovanni Noselli and Daniele Zaccaria.

Chapter 2

WKB Solutions Near an Unstable Equilibrium and Applications

In this chapter, we present some precise results concerning spectral and scattering problems for the Schrödinger equation in the semi-classical regime, which we have obtained in a series of papers [ALE 08, BON 07, BON 11, BON]. As we can expect, properties of the underlying classical system play a crucial role in this regime, and we have studied the case where there exists one hyperbolic fixed point for the associated Hamiltonian flow. This occurs, for example, when the potential has a local maximum. Much is encoded in what we call a microlocal Cauchy problem at the fixed point, which we describe here in detail. In a physicist’s language, the study of this microlocal Cauchy problem is that of the n-dimensional tunneling effect at the hyperbolic fixed point.

2.1. Introduction

In this chapter, we sum up different results obtained in a series of paper [BON 07, BON 11, BON, ALE 08] concerning spectral or scattering quantities attached to the semi-classical Schrödinger operator on

[2.1]

and the corresponding classical Hamiltonian

[2.2]

[2.3]

when the spectral parameter E is in a vicinity of size of E0. Of course, we are in a setting where the tunnel effect occurs at the barrier top. We will see quantitatively that, for such energies, tunneling governs the behavior of the physical quantities we are interested in.

Here, we have chosen to concentrate on a scattering situation, namely we assume that E0 > 0 and V(x) → 0 as |x| → +∞. In this setting, we will describe some results concerning resonances for the Schrödinger operator P.

In physics, the notion of quantum resonance appeared at the beginning of quantum mechanics. Its introduction was motivated by the behavior of various quantities related to scattering experiments, such as the scattering amplitude, the scattering cross-section or the time-delay (the derivative of the spectral shift function). At certain energies, these quantities present peaks (now called Breit–Wigner peaks), which were modelized by a Lorentzian-shaped function