A Comprehensive Stability Criterion for Forced Vibrations in Nonlinear Systems

1953 ◽  
Vol 20 (1) ◽  
pp. 9-12
Author(s):  
K. Klotter ◽  
E. Pinney
Keyword(s):  
Differential Equation ◽  
Nonlinear Systems ◽  
Stability Criterion ◽  
Practical Importance ◽  
Nonlinear Function ◽  
Forced Vibrations ◽  
Maximum Value ◽  
The Stability

Abstract This paper deals with the forced vibrations described by the differential equation a q .. + c q + c Φ ( q , q . ) = P cos Ω t wherein Φ denotes a nonlinear function of q and/or q̇. It presents a criterion for determining their stability. It is shown that under very weak restrictions, which equivalently means, for a large variety of cases (including all of practical importance) the stability depends on the sign of ∂q*/∂P (q* denoting the maximum value of q(t) within a period). The motion is stable if this derivative is positive; it is unstable if it is negative.

THE STUDY OF TAPERED MAGNETIC BEARING

2017 ◽  
pp. 129-139
Author(s):  
Konstantin Kim ◽  
Konstantin Kim
Keyword(s):  
High Speed ◽  
Magnetic Levitation ◽  
Practical Importance ◽  
Nonlinear Function ◽  
Magnetic Bearing ◽  
Electric Engineering ◽  
Controlling System ◽  
Start Up ◽  
The Stability ◽  
Methods Analytical

Objective: To justify the choice of tapered magnetic bearing parameters (with combined radial and axial control), to elaborate recommendations, the fulfillment of which will make it possible to significantly improve the characteristics of rotating rotor magnetic levitation of an electric machine. Methods: Analytical methods of traditional electric engineering, as well as the laboratory-based method. Results: It was established that: the control along radial and axial coordinates is independent, despite the fact that steering forces are created by common electromagnets; linear approximation allowing for small oscillations in radial and axial directions the stability of holding rotating rotor is provided by the same laws of control as for a standard circular magnetic bearing; the latter are characterized by a disadvantage, connected with the loss (or deterioration) of radial control in the start-up mode, caused by the common core in axial and radial channels of control; the significant stiffness and steering force may be achieved in the operating area of axial coordinate variation by simultaneously keeping the maximum radial stiffness, sufficient for the start-up mode, by means of purpose-built nonlinear function generators in an electronic circuit diagram of the controlling system; ease of adjustment highly determines the successful bearing setting. Practical importance: The elaborated recommendations will make it possible to design tapered magnetic bearings for high-speed electric machines.

On Steady-State Harmonic Vibrations of Nonlinear Systems With Many Degrees of Freedom

1966 ◽  
Vol 33 (2) ◽  
pp. 406-412 ◽  
Author(s):  
W. M. Kinney ◽  
R. M. Rosenberg
Keyword(s):  
Steady State ◽  
Nonlinear Systems ◽  
Degrees Of Freedom ◽  
Forced Vibrations ◽  
Geometrical Method ◽  
Mass System ◽  
Harmonic Vibrations ◽  
Forcing Functions ◽  
Exciting Forces ◽  
The Stability

A nonlinear spring-mass system with many degrees of freedom, and subjected to periodic exciting forces, is examined. The class of admissible systems and forcing functions is defined, and a geometrical method is described for deducing the steady-state forced vibrations having a period equal to that of the forcing functions. The methods used combine the geometrical methods developed earlier in the problem of normal mode vibrations and Rauscher’s method. The stability of these steady-state forced vibrations is examined by Hsu’s method. The results are applied to an example of a system having two degrees of freedom.

scholarly journals Robust Stabilization Approach and Performance via Static Output Feedback for a Class of Nonlinear Systems

2009 ◽  
Vol 2009 ◽  
pp. 1-22 ◽  
Author(s):  
Neila Bedioui ◽  
Salah Salhi ◽  
Mekki Ksouri
Keyword(s):  
Nonlinear Systems ◽  
Output Feedback ◽  
Robust Stabilization ◽  
Nonlinear Function ◽  
Static Output Feedback ◽  
Additional Degree ◽  
Lmi Optimization ◽  
And Performance ◽  
The Stability ◽  
Static Output

This paper deals with the stability and stabilization problems for a class of discrete-time nonlinear systems. The systems are composed of a linear constant part perturbated by an additive nonlinear function which satisfies a quadratic constraint. A new approach to design a static output feedback controller is proposed. A sufficient condition, formulated as an LMI optimization convex problem, is developed. In fact, the approach is based on a family of LMI parameterized by a scalar, offering an additional degree of freedom. The problem of performance taking into account an criterion is also investigated. Numerical examples are provided to illustrate the effectiveness of the proposed conditions.

scholarly journals Theory of Frequency Captured of Eccentric Rotor by Vibration Environment with Same Direction

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Su Ming ◽  
Li Rong ◽  
Xie Zhiping ◽  
Zheng Jiming
Keyword(s):  
Differential Equation ◽  
Stability Criterion ◽  
Periodic Coefficient ◽  
Second Order ◽  
Frequency Synchronization ◽  
Synchronization Problem ◽  
Eccentric Rotor ◽  
Small Parameters ◽  
The Stability ◽  
Rotating Bodies

Aiming at the frequency synchronization phenomena of oscillating or rotating bodies, this paper proposes a novel solution to address the self-synchronization problem of vibration systems. An integral mean method with small parameters and periodic coefficient (IMM-SPPC) is proposed, which converts the relative motion of the electrically driven eccentric rotor and the vibration environment into a second-order periodic coefficient differential equation. Through the calculation of the equilibrium point of the second-order periodic coefficient differential equation and the study of its stability, the synchronization criterion and the stability criterion of the eccentric rotor and the vibration environment are deduced. The simulation results show the validity of the deduced synchronization criterion and stability criterion. The proposed IMM-SPPC provides a new way for studying vibration synchronization.

A new robust observer design for nonlinear systems with application to fault diagnosis

2017 ◽  
Vol 40 (13) ◽  
pp. 3696-3708 ◽  
Author(s):  
Ammar Zemzemi ◽  
Mohamed Kamel ◽  
Ahmed Toumi ◽  
Mondher Farza
Keyword(s):  
Fault Diagnosis ◽  
Nonlinear Systems ◽  
Sliding Mode ◽  
Nonlinear Function ◽  
Design Procedure ◽  
Observer Design ◽  
Linear Matrix ◽  
Time Varying ◽  
System Nonlinearity ◽  
The Stability

This paper presents a robust fault diagnosis scheme for a class of uncertain nonlinear systems whose nonlinear function satisfies the Lipschitz condition with unmatched time-varying uncertainties, external disturbances and perturbed output. The design procedure combines the high robustness of the nonlinear unknown input observer with sliding-mode techniques in order to enhance the estimation qualities. The proposed design is derived and expressed as a linear matrix inequality optimization problem. Additionally, we have provided an approach to reduce conservatism in the derivation of the stability conditions. The effectiveness of this observer and the fault diagnosis scheme are shown by applying them to a single-link manipulator. Simulation results are presented to validate the proposed approach and show the robustness for the system nonlinearity and unmatched time-varying uncertainties.

scholarly journals On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 78
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Linear Ordinary Differential Equation ◽  
Time Interval ◽  
Hyperbolic Partial Differential Equation ◽  
Time Step ◽  
Numerical Examples ◽  
One Dimensional ◽  
The Stability ◽  
The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

scholarly journals Observer-based adaptive neural network backstepping sliding mode control for switched fractional order uncertain nonlinear systems with unmeasured states

2021 ◽  
pp. 002029402110211
Author(s):  
Tao Chen ◽  
Damin Cao ◽  
Jiaxin Yuan ◽  
Hui Yang
Keyword(s):  
Neural Network ◽  
Nonlinear Systems ◽  
Sliding Mode Control ◽  
Fractional Order ◽  
Sliding Mode ◽  
Tracking Error ◽  
Adaptive Neural Network ◽  
Dynamic Surface ◽  
Mode Control ◽  
The Stability

This paper proposes an observer-based adaptive neural network backstepping sliding mode controller to ensure the stability of switched fractional order strict-feedback nonlinear systems in the presence of arbitrary switchings and unmeasured states. To avoid “explosion of complexity” and obtain fractional derivatives for virtual control functions continuously, the fractional order dynamic surface control (DSC) technology is introduced into the controller. An observer is used for states estimation of the fractional order systems. The sliding mode control technology is introduced to enhance robustness. The unknown nonlinear functions and uncertain disturbances are approximated by the radial basis function neural networks (RBFNNs). The stability of system is ensured by the constructed Lyapunov functions. The fractional adaptive laws are proposed to update uncertain parameters. The proposed controller can ensure convergence of the tracking error and all the states remain bounded in the closed-loop systems. Lastly, the feasibility of the proposed control method is proved by giving two examples.

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