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

Abstract

An elastic double pendulum subject to a force acting along a fixed straight line, the socalled "Reut's column problem", is a structure exhibiting flutter and divergence instability, which was never realized in practice and thus debated whether to represent reality or mere speculation. It is shown, both theoretically and experimentally, how to obtain the Reut's loading by exploiting the contact with friction of a rigid blade against a freely-rotating cylindrical constraint, which moves axially at constant speed, an action recalling that of a bow's hair on a violin string. With this experimental set-up, flutter and divergence instabilities, as well as the detrimental effect of viscosity on critical loads, are documented indisputably, thus bringing an end to a long debate. This result opens a new research area, with perspective applications to mechanical actuators, high-precision cutting tools, or energy harvesting devices.

Flutter Instability and Ziegler Destabilization Paradox for Elastic Rods Subject to Non-Holonomic Constraints

“…Finally, the obtained results confirm the validity of the experimental verification of the flutter load proposed in [23,10,11,24] and definitely open the way to exploitation of non-holonomic constraints in the mechanics of deformable structures.…”

Non-holonomic constraints inducing flutter instability in structures under conservative loadings

“…Recently, flutter and divergence instabilities have been shown to occur in structures loaded through nonconservative forces produced with frictional devices by Bigoni and Noselli [5], realizing the tangentially follower load postulated by Ziegler [34], and by Bigoni and Misseroni [8], obtaining the fixed-line load introduced by Reut [30]. The same type of instabilities is disclosed here in structural systems subject to conservative loadings when nonholonomic constraints are applied to the structure.…”

“…In the d'Alembert-Lagrange equation ( 21), the unknown reaction force p(t) transmitted to the column by the non-holonomic constraint plays the role of a Lagrangian multiplier enforcing condition (8).…”

“…Koiter was strongly convinced that follower forces were a sort of 'physical non-sense' (Koiter,[20],[21]), so that Singer et al[31] write 'An example in the field of elastic stability of what Drucker referred to as playing useless games was presented by Koiter, in his 1985 Prandtl lecture, where he discussed the physical significance of instability due to non-conservative, purely configurationdependent, external loads.' Several years after these negative views, Bigoni and Noselli[5] and Bigoni and Misseroni[8] respectively showed how to realize a tangentially follower force(Ziegler, 1952 [34]) and a fixed-line force(Reut, 1939 [30]) with devices involving Coulomb friction. Friction is a dissipative and follower force, in the present article we show how this force can be eliminated thus realizing a purely conservative system of load.…”

Extreme deformations of the cantilever Euler Elastica under transverse aerodynamic load

“…Even though only in recent years researchers framed follower forces into a more general and modern mathematical framework, early theoretical research on follower forces dates back to 1950. The most famous works include the Pflüger column (Pflüger, 1950;Pfluger, 1955;Tommasini et al, 2016;Bigoni et al, 2018b), the Ziegler paradox (Ziegler, 1952(Ziegler, , 1956Bottema, 1956;Ziegler, 1977;Bigoni et al, 2018a), the Beck's column (Beck, 1952), the Reut's column problem (Reut, 1939;Bigoni and Misseroni, 2020), and the first monography on the stability of non-conservative forces due to Bolotin (Bolotin, 1963). One of the most challenging issues related to the framework of follower forces was their actual realisation.…”