Hamilton's Variational Principle Research Articles

Dynamic analysis of co-curing double-layer damping films embedded composite beam with fixed-support boundary

Co-curing double-layer damping films embedded composite structure has many advantages over a tradition composite, such as large damping, high specific strength etc. In order to study the dynamic performance of this composite clamped beams, a theoretical model of the free vibration of the beam was established by the combination of first-order shear deformation theory with the Hamilton principle and variational principle. The Galerkin method was presented to find the free vibration theoretical solution under the fixed-support boundary conditions. The theoretical solution was compared with the experimental results through the established experimental platform. According to the comparison results, the validity of the theoretical model was proved. Finally, the effects of the damping layer distribution and the geometric parameters on the dynamic performance were further explored. It provides a theoretical basis for the optimal configuration of double-layer damping films embedded composite clamped beams structure.

On the impact process and stress field of functionally graded graphene reinforced composite pipes with a viscoelastic interlayer

In the paper, impact process of the fluid-conveying pipes composed of graphene-reinforced composite layers and a viscoelastic interlayer is studied. The bending stress field, midspan displacement and contact force are focused on. Based on the theory of high-order displacement field, the governing equations are derived through the Hamilton variational principle. To obtain the approximate solution for the impact dynamics of pipes, the Galerkin method is extended to expand the generalized displacements into the schemes of triangular series. Further, by utilizing the orthogonality of the trigonometric series, the differential equations of various orders for impact dynamics are established. The fourth-order Runge–Kutta method is introduced to the truncated solutions. Subsequently, the parameter sensitivity of the transient responses in the impact stage is emphatically discussed. It is revealed that the insensitive parameters to contact force and midspan displacement have a great influence on the stress field. Furthermore, the evolution characteristics of the bending normal stress field are highlighted. Numerical results illustrate that the attenuation characteristics of bending stress field are determined by the coupling effects of internal flow and structural features on structural stiffness and damping.

Active vibration optimal control of piezoelectric cantilever beam with uncertainties

Considering the stiffness characteristics of piezoelectric layer, the bending stiffness of piezoelectric cantilever beam is obtained by applying the first-order shear deformation theory. The finite element model of piezoelectric cantilever beam is established by Hamilton variation principle, and the modal superposition method is employed to reduce the order of the finite element model. At the maximum strain point, the sensors/actuators are equipped in pairs. Based on the uncertain dynamic model of piezoelectric cantilever beam, the independent modal space control method based on LQR (linear quadratic regulator) control is employed for the active control of the smart beam structure, and the weighted matrices Q and R are selected according to the energy criterion. The numerical simulations and experiments verify the effectiveness of the proposed finite element model and the active vibration optimal control.

Buckling of regular and auxetic honeycombs under a general macroscopic stress state in symplectic system

A symplectic stiffness method, based on Hamiltonian variational principle and symplectic eigen expansion, is presented to investigate the buckling of honeycomb structures under a general macroscopic stress state. The analytical closed-form expressions of critical forces are obtained for the possible buckling modes, which satisfy the deformation compatibility conditions for the cell beams, and the lowest value in the stability surface corresponds to the actual buckling force for the honeycomb. The results show that the buckling strengths and modes for the regular and auxetic honeycomb structures depend strongly on the biaxial stress state but weakly on the relative density. For some complex honeycombs, e.g., the triangular and re-entrant cell, the buckling modes change with the increasement of the orders. Considerable attention is focused on the physical mechanism of phase transformation. There is no possibility of a swaying buckling mode when the beam buckles symmetrically, and the re-entrant geometry has an important influence on the auxetic honeycomb buckling patterns. The present symplectic stiffness method is believed to be beneficial for the stability analysis of some frontier engineering fields such as flexible electronics and soft robots.

An extension of the Hamilton variational principle for piezoelectric bodies with dipolar structure

Our paper deals with a mixed initial-boundary value problem in thermoelasticity of piezoelectric bodies with dipolar structure. It is considered the general case of a thermoelastic body which inhomogeneous and also anisotropic and we add the effect of piezoelectricity. We take into account the main equations and basic conditions of a temperature rate dependent theory which predict the waves propagation with a finite speed. Using some middle conditions imposed on the boundary conditions and initial data, we obtain that our mixed problem admits one solution. A generalized form of the Hamilton’s principle is obtained to cover this context.

Three-dimensional nonlinear vibration model and fatigue failure mechanism of deepwater test pipe

In deepwater test condition, the riser–test pipe (tubing string) system (RTS) is subject to the vortex-induced effect on riser, flow-induced effect on test pipe and longitudinal/transverse coupling effect, which is prone to buckling deformation, fatigue fracture and friction perforation. To resolve this, the three-dimensional (3D) nonlinear vibration model of deepwater RTS is established using the micro-finite method, energy method and Hamilton variational principle. Based on the elastic–plastic contact collision theory, the nonlinear contact load calculation method between riser and test pipe is proposed. Compared with experimental measurement results, calculation results using the proposed vibration model in this study and the single tubing vibration model in our recent work, the correctness and effectiveness of the proposed vibration model of the deepwater RTS are verified. Meanwhile, the cumulative damage theory is used to establish the fatigue life prediction method of test pipe. Based on that, the influences of outflow velocity, internal flow velocity, significant wave height, as well as top tension coefficient on the fatigue life of test pipe are systematically analyzed. The results demonstrate that, first, with the increase in outflow velocity, the maximum alternating stress and the annual fatigue damage rate increased. The location where fatigue failure of the test pipe is easy to occur at the upper “one third” and the bottom of test pipe are easy to occur fatigue failure. Second, with the increase in internal flow velocity, the “one third damage effect” of the test pipe will decrease, and the “bottom damage effect” of the test pipe increased that needs the attention of field operators. Third, during field operation, it is necessary to properly configure the top tension coefficient so that there can be a certain relaxation between the riser and the test pipe, so as to cause transverse vibration and consume some axial energy and load. The study led to a theoretical method for safety evaluation and a practical approach for effectively improving the fatigue life of deepwater test pipe.

Joint quantum–classical Hamilton variational principle in the phase space

We show that the dynamics of a closed quantum system obeys the Hamilton variational principle. Even though quantum particles lack well-defined trajectories, their evolution in the Husimi representation can be treated as a flow of multidimensional probability fluid in the phase space. By introducing the classical counterpart of the Husimi representation in a close analogy to the Koopman–von Neumann theory, one can largely unify the formulations of classical and quantum dynamics. We prove that the motions of elementary parcels of both classical and quantum Husimi fluid obey the Hamilton variational principle, and the differences between associated action functionals stem from the differences between classical and quantum pure states. The Husimi action functionals are not unique and defined up to the Skodje flux gauge fixing (Skodje et al 1989 Phys. Rev. A 40 2894). We demonstrate that the gauge choice can dramatically alter flux trajectories. Applications of the presented theory for constructing semiclassical approximations and hybrid classical–quantum theories are discussed.

Investigation on three-dimensional vibration model and response characteristics of deep-water riser-test pipe system

Under deep-water test conditions, a riser-test pipe system (RTS) is subject to the vortex-induced effect on the riser, flow-induced effect on the test pipe, and longitudinal–transverse coupling effect. Further, the system is prone to buckling deformation, fatigue fracture, and friction perforation. A three-dimensional (3D) nonlinear vibration model of deep-water RTS was established using the micro-finite method, energy method, and Hamilton variational principle. Based on the elastic–plastic contact collision theory, a nonlinear contact load calculation method between the riser and test pipe was proposed. According to field parameters in the South China Sea and the similarity principle, a vibration test bench for the RTS was designed. The experimental measurement results and the calculation results obtained from the proposed vibration model and the single tubing vibration model developed in our recent study were compared. Based on this comparison, the correctness and effectiveness of the proposed vibration model of the deep-water RTS were verified. Then, the vibration characteristics of the RTS for the BY5-2-1 well in the South China Sea were analysed. The results demonstrate that, first, in the vibration fatigue analysis of the test pipe, the influence of its own local high-frequency vibration cannot be neglected. Second, long-term collisions will cause friction and wear failure in the RTS. In the safety analysis of the RTS, the wear problem of the RTS caused by the collision between the riser and test pipe cannot be neglected. Third, the position where the test pipe is prone to strength failure is mainly located in the upper and lower parts. The field designer should focus on the safety of the pipe at these locations.

Multi-field coupling nonlinear vibration characteristics of hydraulic lifting pipe in deep-ocean mining

To address the problem of flow-induced vibration failure of lifting pipe in deep-ocean hydraulic mining, a multi-field coupling nonlinear vibration model of deep-ocean mining lifting pipe was established by using finite element method, energy method and Hamiltonian variational principle. The longitudinal-lateral coupling effect, external current vortex-induced effect and internal fluid pulsation effect were considered, and the numerical solution of the nonlinear vibration model of lifting pipe was realized with the Newmark-β method, Newton Raphson method and finite volume method. Using the same structural parameters, the correctness and effectiveness of the proposed model were verified by comparing with the simulated experimental measurement data in the literature. On this basis, the influence of the buffer station mass and wave parameter on the vibration response characteristics and nonlinear behavior of the lifting pipe were explored. The results show that the vibration frequency of the lifting pipe near the upper end is higher, and the vibration of the pipe string near the lower end is chaotic. With the increase of the buffer station mass, the in-line flow offset and root mean square stress of the lifting pipe decrease, and the axial force on its top end is greater. Under the higher mass of buffer station, the longitudinal vibration of the lifting pipe tends to quasi-periodic motion of the ring, but the vibration amplitude is larger. Small period and large amplitude waves have obvious influences on the vibration, stress and top axial force of the lifting pipe, and make its longitudinal vibration gradually transition from chaotic motion to quasi-periodic motion.

Research on dynamics theory based on the framework of energy conservation

Based on the analysis of establishing dynamic equations by using Newton mechanics, Lagrange and Hamilton mechanics, a new idea of establishing dynamic equations under the framework of energy conservation is proposed. Firstly, the use of Newton’s second law to derive wave equations of motion is introduced. Secondly, the Lagrange equation, Hamilton canonical equation, and the corresponding dynamical equations in a continuum are derived by using Hamilton’s variational principle. Thirdly, under the framework of energy conservation, the Lagrange equation, Hamilton canonical equation, and acoustic dynamics equation of continuum are established, and the results are proved to be consistent with those derived from classical mechanics. Some of fuzzy understandings when using Hamilton's variational principle to establish Lagrange’s equation and Hamilton’s canonical equation, are clarified. A series of dynamical equations established under the framework of energy conservation provides alternative way to characterize and represent the propagation characteristics of wave motions in various complex media without involving the variational principle of functional extremum. Finally, as an application example, the differential equation of dynamics in viscoelastic medium is given under the framework of energy conservation.

Coriolis Force Sliding Mode Control Method for the Rotary Motion of the Central Rigid Body-Flexible Cantilever Beam System in TBM

Beam slab structure is often encountered in a complex tunnel boring machine. Beam slab structure is subject to dynamic load, which is easy to cause fatigue damage and affect its service life. Therefore, it is necessary to control the vibration of this kind of beam slab structure. In this study, the central rigid body-flexible beam model is established for the rotating beam and plate rotating around the y-axis. Based on the Hamilton variational principle, the dynamic equation of the central rigid body-flexible beam system is established, and the dynamic model of the central rigid body-flexible beam system considering the influence of Coriolis force and centrifugal force is given. The vibration control of the central rigid body-flexible beam system is studied. The vibration mode of the rotating Euler Bernoulli beam is determined by using the elastic wave and vibration mode theory. The influence of the rotating motion on the beam vibration is analyzed, and the variable structure control law is designed to suppress the beam vibration. Numerical simulation results show that the control method can effectively suppress the first-order and second-order vibration of the beam and verify the effectiveness of the control strategy.

Investigation on Vortex Induced Vibration Characteristics of Three-dimensional Marine Riser Considering Cross and Inline Flows Coupling Effect

ABSTRACT Marine risers typically experience nonlinear vibration failure problems caused by the ocean vortex excitation force. To resolve this, the Hamilton variational principle is applied to establish a vortex-induced vibration (VIV) model for a flexible riser. A wake oscillator model is employed to simulate the vortex-induced forces of cross flow (CF) and inline flow (IL) and their coupling, considering the effect of top tension and internal flow on the riser. The VIV model is solved by combining the Newmark–β and Runge–Kutta methods and then verified by comparing the calculation results of the proposed vibration model with data in published literature. Based on the foregoing, the effects of uniform flow, top tension, and shear flow on the VIV behaviour of risers with a large aspect ratio are systematically investigated. Moreover, the VIV characteristics are identified. First, the results demonstrate that, when the top tension is 10 t, the IL vibration is 0.73 Hz and the CF vibration is 0.36 Hz, which shows that the IL vibration is twice as high as the CF vibration. Moreover, the amplitude of the latter is between 7 and 21 times that of the former, which are respectively 0.018 and 0.12 m. Second, the shear velocity and top tension also have considerable influence on the IL vibration under shear flow. Third, the CF vibration has a frequency-locking effect under shear flow and uniform flow conditions. The IL vibration exhibits a frequency-locking effect under uniform flow and a multifrequency effect under shear flow. The study led to the formulation of a theoretical method for safety evaluation and a practical approach for effectively improving the service life of marine risers.

Диссипативные свойства трехслойных композитных структур. 1. Постановка задачи

Object and purpose of research. The object under study is a sandwich plate with two rigid anisotropic layers and a filler of soft isotropic viscoelastic polymer. Each rigid layer is an anisotropic structure formed by a finite number of orthotropic viscoelastic composite plies of arbitrary orientation. The purpose is to develop a mathematical model of sandwich plate. Materials and methods. The mathematical model of sandwich plate decaying oscillations is based on Hamilton variational principle, Bolotin’s theory of multilayer structures, improved theory of the first order plates (Reissner-Mindlin theory), complex modulus model and principle of elastic-viscoelastic correspondence in the linear theory of viscoelasticity. In description of physical relations for rigid layers the effects of oscillation frequencies and ambient temperature are considered as negligible, while for the soft viscoelastic polymer layer the temperaturefrequency relation of elastic-dissipative characteristics are taken into account based on experimentally obtained generalized curves. Main results. Minimization of the Hamilton functional makes it possible to reduce the problem of decaying oscillations of anisotropic sandwich plate to the algebraic problem of complex eigenvalues. As a specific case of the general problem, the equations of decaying longitudinal and transversal oscillations are obtained for the globally orthotropic sandwich rod by neglecting deformations of middle surfaces of rigid layers in one of the sandwich plate rigid layer axes directions. Conclusions. The paper will be followed by description of a numerical method used to solve the problem of decaying oscillations of anisotropic sandwich plate, estimations of its convergence and reliability are given, as well as the results of numerical experiments are presented.

Capture and detumbling control for active debris removal by a dual-arm space robot

Active debris removal (ADR) technology is an effective approach to remediate the proliferation of space debris, which seriously threatens the operational safety of orbital spacecraft. This study aims to design a controller for a dual-arm space robot to capture tumbling debris, including capture control and detumbling control. Typical space debris is considered as a non-cooperative target, which has no specific capture points and unknown dynamic parameters. Compliant clamping control and the adaptive backstepping-based prescribed trajectory tracking control (PTTC) method are proposed in this paper. First, the differential geometry theory is utilized to establish the constraint equations, the dynamic model of the chaser-target system is obtained by applying the Hamilton variational principle, and the compliance clamping controller is further designed to capture the non-cooperative target without contact force feedback. Next, in the post-capture phase, an adaptive backstepping-based PTTC is proposed to detumble the combined spacecraft in the presence of model uncertainties. Finally, numerical simulations are carried out to validate the feasibility of the proposed capture and detumbling control method. Simulation results indicate that the target detumbling achieved by the PTTC method can reduce propellant consumption by up to 24.11%.

Analysis of Dynamic Characteristics of Tristable Piezoelectric Energy Harvester Based on the Modified Model

Taking into account the gravitational potential energy of the piezoelectric energy harvester, the size effect, and the rotary inertia of tip magnet, a more accurate distributed parametric electromechanical coupling equation of tristable cantilever piezoelectric energy harvester is established by using the generalized Hamilton variational principle. The effects of magnet spacing, the mass of tip magnet, the thickness ratio of piezoelectric layer and substrate, and the load resistance and piezoelectric material on the performance of piezoelectric energy capture system are studied by using multiscale method. The results show that the potential well depth can be changed by reasonably adjusting the magnet spacing, so as to improve the energy capture efficiency of the system. Increasing the mass of tip magnet can enhance the output power and frequency bandwidth of the interwell motion. When the thickness of the piezoelectric beam remains unchanged, the optimal load impedance of the system increases along with the increase of thickness ratio of piezoelectric layer and substrate. Compared with the traditional model, which neglects the system gravitational potential energy, the eccentricity, and the rotary inertia of the tip magnet, the calculation results of the frequency bandwidth and the peak power of the modified model have significantly increased.

Three-dimensional thermodynamic behaviors for a FML structure subjected to 3-D moving dual-ellipse distribution laser heat source

In present work, the three-dimensional transient heat transfer equation for a fiber metal laminated (FML) plate is established, and temperature field under dual-ellipse distribution laser processing is obtained through the Newton-Cotes method. Then the three-dimensional constitutive equations are derived, and the transient thermodynamic governing equations are obtained based on the Hamilton variational principle subsequently. Finally, the finite difference method and Newmark method are utilized to solve the thermodynamic equations in both space and time domains, respectively. The research aims at studying three-dimensional thermodynamic behaviors for a FML structure subjected to dual-ellipse distribution laser heat source and giving the effects of Fourier series term, finite difference point, aspect ratio, fiber arrangement angle and laser moving speed on the thermodynamic behaviors for the FML structure.

Wave–current interaction on a free surface

AbstractThe classical water wave equations (CWWEs) comprise two boundary conditions for the two‐dimensional flow on the free surface of a bulk three‐dimensional (3D) incompressible potential flow in the volume bounded by the free surface, which itself moves under the restoring force of gravity. One of these two boundary conditions provides the kinematic definition of the vertical velocity of the surface elevation. The other boundary condition is the dynamic Bernoulli law that governs the evaluation of the bulk velocity potential on the free surface. The present paper applies these two boundary conditions as constraints in the action integral for Hamilton's variational principle, along with a non‐hydrostatic pressure constraint that imposes incompressible flow on the free surface. The stationary variations in Hamilton's principle then yield closed dynamical equations of free surface flow whose divergence‐free velocity admits nonzero vorticity and whose nonhydrostatic pressure matches the pressure of the 3D bulk flow when evaluated on the free surface. A minimal coupling approach is proposed to model the mutual interactions of the waves and currents. The dynamical effects of horizontal buoyancy gradients are also considered in this context. For any combination of these model variables, the resulting system of variational equations admits a Lie–Poisson Hamiltonian formulation. Finally, stochastic versions of these model equations are derived by assuming that the material loop for their Kelvin circulation theorem in each case follows stochastic Lagrangian histories in a Stratonovich sense.

Vibration and Snapthrough of Fluid-Conveying Graphene-Reinforced Composite Pipes Under Low-Velocity Impact

There exists complex fluid–structure coupling vibration in the fluid-conveying pipes under the combined actions of internal flow and external loads in the aircraft. The impact dynamic behaviors of the flow-inducing pre- and postbuckled pipes are studied with the surfaces made of functionally graded graphene-reinforced composites and an interlayer made of isotropic viscoelastic materials. It is assumed that the fluid is nonrotating, nonviscous, incompressible; and steady; and the pipe is simply supported at both ends. Under the framework of refined displacement fields, a reduced constitutive relation is developed. On this basis, the nonlinear governing equations are derived through the Hamilton variational principle. The two-step perturbation–Galerkin integral is extended to obtain the equilibrium paths as the initial configurations of the pipes for impact studies. The fourth-order Runge–Kutta method is employed to numerically obtain the contact force and the pipe vibration responses. The effects of flow velocity, impact velocity, structural material, and geometric factors on the dynamic responses are discussed. The numerical results reveal that even the parameters which have little influence on the contact force may have a great effect on the vibration response of the fluid-conveying pipe.

Nonlinear flow-induced vibration response characteristics of leaching tubing in salt cavern underground gas storage

To address flow-induced vibration failure problems of leaching tubing in salt cavern underground gas storage, a nonlinear flow-induced vibration model of the leaching tubing in salt cavern gas storage was established using the energy method, element method, and Hamilton variational principle. In the model, nonlinear factors such as the longitudinal and lateral coupling effect and contact/collision between the center tube and middle tube (tube in tube) were considered. Based on the on-site M example well parameters in Jianghan Oilfield, an experiment to simulate leaching tubing vibration was designed using the similarity principle and carried out to verify the validity and correctness of the proposed model. Thus, the effects of the water injection flow rate, two-port distance, and tubing diameter on the vibration response characteristics of the leaching tubing were analyzed systematically. The vibration failure mechanism of the leaching tubing was revealed, and vibration control methods of the leaching tubing were proposed. First, with the increase in water injection flow rate, the lateral displacement and bending stress of the lower end of the leaching tubing increase, and when the injection flow rate was 50–70 m3/h, the vibration intensity of string increased significantly; therefore, the on-site water injection flow rate should be avoided using this interval operation. Second, with the increase in the two-port distance, the vibration response of the center tube increased; when the two-port distance was greater than 40 m, the vibration amplitude and energy of the tubing string increased significantly, and thus the two-port distance should be less than 40 m to the greatest extent possible in on-site production. Third, when the diameter of the leaching tubing decreased, the vibration amplitude and frequency increased, and the corresponding probability of bending deformation and fatigue failure increased significantly; therefore, provided that the other conditions are met, selecting a large tubing string can effectively improve the safety of the leaching tubing string.

Design and modeling a frequency self-tuning vibration energy harvester for rotational applications

Vibration energy harvester (VEH) for rotational applications has attracted increasing attentions in the last decade, and one of the major challenges is still the mismatch between the resonant frequency of the VEH and the rotational frequency. To solve this issue, the frequency matching mechanism of the VEH applied to rotating environment is studied and a self-tuning piezoelectric VEH is proposed. The main advantage of the proposed VEH is that the effective beam length is changed by centrifugal force while the stiffness of the beam is changed by rigid-flexible coupling effect, which is the key to realize frequency matching in a wide frequency range. Then an accurate rigid-flexible coupling model for the proposed VEH is established with Hamiltonian variational principle, and a finite element model is built for numerical analysis. Finally, the established model is used to study the proposed VEH. Numerical analysis results show that the proposed VEH can achieve frequency matching within the frequency range of 8 ∼28 Hz, and the power output is 0.42 ∼0.74 mW. Numerical analysis results also show that the frequency tuning mass introduced as an independent parameter in VEH is of great significance to improve the frequency matching range, and also makes the structure design flexible.