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This paper considers the problem of stable limit cycles generating in a class of uncertain nonlinear systems which leads to stable oscillations in the system’s output.This is a wanted behavior in many practical engineering problems. For this purpose, first the equation of the desirable limit cycle is achieved according to shape, amplitude and frequency of the required output oscillations. Then, the nonlinear control law is designed such that the phase portrait of the closed-loop system includes this stable limit cycle. The design of controller is based on the Lyapunov stability theorem which is suitable for stability analysis of the positive limit sets (the stable limit cycle is a positive limit set for the nonlinear dynamicl system). The proposed robust controller consists of two parts: nominal control law and additional term which guarantees the robust performance and vanishing the effect of uncertain terms. Finally, to show the applicability of the proposed method, an inertia pendulum system (with parametric uncertainties in its dynamical equations) is considered and the robust output oscillations are achieved by creating the desirable limit cycle in the close-loop system.

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The application of wing and stabilizer in aerospace vehicle is most important to stability and flight motion. Nonlinear 2D wing is estimated. Nonlinear damping and stiffness with freeplay in plunging and pitching motion is assumed. 2nd order Damping nonlinearity and 3rd order stiffness nonlinearity in pitching and plunging motion is assumed. Fully nonlinear structure with nonlinear 3rd order piston theory aerodynamic is assumed for the first time and result evaluated with different references. The equations are defined with Hamilton principle with the use of kinetic and potential energy and virtual work. They are solved in the state space via the ruge-kuta numerical method to determine chaotic and limit cycle oscillation motion of supersonic airfoil. The result show that as the speed increases, the behavior of 2D wing is softening type with the use of nonlinear rotational stiffness. But, It shows hardening type with the use of transversal nonlinear stiffness. The effect of transversal and rotational freeplay is more complicated than other parameters and increases instability in low speed. In other hand the stability increases with freeplay in high speed. As shown, increase velocity decrease damping effect in post flutter behavior.

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In this paper, by defining a new paradigm for nonlinear aerodynamic equations of flow separation and static stall, a new form of nonlinear aeroelastic equations for two degrees of freedom airfoils (torsional and bending) are presented. Structural equations are based on the nonlinear mass-spring model; include the nonlinear quadratic and cubic terms. Aerodynamic equations are obtained by combining the unsteady Wagner model and the nonlinear lift coefficient-angle of attack for simulating stall using a cubic approximation. Hamilton’s principle and Lagrange equations were used to derive the aeroelastic equations. The obtained integro-differential nonlinear aeroelastic equations are solved using a new time-history integration method. The aeroelastic behavior of the airfoil is compared in both unsteady and quasi-steady flow. Using the time-history method compared to the phase space method leads to fewer equations. The results show that the aeroelastic behavior of airfoil with a linear structure, using a nonlinear aerodynamic theory for the stall, causes oscillations with a limit cycle in unsteady and quasi-steady flow compared to other linear aerodynamic theories. Also, the use of the cubic curve instead of the piecewise linear curves which is commonly used in other references, although, causes an apparent complication of the equations, reduces the computational time due to faster convergence in solution and makes the reduction in errors. The results show that the use of nonlinear aerodynamic static stall not only reduces the instability velocity, but also reduces the amplitude of limit cycle oscillations in both unsteady and quasi-steady regimes.

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Passive limit cycle walking is a special type of walking happening on a flat and slight downhill surface, without any energy injection and control, and in a cyclic manner. Compensation of energy lost through every heel strike by gravity effect, creates the cyclic behavior for the walking. The main advantage of this type of walking is getting higher efficiency, leading researchers to extend their studies in order to make passive dynamic based walkers. These bipeds can walk on level ground surface by little energy injection, instead of the gravity effect. This fact describes the standpoint of this article. In this research, with impulsive push-off actuation in hand and developing the related models, the walking of an actuated planar parametric model on level ground surface is simulated. Also the stability (with respect to the area of basin of attraction) and gait length has been analyzed by changing design parameters such as actuator’s location and foot shape. The results of this investigation indicates increase in relative stability and gait length for larger foot’s radius and of symmetrical shape.