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.

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.

scholarly journals Experimental Study on Nonlinear Vibrations of Fixed-Fixed Curved Beams

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Ajay Kumar ◽  
B. P. Patel
Keyword(s):  
Experimental Study ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Excitation Amplitude ◽  
Curved Beam ◽  
Symmetric Mode ◽  
Curved Beams ◽  
Direct Excitation ◽  
First Time ◽  
Good Agreement

AbstractNonlinear dynamic behavior of fixed-fixed shallow and deep curved beams is studied experimentally using non-contact type of electromagnetic shaker and acceleration measurements. The frequency response obtained from acceleration measurements is found to be in fairly good agreement with the computational response. The travellingwave phenomenon along with participation of higher harmonics and softening nonlinearity are observed. The experimental results on the internal resonance of curved beams due to direct excitation of anti-symmetric mode are reported for the first time. The deep curved beam depicts chaotic response at higher excitation amplitude.

scholarly journals Nonlinear Dynamics of Imperfect FGM Conical Panel

2018 ◽  
Vol 2018 ◽  
pp. 1-20 ◽  
Author(s):  
Yan Niu ◽  
Yuxin Hao ◽  
Minghui Yao ◽  
Wei Zhang ◽  
Shaowu Yang
Keyword(s):  
Nonlinear Dynamics ◽  
Equations Of Motion ◽  
Engineering Application ◽  
Shear Deformation Theory ◽  
Harmonic Excitation ◽  
Functionally Graded ◽  
Dynamic Responses ◽  
Phase Portraits ◽  
Power Law Distribution ◽  
Geometric Imperfection

Structures composed of functionally graded materials (FGM) can satisfy many rigorous requisitions in engineering application. In this paper, the nonlinear dynamics of a simply supported FGM conical panel with different forms of initial imperfections are investigated. The conical panel is subjected to the simple harmonic excitation along the radial direction and the parametric excitation in the meridian direction. The small initial geometric imperfection of the conical panel is expressed by the form of the Cosine functions. According to a power-law distribution, the effective material properties are assumed to be graded along the thickness direction. Based on the first-order shear deformation theory and von Karman type nonlinear geometric relationship, the nonlinear equations of motion are established by using the Hamilton principle. The nonlinear partial differential governing equations are truncated by Galerkin method to obtain the ordinary differential equations along the radial displacement. The effects of imperfection types, half-wave numbers of the imperfection, amplitudes of the imperfection, and damping on the dynamic behaviors are studied by numerical simulation. Maximum Lyapunov exponents, bifurcation diagrams, time histories, phase portraits, and Poincare maps are obtained to show the dynamic responses of the system.

scholarly journals Nonlinear pre and post-buckled analysis of curved beams using differential quadrature element method

2019 ◽  
Vol 14 (1) ◽  
Author(s):  
M. Zare ◽  
A. Asnafi
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Mode Shapes ◽  
Curved Beams ◽  
Buckling Loads ◽  
Differential Quadrature Element Method ◽  
Quadrature Element Method ◽  
Post Buckling

AbstractThis paper studied the in-plane elastic stability including pre and post-buckling analysis of curved beams considering the effects of shear deformations, rotary inertia, and the geometric nonlinearity due to large deformations. Firstly, the governing nonlinear equations of motion were derived. The problem was solved performing both the static and dynamic analysis using the numerical method of differential quadrature element method (DQEM) which is a new and efficient numerical method for rapidly solving linear and nonlinear differential equations. Firstly, the method was applied to the equilibrium equations, leading to a nonlinear algebraic system of equations that would be solved utilizing an arc length strategy. Secondly, the results of the static part were employed to linearize the dynamic differential equations of motion and their corresponding boundary and continuity conditions. Without any loss of generality, a clamped-clamped curved beam under a concentrated load was considered to obtain the buckling loads, natural frequencies, and mode shapes of the beam throughout the method. To validate the proposed method, the beam was modeled using a finite element simulation. A great agreement between the results was seen that showed the accuracy of the proposed method in predicting the pre and post-buckling behavior of the beam. The investigation also included an examination of the curvature parameter influencing the dynamic behavior of the problem. It was shown that the values of buckling loads were completely influenced by the curvature of the beam; also, due to the sharp change of longitudinal stiffness after bucking, the symmetric mode shapes changed more than it was expected.

On the Nonlinear Vibrations of Clamped Oval Shells

2008 ◽  
Author(s):  
S. M. Ibrahim ◽  
B. P. Patel ◽  
Y. Nath
Keyword(s):  
Shear Deformation Theory ◽  
Major Axis ◽  
Harmonic Excitation ◽  
Algebraic Equations ◽  
Computationally Efficient ◽  
Semi Major Axis ◽  
Finite Strip Method ◽  
Free Vibration Frequency ◽  
Time Domains ◽  
Spectral Neighborhood

In the present study, the nonlinear dynamic response of clamped immovable oval cylindrical shells subjected to radial harmonic excitation in the spectral neighborhood of the free vibration frequency is investigated. The formulation is based on the first order shear deformation theory. Geometric nonlinearity is inducted in the formulation considering moderately large deformation effects employing the Sanders type kinematic relations. Governing equations are discretized in space and time domains, respectively, employing computationally efficient finite-strip method and Newmark time marching scheme. Resulting nonlinear algebraic equations are solved using Newton-Raphson iterative technique. A detailed parametric study is conducted to bring out the influence of ovality parameter on the nonlinear vibration characteristics of different modes of vibrations of isotropic and angle-ply oval shells. For isotropic oval shells, it is observed that moderately oval shells show softening type nonlinearity whereas shells of large ovality show hardening type nonlinearity. The response of oval shells with large ovality reveals different temporal variation near to the semi-major axis region compared to that in the semi-minor axis region.

Nonlinear Dynamics of FGM Conical Panel With Initial Imperfection in Thermal Environment

2017 ◽  
Author(s):  
Ming Hui Yao ◽  
Yan Niu ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Thermal Environment ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Initial Imperfection ◽  
Harmonic Excitation ◽  
Functionally Graded ◽  
Phase Portraits ◽  
Power Law Distribution ◽  
Geometric Imperfection

In this paper, the nonlinear dynamics of a simply supported functionally graded materials (FGM) conical panel with different forms of initial imperfections is investigated. The conical panel is subjected to the simple harmonic excitation along the radial direction and the parametric excitation in the meridian direction. The small initial geometric imperfection of the conical panel is expressed by the form of the Cosine functions. According to a power-law distribution, the effective material properties are assumed to be graded along the thickness direction. Based on the first-order shear deformation theory and von Karman type nonlinear geometric relationship, the nonlinear equations of motion are established by using the Hamilton principle. The nonlinear partial differential governing equations are truncated by Galerkin’s method to obtain the ordinary differential equations along the radial displacement. The effects of imperfection types, half-wave numbers and amplitudes on the dynamic behaviors are studied by numerical simulation. Maximum Lyapunov exponents, bifurcation diagrams, time histories and phase portraits are obtained to show the dynamic response.

Nonlinear Vibrations of Thermo-Hyperelastic Moderately Thick Cylindrical Shells with 2:1 Internal Resonance

2020 ◽  
Vol 20 (05) ◽  
pp. 2050067
Author(s):  
Jie Xu ◽  
Xuegang Yuan ◽  
Hongwu Zhang ◽  
Fei Zheng ◽  
Liqun Chen
Keyword(s):  
Cylindrical Shell ◽  
Temperature Field ◽  
Nonlinear Vibration ◽  
Internal Resonance ◽  
Order Approximation ◽  
Shear Deformation Theory ◽  
Harmonic Excitation ◽  
Structural Parameter ◽  
Hyperelastic Materials ◽  
Materials Used

The nonlinear vibration of a hyperelastic moderately thick cylindrical shell with 2:1 internal resonance in a temperature field is investigated based on the third-order shear deformation theory. A radial harmonic excitation is applied to the shell. First, by employing the higher-order approximation for the curvature-related expansion, the displacement field of the moderately thick cylindrical shell with improved coefficients is derived. For the temperature field with gradient, the shear modulus of the shell is thickness dependent because of the temperature-dependent nature of the hyperelastic materials used. The graphical results manifest that the temperature gradient has a significant impact on the nonlinear vibration of the shell. In addition, the separation of the resonance peak caused by the variations of the structural parameter and temperature will result in a bubble shaped response curve for the shell.

Vibration Suppression of Flexible Structures Utilizing Internal Resonance of Liquid Sloshing in a Nearly Square Tank

2010 ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Vibration Suppression ◽  
Equations Of Motion ◽  
Flexible Structures ◽  
Harmonic Excitation ◽  
Liquid Sloshing ◽  
Maximum Efficiency ◽  
Tuned Liquid Damper ◽  
Response Curves

This paper proposes a new idea to utilize the internal resonance of two different sloshing modes in a nearly square tank when used as a tuned liquid damper (TLD). This idea results in achieving higher efficiency of vibration suppression for flexible structures subjected to horizontal harmonic excitation. Namely, the two sloshing modes (1, 0) and (0, 1) in a nearly square tank are degenerated and hence their natural frequencies are nearly equal with each other. Because the two predominant sloshing modes are nonlinearly coupled, internal resonance is expected to occur. Galerkin’s method is used to determine the modal equations of motion for liquid sloshing. Then, van der Pol’s method is used to determine the expressions of the frequency response curves. Frequency response curves and bifurcation sets are numerically calculated. From these results, the optimal values of the size and instillation angle of the tank can be determined in order to achieve maximum efficiency of vibration suppression in a flexible structure. Experiments confirmed the validity of the theoretical analysis.

Nonplanar Dynamics of Fixed-Free Beams With Low Torsional Stiffness

2012 ◽  
Author(s):  
Eulher Chaves Carvalho ◽  
Paulo Batista Gonçalves ◽  
Zenon J. G. N. del Prado
Keyword(s):  
Nonlinear Dynamics ◽  
Internal Resonance ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Torsional Stiffness ◽  
Complex Dynamic ◽  
Runge Kutta Method ◽  
The Galerkin Method

The three-dimensional motions of a clamped-free, inextensible beam subject to lateral harmonic excitation are investigated in this paper. Special attention is given to the nonlinear oscillations of beams with low torsional stiffness and its influence on the bifurcations and instabilities of the structure, a problem not tackled in the previous literature on this subject. For this, the nonlinear integro-differential equations describing the flexural-flexural-torsional couplings of the beam are used, together with the Galerkin method, to obtain a set of discretized equations of motion, which are in turn solved by numerical integration using the Runge-Kutta method. Both inertial and geometric nonlinearities are considered in the present analysis. By varying the beam stiffness parameters, and using several tools of nonlinear dynamics, a complex dynamic behavior of the beam is observed near the region where a 1:1:1 internal resonance occurs. In this region several bifurcations leading to multiple coexisting solutions, including planar and nonplanar motions are obtained. Finally, the paper shows how the tools of nonlinear dynamics can help in the understanding of the global integrity of the model, thus leading to a safe design.

Prediction and Elimination of Subharmonic Resonances in Mechanical Systems With Nonlinear Foundations

1995 ◽  
Author(s):  
Sotirios Natslavas ◽  
Petros Tratskas
Keyword(s):  
Internal Resonance ◽  
Equations Of Motion ◽  
Single Mode ◽  
Harmonic Excitation ◽  
Multiple Time Scales ◽  
Model Parameters ◽  
Multiple Time ◽  
Algebraic Equations ◽  
Existence And Stability ◽  
Mode Response

Abstract In the first part of this work an analysis is presented on the dynamics of a two degree of freedom nonlinear mechanical oscillator. The model consists of a rigid body which rests on a foundation with nonlinear stiffness. This body can exhibit both vertical and rocking motions, which are coupled through the nonlinearities only. In the present study, attention is focused on the response of the system under external harmonic excitation of the vertical translation only, leading to conditions of subharmonic resonance of order three. Also, the model parameters are chosen so that its two linear natural frequencies are almost identical (1:1 internal resonance). For this case, the method of multiple time scales is first applied and a set of four coupled odes is derived, governing the amplitudes and phases of approximate motions of the system. Then, determination of approximate periodic steady state response of the oscillator is reduced to solving a set of four nonlinear algebraic equations. It is shown that besides linear and nonlinear single-mode response, two-mode response is also possible, due to the internal resonance. In addition, the stability of the various single- and two-mode periodic responses of the system is analyzed. In the last part of the work, the analytical findings are verified and complemented by numerical results. The main interest lies on identifying the effect of system parameters on the existence and stability of the predicted motions. The results of this study reveal patterns of appearance of these motions, which provide valuable help in the efforts to eliminate them. Finally, direct integration of the original equations of motion reveals the existence of other more complex motions, which coexist with the analytically predicted motions within the frequency ranges of interest.

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