scholarly journals Three-Dimensional Oscillations of Suspended Cables Involving Simultaneous Internal Resonances

1995 ◽  
pp. 45-63 ◽  
Author(s):  
Christopher Lee ◽  
Noel C. Perkins
Keyword(s):  
Three Dimensional ◽  
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.

scholarly journals Nonlinear Mechanics of Beams With Partial Piezoelectric Layers

2019 ◽  
Vol 86 (10) ◽  
Author(s):  
Hamed Farokhi ◽  
Mergen H. Ghayesh
Keyword(s):  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Coupled Model ◽  
Beam Theory ◽  
Internal Resonances ◽  
Element Analysis ◽  
Nonlinear Dynamical ◽  
Dimensional Finite Element ◽  
Resonant Region ◽  
Three Dimensional Finite Element

Abstract This paper investigates the nonlinear static response as well as nonlinear forced dynamics of a clamped–clamped beam actuated by piezoelectric patches partially covering the beam from both sides. This study is the first to develop a high-dimensional nonlinear model for such a piezoelectric-beam configuration. The nonlinear dynamical resonance characteristics of the electromechanical system are examined under simultaneous DC and AC piezoelectric actuations, while highlighting the effects of modal energy transfer and internal resonances. A multiphysics coupled model of the beam-piezoelectric system is proposed based on the nonlinear beam theory of Bernoulli–Euler and the piezoelectric constitutive equations. The discretized model of the system is obtained with the help of the Galerkin weighted residual technique while retaining 32 degrees-of-freedom. Three-dimensional finite element analysis is conducted as well in the static regime to validate the developed model and numerical simulation. It is shown that the response of the system in the nonlinear resonant region is strongly affected by a three-to-one internal resonance.

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.

scholarly journals Nonlinear interactions in deformable container cranes

2015 ◽  
Vol 230 (1) ◽  
pp. 5-20 ◽  
Author(s):  
Andrea Arena ◽  
Walter Lacarbonara ◽  
Matthew P Cartmell
Keyword(s):  
Rigid Body ◽  
Internal Resonance ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Higher Order ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Internal Resonances ◽  
Container Cranes ◽  
Bending Mode

Nonlinear dynamic interactions in harbour quayside cranes due to a two-to-one internal resonance between the lowest bending mode of the deformable boom and the in-plane pendular mode of the container are investigated. To this end, a three-dimensional model of container cranes accounting for the elastic interaction between the crane boom and the container dynamics is proposed. The container is modelled as a three-dimensional rigid body elastically suspended through hoisting cables from the trolley moving along the crane boom modelled as an Euler-Bernoulli beam. The reduced governing equations of motion are obtained through the Euler-Lagrange equations employing the boom kinetic and stored energies, derived via a Galerkin discretisation based on the mode shapes of the two-span crane boom used as trial functions, and the kinetic and stored energies of the rigid body container and the elastic hoisting cables. First, conditions for the onset of internal resonances between the boom and the container are found. A higher order perturbation treatment of the Taylor expanded equations of motion in the neighbourhood of a two-to-one internal resonance between the lowest boom bending mode and the lowest pendular mode of the container is carried out. Continuation of the fixed points of the modulation equations together with stability analysis yields a rich bifurcation behaviour, which features Hopf bifurcations. It is shown that consideration of higher order terms (cubic nonlinearities) beyond the quadratic geometric and inertia nonlinearities breaks the symmetry of the bifurcation equations, shifts the bifurcation points and the stability ranges, and leads to bifurcations not predicted by the low order analysis.

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.

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.

Experimental Investigation of Isolated and Simultaneous Internal Resonances in Suspended Cables

1993 ◽  
Author(s):  
Christopher L. Lee ◽  
Noel C. Perkins
Keyword(s):  
Experimental Investigation ◽  
Large Amplitude ◽  
Resonant Response ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Theoretical Predictions ◽  
Suspended Cables ◽  
Elastic Cables

Abstract 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 non-planar response 3) 1:1 internally resonant non-planar response, and 4) simultaneous, 2:2:1 internally resonant non-planar response. Quasi-periodic responses are also observed.

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