Elastica catastrophe machine: theory, design and experiments

Abstract

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.

Mechanics of High-Flexible Beams Under Live Loads

“…Substituting (3) into (19), calculating the material divergence of T R and supposing that the body forces b are null, the following system of three partial differential equations is obtained: (20) The expressions of the derivatives S X,X , S Y,Y and S Z,Z are given in the Appendix. The first equation of the system (20) governs the equilibrium in the X direction, the second along Y and the third along the Z axis of the undeformed beam.…”

“…Similarly, the geometrical dimensions of the beam have been rendered dimensionless, dividing them by the height H. In the same way, the variables X, Y and Z are normalized. Since these variables and stresses have been normalized, equilibrium equations (20) are also dimensionless, so that their comparison with the scalar zero takes full meaning. Evaluating therefore how much the three equations (20) deviate from zero, a measure of the accuracy of the solution obtained can be performed.…”

“…Given the above, a numerical analysis is performed below, evaluating the equations (20) in all the points of some discrete cross sections of the examples carried out in the previous Sections. The results of computations are delivered in two-dimensional diagrams with the shape similar to the cross section of the beam.…”

“…The results of computations are delivered in two-dimensional diagrams with the shape similar to the cross section of the beam. In these diagrams some contour lines, which join the points where the equations (20) give the same numeral values, are shown. Fig.…”

“…The scientific interest for the Elastica theory has never diminished, even today numerous papers can be found in the Literature (see, for example, [16], [17] and [18]). In addition, the Elastica has been used to simulate a wide range of new practical problems (see, for example, [19], [20], [21], [22], [23] and [24]). The state of the art of the Elastica theory can be found in the fundamental works by Bigoni [25] and O'Reilly [26].…”