Snapping mechanisms are investigated for an elastic strip with ends imposed to move and rotate in time. Attacking the problem analytically via Euler's elastica and the second variation of the total potential energy, the number of stable equilibrium configurations is disclosed by varying the kinematics of the strip ends. This result leads to the definition of a 'universal snap surface', collecting the sets of critical boundary conditions for which the system snaps. The elastic energy release at snapping is also investigated, providing useful insights for the optimization of impulsive motion. The theoretical predictions are finally validated through comparisons with experimental results and finite element simulations, both fully confirming the reliability of the introduced universal surface. The presented analysis may find applications in a wide range of technological fields, as for instance energy harvesting and jumping robots.

Non-conservative loads of the follower type are usually believed to be the source of dynamic instabilities such as flutter and divergence. It is shown that these instabilities (including Hopf bifurcation, flutter, divergence, and destabilizing effects connected to dissipation phenomena) can be obtained in structural systems loaded by conservative forces, as a consequence of the application of non-holonomic constraints. These constraints may be realized through a 'perfect skate' (or a non-sliding wheel), or, more in general, through the slipless contact between two circular rigid cylinders, one of which is free of rotating about its axis. The motion of the structure produced by these dynamic instabilities may reach a limit cycle, a feature that can be exploited for soft robotics applications, especially for the realization of limbless locomotion. . 1 Bolotin [9] writes 'The Euler method is [only] applicable if the external forces have a potential (i.e. if they are conservative forces), and in general is not applicable if they do not.' 2 The attempt by Willems [33] of producing a tangential follower load was indicated as misleading by Huang et al. [13]. Elishakoff [10] reports that 'Bolotin felt -if my memory serves me well!-that it should be impossible to produce Becks column experiment via a conservative system of forces.' Anderson and Done [2] write 'Sometimes, the creation of a force like [a follower force] in the laboratory presents awkward practical problems, and the simulation of this force wherever possible by a conservative force would be very convenient. However, because of the differing nature of the fundamental properties of the conservative and non-conservative systems, the simulation could only work in a situation where the two systems behave in similar ways; that is when the conservative system is not operating in a regime of oscillatory instability. (The conservative system can not become dynamically instable since, by definition, it has no energy source from which to supply the extra kinetic energy involved in the instability).' Koiter [19] states that '[...] it appears impossible to achieve any non-conservative loading conditions in the laboratory by purely mechanical means', because 'non-conservative external loads always require an external energy source, much as a fluid flow or an interaction with electro-dynamic phenomenon'. 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 fr...

Dedicated to our mentor and friend Davide Bigoni in honour of his 60 th birthday, for all of his invaluable teaching throughout these years and for many more to come AbstractThe theory, the design and the experimental validation of a catastrophe machine based on a flexible element are addressed for the first time. A general theoretical framework is developed by extending that of the classical catastrophe machines made up of discrete elastic systems. The new formulation, based on the nonlinear solution of the elastica, is enhanced by considering the concept of the universal snap surface. Among the infinite set of elastica catastrophe machines, two families are proposed and investigated to explicitly assess their features. The related catastrophe locus is disclosed in a large variety of shapes, very different from those generated by the classical counterpart. Substantial changes in the catastrophe locus properties, such as convexity and number of bifurcation points, are achievable by tuning the design parameters of the proposed machines towards the design of very efficient snapping devices. Experiments performed on the physical realization of the elastica catastrophe machine fully validate the present theoretical approach. The developed model can find applications in mechanics at different scales, for instance, in the design of new devices involving actuation or hysteresis loop mechanisms to achieve energy harvesting, locomotion, and wave mitigation.

Two types of non-holonomic constraints (imposing a prescription on velocity) are analyzed, connected to an end of a (visco)elastic rod, straight in its undeformed configuration. The equations governing the nonlinear dynamics are obtained and then linearized near the trivial equilibrium configuration. The two constraints are shown to lead to the same equations governing the linearized dynamics of the Beck (or Pflüger) column in one case and of the Reut column in the other. Therefore, although the structural systems are fully conservative (when viscosity is set to zero), they exhibit flutter and divergence instability and the Ziegler destabilization paradox. It follows that these features are proven to be not only a consequence of ‘unrealistic non-conservative loads’ (as often stated in the literature), rather, the models proposed by Beck, Reut, and Ziegler can exactly describe the linearized dynamics of structures subject to non-holonomic constraints, which are made now fully accessible to experiments.