Hamilton's Variational Principle Research Articles

Covariant Hamiltonian boundary conditions in General Relativity for spatially bounded space–time regions

We investigate the covariant Hamiltonian symplectic structure of General Relativity for spatially bounded regions of space–time with a fixed time-flow vector. For existence of a well-defined Hamiltonian variational principle taking into account a spatial boundary, it is necessary to modify the standard Arnowitt–Deser–Misner Hamiltonian by adding a boundary term whose form depends on the spatial boundary conditions for the gravitational field. The most general mathematically allowed boundary conditions and corresponding boundary terms are shown to be determined by solving a certain equation obtained from the symplectic current pulled back to the hypersurface boundary of the space–time region. A main result is that we obtain a covariant derivation of Dirichlet, Neumann, and mixed type boundary conditions on the gravitational field at a fixed boundary hypersurface, together with the associated Hamiltonian boundary terms. As well, we establish uniqueness of these boundary conditions under certain assumptions motivated by the form of the symplectic current. Our analysis uses a Noether charge method which extends and unifies several results developed in recent literature for General Relativity. As an illustration of the method, we apply it to the Maxwell field equations to derive allowed boundary conditions and boundary terms for the existence of a well-defined Hamiltonian variational principle for an electromagnetic field in a fixed spatially bounded region of Minkowski space–time.

Classical Electrodynamics of Point Charges

A simple mathematical procedure is introduced which allows redefining in an exact way divergent integrals and limits that appear in the basic equations of classical electrodynamics with point charges. In this way all divergences are at once removed without affecting the locality and the relativistic covariance of the theory, and with no need for mass renormalization. The procedure is first used to obtain a finite expression for the electromagnetic energy-momentum of the system. We show that the relativistic Lorentz-Dirac equation can be deduced from the conservation of this electromagnetic energy-momentum plus the usual mechanical term. Then we derive a finite lagrangian, which depends on the particle variables and on the actual electromagnetic potentials at a given time. From this lagrangian the equations of motion of both particles and fields can be derived via Hamilton's variational principle. The hamiltonian formulation of the theory can be obtained in a straightforward way. This leads to an interesting comparison between the resulting divergence-free expression of the hamiltonian functional and the standard renormalization rules for perturbative quantum electrodynamics.

Inverse problem of the variational calculus for higher KdV equations

It is shown that the usual Hamilton's variational principle supplemented by the methodology of the integer-programming problem can be used to construct expressions for the Lagrangian densities of higher KdV fields. This is demonstrated with special emphasis on the second and third members of the hierarchy. However, the method is general enough for applications to equations of any order. The expressions for Lagrangian densities are used to calculate results for Hamiltonian densities that characterize Zakharov-Faddeev-Gardner equation.

DYNAMIC CHARACTERISTICS OF STEPPED CANTILEVER BEAMS CONNECTED WITH A RIGID BODY

The object of this work is to analyze the dynamic characteristics of the portal frame which consists of two stepped beams including a thin plate and a torsional spring at the discontinuous point and a rigid body connecting each beam tip. This structure is available in a lot of cases that need higher stiffness and linear motion of the tip mass. For example, it might be used for an optical pick-up actuator, using piezoelectric materials, for the high area density CD, DVD or the next generation of optical memory devices, which require superrigidity and linear motion in focusing. The mathematical modelling and the derivation of the equation of motion are given for the cantilevers with identically paralleled and stepped beams. The equation of motion and the associated boundary and the continuous conditions are analytically obtained by using Hamilton's variational principle. The exact solutions are presented and compared with the results obtained by FEM Tool (IDEAS).

Modal test and analysis of cantilever beam with tip mass

The phenomenon of dynamic stiffening is a research field of general interest for flexible multi-body systems. In fact, there are not only dynamic stiffening but also dynamic softening phenomenon in the flexible multi-body systems. In this paper, a non-linear dynamic model and its linearization characteristic equations of a cantilever beam with tip mass in the centrifugal field are established by adopting the general Hamilton Variational Principle. Then, the problems of the dynamic stiffening and the dynamic softening are studied by using numerical simulations. Meanwhile, the modal test is carried out on our centrifuge. The numerical results show that the system stiffness will be strengthened when the centrifugal tension force acts on the beam (i.e. the dynamic stiffening). However, the system stiffness will be weakened when the centrifugal compression force acts on the beam (i.e. the dynamic softening). Furthermore, the equilibrium position of the system will lose its stability when the inertial force reaches a critical value. Through theoretical analysis, we find that this phenomenon comes from the effect of dynamic softening resulting from the centrifugal compression force. Our test results verify the above conclusions and confirm that both dynamic stiffening and softening phenomena exist in flexible multi-body systems.

Dynamics of anisotropic composite cantilevers weakened by multiple transverse open cracks

In this paper an exact solution methodology, based on Laplace transform technique enabling one to analyze the bending free vibration of cantilevered laminated composite beams weakened by multiple non-propagating part-through surface cracks is presented. Toward determining the local flexibility characteristics induced by the individual cracks, the concept of the massless rotational spring is applied. The governing equations of the composite beam with open cracks as used in this paper have been derived via Hamilton's variational principle in conjunction with Timoshenko's beam model. As a result, transverse shear and rotatory inertia effects are included in the model. The effects of various parameters such as the ply-angle, fiber volume fraction, crack number, position and depth on the beam free vibration are highlighted. The extensive numerical results show that the existence of multiple cracks in anisotropic composite beams affects the free vibration response in a more complex fashion than in the case of beam counterparts weakened by a single crack. It should be mentioned that to the best of the authors' knowledge, with the exception of the present study, the problem of free vibration of composite beams weakened by multiple open cracks was not yet investigated.

A general vibration model of angle-ply laminated plates that accounts for the continuity of interlaminar stresses

Abstract This paper presents the generalisation of a well documented two-dimensional shear deformable laminated shell theory [Compos. Struct. 25 (1993) 165] that, based on a fixed number of unknown variables, was initially proposed for laminates made of specially orthotropic layers only. The theory is here specialised for laminated plates but is able to encompass monoclinic layers in a general multilayered configuration. Moreover, it is able to account for the interlaminar continuity of both displacements and transverse shear stresses. Higher-order effects, as shear deformation and rotary inertia, are naturally included into the formulation. In order to obtain the relevant governing differential equations, both Hamilton's variational principle and a recently proposed vectorial approach [Compos. Engng. 3 (1993) 3] have been independently used. The effectiveness of the present model is tested numerically by comparing its results with exact three-dimensional elasticity results obtained under the particular condition that the plates vibrate in cylindrical bending.

Generalized theory of multilayer plates

The layer-wise generalized theory of elastodynamic of multilayer plates is presented in this paper. This theory is based on expanding the displacement vector components of each layer into power series about the transverse coordinate. The number of terms retained in the power series is arbitrary and it is chosen depending on the problem being considered and the solution accuracy required. The system of governing equations is obtained by Hamilton's variation principle. The possibilities of the theory proposed and validity of results obtained are illustrated by examples of investigating the strain-stressed state of one- and three-layer structures. The issues of applicability of two-dimensional approximations built on the basis of the power series method are considered with respect to calculation of displacements, inplane and transverse stresses in multilayer plates under dynamic loading. Calculation results are compared with data obtained from Ambartsumyan's theory (the hypothesis of a unique non-strained normal for the pack), the layer-wise theory based on the broken line hypothesis as well as the three-dimensional elasticity theory.

About the Paper by A.I. Lurie on Synchronization Theory

This presentation is devoted to the history of the creation and the contents of not very well-known paper by A.I. Lurie "Certain Problems of Self-synchronization" published in 1970. By this time, some investigations of self-synchronization of unbalanced rotors and pendulum clocks had been carried out mainly by Lurie's pupils. They used the techniques by Poincare, Lyapunov, and asymptotic methods in their investigations. To study this problem A.I. Lurie applied the Hamilton variation principle in conjunction with the method of small parameter and Galerkin's method. Main results were obtained by Lurie in very laconic and elegant form very characteristic for him. This presentation emphasizes the role of Lurie's paper to understand the regularities of frequency self-synchronization and for further development of its theory.

TWO GENERALIZED HIGHER ORDER THEORIES IN FREE VIBRATION STUDIES OF MULTILAYERED PLATES

This paper presents an extension of two-dimensional models for the analysis of freely vibrating laminated plates. The extension concerns the enlargement of higher order theories, recently introduced by different authors in several forms, to encompass higher order terms over the cubic one usually taken into consideration. Higher order effects such as rotatory inertia and transverse shear stress are naturally included without any shear correction factors. Namely, two different models are introduced by expanding, on different functional bases, displacements (D2D) and transverse shear stresses in conjunction with displacements (M2D). The expansion is considered to be consistent with the traction-type boundary condition on the external surfaces of the plate. The governing equations and associated boundary conditions are consistently obtained by the classical Hamilton's variational principle and Reissner's mixed variational theorem. Both models are equivalent single layer type and, therefore, differ according to the layer-wise descriptions, preserve the independence of the number of unknown variables on the number of layers. However, this feature is presented together with intrinsic physical violations for both models. Model D2D violates the interlaminar stress continuity requirement and model M2D violates in a weaker from the same requirement (derivatives are not piecewise continuous), besides neglecting the transverse normal stress. The importance of completely fulfilling the mentioned continuity is then discussed once the relevant governing equations are tailored for the cylindrical bending condition. The effectiveness of the models is indicated by making numerical comparisons with the exact three-dimensional theory of the elasticity for several lamination schemes, angle/cross-ply lay-ups, and characteristic geometric ratios for low and higher frequencies.

Transient waves in a functionally graded cylinder

A hybrid numerical method (HNM) is presented for analyzing transient waves in a cylinder made of functionally graded material (FGM). In the HNM, the FGM cylinder is divided into N cylindrical elements with three-nodal line in the wall thickness. The elemental material properties are assumed to vary linearly in the thickness direction to better model the spatial variation of material properties of FGM. The Hamilton variational principle is used to develop governing equations of the cylinder. The displacement responses are determined by employing the Fourier transformations together with the modal analysis. The HNM is applied to analyze a number of FGM cylinders, and its efficiency is demonstrated.

A method of determining the extraneous unknowns in problems of the stability and vibrations of rods

An approximate method of determining the critical loads in problems of the stability of compressed rods has been extended to statically indeterminate systems. For this purpose, a method has been developed for solving stability problems when there are extraneous unknowns defined by the stationarity condition for the potential energy of the system. It is shown that, combined with Grammel's method and Hamilton's variational principle, the method described for determining the extraneous unknowns in statically indeterminate systems can also be used in problems of finding the natural frequencies of vibrations of rods.

Routh's equations and variational principles

Holonomic mechanical systems with n degrees of freedom are considered in Routh variables, and the equations of motion consist of k < n Lagrange-type equations and 2( n− k) Hamilton-type equations. Expressions are presented in Routh variables for the D'Alembert-Lagrange, Hamilton-Ostrogradskii and Hamilton (third form) variational principles, as well as the Hölder principle and the principle of least action in its Lagrange and Jacobi forms.

Aeroelastic Tailoring of Composite Wings Exhibiting Nonclassical Effects and Carrying External Stores

Structural and aeroelastic tailoring applied to advanced straight and swept aircraft wings carrying external stores is addressed. The wing structure is modeled as a laminated composite plate exhibiting flexibility in transverse shear and warping restraint effects. The equations of motion and boundary conditions are obtained via Hamilton's variational principle and application of generalized function theory. For a comprehensive representation of the stores, their static weights and inertia terms are considered. Three-dimensional modified strip theory aerodynamics is employed, and the obtained eigenvalue/boundary value problems are solved using the extended Galerkin method. The model is used to investigate the implications of external stores and ply angle orientation on divergence, free vibration, and flutter. Within the context of aeroelastic tailoring, the influence of external stores attached to the wing structure has to be considered during the preliminary aircraft design phases.

NON-LINEAR COUPLED TRANSVERSE AND AXIAL VIBRATION OF A COMPLIANT STRUCTURE, PART 1: FORMULATION AND FREE VIBRATION

A compliant tower in the ocean environment is modelled as a beam undergoing coupled transverse and axial motion. The equations of motion are non-linear and coupled and are derived here. The beam is assumed to be supported by a linear-elastic torsional spring at one end and with a point mass at the other end. Such a model is representative of numerous applications. The equations of motion and boundary conditions are obtained using Hamilton's variational principle. It is assumed that strains are small but the rotation is moderate compared to the strain so that the equations of motion for the axial and transverse motion are non-linearly coupled. The free response in vacua and the free response in water are considered in particular. The fluid forces, the added mass and drag forces, are modelled using a semi-empirical Morison equation. The resulting non-linear coupled partial differential equations are solved numerically using the finite difference approach. In Part 2 of this work, various forced responses are studied.

Reduction of Hamilton's variational principle

This paper builds on the initial work of Marsden and Scheurle on nonabelian Routh reduction. The main objective is to carry out the reduction of variational principles in further detail. In particular, we obtain reduced variational principles which are the symplectic analogue of the well-known reduced variational principles for the Euler-Poincare equations and the Lagrange-Poincare equations. On the Lagrangian side, the symplectic analogue is obtained by suitably imposing the constraints of preservation of the momentum map.

Three-dimensional variational impact contact analysis of composite bars and plates

Three-dimensional (3D) variational dynamic contact analysis approach is developed for predicting 3D transient deformation of composite plates exposed to a rigid body impact. The Hamilton variational principle is applied using the Bernstein basis functions of an arbitrary degree for the displacement approximation in the three coordinate directions. The Lagrange multiplier technique is employed in the formulation of the variational dynamic contact problem. The developed computational algorithm and computer code is validated on the example problems of a longitudinal impact on bi-material composite prismatic rectangular bar and transverse impact on rectangular composite plates. A comparison of the impact performance of 3D woven and traditional laminated composite plates is presented.

Forced stationary vibrations of flexible plates induced by random narrow-band excitations

We consider some problems related to the analysis of the modes of forced vibrations of elastic plates induced by random narrow-band excitations. The investigation of nonlinear vibrations of a plate is performed within the framework of a reduced discrete model constructed by using the Hamilton variational principle, finite-element methods, and generalized coordinates. To study the dynamical state of the analyzed structure, we perform numerical simulation of the vibrations induced by a particular realization of narrow-band excitations. Each realization of random narrow-band excitations is obtained with the help of a shaping filter of the second order and describes harmonic oscillations with slowly varying amplitude and phase. We also consider the scenarios of transitions between different modes of vibrations of the plate and estimate the time of its residence in different dynamical states.

On Thermal Dynamic Buckling Analysis of Imperfect Laminated Cylindrical Shells

Based on the general form of Green's strain tensor in curvilinear coordinates, the exact strain-displacement expressions of an imperfect laminated composite circular cylindrical shell undergoing large deflections are developed. The resulted relations may also be used for postbuckling analysis. Employing Hamilton's variational principle, the most general three dimensional and exact integral equations of motion are introduced in a hybrid form. No assumption or simplification is made in deriving the formulations. The resulting equations are solved using a Kantorovich type power series. Dynamic buckling loads treated in this paper, include mechanical loads (axial compression, external pressure, external fluid pressure, and torsion), thermal loads, or a combination of them. The resulted equations are then solved by means of an efficient solution procedure. In contrast to the well-known higher-order and layer-wise theories which are displacement-based, due to the hybrid form of the final equations, various edge conditions (displacement, stress, force, and moment boundary conditions) can be accurately incorporated. Furthermore, in contrast to the existing 3–D elasticity approaches, the solution procedure of the present method is self-started. Finally, few examples of the well-known references done in the mechanical and thermal buckling of the composite circular cylindrical shells, which are based on other theories, are reexamined for comparison purposes. The results are extended to check the capability of the present theory and to enable a sensitivity analysis.

Wave propagation in complex structures modelled by medium with internal variables

AbstractGoverning equations for complex engineering structures are derived by means of the Hamilton variational principle under the assumption that the vibration field has a slowly varying component of an evolutionary type. This function is identified as the displacement of the primary structure, whereas the internal degrees of freedom model substructural vibrations. In the frequency domain of high modal overlap the secondary systems act as dynamic vibration absorbers with respect to primary structure. Wave propagation in the primary structures of complex structure is shown to be modelled by hyperbolic equations with source terms, the latter describing the backward effect of vibrating substructures on the primary structure. In order to take into account uncertain mechanical and spectral properties of real structures propagation of a wave packet in the structure is considered. The evolution of the amplitude of the propagating wave packet is shown to be governed by a boundary value problem of parabolic type. The approach developed presents an alternative to the vibrational conductivity approach and other techniques utilising various sorts of parabolic equations for describing energy propagation in complex structures.