scholarly journals Modeling and 1 : 1 Internal Resonance Analysis of Cable-Stayed Shallow Arches

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Jiangen Lv ◽  
Zhicheng Yang ◽  
Xuebin Chen ◽  
Quanke Wu ◽  
Xiaoxia Zeng
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Period Doubling ◽  
Dynamic Responses ◽  
Shallow Arch ◽  
Response Curves ◽  
Internal Resonances ◽  
Essential Information ◽  
Modulation Equations ◽  
Inclined Angle

In this paper, an analytical model of a cable-stayed shallow arch is developed in order to investigate the 1 : 1 internal resonance between modes of a cable and a shallow arch. Integrodifferential equations with quadratic and cubic nonlinearities are used to model the in-plane motion of a simple cable-stayed shallow arch. Nonlinear dynamic responses of a cable-stayed shallow arch subjected to external excitations with simultaneous 1 : 1 internal resonances are investigated. Firstly, the Galerkin method is used to discretize the governing nonlinear integral-partial-differential equations. Secondly, the multiple scales method (MSM) is used to derive the modulation equations of the system under external excitation of the shallow arch. Thirdly, the equilibrium, the periodic, and the chaotic solutions of the modulation equations are also analyzed in detail. The frequency- and force-response curves are obtained by using the Newton–Raphson method in conjunction with the pseudoarclength path-following algorithm. The cascades of period-doubling bifurcations leading to chaos are obtained by applying numerical simulations. Finally, the effects of key parameters on the responses are examined, such as initial tension, inclined angle of the cable, and rise and inclined angle of shallow arch. The comprehensive numerical results and research findings will provide essential information for the safety evaluation of cable-supported structures that have widely been used in civil engineering.

Two-to-One Internal Resonances in Buckled Beams

1995 ◽  
Author(s):  
Wayne Kreider ◽  
Ali H. Nayfeh ◽  
Char-Ming Chin
Keyword(s):  
Multiple Scales ◽  
Primary Resonance ◽  
Period Doubling ◽  
Equilibrium Solutions ◽  
Internal Resonances ◽  
Modulation Equations ◽  
Buckled Beams ◽  
Second Mode ◽  
Quadratic Nonlinearities ◽  
Period Doubling Bifurcations

Abstract The vibrations of buckled beams with two-to-one internal resonances (ω2 ≈ 2ω1) about a static buckled position are analyzed. General boundary conditions and harmonic excitations (frequency Ω) in both the transverse and axial directions are considered. The analysis assumes a unimodal static buckled deflection, considers quadratic nonlinearities only, and determines the amplitude and phase modulation equations via the method of multiple scales. The following specific cases are treated: Ω ≈ 2ω2, Ω ≈ ω1 + ω2, and Ω ≈ ω1. From the modulation equations for a primary resonance of the second mode (i.e., Ω ≈ ω2), one-mode and two-mode stable equilibrium solutions are found in addition to dynamic solutions caused by Hopf bifurcations. In the region of dynamic solutions, a variety of phenomena are documented, including period-doubling bifurcations, intermittency, chaos, and crises.

Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams

1997 ◽  
Author(s):  
Char-Ming Chin ◽  
Ali H. Nayfeh
Keyword(s):  
Natural Frequency ◽  
Parametric Resonance ◽  
Axial Load ◽  
Multiple Scales ◽  
Period Doubling ◽  
Equilibrium Solutions ◽  
Internal Resonances ◽  
Modulation Equations ◽  
Modal Damping ◽  
Second Mode

Abstract The nonlinear planar response of a hinged-clamped beam to a parametric excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact via a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of principal parametric resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of parametric resonance of the first mode, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single-, and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. In the region of dynamic solutions, some phenomena are documented, including period-doubling bifurcations and blue-sky catastrophes.

Effects of nonlinear interactions of flexural modes on the performance of a beam autoparametric vibration absorber

2019 ◽  
Vol 26 (7-8) ◽  
pp. 459-474
Author(s):  
Saeed Mahmoudkhani ◽  
Hodjat Soleymani Meymand
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Vibration Absorber ◽  
Primary System ◽  
Lumped Mass ◽  
Response Curves ◽  
Internal Resonances ◽  
The One

The performance of the cantilever beam autoparametric vibration absorber with a lumped mass attached at an arbitrary point on the beam span is investigated. The absorber would have a distinct feature that in addition to the two-to-one internal resonance, the one-to-three and one-to-five internal resonances would also occur between flexural modes of the beam by tuning the mass and position of the lumped mass. Special attention is paid on studying the effect of these resonances on increasing the effectiveness and extending the range of excitation amplitudes at which the autoparametric vibration absorber remains effective. The problem is formulated based on the third-order nonlinear Euler–Bernoulli beam theory, where the assumed-mode method is used for deriving the discretized equations of motion. The numerical continuation method is then applied to obtain the frequency response curves and detect the bifurcation points. The harmonic balance method is also employed for detecting the type of internal resonances between flexural modes by inspecting the frequency response curves corresponding to different harmonics of the response. Parametric studies on the performance of the absorber are conducted by varying the position and mass of the lumped mass, while the frequency ratio of the primary system to the first mode of the beam is kept equal to two. Results indicated that the one-to-five internal resonance is especially responsible for the considerable enhancement of the performance.

scholarly journals Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Bamadev Sahoo ◽  
L. N. Panda ◽  
G. Pohit
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Mean Value ◽  
Phase Portraits ◽  
Axially Moving Beam ◽  
Internal Resonances ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

Nonlinear Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances

1997 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Walter Lacarbonara ◽  
Char-Ming Chin
Keyword(s):  
Boundary Conditions ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Internal Resonances ◽  
Buckling Mode ◽  
Stable Mode ◽  
One To One ◽  
Nonlinear Normal

Abstract Nonlinear normal modes of a buckled beam about its first buckling mode shape are investigated. Fixed-fixed boundary conditions are considered. The cases of three-to-one and one-to-one internal resonances are analyzed. Approximate expressions for the nonlinear normal modes are obtained by applying the method of multiple scales to the governing integro-partial-differential equation and boundary conditions. Curves displaying variation of the amplitude with the internal resonance detuning parameter are generated. It is shown that, for a three-to-one internal resonance between the first and third modes, the beam may possess either one stable mode, or three stable normal modes, or two stable and one unstable normal modes. On the other hand, for a one-to-one internal resonance between the first and second modes, two nonlinear normal modes exist. The two nonlinear modes are either neutrally stable or unstable. In the case of one-to-one resonance between the third and fourth modes, two neutrally stable, nonlinear normal modes exist.

Multimode Dynamics of Inextensional Beams on the Elastic Foundation With Two-to-One Internal Resonances

2013 ◽  
Vol 80 (6) ◽  
Author(s):  
Lianhua Wang ◽  
Jianjun Ma ◽  
Minghui Yang ◽  
Lifeng Li ◽  
Yueyu Zhao
Keyword(s):  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Continuation Methods ◽  
Nonlinear Phenomena ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Modulation Equations ◽  
Nonlinear Responses ◽  
Methods Numerical

The modal interactions and nonlinear responses of inextensional beams resting on elastic foundations with two-to-one internal resonances are investigated and the primary resonance excitations are considered. The multimode discretization and the method of multiple scales are applied to obtain the modulation equations. The equilibrium and dynamic solutions of the modulation equations are examined by the Newton–Raphson, shooting, and continuation methods. Numerical simulations are performed to investigate the chaotic dynamics of the beam. It is shown that the nonlinear responses may undergo different bifurcations and exhibit rich nonlinear phenomena. Finally, the effects of the foundation models on the nonlinear interactions of the beam are examined.

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.

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 One-to-One and Three-to-One Internal Resonances in MEMS Shallow Arches

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Hassen M. Ouakad ◽  
Hamid M. Sedighi ◽  
Mohammad I. Younis
Keyword(s):  
Internal Resonance ◽  
Microelectromechanical Systems ◽  
Multiple Scales ◽  
Internal Resonances ◽  
Interesting Phenomenon ◽  
Shallow Arches ◽  
Energy Harvesters ◽  
Modal Coupling ◽  
One To One ◽  
Direct Attack

The nonlinear modal coupling between the vibration modes of an arch-shaped microstructure is an interesting phenomenon, which may have desirable features for numerous applications, such as vibration-based energy harvesters. This work presents an investigation into the potential nonlinear internal resonances of a microelectromechanical systems (MEMS) arch when excited by static (DC) and dynamic (AC) electric forces. The influences of initial rise and midplane stretching are considered. The cases of one-to-one and three-to-one internal resonances are studied using the method of multiple scales and the direct attack of the partial differential equation of motion. It is shown that for certain initial rises, it is possible to activate a three-to-one internal resonance between the first and third symmetric modes. Also, using an antisymmetric half-electrode actuation, a one-to-one internal resonance between the first symmetric and the second antisymmetric modes is demonstrated. These results can shed light on such interactions that are commonly found on micro and nanostructures, such as carbon nanotubes.

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