The general equations of analytical dynamics

It is shown that the generalized Poincaré and Chetayev equations, which represent the equations of motion of mechanical systems using a certain closed system of infinitesimal linear operators, are related to the fundamental equations of analytical dynamics. Equations are derived in quasi-coordinates for the case of redundant variables; it is shown that when an energy integral exists the operator X₀ = ∂/∂t satisfies the Chetayev cyclic-displacement conditions. Using the energy integral the order of the system of equations of motion is reduced, and generalized Jacobi-Whittaker equations are derived from the Chetayev equations. It is shown that the Poincaré-Chetayev equations are equivalent to a number of equations of motion of non-holonomic systems, in particular, the Maggi, Volterra, Kane, and so on, equations. On the basis of these, and also of other previously obtained results, the Poincaré and Chetayev equations in redundant variables, applicable both to holonomic and non-holonomic systems, can be regarded as general equations of classical dynamics, equivalent to the well-known fundamental forms of the equations of motion, a number of which follow as special cases from the Poincaré and Chetayev equations.     

Asymptotic analysis of compound liquid jets at low Reynolds numbers

Asymptotic methods based on the slenderness ratio are used to obtain the leading-order equations which govern the fluid dynamics of both hollow and solid, compound jets such as those employed in the manufacture of textile fibers, composite fibers and optical fibers. These fibers consist of an inner material which may be a round jet or an annular one and which, in turn, is surrounded by an annular jet in contact with ambient air. It is shown that the leading-order equations are one-dimensional and that it is possible to obtain analytical solutions for several flow regimes for steady, compound jets. These analytical solutions are presented here. The one-dimensional leading-order equations for the fluid dynamics of annular liquid jets at low Reynolds numbers are also derived here.     

A variational method of deriving the equations of the non-linear mechanics of liquid crystals

The non-linear equations of the dynamics of liquid crystals [1], derived previously by the Poisson brackets method, are derived from the Hamilton-Ostrogradskii variational principle. The variational problem of an unconditional extremum of the action functional in Lagrange variables is investigated. The difference between the volume densities of the kinetic and free energy of the liquid crystal is used as the Lagrangian. It is shown that the variational equations obtained are equivalent to the differential laws of conservation of momentum and the kinetic moment of the liquid crystal in Euler variables, while the Ericksen stress tensor and the molecular field are defined in terms of the derivatives of the free energy.     

Asymptotics for thin elastic fibers in unilateral contact

What is the contact condition in a 1D beam-model and is it possible to obtain the frictional moments and forces from the 3D traction? If it is possible does the cross-section of the beams influence these values? These questions motivate to study the dimension reduction of a 3D contact problem for beams. This paper is a continuation of [1]. In [1] the asymptotic dimension reduction of a Robin-type elasticity boundary value problem was presented. In this work the explicit relation between a 3D contact problem and a 3D Robin-type elasticity boundary value problem are established and the 1D equations derived in [1] are interpreted as 1D contact conditions, further some numerical examples are shown. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model Equations for Gravity-Capillary Waves in Deep Water

The Euler equations for water waves in any depth have been shown to have solitary wave solutions when the effect of surface tension is included. This paper proposes three quadratic model equations for these types of waves in infinite depth with a two-dimensional fluid domain. One model is derived directly from the Euler equations. Two further simpler models are proposed, both having the full gravity-capillary dispersion relation, but preserving exactly either a quadratic energy or a momentum. Solitary wavepacket waves are calculated for each model. Each model supports the elevation and depression waves known to exist in the Euler equations. The stability of these waves is discussed, as is the dynamics resulting from instabilities and solitary wave collisions.     

Stable Boundary Conditions and Discretization for $P_N$ Equations

A solution to the linear Boltzmann equation satisfies an energy bound, which reflects a natural fact: The energy of particles in a finite volume is bounded in time by the energy of particles initially occupying the volume augmented by the energy transported into the volume by particles entering the volume over time. In this paper, we present boundary conditions (BCs) for the spherical harmonic $(P_N)$ approximation, which ensure that this fundamental energy bound is satisfied by the $P_N$ approximation. Our BCs are compatible with the characteristic waves of $P_N$ equations and determine the incoming waves uniquely. Both, energy bound and compatibility, are shown on abstract formulations of $P_N$ equations and BCs to isolate the necessary structures and properties. The BCs are derived from a Marshak type formulation of BC and base on a non-classical even/odd-classification of spherical harmonic functions and a stabilization step, which is similar to the truncation of the series expansion in the $P_N$ method. We show that summation by parts (SBP) finite difference on staggered grids in space and the method of simultaneous approximation terms (SAT) allows to maintain the energy bound also on the semi-discrete level.     

A demonstration of consistency of an entropy balance with balance of energy

It is shown in this note that a general balance of entropy postulated previously with only a limited motivation (based on the form of the energy equation for an inviscid fluid) is consistent with, and can be derived from, a general balance of energy. In this derivation, an early form of entropy balance does not make use of invariance conditions under superposed rigid body motions. However, with the help of the latter invariance conditions, additional results are also derived which provide some insight on the structure of the basic equations in thermomechanics.     

Contact of ropes and orthotropic rough surfaces

Contact between arbitrary curved ropes and arbitrary curved rough orthotropic surfaces has been revised from the geometrical point of view. Variational equations for the equilibrium of ropes on orthotropic rough surfaces are derived, first, using the consistent variational inclusion of frictional contact constraints via Karush-Kuhn-Tucker conditions expressed in Darboux basis. Then, the systems of differential equations are derived for both statics and dynamics of ropes on a rough surface depending on the sticking-sliding condition for orthotropic Coulomb's friction. Three criteria are found to be fulfilled during the static equilibrium of a rope on a rough surface: “no separation”, condition for dragging coefficient of friction and inequality for tangential forces at the end of the rope. The limit tangential loads still preserve the famous “Euler view” T = T₀e^ωs for the curves and surfaces of constant curvature. It is shown that the curve of the maximum tension of a rough orthotropic surface is geodesic. Equations of motion are derived in the case if the sliding criteria is fulfilled and there is “no separation”. Various cases possessing analytical solutions of the derived system, including Euler case and a spiral rope on a cylinder are shown as examples of application of the derived theory. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gradient stability of numerical algorithms in local nonequilibrium problems of critical dynamics

The critical dynamics of a spatially inhomogeneous system are analyzed with allowance for local nonequilibrium, which leads to a singular perturbation in the equations due to the appearance of a second time derivative. An extension is derived for the Eyre theorem, which holds for classical critical dynamics described by first-order equations in time and based on the local equilibrium hypothesis. It is shown that gradient-stable numerical algorithms can also be constructed for second-order equations in time by applying the decomposition of the free energy into expansive and contractive parts, which was suggested by Eyre for classical equations. These gradient-stable algorithms yield a monotonically nondecreasing free energy in simulations with an arbitrary time step. It is shown that the gradient stability conditions for the modified and classical equations of critical dynamics coincide in the case of a certain time approximation of the inertial dynamics relations introduced for describing local nonequilibrium. Model problems illustrating the extended Eyre theorem for critical dynamics problems are considered.     

Analysis of contact-impact dynamics of soft finger tapping system by using hybrid computational model

The aim of this paper is to develop an efficient hybrid computational model to analyze the frictional impact dynamic responses of soft finger during the tapping event. The large deformation field and inertial field of soft structure are discretized by absolute nodal coordinate formulation. Lagrange multiplier method is adopted to account for the constraints between neighbor phalanges. Considering tangential contact compliance, a lumped-parameter model at the local contact zone is presented to calculate contact forces. The governing equations of soft finger tapping system expressed by generalized coordinates are derived. An event driven scheme is given to analyze and evaluate the stick–slip transitions. The governing equation is integrated by generalized-α method. The applications of the hybrid computational model are demonstrated using various soft finger tapping systems acted by different driven moments. The feasibility of the proposed model is validated by comparing with the LS-DYNA solution. The error of the solution calculated by the proposed model for the peak value of contact force is less than 8%. Furthermore, numerical results show that the large structural compliance and driven moments have significant effect on the frictional impact responses. The normal relative motion between the fingertip and the rough target surface will experience 1∼4 compression-restitution transitions when Young's Modulus is from 0.01 GPa to 1 GPa or the slenderness ratio of phalanx is from 10.7 to 32. When the posture of soft finger is convex at impact instance, the tangential relative motion will experience 3 slip–stick transitions during the contact process. In addition, it also can be found that the tangential contact compliance can reverse the direction of slip (i.e. ‘reverse slip’ phenomenon).     

New origins of hidden symmetries for partial differential equations

Hidden symmetries of differential equations are point symmetries that arise unexpectedly in the increase (equivalently decrease) of order, in the case of ordinary differential equations, and variables, in the case of partial differential equations. The origins of Type II hidden symmetries (obtained via reduction) for ordinary differential equations are understood to be either contact or nonlocal symmetries of the original equation while the origin for Type I hidden symmetries (obtained via increase of order) is understood to be nonlocal symmetries of the original equation. Thus far, it has been shown that the origin of hidden symmetries for partial differential equations is point symmetries of another partial differential equation of the same order as the original equation. Here we show that hidden symmetries can arise from contact and nonlocal/potential symmetries of the original equation, similar to the situation for ordinary differential equations.     

Convergence and Continuous Dependence for the Brinkman–Forchheimer Equations

The Brinkman–Forchheimer equations for non-slow flow in a saturated porous medium are analyzed. It is shown that the solution depends continuously on changes in the Forchheimer coefficient, and convergence of the solution of the Brinkman–Forchheimer equations to that of the Brinkman equations is deduced, as the Forchheimer coefficient tends to zero. The next result establishes continuous dependence on changes in the Brinkman coefficient. Following this, a result is proved establishing convergence of a solution of the Brinkman–Forchheimer equations to a solution of the Darcy–Forchheimer equations, as the Brinkman coefficient (effective viscosity) tends to zero. Finally, upper and lower bounds are derived for the energy decay rate which establish that the energy decays exponentially, but not faster than this.     

On the equations of motion of the folded inextensible string

In a number of excellent, even recently published, books in Theoretical Mechanics the equations of motion of a folded inextensible string, for example to explain the crack of a whip, are derived using either the energy principle, if a system of one degree of freedom is considered, or Lagrange's equations for conservative systems, if a system of more than one degree of freedom is studied. However, it will be shown in this paper that the resulting equations are incorrect because physically a nonconservative system is given. Hence, both methods mentioned above based on energy conservation must not be used. A correct derivation of the equations of motion is given by means of the linear momentum balance.     

Geometry of conservation laws for a class of parabolic PDE's, II: Normal forms for equations with conservation laws

We consider conservation laws for second-order parabolic partial differential equations for one function of three independent variables. An explicit normal form is given for such equations having a nontrivial conservation law. It is shown that any such equation whose space of conservation laws has dimension at least four is locally contact equivalent to a quasi-linear equation. Examples are given of nonlinear equations that have an infinite-dimensional space of conservation laws parameterized (in the sense of Cartan-K?hler) by two arbitrary functions of one variable. Furthermore, it is shown that any equation whose space of conservation laws is larger than this is locally contact equivalent to a linear equation.     

An adaptive NS/ES-FEM approach for 2D contact problems using triangular elements

An adaptive contact analysis approach is presented for 2D solid mechanics problems using only triangular elements and the subdomain parametric variational principle (SPVP). The present approach is implemented for the node-based smoothed FEM (or NS-FEM), the edge-based smoothed FEM (ES-FEM) and the standard FEM models with automatically adaptive refinement scheme. A modified Coulomb frictional contact model and its corresponding discrete equations are introduced. The global discretized system equations are then formulated in an incremental form with the aid of the basic boundary value equations for friction contact and the subdomain parametric variational principle. A simple adaptive refining scheme is presented, and the Voronoi vertices are taken as candidate points to become new nodes because of duality property between the Voronoi diagrams and Delaunay triangulation. The present adaptive approach can properly simulate variable behaviors of a contact interface such as bonding/debonding, contacting/departing, and sticking/slipping. Several examples are presented to numerically validate the proposed approach via the comparison with reference solutions obtained by ABAQUS^®, and to investigate the effects of the various parameters used in the computations on the response of the contact system. The numerical results have demonstrated that the present adaptive contact analysis approach using the ES-FEM has higher accuracy and convergence rate in the strain energy than that using FEM and NS-FEM. However, the latter two methods can provide the lower and upper bound solution for the system strain energy, respectively.     

On using the more-accurate equations of thin coatings in the theory of axisymmetric contact problems for composite foundations

More-accurate equations describing the axisymmetric deformations of elastic, thin-walled elements (coatings) are derived using the asymptotic analysis of the solution to the first fundamental problem of the theory of elasticity for a layer. The notable difference distinguishing these relations from the classical, Kirchhoff-Love and Reissner-Timoshenko equations of flexure of plates, and their modifications /1/, is, that there are no concentrated forces at the edges of the stamp when the corresponding contact problems are solved. Moreover, the formulas obtained contain the equations of classical theory as a special case. The solutions obtained using various applied theories are compared with the corresponding solution obtained using the equations of the theory of elasticity, using the example of the axisymmetric contact problem of impressing a plane circular stamp into a layer lying on a Fuss-Winkler foundation. The characteristic parameters of the problem in question are computed by numerical methods.     

受冲击性约束作用的系统的运动方程

当具有n个自由度的系统加有P个冲击性的约束时,要求解系统的运动,一般都需要解含n+P个方程的方程组.本文提出以待定乘子法为基础,分别就取广义坐标和伪坐标的二种情况,从n个碰撞方程中消去未知的待定乘子,将碰撞方程简化为n-P个,它和P个冲击性约束方程一起组成了含n个方程的方程组,就能求解具有冲击性约束的碰撞问题,这比一般方法更为简便.     

Global stability of viscous contact wave for 1-D compressible Navier–Stokes equations

The viscous contact waves for one-dimensional compressible Navier–Stokes equations has recently been shown to be asymptotically stable. The stability results are called local stability or global stability depending on whether the norms of initial perturbations are small or not. Up to now, local stability results toward viscous contact waves of compressible Navier–Stokes equations have been well established (see Huang et al., 2006, 2008, 2009 [9], [10], [7]), but there are few results for the global stability in the case of Cauchy problem which is the purpose of this paper. The proof is based on an elementary energy method using an inequality concerning the heat kernel (see Lemma 1 of Huang et al., 2010 [7]).     

Resonant Transfer of Energy between Nonlinear Waves in Neighboring Pycnoclines

The interaction of weakly nonlinear, long internal gravity waves in neighboring pycnoclines is studied. Two coupled equations which describe the evolution of the wave amplitudes are derived. These equations are shown to possess three conserved quantities. The numerical results demonstrate the existence of two types of periodic nonlinear wave solutions when mode-two waves propagate along each pycnocline with nearly equal speeds. The energy transfer between these resonant waves is discussed for two pycnocline separations.     

Stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations

This paper is devoted to the investigation of stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations. For a Leray weak solution of the Navier–Stokes equations in a critical Besov space, it is shown that the Leray weak solution is uniformly stable with respect to a small perturbation of initial velocity and external forcing. If the perturbation is not small, the perturbed weak solution converges asymptotically to the original weak solution as the time tends to the infinity. Additionally, an energy equality and weak–strong uniqueness for the three-dimensional Navier–Stokes equations are derived. The findings are mainly based on the estimations of the nonlinear term of the Navier–Stokes equations in a Besov space framework, the use of special test functions and the energy estimate method.