Pattern evocation and geometric phases in mechanical systems with symmetry

This paper is concerned with the relation between the dynamics of a given Hamiltonian system with a given symmetry group and its reduced dynamics. We illustrate the process of visualization of reduced orbits using the double spherical pendulum. In this process of visualization, one sees certain patterns when the dynamics is viewed relative to rotating frames with certain critical angular velocities. By using the reduced dynamics, we also explain these patterns. We show that if the motion on the phase space reduced by a continuous symmetry group at a given momentum level is periodic, then there is a uniformly rotating frame, that is, a one-parameter group motion, relative to which the unreduced trajectory is periodic with the same period. If the continuous symmetry group of the system is Abelian, which corresponds to the system having cyclic variables, we derive an explicit expression for the required angular velocity in terms of the dynamic phase (an average of the mechanical connection) and the geometric phase (the holonomy of the mechanical connection). We show that one can also find such a frame if the reduced orbit is quasi-periodic and a KAM (Kolmogorov-Arnold-Moser) condition is satisfied The almost periodic case is also discussed. An important aspect of this procedure is how to use it in the presence of discrete symmetries. We show that, under appropriate conditions, the visualized orbit has, relative to a suitable uniformly rotating frame, the same temporal behavior and discrete symmetries as the reduced orbit. Since these spatio-temporal patterns are not apparent with repect to most frames, we call the phenomenon pattern evocation     

Bifurcations in dynamical systems with interior symmetry

We propose a definition of interior symmetry in the context of general dynamical systems. This concept appeared originally in the theory of coupled cell networks, as a generalization of the idea of symmetry of a network. The notion of interior symmetry introduced here can be seen as a special form of forced symmetry breaking of an equivariant system of differential equations. Indeed, we show that a dynamical system with interior symmetry can be written as the sum of an equivariant system and a ‘perturbation term’ which completely breaks the symmetry. Nonetheless, the resulting dynamical system still retains an important feature common to systems with symmetry, namely, the existence of flow-invariant subspaces. We define interior symmetry breaking bifurcations in analogy with the definition of symmetry breaking bifurcation from equivariant bifurcation theory and study the codimension one steady-state and Hopf bifurcations. Our main result is the full analogues of the well-known Equivariant Branching Lemma and the Equivariant Hopf Theorem from the bifurcation theory of equivariant dynamical systems in the context of interior symmetry breaking bifurcations.     

Geometric phases and criticality in spin systems

A general formalism of the relation between geometric phases produced by circularly evolving interacting spin systems and their criticality behaviour is presented. This opens up the way for the use of geometric phases as a tool to probe regions of criticality without having to undergo a quantum phase transition. As a concrete example, a spin-1/2 chain with XY interactions is considered and the corresponding geometric phases are analysed. Finally, a generalization of these results to the case of an arbitrary spin system is presented.     

Conservation of generalized momentum maps in mechanical optimal control problems with symmetry

In this paper, a structure‐preserving direct method for the optimal control of mechanical systems is developed. The new method accommodates a large class of one‐step integrators for the underlying state equations. The state equations under consideration govern the motion of affine Hamiltonian control systems. If the optimal control problem has symmetry, associated generalized momentum maps are conserved along an optimal path. This is in accordance with an extension of Noether's theorem to the realm of optimal control problems. In the present work, we focus on optimal control problems with rotational symmetries. The newly proposed direct approach is capable of exactly conserving generalized momentum maps associated with rotational symmetries of the optimal control problem. This is true for a variety of one‐step integrators used for the discretization of the state equations. Examples are the one‐step theta method, a partitioned variant of the theta method, and energy‐momentum (EM) consistent integrators. Numerical investigations confirm the theoretical findings. Copyright © 2016 John Wiley & Sons, Ltd.     

机械制造工艺对称性的概念体系及其应用思路

为系统地研究对称性在机械系统制造工艺中的存在、功效及应用规律,提出工艺对称性的概念,构建由工艺方法对称性、工艺过程对称性和工艺设备对称性组成的多层次分类体系.进而对工艺对称性中的组合/分解对称性和时空对称性加以细分,并举例说明了它们之间的区别.机械制造工艺对称性不仅能被用于提高机械制造工艺水平,而且能被用于提高机械施工设计水平,说明了这两者之间的联系、区别和互动.举例分析表明,提出的概念体系完整地体现了对称性在机械系统制造工艺过程中的各种存在,为进一步研究工艺对称性设计规律、建立机械工艺对称性设计方法提供了理论基础.     

A possible crystallographic explanation for the five-fold diffraction symmetry in icosahedral phases

X-ray and electron diffraction data from the Al-Cu-Fe icosahedral phase are compared and analysed on the basis of the microcrystalline and multi-domain model developed by the author. It is shown that a crystallographic explanation is now possible for both the enigmatic five-fold symmetry and non-periodicity of reflections observed in electron diffraction patterns of icosahedral phases.     

Machine learning of phases and mechanical properties in complex concentrated alloys

The mechanical properties of complex concentrated alloys (CCAs) depend on their formed phases and corresponding microstructures.The data-driven prediction of the phase formation and associated mechanical properties is essential to discovering novel CCAs.The present work collects 557 samples of various chemical compositions,comprising 61 amorphous,167 single-phase crystalline,and 329 multi-phases crystalline CCAs.Three classification models are developed with high accuracies to category and understand the formed phases of CCAs.Also,two regression models are constructed to predict the hard-ness and ultimate tensile strength of CCAs,and the correlation coefficient of the random forest regression model is greater than 0.9 for both of two targeted properties.Furthermore,the Shapley additive expla-nation (SHAP) values are calculated,and accordingly four most important features are identified.A significant finding in the SHAP values is that there exists a critical value in each of the top four fea-tures,which provides an easy and fast assessment in the design of improved mechanical properties of CCAs.The present work demonstrates the great potential of machine learning in the design of advanced CCAs.     

Vibrational processes in mechanical systems

A method is presented and mathematically rigorously justified for calculating vibrational processes in mechanical systems described by nonlinear differential equations. An example of its implementation is considered and it is shown that this method leads to known self-oscillation conditions.     

Chaotic oscillations in mechanical systems

Chaotic oscillations have now been observed in non-linear mechanical systems by analytical, numerical and experimental methods. Nevertheless a more fundamental understanding of why and when such oscillations occur is of great importance. This goal will be pursued here by considering the relationship between chaos induced by forced oscillations versus self-excited oscillations, the relationship of indeterminancy of the final equilibrium state in the initial value problem to chaos in the sustained oscillation problem, comparison of theory to physical experiment, necessary and sufficient conditions for chaos to occur, and the question of convergence of systems of modal ordinary differential equations which derive from partial differential equations.     

The pancharatnam and the berry phases in three-level photonic systems

Abstract

A theoretical analysis of Pancharatnam and Berry phases is made for biphoton three-level systems, which are produced via frequency degenerate collinear spontaneous parametric down conversion (SPDC). The general theory of Pancharatnam phases is discussed with a special emphasis on geodesic ‘curves' in Hilbert space. Explicit expressions for Pancharatnam, dynamic and geometric phases are derived for the transformations produced by linear phase converters. The problem of gauge invariance is treated in this article.     

Stress analysis for multilayered coating systems using semi-analytical BEM with geometric non-linearities

For a long time, most of the current numerical methods, including the finite element method, have not been efficient to analyze stress fields of very thin structures, such as the problems of thin coatings and their interfacial/internal mechanics. In this paper, the boundary element method for 2-D elastostatic problems is studied for the analysis of multi-coating systems. The nearly singular integrals, which is the primary obstacle associated with the BEM formulations, are dealt with efficiently by using a semi-analytical algorithm. The proposed semi-analytical integral formulas, compared with current analytical methods in the BEM literature, are suitable for high-order geometry elements when nearly singular integrals need to be calculated. Owing to the employment of the curved surface elements, only a small number of elements need to be divided along the boundary, and high accuracy can be achieved without increasing more computational efforts. For the test problems studied, very promising results are obtained when the thickness of coated layers is in the orders of 10⁻⁶–10⁻⁹, which is sufficient for modeling most coated systems in the micro- or nano-scales.     

Stability of some mechanical systems with dry friction

Mechanical systems subjected to dry friction forces can be studied by differential equations with discontinuous vector fields. The stability of equilibria of some such systems is considered     

Evolution of phases during mechanochemical activation in magnetite-containing systems

In a first experiment, we subjected a mixture of Fe₃O₄ and Fe (80–20 wt.%) to mechanochemical activation by high-energy ball milling, for time periods ranging from 0.5 to 14 h. Complementary X-ray diffraction (XRD) and Mössbauer spectroscopy data demonstrated a phase transformation of magnetite to hematite, accompanied by a partial oxidation of iron to hematite. The reaction can be used to obtain nanometer-size magnetite by ball milling, due to the inhibition of its transformation to hematite, caused by the presence of iron atoms. In a second experiment, we ball-milled Fe₃O₄:Co²⁺ (Fe_3−xCo_xO₄ with x=0.1) for time intervals between 2.5 and 17.5 h. Our XRD and Mössbauer measurements showed that the cobalt-doped magnetite undergoes a phase transformation to hematite, which is actually cobalt-doped hematite. We were able to show herewith that the Co ion is not kicked out of the lattice during the milling process, but undergoes the phase transformation inside the hematite lattice. Finally, we exposed a mixture of Fe₃O₄ and Co (80–20 wt.%) to mechanochemical activation for time periods ranging from 0.5 to 10 h. The XRD and Mössbauer results are consistent with the formation of cobalt ferrite (a strongly Co-substituted magnetite), with the occurrence of hematite as an intermediate product. In all three cases, the milling-induced phase transformations started with a considerable disorder of the octahedral sublattice of magnetite.     

Chiral symmetry breaking in two-dimensional C60-ACA intermixed systems

We have demonstrated a method for fabricating C60 overlayers with controlled spacing and chirality by reactive coadsorption with the aromatic molecule acridine-9-carboxylic acid (ACA). Structural control is achieved by the mismatched symmetries of the coadsorbates, as well as specific intermolecular and adsorbate-substrate interactions. The resulting supramolecular structure has a C60 period nearly three times as large as the normal C60 2D packing of 1 nm and exists in enantiopure domains with robust chirality.     

Berry phases, pairing symmetry, and nodelessd-wave states in a strongly correlated polaron model

We use path integral and exact diagonalization techniques to study the tunneling of dopant-induced hole polarons which are self-localized by electron-phonon coupling in a two-dimensional anti-ferromagnet. Their low-energy dynamics is found to be dominated by 2nd and 3rd neighbor hopping processes which carry a (–1) Berry phase factor and lead to non-s-wave hole pairing symmetries, including possiblenodeless pair wavefunctions. The polarons obey effective spin-1/2-fermion quasi-particle statistics. The nodeless -wave pairing state can exhibit a transition from gapless to fully gapped behavior in its quasi-particle excitation spectrum, without change of pairing symmetry.This work was supported by the National Science Foundation under Grant Nos. DMR-8913878 and DMR-9215123. Computing support from UCNS at the University of Georgia and from NCSA at the University of Illinois at Urbana-Champaign is gratefully acknowledged.     

Role of sintering time,crystalline phases and symmetry in the piezoelectric properties of lead-free KNN-modified ceramics

Lead-free KNN-modified piezoceramics of the system (Li,Na,K)(Nb,Ta,Sb)O₃ were prepared by conventional solid-state sintering. The X-ray diffraction patterns revealed a perovskite phase, together with some minor secondary phase, which was assigned to K₃LiNb₆O₁₇, tetragonal tungsten–bronze (TTB). A structural evolution toward a pure tetragonal structure with the increasing sintering time was observed, associated with the decrease of TTB phase. A correlation between higher tetragonality and higher piezoelectric response was clearly evidenced. Contrary to the case of the LiTaO₃ modified KNN, very large abnormal grains with TTB structure were not detected. As a consequence, the simultaneous modification by tantalum and antimony seems to induce during sintering a different behaviour from the one of LiTaO₃ modified KNN.     

Formation of nanocrystalline phases in the Cu-Zn system during mechanical alloying

The evolution of various nanocrystalline phases in the Cu-Zn system during the course of mechanical alloying of elemental powder blends, manifests a similar sequence of phase formation in all the compositions. Zinc-rich phases were always first to form, which can be attributed to the diversities of diffusivities and diffusion distances in the constituents. The extent of zinc in -phase becomes significant only after turns nanocrystalline. The crystallite size of reached a minimum (18 nm) near the - phase boundary, while the and phases showed quite coarse crystallite size due to their low melting points. Alloying was sluggish during dry milling compared to wet milling, or at lower milling speed, which demonstrates the effects of oxidation during milling and lower milling energy.     

Response and eigenvalue analysis of stochastic finite element systems with multiple correlated material and geometric properties

The variability of the random response displacements and eigenvalues of structures with multiple uncertain material and geometric properties are studied in this paper using variability response functions. The material and geometric properties are assumed to be described by cross-correlated stochastic fields. Specifically, two types of problems are considered: the response displacement variability of plane stress/plane strain structures with stochastic elastic modulus, Poisson's ratio, and thickness, and the eigenvalue variability of beam and plate structures with stochastic elastic modulus and mass density. The variance of the displacement/eigenvalue is expressed as the sum of integrals that involve the auto-spectral density functions characterizing the structural properties, the cross-spectral density functions between the structural properties, and the deterministic variability response functions. This formulation yields separate terms for the contributions to the response displacement/eigenvalue variability from the auto-correlation of each of the material/geometric properties, and from the cross-correlation between these properties. The variability response functions are used to compute engineering-wise very important spectral-distribution-free realizable upper bounds of the displacement/eigenvalue variability. Using this formulation, it is also possible to compute the displacement/eigenvalue variability for prescribed auto- and cross-spectral density functions.     

Sensitivity analysis for dynamic mechanical systems with finite rotations

This paper presents a sensitivity analysis for dynamic systems with large rotations using a semi‐analytical direct differentiation technique. The choice of a suitable time integration strategy for the rotation group appears to be critical for the development of an efficient sensitivity analysis. Three versions of the generalized‐α time integration scheme are considered: a residual form, a linear form, and a geometric form. Their consistency is discussed, and we show that the residual form, which is the most direct extension of the generalized‐α algorithm defined in structural dynamics, should not be used for problems with large rotations. The sensitivity analysis is performed and close connections are highlighted between the structure of the sensitivity equations and of the linearized dynamic equations. Hence, algorithms developed for the original problem can be directly reused for the sensitivities. Finally, a numerical example is analysed in detail. Copyright © 2007 John Wiley & Sons, Ltd.