scholarly journals Tuning nonlinear damping in graphene nanoresonators by parametric–direct internal resonance

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Ata Keşkekler ◽  
Oriel Shoshani ◽  
Martin Lee ◽  
Herre S. J. van der Zant ◽  
Peter G. Steeneken ◽  
...  
Keyword(s):  
Parametric Resonance ◽  
Internal Resonance ◽  
Microscopic Theory ◽  
Fold Increase ◽  
Nonlinear Damping ◽  
Supporting Evidence ◽  
Nonlinear Dissipation ◽  
Modal Interactions ◽  
Solid State Physics ◽  
Internal Resonances

AbstractMechanical sources of nonlinear damping play a central role in modern physics, from solid-state physics to thermodynamics. The microscopic theory of mechanical dissipation suggests that nonlinear damping of a resonant mode can be strongly enhanced when it is coupled to a vibration mode that is close to twice its resonance frequency. To date, no experimental evidence of this enhancement has been realized. In this letter, we experimentally show that nanoresonators driven into parametric-direct internal resonance provide supporting evidence for the microscopic theory of nonlinear dissipation. By regulating the drive level, we tune the parametric resonance of a graphene nanodrum over a range of 40–70 MHz to reach successive two-to-one internal resonances, leading to a nearly two-fold increase of the nonlinear damping. Our study opens up a route towards utilizing modal interactions and parametric resonance to realize resonators with engineered nonlinear dissipation over wide frequency range.

scholarly journals Computational Synthesis for Nonlinear Dynamics Based Design of Planar Resonant Structures

2012 ◽  
Author(s):  
Astitva Tripathi ◽  
Anil K. Bajaj
Keyword(s):  
Finite Element ◽  
Internal Resonance ◽  
Optimization Techniques ◽  
Mode Shapes ◽  
Nonlinear Phenomena ◽  
Mems Devices ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Element Analysis ◽  
Resonant Structures

Nonlinear phenomena such as internal resonances have significant potential applications in Micro Electro Mechanical Systems (MEMS) for increasing the sensitivity of biological and chemical sensors and signal processing elements in circuits. While several theoretical systems are known which exhibit 1:2 or 1:3 internal resonances, designing systems that have the desired properties required for internal resonance as well as are physically realizable as MEMS devices is a significant challenge. Traditionally, the design process for obtaining resonant structures exhibiting an internal resonance has relied heavily on the designer’s prior knowledge and experience. However, with advances in computing power and topology optimization techniques, it should be possible to synthesize structures with the required nonlinear properties (such as having modal interactions) computationally. In this work, a preliminary method for computer based synthesis of structures consisting of beams for desired internal resonance is presented. The linear structural design is accompalished by a Finite Element Method (FEM) formulation implemented in Matlab to start with a base structure and iteratively modify it to obtain a structure with the desired properties. Possible design criteria are having the first two natural frequencies of the structure in some required ratio (such as 1:2 or 1:3). Once a topology of the structure is achieved that meets the desired criterion (i.e., the program converges to a definite structure), the linear mode shapes of the structure can be extracted from the finite element analysis, and a more complete Lagrangian formulation of the nonlinear elastic structure can be used to develop a nonlinear two-mode model of the structure. The reduced-order model is expected to capture the appropriate resonant dynamics associated with modal interactions between the two modes. The nonlinear response of the structure can be obtained by application of perturbation methods such as averaging on the two-mode model. Many candidate structures are synthesized that meet the desired modal frequency criterion and their nonlinear responses are compared.

scholarly journals Computational Synthesis for Nonlinear Dynamics Based Design of Planar Resonant Structures

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Astitva Tripathi ◽  
Anil K. Bajaj
Keyword(s):  
Finite Element ◽  
Internal Resonance ◽  
Optimization Techniques ◽  
Mode Shapes ◽  
Nonlinear Phenomena ◽  
Mems Devices ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Element Analysis ◽  
Resonant Structures

Nonlinear phenomena such as internal resonances have significant potential applications in micro electro mechanical systems (MEMS) for increasing the sensitivity of biological and chemical sensors and signal processing elements in circuits. While several theoretical systems are known which exhibit 1:2 or 1:3 internal resonances, designing systems that have the desired properties required for internal resonance and that are physically realizable as MEMS devices is a significant challenge. Traditionally, the design process for obtaining resonant structures exhibiting an internal resonance has relied heavily on the designer's prior knowledge and experience. However, with advances in computing power and topology optimization techniques, it should be possible to synthesize structures with the required nonlinear properties (such as having modal interactions) computationally. In this work, a preliminary work on computer based synthesis of structures consisting of beams for desired internal resonance is presented. The linear structural design is accomplished by a Finite Element Method (FEM) formulation implemented in Matlab to start with a base structure and iteratively modify it to obtain a structure with the desired properties. Possible design criteria are having the first two natural frequencies of the structure in some required ratio (such as 1:2 or 1:3). Once a topology of the structure is achieved that meets the desired criterion, the linear mode shapes of the structure can be extracted from the finite element analysis, and a more complete Lagrangian formulation of the nonlinear elastic structure can be used to develop a nonlinear two-mode model of the structure. The reduced-order model is expected to capture the appropriate resonant dynamics associated with modal interactions between the two modes. The nonlinear response of the structure can be obtained by application of perturbation methods such as averaging on the two-mode model. Many candidate structures are synthesized that meet the desired modal frequency criterion and their nonlinear responses are compared.

An Energy-Based Approach to Computing Resonant Nonlinear Normal Modes

1996 ◽  
Vol 63 (3) ◽  
pp. 810-819 ◽  
Author(s):  
M. E. King ◽  
A. F. Vakakis
Keyword(s):  
Internal Resonance ◽  
Natural Frequencies ◽  
Linear Expansion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom

A formulation for computing resonant nonlinear normal modes (NNMs) is developed for discrete and continuous systems. In a canonical framework, internal resonance conditions are immediately recognized by identifying commensurable linearized natural frequencies of these systems. Additionally, a canonical formulation allows for a single (linearized modal) coordinate to parameterize all other coordinates during a resonant NNM response. Energy-based NNM methodologies are applied to a canonical set of equations and asymptotic solutions are sought. In order to account for the resonant modal interactions, it will be shown that high-order terms in the O(1) solutions must be considered (in the absence of internal resonances, a linear expansion at O(1) is sufficient). Two applications (‘3:1’ resonances in a two-degree-of-freedom system and ‘3:1’ resonance in a hinged-clamped beam) are then considered by which to demonstrate the resonant NNM methodology. It is shown that for some responses, nonlinear modal relations do not exist in the context of physical coordinates and thus a transformation to a canonical framework is necessary in order to appropriately define NNM relations.

Design for Internal Resonances in Nonlinear Transverse Vibrations of Hyperelastic Plates

2013 ◽  
Author(s):  
Astitva Tripathi ◽  
Anil K. Bajaj
Keyword(s):  
Finite Element ◽  
Internal Resonance ◽  
Optimization Techniques ◽  
Mode Shapes ◽  
Nonlinear Phenomenon ◽  
Mems Devices ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Element Analysis ◽  
Nonlinear Properties

Nonlinear phenomenon such as internal resonance have significant potential applications in Micro Electro Mechanical Systems (MEMS) for increasing the sensitivity of biological and chemical sensors and signal processing elements in circuits. While several theoretical systems are known which exhibit 1:2 or 1:3 internal resonances, designing systems that have the desired properties required for internal resonance as well as are physically realizable as MEMS devices is a significant challenge. Traditionally, the design process for obtaining resonant structures exhibiting an internal resonance has relied heavily on the designer’s prior knowledge and experience. However, with advances in computing power and topology optimization techniques, it should be possible to synthesize structures with the required nonlinear properties (such as having modal frequencies in certain ratios) computationally. In this work, plate structures which are candidates for internal resonances are obtained using a Finite Element Method (FEM) formulation implemented in Matlab to iteratively modify a base structure to get its first two natural frequencies close to the desired ratio (1:2 or 1:3). Once a structure with desired topology is achieved, the linear mode shapes of the structure can be extracted from the finite element analysis, and a more complete Lagrangian formulation of the Hyperelastic structure can be used to develop a nonlinear two-mode model of the structure. The reduced-order model is expected to capture the appropriate resonant dynamics associated with modal interactions between the two modes, and the nonlinear response can be obtained by application of perturbation methods such as averaging on the two-mode model.

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.

scholarly journals Effect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well Modified Rayleigh–Duffing Oscillator

2015 ◽  
Vol 25 (02) ◽  
pp. 1550024 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
J. B. Chabi Orou
Keyword(s):  
Initial Conditions ◽  
Duffing Oscillator ◽  
Excitation Amplitude ◽  
Nonlinear Damping ◽  
Homoclinic Bifurcation ◽  
Nonlinear Dissipation ◽  
Transition To Chaos ◽  
Melnikov Criterion ◽  
Quadratic Nonlinearities ◽  
Basin Boundaries

This paper considers the effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh–Duffing oscillator where pure cubic, unpure cubic, pure quadratic and unpure quadratic nonlinearities are considered. By analyzing the potential, an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos. Unpure quadratic parameter and parametric excitation amplitude effects are found on the critical Melnikov amplitude μ cr . Finally, the phase space of initial conditions is carefully examined in order to analyze the effect of the nonlinear damping, and particularly how the basin boundaries become fractalized.

Parametrically Excited Vibrations and Auto-Parametric Resonance in Cylindrical Shells

2000 ◽  
Author(s):  
A. A. Popov ◽  
J. M. T. Thompson ◽  
F. A. McRobie
Keyword(s):  
Internal Resonance ◽  
Cylindrical Shells ◽  
Mode Interaction ◽  
Geometric Description ◽  
Modal Interactions ◽  
Structural Behaviour ◽  
Parametrically Excited ◽  
Parametric Resonances ◽  
Shell Vibration ◽  
Low Dimensional

Abstract Vibrations of cylindrical shells parametrically excited by external axial forcing or by internal auto-parametric resonances are considered. A Rayleigh-Ritz discretization of the von Kármán-Donnell equations through symbolic computations leads to low dimensional models of shell vibration. After applying methods and ideas of modern dynamical systems theory, complete bifurcation diagrams are constructed and analyzed with an emphasis on modal interactions and their relevance to structural behaviour. In the case of free shell vibrations, the Hamiltonian and a transformation into action-angle coordinates followed by averaging provides readily a geometric description of the interaction between concertina and chequerboard modes. It was established that the interaction should be most pronounced when there are slightly less than 2 N square chequerboard panels circumferentially, where N is the ratio of shell radius to thickness. The two mode interaction leads to preferred vibration patterns with larger deflection inwards than outwards, and at internal resonance, significant energy transfer occurs between the modes. The regular and chaotic features of this interaction are studied analytically and numerically.

Applications of Time-Frequency Analysis to Structures With Internal Resonance

1995 ◽  
Author(s):  
Kyle D. Dippery ◽  
Suzanne Weaver Smith
Keyword(s):  
Structural Dynamics ◽  
Frequency Analysis ◽  
Internal Resonance ◽  
Signal Analysis ◽  
Frequency Content ◽  
Modal Interactions ◽  
Time Frequency Analysis ◽  
Time Frequency ◽  
Transition To Chaos ◽  
Changes Over Time

Abstract Time-frequency analysis is an approach to characterizing the nature of signals whose frequency content changes over time. Although the primary applications of this field have, to date, been in the area of communications and signal analysis, it is becoming known in the field of structural dynamics. This paper explores the application of two straightforward time-frequency techniques to several structures that exhibit internal resonance. In particular, the systems analyzed exhibit simple modal interactions and, in one case, a transition to chaos. While other methods exist for analysis of these types of behaviors, larger systems with more complex resonances maybe better analyzed with time-frequency techniques.

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.

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