Hamilton's Variational Principle Research Articles

Trapped-Energy Thickness-Extensional Mode of a Partially Electroded ZnO Thin-Film Resonator.

Free and forced vibration analysis of ZnO thin-film resonator operating with trapped-energy thickness-extensional (TE) mode is performed based on the dispersion curves for both the unbounded fully electroded and unelectroded films. The thickness solutions to the free vibration problem consist of the mode branches of the dispersion curves for the electroded and the unelectroded regions, respectively. The mode branches in the unelectroded region that carry energy away from the vibration zone are neglected because this effective energy loss corresponds to a complex frequency in the frequency spectrum. Since the thickness solutions in each region have satisfied the differential equations and the boundary conditions on the major surfaces exactly, the substitution of the thickness solutions into the modified Hamilton variational principle derived by Tiersten gives an approximate continuity condition in the form of integral over the thickness of the resonator at the interface between the electroded and unelectroded regions. The stationary condition of the integral continuity condition leads to a system of homogeneous linear equations, which determines the frequency spectrum ranging from the first TE cutoff frequency of the fully electroded film to that of the unelectroded film where the conventional trapped-energy vibration occurs. Forced vibration analysis of ZnO thin film driven into TE mode by applying a voltage to the top and bottom electrodes is also performed, which further verifies the validity of the obtained results from the free vibration analysis.

Rotor Airload and Acoustics Prediction Based on CFD/CSD Coupling Method

An advanced airload and noise prediction method based on computational fluid dynamics/computational structural dynamics (CFD/CSD) coupling for helicopter rotor has been developed in this paper. In the present method, Navier-Stokes equation is applied as the governing equation, and a moving overset grid system is generated in order to account for the blade motions in rotation, flapping and pitching. The blade structural analysis is based on 14-DOF Euler beam model, and the finite element discretization is conducted on Hamilton's variational principle and moderate deflection theory. Aerodynamic noise is calculated by Farassat 1A formula derived from FW-H equation. Using the developed method, numerical example of UH-60A is performed for aeroelastic loads calculation in a low-speed forward flight, and the calculated results are compared with both those from isolated CFD method and available experimental data. Then, rotor noise is emphatically calculated by CFD/CSD coupling method and compared with the isolated CFD method. The results show that the aerodynamic loads calculated from CFD/CSD method are more satisfactory than those from isolated CFD method, and the exclusion of blade structural deformation in rotor noise calculation may cause inaccurate results in low-speed forward flight state.

Study on Rigid-Liquid Coupling Dynamics of the Spacecraft With Arbitrary Axisymmetrical Tanks

AbstractThis article focus on the rigid-liquid coupling dynamics of the spacecraft with arbitrary axisymmetrical tanks with curved walls and top. The carrier velocity potential function was obtained according to the motion equations of a representative point in the tank of spacecraft, and the relative velocity potential function can be expressed as Gauss hypergeometric series. The Hamilton's variational principle was applied to derive the governing equations of liquid sloshing based on the profile of hydrostatic shape of free liquid surface. The dynamic equations of modal coordinates were established through the Galerkin method. The state equations of coupled motion of main rigid platform of the spacecraft were deduced by using the Lagrange's equations in term of general quasi-coordinates. Thruster firing was actuated to the coupled system to analyze the rigid-liquid coupled dynamic behaviors. Computer numerical simulations was carried out to confirm the validity of the method developed in this paper.

An E-shape broadband piezoelectric energy harvester induced by magnets

We describe in this work a broadband magnetic E-shape piezoelectric energy harvester with wide frequency bandwidth. We develop first a nonlinear electromechanical model of the harvester based on the Hamilton variation principle that simulates the effect of the nonlinear magnetic restoring force at different spacing distances. The model is used to identify the distances existing between two different magnets that enable the system to perform with a specific nonlinearity. The performance of the E-shape piezoelectric energy harvester is also investigated through experiments, with E-shape energy harvesters at different spacing distances tested under several base acceleration excitations. We observe that the frequency domain output voltage of the system shows a general excellent controllable performance, with a widening of the frequency bandwidth. The half-power bandwidth of the linear energy harvester for a distance of 25 mm is 0.8 Hz only, which can be expanded to 2.67 Hz for the larger distance of 11 mm between magnets. The energy harvester presented in this work shows promising performances for broad-spectrum vibration excitations compared to conventional cantilever piezoelectric energy harvester systems with a tip mass.

Energy Dissipation during Vibrations of Nonuniform Composite Structures: 1. Formulation of the Problem

In this paper, a mathematical model is suggested for decaying vibrations of laminated plates formed by a finite number of arbitrarily oriented orthotropic viscoelastic layers of polymer composites arranged into an anisotropic structure with a layer of stiff isotropic viscoelastic material applied on one of its outer surfaces. The model is based on the Hamilton variation principle, the refined Mindlin–Reissner theory for plates, and the elastic–viscoelastic correspondence principle of the linear viscoelasticity theory. In the description of the physical relationships between the materials of the layers forming structural orthotropic polymeric composites, the influence of vibration frequency and ambient temperature is assumed to be negligible, whereas for the stiff viscoelastic polymeric layer, the dependence of elastic dissipation and stiffness properties on temperature and frequency is considered by means of experimentally determined generalized curves. Equations of motion are obtained for the Timoshenko beam with a layer of stiff isotropic viscoelastic polymer on one of its outer surfaces as a specific case of the general problem by neglecting mid-surface strain in the direction of one of the plate axes. Minimization of the Hamiltonian makes it possible to describe the decaying vibrations of anisotropic structures with an algebraic problem of complex eigenvalues. A system of algebraic equations is formed by the Ritz method using the Legendre polynomials as coordinate functions. First, the real solutions are obtained. To derive complex natural frequencies of the system, the obtained real natural frequencies are taken as the input values, and the complex natural frequencies are calculated applying the third-order iteration method.

Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via differential quadrature method

The free vibration analysis of a circular plate made up of a porous material integrated by piezoelectric actuator patches has been studied. The plate is assumed to be thin and its shear deformations have been neglected. The porous material properties vary through the plate thickness according to some given functions. Using Hamilton's variational principle and the classical plate theory (CPT) the governing motion equations have been obtained. Simple and clamped supports have been considered for the boundary conditions. The differential quadrature method (DQM) has been used for the discretizations required for numerical analysis. The effect of some parameters such as thickness ratio, porosity, piezoelectric actuators, variation of piezoelectric actuators-to-porous plate thickness ratio, pores distribution and pores compressibility on the natural frequency, radial and circumferential stresses has been illustrated. The results have been compared with the similar ones in the literature.

Effect of Winkler and Pasternak elastic foundation on the vibration of rotating functionally graded material cylindrical shell

In this paper, vibration characteristics of rotating functionally graded cylindrical shell resting on Winkler and Pasternak elastic foundations have been investigated. These shells are fabricated from functionally graded materials. Shell dynamical equations are derived by using the Hamilton variational principle and the Langrangian functional framed from the shell strain and kinetic energy expressions. Elastic foundations, namely Winkler and Pasternak moduli are inducted in the tangential direction of the shell. The rotational motions of the shells are due to the Coriolis and centrifugal acceleration as well as the hoop tension produced in the rotating case. The wave propagation approach in standard eigenvalue form has been employed in order to derive the characteristic frequency equation describing the natural frequencies of vibration in rotating functionally graded cylindrical shell. The complex exponential functions, with the axial modal numbers that depend on the boundary conditions stated at edges of a cylindrical shell, have been used to compute the axial modal dependence. In our new investigation, frequency spectra are obtained for circumferential wave number, length-to-radius ratio, height-to-radius ratio with simply supported–simply supported and clamped–clamped boundary conditions without elastic foundation. Also, the effect of elastic foundation on the rotating cylindrical shells is examined with the simply supported–simply supported edge. To check the validity of the present method, the fundamental natural frequencies of non-rotating isotropic and functionally graded cylindrical shells are compared with the open literature. Also, a comparison is made for infinitely long rotating with the earlier published paper.

On vibrations of porous nanotubes

In this study, the vibration behaviors of porous nanotubes are investigated for the first time. The nonlocal strain gradient theory in conjunction with a refined beam model are employed to formulate the size-dependent model. It is presumed that the porous nanotubes are made from functionally graded materials, and the material parameters of nanotubes relate to temperature variation and vary continuously in the radial direction. Employing a refined beam theory which includes the effects of transverse shear deformation, the equations of motion are derived based on Hamilton's variation principle and solved by the Navier solution method. Some comparisons are presented to valid the correctness of present solution method. The effects of the nonlocal parameter, strain gradient parameter, temperature variations, porosity volume fraction and material variation on the vibration characteristic of the nanotubes are discussed in detail.

Neki aspekti u analizi poprečnih slobodnih vibracija prizmatičnih greda opterećenih aksijalnom silom

Influence of constant compressive axial force on free transverse vibrations of uniform beams is considered. Three technical beam theories are applied and appropriate analytical solutions are obtained. Hamilton's variation principle is utilized for derivation of well -known differential equations of motion of Timoshenko beam, while Bernoulli-Euler and Rayleigh theories are considered as special cases. Furthermore, simple two-dimensional plane stress finite element with drilling degree of freedom is employed for numerical analysis. Appropriate code is developed and verified via comparison with commercial finite element software Abaqus, as well as with analytical results. Parametric analysis of influence of the axial force on the modal characteristics of considered beams is performed and discussed in detail.

Transverse Vibration of Rotating Tapered Cantilever Beam with Hollow Circular Cross‐Section

Problems related to the transverse vibration of a rotating tapered cantilever beam with hollow circular cross‐section are addressed, in which the inner radius of cross‐section is constant and the outer radius changes linearly along the beam axis. First, considering the geometry parameters of the varying cross‐sectional beam, rotary inertia, and the secondary coupling deformation term, the differential equation of motion for the transverse vibration of rotating tapered beam with solid and hollow circular cross‐section is derived by Hamilton variational principle, which includes some complex variable coefficient terms. Next, dimensionless parameters and variables are introduced for the differential equation and boundary conditions, and the differential quadrature method (DQM) is employed to solve this differential equation with variable coefficients. Combining with discretization equations for the differential equation and boundary conditions, an eigen‐equation of the system including some dimensionless parameters is formulated in implicit algebraic form, so it is easy to simulate the dynamical behaviors of rotating tapered beams. Finally, for rotating solid tapered beams, comparisons with previously reported results demonstrate that the results obtained by the present method are in close agreement; for rotating tapered hollow beams, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross‐section on the first three‐order dimensionless natural frequencies are more further depicted.

Coupling Dynamic Behavior Characteristics of a Spacecraft Beam with Composite Laminated Structures and Large-Scale Motions

A nonlinear dynamic modeling method for a spacecraft body composed of a laminated composite beam undergoing large rotation is proposed in this paper. To study the characteristics of a laminated composite beam attached to a spacecraft body for the dynamic systems, the deformation description of a laminated beam is established with the consideration of laying angles and laying layers, and the displacement-strain relations is acquired based on the global-local higher-order shear deformation theory. Accordingly, a nonlinear dynamic model of the spacecraft body composed of a laminated composite beam is deduced using Hamilton variational principle. And the complete coupling terms for the laminated material properties are considered unlike any other singular or unidirectional materials. Then, the dynamic behavior of the spacecraft system is analyzed by comparison of an orthogonal-symmetric, singular, and unidirectional laminated beam. The results show that the laminated composite structures have significant influences on the dynamics properties of spacecraft compared with conventional equivalent singular or unidirectional materials. Hence, the nonlinear model is well suitable for approaching the problem of coupling relationship between geometric nonlinearity and large rotation motions. These conclusions will have significant theory and engineering practice values for coupling dynamics properties of laminated beams.

The effect of gas and fluid flows on nonlinear lateral vibrations of rotating drill strings

In this work we develop nonlinear mathematical models describing coupled lateral vibrations of a rotating drill string under the effect of external supersonic gas and internal fluid flows. An axial compressive load and a torque also affect the drill string. The mathematical models are derived by the use of Novozhilov's nonlinear theory of elasticity with implementation of Hamilton's variation principle. Expressions for the gas flow pressure are determined according to the piston theory. The fluid flow is considered as added mass inside the curved tube of the drill string. Using an algorithm developed in the Mathematica computation program on the basis of the Galerkin approach and the stiffness switching method the numerical solution of the obtained approximate differential equations is found. Influences of the external loads, drill string angular speed of rotation, parameters of the gas and fluid flows on the drill string vibrations are shown.

Dynamic response and limit analysis of buried gas pipeline under ground consolidation load

Currently, the significant dynamic plastic deformation of a buried gas pipeline frequently occurs due to the ground construction process that acts as a direct threat to the operation security of a buried gas transmission system. In this study, the pipe-soil interaction structure under a dynamic consolidation load, such as high energy dynamic compaction load, was considered as a non-conservative system in the work. Two parts of structure dissipation energy were introduced into the Lagrange function, and the elastoplastic dynamic equations of a non-conservative system based on the Hamilton Variation Principle (HVP) and the finite element (FE) theory were established. Implicit solution schemes were proposed based on the dynamic equations, and a steel weight-soil-buried pipeline finite element model was developed by performing a dynamic analysis in the LS-DYNA software with an explicit format. Vivid impact responses of an underground pipeline associated with the buried depth, wall thickness, and tamping energy were simulated. The plastic failure criterion of high toughness pipeline steel indicates that treated pipeline buried depth, wall thickness, and tamping energy corresponded to the generalized loads, and limit state of a specific case. So, they were recognized via the relationship of generalized load in relation to the total strain of pipelines. This was performed by using tangent intersection criteria, two elastic slope criteria, and zero curvature criteria. Additionally, the von Mises yield stress criterion was also applied as a traditional approach. The study potentially offers significant references on the quantitative pre-evaluation of a buried gas pipeline that poses as a threat due to the occurrence of third-party damage such as extreme strong ground interference.

Energy dissipation/transfer and stable attitude of spatial on-orbit tethered system

For the Tethered Satellite System, the coupling between the platform system and the solar panel is a challenge in the dynamic analysis. In this paper, the coupling dynamic behaviors of the Tethered Satellite System that is idealized as a planar flexible damping beam-spring-mass composite system are investigated via a structure-preserving method. Considering the coupling between the plane motion of the system, the oscillation of the spring and the transverse vibration of the beam, the dynamic model of the composite system is established based on the Hamiltonian variational principle. A symplectic dimensionality reduction method is proposed to decouple the dynamic system into two subsystems approximately. Employing the complex structure-preserving approach presented in our previous work, numerical iterations are performed between the two subsystems with weak damping to study the energy dissipation/transfer in the composite system, the effect of the spring stiffness on the energy distribution and the effect of the particle mass on the stability of the composite system. The numerical results show that: the energy transfer approach is uniquely determined by the initial attitude angle, while the energy dissipation speed is mainly depending on the initial attitude angle and the spring stiffness besides the weak damping. In addition, the mass ratio between the platform system and the solar panel determines the stable state as well as the time needed to reach the stable state of the composite system. The numerical approach presented in this paper provides a new way to deal with the coupling dynamic system and the conclusions obtained give some useful advices on the overall design of the Tethered Satellite System.

Three-dimensional vibration of a ring with a noncircular cross-section on an elastic foundation

The three-dimensional dynamic equations of a ring with a noncircular cross-section on an elastic foundation are obtained using the Hamilton variation principle. In contrast to the previous rings on elastic foundation model, the developed model incorporates both the in-plane and out-of-plane bend and the out-of-plane torsion in displacement fields. The errors in the derivation of the initial stress and the work of the internal pressure in previous rings on elastic foundation models have been corrected. The mode expansion was used to obtain the analytical solution of the natural frequency. The initial motivation is to develop a theoretical model for car tire dynamics. Therefore, to validate the proposed model, the in-plane and out-of-plane vibrations of a truck tire have been analyzed using the proposed method. To further verify the accuracy of the model, the results of the theoretical formula are compared with the finite element analysis and modal test, and good agreement can be found.

Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity

Key aspects of the manifestly-covariant theory of quantum gravity (Cremaschini and Tessarotto 2015–2017) are investigated. These refer, first, to the establishment of the four-scalar, manifestly-covariant evolution quantum wave equation, denoted as covariant quantum gravity (CQG) wave equation, which advances the quantum state ψ associated with a prescribed background space-time. In this paper, the CQG-wave equation is proved to follow at once by means of a Hamilton–Jacobi quantization of the classical variational tensor field g ≡ g μ ν and its conjugate momentum, referred to as (canonical) g-quantization. The same equation is also shown to be variational and to follow from a synchronous variational principle identified here with the quantum Hamilton variational principle. The corresponding quantum hydrodynamic equations are then obtained upon introducing the Madelung representation for ψ , which provides an equivalent statistical interpretation of the CQG-wave equation. Finally, the quantum state ψ is proven to fulfill generalized Heisenberg inequalities, relating the statistical measurement errors of quantum observables. These are shown to be represented in terms of the standard deviations of the metric tensor g ≡ g μ ν and its quantum conjugate momentum operator.

Free vibration analysis of nanocomposite sandwich plates reinforced with CNT aggregates

AbstractIn this article, free vibration analysis of simply supported sandwich plates with a flexible core and two functionally graded nanocomposite face sheets resting on two‐parameter elastic foundation is carried out using Navier's solution. The nanocomposite face sheets are reinforced by randomly oriented and aggregated carbon nanotube (CNT). The material properties of the nanocomposites plates are graded along the thickness and are estimated though the Eshelby–Mori–Tanaka approach. In the proposed theory, 3‐D elasticity theory and first order shear deformation theory are used for core and face sheets, respectively. Also, Hamilton's variational principle is used to derive the equations of motion.

A shallow water equation based on displacement and pressure and its numerical solution

The primary purpose of this paper is to develop an efficient numerical scheme for solving the shallow water wave problem with a sloping water bottom and wet-dry interface. For this purpose, the Lagrange method and the constrained Hamilton variational principle are used to solve the shallow water wave problem. According to the constrained Hamilton variational principle, a shallow water equation based on the displacement and pressure (SWE-DP) is derived. Based on the discretized constrained Hamilton variational principle, a numerical scheme is developed for solving the SWE-DP. The proposed scheme combines the finite element method for spatial discretization and the simplectic Zu-class method for time integration. The correctness of the SWE-DP and the effectiveness of the proposed scheme are verified by three classical numerical examples. Numerical examples show that the proposed method performs well in the simulation of the shallow water problem with a sloping water bottom and wet-dry interface.

The multiscale eigenelement method in dynamic analyses of periodical composite structures

Multiscale eigenelement method (MEM), which was presented by the author and co-authors, is based on the idea of eigenvector expansion, and the shape functions in MEM bridge the information between macro and micro scales. In previous works, the shape functions in the classical MEM are solved from the self-equilibrium equation of unit cell, while in the improved MEM they are from the self-equilibrium equation as well as the eigenvalue problem of clamped unit cell. In this work, MEM is compared with substructure method for static problem, two effective correction terms are suggested to improve accuracy if necessary. Another contribution of the present work is, the dynamic governing differential equation of MEM is derived for periodical composite structures according to the generalized Hamilton variational principle, and the details about efficient and accurate use of mode superposition method and Newmark integration method are presented. The numerical comparisons with the standard FEM using fine meshes are conducted for free and forced vibration analyses of one-dimensional and two-dimensional problems. And it is concluded that the improved MEM is suitable for the analyses of both static and dynamic problems, and the classical MEM is suitable for the static problems, especially the one with correction term.

Nonlinear harmonic response characteristics and experimental investigation of cantilever hard-coating plate

The nonlinear harmonic response of a cantilever hard-coating plate which is made of a layer of anisotropic hard-coating material and isotropic metal substrate is investigated based on the theory of high-order shear deformation of plate. Firstly, based on the theories of von Karman and Reddy’s three-order shear deformation, the nonlinear dynamic equations of hard-coating plate are built by Hamilton variation principle. Secondly, to obtain nonlinear governing equation of hard-coating plate under transverse load, these equations are discretized in Galerkin method. The system averaged equations with 1:3 internal resonances are obtained by the method of multiple scales, and the multi-periodic responses behavior of cantilever hard-coating plate under transverse loading could be presented. Finally, the vibration response experiment of hard-coating plate is conducted, and the multi-periodic responses are also present for the hard-coating plate with three-to-one internal resonance. Besides, through the vibration response experiment of uncoated titanium alloy plate, the damping characteristic of hard coating is further analyzed.