Hamilton's Variational Principle Research Articles

Linearized propagation equations for metric fluctuations in a general (non-vacuum) background geometry

The linearized dynamical equation for metric perturbations in a fully general, non-vacuum, background geometry is obtained from the Hamilton variational principle applied to the action up to second order. We specialize our results to the case of traceless and transverse metric fluctuations, and we discuss how the intrinsic properties of the matter stress tensor can affect (and modify) the process of gravity wave propagation even in most conventional geometric scenarios, like (for instance) those described by a FLRW metric background. We provide explicit examples for fluid, scalar field and electromagnetic field sources.

Fatigue failure mechanism of 3D tubing strings used in high-pressure, high-temperature and high-yield curved gas wells

In view of the fatigue failure of tubing strings by flow-induced nonlinear vibration (FINV) in high-temperature, high-pressure and high-yield (3H) curved gas wells, a three-dimensional (3D) FINV model of the tubing string is established using the micro-finite method, energy method and Hamilton variational principle, and verified by a similar experiment of tubing vibration. Meanwhile, the cumulative damage theory is used to establish the fatigue life prediction method of tubing string combined with the stress response which was determined by the proposed FINV model and the S-N curve of the tubing material (13Cr-L80) which was measured by fatigue test. Thus, the influences of production rate, well trajectory parameters (inclination angle and well section length), as well as the installation position of downhole tools (packer and centralizer) on the fatigue life of the tubing string are systematically analyzed. The results obtained demonstrate that, first, with the increase of production rate, the fatigue life of the tubing string decreases, and the field production allocation should be far away from the sudden production rate which can be determined using the proposed analysis method. Second, in the well trajectory design, the vertical section length should be reduced, while that of build-up section and the angle of stable inclined section should be increased. Third, the packer and centralizer should be placed in the optimal position, which depends on the well structure, downhole tool size, etc. and can be determined using the proposed analysis method. The research results provide a theoretically sound guidance for designing and practically sound approach for effectively improving the fatigue life of tubing string in 3H curved gas wells.

Coupling of quantum gravitational field with Riemann and Ricci curvature tensors

The theoretical problem of establishing the coupling properties existing between the classical and quantum gravitational field with the Ricci and Riemann curvature tensors of General Relativity is addressed. The mathematical framework is provided by synchronous Hamilton variational principles and the validity of classical and quantum canonical Hamiltonian structures for the gravitational field dynamics. It is shown that, for the classical variational theory, manifestly-covariant Hamiltonian functions expressed by either the Ricci or Riemann tensors are both admitted, which yield the correct form of Einstein field equations. On the other hand, the corresponding realization of manifestly-covariant quantum gravity theories is not equivalent. The requirement imposed is that the Hamiltonian potential should represent a positive-definite quadratic form when performing a quadratic expansion around the equilibrium solution. This condition in fact warrants the existence of positive eigenvalues of the quantum Hamiltonian in the harmonic-oscillator representation, to be related to the graviton mass. Accordingly, it is shown that in the background of the deSitter space-time, only the Ricci tensor coupling is physically admitted. In contrast, the coupling of quantum gravitational field with the Riemann tensor generally prevents the possibility of achieving a Hamiltonian potential appropriate for the implementation of the quantum harmonic-oscillator solution.

A higher-order size-dependent beam model for nonlinear mechanics of fluid-conveying FG nanotubes incorporating surface energy

Fluid-conveying tubes in nano scale present different mechanical behaviors from the macro tubes. A higher-order size-dependent beam model is developed coupling with the nonlocal stress, strain gradient effects, surface energy effects for flow-inducing post-buckling and free vibration analysis of the functionally graded nanotubes. Also, the slip flow effect is considered to modify the kinetic energy of the fluid. The governing equations of post-buckling and vibration are derived by the Hamilton variational principle in which the obtained post-buckling tube configuration is also the initial configuration for vibration analysis. The two-step perturbation method is extended to the nonlinear analysis of the fluid-conveying tubes, and a perturbation scheme is proposed. Subsequently, the influences of nano effects on post buckling, natural frequency and nonlinear vibration of the fluid-conveying nanotubes are studied. The imperfection sensitivity of the static and dynamic behaviors is also discussed. Numerical results reveal the dual influences of nano effects on the nonlinear behaviors and indicate the significance to consider the effects of surface energy, small scale and slip flow in nonlinear analyses of the fluid-conveying nanotubes in a comprehensive way.

Особенности сложных колебаний гибких микрополярных сетчатых панелей

In this paper, a mathematical model of complex oscillations of a flexible micropolar cylindrical mesh structure is constructed. Equations are written in displacements. Geometric nonlinearity is taken into account according to the Theodore von Karman model. A non-classical continual model of a panel based on a Cosserat medium with constrained particle rotation (pseudocontinuum) is considered. It is assumed that the fields of displacements and rotations are not independent. An additional independent material parameter of length associated with a symmetric tensor by a rotation gradient is introduced into consideration. The equations of motion of a panel element, the boundary and initial conditions are obtained from the Ostrogradsky – Hamilton variational principle based on the Kirchhoff – Love’s kinematic hypotheses. It is assumed that the cylindrical panel consists of n families of edges of the same material, each of which is characterized by an inclination angle relative to the positive direction of the axis directed along the length of the panel and the distance between adjacent edges. The material is isotropic, elastic and obeys Hooke’s law. To homogenize the rib system over the panel surface, the G. I. Pshenichnov continuous model is used. The dissipative mechanical system is considered. The differential problem in partial derivatives is reduced to an ordinary differential problem with respect to spatial coordinates by the Bubnov – Galerkin method in higher approximations. The Cauchy problem is solved by the Runge – Kutta method of the 4th order of accuracy. Using the establishment method, a study of grid geometry influence and taking account of micropolar theory on the behavior of a grid plate consisting of two families of mutually perpendicular edges was conducted.

The Principle of Covariance and the Hamiltonian Formulation of General Relativity.

The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out.

NEWMARK ALGORITHM FOR DYNAMIC ANALYSIS WITH MAXWELL CHAIN MODEL

This paper investigates a time-stepping procedure of the Newmark type for dynamic analyses of viscoelastic structures characterized by a generalized Maxwell model. We depart from a scheme developed for a three-parameter model by Hatada et al. [1], which we extend to a generic Maxwell chain and demonstrate that the resulting algorithm can be derived from a suitably discretized Hamilton variational principle. This variational structure manifests itself in an excellent stability and a low artificial damping of the integrator, as we confirm with a mass-spring-dashpot example. After a straightforward generalization to distributed systems, the integrator may find use in, e.g., fracture simulations of laminated glass units, once combined with variationally-based fracture models.

Double Resonance of Magnetism-Solid Coupling of in-Plane Moving Thin Plates With Linear Loads and Elastic Supports

For the in-plane moving thin plates with linear loads and elastic supports in magnetic field, the potential energy, the kinetic energy and the electromagnetic force expressions of the system were given. Based on the Hamiltonian variational principle, the magnetism-solid coupling nonlinear vibration equation for the in-plane moving strip plate was deduced. For the clamped-hinged boundary condition, the variable separation method and the Galerkin method were employed to obtain the 2DOF nonlinear vibration differential equations containing the simple harmonic linear load and the electromagnetic damping force terms. The multiscale method was used to analytically solve the principal-internal resonance problem, and the 1st-order state equation and the resonance response characteristic equation for the system under the double joint resonance were obtained. Through numerical examples, the 1st- and 2nd-order resonance amplitude curves of the in-plane moving thin plate were obtained. The effects of different parameters and load positions on vibration characteristics of the system were analyzed. The results show that, for the principal-internal resonance occurring in the system, the multivaluedness and jumping phenomenon of the solution are obvious, and the effects of the elastic support and the linear load position on the resonance are significant. Additionally, the 1st- and 2nd-order resonance multivalued solution areas appear and disappear simultaneously, which reflects obvious internal resonance characteristics.

Friction-wear failure mechanism of tubing strings used in high-pressure, high-temperature and high-yield gas wells

This study establishes a flow-induced nonlinear vibrations (FINV) model of tubing string using micro-finite element and energy methods along with the Hamilton variational principle. This model helps address the wear failure of tubing strings caused by FINV in high-pressure, high-temperature, and high-yield (3H) gas wells and considers the changes in wellbore trajectory, wellbore temperature/pressure, and contact/collision of the tubing-casing. To validate the proposed model, a simulation experiment of tubing string vibration was conducted. Based on the White-Fleisher wear theory, a method is proposed for calculating wear amount and depth of tubing string in 3H gas wells; the friction coefficient and wear rate are determined via a wear experiment. Thus, the influences of production rate, inclination angle, well section length, packer position, and centralizer position on the wear characteristics of tubing string are systematically analyzed. The results obtained demonstrate that, first, with the increase of production rate, the wear life of the tubing string decreases, and there exists a sudden value of production rate, which can be determined using the proposed analysis method, such that the field production allocation should be far away from the sudden production rate. Second, in the well trajectory design, the vertical section length should be small, while that of build-up section and the angle of stable inclined section should be large. Third, the optimal positions for the packer and centralizer in the field depend on the well structure, downhole tool size, etc. and can be determined using the proposed analysis method; these positions help design field tools. The research results provide a theoretically sound guidance for designing and practically sound approach for effectively improving the service life of tubing string in 3H gas wells.

Nonlinear flow-induced vibration response characteristics of a tubing string in HPHT oil&gas well

This study has established a nonlinear flow-induced vibration (FIV) model of tubing string by using the micro-element method and energy method with the Hamiltonian variational principle, in order to address the vibration failure problem of the tubing string induced by high-speed fluid flow in the tubing of high temperature and high pressure (HPHT) oil&gas well. This model considered factors such as borehole trajectory changes, wellbore temperature and pressure variations, self-weight of the tubing string, and the contact-impact of the tubing-casing. Based on the structural parameters of the M high-yield gas well in the South China Sea, a simulation experiment of the tubing string vibration was conducted according to the similarity principle. A comparison between the experiment data and theoretical data showed that the calculation accuracy exceeded 84%, which verified the correctness and validity of the nonlinear vibration model established in this study. On this basis, the influence of the production rate, well inclination angle, well section length, and packer position on the vibration response characteristics of the tubing string were analyzed systematically. It was found that: ①with the increase of the production rate, the vibration response of the tubing string became clearer. The production rate interval that showed abrupt changes was from 0.9 ~1.2 million m3/day. Thus, during designing, such production rate intervals with abrupt changes should be avoided to ensure the safe usage of the tubing strings; ②with the increase of the well inclination angle, the vibration response of the tubing string decreased to a certain extent. Regarding the influence of the well section length, the longitudinal vibration response of the tubing string was most sensitive to the length of the vertical section, followed by the steady inclined section, and finally the angle building section. Thus, during designing, the length of the vertical section should be reduced, while the length of the angle building section and the well inclination angle of the steady inclined section should be increased; ③As the position of the production packer moved up, the vibration responses of the lower and middle tubing strings both decreased, and the optimal packer position was between 3,549 and 3,666 m. The research results can serve as an effective analysis tool and provide a theoretical basis for the safety design of the tubing strings in the HPHT oil&gas well site.

Safety evaluation method of tubing strings in high-pressure, high-temperature and high-yield gas wells based on FIV analysis

In view of the failures of tubing string caused by the flow-induced nonlinear vibration (FINV) in high-pressure, high-temperature and high-yield (3H) gas wells, a FINV model of tubing string is established using micro-finite element and energy methods, as well as Hamilton variational principle, and verified by a similar experiment of tubing vibration. Moreover, in order to effectively evaluate the safety of the tubing string, the buckling stability and strength damage evaluation method of tubing string are proposed. Thus, the influences of production rate, inclination angle, well section length, and centralizer position on the tubing string safety were systematically analyzed. The failure mechanism of tubing string in 3H gas wells is elucidated, and the safety control methods of field tubing string are proposed; which suggest that, firstly, with the increase of production rate, the safety of the tubing string decreases, and there exists a sudden value, which can be determined using the proposed analysis method, such that the field production allocation should be far away from the sudden production rate. Secondly, in the well trajectory design, the vertical section length should be reduced, while that of kick-off section and the angle of stable inclined section should be increased. Thirdly, there is an optimal position for the centralizer in the field, which is related to the well structure, downhole tool size, etc. and should be determined by the proposed analysis method to enable the design of the centralizer on-site. The research results provide a theoretically method for safety evaluation and practically sound approach for effectively improving the service life of tubing string in 3H gas wells.

Coupled Orbit-Attitude Dynamics of Tethered-SPS

In this paper, a dynamic model for long-term on-orbit operation of the tethered solar power satellite (Tethered-SPS) is established. Because the Tethered-SPS is a super-large and super-flexible structure, the coupling among the orbit, attitude, and structure vibration of the system should be considered in comparison with the traditional spacecraft dynamic models. Based on the absolute nodal coordinate formulation (ANCF), the dynamic equation of the Tethered-SPS is established by Hamiltonian variational principle, and the dual equations of the Hamiltonian system are established by introducing generalized momentum through Legendre transformation. The symplectic Runge–Kutta method is used for numerical simulation, and the validation of the modeling method is verified by a numerical example. The effects of orbital altitude, initial attitude angle, length of solar panel, and orbital eccentricity on the orbit and attitude of the Tethered-SPS are analyzed. The numerical simulation results show that the effect of orbital altitude and length of solar panel on the orbital error of midpoint of beam is small. However, the initial attitude angle has a significant effect on the orbital error of midpoint of solar panel. The effect of the length of solar panel on the attitude angle of the system is not significant, but orbital altitude, orbital eccentricity, and initial attitude angle of the system severely affect the attitude angle of the system. Then, the stability of the system is affected.

Estimation of average roughness of X200Cr12 steel using displacement from the tool nozzle

In this work, we present a modeling of the dynamic behavior of a turning tool. The goal is to show that it is possible to estimate the roughness average of the workpiece starting from resulting displacement from the tool nozzle. The tool is modeled by the Euler beam clamped-free, excited by the cutting force at its free end. The formulation of the motion equations of the tool is based on the Hamilton variational principle and their resolutions are carried out by the modal method and the Duhamel integral. As the excitation is unspecified, the integrals are evaluated numerically. The three components of the cutting force, obtained experimentally, permit to calculate the tool nozzle displacements. The comparison of the calculated average roughness with the measured one shows a good agreement. We present also a validation of the experimental results of the resulting displacement of the tool nozzle with the calculated displacement.

Eğrisel Fiber Yollu İkili Motorlu-Kanat Sisteminin Aeroelastik Enerji Optimizasyonu

In the present study, the aeroelastic energy response of a twin-engine composite wing system is optimized based on sequential quadratic programming (SQP) method. The variable stiffness is acquired by constructing laminates of thin wall beam (TWB) with curvilinear fibers having prescribed paths. In order to account the effect of spanwise locations and mass of the engines on the aeroelastic characteristics of TWB, the novel governing equations of motion are obtained using Hamilton's variational principle. The paper aims to exploit desirable fiber paths with improved aeroelastic properties for different twin-engine wing configuration. Ritz based solution methodology is employed to solve the equations with coupled incompressible unsteady aerodynamic model based on Wagner’s function. A novel optimization strategy based on the total energy of the aeroelastic system is introduced. The proposed total energy, as a cost function, is minimized in terms of four optimization variables of two engine’s locations and wing structure curvilinear fiber angle with two design parameters. The total energy is obtained by integrating responses of kinetic and potential energy in a specific time interval. The minimum total energy is an indication of ideal optimization variables which leads to the optimum flutter performance. Numerical results demonstrate the effectiveness of the optimization variables on the total energy of the aeroelastic system and determine the optimal values of introduced variables in case of minimum total energy and improved aeroelastic characteristics.

Асимптотическое поведение решения осесимметричной динамической задачи теории упругости для трансверсально-изотропного сферического слоя малой толщины

In this paper, we study the axisymmetric dynamic problem of the theory of elasticity for the transversely isotropic spherical layer of small thickness that does not contain any of the poles 0 and π. It is assumed that the lateral surface of the sphere is free of stresses, and boundary conditions are set on conical sections. Using the method of asymptotic integration of equations of the theory of elasticity, the dynamic problem of this theory is analyzed for the transversely isotropic spherical layer as the thin-walled parameter tends to zero. A possible form of wave formation in the transversely isotropic spherical layer has been studied depending on the frequency of the influencing forces. Homogeneous solutions are constructed and their classification is given. Asymptotic expansions of the homogeneous solutions are obtained, which make possible to calculate the stress-strain state for various values of the frequency of the influencing forces. It is shown that for the high-frequency oscillations in the first term of the asymptotics, the dispersion equation coincides with the well-known Rayleigh-Lamb equation for the elastic band. In the general case of loading on the sphere using the Hamilton variational principle, the boundary-value problem is reduced to the solving infinite systems of linear algebraic equations.

DETERMINING THE BALANCE CONFIGURATION, IN CASE OF THE OSCILLATING MOVEMENT OF THE MAIN SPINDLE AT CNC LATHE

In the paper we present a mathematical model through which are determined the balance conditions, needed for the stability analysis of the oscillating movement of the main spindle at CNC lathe. We take into account Hamilton's variation principle, the axiom of impulse derivative and the axiom of kinetic moment derivative. We present the general movement equations that generate the oscillations based on the calculus hypotheses, performing the introduction of the external solicitations. Establishment of the balance configuration is done by imposing the conditions that the system of forces that act upon the ensemble spindle – bearings - tool causes a deformation of the spindle, without producing spindle vibration. We obtain the new differential equations of the movement, in which the forces and moments are determined from the static case, based on which we can determine the integration constants in the characteristic points of the main spindle.

DETERMINING THE BALANCE CONFIGURATION, IN CASE OF THE OSCILLATING MOVEMENT OF THE MAIN SPINDLE AT CNC LATHE

In the paper we present a mathematical model through which are determined the balance conditions, needed for the stability analysis of the oscillating movement of the main spindle at CNC lathe. We take into account Hamilton's variation principle, the axiom of impulse derivative and the axiom of kinetic moment derivative. We present the general movement equations that generate the oscillations based on the calculus hypotheses, performing the introduction of the external solicitations. Establishment of the balance configuration is done by imposing the conditions that the system of forces that act upon the ensemble spindle – bearings - tool causes a deformation of the spindle, without producing spindle vibration. We obtain the new differential equations of the movement, in which the forces and moments are determined from the static case, based on which we can determine the integration constants in the characteristic points of the main spindle.

Buckling and post-buckling of FRC laminated beams in thermal environment using a generalized higher-order shear deformation zig-zag beam model

The thermal buckling and post-buckling of fibre-reinforced composite (FRC) beams are analyzed based on a generalized higher-order shear deformation zig-zag beam theory (RHDZT). The current zig-zag beam model satisfies the stress boundary conditions on the upper and lower surfaces and the reality that the shear strain is discontinuous at the layer interfaces. The material properties of FRCs are assumed to be temperature independent and the in-plane boundary conditions supposed immovable. By using the Hamilton variation principle, the governing equations are established based on the mid-plane of the beam. A two-step perturbation method is employed to obtain the approximate analytical solutions of the critical thermal buckling loads and post-buckling behaviors with clamped-clamped boundary conditions. The critical buckling results based on the zig-zag model and its degenerate equivalent single-layer models are compared and verified with those in the literatures. The post-buckling results of the FRC beams with different slenderness ratios, elastic modulus ratios and ply angles are investigated. Numerical results show the remarkable accuracy of the current model and also demonstrate the significant roles the above-mentioned effects play in the buckling and post-buckling problem of FRC laminated beams.

Imperfection sensitivity of free vibration of symmetrically/anti-symmetrically laminated FRC beams in thermally pre-and post-buckling equilibrium states

Linear/nonlinear free vibration of imperfect symmetrically/anti-symmetrically laminated fibre-reinforced composite (FRC) beams in the pre-and post-buckling equilibrium states are studied. It is assumed that the initial defect has the same shape as the first-order buckling mode and the in-plane boundaries are immovable. A modified higher-order shear deformation zig-zag beam model is applied for refined modeling of the laminated beams. The Hamilton variational principle is utilized to obtain the governing equations of motion. The two-step perturbation technique is combined with the harmonic balance method to obtain the asymptotic analytic solutions. The results based on the zig-zag model and its degenerate equivalent single-layer models are compared and verified. Further, the sensitivity of vibration characteristics to the size of imperfections at different temperature rise is investigated. Numerical results suggest that there exists a load range near the critical buckling load of the perfect beams where the linear/nonlinear frequency is sensitive to the initial imperfections.

Classical Variational Theory of the Cosmological Constant and Its Consistency with Quantum Prescription

The manifestly-covariant Hamiltonian structure of classical General Relativity is shown to be associated with a path-integral synchronous Hamilton variational principle for the Einstein field equations. A realization of the same variational principle in both unconstrained and constrained forms is provided. As a consequence, the cosmological constant is found to be identified with a Lagrange multiplier associated with the normalization constraint for the extremal metric tensor. In particular, it is proved that the same Lagrange multiplier identifies a 4-scalar gauge function generally dependent on an invariant proper-time parameter s. Such a result is shown to be consistent with the prediction of the cosmological constant based on the theory of manifestly-covariant quantum gravity.