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.

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.

Limit cycle oscillation and multiple entrainment phenomena in a duffing oscillator under time-delayed displacement feedback

2015 ◽  
Vol 23 (17) ◽  
pp. 2742-2756 ◽  
Author(s):  
RK Mitra ◽  
S Chatterjee ◽  
AK Banik
Keyword(s):  
Frequency Response ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Duffing Oscillator ◽  
A Priori ◽  
High Amplitude ◽  
Limit Cycle Oscillation ◽  
Global Dynamics ◽  
Response Curves ◽  
Damped System

The Duffing oscillator under time-delayed displacement feedback is investigated to study the effect of intentional time-delay on the global dynamics of the oscillator. From the free vibration study performed by employing the describing function method it is observed that for the undamped oscillator, an infinite number of limit cycles is present for all possible values of gain and delay. The number of stable and unstable limit cycles in the gain versus delay plane is studied region wise with the help of limit cycle stability lines. Secondly, in a damped system, the number of limit cycles is finite and depends upon the values of gain, delay and damping coefficient from which the maximum number of limit cycles, their frequencies and amplitudes are obtained. When the system is excited by harmonic forcing, these limit cycles exhibit the phenomena of multiple entrainments and their frequency response curves become very complex and most often results in the very high amplitude oscillations. The study of the forced damped oscillator is therefore carried out by applying the method of slowly varying parameter and the frequency response curves for period-1 responses are analyzed. Further, with the a priori knowledge of possible stable and unstable limit cycles obtained by the application of semi-analytical methods, the various instability phenomena due to subharmonic and quasiperiodic responses have also been investigated by numerical simulation using Simulink in the different parametric ranges.

scholarly journals Integration of Freeplay-Induced Limit Cycles Based On a State Space Iterating Scheme

2021 ◽  
Vol 11 (2) ◽  
pp. 741
Author(s):  
Xiangyu Wang ◽  
Zhigang Wu ◽  
Chao Yang
Keyword(s):  
Fixed Points ◽  
State Space ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Initial Conditions ◽  
Time Integration ◽  
Integration Time ◽  
Aeroelastic System ◽  
System States ◽  
Short Integration Time

Time integration is commonly used to obtain accurate system responses, such as the limit cycle oscillations (LCOs) for an aeroelastic system with freeplay. However, the integrations that start with various initial conditions (I.C.s) are usually studied case by case, so only a few system states can possibly be focused on. This paper proposes a state space iterating (SSI) scheme to find LCO solutions using time integration by using another method. First, a large number of arbitrary I.C. cases are used for time integrations, but only a very short integration time is required for each I.C. case. Second, system behaviors are depicted visually through a method that combines a modified Poincaré map and Lorenz map, in which the LCO solutions are found as fixed points via visual inspections. To verify the SSI scheme’s ability to find LCOs, a typical plunge–pitch wing section is established numerically. Time integrations with both the classic scheme and the proposed SSI scheme are carried out. The LCO results of the SSI scheme are well-aligned with those from the classic scheme. The SSI scheme visualizes the patterns of system responses using arbitrary I.C. cases and analyzes the LCO stability, which provides more mathematical insights into an aeroelastic system with freeplay.

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.

Buckling of Circular Cylindrical Shells Using a Displacement Function

1974 ◽  
Vol 96 (4) ◽  
pp. 1322-1327
Author(s):  
Shun Cheng ◽  
C. K. Chang
Keyword(s):  
Boundary Conditions ◽  
Axial Compression ◽  
Cylindrical Shells ◽  
External Pressure ◽  
Displacement Function ◽  
Combined Loading ◽  
Trial Solution ◽  
Governing Differential Equation ◽  
Circular Cylindrical Shells ◽  
The Stability

The buckling problem of circular cylindrical shells under axial compression, external pressure, and torsion is investigated using a displacement function φ. A governing differential equation for the stability of thin cylindrical shells under combined loading of axial compression, external pressure, and torsion is derived. A method for the solutions of this equation is also presented. The advantage in using the present equation over the customary three differential equations for displacements is that only one trial solution is needed in solving the buckling problems as shown in the paper. Four possible combinations of boundary conditions for a simply supported edge are treated. The case of a cylinder under axial compression is carried out in detail. For two types of simple supported boundary conditions, SS1 and SS2, the minimum critical axial buckling stress is found to be 43.5 percent of the well-known classical value Eh/R3(1−ν2) against the 50 percent of the classical value presently known.

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