Hamilton's Variational Principle Research Articles

Parametric-forced coupling resonance of core-shell nanowires with interfacial damage under weak viscoelastic boundary constraint

Revealing the synergistic effects of size-dependency and nonlinearity on dynamics of core-shell nanowires plays an important role in nonlinear design of nanoelectromechanical system (NEMS) maintaining stable working state. The nonlinear principal resonance of core-shell nanowires with weak interface under parametric-forced coupling excitations is studied. The nanowire is simply supported at both ends, with one end movable and under a weak viscoelastic boundary described by Kelvin-Voight model. The weak interface obeys the interfacial cohesive law coupled into the refined displacement field. Geometric nonlinearity due to large deformation is reflected in three aspects. Stiffness nonlinearity is modeled by nonlinear curvature. Longitudinal inertia nonlinearity is modeled based on an in-extensive beam assumption. Nonlinear constraint force provided by the viscoelastic boundary is reflected in nonlinear damping and stiffness. Size-dependency including nonlocal stress, strain gradient and surface elasticity are all incorporated. The governing equations are derived by Hamilton's variational principle and solved via a perturbation-incremental harmonic balance method (IHBM). Results provide the bifurcation diagram of coupled resonance and the stability boundary of parametric resonance for analyzing the change mechanism of bifurcation topology. Decisive ways are revealed that weak interface, size dependency and constraints from foundation and boundary affect the linear and nonlinear properties of the system.

Campbell diagrams, dynamics and instability zones of graphene-based spinning shafts

Carbon-based nanocomposite drive shafts have received considerable attention because of their excellent material and mechanical properties, with applications in modern aerospace, automotive industries and others. Therefore, for the first time, new Campbell diagrams, vibration and bifurcation points for multilayer shafts reinforced with three technologically justified distribution patterns of graphene nanoplatelets (GPLs) in the polymer matrix are presented and examined. A new formulation in the framework of a modified Timoshenko beam theory making use of the rotation angles of the cross sections of the shaft with five degrees of freedom is utilized. The modulus of elasticity of the nanocomposite material is estimated using the Halpin-Tsai micromechanical model, whereas the mass density and Poisson's ratio are computed by means of the law of mixtures. Hamilton's variational principle is used to derive weak and strong forms of motion equations of the shaft with classical and mixed boundary conditions. Assumptions for the derived model of the nanocomposite shaft are presented to clearly describe the mechanical state of the system. The Ritz-Chebyshev method is applied to discretize the governing equations and obtain the generalized nonlinear characteristic equations in matrix form including the gyroscopic effect and elastic stiffness caused by the rotation of the shaft. Verification of obtained results and convergence study were performed to show the correctness and advantages of the proposed solution method. The effect of different key parameters such as GPLs distribution and weight percentage, axial speed of the shaft, and boundary conditions, on its global damping, dynamics and instability zones are shown. For the first time, it is clearly shown how studied factors affect the shape of Campbell diagrams as well as the shift of natural frequency and bifurcation point of the graphene-based nanocomposite shafts.

Dynamic behaviors of hierarchical-tethered towing system for space debris removal

The debris-tether-tug (DTT) system, has been considered as one of the most promising techniques for the space debris removal. In order to decrease the control difficulty of the DTT system, the DTT system was usually designed with a single tether. However, when the target owns the complex attitude motion, the removal efficiency of the classic DTT system (towing the target by a single tether) is limited. A modified DTT model with the hierarchical-tethered architecture is proposed to improve the removal efficiency in this paper. Based on the Hamiltonian variational principle, the coupling non-smooth dynamic model for the hierarchical-tethered towing system is established, in which, four tethers are divided into two groups to describe the asymmetry of the DTT model well. The symplectic Runge-Kutta method is employed to simulate the evolution of the orbit radius of the target, the length of the tethers, the position of the tethers’ mass center and the target attitude angle. The numerical results reported show that, the competition relationship between the attitude stability of the target and the stability of the variation of the sub-tethers is affected by the relative stiffness of the tethers. In addition, the capture ability of the modified DTT system for the space target with a certain spinning speed is illustrated, which provides a theoretical basis for the design of the DTT system in future.

Magnetic-Solid Coupling Nonlinear Vibration of an Axially Moving Thin Plate Under Air-Gap Magnetic Field

The nonlinear inherent vibration of an axially moving ferromagnetic thin plate under the action of the armature air-gap magnetic field is investigated. Based on the nonlinear elasticity theory, the energy relationship equations of the thin plate are given. Based on the electromagnetic theory and the solution of Laplace’s magnetic potential equation, magnetic force on the ferromagnetic rectangular plate under air-gap magnetic field environment is deduced. The Hamilton variational principle is applied to derive the magneto-solid coupled nonlinear vibration equation of the axially moving ferromagnetic thin plate. The two-degree-of-freedom nonlinear vibration differential equations containing static load terms with boundary conditions of SSSS and SSCC are obtained by discretizing through the Galerkin method. The multi-scale method is applied to solve the second-order approximation for deriving the first two orders’ intrinsic frequency of the nonlinear system. The variation laws of first two orders’ nonlinear inherent vibration with axial velocity, magnetic potential, initial air-gap thickness, and initial value are given through numerical examples, and the comparative analysis is performed. The results show that the first-order and second-order inherent vibration frequencies decrease with the increase of axial velocity and magnetic potential and increase with the increase of initial air-gap magnetic field thickness. Different materials and different boundary conditions have greater influence on the first-order and second-order inherent frequencies, and show obvious nonlinear characteristics. The results can provide references for analyzing and controlling the vibration behavior of moving structures in electromagnetic environment.

Torsional vibration of a flexoelectric nanotube with micro-inertia effect

The modified strain gradient theory (MSGT) with flexoelectric and micro-inertia effects is developed to study the free vibration of tube made of elastic centrosymmetric materials. Within this theory, the internal energy density is the function of the strain tensor, deviatoric part of stretch gradient and rotation gradient tensors, dilatation gradient, polarization vector, and electric quadrupoles. The linear coupled governing equations and boundary conditions are derived from the Hamilton variation principle. The torsional vibration problem is analytically solved for non-piezoelectric nanotube with fixed edges. The effect of the nanotube geometry, flexoelectric coefficient, micro-inertia, and micro-stiffness parameters on the natural frequency is studied.

Wave propagation in piezoelectric doubly-curved panels considering thermal effects: Piezoelectricity-based synergistic effect analysi

Doubly curved panels are widely applied in the spacecraft structural system. In the impact environment, the damage phenomenon may appear in doubly curved panels. The studies of wave propagation of piezoelectric structures effectively promote nondestructive testing technology intelligently. The wave propagation of piezoelectric doubly curved panels assembled on the visco-Pasternak foundation is investigated for the first time. Under the plane bending assumption, the reduced constitutive equations considering piezoelectric effects and thermal effects are established for the FGM core and piezoelectric layers. Within the framework of first-order shear deformation theory, the displacement fields and strain fields of spherical panels and hyperbolic parabolic panels are modeled respectively. The dynamic governing equations are derived through Hamilton's variational principle. By extending the harmonic method to the construction of displacement fields and potential fields, the dispersion relations of doubly curved panels are obtained analytically. Subsequently, the multi-order phase velocities of the spherical panels and hyperbolic parabolic panels are solved. Ultimately, the synergistic effects of piezoelectricity under different temperatures, material properties, and geometric characteristics on the first three order phase velocities are systematically discussed. In the first-order phase velocity curves, the contribution of the piezoelectric effect interacts with that of material parameters, geometric parameters, viscoelastic foundation and temperature variation, respectively. From the perspective of energy transfer, the synergistic effect mechanism induced by the piezoelectric effect is revealed. Moreover, the attenuation of flexural wave induced by the higher temperature and viscous coefficient results in the cutoff wave number of the hyperbolic parabolic panel.

Research on elastodynamic theory based on the framework of energy conservation

Based on the analysis of establishing dynamic equations by using Newton's mechanics, Lagrange's, and Hamilton's mechanics, a new idea of establishing elastodynamic equations under the framework of energy conservation is proposed. Firstly, Newton’s second law is used to derive wave equations of motion. Secondly, Lagrange's equation, Hamilton's canonical equations, and the corresponding dynamical equations in a continuum medium are derived by using Hamilton’s variational principle. Thirdly, under the framework of energy conservation, Lagrange's equation, Hamilton's canonical equations, and the acoustic dynamic equations of the continuum are established, and the results are proved to be consistent with those derived from classical mechanics. Some 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 an 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 elastodynamics in a viscoelastic medium is given under the framework of energy conservation.

Nonlinear Frequency Analysis of FGM Pipes Based on the Homotopy Method

Based on the homotopy analysis method, this paper studies the nonlinear vibration of FGM pipeline with pores under generalized boundary conditions. Based on the power-law distribution of FGM, the physical properties of the material are distributed along the radial direction of the pipe. For the mathematical model of the pores of the pipe, we choose Voigt model to describe the material properties of FGM pipes with pores. Based on Euler Bernoulli beam theory and the influence of von-Kármán nonlinearity, using Hamilton variational principle, the dynamic control equations and generalized boundary conditions of functionally graded fluid transmission pipelines with pores are established. The homotopy analysis method is used to solve the nonlinear vibration characteristics of functionally graded flow pipelines under generalized boundary conditions. The numerical results show that the translation spring has little effect on the critical velocity of instability, while the rotation spring can increase the critical velocity of instability, making the system more stable;In the nonlinear system, the viscoelastic coefficient does not change the critical velocity; pipe length, power-law exponent and porosity all affect the nonlinear free vibration of FGM porous flow pipeline.

Dynamic analysis of a laterally loaded rectangular pile in multilayered viscoelastic soil

Rectangular piles are usually used as the foundation for many infrastructures, but there are fewer theoretical solutions for the lateral vibration. This study presents a semi-analytical methodology for the dynamic analysis of laterally loaded rectangular pile in multi-layered viscoelastic soils. The pile-soil system is modeled as a continuum, the horizontal continuous displacement model is proposed as products of the separating variables function. The coupled governing equations for rectangular pile-soil system in the frame of Euler-Bernoulli beam and modified Vlasov model are derived with the aid of Hamilton's Variation principle and calculus of variations. The physical mechanisms of soil reaction and dynamic equilibrium equation are interpreted by employing thin-layer element of soil. The dynamic responses are solved analytically and numerically by an iterative procedure. The verification and accuracy of the research are obtained thorough comparison with the relevant examples from the literatures and finite element analysis. The effects of cross-section shapes and soil stratification on the complex dynamic response are investigated and discussed in detail.

Micro-structural effects in phononic dielectric structures

Based on the higher-grade continuum theory, propagation of longitudinal as well as transverse anti-plane elastic waves in normal direction to nanoscale periodic laminates of piezoelectric dielectrics is studied in this paper. The strain gradients, micro-inertia and direct flexoelectricity phenomena are incorporated into the phenomenological description. The problem is analysed as one-dimensional and the governing equations together with possible boundary conditions are derived from the Hamilton variation principle. It is shown that the transverse waves are not affected by the electric polarization in contrast to the longitudinal waves. The developed formulation is applied to 1D Bloch waves obeying perfect boundary conditions periodical in bi-material laminates and the transfer matrix method is used for derivation of dispersion equation. An analytical method is developed for solution of the dispersion equation. The influence of the micro-stiffness and micro-inertial length scale parameters as well as flexoelectric coefficients on the dispersion curves and the frequency gaps is investigated in parametric study.

Coupled cross-flow and in-line vibration characteristics of frequency-locking of marine composite riser subjected to gas-liquid multiphase internal flow

In this paper, the frequency locking characteristics of marine composite riser coupled with cross-flow (CF) and in-line (IL) under the combined action of internal and external ocean currents in gas-liquid multiphase transportation are studied. Based on Hamilton's variational principle, the coupled vibration control equations of composite marine riser in CF and IL directions were established considering the combined action of internal and external ocean currents in gas-liquid multiphase transport. The Van der Pol wake oscillator model was used to simulate the flow force of the ocean current on the CF and IL directions of the slender riser. Newmark-β and fourth-order Runge-Kutta coupling iterative method were used to solve the coupled dynamics equation. The effectiveness and accuracy of the proposed method were verified by comparing with the results in literature. Subsequently, the effects of liquid phase velocity, fiber orientation angle and top axial tension on the frequency-locking characteristics of marine riser were given. The research results show that the internal flow velocity, external ocean current velocity, fiber orientation angle and top axial tension have a great influence on the coupled vibration stability and frequency locking characteristics of the riser. In addition, compared with the uniform single internal flow, the critical value of the velocity of the gas-liquid multiphase mixed internal flow is larger; The frequency-locking interval of the riser “shifts to the right” with the increase of liquid flow velocity and fiber orientation angle, and “shifts to the left” with the increase of top axial tension. It is also found that the frequency locking range of the riser decreases with the increase of fiber orientation angle and top axial tension, and expands with the increase of liquid flow velocity.

Parametric Analysis of Nonlinear Bi-Stable Piezoelectric Energy Harvester Based on Multi-Scale Method

Given the geometric nonlinearity of the piezoelectric cantilever beam, this study establishes a distributed parameter model of the nonlinear bi-stable cantilever piezoelectric energy harvester, following the generalized Hamilton variational principle. The analytical expressions of the dynamic response were obtained for the energy harvesting system using Galerkin modal decomposition and the multi-scale method. The investigation focuses on how the performance of the energy harvesting system is influenced by the gap distance between magnets, external excited amplitude, mechanical damping ratio and external load resistance. The calculation results were compared with those obtained neglecting the geometric nonlinearity of the beam. The results show that the system responses contain jump and multiple solutions. The consideration of the geometrical nonlinearity significantly amplified the peak displacement and peak output power of the intra-well and inter-well motions. There is an evident hardening effect of the inter-well motion frequency response curve. By reasonable adjusting the parameters, it is possible to improve the output power of the piezoelectric energy harvesting system and broaden the operating frequency of the system.

Hamiltonian Variational Formulation of Three-Dimensional, Rotational Free-Surface Flows, with a Moving Seabed, in the Eulerian Description

Hamiltonian variational principles have provided, since the 1960s, the means of developing very successful wave theories for nonlinear free-surface flows, under the assumption of irrotationality. This success, in conjunction with the recognition that almost all flows in the sea are not irrotational, raises the question of extending Hamilton’s principle to rotational free-surface flows. The Euler equations governing the bulk fluid motion have been derived by means of Hamilton’s principle since the late 1950s. Nevertheless, a complete variational formulation of the rotational water-wave problem, including the derivation of the free-surface boundary conditions, seems to be lacking until now. The purpose of the present work is to construct such a missing variational formulation. The appropriate functional is the usual Hamilton’s action, constrained by the conservation of mass and the conservation of fluid parcels’ identity. The differential equations governing the bulk fluid motion are derived as usually, applying standard methods of the calculus of variations. However, the standard methodology does not provide enough structure to obtain the free-surface boundary conditions. To overcome this difficulty, differential-variational forms of the aforementioned constraints are introduced and applied to the boundary variations of the Eulerian fields. Under this transformation, both kinematic and dynamic free-surface conditions are naturally derived, ensuring the Hamiltonian variational formulation of the complete problem. An interesting feature, appearing in the present variational derivation, is a dual possibility concerning the tangential velocity on the boundary; it may be either the same as in irrotational flow (no condition) or zero, corresponding to the small-viscosity limit. The deeper meaning and the significance of these findings seem to deserve further analysis.

Size-dependent nonlinear bending analysis of nonlocal magneto-electro-elastic laminated nanobeams resting on elastic foundation

Employing Reddy’s third-order shear deformation theory (RTSDT) and nonlocal elasticity theory, a nonlinear bending model of the nonlocal three-layer magneto-electro-elastic (MEE) laminated nanobeam resting on elastic foundation is established by considering von Karman’s geometrically nonlinear equations. Nonlinear higher order partial differential governing equations of MEE laminated nanobeams can be obtained employing Hamilton variational principle. The three-layered MEE laminated nanobeam is considered to have simply supported boundary condition in the present paper. Meanwhile, the electric and magnetic potential distributions in the laminated nanobeam are determined through Maxwell’s magnetic-electro equations and boundary conditions. The governing equations of laminated nanobeams are re-expressed in the dimensionless form by introducing the non-dimensional terms. Employing Galerkin method, the nonlinear higher order partial differential governing equations are simplified into lower order equations. Several cases are explored to indicate the effects of foundation parameters, nonlocal parameter, stacking sequence, external electric voltage and external magnetic potential on bending behaviors of MEE laminated nanobeams.

Nonlinear dynamics modeling and analysis of a marine buoy single-point mooring system

We investigate the oscillation of a marine buoy in a single-point mooring system under the action of the marine environmental load and the work done by the mooring system gravity. In particular, a nonlinear dynamic model of the single-point marine buoy mooring system was established based on the Hamilton variational principle. The fourth-order Runge-Kutta integral method was employed to numerically solve and analyze the proposed nonlinear dynamics model. Results demonstrate that under the action of external excitation, the longitudinal displacement, velocity and acceleration of the cables within the marine buoy single-point mooring system change significantly and periodically with time. The frequency change of the three variables in the lateral is rapid yet the fluctuation range is small. This indicates the longitudinal motion pattern of the cables within the marine buoy single-point mooring system to be relatively stable under the influence of external excitation, while the lateral motion pattern of the cables is generally unstable. This work can provide important reference for future research on the safety and reliability of the entire buoy system.

Vibration analysis of piezoelectric Kirchhoff–Love shells based on Catmull–Clark subdivision surfaces

AbstractAn isogeometric Galerkin approach for analysing the free vibrations of piezoelectric shells is presented. The shell kinematics is specialized to infinitesimal deformations and follow the Kirchhoff–Love hypothesis. Both the geometry and physical fields are discretized using Catmull–Clark subdivision bases. This provides the required ‐continuous discretization for the Kirchhoff–Love theory. The crystalline structure of piezoelectric materials is described using an anisotropic constitutive relation. Hamilton's variational principle is applied to the dynamic analysis to derive the weak form of the governing equations. The coupled eigenvalue problem is formulated by considering the problem of harmonic vibration in the absence of external load. The formulation for the purely elastic case is verified using a spherical thin shell benchmark. Thereafter, the piezoelectric shell formulation is verified using a one dimensional piezoelectric beam. The piezoelectric effect and vibration modes of a transverse isotropic curved plate are analyzed and evaluated for the Scordelis–Lo roof problem. Finally, the eigenvalue analysis of a CAD model of a piezoelectric speaker shell structure showcases the ability of the proposed method to handle complex geometries.

Generalized nonconservative gravitational field equations from Herglotz action principle

We formulate a nonconservative gravitational theory based on the Herglotz variational principle in a tensorial covariant form. The model presented here may be seen as an improvement of the theory proposed in [ et al., Phys. Rev. D 95, 101501 (2017)], whose resulting field equations are meaningful just in particular coordinate systems. The new theory we report in this work is free from such a restriction. In comparison to the standard general relativity, both theories based on the Herglotz principle contain an extra vector field, the Herglotz field, but the new theory is formulated by taking advantage of the restricted equivalence between Lagrangian functions in the scope of the Herglotz action principle. The more restricted class of equivalent Lagrangian functions, in comparison with the Hamilton variational principle, is the key point to finding a Lagrangian that generates a new alternative gravitational theory in a covariant form. Once the equations that govern the dynamics of the gravitational field are obtained, a few simple cosmological models are investigated. It is found that the Herglotz vector field reduces to a single function that, under certain conditions, plays the role of the cosmological constant in general relativity, turning unnecessary the introduction of dark energy to explain the accelerated expansion of the Universe. The linearized version of the theory is also investigated and it is verified that the theory shows a dissipative character regarding the propagation of gravitational perturbations. From observational data, in both scenarios, the magnitude of the Herglotz field is estimated.

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.