A van der Pol Based Reduced-Order Model for Non-Synchronous Vibration (NSV) in Turbomachinery

2013 ◽  
Author(s):  
Stephen T. Clark ◽  
Robert E. Kielb ◽  
Kenneth C. Hall
Keyword(s):  
Design Tool ◽  
Van Der Pol Oscillator ◽  
Future Research ◽  
Preliminary Design ◽  
Stable Limit ◽  
Coupling Coefficients ◽  
Van Der Pol ◽  
Reduced Order ◽  
Coupling Parameters ◽  
Van Der Pol Model

This paper demonstrates the potential of using a multi-degree-of-freedom, traditional van der Pol oscillator to model non-synchronous vibration (NSV) in turbomachinery. It is shown that the two main characteristics of NSV are captured by the reduced-order, van der Pol model. First, a stable limit cycle oscillation (LCO) is maintained for various conditions. Second, the lock-in phenomenon typical of NSV is captured for various fluid-structure frequency ratios. This research identifies values and significance of the coupling parameters used in the van der Pol model. These coefficients are chosen to model confirmed instances of experimental NSV, and to develop a preliminary design tool that engineers can use to better design turbomachinery for NSV. Specifically, coefficient tuning from experimental instances of NSV are considered to identify the unknown coupling coefficients in the van der Pol model. The goal of future research will be to identify values and significance of the coupling parameters used in the van der Pol model, to match these coefficients with confirmed instances of experimental NSV, and to develop a preliminary design tool that engineers can use to better design turbomachinery for NSV. Proper orthogonal decomposition (POD) CFD techniques and coefficient tuning from experimental instances of NSV have been considered to identify the unknown coupling coefficients in the van der Pol model. The finalization of this preliminary-design research will be completed in future research.

Developing a Reduced-Order Model to Understand Non-Synchronous Vibration (NSV) in Turbomachinery

2012 ◽  
Author(s):  
Stephen T. Clark ◽  
Robert E. Kielb ◽  
Kenneth C. Hall
Keyword(s):  
Stable Limit Cycle ◽  
Forced Response ◽  
Van Der Pol Oscillator ◽  
Future Research ◽  
Limit Cycle Oscillation ◽  
Preliminary Design ◽  
Stable Limit ◽  
Van Der Pol ◽  
Reduced Order ◽  
Van Der Pol Model

This paper demonstrates the potential of using a multi-degree-of-freedom, traditional van der Pol oscillator to model Non-Synchronous Vibration (NSV) in turbomachinery. It is shown that the two main characteristics of NSV are captured by the reduced-order, van der Pol model. First, a stable limit cycle oscillation (LCO) is maintained for various conditions. Second, the lock-in phenomenon typical of NSV is captured for various fluid-structure frequency ratios. The results also show the maximum amplitude of the LCO occurs at an off-resonant condition, i.e., when the natural shedding frequency of the aerodynamic instability is not coincident with the natural modal frequency of the structure. This conclusion is especially relevant in preliminary design in industry because it suggests that design engineers cannot treat NSV as a normal Campbell-diagram crossing as they would for preliminary design for forced response; it is possible that by redesigning the blade, the response amplitude of the blade may actually be higher. The goal of future research will be to identify values and significance of the coupling parameters used in the van der Pol model, to match these coefficients with confirmed instances of experimental NSV, and to develop a preliminary design tool that engineers can use to better design turbomachinery for NSV. Proper Orthogonal Decomposition (POD) CFD techniques and coefficient tuning from experimental instances of NSV have been considered to identify the unknown coupling coefficients in the van der Pol model. Both the modeling of experimental NSV and preliminary design development will occur in future research.

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.

EXTENDING A VAN DER POL BASED REDUCED-ORDER MODEL FOR FLUID-STRUCTURE INTERACTION APPLIED TO NON-SYNCHRONOUS VIBRATIONS IN TURBOMACHINERY

2021 ◽  
pp. 1-14
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.

DYNAMICS AND ACTIVE CONTROL OF MOTION OF A DRIVEN MULTI-LIMIT-CYCLE VAN DER POL OSCILLATOR

2007 ◽  
Vol 17 (04) ◽  
pp. 1343-1354 ◽  
Author(s):  
R. YAMAPI ◽  
B. R. NANA NBENDJO ◽  
H. G. ENJIEU KADJI
Keyword(s):  
Limit Cycle ◽  
Active Control ◽  
Coupling Parameter ◽  
Van Der Pol Oscillator ◽  
External Excitation ◽  
Van Der Pol ◽  
Chaotic Vibrations ◽  
Van Der Pol Model ◽  
Nonautonomous Case ◽  
Selection Of

This paper deals with the dynamics and active control of a driven multi-limit-cycle Van der Pol oscillator. The amplitude of the oscillatory states both in the autonomous and nonautonomous case are derived. The interaction between the amplitudes of the external excitation and the limit-cycles are also analyzed. The domain of the admissible values on the amplitude for the external excitation is found. The effects of the control parameter on the behavior of a driven multi-limit-cycle Van der Pol model are analyzed and it appears that with the appropriate selection of the coupling parameter, the quenching of chaotic vibrations takes place.

Bonhoeffer-van der Pol Oscillator Model of the Sino-Atrial Node: A Possible Mechanism of Heart Rate Regulation

1994 ◽  
Vol 33 (01) ◽  
pp. 116-119 ◽  
Author(s):  
S. Sato ◽  
S. Doi ◽  
T. Nomura
Keyword(s):  
Heart Rate ◽  
Stable Limit Cycle ◽  
Nonlinear Dynamic System ◽  
Van Der Pol Oscillator ◽  
Transition Curve ◽  
Stable Limit ◽  
Van Der Pol ◽  
Rate Regulation ◽  
Dynamic System Theory ◽  
Heart Rate Regulation

Abstract:A Bonhoeffer-van der Pol equation with a stable limit cycle is proposed as a model of the pacemaker in the sino-atrial node to exptain heart rate regulation. Standard tools, such as the phase transition curve in nonlinear dynamic system theory, are used to analyze the model and results are compared with other studies on experiments with dogs.

scholarly journals Design of a New Chaotic System Based on Van Der Pol Oscillator and Its Encryption Application

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 743 ◽  
Author(s):  
Jianbin He ◽  
Jianping Cai
Keyword(s):  
Control Method ◽  
Variable Parameter ◽  
Stable Limit Cycle ◽  
Simple Algorithm ◽  
Van Der Pol Oscillator ◽  
Stable Limit ◽  
Van Der Pol ◽  
Dynamical Characteristics ◽  
New Chaotic System ◽  
Exclusive Or

The Van der Pol oscillator is investigated by the parameter control method. This method only needs to control one parameter of the Van der Pol oscillator by a simple periodic function; then, the Van der Pol oscillator can behave chaotically from the stable limit cycle. Based on the new Van der Pol oscillator with variable parameter (VdPVP), some dynamical characteristics are discussed by numerical simulations, such as the Lyapunov exponents and bifurcation diagrams. The numerical results show that there exists a positive Lyapunov exponent in the VdPVP. Therefore, an encryption algorithm is designed by the pseudo-random sequences generated from the VdPVP. This simple algorithm consists of chaos scrambling and chaos XOR (exclusive-or) operation, and the statistical analyses show that it has good security and encryption effectiveness. Finally, the feasibility and validity are verified by simulation experiments of image encryption.

Parameter Estimation of a Fluttering Aeroelastic System in the Transitional Reynolds Number Regime

2010 ◽  
Author(s):  
Mohammad Khalil ◽  
Abhijit Sarkar ◽  
Dominique Poirel
Keyword(s):  
Reynolds Number ◽  
Parameter Estimation ◽  
Wind Tunnel ◽  
Van Der Pol Oscillator ◽  
Limit Cycle Oscillation ◽  
Parameter Estimates ◽  
Stable Limit ◽  
Van Der Pol ◽  
Aeroelastic System ◽  
Rigid Wing

We report the parameter estimation results of a self-sustaining aeroelastic oscillator. The system is composed of a rigid wing that is elastically mounted on a rig, which in turn is fixed in a wind tunnel. For certain flow conditions, in particular dictated by the Reynolds number in the transitional regime, the wing extracts energy from the flow leading to a stable limit cycle oscillation. The basic physical mechanism at the origin of the oscillations is laminar boundary layer separation, which leads to negative aerodynamic damping. An empirical model of the aeroelastic system is proposed in the form of a generalized Duffing-van der Pol oscillator, whereby the linear and nonlinear aeroelastic terms are unknowns to be estimated. The model (input) noise process accounting for the amplitude modulation observed from experiments will also be estimated. We apply a Bayesian inference based batch data assimilation method in tackling this strongly nonlinear and non-Gaussian model. In particular, Markov Chain Monte Carlo sampling technique is used to generate samples from the joint distribution of the unknown parameters given noisy measurement data. The extended Kalman filter is utilized to obtain the conditional distribution of the model state given the noisy measurements. The parameter estimates for a third order generalized Duffing-van der Pol oscillator are obtained and marginal and joint probability density functions for the parameters will be presented for both a numerical model and a rigid wing that is elastically mounted on a rig in a wind tunnel.

NUMERICAL SOLUTION FOR DUFFING-VAN DER POL OSCILLATOR VIA BLOCK METHOD

2020 ◽  
Vol 10 (1) ◽  
pp. 1857-8365
Author(s):  
A. F. Nurullah ◽  
M. Hassan ◽  
T. J. Wong ◽  
L. F. Koo
Keyword(s):  
Numerical Solution ◽  
Van Der Pol Oscillator ◽  
Block Method ◽  
Van Der Pol

scholarly journals Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

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