Internal Resonance and Nonlinear Vibrations of Eccentric Rotating Composite Laminated Circular Cylindrical Shell

2019 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yan Zheng ◽  
Yufei Zhang
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Shear Deformation Theory ◽  
Internal Resonances

Abstract This paper is focused on the internal resonances and nonlinear vibrations of an eccentric rotating composite laminated circular cylindrical shell subjected to the lateral excitation and the parametric excitation. Based on Love thin shear deformation theory, the nonlinear partial differential equations of motion for the eccentric rotating composite laminated circular cylindrical shell are established by Hamilton’s principle, which are derived into a set of coupled nonlinear ordinary differential equations by the Galerkin discretization. The excitation conditions of the internal resonance is found through the Campbell diagram, and the effects of eccentricity ratio and geometric papameters on the internal resonance of the eccentric rotating system are studied. Then, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equations in the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. Finally, we study the nonlinear vibrations of the eccentric rotating composite laminated circular cylindrical shell systems.

Multimode Interactions in Suspended Cables

2002 ◽  
Vol 8 (3) ◽  
pp. 337-387 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat ◽  
Char-Ming Chin ◽  
Walter Lacarbonara
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Internal Resonances ◽  
Out Of Plane ◽  
Space Formulation ◽  
Suspended Cables ◽  
One To One ◽  
Plane Mode

We investigate the nonlinear nonplanar responses of suspended cables to external excitations. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The sag-to-span ratio of the cable considered is such that the natural frequency of the first symmetric in-plane mode is at first crossover. Hence, the first symmetric in-plane mode is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and, simultaneously, in a two-to-one internal resonance with the first symmetric out-of-plane mode. Under these resonance conditions, we analyze the response when the first symmetric in-plane mode is harmonically excited at primary resonance. First, we express the two governing equations of motion as four first-order (i.e., state-space formulation) partial-differential equations. Then, we directly apply the methods of multiple scales and reconstitution to determine a second-order uniform asymptotic expansion of the solution, including the modulation equations governing the dynamics of the phases and amplitudes of the interacting modes. Then, we investigate the behavior of the equilibrium and dynamic solutions as the forcing amplitude and resonance detunings are slowly varied and determine the bifurcations they may undergo.

Nonlinear Breathing Vibrations and Chaos of a Circular Truss Antenna with 1:2 Internal Resonance

2016 ◽  
Vol 26 (05) ◽  
pp. 1650077 ◽  
Author(s):  
W. Zhang ◽  
J. Chen ◽  
Y. Sun
Keyword(s):  
Differential Equation ◽  
Cylindrical Shell ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Thermal Environment ◽  
Circular Cylindrical Shell ◽  
Numerical Results ◽  
Thermal Excitation ◽  
Response Curves ◽  
Chaotic Motions

This paper investigates the nonlinear breathing vibrations and chaos of a circular truss antenna under changing thermal environment with 1:2 internal resonance for the first time. A continuum circular cylindrical shell clamped by one beam along its axial direction on one side is proposed to replace the circular truss antenna composed of the repetitive beam-like lattice by the principle of equivalent effect. The effective stiffness coefficients of the equivalent circular cylindrical shell are obtained. Based on the first-order shear deformation shell theory and the Hamilton’s principle, the nonlinear governing equations of motion are derived for the equivalent circular cylindrical shell. The Galerkin approach is utilized to discretize the nonlinear partial governing differential equation of motion to the ordinary differential equation for the equivalent circular cylindrical shell. The case of the 1:2 internal resonance, primary parametric resonance and 1/2 subharmonic resonance is taken into account. The method of multiple scales is used to obtain the four-dimensional averaged equation. The frequency-response curves and force-response curves are obtained when considering the strongly coupled of two modes. The numerical results indicate that there are the hardening type and softening type nonlinearities for the circular truss antenna. Numerical simulation is used to investigate the influences of the thermal excitation on the nonlinear breathing vibrations of the circular truss antenna. It is demonstrated from the numerical results that there exist the bifurcation and chaotic motions of the circular truss antenna.

Experiments and Simulations in Travelling Wave and Non-Stationary Nonlinear Vibrations of Circular Cylindrical Shells

2016 ◽  
Author(s):  
Marco Amabili ◽  
Prabakaran Balasubramanian ◽  
Giovanni Ferrari
Keyword(s):  
Cylindrical Shell ◽  
Traveling Wave ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Cylindrical Shells ◽  
Excitation Amplitude ◽  
Added Mass ◽  
Travelling Wave ◽  
Wave Response

The nonlinear vibrations of a water-filled circular cylindrical shell subjected to radial harmonic excitation in the spectral neighborhood of the lowest resonances are investigated numerically and experimentally by using a seamless aluminum sample. The experimental boundary conditions are close to a simply supported circular cylindrical shell. Modal analysis reveals the presence of predominantly radial driven and companion modes in the low frequency range, implying the existence of a traveling wave phenomenon in the nonlinear field. Experimental studies previously carried out on cylindrical shells did not permit the complete identification of the characteristic traveling wave response and of its non-stationary nature. The added mass of the internal quiescent, incompressible and inviscid fluid results in an increase of the weakly softening behavior of the shell, as expected. The minimization of the added mass due to the excitation system and the negligible entity of the geometric imperfections of the shell allow the appearance of an exact one-to-one internal resonance between driven and companion modes. This internal resonance gives rise to a travelling wave response around the shell circumference and non-stationary, quasi-periodic vibrations, which are experimentally verified by means of stepped-sine testing with feedback control of the excitation amplitude. The same phenomenon is observed in the nonlinear response obtained numerically. The traveling wave is measured by means of state-of-the-art laser Doppler vibrometry applied to multiple points on the structure simultaneously. Previous studies present in literature did not show if this vibration can be chaotic for relatively small vibration amplitudes. Chaos is here observed in the frequency region where the travelling wave response is present for vibrations amplitudes smaller than the thickness of the shell. The relevant nonlinear reduced order model of the shell is based on the Novozhilov nonlinear shell theory retaining in-plane inertia and on an expansion of the displacements in terms of a properly chosen base of linear modes. An energy approach is used to obtain the nonlinear equations of motion, which are numerically studied (i) by using a code based on arc-length continuation and collocation method that allows bifurcation analysis in case of stationary vibrations, (ii) by a continuation code based on direct integration and Poincaré maps, which also evaluates the maximum Lyapunov exponent in case of non-stationary vibrations. The comparison of experimental and numerical results is particularly satisfactory throughout the various excitation amplitude levels considered. The two methods concur in describing the progressive development of the companion mode into a fully developed traveling wave and the subsequent appearance of quasi-periodic and eventually chaotic vibrations.

A beam–ring circular truss antenna restrained by means of the negative speed feedback procedure

2021 ◽  
pp. 107754632110036
Author(s):  
Ashraf T EL-Sayed Taha ◽  
Hany S Bauomy
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Nonlinear Partial Differential Equations ◽  
Time Frame ◽  
Ring Structure ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Internal Resonances ◽  
Averaged Equations ◽  
Fractional Differential

The present article contemplates the nonlinear powerful exhibitions of affecting dynamic vibration controller over a beam–ring structure for demonstrating the circular truss antenna exposed to mixed excitations. The dynamic controller comprises the included negative speed input added to the framework’s idea. By using the statue, Hamilton, the nonlinear fractional differential administering conditions of movement and the limit conditions have inferred for the shaft ring structure. Through Galerkin’s method, the nonlinear partial differential equations referred to overseeing the movement of the shaft ring structure have diminished to a coupled normal differential equations extending the nonlinearities square terms. Multiple time scales have helped in acquiring (getting) the four-dimensional averaged equations for measuring the primary and 1:2 internal resonances. This article’s controlled assessment is useful for controlling the nonlinear vibrations of the considered framework. Likewise, the controller dispenses with the framework’s oscillations in a brief time frame. The demonstrations of the numerous coefficients and the framework directed at the examined resonance case have been determined. Using MATLAB 7.0 programs has aided in completing the simulation results. At last, the numerical outcomes displayed an admirable concurrence with the methodical ones. A comparison with recently available articles has also indicated good results through using the presented controller.

Nonlinear Vibrations of Elastically-Constrained Rotating Disks

1998 ◽  
Vol 120 (2) ◽  
pp. 475-483 ◽  
Author(s):  
L. Yang ◽  
S. G. Hutton
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Rotating Disk ◽  
Transverse Displacement ◽  
Rotating Disks ◽  
Thin Plate Theory ◽  
Force Equilibrium

An analysis of nonlinear vibrations of an elastically-constrained rotating disk is developed. The equations of motion, which are two coupled nonlinear partial differential equations corresponding to the transverse force equilibrium and to the deformation compatibility, are first developed by using von Karman thin plate theory. Then the stress function is analytically solved from the compatibility equation by assuming a multi-mode transverse displacement field. Galerkin’s method is applied to transform the force equilibrium equation into a set of coupled nonlinear ordinary differential equations in terms of time functions. Finally, numerical integration is used to solve the time governing equations, and the effects of nonlinearity on the vibrations of a rotating disk are discussed.

The Magneto-Elastic Internal Resonances of Rectangular Conductive Thin Plate With Different Size Ratios

2017 ◽  
Vol 34 (5) ◽  
pp. 711-723
Author(s):  
J. Li ◽  
Y. D. Hu ◽  
Y. N. Wang
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Thin Plate ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Transverse Magnetic ◽  
Phase Portraits ◽  
Internal Resonances ◽  
Basic Equations

AbstractBased on the basic equations of electromagnetic elastic motion and the expression of electromagnetic force, the electromagnetic vibration equation of the rectangular thin plate in transverse magnetic field is obtained. For a rectangular plate with one side fixed and three other sides simply supported, its time variable and space variable are separated by the method of Galerkin, and the two-degree-of-freedom nonlinear Duffing vibration differential equations are proposed. The method of multiple scales is adopted to solve the model equations and obtain four first-order ordinary differential equations governing modulation of the amplitudes and phase angles involved via 1:1 or 1:3 internal resonances with different size ratios. With a numerical example, the time history response diagrams, phase portraits and 3-dimension responses of two order modal amplitudes are respectively captured. And the effects of initial values, thickness and magnetic field intensities on internal resonance characteristics are discussed respectively. The results also present obvious characteristics of typical nonlinear internal resonance in this paper.

scholarly journals Reduced-order models for nonlinear vibrations, based on natural modes: the case of the circular cylindrical shell

2013 ◽  
Vol 371 (1993) ◽  
pp. 20120474 ◽  
Author(s):  
Marco Amabili
Keyword(s):  
Cylindrical Shell ◽  
Degrees Of Freedom ◽  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Shear Deformation Theory ◽  
Natural Mode ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Natural Modes ◽  
Admissible Functions

Reduced-order models are essential to study nonlinear vibrations of structures and structural components. The natural mode discretization is based on a two-step analysis. In the first step, the natural modes of the structure are obtained. Because this is a linear analysis, the structure can be discretized with a very large number of degrees of freedom. Then, in the second step, a small number of these natural modes are used to discretize the nonlinear vibration problem with a huge reduction in the number of degrees of freedom. This study finds a recipe to select the natural modes that must be retained to study nonlinear vibrations of an angle-ply laminated circular cylindrical shell that the author has previously studied by using admissible functions defined on the whole structure, so that an accuracy analysis is performed. The higher-order shear deformation theory developed by Amabili and Reddy is used to model the shell.

Numerical Formulation of Nonlinear Vibrations of Elastically-Constrained Rotating Disks

1995 ◽  
Author(s):  
Longxiang Yang ◽  
Stanley G. Hutton
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Rotating Disk ◽  
Transverse Displacement ◽  
Rotating Disks ◽  
Numerical Formulation ◽  
Force Equilibrium

Abstract An analysis of nonlinear vibrations of an elastically-constrained rotating disk is developed. The equations of motion, which are two coupled nonlinear partial differential equations corresponding to the transverse force equilibrium and to the deformation compatibility, are first developed by using von Karman thin plate theory. Then the stress function is analytically solved from the compatibility equation by assuming a multi-mode transverse displacement field. Galerkin’s method is applied to transform the force equilibrium equation into a set of coupled nonlinear ordinary differential equations in terms of time functions. Finally, numerical integration is used to solve the time governing equations, and the effects of nonlinearity on the vibrations of a rotating disk are discussed.

Nonlinear Periodic Response of Composite Curved Beam Subjected to Symmetric and Antisymmetric Mode Excitation

2010 ◽  
Vol 5 (2) ◽  
Author(s):  
S. M. Ibrahim ◽  
B. P. Patel ◽  
Y. Nath
Keyword(s):  
Differential Equations ◽  
Internal Resonance ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Harmonic Excitation ◽  
Symmetric Mode ◽  
Curved Beams ◽  
Computationally Efficient ◽  
Antisymmetric Mode ◽  
Periodic Response

The periodic response of cross-ply composite curved beams subjected to harmonic excitation with frequency in the neighborhood of symmetric and antisymmetric linear free vibration modes is investigated. The analysis is carried out using higher-order shear deformation theory based finite element method (FEM). The governing equations are integrated using Newmark’s time marching coupled with shooting technique and arc-length continuation. Shooting method is used to solve the second-order differential equations of motion directly without converting them to the first-order differential equations. This approach is computationally efficient as the banded nature of equations is retained. A detailed study revealed that the response of antisymmetrically excited beams has contribution of higher antisymmetric as well as symmetric modes whereas the response of symmetrically excited beams has the significant participation of the higher symmetric modes except for the excitation in the neighborhood of first symmetric mode. The beam excited in the neighborhood of first symmetric mode has an additional branch corresponding to significant participation of first antisymmetric mode due to two-to-one internal resonance. Furthermore, for the beams excited in the neighborhood of higher modes, the peak response amplitude becomes less than that of the beam excited in the neighborhood of first mode but vibration behavior is drastically different due to the presence of subharmonics and higher harmonics. Two-to-one internal resonance between second antisymmetric mode and first symmetric mode is predicted for the first time.

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