The effect of small internal and external damping on the stability of distributed non-conservative systems

Kirillov, Oleg N.; Seyranian, Alexander P.

doi:10.1016/j.jappmathmech.2005.07.004

Cited by 73 publications

(77 citation statements)

References 23 publications

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“…The vibration energy dissipation during the radius change arises due to the internal damping of the viscoelastic material of the system. Similar solutions are presented in works [5,6]. In the work [5] an impact of small internal and outer damping on the stability of non-conservative beam systems has been revealed.…”

Section: Introductionsupporting

confidence: 67%

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Vibration of crane radius change system with internal damping

Sochacki¹,

Bold²

2013

J. Appl. Math. Comput. Mech.

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Abstract. In the present work a problem pertaining to the damped lateral vibrations of the truck crane radius change system with the developed hydraulic cylinder model that changes the radius has been formulated and solved. In the adopted model the vibration energy dissipation derives from the internal damping of the viscoelastic material (the Kelvin-Voigt rheological model) of beams that model the system. Damped vibration frequency and the vibration amplitude decay level have been calculated. Changes of the eigenvalues of system vibrations with the damping ratio change and the change of the system geometry with different loads observed on it have been presented.

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Section: Introductionsupporting

confidence: 67%

“…Similar solutions are presented in works [5,6]. In the work [5] an impact of small internal and outer damping on the stability of non-conservative beam systems has been revealed. Studies included in the work [6] depict the influence of internal damping on the vibrations of the cantilever beam with mass attached at the free end of the beam.…”

Section: Introductionsupporting

confidence: 67%

Vibration of crane radius change system with internal damping

Sochacki¹,

Bold²

2013

J. Appl. Math. Comput. Mech.

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“…We did not even consider the effect of nonlinearites, see however [1,15,29]. We mention the closely related effect of discontinuous change of the critical flutter frequency due to small dissipation [14,51] and its connection to the Whitney umbrella singularity at the exceptional points on the eigenvalue surfaces [96]. Another related topic is the role of the spectral exceptional points in modern non-Hermitian physics including crystal optics, open quantum systems, and PT -symmetric quantum mechanics [10].…”

Section: Resultsmentioning

confidence: 99%

“…For the Beck column with the operators (63) and boundary conditions (64) we calculate the critical load as [50,51] The form of the stability boundary with the Whitney umbrella singularity approximated by Eq. (78) was confirmed later by numerical computations in [42].…”

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

confidence: 99%

“…13 (right) reproduces the original drawing from [87] showing the typical imperfect merging of modes [36] that substitutes the 'perfect' Krein collision in near-Hamiltonian and near-reversible systems. Approximation to the eigenvalue branches in imperfect merging can be efficiently calculated by means of the perturbation theory of multiple eigenvalues for a wide class of non-conservative systems [46,48,49,51,52,59]. …”

Section: Near-hamiltonian Case: the Instability Of Baroclinic Zonal Cmentioning

confidence: 99%

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

Kirillov¹,

Verhulst

2010

Z Angew Math Mech

Self Cite

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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A rod‐like piezoelectric controller for the improvement of the visco‐elastic Beck's beam linear stability

Casalotti

D’Annibale

2021

Structural Contr & Hlth

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The linear stability of a cantilever beam subject to a follower force is investigated. A system of distributed piezoelectric devices is attached to the beam with the purpose of improving the dynamic stability of the structure. They are connected to a second-order network and second-order analog electrical circuit. The assembly of the mechanical and piezoelectric subsystems constitutes a coupled Piezo-Electro-Mechanical (PEM) system, whose equations of motion, derived within a variational approach, are discretized via the Galerkin weighted residual method and the stability of the trivial equilibrium is addressed by numerically solving the associated eigenvalue problem. A sensitivity analysis, carried out on a numerical ground, is thus conducted on a wide range of the electrical parameters to investigate the effectiveness of the proposed controller.

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The effect of small internal and external damping on the stability of distributed non-conservative systems

Cited by 73 publications

References 23 publications

Vibration of crane radius change system with internal damping

Vibration of crane radius change system with internal damping

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

A rod‐like piezoelectric controller for the improvement of the visco‐elastic Beck's beam linear stability

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