Parametrically Excited Vibrations and Auto-Parametric Resonance in Cylindrical Shells

2000 ◽  
Author(s):  
A. A. Popov ◽  
J. M. T. Thompson ◽  
F. A. McRobie
Keyword(s):  
Internal Resonance ◽  
Cylindrical Shells ◽  
Mode Interaction ◽  
Geometric Description ◽  
Modal Interactions ◽  
Structural Behaviour ◽  
Parametrically Excited ◽  
Parametric Resonances ◽  
Shell Vibration ◽  
Low Dimensional

Abstract Vibrations of cylindrical shells parametrically excited by external axial forcing or by internal auto-parametric resonances are considered. A Rayleigh-Ritz discretization of the von Kármán-Donnell equations through symbolic computations leads to low dimensional models of shell vibration. After applying methods and ideas of modern dynamical systems theory, complete bifurcation diagrams are constructed and analyzed with an emphasis on modal interactions and their relevance to structural behaviour. In the case of free shell vibrations, the Hamiltonian and a transformation into action-angle coordinates followed by averaging provides readily a geometric description of the interaction between concertina and chequerboard modes. It was established that the interaction should be most pronounced when there are slightly less than 2 N square chequerboard panels circumferentially, where N is the ratio of shell radius to thickness. The two mode interaction leads to preferred vibration patterns with larger deflection inwards than outwards, and at internal resonance, significant energy transfer occurs between the modes. The regular and chaotic features of this interaction are studied analytically and numerically.

scholarly journals Tuning nonlinear damping in graphene nanoresonators by parametric–direct internal resonance

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Ata Keşkekler ◽  
Oriel Shoshani ◽  
Martin Lee ◽  
Herre S. J. van der Zant ◽  
Peter G. Steeneken ◽  
...  
Keyword(s):  
Parametric Resonance ◽  
Internal Resonance ◽  
Microscopic Theory ◽  
Fold Increase ◽  
Nonlinear Damping ◽  
Supporting Evidence ◽  
Nonlinear Dissipation ◽  
Modal Interactions ◽  
Solid State Physics ◽  
Internal Resonances

AbstractMechanical sources of nonlinear damping play a central role in modern physics, from solid-state physics to thermodynamics. The microscopic theory of mechanical dissipation suggests that nonlinear damping of a resonant mode can be strongly enhanced when it is coupled to a vibration mode that is close to twice its resonance frequency. To date, no experimental evidence of this enhancement has been realized. In this letter, we experimentally show that nanoresonators driven into parametric-direct internal resonance provide supporting evidence for the microscopic theory of nonlinear dissipation. By regulating the drive level, we tune the parametric resonance of a graphene nanodrum over a range of 40–70 MHz to reach successive two-to-one internal resonances, leading to a nearly two-fold increase of the nonlinear damping. Our study opens up a route towards utilizing modal interactions and parametric resonance to realize resonators with engineered nonlinear dissipation over wide frequency range.

scholarly journals Dynamical analysis of multilayered reinforced composite cylindrical shells

2005 ◽  
Vol 27 (4) ◽  
pp. 220-228
Author(s):  
Dao Huy Bich ◽  
Tran Thanh Tuan
Keyword(s):  
Linear Problem ◽  
Cylindrical Shells ◽  
Dynamical Analysis ◽  
Dynamical Equations ◽  
Fundamental Frequencies ◽  
Reinforced Composite ◽  
Longitudinal Stiffeners ◽  
Shell Vibration ◽  
Shell Geometry ◽  
Composite Cylindrical Shells

In the present paper the governing dynamical equations for multilayered reinforced composite cylindrical shells based on Kirchhoff-Love's theory and Lekhnitsky's smeared stiffeners technique are derived. The shell is reinforced by longitudinal and ring stiffeners. The longitudinal stiffeners may be composite or sleeves with SMA wire. The linear problem of shell vibration is considered for illustrating the effects of the stiffeners, the shell geometry and altering the lamination scheme on fundamental frequencies of the shell.

Applications of Time-Frequency Analysis to Structures With Internal Resonance

1995 ◽  
Author(s):  
Kyle D. Dippery ◽  
Suzanne Weaver Smith
Keyword(s):  
Structural Dynamics ◽  
Frequency Analysis ◽  
Internal Resonance ◽  
Signal Analysis ◽  
Frequency Content ◽  
Modal Interactions ◽  
Time Frequency Analysis ◽  
Time Frequency ◽  
Transition To Chaos ◽  
Changes Over Time

Abstract Time-frequency analysis is an approach to characterizing the nature of signals whose frequency content changes over time. Although the primary applications of this field have, to date, been in the area of communications and signal analysis, it is becoming known in the field of structural dynamics. This paper explores the application of two straightforward time-frequency techniques to several structures that exhibit internal resonance. In particular, the systems analyzed exhibit simple modal interactions and, in one case, a transition to chaos. While other methods exist for analysis of these types of behaviors, larger systems with more complex resonances maybe better analyzed with time-frequency techniques.

The Role of Modal Coupling on the Non-Linear Response of Cylindrical Shells Subjected to Dynamic Axial Loads

2000 ◽  
Author(s):  
Paulo B. Gonçalves ◽  
Zenón J. G. N. Del Prado
Keyword(s):  
Dynamic Instability ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Parametric Instability ◽  
Compressive Load ◽  
Basins Of Attraction ◽  
Dynamic Component ◽  
Linear Ordinary Differential Equations ◽  
Non Linear ◽  
Low Dimensional

Abstract This paper discusses the dynamic instability of circular cylindrical shells subjected to time-dependent axial edge loads of the form P(t) = P0+P1(t), where the dynamic component p1(t) is periodic in time and P0 is a uniform compressive load. In the present paper a low dimensional model, which retains the essential non-linear terms, is used to study the non-linear oscillations and instabilities of the shell. For this, Donnell’s shallow shell equations are used together with the Galerkin method to derive a set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge-Kutta method. To study the non-linear behavior of the shell, several numerical strategies were used to obtain Poincaré maps, stable and unstable fixed points, bifurcation diagrams and basins of attraction. Particular attention is paid to two dynamic instability phenomena that may arise under these loading conditions: parametric instability and escape from the pre-buckling potential well. The numerical results obtained from this investigation clarify the conditions, which control whether or not instability may occur. This may help in establishing proper design criteria for these shells under dynamic loads, a topic practically unexplored in literature.

Combination Resonances of a Circular Plate With Three-Mode Interaction

1995 ◽  
Vol 62 (4) ◽  
pp. 1015-1022 ◽  
Author(s):  
Won Kyoung Lee ◽  
Cheol Hong Kim
Keyword(s):  
Circular Plate ◽  
Internal Resonance ◽  
Natural Frequencies ◽  
Initial Conditions ◽  
Resonance Condition ◽  
Equilibrium Points ◽  
Nonautonomous System ◽  
Mode Interaction ◽  
Resonance Frequencies

A nonlinear analysis is presented for combination resonances in the symmetric responses of a clamped circular plate with the internal resonance, ω3≈ω1+2ω2. The combination resonances occur when the frequency of the excitation are near a combination of the natural frequencies, that is, when Ω≈2ω1+ω2. By means of the internal resonance condition, the frequency of the excitation is also near another combination of the natural frequencies, that is, Ω≈ω1−ω2+ω3. The effect of two near combination resonance frequencies on the response of the plate is examined. The method of multiple scales is used to solve the nonlinear nonautonomous system of equations governing the generalized coordinates in Galerkin’s procedure. For steady-state responses, we determine the equilibrium points of the autonomous system transformed from the nonautonomous system and examine their stability. It has been found that in some cases resonance responses with nonzero-amplitude modes don’t exist, and the amplitudes of the responses decrease with the excitation amplitude. We integrate numerically the nonautonomous system to find the long-term behaviors of the plate and to check the validity of the analytical solution. It is found that there exist multiple stable responses resulting in jumps. In this case the long-term response of the plate depends on the initial condition. In order to visualize total responses depending on the initial conditions, we draw the deflection curves of the plate.

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