scholarly journals Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances

2006 ◽  
Vol 43 (25-26) ◽  
pp. 7800-7819 ◽  
Author(s):  
Lianhua Wang ◽  
Yueyu Zhao
Keyword(s):  
Chaotic Dynamics ◽  
Nonlinear Interactions ◽  
Internal Resonances ◽  
Suspended Cables

Experimental Investigation of Isolated and Simultaneous Internal Resonances in Suspended Cables

1995 ◽  
Vol 117 (4) ◽  
pp. 385-391 ◽  
Author(s):  
C. L. Lee ◽  
N. C. Perkins
Keyword(s):  
Experimental Investigation ◽  
Large Amplitude ◽  
Resonant Response ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Theoretical Predictions ◽  
Suspended Cables ◽  
Elastic Cables

The near resonant response of suspended elastic cables driven by harmonic, planar excitation is investigated experimentally. Measurements of large amplitude cable motions confirm previous theoretical predictions of fundamental classes of internally-resonant responses. For particular magnitudes of equilibrium curvature, strong modal interactions arise through isolated (two-mode) or simultaneous (three-mode) internal resonances. Four qualitatively different periodic responses are observed: (1) pure planar response, (2) 2:1 internally resonant nonplanar response, (3) 1:1 internally resonant nonplanar response, and (4) simultaneous, 2:2:1 internally resonant nonplanar response. Quasiperiodic responses are also observed.

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.

Multimode Dynamics and Out-of-Plane Drift in Suspended Cable Using the Kinematically Condensed Model

2009 ◽  
Vol 131 (6) ◽  
Author(s):  
Lianhua Wang ◽  
Yueyu Zhao ◽  
Giuseppe Rega
Keyword(s):  
Primary Resonance ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Modulation Equations ◽  
Plane Vibration ◽  
Suspended Cable ◽  
Out Of Plane ◽  
Large Amplitude Vibration ◽  
Suspended Cables ◽  
Plane Mode

The large amplitude vibration and modal interactions of shallow suspended cable with three-to-three-to-one internal resonances are investigated. The quasistatic assumption and direct approach are used to obtain the condensed suspended cable model and the corresponding modulation equations for the case of primary resonance of the third symmetric in-plane or out-of-plane mode. The equilibrium, periodic, and chaotic solutions of the modulation equations are studied. Moreover, the nonplanar motion and symmetric character of out-of-plane vibration of the shallow suspended cables are investigated by means of numerical simulations. Finally, the role played by the quasistatic assumption, internal resonance, and static configuration in disrupting the symmetry of the out-of-plane vibration is discussed.

Multi-Pulse Heteroclinic Orbits and Chaotic Dynamics of Laminated Composite Piezoelectric Rectangular Plate

2009 ◽  
Author(s):  
Ming-Hui Yao ◽  
Wei Zhang ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Laminated Composite ◽  
Heteroclinic Orbits ◽  
Global Dynamics ◽  
Rectangular Plates ◽  
Internal Resonances ◽  
Chaotic Motions ◽  
Simply Supported

The multi-pulse orbits and chaotic dynamics of the simply supported laminated composite piezoelectric rectangular plates under combined parametric excitation and transverse loads are studied in detail. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded. The nonlinear equations of motions for the laminated composite piezoelectric rectangular plates are derived from von Karman-type equation and third-order shear deformation laminate theory of Reddy. The four-dimensional averaged equation under the case of primary parametric resonance and 1:2 internal resonances is obtained by directly using the method of multiple scales and Galerkin approach to the partial differential governing equation of motion for the laminated composite piezoelectric rectangular plates. The system is transformed to the averaged equation. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on normal form obtained, the extended Melnikov method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the laminated composite piezoelectric rectangular plates. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the laminated composite piezoelectric rectangular plates are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the simply supported laminated composite piezoelectric rectangular plates.

Nonlinear Dynamics of Suspended Cables under Periodic Excitation in Thermal Environments: Two-to-One Internal Resonance

2021 ◽  
Vol 31 (10) ◽  
pp. 2150153
Author(s):  
Yaobing Zhao ◽  
Henghui Lin
Keyword(s):  
Thermal Effects ◽  
Numerical Solutions ◽  
Time History ◽  
Phase Portraits ◽  
Periodic Excitation ◽  
Internal Resonances ◽  
Chaotic Motions ◽  
Temperature Changes ◽  
Thermal Environments ◽  
Suspended Cables

The temperature change is a non-negligible factor in examining the vibration behaviors of the cable structures. For this reason, the paper aims at investigating the thermal effects on suspended cables’ resonant responses considering two-to-one internal resonances. Firstly, a nonlinear continuous condensed model of the suspended cable under periodic excitation in thermal environments is adopted. Then, a multidimensional discretized model is constructed via the Galerkin method. Following the multiple scaling procedure, the modulation equations with both polar and Cartesian forms are obtained, which are solved numerically. A complete dynamic scenario is presented through bifurcation diagrams, phase portraits, time history curves, Fourier spectra, and Poincaré sections in three internal resonant cases. Numerical examples show that a small change in the static configuration due to thermal effects induces some noticeable changes in dynamic behaviors. The response amplitude, the nonlinear spring behavior, the resonant and stability region, the multi-periodic and chaotic motions are all dependent on temperature changes. Additional Hopf bifurcations might be found due to temperature changes, and it may lead to some more complicated dynamic characteristics. A good agreement between the perturbation and numerical solutions is observed to confirm the results’ correctness and accuracy.

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