Non-Synchronous Vibration and Lock-in Regimes in Coupled Structures Using Reduced Models

2021 ◽  
Author(s):  
Miroslav Byrtus ◽  
Štěpán Dyk ◽  
Michal Hajžman
Keyword(s):  
Elastic Interaction ◽  
Dynamical Behaviour ◽  
Van Der Pol ◽  
Reduced Order ◽  
Fundamental Properties ◽  
Modes Of Vibration ◽  
Unstable Solutions ◽  
Coupled Structures ◽  
Lock In ◽  
Subharmonic Resonances

Abstract The contribution is aimed at phenomenological modelling and analysis of fundamental properties of non-synchronous vibrations in chosen coupled structures. A van der Pol reduced-order model is employed to simulate the aero-elastic interaction between flexible bodies and fluid. Non-synchronous vibration and frequency lock-in, along with hysteresis, are captured in the main resonance area. Moreover, the model reveals subharmonic resonances accompanied by the existence of unstable solutions and frequency lock-in. The study is further extended to investigate the dynamical behaviour of a coupled cyclic structure, where different modes of vibration due to the aero-elastic interaction are analysed.

BIFURCATION SCENARIO OF A THREE-DIMENSIONAL VAN DER POL OSCILLATOR

1993 ◽  
Vol 03 (02) ◽  
pp. 399-404 ◽  
Author(s):  
T. SÜNNER ◽  
H. SAUERMANN
Keyword(s):  
Bifurcation Diagram ◽  
Bifurcation Analysis ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Global Bifurcation ◽  
Dynamical Behaviour ◽  
Van Der Pol Oscillator ◽  
Two Dimensional ◽  
Van Der Pol ◽  
Dimensional Models

Nonlinear self-excited oscillations are usually investigated for two-dimensional models. We extend the simplest and best known of these models, the van der Pol oscillator, to a three-dimensional one and study its dynamical behaviour by methods of bifurcation analysis. We find cusps and other local codimension 2 bifurcations. A homoclinic (i.e. global) bifurcation plays an important role in the bifurcation diagram. Finally it is demonstrated that chaos sets in. Thus the system belongs to the few three-dimensional autonomous ones modelling physical situations which lead to chaotic behavior.

Extending a Van der Pol Based Reduced-Order Model for Fluid-Structure Interaction Applied to Non-Synchronous Vibrations in Turbomachinery

2021 ◽  
Author(s):  
Richard Hollenbach ◽  
Robert Kielb ◽  
Kenneth Hall
Keyword(s):  
Fluid Structure Interaction ◽  
Reduced Order Model ◽  
Van Der Pol Oscillator ◽  
Previous Model ◽  
Stable Limit ◽  
Order Model ◽  
Van Der Pol ◽  
Fluid Structure ◽  
Reduced Order ◽  
Structure Interaction

Abstract This paper expands upon a multi-degree-of-freedom, Van der Pol oscillator used to model buffet and Nonsynchronous Vibrations (NSV) in turbines. Two degrees-of-freedom are used, a fluid tracking variable incorporating a Van der Pol oscillator and a classic spring, mass, damper mounted cylinder variable; thus, this model is one of fluid-structure interaction. This model has been previously shown to exhibit the two main aspects of NSV. The first is the lock-in or entrainment phenomenon of the fluid shedding frequency jumping onto the natural frequency of the oscillator, while the second is a stable limit cycle oscillation (LCO) once the transient solution disappears. Improvements are made to the previous model to better understand this aeroelastic phenomenon. First, an error minimizing technique through a system identification method is used to tune the coefficients in the Reduced Order Model (ROM) to improve the accuracy in comparison to experimental data. Secondly, a cubic stiffness term is added to the fluid equation; this term is often seen in the Duffing Oscillator equation, which allows this ROM to capture the experimental behavior more accurately, seen in previous literature. The finalized model captures the experimental cylinder data found in literature much better than the previous model. These improvements also open the door for future models, such as that of a pitching airfoil or a turbomachinery blade, to create a preliminary design tool for studying NSV in turbomachinery.

A Reduced Order Model of Unsteady Flows in Turbomachinery

1994 ◽  
Author(s):  
Kenneth C. Hall ◽  
Răzvan Florea ◽  
Paul J. Lanzkron
Keyword(s):  
Fluid Motion ◽  
Unsteady Flows ◽  
Reduced Order Model ◽  
Lanczos Algorithm ◽  
Order Model ◽  
Reduced Order ◽  
Unsteady Aerodynamic ◽  
Natural Modes ◽  
Wide Range ◽  
Modes Of Vibration

A novel technique for computing unsteady flows about turbomachinery cascades is presented. Starting with a frequency domain CFD description of unsteady aerodynamic flows, we form a large, sparse, generalized, non-Hermitian eigenvalue problem which describes the natural modes and frequencies of fluid motion about the cascade. We compute the dominant left and right eigenmodes and corresponding eigenfrequencies using a Lanczos algorithm. Then, using just a few of the resulting eigenmodes, we construct a reduced order model of the unsteady flow field. With this model, one can rapidly and accurately predict the unsteady aerodynamic loads acting on the cascade over a wide range of reduced frequencies and arbitrary modes of vibration. Moreover, the eigenmode information provides insights into the physics of unsteady flows. Finally we note that the form of the reduced order model is well suited for use in active control of aeroelastic and aeroacoustic phenomena.

Voltage Response of Superharmonic Resonance of Electrostatically Actuated MEMS Resonators: Comparison Between Boundary Value Problem Model and Reduced Order Model

2018 ◽  
Author(s):  
Julio Beatriz ◽  
Martin Botello ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value ◽  
Mode Shapes ◽  
Voltage Response ◽  
Order Model ◽  
Partial Differential ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Modes Of Vibration

This paper deals with the voltage response of electrostatically actuated NEMS resonators at superharmonic resonance. In this work a comparison between Boundary Value Problem (BVP) model, and Reduced Order Model (ROM) is conducted for this type of resonance. BVP model is developed from the partial differential equation by replacing the time derivatives with finite differences. So, the partial differential equation is replaced by a sequence of boundary value problems, one for each step in time. Matlab’s function bvp4c is used to numerically integrate the BVPs. ROMs are based on Galerkin procedure and use the mode shapes of the resonator as a basis of functions. Therefore, the partial differential equation is replaced by a system of differential equations in time. The number of the equations in the system is equal to the number of mode shapes (or modes of vibration) used in the ROM. One mode of vibration ROM is solved using the method of multiple scales. Two modes of vibration ROM is numerically integrated using Matlab’s function ode15s in order to obtain time responses, and a continuation and bifurcation analysis is conducted using AUTO 07P. The effects of different nonlinearities in the system on the voltage response are reported. This work shows that BVP model is a valid method to predict the voltage response of a micro/nano cantilevers.

Frequency Response of Parametric Resonance of Electrostatically Actuated Circular Plates Under Hard Excitations

2019 ◽  
Author(s):  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Frequency Response ◽  
Parametric Resonance ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Reduced Order Model ◽  
Circular Plates ◽  
Order Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Modes Of Vibration

Abstract In this paper, the Method of Multiple Scales, and the Reduced Order Model method of two modes of vibration are used to investigate the amplitude-frequency response of parametric resonance of electrostatically actuated circular plates under hard excitations. Results show that the Method of Multiple Scales is accurate for low voltages. However, it starts to separate from the Reduced Order Model results as the voltage values are larger. The Method of Multiple Scales is good for low amplitudes and weak non-linearities. Furthermore the Reduced Order Model running with AUTO 07p is validated and calibrated using the 2 Term ROM time responses.

Lock-in and quasiperiodicity in a forced hydrodynamically self-excited jet

2013 ◽  
Vol 726 ◽  
pp. 624-655 ◽  
Author(s):  
Larry K. B. Li ◽  
Matthew P. Juniper
Keyword(s):  
Natural Frequency ◽  
Nonlinear Dynamical Systems ◽  
Three Dimensional ◽  
Van Der Pol Oscillator ◽  
Natural Mode ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Frequency Pulling ◽  
Low Dimensional ◽  
Lock In

AbstractThe ability of hydrodynamically self-excited jets to lock into strong external forcing is well known. Their dynamics before lock-in and the specific bifurcations through which they lock in, however, are less well known. In this experimental study, we acoustically force a low-density jet around its natural global frequency. We examine its response leading up to lock-in and compare this to that of a forced van der Pol oscillator. We find that, when forced at increasing amplitudes, the jet undergoes a sequence of two nonlinear transitions: (i) from periodicity to ${ \mathbb{T} }^{2} $ quasiperiodicity via a torus-birth bifurcation; and then (ii) from ${ \mathbb{T} }^{2} $ quasiperiodicity to 1:1 lock-in via either a saddle-node bifurcation with frequency pulling, if the forcing and natural frequencies are close together, or a torus-death bifurcation without frequency pulling, but with a gradual suppression of the natural mode, if the two frequencies are far apart. We also find that the jet locks in most readily when forced close to its natural frequency, but that the details contain two asymmetries: the jet (i) locks in more readily and (ii) oscillates more strongly when it is forced below its natural frequency than when it is forced above it. Except for the second asymmetry, all of these transitions, bifurcations and dynamics are accurately reproduced by the forced van der Pol oscillator. This shows that this complex (infinite-dimensional) forced self-excited jet can be modelled reasonably well as a simple (three-dimensional) forced self-excited oscillator. This result adds to the growing evidence that open self-excited flows behave essentially like low-dimensional nonlinear dynamical systems. It also strengthens the universality of such flows, raising the possibility that more of them, including some industrially relevant flames, can be similarly modelled.

Flow-Induced Acoustics in Corrugated Pipes

2011 ◽  
Vol 10 (1) ◽  
pp. 120-139 ◽  
Author(s):  
Mihaela Popescu ◽  
Stein Tore Johansen ◽  
Wei Shyy
Keyword(s):  
Vortex Shedding ◽  
Fluid Velocity ◽  
Discrete Distribution ◽  
Structural Vibration ◽  
Gas Flows ◽  
Pressure Waves ◽  
Acoustic Response ◽  
Van Der Pol ◽  
Van Der Pol Oscillators ◽  
Lock In

AbstractWhen gas flows through corrugated pipes, pressure waves interacting with vortex shedding can produce distinct tonal noise and structural vibration. Based on established observations, a model is proposed which couples an acoustic pipe and self-excited oscillations with vortex shedding over the corrugation cavities. In the model, the acoustic response of the corrugated pipe is simulated by connecting the lossless medium moving with a constant velocity with a source based on a discrete distribution of van der Pol oscillators arranged along the pipe. Our time accurate solutions exhibit dynamic behavior consistent with that experimentally observed, including the lock-in frequency of vortex shedding, standing waves and the onset fluid velocity capable of generating the lock-in.

Investigation on Dynamics of the Extended Duffing-Van der Pol System

2009 ◽  
Vol 64 (5-6) ◽  
pp. 341-346 ◽  
Author(s):  
Jun Yu ◽  
Jieru Li
Keyword(s):  
Numerical Simulation ◽  
Fractal Dimension ◽  
Nonlinear System ◽  
Computer Simulations ◽  
Chaotic Motion ◽  
Dynamical Behaviour ◽  
Higher Order ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Melnikov Analysis

Abstract The chaotic motion in periodic self-excited oscillators has been extensively investigated through experiments and computer simulations. However, with the advent of the study of chaotic motion by means of strange attractors, Poincar´e map, fractal dimension, it has become necessary to seek for a better understanding of nonlinear system with higher order nonlinear terms. In this paper we consider an extended Duffing-Van der Pol oscillator by introducing a nonlinear quintic term. The dynamical behaviour of the system is investigated by using Melnikov analysis and numerical simulation. The results can help one to understand the essence of given nonlinear system.

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