Topological mechanical metamaterials are artificial structures whose unusual properties are protected very much like their electronic and optical counterparts. Here, we present an experimental and theoretical study of an active metamaterial-composed of coupled gyroscopes on a lattice-that breaks time-reversal symmetry. The vibrational spectrum displays a sonic gap populated by topologically protected edge modes that propagate in only one direction and are unaffected by disorder. We present a mathematical model that explains how the edge mode chirality can be switched via controlled distortions of the underlying lattice. This effect allows the direction of the edge current to be determined on demand. We demonstrate this functionality in experiment and envision applications of these edge modes to the design of oneway acoustic waveguides.topological mechanics | gyroscopic metamaterial | metamaterial A vast range of mechanical structures, including bridges, covalent glasses, and conventional metamaterials, can be ultimately modeled as networks of masses connected by springs (1-6). Recent studies have revealed that despite its apparent simplicity, this minimal setup is sufficient to construct topologically protected mechanical states (7-11) that mimic the properties of their quantum analogs (12). This follows from the fact that, irrespective of its classic or quantum nature, a periodic material with a gapped spectrum of excitations can display topological behavior as a result of the nontrivial topology of its band structure (13-21).All such mechanical systems, however, are invariant under time reversal because their dynamics are governed by Newton's second law, which, unlike the Schrödinger equation, is second order in time. If time-reversal symmetry is broken, as in recently suggested acoustic structures containing circulating fluids (16), theoretical work (13) has suggested that phononic chiral topological edge states that act as unidirectional waveguides resistant to scattering off impurities could be supported. In this paper, we show that by creating a coupled system of gyroscopes, a "gyroscopic metamaterial," we can produce an effective material with intrinsic time-reversal symmetry breaking. As a result, our gyroscopic metamaterials support topological mechanical modes analogous to quantum Hall systems, which have robust chiral edge states (22)(23)(24). We demonstrate these effects by building a real system of gyroscopes coupled in a honeycomb lattice. Our experiments show long-lived, unidirectional transport along the edge, even in the presence of significant defects. Moreover, our theoretical analysis indicates that direction of edge propagation is controlled both by the gyroscope spin and the geometry of the underlying lattice. As a result, deforming the lattice of gyroscopes allows one to control the edge mode direction, offering unique opportunities for engineering novel materials.Much of the counterintuitive behavior of rapidly spinning objects originates from their large angular momentum, which endows th...

SynopsisAnalytic solutions of the functional equation f[z, φ{g(z)}] = φ(z), in which f(z, w) and g(z) are given analytic functions and φ(z) is the unknown function, are investigated in the neighbourhood of points ζ such that g(ζ) = ζ. Conditions are established under which each solution φ(z) may be given as the limit of a sequence of functions φn(z), defined by the recurrence relation φn+1(Z) = ƒ[z, φn{g(z)}], the function φn(z) being to a large extent arbitrary.

One of the remarkable features of classical complex variable theory is the indispensable part played by integration. As L. V. Ahlfors has emphasised in his Complex Analysis [1], many fundamental properties of analytic functions, such as the existence of higher derivatives, have never been proved without resort to complex integrals or equivalent toolsf. The elementary proof by Eggleston and Ursell [2] that an analytic function is light and open makes certain important theorems, including those of Rouche and Hurwitz, independent of integration [3], and has led more recently to a topological proof of the continuity of the derivative of an analytic function [4]. Other fundamental properties of analytic functions have remained inaccessible by methods of this kind. The present paper gives a development of complex variable theory which leads in particular to the existence of the higher derivatives and to the Taylor exj)ansion without any appeal to complex integration. It will be evident that other standard results can be derived directly from the theorems here proved.Our starting point is a form of the maximum principle used by Whyburn ([1], p. 74) in his presentation of the above-mentioned theorem of Eggleston and Ursell. An adequate form of this result is the following, and in this limited form it can also be proved, with less topology, by keeping closer to the treatment in [2]. LEMMA A. Iff(z) is continuous inside and on a circle C.ivith interior R, if it is differentiate in R except at a finite number of points, and if \f(z)\ ^ M on C, then \f(z)\ < M in R.By applying this result to the function g{z)/z we immediately obtain Schwarz's Lemma:If y( z ) * 5 differentiate in | z | < r with g(0) = 0 and \g(z)\^.M, then | g{z)/z\ < M/r when 0 < | z\ < r. This is sufficient to validate the fundamental theorem on normal families of analytic functions:A family of functions differentiate and locally uniformly bounded in a region R is normal in R.To prove this it will be enough to establish that the family is equicontinuous on every compact subset of R, for once this has been achieved Received 16 July, 1960. f While this paper was in the press non-integrational proofs of some of these properties were announced by P. Porcelli and E.