Instability and Unbalance Response of Dissymmetric Rotor-Bearing Systems

1988 ◽  
Vol 110 (3) ◽  
pp. 288-294 ◽  
Author(s):  
P. M. Guilhen ◽  
P. Berthier ◽  
G. Ferraris ◽  
M. Lalanne
Keyword(s):  
Differential Equations ◽  
Coordinate System ◽  
Transfer Matrix ◽  
Differential System ◽  
Industrial Application ◽  
Floquet Theory ◽  
Unbalance Response ◽  
Stationary Coordinate System ◽  
The Matrix ◽  
The Stability

The study deals with the instability and unbalance response of dissymmetric rotors, when periodic differential equations are impossible to avoid. The method which yields motion instability is based on an extension of the well-known Floquet theory. A transfer matrix over one period of the motion is obtained, and the stability of the system can be tested with the eigenvalues of the matrix. To find the instability and the unbalance response, the Newmark formulation is used. Here, the dissymmetry comes either from the rotor or from the bearings in such a way that it is possible to solve a regular differential system without periodic coefficients, either in the stationary coordinate system or in the rotating one. Three examples are given, including an industrial application. The results show that the method proposed is satisfactory.

Discrete-Element Methods for Stability Analysis of Complex Structures

1968 ◽  
Vol 72 (696) ◽  
pp. 1077-1086 ◽  
Author(s):  
J. S. Przemieniecki
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Load Condition ◽  
Buckling Load ◽  
Complex Structures ◽  
Matrix Methods ◽  
The Matrix ◽  
Deflection Analysis ◽  
The Stability ◽  
Discrete Element Methods

The matrix methods of structural analysis developed specifically for use on modern digital computers have now become universally accepted in structural design. These methods provide a means for rapid and accurate stress and deflection analysis of complex structures under static and dynamic loading conditions and they can also be used very effectively for the stability analysis. In the conventional stability analysis two possible approaches are normally used; either the differential equations describing the structural deflections are formulated and the lowest eigenvalue representing the buckling load condition is found for a given set of boundary conditions, or alternatively, if the differential equations are too difficult to prescribe, approximate deflection shapes are used in the strain energy expression for large deflections which is subsequently minimised, leading to the stability determinant whose lowest root represents the instability condition. When designing complex structures the conventional methods of finding buckling load conditions are extremely difficult to apply, and therefore in such cases we have to rely on the matrix methods of stability analysis.

Chatter Stability Analysis Using the Matrix Lambert Function and Bifurcation Analysis

2006 ◽  
Author(s):  
Sun Yi ◽  
Patrick W. Nelson ◽  
A. Galip Ulsoy
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Lambert Function ◽  
Chatter Stability ◽  
Approximate Methods ◽  
Linear Delay ◽  
The Matrix ◽  
The Stability ◽  
Delay Differential

We investigate the stability of the regenerative machine tool chatter problem, in a turning process modeled using delay differential equations (DDEs). An approach using the matrix Lambert function for the analytical solution to systems to delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert function, known to be useful for solving scalar first order DDEs, has recently been extended to a matrix Lambert function approach to solve systems of DDEs. The essential advantage of the matrix Lambert approach is not only the similarity to the concept of the state transition matrix in linear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy, and certain other advantages, when compared to traditional graphical, computational and approximate methods.

Investigation of Parametric Vibration Stability in Slider-Crank Mechanisms With Elastic Coupler

1991 ◽  
Author(s):  
K. Farhang ◽  
A. Midha
Keyword(s):  
Differential Equations ◽  
Floquet Theory ◽  
Continuous Model ◽  
Parametric Vibration ◽  
Connecting Rod ◽  
Parametric Stability ◽  
Vibration Stability ◽  
Geometric Stiffening ◽  
Instability Regions ◽  
The Stability

Abstract An analytical model for investigating parametric vibration stability of slider-crank mechanisms with flexible coupler is presented. The continuous model is formulated to account for initial curvature as well as internal material damping in the coupler. The governing partial differential equations are reduced to a system of ordinary differential equations in terms of the time-dependent modal coefficients. Floquet theory is employed to determine the effects of geometric stiffening as well as relative component mass on parametric stability of mechanism response. Results indicate the existence of instability regions due to combination resonances of various modes. In addition, the stability characteristics of the mechanism is found to improve when slider forces are directed away from the crank-ground pin (i.e. the connecting rod is in tension), and when a relatively smaller slider mass is used.

scholarly journals Stochastic Stability of Damped Mathieu Oscillator Parametrically Excited by a Gaussian Noise

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Claudio Floris
Keyword(s):  
Differential Equations ◽  
Stochastic Stability ◽  
Stochastic Averaging ◽  
Critical Values ◽  
Second Moments ◽  
Ode System ◽  
Complex Eigenvalues ◽  
The Matrix ◽  
The Stability ◽  
Necessary And Sufficient

This paper analyzes the stochastic stability of a damped Mathieu oscillator subjected to a parametric excitation of the form of a stationary Gaussian process, which may be both white and coloured. By applying deterministic and stochastic averaging, two Itô’s differential equations are retrieved. Reference is made to stochastic stability in moments. The differential equations ruling the response statistical moment evolution are written by means of Itô’s differential rule. A necessary and sufficient condition of stability in the moments of orderris that the matrixArof the coefficients of the ODE system ruling them has negative real eigenvalues and complex eigenvalues with negative real parts. Because of the linearity of the system the stability of the first two moments is the strongest condition of stability. In the case of the first moments (averages), critical values of the parameters are expressed analytically, while for the second moments the search for the critical values is made numerically. Some graphs are presented for representative cases.

Analysis and verification of harmonic interaction between DDWTs based on αβ stationary coordinate system

2020 ◽  
Vol 14 (10) ◽  
pp. 1893-1901
Author(s):  
Pingping Han ◽  
Longjian Wang ◽  
Sheng Dou ◽  
Lei Wang ◽  
Rui Bi ◽  
...  
Keyword(s):  
Coordinate System ◽  
Stationary Coordinate System ◽  
Harmonic Interaction

scholarly journals Implicit nonlinear fractional differential equations of variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Mohammed Said Souid ◽  
Kanokwan Sitthithakerngkiet ◽  
Ali Hakem
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Variable Order ◽  
Stability Of Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

scholarly journals Stability Analysis of Distributed Order Fractional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
H. Saberi Najafi ◽  
A. Refahi Sheikhani ◽  
A. Ansari
Keyword(s):  
Stability Analysis ◽  
Characteristic Function ◽  
Differential Equations ◽  
Density Function ◽  
Robust Stability ◽  
Fractional Differential Equations ◽  
Stability Condition ◽  
Fractional Differential ◽  
The Stability ◽  
Distributed Order

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure.

scholarly journals Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

2010 ◽  
Vol 2010 ◽  
pp. 1-23 ◽  
Author(s):  
Josef Diblík ◽  
Denys Ya. Khusainov ◽  
Irina V. Grytsay ◽  
Zdenĕk Šmarda
Keyword(s):  
Lyapunov Functions ◽  
Critical Case ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Simple Eigenvalue ◽  
Discrete Equations ◽  
The Matrix ◽  
The Stability ◽  
Important Characterization

Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalueλ=1of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

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