A.N. Kounadis

The existence of regions of divergence instability for nonconservative systems under follower forces

Abstract In this investigation, the existence of regions of divergence instability for an elastically restrained column under a follower compressive force at its end, is discussed. Necessary and sufficient conditions for the existence of regions of divergence instability are established. The boundary between flutter and divergence instability passes always through a double critical point, where the first and second static (buckling) eigenmodes coincide. The eigenvalues (buckling loads) of nonself-adjoint problems of this type are positive and distinct for the entire region of divergence instability, except at its boundary, where the first and second eigenvalues coincide. At this boundary, where the buckling mechanism changes from divergence to flutter and vice-versa, a sudden increase in the critical load occurs with the flutter load always being higher than the corresponding divergence load.

On the paradox of the destabilizing effect of damping in non-conservative systems

Abstract The nature of the critical states of a follower-type non-conservative elastic system with or without damping, is reexamined using a non-linear dynamic analysis. It is found that the destabilizing effect due to certain ratios of damping coefficients, regarded for a long time as a “paradox”, is associated with globally stable critical states. However, in the case of equal damping coefficients, the increase of damping increases the critical load (stabilizing effect). It is also established that this system is statically and dynamically stable for the entire region of variation of the non-conservativeness load parameter, contrary to existing analyses. Moreover, it is deduced that the critical states corresponding to both types of instability may become unstable if a slight material non-linearity is included; then, the mechanism of divergence and flutter instability change from stable to unstable and vice versa for a critical value of the material non-linearity which depends on the non-conservativeness loading parameter.

On the failure of static stability analyses of nonconservative systems in regions of divergence instability

Abstract The stability of perfect bifurcational discrete dissipative systems under follower loads in regions of existence/non-existence of adjacent equilibria is thoroughly re-examined in the light of recent progress in nonlinear dynamics. A general theory for such nongradient systems described by autonomous ordinary differential equations is developed. Conditions for the existence of adjacent equilibria, the stability of precritical, critical and postcritical states, as well as for different types of local bifurcations are established. Focusing attention on the interaction of geometric nonlinearities and vanishing damping, new findings contradicting widely accepted results of the classical (linear) analysis are discovered. In a small region of adjacent equilibria near a compound branching point, which is explicitly determined, an interaction of two consecutive postbuckling modes occurs related to the following phenomena : in case of vanishing damping, loss of stability may occur via a Hopf (dynamic) bifurcation prior to static (divergence) buckling. Moreover, the critical states of divergence instability may be associated with a double zero Jacobian eigenvalue satisfying also the conditions of a Hopf (local) bifurcation. Besides local (dynamic) bifurcations, global bifurcations are also found. An example is used to illustrate the qualitative findings.

Divergence and flutter instability of elastically restrained structures under follower forces

Abstract The instability of elastically restrained simple structures acted upon by compressive follower forces is discussed by using a static stability analysis. From this investigation it is concluded that depending on the amount of elastic restraint: (a) divergence or flutter type instability is possible and (b) the critical load of a divergence type non-conservatively loaded structure may be higher or smaller than the critical load of the corresponding structure subjected to a conservative load. Moreover, a lower bound theorem is presented according to which under a certain condition the load carrying capacity of a non-conservatively loaded structure of divergence type is higher than the load carrying capacity of the corresponding conservatively loaded structure. From the foregoing findings a better insight into the actual mechanism of loss of stability of structures under follower forces is gained.

NON-LINEAR PANEL FLUTTER IN REMOTE POST-CRITICAL DOMAINS

Abstract The non-linear behavior of an elastic panel subjected to the combination of supersonic gas flow and quasi-static loading in the middle surface is studied. Particular attention is focused on the bifurcation paths in the flutter domain, including their remote parts. A variety of attractors, both periodic and chaotic, are observed. Moreover, non-symmetric stable periodic vibrations have been explored in regions far away from areas of divergence (buckling) instability. The transition from one attractor to the next one is discussed when the control parameters vary continuously. Hysteretic phenomena are studied by comparing the system’s behavior in direct and return paths on the control parameter plane.

An efficient and simple approximate technique for solving nonlinear initial and boundary-value problems

An efficient and easily applicable, approximate analytic technique for the solution of nonlinear initial and boundary-value problems associated with nonlinear ordinary differential equations (O.D.E.) of any order and variable coefficients, is presented. Convergence, uniqueness and upper bound error estimates of solutions, obtained by the successive approximations scheme of the proposed technique, are thoroughly established. Important conclusions regarding the improvement of convergence for large time and large displacement solutions in case of nonlinear initial-value problems are also assessed. The proposed technique is much more efficient than the perturbations schemes for establishing the large postbuckling response of structural systems. The efficiency, simplicity and reliability of the proposed technique is demonstrated by two illustrative examples for which available numerical results exist.

Some new instability aspects for nonconservative systems under follower loads

Abstract The mechanism of instability of nonlinear nonconservative discrete systems under follower loads with or without pre-critical deformation is thoroughly re-examined with the aid of a complete nonlinear dynamic analysis. Considering the stability of motion in the large, in the sense of Lagrange, the critical (divergence or dynamic) load is defined as the minimum load for which an unbounded (divergent) motion is initiated. Regions which have been considered (on the basis of linearized analyses) as of flutter instability are found (using a nonlinear dynamic analysis) dynamically stable. Some new instability phenomena contradict existing findings which have been widely accepted. Moreover, it is established that the divergence buckling loads, obtained by static methods of analysis, coincide with the nonlinear dynamic loads only in the case of no pre-critical deformation. Cases of random-like (or chaotic-like) motions for certain values of the nonconservativeness loading parameter are also revealed for autonomous non-dissipative structural systems.

On the discontinuity of the flutter load for various types of cantilevers

Abstract In this investigation, using an energy (variational) approach, the flutter instability for various types of elastically restrained uniform cantilevers carrying up to three concentrated masses and subjected to a follower compressive force, is presented. The effects of transverse shear deformation and rotatory inertia of the mass of the column and of the positioning of the concentrated masses with or without their rotational inertia, are also included in the analysis. In all cases, where the flutter load is obtained from the coincidence of the second and third flexural eigenfrequencies a discontinuity with a finite jump in this load is possible; the lower value of the flutter load at this discontinuity is obtained from the coincidence of the first and second eigenfrequencies, while the upper value of this load is obtained from the coincidence of the second and third eigenfrequencies. Subsequently, it is found that the flutter load is a sectionally continuous function of certain of the varying parameters. Finally, the effect of several parameters upon the magnitude of the jump in the flutter load, is also discussed.

Use of non-linear localization for isolating structures from earthquake-induced motions

The dynamic response due to earthquake-induced excitations of multi-storey buildings simulated by a cantilever (with attached concentrated masses) supported on a flexible foundation, is reconsidered when stiffness non-linearities are included. To this end, a suitable non-linear spring-mass device is placed between the ground and the mass of the foundation, which under certain conditions can absorb a significant amount of seismic energy over a large frequency range, thus drastically reducing the seismic response of the foundation. This is achieved by the stiffness non-linearity that gives rise to a localization phenomenon, according to which motions generated by external disturbances remain passively localized close to the point of seismic excitation instead of ‘spreading’ to the entire structure. The implications of these findings to the design of earthquake-resistant structures are discusssed. Copyright

Chaoslike phenomena in the non-linear dynamic stability of discrete damped or undamped systems under step loading

Abstract The non-linear dynamic buckling of a two-degree-of-freedom, damped or undamped, multiple-parameter, structural system under step loading of infinite duration, is thoroughly examined. Attention is mainly focused on the dynamic stability of imperfect multiple-parameter systems which under static loading may either lose their stability through a limit point or exhibit a continuously rising equilibrium path. Considering the stability of motion in the large, in the sense of Lagrange, the dynamic buckling load is defined as the smallest load for which an escaped (leading to an unbounded) motion is initiated. A lower bound estimate very close to the exact dynamic buckling load is also established. The discussion of the individual and combined effect on the dynamic buckling load of this autonomous system reveals some interesting phenomena such as: discontinuity in the dynamic buckling load and considerable sensitivity to damping and to initial conditions. Chaoslike and metastability phenomena are also revealed for a simple deterministic system of structural engineering importance.