Influence of Bistable Plunge Stiffness On Nonlinear Airfoil Flutter

2021 ◽  
Author(s):  
Renan F. Corrêa ◽  
Flávio D. Marques
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Nonlinear Behavior ◽  
Limit Cycle Oscillation ◽  
Restoring Force ◽  
Aeroelastic System ◽  
Technical Literature ◽  
Practical Applications

Abstract Aeroelastic systems have nonlinearities that provide a wide variety of complex dynamic behaviors. Nonlinear effects can be avoided in practical applications, as in instability suppression or desired, for instance, in the energy harvesting design. In the technical literature, there are surveys on nonlinear aeroelastic systems and the different manners they manifest. More recently, the bistable spring effect has been studied as an acceptable nonlinear behavior applied to mechanical vibration problems. The application of the bistable spring effect to aeroelastic problems is still not explored thoroughly. This paper contributes to analyzing the nonlinear dynamics of a typical airfoil section mounted on bistable spring support at plunging motion. The equations of motion are based on the typical aeroelastic section model with three degrees-of-freedom. Moreover, a hardening nonlinearity in pitch is also considered. A preliminary analysis of the bistable spring geometry’s influence in its restoring force and the elastic potential energy is performed. The response of the system is investigated for a set of geometrical configurations. It is possible to identify post-flutter motion regions, the so-called intrawell, and interwell. Results reveal that the transition between intrawell to interwell regions occurs smoothly, depending on the initial conditions. The bistable effect on the aeroelastic system can be advantageous in energy extraction problems due to the jump in oscillation amplitudes. Furthermore, the hardening effect in pitching motion reduces the limit cycle oscillation amplitudes and also delays the occurrence of the snap-through.

scholarly journals Nonlinear Behavior of a Magnetic Bearing System

1995 ◽  
Vol 117 (3) ◽  
pp. 582-588 ◽  
Author(s):  
L. N. Virgin ◽  
T. F. Walsh ◽  
J. D. Knight
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Nonlinear Behavior ◽  
Analytical Techniques ◽  
Magnetic Bearing ◽  
Forced Response ◽  
The Mathematical Model ◽  
Two Degree Of Freedom ◽  
Sensitivity To Initial Conditions ◽  
Bearing System

This paper describes the results of a study into the dynamic behavior of a magnetic bearing system. The research focuses attention on the influence of nonlinearities on the forced response of a two-degree-of-freedom rotating mass suspended by magnetic bearings and subject to rotating unbalance and feedback control. Geometric coupling between the degrees of freedom leads to a pair of nonlinear ordinary differential equations, which are then solved using both numerical simulation and approximate analytical techniques. The system exhibits a variety of interesting and somewhat unexpected phenomena including various amplitude driven bifurcational events, sensitivity to initial conditions, and the complete loss of stability associated with the escape from the potential well in which the system can be thought to be oscillating. An approximate criterion to avoid this last possibility is developed based on concepts of limiting the response of the system. The present paper may be considered as an extension to an earlier study by the same authors, which described the practical context of the work, free vibration, control aspects, and derivation of the mathematical model.

On establishing equations of motion of mechanical vibration systems placed on moving bases

2017 ◽  
Vol 45 (3) ◽  
pp. 209-227
Author(s):  
M Gürgöze ◽  
F Terzioğlu
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Main Idea ◽  
Vertical Axis ◽  
Constant Angular Velocity ◽  
Vibration System ◽  
Reaction Forces ◽  
The Subject ◽  
Vibrating Systems

The first author has been teaching the postgraduate course, “The Dynamics of Mechanical Systems” in The ITU Faculty of Mechanical Engineering for nearly 20 years. He has observed that students frequently have problems in obtaining the equations of motion of the vibrating systems which were placed on moving bases. Starting from this observation, he has found that the homework stated below, which was given to the students occasionally, was very helpful in learning the subject. The main idea of the homework is the derivation of the equations of motion, with the help of formulating the Lagrange’s equations with respect to a moving set of axis for a vibration system with two degrees of freedom which consists of a horizontal table rotating with a constant angular velocity around a vertical axis. The students were also asked to solve the same problem with a different method of their choice and to determine the reaction forces as well. We want to share this problem with the reader, which we have assessed as very instructive and appropriate from the viewpoint of applicability of different methods.

scholarly journals A Bifurcation Analysis and Sensitivity Study of Brake Creep Groan

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Shaun Smith ◽  
James Knowles ◽  
Byron Mason ◽  
Sean Biggs
Keyword(s):  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Period Doubling ◽  
Nonlinear Behavior ◽  
Sensitivity Study ◽  
Friction Model ◽  
Model Parameters ◽  
Period Doubling Bifurcations ◽  
Simple Models

Creep groan is the undesirable vibration observed in the brake pad and disc as brakes are applied during low-speed driving. The presence of friction leads to nonlinear behavior even in simple models of this phenomenon. This paper uses tools from bifurcation theory to investigate creep groan behavior in a nonlinear 3-degrees-of-freedom mathematical model. Three areas of operational interest are identified, replicating results from previous studies: region 1 contains repelling equilibria and attracting periodic orbits (creep groan); region 2 contains both attracting equilibria and periodic orbits (creep groan and no creep groan, depending on initial conditions); region 3 contains attracting equilibria (no creep groan). The influence of several friction model parameters on these regions is presented, which identify that the transition between static and dynamic friction regimes has a large influence on the existence of creep groan. Additional investigations discover the presence of several bifurcations previously unknown to exist in this model, including Hopf, torus and period-doubling bifurcations. This insight provides valuable novel information about the nature of creep groan and indicates that complex behavior can be discovered and explored in relatively simple models.

scholarly journals Taming the Beast: Diffusion Method in Nonlocal Gravity

Universe ◽  
2018 ◽  
Vol 4 (9) ◽  
pp. 95 ◽  
Author(s):  
Gianluca Calcagni
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Form Factors ◽  
Specific Form ◽  
Diffusion Method ◽  
Dynamical Equations ◽  
Nonlinear Dynamical ◽  
Gravitational Theories

We present a method to solve the nonlinear dynamical equations of motion in gravitational theories with fundamental nonlocalities of a certain type. For these specific form factors, which appear in some renormalizable theories, the number of field degrees of freedom and of initial conditions is finite.

Approximate Floquet Analysis of Parametrically Excited Multi-Degree-of-Freedom Systems With Application to Wind Turbines

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
Gizem D. Acar ◽  
Brian F. Feeny
Keyword(s):  
Eigenvalue Problem ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
System Response ◽  
Periodic Part ◽  
Mdof Systems ◽  
Floquet Modes ◽  
Problem Frequency ◽  
Exponential Part

General responses of multi-degrees-of-freedom (MDOF) systems with parametric stiffness are studied. A Floquet-type solution, which is a product between an exponential part and a periodic part, is assumed, and applying harmonic balance, an eigenvalue problem is found. Solving the eigenvalue problem, frequency content of the solution and response to arbitrary initial conditions are determined. Using the eigenvalues and the eigenvectors, the system response is written in terms of “Floquet modes,” which are nonsynchronous, contrary to linear modes. Studying the eigenvalues (i.e., characteristic exponents), stability of the solution is investigated. The approach is applied to MDOF systems, including an example of a three-blade wind turbine, where the equations of motion have parametric stiffness terms due to gravity. The analytical solutions are also compared to numerical simulations for verification.

scholarly journals Nonlinear Behavior of a Magnetic Bearing System

1994 ◽  
Author(s):  
Lawrence N. Virgin ◽  
Thomas F. Walsh ◽  
Josiah D. Knight
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Nonlinear Behavior ◽  
Analytical Techniques ◽  
Magnetic Bearing ◽  
Forced Response ◽  
The Mathematical Model ◽  
Two Degree Of Freedom ◽  
Sensitivity To Initial Conditions ◽  
Bearing System

This paper describes the results of a study into the dynamic behavior of a magnetic bearing system. The research focuses attention on the influence of nonlinearities on the forced response of a two-degree-of-freedom rotating mass suspended by magnetic bearings and subject to rotating unbalance and feedback control. Geometric coupling between the degrees of freedom leads to a pair of nonlinear ordinary differential equations which are then solved using both numerical simulation and approximate analytical techniques. The system exhibits a variety of interesting and somewhat unexpected phenomena including various amplitude driven bifurcational events, sensitivity to initial conditions and the complete loss of stability associated with the escape from the potential well in which the system can be thought to be oscillating. An approximate criterion to avoid this last possibility is developed based on concepts of limiting the response of the system. The present paper may be considered as an extension to an earlier study by the same authors which described the practical context of the work, free vibration, control aspects and derivation of the mathematical model.

An Introduction to Self-Excited Oscillations

2009 ◽  
Author(s):  
Riccardo Panciroli ◽  
Serge Abrate
Keyword(s):  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Nonlinear Problems ◽  
Spreadsheet Program ◽  
Practical Applications ◽  
Euler’S Method ◽  
Complicating Factors ◽  
High Frequency Excitation ◽  
Friction Models

This paper describes an approach used to introduce a type of nonlinear problems in an undergraduate class on mechanical vibrations. Self-excited oscillations are encountered in a number of practical applications including brakes, clutches, belts, tires, and violins. To go beyond the derivation of the equations of motion for simplified models and examine the effect of various parameters requires the ability to find numerical solutions. It was found that developing numerical solutions using a simple integration technique such as Euler’s method with a spreadsheet program like Excel was most effective because: (1) Euler’s method is easy to implement; (2) Excel is widely available; (3) students are able to develop the solution themselves; (4) it can be done quickly. In this case students were able to explore problems with one or more degrees of freedom and compare their results with those found in recent publications which presents several advantages: students develop confidence in their ability to explore different models and examine the effects of different complicating factors, they develop their own solutions and are able to focus on understanding the physics of the problem, and they develop a sense that they are working on problems of current interest instead of some overly simplified textbook problem. Examples dealing with brake squeal problem were used and the effects of mass, stiffness, damping and friction were studied. Many different friction models are available and several of them were used to determine the effect of friction on the appearance of self-excited vibrations. The appearance of a limit cycle in the phase portrait is discussed along with the dynamics of the system. It is also shown that a short high frequency excitation can be used to squelch those self-excited oscillations.

scholarly journals Control of Limit Cycle Oscillation in a Three Degrees of Freedom Airfoil Section Using Fuzzy Takagi-Sugeno Modeling

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Douglas Domingues Bueno ◽  
Luiz Carlos Sandoval Góes ◽  
Paulo José Paupitz Gonçalves
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Control Surface ◽  
Space Representation ◽  
Linear Matrix ◽  
Limit Cycle Oscillation ◽  
Aerodynamic Forces ◽  
Control Gain ◽  
Takagi Sugeno ◽  
Three Degrees Of Freedom

This work presents a strategy to control nonlinear responses of aeroelastic systems with control surface freeplay. The proposed methodology is developed for the three degrees of freedom typical section airfoil considering aerodynamic forces from Theodorsen’s theory. The mathematical model is written in the state space representation using rational function approximation to write the aerodynamic forces in time domain. The control system is designed using the fuzzy Takagi-Sugeno modeling to compute a feedback control gain. It useds Lyapunov’s stability function and linear matrix inequalities (LMIs) to solve a convex optimization problem. Time simulations with different initial conditions are performed using a modified Runge-Kutta algorithm to compare the system with and without control forces. It is shown that this approach can compute linear control gain able to stabilize aeroelastic systems with discontinuous nonlinearities.

Reduced Order Modeling of Limit Cycle Oscillation for Aeroelasticity Systems

2002 ◽  
Author(s):  
Philip S. Beran ◽  
David J. Lucia ◽  
Chris L. Pettit
Keyword(s):  
Limit Cycle ◽  
Degrees Of Freedom ◽  
Galerkin Approximation ◽  
Computational Cost ◽  
Coupled System ◽  
Low Rank ◽  
Limit Cycle Oscillation ◽  
Aeroelastic System ◽  
Reduced Order ◽  
Cycle Oscillation

Limit-cycle oscillations of a nonlinear panel in supersonic flow are computed using a reduced-order aeroelastic model. Panel dynamics are governed by the large-deflection, nonlinear, von Ka´rma´n equation as expressed in low-order form through a Galerkin approximation. The aerodynamics are described by the Euler equations, which are reduced in order using proper orthogonal decomposition. The coupled system of equations is implicitly time integrated with second-order temporal accuracy to predict limit-cycle oscillation (LCO) amplitude, and linearly analyzed to predict LCO onset. The fluid is synchronized with the structure in time through subiteration, using only 18 degrees of freedom to describe the aeroelastic system. The Jacobian employed in the fully implicit analysis is of equivalently low rank, enabling rapid analysis. Using the reduced order model, LCO onset is predicted directly at a computational cost of approximately 400 time steps with a high accuracy verified by full-order analysis.

Assessing the influence of the road-tire friction coefficient on the yaw and roll stability of articulated vehicles

2018 ◽  
Vol 233 (12) ◽  
pp. 2987-2999 ◽  
Author(s):  
André de Souza Mendes ◽  
Agenor de Toledo Fleury ◽  
Marko Ackermann ◽  
Fabrizio Leonardi ◽  
Roberto Bortolussi
Keyword(s):  
Friction Coefficient ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Lateral Force ◽  
Longitudinal Force ◽  
Convergence Region ◽  
The Road ◽  
Articulated Vehicles ◽  
Tire Friction

This article addresses the yaw stability of articulated vehicles by assessing the influence of the road-tire friction coefficient on the convergence region of a particular equilibrium condition. In addition, the boundaries of this region are compared to the boundaries of the non-jackknife and non-rollover regions to distinguish the instability phenomenon, jackknife or roll-over, responsible for this delimitation. The vehicle configuration considered in this analysis is composed by one tractor unit and one towed unit connected through an articulation point, for instance, a tractor-semitrailer combination. A nonlinear articulated bicycle model with four degrees of freedom is used together with a nonlinear lateral force tire model. To estimate the convergence region, the phase trajectory method is used. The equations of motion of the mathematical model are numerically integrated for different initial conditions in the phase plane, and the state orbits are monitored in order to verify the convergence point and the occurrence of instability events. In all cases, the longitudinal force on each tire, such as traction and braking, is not considered. The results show the existence of convergence regions delimited only by jackknife events, for low values of the friction coefficient, and only by rollover events, for high values of the friction coefficient. Moreover, the transition between these two conditions as the friction coefficient is changed is graphically presented. The main contributions of this article are the identification of the abrupt reduction of the convergence region as the value of the friction coefficient increases and the distinction of the instability events, jackknife or rollover, that define the boundaries of the convergence region.

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