Hamilton's Variational Principle Research Articles

EXACT-VOLUME DIFFERENTIAL FORM DE RHAM COHOMOLOGY AND HAMILTON VARIATIONAL PRINCIPLE

In this paper, we suggested exact-volume differential form (for short: EVDF) and proved four theorems correlative with them: 1. existence theorem, 2. cohomology theorem, 3. constant multiple theorem, and 4. equal gauge theorem. And their application were discussed also. For examlpe, we deduced the particle dynamic equation of the special theory of relativity. At the same time we analyzed and contrasted cohomology theory with Hamilton's variational principle. The contrast showed the superiority of cohomology theory. Moreover, we gave a more complete classification list of differential forms.

Approximation of the solutions in boundary-value problems in Lagrangian mechanics

A projection method is described for constructing motions in Lagrangian mechanics. A Galerkin scheme is constructed for the trajectory in configuration space, using Hamilton's variational principle. The class of admissible paths for which the principle is considered consists of motions with fixed initial and terminal positions.

Hydrodynamics of superfluid turbulence in He II: three-dimensional theory

The one-dimensional hydrodynamic theory for superfluid turbulent flow of He II developed recently by one of the authors is extended to the three-dimensional case. The impulse density of the entangled mass of quantised vortices continues to play an important part in the analysis. The non-dissipative equations of motion are derived from an appropriate form of Hamilton's variational principle, while the dissipative effects are taken into account systematically according to the thermodynamics of irreversible processes. A number of helicity conservation theorems appears to be valid in the case of non-dissipative flow. Particular attention is paid to homogeneous superfluid turbulence. It turns out that the two states allowed in one-dimensional homogeneous flow, viz. the state in which the direction of the vortex tangle impulse is parallel and the one in which it is anti-parallel to the counterflow velocity, are, respectively, stable and metastable. The analysis might assist in clarifying the one-dimensional free decay of homogeneous superfluid turbulence.

Stress-based elastodynamic discrete laminated plate theory

A static laminated plate theory based on an assumed piecewise linear through-the-thickness in-plane stress distribution has been extended to include inertia effects. Based on this in-plane stress distribution assumption, outof-plane shear and normal stress component distributions were derived from the three-dimensional equations of motion, resulting in six nonzero stress components. Hamilton's variational principle was used to derive the plate equations of motion, the plate constitutive relationships, and the interface continuity equations. The governing equations were written in a form that is self-adjoint with respect to the convolution bilinear mapping. The resulting system of equations for a single lamina consists of 25 field equations in terms of 9 weighted displacement field variables, 10 stress and moment resultant field variables, and 6 out-of-plane shear and normal stress boundary field variables. For the laminated system, the mixed formulation enforces both traction and displacement continuity at lamina interfaces as it satisfies layer equilibrium. A finite element formulation based on a specialized form of the governing functional was developed. The method is illustrated with results of a free vibration analysis of sandwich and homogeneous plates for which exact solutions are available.

Comprehensive model of anisotropic composite aircraft wings suitable for aeroelastic analyses

A comprehensive plate-beam structural model suitable for aeroelastic analyses of aircraft wings made of anisotropic composite materials is developed. The equations governing the static and dynamic aeroelastic equilibrium of cantilevered swept-wing structures and the associated boundary conditions are derived by means of the Hamilton variational principle. These equations incorporate a number of effects: 1) anisotropy of the materials of constituent layers, 2) warping inhibition, 3) transverse shear flexibility, and 4) rotatory inertias. A uniform swept-wing model composed of a transversely isotropic material is considered to illustrate the coupled and separate effects of transverse shear deformation and warping restraint upon its divergence and static aeroelastic load distribution. An exact method based upon the Laplace integral transform technique is used to solve the above mentioned problems. The results displayed in this article reveal the importance of transverse shear and warping restraint effects in predicting more accurately the static aeroelastic response of swept-forward wings. However, for swept-back wings, these effects represent higher-order corrections to the classical theory.

Formulation of General Higher-Order Theory for Coupled Thermoelasto-Dynamic Plates by Use of Biot's Variational Principle.

Using Biot's variational principle, two governing equations of general higher-order theory are formulated for the coupled thermoelasto-dynamic plates. The derived governing equations are two equations for coupled thermoelastic vibration and for the law of heat conduction. As for the law of heat conduction, a classical and a modified Fourier's law are adopted in Biot's variational principle. The two governing equations for coupled thermoelastic vibrations and the two boundary conditions are the same equations, respectively. However, the two governing equations obtained for the law of heat conduction are different from each other. Lastly, relations between the governing equations of general higher-order theory derived from Biot's variational principle and those derived from Hamilton's variational principle are considered

Forces of reaction and neighbouring Hamilton's principle in the tracking control of manipulators via a sliding scheme

SUMMARYIt is shown that if one comes back to the formulation of the Hamilton's variational principle, it is then possible to obtain new viewpoints on the tracking control of robot manipulators. First, the Lagrange multiplier associated to the sliding surface can be interpretated in terms of control effort and/or forces of reaction of the mechanical system. Secondly, one can use the Taylor expansion of the mechanical Lagrangian, combined with a neighbour- ing Hamilton's principle, to obtain control schemes via sliding surfaces. Thirdly, a perturbation approach combined with the neighbouring Hamilton's principle provides results on the robustness of the control.

Vibration Control of Conical Shells Using Viscoelastic Materials

The vibration and damping analysis of multi-layered conical shells incorporating layers of viscoelastic materials in addition to elastic ones, the former causing dissipation of vibratory energy, is the subject matter of this paper. The analysis given herein uses Hamilton's variational principle for deriving equations of motion of a general multi-layered conical shell. In view of the correspondence principle of linear viscoelasticity which is valid for harmonic vibrations, the solution is obtained by replacing the moduli of viscoelastic layers by complex moduli. An approximate solution for axisymmetric vibrations of multi-layered conical shells with two end conditions—simply supported edges and clamped edges—is obtained by utilizing the Galerkin procedure. The damping effectiveness in terms of the system loss factor for all families of modes of vibrations for three-, five- and seven-layered shells is evaluated and its variation with geometrical parameters is investigated.

Formulation of boundary-value problems for the dynamics of two-dimensional systems with moving loads and fastenings

A load or fastening is moving in a two-dimensional system without separation. The interdependent dynamical behaviour of these two elements is investigated. The Hamilton variational principle is used to formulate a self-consistent boundary-value problem which correctly incorporates the forces of interaction in the moving contact, including those due to the relative motion and wave pressure. The equations of energy and momentum transport are derived. It is shown that the intermediary through which the vibrational energy of the two-dimensional system is converted into the kinetic energy of the one-dimensional object is the wave pressure force. As an example, a boundary-value problem is formulated for the motion of a beam along a Kirchhoff model plate.

Dynamic equations for time-averaged coordinates at thermal equilibrium

The Hamilton variational principle with “coarse” variation (over a limited number of parameters introduced into the coordinates) is used to find relations for the time-averages of defined functions of the coordinates and velocities. The dynamic equations for the parameters characterizing the average behavior of the coordinates are then found by orthogonalization of the coordinates. It is shown that the entropy of the dynamic system is expressed in terms of these averaged parameters, and the dynamic equations and stability conditions may be found by minimization of the defined function of the motion, which is constructed like the thermodynamic potential. A perturbation procedure is suggested for solving the dynamic equations.

Weak Hamiltonian finite element method for optimal control problems

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

Finite element analysis based on stochastic Hamilton variational principle

A stochastic Hamilton variational principle (SHVP) is formulated for dynamic problems of linear continuum. The SHVP allows incorporation of probabilistic distributions into the finite element analysis. The formulation is simplified by transformation of correlated random variables to a set of uncorrelated random variables through a standard eigenproblem. A procedure based on the Fourier analysis and synthesis is presented for eliminating secularities from the perturbation approach. The algorithms developed can readily be adapted to existing deterministic finite element codes.

Momentum of elementary excitations revisited: New variational approach to superfluid hydrodynamics of 4He

The principal objectives of the paper are (i) to clarify the two-fluid picture of superfluid 4He by distinguishing and analysing the London-Tisza and Landau two-fluid models, (ii) to present a new derivation of the two-fluid equations within the framework of the Landau model by starting from Hamilton's variational principle in Lagrange form, and (iii) to critically discuss some generally accepted notions concerning momentum densities in the two-fluid models after characterizing the effective mass of the elementary excitations. It turns out that the effective mass density m ∗ of the elementary excitations is determined by m ∗ = ρρ n/ρ s . The superfluid velocity v s is shown to allow a dynamic, but not a kinematic interpretation in the Landau two-fluid model.

Drift mass, multifluid modelling of two-phase bubbly flow and superfluid hydrodynamics

Hamilton's variational principle of least action for the two-fluid hydrodynamics of bubbly liquid/gas mixtures is extended by adding the virtual mass of the gas bubbles as an independent variable. The additional Euler-Lagrange equation, which determines the virtual-mass coefficient, expresses some local internal equilibrium of the two-phase medium. The extended two-fluid model is transformed analytically into a three-fluid model. The drift-mass coefficient, which is defined in a natural way within the frame of the three-fluid model, turns out to be a simple function of the virtual-mass coefficient. In the limit of small void fractions Darwin's thoerem is recovered. Some striking similarities with the two-fluid hydrodynamics of superfluid 4He are pointed out. The new three-fluid model is compared with the models that were proposed recently by Cook and Harlow and by Wallis.

Nonlinear void-fraction waves in two-phase bubbly flow

It is shown that nonlinear void-fraction waves constitute exact solutions of the two-fluid equations for bubbly flow, if and only if the virtual-mass coefficientm(α) is a definite function of the void fractionα. In the case of spherical bubbles the functional dependence is given bym(α)=(1/2)α(1−a)(1−3α). A similar expression was obtained recently by means of an elaborate linear stability analysis of a uniform two-phase flow. It is remarkable that the greater part of the physical quantities of a nonlinear void-fraction wave depends linearly onα. The virtual-mass coefficient may be derived directly from a special form of Hamilton's variational principle.

Static and dynamic analysis of a viscoelastic plate by the finite element method

In this paper the finite element method (FEM) is employed to analyse the quasi-static and dynamic response of the linear viscoelastic plate. The temperature field is assumed to be constant and homogeneous. The constitutive law is of hereditary integral type with constant Poisson's ratio. The relaxation modulus is assumed to have the form of a Prony series. In order to save computer memory space, the integral equation is transformed to algebraic equations through a finite difference approach. Based on Mindlin plate theory, the finite element model is developed by applying Hamilton's variational principle without any integral transformation. The models of Maxwell type and three-parameter-solid type are analysed and discussed in several examples.

Refined theories of elastic anisotropic shells accounting for small strains and moderate rotations

A geometrically non-linear theory of elastic anisotropic shells based upon discarding Kirchhoff-Love assumptions is developed in this paper. The theory incorporates transverse (normal and shear) deformation as well as the higher-order effects and accounts for small strains and moderate rotations of the normal. The theory is given in a complete Lagrangian description while the equations of motion and the boundary conditions are obtained through the Hamilton variational principle. Appropriate specializations of the obtained results are performed revealing that they cover a large number of cases separately treated in the specialized literature but considered here in a unified manner.

Curved beam elements with penalty relaxation

AbstractShallowly curved beam elements, including shear deformation and rotary inertia effects, are derived from Hamilton's variational principle. Different degree polynomials, labelled ‘anisoparametric’, are used to interpolate the kinematic variables, instead of uniform interpolations as in the conventional isoparametric procedure. This approach yields a correct representation of the bending strain and, importantly, the membrane and transverse shear strains. Consequently, the severe shortcomings of the exactly integrated isoparametric elements, characterized by excessively stiff solutions in the thin regime (a phenomenon often referred to as membrane and shear locking), are overcome. Uniform (isoparametric‐like) nodal patterns are achieved by explicitly enforcing higher‐degree penalty modes in the membrane and shear strains. This procedure preserves the compatibility of the kinematic field and the capability of the element to move rigidly without straining. Exact quadratures are used on all element matrices, producing a correct rank stiffness matrix, a consistent load vector and a consistent mass matrix. The elements suffer no limitations over the entire theoretical range of the slenderness ratio. For further enhancement and, particularly, in coarse‐mesh situations, an effective relaxation of penalty constraints at the local element level is introduced. This technique ensures a well‐conditioned stiffness matrix. Although the element penalty constraints are relaxed, the corresponding global structure constraints are enforced as is required by the analytic theory. Particular attention is given to the simplest element—a two‐node, six degree‐of‐freedom beam in which all strains are constant. Solutions to static and free vibration arch and ring problems are presented, demonstrating the exceptional modelling capabilities of this element.

Analysis of shells of revolution under earthquake motion

A generalization of the solution of any shell of revolution under earthquake motion and dynamic loading is presented. The equations of motion are derived with the aid of Hamilton's variational principle. A numerical solution is obtained by expending the variables in a Fourier series in the circumferential and in finite elements in the meridional direction. For earthquake analysis an exact method based on step-by-step direct integration and an approximate one involving the spectral velocity, are compared. A case of a nuclear containment under a horizontal earthquake is considered by both methods as an example.

Acceleration waves in a diatomic solid

Utilizing Hamilton's variational principle, the nonlinear field equations and the associated boundary conditions are obtained for an elastic diatomic solid. The possibility of propagation of a singular surface in the form of an acceleration wave is investigated, and the explicit expressions of the acoustical tensors and their strong ellipticity conditions are presented. The existence of six independent waves in such a medium is shown under the conditions stated. Finally, as a simple application, the propagation of acceleration waves in a linear and isotropic diatomic solid is studied and the expressions of wave speeds are obtained.