Subharmonic responses in harmonically excited rectangular plates with one-to-one internal resonance

1997 ◽  
Vol 8 (4) ◽  
pp. 479-498 ◽  
Author(s):  
Seo Il Chang ◽  
J.M. Lee ◽  
A.K. Bajaj ◽  
C.M. Krousgrill
Keyword(s):  
Internal Resonance ◽  
Rectangular Plates ◽  
One To One

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 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 Vibrations in a Parametrically Excited Fluid-Structure Interaction System With One-to-One Internal Resonance

2003 ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Natural Frequency ◽  
Internal Resonance ◽  
Fluid Structure Interaction ◽  
Liquid Sloshing ◽  
Fluid Structure ◽  
Inertia Effects ◽  
Structure Interaction ◽  
Interaction System ◽  
Resonance Curves ◽  
One To One

Theoretical resonance curves prove that a structure’s resonance can facilitate liquid sloshing even when the internal resonance ratio is one-to-one. An investigation of nonlinear sloshing liquid vibrations in a rectangular tank supported by an elastic structure that is subjected to a vertical and sinusoidal excitation reveals that liquid sloshing occurs when the structure’s natural frequency is approximately equal to the natural frequency of sloshing, that is, in the state of one-to-one internal resonance, and that amplitude-modulated motions appear when the condition of the internal resonance deviates to some extent. A special consideration of the nonlinear inertia effects of liquid force and the use of Galerkin’s method help derive the differential (modal) equations governing the dynamic behaviors of the fluid-structure interaction system, while van der Pol’s method helps express the theoretical resonance curves. These theoretical results are in quantitative agreement with the experimental data.

scholarly journals ANALYSIS OF NONLINEAR FORCED VIBRATIONS OF FRACTIONALLY DAMPED SUSPENSION BRIDGES SUBJECTED TO THE ONE-TO-ONE INTERNAL RESONANCE

2020 ◽  
Vol 16 (2) ◽  
pp. 113-131
Author(s):  
Marina Shitikova ◽  
Aleks Katembo
Keyword(s):  
Internal Resonance ◽  
Fractional Power ◽  
Suspension Bridge ◽  
Multiple Time Scales ◽  
Suspension Bridges ◽  
Multiple Time ◽  
Method Of Solution ◽  
Nonlinear Force ◽  
One To One ◽  
The One

Nonlinear force driven coupled vertical and torsional vibrations of suspension bridges, when the frequency of an external force is approaching one of the natural frequencies of the suspension system, which, in its turn, undergoes the conditions of the one-to-one internal resonance, are investigated. The method of multiple time scales is used as the method of solution. The damping features are described by the fractional derivative, which is interpreted as the fractional power of the differentiation operator. The influence of the fractional parameters (orders of fractional derivatives) on the motion of the suspension bridge is investigated.

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.

Identification of Non-Linear Damping of Nuclear Reactor Components in Case of One-to-One Internal Resonance

2016 ◽  
Author(s):  
Joachim Delannoy ◽  
Marco Amabili ◽  
Brett Matthews ◽  
Brian Painter ◽  
Kostas Karazis
Keyword(s):  
Fuel Assembly ◽  
Internal Resonance ◽  
Fluid Structure Interaction ◽  
Fundamental Parameter ◽  
Pressurized Water ◽  
Fluid Structure ◽  
Structure Interaction ◽  
Safety Margins ◽  
Non Linear ◽  
One To One

In Pressurized Water Reactors (PWR) assemblies are exposed to challenging thermal, mechanical, and irradiation loads during operation. Global core and local fuel assembly flow fields coupled with seismic excitation result in fuel assembly and fuel rod vibrations. The fact that vibrations may become excessive in certain conditions has consequences on operational safety margins in fuel assemblies designs. In order to understand how the fuel assembly responds dynamically to an external excitation, it is important to identify the main characteristics of the structures. Among them, the fuel assembly system damping is a fundamental parameter that is usually identified by a number of experiments involving fluid-structure interaction. Recent studies have shown that the damping ratio increases with the excitation force when the structure is entering large-amplitude vibrations, in which case the geometric non-linearities have to be taken into account. The present paper presents an advanced identification procedure developed to identify the system characteristics from experimental non-linear response curves obtained from forced vibration tests, accounting for fluid-structure interaction, at different excitation levels. Furthermore, the numerical tool developed in this analysis is capable of working with systems presenting one-to-one internal resonance, i.e. systems with symmetry such as circular tubes and circular cylindrical shells. The method relies on a harmonic decomposition of the displacement to cope with the data usually available by vibration measurements.

Sign in / Sign up

Export Citation Format

Share Document