Stability and vibration of a helical rod constrained by a cylinder

The stability and vibration of an elastic rod with a circular cross section under the constraint of a cylinder is discussed. The differential equations of dynamics of the constrained rod are established with Euler＇s angles as variables describing the attitude of the cross section. The existence conditions of helical equilibrium under constraint are discussed as a special configuration of the rod. The stability of the helical equilibrium is discussed in the realms of statics and dynamics, respectively. Necessary conditions for the stability of helical rod are derived in space domain and time domain, and the difference and relationship between Lyapunov＇s and Euler＇s stability concepts are discussed. The free frequency of flexural vibration of the helical rod with cylinder constraint is obtained in analytical form.

On stability of elastic domain during isothermal solid-solid phase transformationin a tube configuration

Under isothermal quasi-static stretching the phase transition of a superelastic NiTi tube involves the formation （during loading） and vanishing （in unloading） of a high strain （martensite） domain. The two events are accompanied by a rapid stress drop/rise due to the formation/vanishing of do- main fronts. From a thermodynamic point of view, both are instability phenomena that occur once the system reaches its critical state. This paper investigates the stability of a shrink- ing cylindrical domain in a tube configuration during unload- ing. The energetics and thermodynamic driving force of the cylindrical domain are quantified by using an elastic inclu- sion model. It is demonstrated that the two domain fronts ex- hibit strong interaction when they come close to each other, which brings a peak in the total energy and a sign change in the thermodynamic driving force. It is proved that such domain front interaction plays an important role in control- ling the stability of the domain and in the occurrence of stress jumps during domain vanishing. It is also shown that the pro- cess is governed by two nondimensional length scales （the normalized tube length and normalized wall-thickness） and that the length scale dependence of the critical domain length and stress jump for the domain vanishing can be quantified by the elastic inclusion model.

Elastic Dynamic Stability of Big-Span Power Transmission Tower Subjected to Seismic Excitations

By combining the time-history response analysis and the eigenvalue buckling analysis, this paper developed a computational procedure to study the elastic dynamic stability of a transmission tower by APDL language in ANSYS. The influences of different input directions of seismic excitations and damping ratio on the elastic dynamic stability of tower were discussed. The following conclusions were obtained： （ 1 ） Longitudinal direction of the transmission lines is the worst input direction of seismic excitation for the transmission tower. （2） Dead load has no significant effect on the critical load and the occurrence time of buckling. （3） Vertical input of seismic excitations has no great effect on the dynamic stability of the transmission tower. （4） Damping effect has an influence on the dynamic stability of the transmission tower; however, the inherent characteristics of dynamic buckling is not changed.

A simple criterion for finite time stability with application to impacted buckling of elastic columns

The existing theories of finite-time stability depend on a prescribed bound on initial disturbances and a prescribed threshold for allowable responses. It remains a challenge to identify the critical value of loading parameter for finite time instability observed in experiments without the need of specifying any prescribed threshold for al- lowable responses. Based on an energy balance analysis of a simple dynamic system, this paper proposes a general criterion for finite time stability which indicates that finite time stability of a linear dynamic system with constant coefficients during a given time interval [0, tf] is guaranteed provided the product of its maximum growth rate （determined by the maximum eigen-root pl 〉0） and the duration tf does not exceed 2, i.e., pltf 〈2. The proposed criterion （pltf=2） is applied to several problems of impacted buckling of elastic columns： （i） an elastic column impacted by a striking mass, （ii） longitudinal impact of an elastic column on a rigid wall, and （iii） an elastic column compressed at a constant speed （＂Hoff problem＂）, in which the time-varying axial force is replaced approximately by its average value over the time duration. Comparison of critical parameters predicted by the proposed criterion with available experimental and simulation data shows that the proposed criterion is in robust reasonable agreement with the known data, which suggests that the proposed simple criterion （pltf---2） can be used to estimate critical parameters for finite time stability of dynamic systems governed by linear equations with constant coefficients.

BOUNDARY FEEDBACK CONTROL OF ELASTIC BEAM EQUATION WITH STRUCTURAL DAMPING AND STABILITY

In this paper, we consider the partial differential equation of an elastic beam with structuraldamping by boundary feedback control. First, we prove this closed system is well--posed; then weestablish tbe exponential stability for this elastic system by using a theorem whichbelongs to F. L.Huang; finally, we discuss the distribution and multiplicity of the spectrum of this system. Theseresults are very important and useful in practical applications.