scholarly journals Theoretical and experimental investigation of a 1:3 internal resonance in a beam with piezoelectric patches

2020 ◽  
Vol 26 (13-14) ◽  
pp. 1119-1132 ◽  
Author(s):  
Vinciane Guillot ◽  
Arthur Givois ◽  
Mathieu Colin ◽  
Olivier Thomas ◽  
Alireza Ture Savadkoohi ◽  
...  
Keyword(s):  
Harmonic Balance ◽  
Internal Resonance ◽  
Harmonic Balance Method ◽  
Mode Shapes ◽  
Governing Equations ◽  
Internal Resonances ◽  
Thin Beam ◽  
Beam Focusing ◽  
Theoretical Results ◽  
Theoretical Developments

Experimental and theoretical results on the nonlinear dynamics of a homogeneous thin beam equipped with piezoelectric patches, presenting internal resonances, are provided. Two configurations are considered: a unimorph configuration composed of a beam with a single piezoelectric patch and a bimorph configuration with two collocated piezoelectric patches symmetrically glued on the two faces of the beam. The natural frequencies and mode shapes are measured and compared with those obtained by theoretical developments. Ratios of frequencies highlight the realization of 1:2 and 1:3 internal resonances, for both configurations, depending on the position of the piezoelectric patches on the length of the beam. Focusing on the 1:3 internal resonance, the governing equations are solved via a numerical harmonic balance method to find the periodic solutions of the system under harmonic forcing. A homodyne detection method is used experimentally to extract the harmonics of the measured vibration signals, on both configurations, and exchanges of energy between the modes in the 1:3 internal resonance are observed. A qualitative agreement is obtained with the 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.

A theoretical and experimental investigation into the vibration response of a flexible rotor and squeeze-film damper assembly

2001 ◽  
Vol 215 (10) ◽  
pp. 1251-1269 ◽  
Author(s):  
C-C Siew ◽  
M Hill ◽  
R Holmes ◽  
M Brennan
Keyword(s):  
Harmonic Balance ◽  
Three Dimensional ◽  
Harmonic Balance Method ◽  
Vibration Response ◽  
Mode Shapes ◽  
Squeeze Film ◽  
Flexible Rotor ◽  
Linear Vibration ◽  
Squeeze Film Damper ◽  
Good Agreement

This paper presents two efficient methods to calculate the unbalance vibration response of a flexible rotor provided with a squeeze-film damper (SFD) with retainer springs. Both methods are iterative and combine the harmonic balance and receptance approaches. The first method, called the modified iteration method (MIM), is suitable for predicting the three-dimensional mode shapes of a concentric SFD-rotor system. The second method, called the modified harmonic balance method (MHBM), is developed to calculate the non-linear vibration response of a flexible shaft provided with either a concentric or eccentric SFD. The system is also investigated experimentally under different conditions. The predictions computed by these methods are compared with experimental measurements and reasonably good agreement is obtained.

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.

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 A Study of the Influence of Rubbing on the Dynamics of a Flexible Disk Rotor System

1993 ◽  
Author(s):  
Fangsheng Wu ◽  
George T. Flowers
Keyword(s):  
Harmonic Balance ◽  
Dynamical Behavior ◽  
Harmonic Balance Method ◽  
Rotor System ◽  
Governing Equations ◽  
System A ◽  
Jeffcott Rotor ◽  
Inertial Moment ◽  
Flexible Disk ◽  
Rotor Model

This study is concerned with investigating the influence of lateral disk flexibility on the dynamics of a rotor system experiencing rub. A rotating, flexible continuous disk/shaft model was developed and the dynamical behavior of this system with and without rubbing was studied. The model developed in this study is similar to the Jeffcott rotor model except that the disk is treated as a laterally flexible continuous circular plate. The motion of the disk was transformed from physical coordinates to a set of generalized coordinates under which the generalized motion was uncoupled and the responses were calculated. Then the inertial moment acting on the shaft was computed and introduced into the governing equations of the shaft motion. Direct integration and the harmonic balance method were used to study the steady state motion of the system. A number of parameter variation studies were performed for varied rub clearances and disk mass influence ratios. The system responses to the rub, its occurrence and development, and the global stability of the observed responses were studied. The results show that rub can be classified into two types: light rub and heavy rub, and the light rub has the forms of forward, backward, or mixed whirling motion. The results also show that the disk flexibility may alter the critical speed to some degree and may also significantly affect the amplitude and stability of the rotor vibration.

Vibration Analysis of Nano-Beam with Consideration of Surface Effects and Damage Effects

2015 ◽  
Vol 4 (1) ◽  
Author(s):  
Fan Yin ◽  
Chang Ping Chen ◽  
De Liang Chen
Keyword(s):  
Harmonic Balance ◽  
Couple Stress Theory ◽  
Harmonic Balance Method ◽  
Beam Theory ◽  
Governing Equations ◽  
Bernoulli Beam ◽  
Stress Theory ◽  
Beam Thickness ◽  
The Galerkin Method ◽  
Euler Bernoulli Beam

AbstractOn the basis of Euler-Bernoulli beam theory, surface elastic theory, the strain equivalent assumption and modiffed couple stress theory, the nonlinear governing equations of the nano-beam are derived. In addition, the Galerkin method and the Harmonic Balance Method are adopted so as to give a solution to the equations. In the example, the effects of nano-beam length, nano-beam thickness, damage factor and surface efect to curves of amplitude-frequency response of the nano-beam are discussed. The results show that damage effects should be taken into consideration and the frequency can be controlled by load and structure size of nano-beam.

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.

scholarly journals Floquet analysis of linear dynamic RLC circuits

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 264-277
Author(s):  
Mohamed El-Borhamy ◽  
Essam Eddin M. Rashad ◽  
Ismail Sobhy
Keyword(s):  
Dynamic Analysis ◽  
Harmonic Balance ◽  
Necessary Conditions ◽  
Harmonic Balance Method ◽  
Approximate Expression ◽  
Time Varying ◽  
Linear Dynamic ◽  
Rlc Circuits ◽  
Transition Curves ◽  
Theoretical Results

AbstractIn this article, the linear dynamic analysis of AC generators modeled as RLC circuits with periodically time-varying inductances via Floquet’s theory is considered. Necessary conditions for the dynamic stability are derived. The harmonic balance method is employed to predict the transition curves and stability domains. An approximate expression for the Floquet form of solution is constructed using Whittaker’s method in the neighborhood of transition curves. Numerical verifications for the obtained theoretical results are considered. In accordance with the experimental results, a satisfactory agreement is relatively achieved with the closed experimental literature of the problem.

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.

scholarly journals Nonlinear Modes of Vibration and Internal Resonances in Nonlocal Beams

2017 ◽  
Vol 12 (3) ◽  
Author(s):  
Pedro Ribeiro ◽  
Olivier Thomas
Keyword(s):  
Harmonic Balance ◽  
Rectangular Cross Section ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Published Data ◽  
Internal Resonances ◽  
Nonlocal Effects ◽  
Nonlinear Modes ◽  
Modes Of Vibration

A nonlocal Bernoulli–Euler p-version finite-element (p-FE) is developed to investigate nonlinear modes of vibration and to analyze internal resonances of beams with dimensions of a few nanometers. The time domain equations of motion are transformed to the frequency domain via the harmonic balance method (HBM), and then, the equations of motion are solved by an arc-length continuation method. After comparisons with published data on beams with rectangular cross section and on carbon nanotubes (CNTs), the study focuses on the nonlinear modes of vibration of CNTs. It is verified that the p-FE proposed, which keeps the advantageous flexibility of the FEM, leads to accurate discretizations with a small number of degrees-of-freedom. The first three nonlinear modes of vibration are studied and it is found that higher order modes are more influenced by nonlocal effects than the first mode. Several harmonics are considered in the harmonic balance procedure, allowing us to discover modal interactions due to internal resonances. It is shown that the nonlocal effects alter the characteristics of the internal resonances. Furthermore, it is demonstrated that, due to the internal resonances, the nonlocal effects are still noticeable at lengths that are longer than what has been previously found.

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