Flutter and divergence instability in the Pflüger column: Experimental evidence of the Ziegler destabilization paradox

Abstract

Flutter instability in elastic structures subject to follower load, the most important cases being the famous Beck's and Pflüger's columns (two elastic rods in a cantilever configuration, with an additional concentrated mass at the end of the rod in the latter case), have attracted, and still attract, a thorough research interest. In this field, the most important issue is the validation of the model itself of follower force, a nonconservative action which was harshly criticized and never realized in practice for structures with diffused elasticity. An experimental setup to introduce follower tangential forces at the end of an elastic rod was designed, realized, validated, and tested, in which the follower action is produced by exploiting Coulomb friction on an element (a freely-rotating wheel) in sliding contact against a flat surface (realized by a conveyor belt). It is therefore shown that follower forces can be realized in practice and the first experimental evidence is given for both the flutter and divergence instabilities occurring in the Pflüger's column. In particular, load thresholds for the two instabilities are measured and the detrimental effect of dissipation on the critical load for flutter is experimentally demonstrated, while a slight increase in load is found for the divergence instability. The presented approach to follower forces discloses new horizons for testing self-oscillating structures and for exploring and documenting dynamic instabilities possible when nonconservative loads are applied.

Structures loaded with a force acting along a fixed straight line, or the “Reut’s column problem”

“…so that, since the mass matrix M is real, symmetric and positive definite, its square root M 1/2 is invertible. Therefore, the generalized eigenvalue problem (16) can be recast in a standard eigenvalue form…”

“…cylinder (detail c), moved against the structure at fixed velocity by a testing machine (its load frame and loading cell are visible, part a) turned horizontally. Note that the system used to load the head of the Reut's column is the same as that developed in [16].…”

“…As a system to vary the vertical force transmitted between the blade and the cylinder (so to vary the Reut's force via Coulumb friction) the same apparatus developed for the flutter machine was used (a detailed description is reported in [16]). The friction coefficient measured at the blade/cylinder contact decreases with the vertical load within the interval 0.71 ± 0.15.…”

“…a boundary condition with nothing but a mathematical meaning, never observed in the real world (see the vivid description of the state-of-the-art given in [14]). A similar difficulty arose in the realization of the tangentially follower force introduced by Ziegler, which, considered 'unrealistic' by Koiter [13,14], was only recently realized by Bigoni and Noselli [15] (see also [16,17]) through the introduction of a special constraint, namely, a freely rotating wheel sliding with friction against a moving rigid plate. In this article it is shown that the Reut's load can be obtained through the sliding, with constant Coulomb friction, of a cylindrical surface free of rotating about its axis against a rigid blade, as shown from two equivalent views (but taken with a different perspective) in Fig.…”

Harnessing elasticity to generate self-oscillation via an electrohydrodynamic instability

“…Because the follower force was initially invented theoretically and assumed to be always tangential to the slender structure regardless of its time-dependent deformation, it was demonstrated only mathematically and has been considered impractical 22 . However, it was recently realised experimentally on a metre-scale rod 23 .…”