Numerical-Analytical Stability Investigationof Beam With Servo Force Fixed as Cantilever at Free End

Kapitanov, D. V.; Ovchinnikov, V. F.; Смирнов, Л. В.

doi:10.32326/1814-9146-2007-69-1-177-184

Cited by 3 publications

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“…10(c The form of the stability boundary with the Whitney umbrella singularity approximated by Eq. (78) was confirmed later by numerical computations in [42]. The limit in the critical load following from (78) agrees well with the numerical data of [2].…”

Section: Near-reversible Case: Beck's Column With External and Internsupporting

confidence: 84%

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

Kirillov¹,

Verhulst

2010

Z Angew Math Mech

100

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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.

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Section: Near-reversible Case: Beck's Column With External and Internsupporting

confidence: 84%

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

Kirillov¹,

Verhulst

2010

Z Angew Math Mech

100

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“…10. The form of the stability boundary with the Whitney umbrella singularity approximated by equation ( 73) was confirmed later by numerical computations in [33]. The limit in the critical load following from (73) agrees well with the numerical data of [1].…”

Section: Circulatory Systems Of Rotor Dynamicssupporting

confidence: 79%

Authors' reply to “A remark to the paper by O. N. Kirillov and F. Verhulst “Paradoxes of dissipation‐induced destabilization or who opened Whitney's umbrella?” [Zamm 90, No. 6, 462–488 (2010)]

Kirillov¹,

Verhulst²

2012

Z Angew Math Mech

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We are glad to see from the remark that the paradoxical influence of small dissipation emphasized in the seminal work by Hans Ziegler of 1952 and resolved in 1956 by Oene Bottema continues to attract an attention of researchers, confirming that our efforts in writing the survey were not useless.The remark states that the work by Bottema is "a study of the singularity in a simple specific case" while the authors of the remark "provided a final step in the qualitative understanding of Ziegler's paradox as a singularity". Note, however, that the "final step" was already taken by Bottema in 1956; since then it was re-iterated independently by a number of researchers.Indeed, we quote from Bottema (1956) where he studies the stability of a general two-degrees-of-freedom non-conservative system with damping and compares its domain of asymptotic stability (A) with the domain of marginal stability in the absence of damping (B) in the space of the coefficients of the characteristic polynomial (a1, a2, a3); he directly points out that a1 and a3 depend on damping:"One could expect B to be a limit of A, so that for a1 → 0, a3 → 0 the set A would continuously tend to B. That is not the case. . . . Here is the discontinuity we mentioned above. It plays a part in questions regarding the stability of equilibrium. The coefficients a1 and a3 depend on the linear damping forces and it is well known that the stability condition may change in a discontinuous way if a very small damping vanishes at all [1]. The phenomenon may be illustrated by a geometrical diagram.[1] Ziegler, Die Stabilitätskriterien der Elastomechanik, Ing. Arch. 20, 49-56 (1952); Bottema, On the stability of the equilibrium of a linear mechanical system, Journal of Appl. Math. Phys. (ZAMP) 6, 97-104 (1955)."The geometrical diagram plotted by Bottema in 1956 is exactly the Whitney umbrella surface, one half of which bounds the domain of asymptotic stability. Bottema, an established geometer, correctly classifies it as a ruled surface and describes in detail its properties in terms of the generators. Therefore, it was Bottema who in 1956 resolved the Ziegler's paradox by discovering the Whitney umbrella singularity on the stability boundary of two-degrees-of freedom non-conservative systems that include Ziegler's pendulum as a particular case.The results of Arnold on singularity theory of 1970s were quickly known in the Western dynamical systems community. For example, already in 1995, in the work [I. Hoveijn and M. Ruijgrok, The stability of parametrically forced coupled oscillators in sum resonance, Z. Angew. Math. Phys. 46, 384-392 (1995)] the Arnold's method of versal deformation of matrix families was used in order to describe the destabilizing effect of damping in a general four dimensional dynamical system in sum resonance with application to rotating shafts. In the space of parameters of the versal deformation, Hoveijn and Ruijgrok found a generic Whitney umbrella singularity from the list of Arnold and plotted the corresponding singular stability threshold....

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Sensitivity Analysis of Dissipative Reversible and Hamiltonian Systems: A Survey

Kirillov

Verhulst

2009

Volume 10: Mechanical Systems and Control, Parts a and B

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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.

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Numerical-Analytical Stability Investigationof Beam With Servo Force Fixed as Cantilever at Free End

Abstract: *) Ðàáîòà âûïîëíåíà ïðè ïîääåðaeêå ÐÔÔÈ (ãðàíò 05-08-50187à) è Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ (ãðàíò Ïðåçèäåíòà ÐÔ íà ïîääåðaeêó âåäóùèõ íàó÷íûõ øêîë ÍØ-6391.2006.8).

Cited by 3 publications

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

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

Authors' reply to “A remark to the paper by O. N. Kirillov and F. Verhulst “Paradoxes of dissipation‐induced destabilization or who opened Whitney's umbrella?” [Zamm 90, No. 6, 462–488 (2010)]

Sensitivity Analysis of Dissipative Reversible and Hamiltonian Systems: A Survey

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