Parametric Instability of Circular Cylindrical Shells

1967 ◽  
Vol 34 (4) ◽  
pp. 985-990 ◽  
Author(s):  
A. Vijayaraghavan ◽  
R. M. Evan-Iwanowski
Keyword(s):  
Longitudinal Wave ◽  
Cylindrical Shells ◽  
Parametric Instability ◽  
Excitation Frequency ◽  
Instability Region ◽  
Driving Frequency ◽  
Base Excitation ◽  
Frequency Range ◽  
Lateral Vibrations ◽  
Circular Cylindrical Shells

Parametric instability of thin, circular cylindrical shells subjected to in-plane longitudinal inertia loading arising from sinusoidal base excitation has been investigated analytically and experimentally. The shell under consideration was rigidly clamped at the base and free at the upper edge. In the applied excitation frequency range, the test specimens exhibited lateral vibrations, at half the driving frequency, with one half longitudinal wave and three full circumferential waves. The linear bending theory used in the analysis was adequate in predicting the incipience of instability, just as in the case of slender rods. Attention has been confined to investigating only the principal instability region, as observed during the experiments. Excellent agreement was obtained between the analytical and experimental results.

Imperfection Sensitivity of Compressed Circular Cylindrical Shells Under Periodic Axial Loads

2009 ◽  
Author(s):  
Francesco Pellicano
Keyword(s):  
Differential Equations ◽  
Dynamic Stability ◽  
Cylindrical Shells ◽  
Nonlinear Partial Differential Equations ◽  
Excitation Frequency ◽  
Finite Amplitude ◽  
Axial Loads ◽  
Imperfection Sensitivity ◽  
Lagrange Equations ◽  
Circular Cylindrical Shells

In the present paper the dynamic stability of circular cylindrical shells is investigated; the combined effect of compressive static and periodic axial loads is considered. The Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration; Lagrange equations are used to reduce the nonlinear partial differential equations to a set of ordinary differential equations. The dynamic stability is investigated using direct numerical simulation and a dichotomic algorithm to find the instability boundaries as the excitation frequency is varied; the effect of geometric imperfections is investigated in detail. The accuracy of the approach is checked by means of comparisons with the literature.

Dynamics of a Vibro-Impacting Hinged Beam: Rigid and Flexible Models

1999 ◽  
Author(s):  
Karen J. L. Fegelman ◽  
Karl Grosh
Keyword(s):  
Model Problem ◽  
Piecewise Linear ◽  
Excitation Frequency ◽  
Base Excitation ◽  
Frequency Range ◽  
Piecewise Linear System ◽  
Constant Displacement ◽  
The One ◽  
Future Work ◽  
Flexible Models

Abstract This study investigates the behavior of a vibro-impacting system with a rigid body mode. The model problem has been formulated as both a one-DOF and a multi-DOF piecewise linear system. The behavior of the models for a base excitation frequency range of 50–200 Hz (4.9–19.5% of the fundamental natural frequency of the in-contact case) is shown for constant base motion acceleration and displacement. It is seen that the models do not predict the same motion for the constant acceleration case, but agree quite closely for the constant displacement case. The system is capable of chaotic motion and several examples of Poincaré sections are shown which suggest the presence of a strange attractor. Future work including analytical methods of analysis for both the one- and multi-DOF systems is discussed.

Nonlinear Response of Infinitely Long Circular Cylindrical Shells to Subharmonic Radial Loads

1991 ◽  
Vol 58 (4) ◽  
pp. 1033-1041 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Raouf A. Raouf ◽  
Jamal F. Nayfeh
Keyword(s):  
Fixed Points ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Nonlinear Response ◽  
Cylindrical Shells ◽  
Excitation Frequency ◽  
Hopf Bifurcations ◽  
Response Curves ◽  
Modulation Equations ◽  
Circular Cylindrical Shells

The method of multiple scales is used to analyze the nonlinear response of infinitely long, circular cylindrical shells (thin circular rings) in the presence of a two-to-one internal (autoparametric) resonance to a subharmonic excitation of order one-half of the higher mode. Four autonomous first-order ordinary differential equations are derived for the modulation of the amplitudes and phases of the interacting modes. These modulation equations are used to determine the fixed points and their stability. The fixed points correspond to periodic oscillations of the shell, whereas the limit-cycle solutions of the modulation equations correspond to amplitude and phase-modulated oscillations of the shell. The force response curves exhibit saturation, jumps, and Hopf bifurcations. Moreover, the frequency response curves exhibit Hopf bifurcations. For certain parameters and excitation frequencies between the Hopf values, limit-cycle solutions of the modulation equations are found. As the excitation frequency changes, all limit cycles deform and lose stability through either pitchfork or cyclic-fold (saddle-node) bifurcations. Some of these saddlenode bifurcations cause a transition to chaos. The pitchfork bifurcations break the symmetry of the limit cycles.

Parametric Instability of Beams with Transverse Cracks Subjected to Harmonic In-Plane Loading

2015 ◽  
Vol 15 (01) ◽  
pp. 1540006 ◽  
Author(s):  
Uttam Kumar Mishra ◽  
Shishir Kumar Sahu
Keyword(s):  
Structural Integrity ◽  
Dynamic Instability ◽  
Parametric Instability ◽  
Excitation Frequency ◽  
Small Deformation ◽  
Instability Region ◽  
Modal Testing ◽  
Open Crack ◽  
Transverse Cracks ◽  
Fixed End

Cracks in structural members lead to local changes in their stiffness and consequently their static, dynamic and stability behavior is altered. The influence of cracks on dynamic characteristics like free vibration, buckling and parametric resonance characteristics of a cracked beam with a transverse crack using finite element method (FEM) is investigated in the present work. Modal testing of beams with transverse open crack is conducted using FFT analyzer to verify the frequencies of vibration of beams. The crack is assumed to be open type and the analysis is linear based on small deformation theory neglecting damping. The loading on the beam is considered as axial with a simple harmonic fluctuation with respect to time. A two-noded Timoshenko beam element with provision of crack is used in this study. The equation of motion represents a system of second-order differential equations with periodic coefficients of the Mathieu–Hill type. The development of the regions of instability arises from Floquet's theory and the periodic solution is obtained by Bolotin's approach using FEM. It is observed that the frequencies of vibration and buckling load of the beam are influenced significantly by location and depth of cracks. It is observed that, for a given location of crack, the onset of instability occurs earlier with increase in depth of crack. As the location of crack moves from the fixed end to the free end the excitation frequency increases. The instability occurs later and the width of the instability regions reduces. When the damage is near to the free end, the instability region almost coincides with the instability region for the undamaged beam. This means that the damage near the fixed end is more severe on the dynamic instability behavior than that of the crack located at other positions. The vibration and instability results can be used as a technique for structural health monitoring or testing of structural integrity, performance and safety.

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