Conformal invariance for nonholonomic system of Chetaev’s type with variable mass

Conformal invariance and conserved quantities for a nonholonomic system of Chetaev’s type with variable mass are studied. The conformal factor expressions are derived. The necessary and sufficient conditions are obtained to make the system’s conformal invariance Lie symmetrical. The conformal invariance of the weak and strong Lie symmetries for the system is given. The corresponding conserved quantities of the system are derived. Finally, an application of the result is shown with an example.     

Conformal invariance and conserved quantity of Mei symmetry for Appell equations in a nonholonomic system of Chetaev’s type

For a nonholonomic system of Chetaev’s type, the conformal invariance and the conserved quantity of Mei symmetry for Appell equations are investigated. First, under the infinitesimal one-parameter transformations of group and the infinitesimal generator vectors, Mei symmetry and conformal invariance of differential equations of motion for the system are defined, and the determining equation of Mei symmetry and conformal invariance for the system are given. Then, by means of the structure equation to which the gauge function is satisfied, the Mei-conserved quantity corresponding to the system is derived. Finally, an example is given to illustrate the application of the result.     

On the invariance of time-dependent Hamilton＇s functions

The general framework of Poincaré's formalism is used to establish the connection between conservation laws and invariance properties of Hamilton's function under infinitesimal transformations when these laws and the Hamiltonian are time-dependent. An example illustrative of the theory is also considered. The English text was polished by Yunming Chen     

Stability for the Equilibrium State of Chaplygin’s Systems

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Carlo Cercignani’s interests for the foundations of physics

Carlo Cercignani was known all over the world for his works on the Boltzmann equation and on kinetic theory. There was however another aspect of his scientific life, which is not much known. Namely, his interest for the foundations of physics, in particular for the possibility of understanding quantum mechanics through classical mechanics, which he shared with several people in Milan. A review of such researches is given here, together with some personal recollections of him.     

Prandtl’s formulation for the Saint–Venant’s torsion of homogeneous piezoelectric beams

The Saint–Venant torsional problem for homogeneous, monoclinic piezoelectric beams is formulated in terms of Prandtl’s stress function and electric displacement potential function. The analytical approach presented in this paper generalizes the known formulation of Prandtl’s solution which refers to homogeneous elastic beams. The Prandtl’s stress function and electric displacement potential function satisfy the so called coupled Dirichlet problem (CDP) in the cross-sectional domain. A direct and a variational formulation are developed. Exact analytical solutions for solid elliptical cross-section and hollow circular cross-section and an approximate solution based on a variational formulation for thin-walled closed cross-section are presented.     

Mei’s symmetry theorem for time scales nonshifted mechanical systems

We focus on Mei symmetry for time scales nonshifted mechanical systems within Lagrangian framework and its resulting new conserved quantities. Firstly, the dynamic equations of time scales nonshifted holonomic systems and time scales nonshifted nonholonomic systems are derived from the generalized Hamilton’s principle. Secondly, the definitions of Mei symmetry on time scales are given and its criterions are deduced. Finally, Mei’s symmetry theorems for time scales nonshifted holonomic conservative systems, time scales nonshifted holonomic nonconservative systems and time scales nonshifted nonholonomic systems are established and proved, and new conserved quantities of above systems are obtained. Results are illustrated with two examples.     

Derivatives of the three-dimensional Green’s functions for anisotropic materials

A concise formulation is presented for the derivatives of Green’s functions of three-dimensional generally anisotropic elastic materials. Direct calculation for derivatives of the Green’s function on the Cartesian coordinate system is a common practice, which, however, usually leads to a complicated course. In this paper the Green’s function derived by Ting and Lee [Ting, T.C.T., Lee, V.G., 1997. The three-dimensional elastostatic Green’s function for general anisotropic linear elastic solids. The Quarterly Journal of Mechanics and Applied Mathematics 50 (3) 407–426] is extended to obtain the derivatives. Using a spherical coordinate system, the Green’s function can be shown as the composition of two independent functions, one depends only on the radial distance of the field point to the origin and the other is in spherical angles. The method of derivation is based on the total differential scheme and then takes its partial differentiation accordingly. With the application of Cauchy residue theorem, the contour integral can be evaluated in terms of the Stroh eigenvalues of a sextic equation. For the degenerate case, evaluation of residues at multiple poles is also given. Applications of the present result are made to examine the Green’s functions and stress components for isotropic and transversely isotropic materials. The results are in exact agreement with existing solutions.     

Mathematical modeling of Maklakoff’s method for measuring the intraocular pressure

The physical content of Maklakoffs tonometric (based on the loading of the cornea) method of measuring the intraocular pressure, widely used in medical practice, is discussed. For this purpose, we employ both the results of physical modeling of the eye described in the literature and the results of our own mathematical modeling based on the representation of the eyeball as a thin shell. The effect of the physical properties of the shell on the results of the modeling is investigated. Qualitative conclusions that follow from our study and may be of practical interest in measuring the intraocular pressure are discussed.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 24–39. Original Russian Text Copyright © 2005 by Bauer, Lyubimov, and Tovstik.     

Change of Sign of the Corrector’s Determinant for Homogenization in Three-Dimensional Conductivity

In this paper we study the positivity of the determinant of the local electric field in a conducting composite. We know by [1] that the positivity holds true in two dimensions for any periodic structure. Using a different approach from [11] we prove that is also the case for a laminate microstructure in any dimension. However, and this is the main result of the paper, we provide an example of a two-phase three-dimensional periodic composite for which the determinant changes sign.     

A feedback linearization design for the control of vehicle’s lateral dynamics

In this paper, the feedback linearization scheme is applied to the control of vehicle’s lateral dynamics. Based on the assumption of constant driving speed, a second-order nonlinear lateral dynamical model is adopted for controller design. It was observed in (Liaw, D.C., Chung, W.-C. in 2006 IEEE International Conference on Systems, Man, and Cybernetics, 2006) that the saddle-node bifurcation would appear in vehicle dynamics with respect to the variation of the front wheel steering angle, which might result in spin and/or system instability. The vehicle dynamics at the saddle node bifurcation point is derived and then decomposed as an affine nominal model plus the remaining term of the overall system dynamics. Feedback linearization scheme is employed to construct the stabilizing control laws for the nominal model. The stability of the overall vehicle dynamics at the saddle-node bifurcation is then guaranteed by applying Lyapunov stability criteria. Since the remaining term of the vehicle dynamics contains the steering control input, which might change system equilibrium except the designed one. Parametric analysis of system equilibrium for an example vehicle model is also obtained to classify the regime of control gains for potential behavior of vehicle’s dynamical behavior.     

Superposing scheme for the three-dimensional Green’s functions of an anisotropic half-space

This paper presents a method of superposition for the half-space Green’s functions of a generally anisotropic material subjected to an interior point loading. The mathematical concept is based on the addition of a complementary term to the Green’s function in an anisotropic infinite domain. With the two-dimensional Fourier transformation, the complementary term is derived by solving the generalized Stroh eigenrelation and satisfying the boundary conditions on the free surface with the use of Green’s functions in the full-space case. The inverse Fourier transform leads to the contour integrals, which can be evaluated with the application of Cauchy residue theorem. Application of the present results is made to obtain analytical expression for the orthotropic materials which were not reported previously. The closed-form solutions for the transversely isotropic and isotropic materials derived directly from the solutions as being a special case are also given in this paper.     

On the Domain of Validity of Brinkman’s Equation

An increasing number of articles are adopting Brinkman’s equation in place of Darcy’s law for describing flow in porous media. That poses the question of the respective domains of validity of both laws, as well as the question of the value of the effective viscosity μ ^e which is present in Brinkman’s equation. These two topics are addressed in this article, mainly by a priori estimates and by recalling existing analyses. Three main classes of porous media can be distinguished: “classical” porous media with a connected solid structure where the pore surface S _p is a function of the characteristic pore size l _p (such as for cylindrical pores), swarms of low concentration fixed particles where the pore surface is a function of the characteristic particle size l _s , and fiber-made porous media at low solid concentration where the pore surface is a function of the fiber diameter. If Brinkman’s 3D flow equation is valid to describe the flow of a Newtonian fluid through a swarm of fixed particles or fibrous media at low concentration under very precise conditions (Lévy 1983), then we show that it cannot apply to the flow of such a fluid through classical porous media.     

Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations

For every ${\varepsilon > 0}$ , we consider the Green’s matrix ${G_{\varepsilon}(x, y)}$ of the Stokes equations describing the motion of incompressible fluids in a bounded domain ${\Omega_{\varepsilon} \subset \mathbb{R}^d}$ , which is a family of perturbation of domains from ${\Omega\equiv \Omega_0}$ with the smooth boundary ${\partial\Omega}$ . Assuming the volume preserving property, that is, ${\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}$ for all ${\varepsilon > 0}$ , we give an explicit representation formula for ${\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}$ in terms of the boundary integral on ${\partial \Omega}$ of ${G_0(x, y)}$ . Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.     

On application of Newton’s law to mechanical systems with motion constraints

This work is devoted to deriving and investigating conditions for the correct application of Newton’s law to mechanical systems subjected to motion constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and nonholonomic constraints. This approach is convenient since it permits one to view the motion of any dynamical system as a path of a point on a manifold. In particular, the main focus is on the establishment of appropriate conditions, so that the form of Newton’s law of motion remains invariant when imposing an additional set of motion constraints on a mechanical system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold, which results after enforcing the additional constraints. The latter is weaker than a similar condition obtained by imposing a metric compatibility condition holding on Riemannian manifolds and employed frequently in the literature. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton’s law on the tangent space and not on the dual space of a manifold, which is the natural geometrical space for this. Finally, the Euler–Lagrange operator is examined and issues related to equations of motion for anholonomic and vakonomic systems are investigated and clarified further.     

Hyperbolicity of the Nonlinear Models of Maxwell’s Equations

We consider the class of nonlinear models of electromagnetism that has been described by Coleman & Dill [7]. A model is completely determined by its energy density W(B,D). Viewing the electromagnetic field (B,D) as a 3×2 matrix, we show that polyconvexity of W implies the local well-posedness of the Cauchy problem within smooth functions of class H^s with s>1+d/2. The method follows that designed by Dafermos in his book [9] in the context of nonlinear elasticity. We use the fact that B×D is a (vectorial, non-convex) entropy, and we enlarge the system from 6 to 9 equations. The resulting system admits an entropy (actually the energy) that is convex. Since the energy conservation law does not derive from the system of conservation laws itself (Faradays and Ampères laws), but also needs the compatibility relations divB=divD=0 (the latter may be relaxed in order to take into account electric charges), the energy density is not an entropy in the classical sense. Thus the system cannot be symmetrized, strictly speaking. However, we show that the structure is close enough to symmetrizability, so that the standard estimates still hold true.     

FE analysis of the in-plane mechanical properties of a novel Voronoi-type lattice with positive and negative Poisson’s ratio configurations

This work presents a novel formulation for a Voronoi-type cellular material with in-plane anisotropic behaviour, showing global positive and negative Poisson’s ratio effects under uniaxial tensile loading. The effects of the cell geometry and relative density over the global stiffness, equivalent in-plane Poisson’s ratios and shear modulus of the Voronoi-type structure are evaluated with a parametric analysis. Empirical formulas are identified to reproduce the mechanical trends of the equivalent homogeneous orthotropic material representing the Voronoi-type structure and its geometry parameters.     

Green’s functions for anti-plane deformations of a circular arc-crack at the interface of piezoelectric materials

Summary The anti-plane deformation problem of an interfacial debounding crack between a circular piezoelectric inclusion and a piezoelectric matrix is investigated by means of the complex variables method. For a line load applied within the matrix or inside the inclusion, Greens functions are presented for the complex potentials, intensity factors and electric fields on the crack faces, respectively, in closed and explicit form. The solutions are valid for both permeable and impermeable crack models. It is shown that, in the general case of permeable cracks, the electric field singularity is always proportional to the stress singularity.The first author (C.F.Gao) would like to express his gratitude for the support of the Alexander von Humboldt Foundation (Germany).     

A necessary and sufficient condition for the oscillation of solutions of Liénard type system with multiple singular points

In this paper, a necessary and sufficient condition for the solution of Liénard type system with multiple singular points to oscillation under the more general assumption is given. Results of the papers [14] are also extended and improved in this paper.Project supported by Science Foundation of Education Bureau of the Metallurgy Ministry, P. R. China