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.

scholarly journals Geometrical Non-Linear Periodic Vibration of Plates

2000 ◽  
Author(s):  
M. Petyt ◽  
P. Ribeiro
Keyword(s):  
Steady State ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Internal Resonances ◽  
Resonance Frequencies ◽  
Modal Coupling ◽  
Non Linear ◽  
The Stability

Abstract Periodic, geometrically non-linear free and steady-state forced vibrations of fully clamped plates are investigated. The hierarchical finite element method (HFEM) and the harmonic balance method are used to derive the equations of motion in the frequency domain, which are solved by a continuation method. It is demonstrated that the HFEM requires far fewer degrees of freedom than the h-version of the FEM. Internal resonances due to modal coupling between modes with resonance frequencies related by a rational number, are discovered. In free vibration, internal resonances cause a very significant variation of the mode shape during the period of vibration. A similar behaviour is observed in steady-state forced vibration. The stability of the steady-state solutions is studied by Floquet’s theory and it is shown that stable multi-modal solutions occur.

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.

The effects of nonlinear energy sink and piezoelectric energy harvester on aeroelastic instability of an airfoil

2021 ◽  
pp. 107754632199358
Author(s):  
Ali Fasihi ◽  
Majid Shahgholi ◽  
Saeed Ghahremani
Keyword(s):  
Energy Harvester ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Dynamical Behavior ◽  
Harmonic Balance Method ◽  
Nonlinear Energy Sink ◽  
Piezoelectric Energy ◽  
Energy Sink ◽  
Two Degree Of Freedom ◽  
Nonlinear Energy

The potential of absorbing and harvesting energy from a two-degree-of-freedom airfoil using an attachment of a nonlinear energy sink and a piezoelectric energy harvester is investigated. The equations of motion of the airfoil coupled with the attachment are solved using the harmonic balance method. Solutions obtained by this method are compared to the numerical ones of the pseudo-arclength continuation method. The effects of parameters of the integrated nonlinear energy sink-piezoelectric attachment, namely, the attachment location, nonlinear energy sink mass, nonlinear energy sink damping, and nonlinear energy sink stiffness on the dynamical behavior of the airfoil system are studied for both subcritical and supercritical Hopf bifurcation cases. Analyses demonstrate that absorbing vibration and harvesting energy are profoundly affected by the nonlinear energy sink parameters and the location of the attachment.

A study on nonlinear, parametric aeroelastic energy harvesters under oscillatory airflow

2020 ◽  
pp. 107754632097447
Author(s):  
Mohammad Mehdi Meshki ◽  
Ali Salehzadeh Nobari ◽  
Mohammad Homayoune Sadr
Keyword(s):  
Parametric Excitation ◽  
Indirect Effects ◽  
Harmonic Balance ◽  
Energy Harvester ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Direct And Indirect Effects ◽  
Flow Oscillation ◽  
Energy Harvesters ◽  
Wide Range

In this study, based on parametric excitation originating from airflow oscillation, a novel nonlinear aeroelastic energy harvester is proposed. In this respect, first, the governing equation of the system is derived and studied thoroughly to understand the direct and indirect effects of airflow oscillation on the local and global responses of the system. Then, by using a pseudo-arclength continuation method based on the harmonic balance method, the stable and unstable periodic and quasi-periodic responses of the system are tracked and analyzed. It is demonstrated that the proposed self-parametric (combination parametric and self-excitation) energy harvester can extract more power than the respective nonparametric system for a wide range of amplitudes and frequencies. The gained knowledge of parametric, aeroelastic systems is applicable for both aero-harvesters and other aeroelastic systems undergoing flow oscillation.

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.

Misalignment Modeling in Rotating Systems

2009 ◽  
Author(s):  
Hassan Bahaloo ◽  
Alireza Ebrahimi ◽  
Mostafa Samadi
Keyword(s):  
Harmonic Balance ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Rotating Machinery ◽  
Rotor System ◽  
Common Source ◽  
Series Method ◽  
Coupling Element ◽  
Harmonic Response ◽  
Rotating Systems

Misalignment is a common source of high vibration and malfunction in rotating machinery. Despite its importance and prevalence, no sufficient documentation exists treating this problem. In this paper, a method is introduced for modeling a continuous rotor system which incorporates a misaligned coupling element. It is assumed that both the angular and parallel misalignments are present in the coupling location. The energy expressions are derived and then, applying the Ritz series method, the equations of motion are constructed in matrix form. Because of the special characteristics of the system due to misalignment, a Harmonic Balance Method (HBM) is utilized to obtain the multi harmonic response to an unbalance excitation in disk location. A study on shaft center orbits is also provided and the effect of misalignment type and severity on the orbits is analyzed.

Blade Vibration Suppression Using Friction Elements in Shrouding

2011 ◽  
Author(s):  
Michal Hajzˇman ◽  
Miroslav Byrtus ◽  
Vladimi´r Zeman
Keyword(s):  
Numerical Integration ◽  
Nonlinear Equations ◽  
Harmonic Balance ◽  
Vibration Suppression ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Blade Vibration ◽  
Friction Element ◽  
Nonlinear Equations Of Motion

The problem of two blades with a friction element is studied in order to analyze the effects of the friction on the undesirable vibration suppression. The simplified contact model between friction planes of the blade shrouding and the friction element is derived to be a fast computational tool comparing with a time-consuming finite element solution. The harmonic balance method is suitable for the linearization of originally nonlinear equations of motion under certain assumptions given on the excitation of the system and on its dynamic response. On the other hand the nonlinear equations of motion can be solved directly by their numerical integration, which is more time-consuming but it is not limited by given assumptions. The comparison of results of the harmonic balance method and of the numerical integration of motion equations is given in the paper.

scholarly journals Application of harmonic balance method for solving non-linear equations of motion

2018 ◽  
Vol 182 ◽  
pp. 02024
Author(s):  
Robert Kostek
Keyword(s):  
Harmonic Balance ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Harmonic Balance Method ◽  
Time History ◽  
Balance Method ◽  
Non Linear ◽  
Time Histories ◽  
History Of ◽  
Non Linear Equations

This article presents the advantages and limitations of a harmonic balance method applied for solving non-linear equations of monition. This method provides an opportunity to find stable and unstable periodic solutions, which was demonstrated for a few equations. An error of solution decreases rapidly with increase of number of harmonics for smooth time history of acceleration, which shows convergence; whereas, for discontinuous time histories, this method is not effective.

A Study of Dynamic Instability of Plates by an Extended Incremental Harmonic Balance Method

1985 ◽  
Vol 52 (3) ◽  
pp. 693-697 ◽  
Author(s):  
C. Pierre ◽  
E. H. Dowell
Keyword(s):  
Harmonic Balance ◽  
Dynamic Instability ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Mathieu Equation ◽  
Equation System ◽  
Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Incremental Harmonic Balance ◽  
Nonlinear Equations Of Motion

The dynamic instability of plates is investigated with geometric nonlinearities being included in the model, which allows one to determine the amplitude of the parametric vibrations. A modal analysis allowing one spatial mode is performed on the nonlinear equations of motion and the resulting nonlinear Mathieu equation is solved by the incremental harmonic balance method, which takes several temporal harmonics into account. When viscous damping is included, a new algorithm is proposed to solve the equation system obtained by the incremental method. For this purpose, a new characterization of the parametric vibration by its total amplitude—or Euclidian norm—is introduced. This algorithm is particularly simple and convenient for computer implementation. The instability regions are obtained with a high degree of accuracy.

Analysis of the Primary and Secondary Resonances of Viscoelastic Beams Made of Zener Material

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
Przemysław Wielentejczyk ◽  
Roman Lewandowski
Keyword(s):  
Steady State ◽  
Dynamic Behavior ◽  
Harmonic Balance ◽  
Virtual Work ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Steady State Solution ◽  
Response Curves ◽  
Zener Model ◽  
Secondary Resonances

The problem of geometrically nonlinear, steady-state vibrations of beams made of viscoelastic (VE) materials is considered in this paper. The Euler–Bernoulli and the von Kármán theories are used to describe the dynamic behavior of beams. The VE material of the beams is modeled using the Zener model. Two harmonics are present in the assumed steady-state solution of the problem at hand, which enables an analysis of both the primary and secondary resonances. The virtual work equation and the harmonic balance method are used to derive the amplitude equations in the explicit form. The response curves are determined using the continuation method and treating the frequency of excitation as the main parameter. The results of several examples, which illustrate the dynamic behavior of the considered beams, are presented and discussed.

Sign in / Sign up

Export Citation Format

Share Document