Paradoxes of dissipation‐induced destabilization or who opened Whitney's umbrella?

The paradox of destabilization of a conservative or non-conservative system by small dissipation, or Ziegler’s paradox (1952), has stimulated an ever growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations. Since the last decade it has been widely accepted that dissipation-induced instabilities are closely related to singularities arising on the stability boundary. What is less known is that the first complete explanation of Ziegler’s paradox by means of the Whitney umbrella singularity dates back to 1956. We revisit this undeservedly forgotten pioneering result by Oene Bottema that outstripped later findings for about half a century. We discuss subsequent developments of the perturbation analysis of dissipation-induced instabilities and applications over this period, involving structural stability of matrices, Krein collision, Hamilton-Hopf bifurcation, and related bifurcations. c

[1]  Arun K. Samantaray,et al.  On the stability of Crandall gyropendulum , 2008 .

[2]  G.T.S. Done,et al.  Damping configurations that have a stabilizing influence on nonconservative systems , 1973 .

[3]  Oleg N. Kirillov.  Gyroscopic stabilization of non-conservative systems , 2006 .

[4]  D. Decker,et al.  Topics in bifurcation theory , 1978 .

[5]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[6]  Louis Jezequel,et al.  Mode coupling instability in friction-induced vibrations and its dependency on system parameters including damping , 2007 .

[7]  Oleg N. Kirillov.  Bifurcation of the roots of the characteristic polynomial and the destabilization paradox in friction induced oscillations , 2007 .

[8]  V. A. I︠A︡kubovich,et al.  Linear differential equations with periodic coefficients , 1975 .

[9]  Wanda Szemplińska-Stupnicka,et al.  The Behavior of Nonlinear Vibrating Systems , 1990 .

[10]  A. Milanese,et al.  On the Magnetic Field Effect in Electroconductive Plates Under Nonconservative Loading , 2009 .

[11]  V. V. Bolotin,et al.  The Dynamic Stability of Elastic Systems , 1966 .

[12]  Oleg N. Kirillov.  A theory of the destabilization paradox in non-conservative systems , 2005 .

[13]  Mikael A. Langthjem,et al.  Physical mechanism of the destabilizing effect of damping in continuous non-conservative dissipative systems , 2007 .

[14]  V. V. Bolotin,et al.  Effect of damping on the postcritical behaviour of autonomous non-conservative systems , 2002 .

[15]  David Macleish Smith,et al.  The motion of a rotor carried by a flexible shaft in flexible bearings , 1933 .

[16]  Johan Hultén,et al.  Brake Squeal - A Self-Exciting Mechanism with Constant Friction , 1993 .

[17]  J. Carr,et al.  Beck’s Problem , 1979 .

[18]  N. Namachchivaya,et al.  Some aspects of destabilization in reversible dynamical systems with application to follower forces , 1996 .

[19]  COMMUNICATIONS OF THE MOSCOW MATHEMATICAL SOCIETY: On the destabilizing effect of small damping on abstract non-conservative systems , 1986 .

[20]  Peter Hagedorn,et al.  Modeling and Stability Analysis of an Axially Moving Beam With Frictional Contact , 2008 .

[21]  Oliver M. O’Reilly,et al.  On dissipation-induced destabilization and brake squeal: A perspective using structured pseudospectra , 2007 .

[22]  Oleg N. Kirillov.  Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices , 2010 .

[23]  H. Ziegler,et al.  Die Stabilitätskriterien der Elastomechanik , 1952 .

[24]  Michael L. Overton,et al.  Stability theory for dissipatively perturbed hamiltonian systems , 1995 .

[25]  I. Hoveijn,et al.  The stability of parametrically forced coupled oscillators in sum resonance , 1995 .

[26]  Yoshihiko Sugiyama,et al.  Dynamic stability of columns subjected to follower loads : A survey , 2000 .

[27]  M. P. Païdoussis,et al.  A Physical Explanation of the Destabilizing Effect of Damping , 1998 .

[28]  Anthony N. Kounadis,et al.  On the paradox of the destabilizing effect of damping in non-conservative systems , 1992 .

[30]  Jerrold E. Marsden,et al.  Dissipation-induced instabilities in finite dimensions , 2007 .

[31]  J. Maxwell.  Treatise on Natural Philosophy , 2022, Nature.

[32]  H. Leipholz.  Über den Einfluß der Dämpfung bei nichtkonservativen Stabilitätsproblemen elastischer Stäbe , 1964 .

[33]  G. Iooss,et al.  Topics in bifurcation theory and applications , 1999 .

[34]  Anthony Bloch P.S.Krishnaprasad,et al.  Dissipation Induced Instabilities , 1993, chao-dyn/9304005.

[35]  Oleg N. Kirillov,et al.  The effect of small internal and external damping on the stability of distributed non-conservative systems , 2005 .

[36]  A. Eliassen,et al.  A Numerical Method for Predicting the Perturbations of the Middle Latitude Westerlies , 1949 .

[37]  Jon Juel Thomsen,et al.  Chaotic Dynamics of the Partially Follower-Loaded Elastic Double Pendulum , 1995 .

[38]  S. Agafonov.  Stability and Motion Stabilization of Nonconservative Mechanical Systems , 2002 .

[39]  H. P..  Annales de l'Institut Henri Poincaré , 1931, Nature.

[40]  William Thomson Baron Kelvin,et al.  Treatise on Natural Philosophy , 1867 .

[41]  Oleg N. Kirillov,et al.  Stabilization and destabilization of a circulatory system by small velocity-dependent forces , 2005 .

[42]  O. Bottema.  On the stability of the equilibrium of a linear mechanical system , 1955 .

[43]  H. Whitney.  The singularities of a smooth n-manifold in (2n-1)-space , 1944 .

[44]  George Herrmann,et al.  Some general considerations concerning the destabilizing effect in nonconservative systems , 1965 .

[45]  V. F. Ovchinnikov,et al.  NUMERICAL-ANALYTICAL STABILITY INVESTIGATIONOF BEAM WITH SERVO FORCE FIXED AS CANTILEVER AT FREE END , 2007 .

[46]  W. Hauger.  Stability of a Gyroscopic Nonconservative System , 1975 .

[47]  Ing-Chang Jong,et al.  On the destabilizing effect of damping in nonconservative elastic systems. , 1965 .

[48]  R. Romea.  The Effects of Friction and β on Finite-Amplitude Baroclinic Waves , 1977 .

[49]  O. N. Kirillov,et al.  Coupling of eigenvalues of complex matrices at diabolic and exceptional points , 2005 .

[50]  H. Whitney.  The general type of singularity of a set of $2n-1$ variables , 1943 .

[51]  M. Berry.  Physics of Nonhermitian Degeneracies , 2004 .

[52]  H. Troger,et al.  Zur korrekten Modellbildung in der Dynamik diskreter Systeme , 1981 .

[53]  L. G. Lobas.  The Dynamics of Finite-Dimensional Systems Under Nonconservative Position Forces , 2001 .

[54]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[55]  C. D. Mote,et al.  Stability of non-conservative linear discrete gyroscopic systems , 1991 .

[56]  R. Plaut.  A New Destabilization Phenomenon in Nonconservative Systems , 1971 .

[57]  Adrian S. Lewis,et al.  Stabilization via Nonsmooth, Nonconvex Optimization , 2006, IEEE Transactions on Automatic Control.

[58]  V. V. Bolotin,et al.  Nonconservative problems of the theory of elastic stability , 1963 .

[59]  W. Langford,et al.  Hopf Meets Hamilton Under Whitney’s Umbrella , 2003 .

[60]  Hartmut Hetzler,et al.  On moving continua with contacts and sliding friction: Modeling, general properties and examples , 2009 .

[61]  Earl H. Dowell,et al.  Effect of structural damping on flutter of plates with a follower force , 1992 .

[62]  Oleg N. Kirillov,et al.  ON THE RELATION OF STANDARD AND HELICAL MAGNETOROTATIONAL INSTABILITY , 2009, 0911.0067.

[63]  V. Arnold.  ON MATRICES DEPENDING ON PARAMETERS , 1971 .

[64]  L. Fang,et al.  VORTICITY OF INTERGALACTIC MEDIUM VELOCITY FIELD ON LARGE SCALES , 2010, 1001.4127.

[65]  E. O. Holipainen.  On the Effect of Friction in Baroclinic Waves , 1961 .

[66]  Alexander S. Bratus,et al.  Stabilizing and destabilizing effects in non-conservative systems☆ , 1989 .

[67]  Oleg N. Kirillov,et al.  Singularities on the boundary of the stability domain near 1: 1-resonance , 2009, 0911.1224.

[68]  Hans Ziegler.  Linear elastic stability , 1953 .

[69]  Jerrold E. Marsden,et al.  Dissipation-Induced Heteroclinic Orbits in Tippe Tops , 2008, SIAM Rev..

[70]  T. K. Caughey,et al.  Stabilität und Matrizen , 1977 .

[71]  Paolo Gallina,et al.  On the stabilizing and destabilizing effects of damping in wood cutting machines , 2003 .

[73]  Zhaoyan Zhang,et al.  Physical mechanisms of phonation onset: a linear stability analysis of an aeroelastic continuum model of phonation. , 2007, The Journal of the Acoustical Society of America.

[74]  Oleg N. Kirillov.  Destabilization paradox , 2004 .

[75]  O. Bottema.  The Routh-Hurwitz Condition for the Biquadratic Equation , 1956 .

[76]  Oleg N. Kirillov.  Destabilization paradox due to breaking the Hamiltonian and reversible symmetry , 2007 .

[77]  H. Leipholz,et al.  Stability of Elastic Systems , 1980 .

[78]  Effect of small dissipative and gyroscopic forces on the stability of nonconservative systems , 2003 .

[79]  L. Gaul,et al.  Effects of damping on mode‐coupling instability in friction induced oscillations , 2003 .

[80]  R. Knops,et al.  The Dynamic Stability of Elastic Systems. By V. V. Bolotin. 1964. (Holden-Day)Non-Conservative Problems of the Theory of Elastic Stability. V. V. Bolotin. 1963. (Pergamon) , 1966, The Mathematical Gazette.

[81]  Didier Sornette,et al.  Encyclopedia of Complexity and Systems Science , 2009 .

[82]  Ferdinand Verhulst,et al.  Parametric and Autoparametric Resonance , 1996 .

[83]  Ferdinand Verhulst,et al.  Resonance in a rigid rotor with elastic support , 1993 .

[84]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[85]  Movement of eigenvalues of Hamiltonian equilibria under non-Hamiltonian perturbation , 1991 .

[86]  Oleg N. Kirillov.  Subcritical flutter in the acoustics of friction , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[87]  V. V. Bolotin,et al.  Effects of damping on stability of elastic systems subjected to nonconservative forces , 1969 .

[88]  E. F. Infante,et al.  The Effect of external Damping on the Stability of Beck's Column , 1970 .

[89]  How do small velocity-dependent forces (de)stabilize a non-conservative system? , 2003, 2003 IEEE International Workshop on Workload Characterization (IEEE Cat. No.03EX775).

[90]  Ferdinand Verhulst,et al.  Perturbation Analysis of Parametric Resonance , 2009, Encyclopedia of Complexity and Systems Science.

[91]  H. Whitney.  The General Type of Singularity of a Set of 2n − 1 Smooth Functions of n Variables , 1992 .

[92]  J. A. Kuznecov.  Elements of applied bifurcation theory , 1998 .

[93]  Stephen H. Crandall,et al.  The effect of damping on the stability of gyroscopic pendulums , 1995 .

[94]  J. Marsden,et al.  Dissipation-Induced Instability Phenomena in Infinite-Dimensional Systems , 2009 .

[95]  Oleg N. Kirillov.  Gyroscopic stabilization in the presence of nonconservative forces , 2007 .

[96]  Peter Hagedorn,et al.  On the destabilizing effect of non-linear damping in non-conservative systems with follower forces , 1970 .

[97]  L. V. Levantovskii.  Communications of the Moscow Mathematical Society: the Boundary of a Set of Stable Matrices , 1980 .

[98]  Oleg N. Kirillov.  Perturbation of multiparameter non-self-adjoint boundary eigenvalue problems for operator matrices , 2008, 0803.2248.

[99]  G. Haller.  Gyroscopic stability and its loss in systems with two essential coordinates , 1992 .

[100]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[101]  J. A. Walker.  A note on stabilizing damping configurations for linear nonconservative systems , 1973 .

[102]  Jerrold E. Marsden,et al.  On destabilizing effects of two fundamental non-conservative forces , 2006 .