Design projects

M 5740 Math Modeling Fall 2017

November 29, 2017

1 Transformer of vibrations

A simple model of a device (a metamaterial) that transforms the external periodic force F_{e} = A cos(ωt) into electricity is shown in the Figure. It consists of the external massless and rigid plates (two rods), the inclined massless rigid joints that transform vertical vibrations into a horizontal motion x(t) of the central rigid rod with the mass m, a spring with constant C (F_{spring} = −C x), and the dissipative element (or an electrical generator) with resistance force F is F = −γẋ(t).

Model the motion of the rod. Derive the differential equation for x and solve it. Find optimal value of m and angle α that maximize the average dissipated energy

D = \frac{ω}{2π} ∫0^{\frac{2π}{ω}} ẋ^{2}(t) dt

2 Metamaterials: Negative Poisson’s ratio materials

Investigate the mode of deformation of the two structures and compute their Poisson ratio ν in dependence of the angle θ. http://silver.neep.wisc.edu/ lakes/Poisson.html http://en.wikipedia.org/wiki/Auxetics

Find other models of Auxetics and discuss their common properties. Suggest a 3d model of auxetic material. What are natural boundaries

for the Poisson ratio?

A natural continuation of the auxetics: Theory GW.Milton and A.Cherkaev. Which Elasticity Tensors are Realizable? J. Eng. Mater. Technol 117(4), 483-493 (Oct 01, 1995) Realization M.Kadic, T.Bckmann, N.Stenger, M.Thiel, M.Wegener. On the feasibility of pentamode mechanical metamaterials. https://arxiv.org/abs/1203.1481