scholarly journals On the Destabilizing Effect of Damping in Nonconservative Elastic Systems

1965 ◽  
Vol 32 (3) ◽  
pp. 592-597 ◽  
Author(s):  
G. Herrmann ◽  
Ing-Chang Jong
Keyword(s):  
Critical Load ◽  
Characteristic Equation ◽  
Elastic System ◽  
Viscous Damping ◽  
Stability Criteria ◽  
Elastic Systems ◽  
The Stability ◽  
Proper Interpretation ◽  
Degree Of Instability

The destabilizing effect of linear viscous damping in a nonconservative elastic system is investigated by studying the roots of the characteristic equation in addition to the stability criteria and by introducing the concept of degree of instability. A generic relationship between critical loadings for no damping and for slight damping as well as vanishing damping is established. It is found that while the presence of small damping may have a destabilizing effect, proper interpretation of the limiting process of vanishing damping leads to the same critical load as for no damping.

scholarly journals Predicting Collapse of Complex Ecological Systems: Quantifying the Stability-Complexity Continuum

2019 ◽  
Author(s):  
Susanne Pettersson ◽  
Van M. Savage ◽  
Martin Nilsson Jacobi
Keyword(s):  
Single Species ◽  
Ecological Systems ◽  
Ecological Research ◽  
Stability Criteria ◽  
Numerical Techniques ◽  
Intermediate Regime ◽  
Limits Of Stability ◽  
Species Abundances ◽  
The Stability ◽  
Degree Of Instability

Dynamical shifts between the extremes of stability and collapse are hallmarks of ecological systems. These shifts are limited by and change with biodiversity, complexity, and the topology and hierarchy of interactions. Most ecological research has focused on identifying conditions for a system to shift from stability to any degree of instability—species abundances do not return to exact same values after perturbation. Real ecosystems likely have a continuum of shifting between stability and collapse that depends on the specifics of how the interactions are structured, as well as the type and degree of disturbance due to environmental change. Here we map boundaries for the extremes of strict stability and collapse. In between these boundaries, we find an intermediate regime that consists of single-species extinctions, which we call the Extinction Continuum. We also develop a metric that locates the position of the system within the Extinction Continuum—thus quantifying proximity to stability or collapse—in terms of ecologically measurable quantities such as growth rates and interaction strengths. Furthermore, we provide analytical and numerical techniques for estimating our new metric. We show that our metric does an excellent job of capturing the system behaviour in comparison with other existing methods—such as May’s stability criteria or critical slowdown. Our metric should thus enable deeper insights about how to classify real systems in terms of their overall dynamics and their limits of stability and collapse.

scholarly journals Predicting collapse of complex ecological systems: quantifying the stability–complexity continuum

2020 ◽  
Vol 17 (166) ◽  
pp. 20190391 ◽  
Author(s):  
Susanne Pettersson ◽  
Van M. Savage ◽  
Martin Nilsson Jacobi
Keyword(s):  
Single Species ◽  
Ecological Systems ◽  
Ecological Research ◽  
Stability Criteria ◽  
Numerical Techniques ◽  
Intermediate Regime ◽  
Limits Of Stability ◽  
Species Abundances ◽  
The Stability ◽  
Degree Of Instability

Dynamical shifts between the extremes of stability and collapse are hallmarks of ecological systems. These shifts are limited by and change with biodiversity, complexity, and the topology and hierarchy of interactions. Most ecological research has focused on identifying conditions for a system to shift from stability to any degree of instability—species abundances do not return to exact same values after perturbation. Real ecosystems likely have a continuum of shifting between stability and collapse that depends on the specifics of how the interactions are structured, as well as the type and degree of disturbance due to environmental change. Here we map boundaries for the extremes of strict stability and collapse. In between these boundaries, we find an intermediate regime that consists of single-species extinctions, which we call the extinction continuum. We also develop a metric that locates the position of the system within the extinction continuum—thus quantifying proximity to stability or collapse—in terms of ecologically measurable quantities such as growth rates and interaction strengths. Furthermore, we provide analytical and numerical techniques for estimating our new metric. We show that our metric does an excellent job of capturing the system's behaviour in comparison with other existing methods—such as May’s stability criteria or critical slowdown. Our metric should thus enable deeper insights about how to classify real systems in terms of their overall dynamics and their limits of stability and collapse.

scholarly journals RESTABILIZATION OF THE POSTCRITICAL EQUILIBRIUM OF ELASTIC SYSTEMS

2019 ◽  
Vol 15 (1) ◽  
pp. 98-109
Author(s):  
Gaik Manuylov ◽  
Sergey Kosytsyn ◽  
Maxim Begichev
Keyword(s):  
Bifurcation Diagram ◽  
Elastic System ◽  
Critical Force ◽  
Load Parameter ◽  
First Eigenvalue ◽  
Bifurcation Points ◽  
The First Eigenvalue ◽  
Elastic Systems ◽  
Hesse Matrix ◽  
The Stability

The application of the Appel-Vozlinsky theorem on the stability or instability conditions for bifurcation points of conservative elastic systems with a symmetric bifurcation diagram to evaluate restabilization possibility of structures under loads substantially larger than the first critical force. It is shown that restabilization is possible if the first eigenvalue of the Hesse matrix is a continuous alternating function of the load parameter, and the remaining eigenvalues are sign-definite quantities. The examples of the systems with restabilization are given: a high Mises girder and an elastic system composed of compressible rods.

Some Observations on Nonconservative Problems of Elastic Stability

1970 ◽  
Vol 37 (3) ◽  
pp. 671-676 ◽  
Author(s):  
C. Oran
Keyword(s):  
Elastic System ◽  
Elastic Stability ◽  
Stability Behavior ◽  
Elastic Systems ◽  
Concentrated Masses ◽  
The Stability ◽  
The Masses ◽  
Special Case ◽  
Infinitesimal Mass

With reference to undamped elastic systems subjected to nonconservative forces, the possibility of a transition from stability to divergence through infinitely large values of the frequency is considered. Examined in detail is the special case of a cantilever column with two concentrated masses and subjected to a follower end force. It is shown that either flutter or divergence may first set in, depending on the relative magnitudes and locations of the masses. The results obtained by letting one of the masses tend to zero are not necessarily consistent with those obtained directly by setting that mass equal to zero; alternately, it may be stated that an additional infinitesimal mass may lead to drastic changes in the stability behavior of a given elastic system subjected to nonconservative forces.

The Equilibrium Stability of a System of Disk Dynamos

1960 ◽  
Vol 56 (2) ◽  
pp. 154-173 ◽  
Author(s):  
Norman R. Lebovitz
Keyword(s):  
Equilibrium State ◽  
Viscous Damping ◽  
Equilibrium Stability ◽  
Stability Criteria ◽  
Equilibrium Solutions ◽  
Current Case ◽  
Zero Current ◽  
Single Disk ◽  
The Stability ◽  
Disk Dynamo

ABSTRACTThe stability of the equilibrium solutions of the equations describing the behaviour of a system of coupled disk dynamos is investigated. In the absence of viscous damping, it is found that systems consisting of more than two dynamos are unstable. That a single disk dynamo is stable is known. The stability of the undamped two-dynamo system has not been ascertained. When viscous damping is present, there are two equilibrium solutions, one in which all the currents are zero, and one in which they are finite. In the zero-current case, a stability criterion is found. Stability criteria are also found in the finite-current case. Further, the existence of the finite-current equilibrium state excludes the stability of the zero-current equilibrium state.

Gravity-Gradient Excitation of a Rotating Cable-Counterweight Space Station in Orbit

1963 ◽  
Vol 30 (4) ◽  
pp. 547-554 ◽  
Author(s):  
V. Chobotov
Keyword(s):  
Parametric Excitation ◽  
Equations Of Motion ◽  
Space Station ◽  
Type A ◽  
Gravity Gradient ◽  
Viscous Damping ◽  
Stability Criteria ◽  
Axial Vibrations ◽  
The Stability ◽  
Transverse Oscillations

The gravity-gradient excitation of a whirling cable-counterweight space station in orbit is investigated. The Lagrange’s equations of motion for transverse oscillations of the system are derived and shown to be of the Mathieu type. A few representative cases are investigated analytically and on an IBM 7090 computer. The stability criteria for the axial vibrations are also considered and shown to be of the same kind as those for the transver se vibrations of the cable. Viscous damping is included in the analyses and found to be effective and essential for prevention of parametric excitation instability of the system.

Stability criteria for linear systems and realizability criteria for RC networks

1957 ◽  
Vol 53 (4) ◽  
pp. 878-896 ◽  
Author(s):  
A. T. Fuller
Keyword(s):  
Linear Systems ◽  
Characteristic Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Equations ◽  
Stability Criteria ◽  
Polynomial Coefficients ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Rc Networks

ABSTRACTA new set of stability criteria for linear systems is derived. This shows that about half of the Hurwitz criteria are redundant when certain of the coefficients of the characteristic equation are known to be positive. The theory is applied to obtain a very short derivation of the known aperiodicity criteria. The conditions for realizability of RC networks are shown to be closely related to the stability and aperiodicity criteria, and are stated as sets of criteria in terms of the polynomial coefficients. Two basic theorems are involved which give the necessary and sufficient conditions for the roots of two polynomial equations to be real and separated.

Stability Criteria for Linear Equations with Time-Varying Coefficients

1956 ◽  
Vol 60 (543) ◽  
pp. 205-208 ◽  
Author(s):  
P. E. W. Grensted
Keyword(s):  
Characteristic Equation ◽  
Linear Equations ◽  
Time Varying ◽  
Stability Criteria ◽  
Characteristic Equations ◽  
Stability Problems ◽  
Varying Coefficients ◽  
Recent Interest ◽  
Time Varying Coefficients ◽  
The Stability

Recent interest has been shown in stability problems in which the parameters in the relevant characteristic equations are not constant. For instance, the coefficients in the characteristic equation for the stability of flutter of an aeroplane's wings are functions of the speed of the aircraft. When the aircraft is accelerating or decelerating it is therefore necessary to consider whether the resulting variation of the coefficients is sufficiently rapid to invalidate a calculation of stability which is based on the assumption that the coefficients are constant. Similar stability problems arise in connection with guided missiles.

Dissipative instability of the MHD tangential discontinuity in magnetized plasmas with anisotropic viscosity and thermal conductivity

1996 ◽  
Vol 56 (2) ◽  
pp. 285-306 ◽  
Author(s):  
M. S. Ruderman ◽  
E. Verwichte ◽  
R. Erdélyi ◽  
M. Goossens
Keyword(s):  
Thermal Conductivity ◽  
Dispersion Equation ◽  
Viscosity Coefficient ◽  
Tangential Discontinuity ◽  
Stability Criteria ◽  
Threshold Velocity ◽  
Cold Plasmas ◽  
The Difference ◽  
The Stability ◽  
Anisotropic Viscosity

The stability of the MHD tangential discontinuity is studied in compressible plasmas in the presence of anisotropic viscosity and thermal conductivity. The general dispersion equation is derived, and solutions to this dispersion equation and stability criteria are obtained for the limiting cases of incompressible and cold plasmas. In these two limiting cases the effect of thermal conductivity vanishes, and the solutions are only influenced by viscosity. The stability criteria for viscous plasmas are compared with those for ideal plasmas, where stability is determined by the Kelvin—Helmholtz velocity VKH as a threshold for the difference in the equilibrium velocities. Viscosity turns out to have a destabilizing influence when the viscosity coefficient takes different values at the two sides of the discontinuity. Viscosity lowers the threshold velocity V below the ideal Kelvin—Helmholtz velocity VKH, so that there is a range of velocities between V and VKH where the overstability is of a dissipative nature.

Sign in / Sign up

Export Citation Format

Share Document