Postbuckling Analysis of Continuous, Elastic Systems Under Multiple Loads—Part 1: Theory

1979 ◽  
Vol 46 (2) ◽  
pp. 393-397 ◽  
Author(s):  
R. H. Plaut
Keyword(s):  
Perturbation Technique ◽  
Type Systems ◽  
Parameter Perturbation ◽  
Postbuckling Behavior ◽  
Imperfection Sensitivity ◽  
Finite Degree ◽  
Elastic Systems ◽  
Multiple Parameter ◽  
Nonlinear Equilibrium ◽  
The Stability

The stability of continuous, elastic systems subjected to multiple independent loads is considered. The analysis includes nonconservative loads as well as conservative loads, provided that instability is of the static, type. Systems exhibiting prebuckling deformations are included. A multiple-parameter perturbation technique is applied to the nonlinear equilibrium equation in the neighborhood of a critical point, and the postbuckling behavior and imperfection-sensitivity of the system are investigated. Critical points are classified as “general” or “special”, in analogy with Huseyin’s definitions for finite-degree-of-freedom, conservative systems. The results can be applied to study the interaction effects of the independent loads on stability. The theory is given in the present paper, while applications to columns and arches will be presented in the sequel.

Branching Analysis at Coincident Buckling Loads of Nonconservative Elastic Systems

1977 ◽  
Vol 44 (2) ◽  
pp. 317-321 ◽  
Author(s):  
R. H. Plaut
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Power Laws ◽  
Postbuckling Behavior ◽  
Imperfection Sensitivity ◽  
Buckling Loads ◽  
Elastic Systems

Discrete nonconservative elastic systems which lose stability by buckling (divergence) are considered. Simple (distinct) critical points were treated previously, and the case of coincident buckling loads is analyzed here. An asymptotic procedure in the neighborhood of the critical point is used to determine postbuckling behavior and imperfection-sensitivity. It is shown that the system may exhibit no bifurcation at all. In other cases postbuckling paths may be tangential to the fundamental path at the critical point. The sensitivity to imperfections is shown to be more severe than for systems with distinct buckling loads (e.g., one-third, one-fourth, and one-fifth power laws are obtained for certain cases).

scholarly journals Influence of Uncertainties on the Dynamic Buckling Loads of Structures Liable to Asymmetric Postbuckling Behavior

2008 ◽  
Vol 2008 ◽  
pp. 1-24 ◽  
Author(s):  
Paulo B. Gonçalves ◽  
Donald Mark Santee
Keyword(s):  
Static Load ◽  
Basin Of Attraction ◽  
Dynamic Buckling ◽  
Buckling Load ◽  
Postbuckling Behavior ◽  
Imperfection Sensitivity ◽  
Buckling Loads ◽  
Melnikov Criterion ◽  
Linear Buckling ◽  
The Stability

Structural systems liable to asymmetric bifurcation usually become unstable at static load levels lower than the linear buckling load of the perfect structure. This is mainly due to the imperfections present in real structures. The imperfection sensitivity of structures under static loading is well studied in literature, but little is know on the sensitivity of these structures under dynamic loads. The aim of the present work is to study the behavior of an archetypal model of a harmonically forced structure, which exhibits, under increasing static load, asymmetric bifurcation. First, the integrity of the system under static load is investigated in terms of the evolution of the safe basin of attraction. Then, the stability boundaries of the harmonically excited structure are obtained, considering different loading processes. The bifurcations connected with these boundaries are identified and their influence on the evolution of safe basins is investigated. Then, a parametric analysis is conducted to investigate the influence of uncertainties in system parameters and random perturbations of the forcing on the dynamic buckling load. Finally, a safe lower bound for the buckling load, obtained by the application of the Melnikov criterion, is proposed which compare well with the scatter of buckling loads obtained numerically.

Postbuckling Analysis of Continuous, Elastic Systems Under Multiple Loads—Part 2: Applications

1979 ◽  
Vol 46 (2) ◽  
pp. 398-403 ◽  
Author(s):  
R. H. Plaut
Keyword(s):  
Axial Load ◽  
The Other ◽  
Follower Load ◽  
Axial Loads ◽  
Postbuckling Behavior ◽  
Imperfection Sensitivity ◽  
Shallow Arch ◽  
Multiple Loads ◽  
Elastic Systems ◽  
Nonconservative Loading

In Part 1, a theoretical stability analysis of continuous, elastic systems subjected to multiple independent loads was developed. Postbuckling behavior and imperfection-sensitivity were investigated. Three examples are presented here to illustrate the procedures and results of that study. The first example consists of a cantilever subjected to a pair of concentrated axial loads, one applied at the tip and the other at midspan. In the second example, involving nonconservative loading, an axial load and a follower load are applied at the tip of a cantilever. Finally, a shallow arch under three concentrated vertical loads is considered, in order to demonstrate a case with prebuckling deformations.

scholarly journals Formation Control of Unmanned Vessels with Saturation Constraint and Extended State Observation

2021 ◽  
Vol 9 (7) ◽  
pp. 772
Author(s):  
Huixuan Fu ◽  
Shichuan Wang ◽  
Yan Ji ◽  
Yuchao Wang
Keyword(s):  
Formation Control ◽  
Control Method ◽  
Tracking Error ◽  
External Environment ◽  
Control Force ◽  
Extended State ◽  
Parameter Perturbation ◽  
State Observation ◽  
Saturation Constraint ◽  
The Stability

This paper addressed the formation control problem of surface unmanned vessels with model uncertainty, parameter perturbation, and unknown environmental disturbances. A formation control method based on the control force saturation constraint and the extended state observer (ESO) was proposed. Compared with the control methods which only consider the disturbances from external environment, the method proposed in this paper took model uncertainties, parameter perturbation, and external environment disturbances as the compound disturbances, and the ESO was used to estimate and compensate for the disturbances, which improved the anti-disturbance performance of the controller. The formation controller was designed with the virtual leader strategy, and backstepping technique was designed with saturation constraint (SC) function to avoid the lack of force of the actuator. The stability of the closed-loop system was analyzed with the Lyapunov method, and it was proved that the whole system is uniformly and ultimately bounded. The tracking error can converge to arbitrarily small by choosing reasonable controller parameters. The comparison and analysis of simulation experiments showed that the controller designed in this paper had strong anti-disturbance and anti-saturation performance to the compound disturbances of vessels and can effectively complete the formation control.

Stability of Flexibly Supported Oil Journal Bearings Using Non-Newtonian Lubricants: Linear Perturbation Analysis

2000 ◽  
Vol 123 (3) ◽  
pp. 651-654 ◽  
Author(s):  
K. Raghunandana ◽  
B. C. Majumdar, and ◽  
R. Maiti
Keyword(s):  
Power Law ◽  
Perturbation Analysis ◽  
Perturbation Technique ◽  
Stability Margin ◽  
Journal Bearings ◽  
Diameter Ratio ◽  
Linear Perturbation ◽  
Flexible Support ◽  
The Stability ◽  
Power Law Index

The purpose of this paper is to study the effect of non-Newtonian lubricant on the stability of oil film journal bearings mounted on flexible support using linear perturbation technique. The model of non-Newtonian lubricant developed by Dien and Elrod is taken into consideration. The dynamic co-coefficients are calculated for different values of power law index and length to diameter ratio. These are then used to find stability margin for different support parameters to study the effect of the non-Newtonian lubricant.

Zeros of a Class of Transcendental Equation with Application to Bifurcation of DDE

2016 ◽  
Vol 26 (04) ◽  
pp. 1650062 ◽  
Author(s):  
Kit Ian Kou ◽  
Yijun Lou ◽  
Yong-Hui Xia
Keyword(s):  
Small Parameter ◽  
Climate Changes ◽  
Transcendental Equation ◽  
Bifurcation Parameter ◽  
Parameter Perturbation ◽  
Distribution Of Zeros ◽  
World Model ◽  
The Real ◽  
Stability And Bifurcations ◽  
The Stability

Zeros of a class of transcendental equation with small parameter [Formula: see text] are considered in this paper. There have been many works in the literature considering the distribution of zeros of the transcendental equation by choosing the delay [Formula: see text] as bifurcation parameter. Different from standard consideration, we choose [Formula: see text] as bifurcation parameter (not the delay [Formula: see text]) to discuss the distribution of zeros of such transcendental equation. After mathematical analysis, the obtained results are successfully applied to the bifurcation analysis in a biological model in the real word phenomenon. In the real world model, the effect of climate changes can be seen as the small parameter perturbation, which can induce bifurcations and instability. We present two methods to analyze the stability and bifurcations.

On Post-Critical Behavior of a Beam on an Elastic Foundation

2018 ◽  
Vol 18 (06) ◽  
pp. 1850082 ◽  
Author(s):  
Lidija Z. Rehlicki ◽  
Marko B. Janev ◽  
Branislava N. Novaković ◽  
Teodor M. Atanacković
Keyword(s):  
Total Energy ◽  
Elastic Foundation ◽  
Nonlinear Problem ◽  
Bifurcation Analysis ◽  
Nontrivial Solutions ◽  
Buckling Mode ◽  
Nonlinear Equilibrium ◽  
The Stability ◽  
The One ◽  
Two Parameter

In this paper, we analyze the nonlinear equilibrium equation corresponding to the two-parameter bifurcation problem arising in the stability analysis of an elastic simply supported beam on the Winkler type elastic foundation for the case when bimodal buckling occurs. We perform the bifurcation analysis of the nonlinear problem, by using Lyapunov–Schmidt reduction, thus obtaining the number of the nontrivial solutions to the nonlinear problem and qualitatively characterizing the solution patterns. We also give the formulation of the problem and bifurcation analysis from the total energy viewpoint and determine the energy of each bifurcating solution. We assert that the solution with the smallest energy is the one that will be observed in the post-critical state. For specific choice of parameters, the bifurcating solution in the form of the second buckling mode has the smallest total energy. The numerical results illustrating the theory are also provided.

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