On the Sufficiency of the Energy Criterion for the Stability of Certain Nonconservative Systems of the Follower-Load Type

1972 ◽  
Vol 39 (3) ◽  
pp. 717-722 ◽  
Author(s):  
H. H. E. Leipholz
Keyword(s):  
Lower Bound ◽  
Critical Load ◽  
Energy Criterion ◽  
Conservative System ◽  
Buckling Load ◽  
Follower Load ◽  
Nonconservative System ◽  
Nonconservative Systems ◽  
Divergence Type ◽  
The Stability

Using Galerkin’s method it is shown that in the domain of divergence, the nonconservative system of the follower-load type is always more stable than the corresponding conservative system. Hence, for nonconservative systems of the divergence type, the critical load of the corresponding conservative system becomes a lower bound for the buckling load, and the energy criterion remains sufficient for predicting stability. Moreover, it is proven that even for more general nonconservative systems, the energy criterion is sufficient under certain restrictions.

A Simplified Stability Criterion for Nonconservative Systems

1979 ◽  
Vol 46 (2) ◽  
pp. 423-426 ◽  
Author(s):  
I. Fawzy
Keyword(s):  
Dynamic Stability ◽  
Stability Criterion ◽  
Degrees Of Freedom ◽  
Necessary Condition ◽  
Matrix Methods ◽  
Nonconservative System ◽  
Nonconservative Systems ◽  
The Stability ◽  
Lyapunov’S Theorem ◽  
Sufficient And Necessary Condition

Dynamic stability of a general nonconservative system of n degrees of freedom is investigated. A sufficient and necessary condition for the stability of such a system is developed. It represents a simplified criterion based on the famous Lyapunov’s theorem which is proved afresh using λ-matrix methods only. When this criterion is adopted, it reduces the number of equations in Lyapunov’s method to less than half. A systematic procedure is then suggested for the stability investigation and its use is illustrated through a numerical example at the end of the paper.

A LOWER BOUND FOR THE BUCKLING LOAD OF A DIVERGENT NON-CONSERVATIVE SYSTEM

1988 ◽  
Vol 12 (3) ◽  
pp. 129-132
Author(s):  
B.L. LY
Keyword(s):  
Lower Bound ◽  
Eigenvalue Problem ◽  
Adjoint System ◽  
Conservative System ◽  
Buckling Load ◽  
Smallest Eigenvalue

The divergent non-conservative problems considered in this paper are pseudo self-adjoint. It is shown that a self-adjoint eigenvalue problem is related to the original non-conservative problem. The smallest eigenvalue of this self-adjoint system provides a lower bound for the buckling load of the non-conservative system.

Snapping of Low Pinned Arches on an Elastic Foundation

1973 ◽  
Vol 40 (3) ◽  
pp. 741-744 ◽  
Author(s):  
G. J. Simitses
Keyword(s):  
Critical Load ◽  
Elastic Foundation ◽  
Critical Loads ◽  
Entire Range ◽  
Buckling Load ◽  
Stability Studies ◽  
Deflection Curve ◽  
The Stability ◽  
Load Deflection Curve ◽  
Range Of Values

The problem of a low half-sine pinned arch under a quasi-statically applied half-sine load is considered. The low arch is resting on an elastic foundation. Critical loads are obtained by investigating the stability of the equilibrium positions by considering all possible modes of deformation. It is assumed that the behavior of the arch is linearly elastic up to the critical load. The entire range of values for the modulus of the foundation is considered. The results are presented graphically as either critical load (snap-through) or classical buckling load (stable bifurcation) versus the rise parameters for a large number of values of the modulus of foundations. This investigation presents an interesting model for stability studies, because, depending on the value of the rise parameter and the modulus of the foundation, the load-deflection curve exhibits the possibilities of the top-of-the-knee buckling, snap-through buckling through unstable bifurcation, and classical buckling (stable bifurcation).

scholarly journals Stability analysis of Beck's column over a fractional-order hereditary foundation

2018 ◽  
Vol 474 (2218) ◽  
pp. 20180315 ◽  
Author(s):  
E. Bologna ◽  
M. Zingales
Keyword(s):  
Stability Analysis ◽  
Fractional Derivative ◽  
Critical Load ◽  
Rheological Behaviour ◽  
Follower Load ◽  
Complex Half Plane ◽  
Linear Spring ◽  
Novel Complex ◽  
The Stability ◽  
Hurwitz Theorem

This paper considers the case of Beck's column resting on a hereditary bed of independent springpots. The springpot possesses an intermediate rheological behaviour among linear spring and linear dashpot. It is defined by means of couple ( C β ,  β ) that characterize the material of the element and is ruled by a Caputo's fractional derivative. In this paper, we investigate the critical load of the column under the action of a follower load by means of a novel complex transform that allows to use the Routh–Hurwitz theorem in the complex half-plane for the stability analysis.

An energy criterion for the linear stability of conservative flows

2000 ◽  
Vol 402 ◽  
pp. 329-348 ◽  
Author(s):  
P. A. DAVIDSON
Keyword(s):  
Variational Principle ◽  
Linear Stability ◽  
Energy Criterion ◽  
Conservative System ◽  
Base Flow ◽  
Recirculating Flow ◽  
Ideal Mhd ◽  
Mhd Equilibria ◽  
Euler Flows ◽  
The Stability

We investigate the linear stability of inviscid flows which are subject to a conservative body force. This includes a broad range of familiar conservative systems, such as ideal MHD, natural convection, flows driven by electrostatic forces and axisymmetric, swirling, recirculating flow. We provide a simple, unified, linear stability criterion valid for any conservative system. In particular, we establish a principle of maximum action of the formformula herewhere η is the Lagrangian displacement,e is a measure of the disturbance energy, T and V are the kinetic and potential energies, and L is the Lagrangian. Here d represents a variation of the type normally associated with Hamilton's principle, in which the particle trajectories are perturbed in such a way that the time of flight for each particle remains the same. (In practice this may be achieved by advecting the streamlines of the base flow in a frozen-in manner.) A simple test for stability is that e is positive definite and this is achieved if L(η) is a maximum at equilibrium. This captures many familiar criteria, such as Rayleigh's circulation criterion, the Rayleigh–Taylor criterion for stratified fluids, Bernstein's principle for magnetostatics, Frieman & Rotenberg's stability test for ideal MHD equilibria, and Arnold's variational principle applied to Euler flows and to ideal MHD. There are three advantages to our test: (i) d2T(η) has a particularly simple quadratic form so the test is easy to apply; (ii) the test is universal and applies to any conservative system; and (iii) unlike other energy principles, such as the energy-Casimir method or the Kelvin–Arnold variational principle, there is no need to identify all of the integral invariants of the flow as a precursor to performing the stability analysis. We end by looking at the particular case of MHD equilibria. Here we note that when u and B are co-linear there exists a broad range of stable steady flows. Moreover, their stability may be assessed by examining the stability of an equivalent magnetostatic equilibrium. When u and B are non-parallel, however, the flow invariably violates the energy criterion and so could, but need not, be unstable. In such cases we identify one mode in which the Lagrangian displacement grows linearly in time. This is reminiscent of the short-wavelength instability of non-Beltrami Euler flows.

Dynamic Stability of an Elastic Beam Subjected to Follower Forces

2012 ◽  
Author(s):  
Katsuhisa Fujita ◽  
Akihide Gotou
Keyword(s):  
Normal Modes ◽  
Elastic Beam ◽  
Conservative System ◽  
Critical Force ◽  
Concentrated Force ◽  
Nonconservative System ◽  
Damped System ◽  
Follower Forces ◽  
The Stability ◽  
The Relationship

The stability of nonconservative system of a beam is investigated when an elastic beam is subjected to follower forces. The mathematical formulations for a conservative system and a nonconservative system are established regarding to a uniform cantilever subjected to a concentrated force and a uniform distributed force axially. The displacement of a uniform cantilever is assumed to be obtained by superposing the modal functions which are normal modes in a vacuum, and is estimated by applying the Galerkin’s method. Changing the forces, the eigenvalue analysis is performed, and the root locus is calculated for the stability analysis. And, the relationship between forces and frequencies for the undamped system and the damped system of the uniform cantilever subjected to a concentrated force and a uniform distributed force is investigated. When the system is considered to be conservative, the divergence phenomenon is confirmed to appear first. On the other hand, when the system is considered to be nonconservative, the flutter phenomenon is confirmed to appear first although the critical force becomes high. And, by changing the structural damping, the destabilized effect due to the damping is confirmed when an elastic beam is subjected to follower forces.

A lower bound estimate of the critical load in bifurcation analysis for incompressible elastic solids

2014 ◽  
Vol 20 (1) ◽  
pp. 53-79 ◽  
Author(s):  
Roger Fosdick ◽  
Pilade Foti ◽  
Aguinaldo Fraddosio ◽  
Salvatore Marzano ◽  
Mario Daniele Piccioni
Keyword(s):  
Lower Bound ◽  
Critical Load ◽  
Bifurcation Analysis ◽  
Elastic Solids ◽  
Lower Bound Estimate

Vibration and Stability Control of Spinning Flexible Shaft via Integration of Smart Materials Technology

2000 ◽  
Author(s):  
Ohseop Song ◽  
Liviu Librescu ◽  
Nam-Heui Jeong
Keyword(s):  
Smart Materials ◽  
Stability Control ◽  
Rotational Axis ◽  
Rotating Shaft ◽  
Nonconservative System ◽  
Gyroscopic Forces ◽  
Points Of View ◽  
The Stability ◽  
Spinning Speed ◽  
Static Character

Abstract Within this paper problems related with the vibration and stability control of circular flexible shafts spinning about their rotational axis are addressed. Due to the occurrence, as a result of the spinning speed, of gyroscopic forces in the system, the rotating shaft can experience, in some conditions, instabilities of the same nature as any nonconservative system, namely divergence and flutter instabilities. Whereas the former instability is of a static character, the latter one is of dynamic character and the results of its occurrence are catastrophic. By including collocated sending and actuating capabilities via integration in the system of piezoelectric devices and of a feedback control law, it is shown that a dramatic enhancement of both the free dynamic response and of the stability behavior from both the divergence and flutter points of view can be achieved. This implies that via the implementation of this technology an increase of the spinning speed can be achieved without the occurrence of these instabilities. Numerical simulations documenting these findings are provided and pertinent conclusions are outlined. It is also worthy to mention that the shaft is modeled as a thin-walled cylinder made of an anisotropic material and incorporating a number of non-classical features.

MODAL DYNAMICS APPROXIMATIONS OF CANTILEVERS UNDER PARTIAL FOLLOWER LOAD

2010 ◽  
Vol 10 (05) ◽  
pp. 1055-1082
Author(s):  
D. S. SOPHIANOPOULOS ◽  
S. KATSAMAGOU ◽  
N. KEFOU
Keyword(s):  
Initial Conditions ◽  
Follower Load ◽  
Shape Functions ◽  
Actual Problem ◽  
Nonconservative System ◽  
Nonhomogeneous Boundary ◽  
Qualitative Dynamic ◽  
Basic Functions ◽  
Modal Dynamics ◽  
Discretization Procedure

Presented herein is a modified Galerkin discretization procedure for determining the qualitative dynamic behavior of elastic cantilevers with internal damping under partial follower step loading at their tips. For this strong nonlinear nonconservative system, the scheme proposed makes use of basic functions that are a product of nonlinear corrections of approximate linear shape functions. These corrected modes are computed in a way that all the nonlinear nonhomogeneous boundary conditions of the actual problem are satisfied throughout the motion. Numerical results obtained using a two-mode approach are found to be in very good qualitative agreement with the finite element results presented in the literature, not only in the vicinity of the critical states, but also in remote unstable domains. The effect of variation of initial conditions is also investigated and the advantages of the proposed procedure compared with conventional ones are discussed. Further research is required for establishing its capabilities and the range of its applicability for a broader class of nonconservative dynamic problems.

The Energy Solutions for the Stability of Tower Crane

2012 ◽  
Vol 170-173 ◽  
pp. 3159-3165
Author(s):  
Ming Xin Huang ◽  
Jian Ping Xu ◽  
Jian Guo Wu
Keyword(s):  
Energy Method ◽  
Buckling Load ◽  
Tower Crane ◽  
The Stability ◽  
Design And Construction

The energy method is used to solve the buckling load of tower crane. It can conclude the effect law on the stability of different section parameters of tower crane and thus provides some references for the design and construction of tower crane.

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