Hamilton's Variational Principle Research Articles

Stability and nonlinear vibration characteristics of cantilevered fluid-conveying pipe with nonlinear energy sink

Considering the midline non-expansion theory and nonlinear coordinate method, this paper mainly investigates the stability and nonlinear vibration characteristics of cantilevered fluid-conveying pipe with energy sink. Based on Hamilton 's variational principle, the dynamic equation of a cantilevered pipe conveying fluid with nonlinear energy sink (NES) under pulsating flow is established. The system equation is processed by Galerkin method, a novel matrix equations of single-degree-of-freedom and multi-degree-of-freedom coupling system is established. The results show that the physical factors in the flow pipe system have a great influence on the critical flow velocity of the whole system, comparing with the case of adding NES. Increasing the proportion of the mass of the fluid will have a greater impact on the real part frequency, thereby increasing the critical velocity of the fluid-conveying pipe system. Addition of nonlinear energy sink can significantly suppress the amplitude and decrease the unstable vibration of the pipe.

Combination resonance of a moving ferromagnetic thin plate under double alternating line loads in a transverse constant magnetic field

The combination resonance of a moving rectangular ferromagnetic thin plate under double alternating line loads in a transverse constant magnetic field is investigated. Initially, the kinetic and potential energies of the system, considering geometrical nonlinearity, are obtained based on Kirchhoff's theory of thin plates. Additionally, the works of external forces are determined using the principle of virtual work. Subsequently, taking into account the nonlinear magnetization phenomenon of ferromagnetic materials, the magnetization force and Lorentz force acting on the moving ferromagnetic plate are calculated according to electromagnetic theory. Finally, the nonlinear magnetoelastic vibration equation is established by applying the Hamiltonian variational principle. The algebraic equation for static deflection and the differential equation for disturbed deflection are derived by applying the Galerkin method to the vibration equation. Then, an analytical solution for the frequency response equation under combination resonance is obtained using the multiple scale method. The proposed model and analytical research methods are validated by comparison with existing literature and numerical methods. Furthermore, the stability of steady-state solutions is determined, and static bifurcation is analyzed through singularity analysis. With the analytical results, numerical examples are plotted with varying frequency tuning values, magnetic field intensities, load amplitudes and load positions. The effects of these parameters on vibration characteristics are examined by comparing amplitude curves of systems with different physical parameters. The results demonstrate significant nonlinear vibration characteristics of the thin plate under the influence of a constant magnetic field and motion effects.

Dynamic response of sandwich beam with porous core excited by two harmonic loads travelling in opposite directions

This paper presents the time history response of a sandwich beam with a porous core subjected to two moving harmonic loads with opposite directions. In the modelling, the beam is combined of two isotropic face sheets and a porous core with symmetric porosity distribution. The quasi-3D shear deformation beam theory in conjunction with Hamilton's variational principle is utilized to set up the governing equations of motion. The Navier solution is used to obtain the displacement field. The accuracy of the study is validated by comparing with existing results in the literature for specific cases. Effects of the velocity and excitation frequency of the moving loads on the deflection-time history are investigated and discussed. Numerical results reveal that in the case of the double-moving harmonic loads in the antiphase, the resonance phenomenon cannot occur when the excitation frequency approaches the natural frequency of the beam.

A new finite element formulation for the dynamic analysis of beams under moving loads

Early studies highlighted the fact that moving loads on beams may induce significantly higher dynamic deflections and stresses than those observed in the quasi-static case. Since then, the dynamics of beams under moving loads has been an active research area, in particular for the past decades, with the rapidly increasing computer power allowing the development of highly robust and sophisticated computational and numerical methods in applied mechanics and engineering. This work introduces a novel finite element formulation for the dynamic analysis of Euler-Bernoulli beams subjected to moving loads. The formulation is consistent with a complementary form of the well known Hamilton's variational principle, and will be used to address some numerical tests in both modal and time domains. The effectiveness and accuracy of the formulation will be assessed and discussed by comparison between the obtained results and those rendered by the standard displacement-based finite element formulation. As it will be demonstrated, the proposed formulation not only renders continuous bending-moment and shear-force distributions, a desired feature in the structural design field, but also has a superior accuracy than that provided by the displacement formulation that uses the same number of nodal degrees-of-freedom.

Analysis of the vortex-induced vibration response characteristics of riser under the soft suspension evacuation condition

This paper establishes a nonlinear dynamic model for a suspended riser under the condition of considering realistic configurations, based on the principles of virtual work and Hamilton's variational principle. The nonlinearity of the model is manifested by considering vortex-induced vibration in the cross flow (CF) and in-line (IL) directions, as well as large deformations, taking into account the variable diameter of the riser in the compliant riser system. The model is discretized using the finite element method and solved using the Newmark-β method. Experimental validation is conducted to verify the effectiveness of the model. Utilizing the parameters of a riser in a well in the South China Sea, the paper explores the influence of different environmental and structural parameters on the VIV response characteristics of a compliant riser. Results show that the vibration frequency is controlled by both the bare tube and buoyant block when the buoyant block coverage rate is 25%, which can significantly inhibit VIV. With the increase of flow velocity, the VIV energy jumps from low order to high order, and the riser mode also transitions from low order to high order. The increase in suspension length transfers the vibration energy from high to low frequency, thus realizing energy transition and mode shape transformation. With the increase in LMRP weight, the vibration and natural frequencies of the riser increase synchronously, the vibration amplitude decreases, and the mode shape remains unchanged.

Dynamic analysis on an asymmetric spatial dumbbell-type model

The structural symmetric breaking of the spatial structure will enhance the coupling dynamic behaviors and bring new challenges for the dynamic analysis on the spatial structure inevitably. The main contribution of this paper is proposing a structure-preserving iteration method to investigate the effects of the dynamic symmetry-breaking factors on the dynamic behaviors of the rigid-flexible coupling systems. Firstly, the spatial flexible damping beam assembled with two particles on both ends is simplified as an asymmetric spatial dumbbell-type model. For this model, the coupling dynamic equations are presented based on the Hamiltonian variational principle. Then, connecting the symplectic precise integration method for the ordinary differential equations mainly controlling the plane motion of the model and the generalized multi-symplectic scheme for the partial differential equation mainly controlling the transverse vibration of the beam, a structure-preserving numerical iteration method is proposed, which provides a new way to analysis the dynamic behaviors of the coupling problems with the symmetric breaking. Finally, the effects of the absolute mass and the mass ratio of the additional particles on the orbit radius, the orbit true anomaly velocity, the attitude angle velocity of the model and the transverse vibration of the flexible beam are reproduced by using the structure-preserving numerical iteration method in the numerical simulations. Particularly, the effects of the additional particles on the attitude stability of the model and the vibration dissipation of the flexible damping beam are investigated, which gives some guidance for the attitude adjustment strategy and the vibration control method for the spatial flexible structure with the structural symmetric breaking.

Time-delay feedback control of a suspended cable driven by subharmonic and superharmonic resonance

The longitudinal time-delay feedback control strategy is implemented to suppress the subharmonic and superharmonic responses of the suspended cable. Formulated based on the Hamilton variational principle and the longitudinal time-delay feedback strategy, the in-plane nonlinear differential equations for a suspended cable are formulated, and Galerkin method is utilized to transform the equations into delayed differential equations. By using the method of multiple scales, approximate solutions for the 1/2-subharmonic and second-order superharmonic resonance responses of the controlled suspended cable were derived. The effects of the time delay and control gain are obtained through numerical examples. The research results demonstrate that through the adjustment of control gain and time delay, resonance regions can be avoided, effectively suppressing large-amplitude vibrations in the suspended cable, thereby achieving favorable control performance.

Three-dimensional active-control hygrothermal dynamic behavior of PFMLs structure subjected to laser heat treatment with dual-ellipse distribution form

Based on three-dimensional transient thermal conduction theory, the refined plate theory and Hamilton variational principle, the constitutive equations, geometrical equations, and governing equations for fiber metal laminates with piezoelectric layer (PFMLs) under hygrothermal environment is presented. Subsequently, the Newmark method and Galerkin method are used to solve the governing equations which in terms of displacement functions. The results show that electric potential function, fiber orientation, external load as well as thickness ratio have great influence on the deflection as well as normal stress. The selection of (45°/0°) fiber orientation is more resistance to deformation and normal stress.

Electro‐mechanical coupling analysis of three‐dimensional embedded braided composite piezoelectric vibration energy harvester

AbstractIn this article, the piezoelectric patches are embedded into three‐dimensional (3‐D) braided composite material (BCM), and a 3‐D embedded braided composite piezoelectric vibration energy harvester (BCPEH) is designed. The 3‐D embedded BCPEH can be applied in more complex working environments to protecting the piezoelectric patches, and has a longer lifespan. The material parameters of the 3‐D BCM are obtained by analyzing the representative volume units (RVU). Using Hamilton's variational principle, the vibration control equation, and the output voltage and power expression are derived. The influence of excitation frequency, load resistance, and external excitation on the output voltage and power are simulated under different braided angles and embedded depths.

On the nonlinear dynamics of a multi-scale flexoelectric cylindrical shell

The flexoelectric property shows the relationship between the strain gradient and the electric polarization and has a significant effect on static and dynamic responses of structures at small scales such as micro and nano. Nowadays, experimental studies for intelligent/composite ultra-small mechanical systems are complex, challenging, and in most cases still impossible, especially for nonlinear dynamic behavior analysis. For the first time, this study presents an advanced and complex generalized model for the nonlinear vibrations of a piezo-flexoelectric mechanical system based on a cylindrical shell. It should be noted, that the formulated boundary value problem is generalized, well-posed, and can be applied to the system at micro and nano scales. Hamilton's variational principle as well as assumptions of the first-order shear deformation theory (FSDT) and reformulated flexoelectric theory were used to derive equations of motion of the multi-scale intelligent shell system. To introduce nonlinear effects, von-Kármán strains are applied. Appropriate classical and non-classical mechanical/electric boundary conditions as well as higher-order forces and moments are determined to fully close the formulated problem. The Galerkin method combined with the GDQM, supported by the displacement control strategy and the weighted residual method, have been used to discretize and solve the nonlinear governing equations of the shell with different boundary conditions. Good convergence between the current results and the results from previous studies for simpler cases of the shell was proved. Furthermore, the results showed that geometric ratios of the structure, gradation of material properties, thickness of the flexoelectric layers, and the size effect parameter significantly influence the nonlinear frequency of the structure under the direct flexoelectric effect.

Symplectic stiffness method for the buckling analysis of hierarchical and chiral cellular honeycomb structures

Structure-function-material integration design based on the buckling instability of flexible cellular structures has attracted much attention. According to the reversible and repeatable characteristics of buckling deformation, phonon bandgap switches, the autonomous drive of soft robots, programmable logic controllers, and reusable spacecraft landing buffer devices can be designed. Based on the Hamiltonian variational principle and symplectic eigen expansion, a theoretical analysis model of the symplectic stiffness method is established, and analytical expressions for the stiffness of several typical buckling modes of elastically supported beams are obtained. The analytical closed-form expressions for the macroscopic buckling strength of honeycombs are obtained according to the bending moment equilibrium equation. The buckling modes of regular hexagonal, hierarchical hexagonal, and tri-chiral honeycombs are obtained by numerical simulation and experimental verification, and some novel higher-order modes are observed. Uniaxial compression experiments reveal that these higher-order buckling modes can be triggered by adjusting the geometric parameters of the unit cells. The high agreement between experimental, numerical, and existing literature results for the failure surfaces of cellular structures verifies the effectiveness of the proposed symplectic stiffness method. In a word, the buckling modes can be designed to meet different functional requirements by changing the initial configuration of flexible cellular structures. This provides a new idea and theoretical guidance for the development of metamaterials and structures with tailored properties.

Coupled five-parameter dynamics of Mindlin and third-order shear deformable FG graphene-platelets reinforced viscoelastic plates with geometric and material imperfections

The present work investigates the coupled five-parameter dynamics, in particular, the coupled five-parameter structural resonance frequency, of functionally graded (FG) graphene-platelets reinforced viscoelastic plates in the presence of geometric and material (porosity) imperfections, based on the Mindlin theory and the third-order shear deformation plate theory (TSDPT). Various imperfections are often found in structures as a result of the challenges in manufacturing and operations over time. Geometric imperfection is considered in the FG reinforced plate structure by modelling an initial curvature in the out-of-plane direction. Material imperfections, in the form of closed-cell voids, are formulated as uniform and non-uniform FG porosity distributions. Uniform and thickness-wise FG gradation effects of graphene-platelets are accounted for in the determination of the effective material properties using a two-dimensional Halpin-Tsai micromechanics model, in conjunction with the rule of mixtures. Viscoelasticity of the FG imperfect plate structure is modelled by employing the Kelvin-Voigt model to consider energy dissipation. Using the Mindlin plate theory and the TSDPT, five equations of motion, which are coupled through the five parameters u, v, w, ϕ1, and ϕ2, are derived using Hamilton's variational principle. An assumed mode method is then employed to obtain the real and the imaginary components of resonance frequencies. The combined influences of the different effects on the structural resonance are examined. The results obtained offer a comprehensive understanding of how the FG graphene-platelets, FG porosity, plate thickness, geometric imperfection, and the utilisation of two different shear deformation plate theories collectively influence the coupled five-parameter dynamics of the FG graphene-platelets reinforced viscoelastic plate structure with imperfections. It was found that the difference in frequency resulted from the use of the two plate theories is most dominant with the FG GX and/or FG PX graphene-platelets reinforced porous plates, and that geometric imperfection was found to alter a graphene-platelets reinforced plate's sensitivity towards material imperfection.

Расчет системы амортизации шасси пассажирского самолета при ударном нагружении

The paper considers the shock absorption system of an advanced domestic mainline passenger aircraft, which includes oleo-pneumatic shock absorbers and wheels equipped with the pneumatic tires. Nonlinear mathematical models of single-active and two-active oleo-pneumatic shock absorbers of telescopic landing gear supports are proposed. The models take into account the process of polytropic gas compression, the forces of hydraulic resistance to the working fluid flow and the forces of dry friction arising in the system moving parts. To describe the vertical compression reaction of the pneumatic tires, the V.L. Biderman structural model was introduced, which parameters were determined by test results using the least squares method. Motion equations of the landing gear elements were obtained by the analytical mechanics methods involving the Ostrogradsky --- Hamilton variational principle of least action. Constraints imposed on the system were introduced using the penalty functions method. The problem statement of carrying out virtual computational experiments on the landing gear dvops was considered. In a wide range of the impact energies, models of shock absorbers of a passenger aircraft landing gear were validated, for which time realizations of the system state vector were determined, and areas of hysteresis loops of the shock absorber load characteristics were evaluated by the numerical integration. Satisfactory compliance of the results of simulated characteristics of the natural objects was shown. The obtained results could be used to improve the absorption quality criteria by numerical optimization methods, calculate the landing impacts and analyze the aircraft oscillations when moving on the runway

Design, manufacture, and clamping operation of a 4-DOF piezoelectric micro-gripper

To improve the operational flexibility of the piezoelectric microgripper, a new four-degree-of-freedom piezoelectric microgripper was designed and fabricated. The clamp fingers can move both along the clamping direction and along its vertical direction. Also, clamping experiments were conducted on a φ 300 μm × 20 mm micro-shaft. Based on the transverse inverse piezoelectric effect of two groups of vertical intersections, a new configuration of a four-degree-of-freedom piezoelectric micro-gripper is designed. It can produce micro-displacement along the clamping direction and vertical clamping direction simultaneously. According to the Euler-Bernoulli beam equation, the Lagrangian function method and Hamilton variational principle are used to model the four-degree-of-freedom piezoelectric micro-gripper. Then, based on the optimization of the geometric parameters of the fingers, the static and dynamic characteristics of the microgripper are analyzed by the finite element method. After that, the micro-gripper is made using lithography, gluing, and laser cutting. Finally, the piezoelectric microgripper's static and dynamic characteristics and the micro-shaft's clamping operation are tested by experiments. The experimental results show that the maximum displacement, response time, and natural frequency of the designed micro-gripper along and perpendicular to the clamping direction agree well with the finite element simulation. The designed microgripper exhibits a promising prospect in practical micromanipulation applications.

On impact process of FGPM plate with the damaged interlayer: piezoelectricity-based interaction effect analysis and stress field prediction using BPNN

Revealing the piezoelectricity-based interaction effect on the impact process of functionally graded piezoelectric material(FGPM) plates with the damaged interlayer promotes the intelligent development of engineering structures. The extended neural network algorithm is conducive to the integrated design of the intelligent structure. The impact process of FGPM plate with damaged interlayer is investigated. In the framework of Reddy's higher-order shear plate theory, the reduced constitutive equations considering the piezoelectric effect and damage effect are established for the FGPM layer and damaged interlayer respectively. The contact forces in the loading and unloading processes are modeled by the modified Hertz contact theory. The impact dynamic equations are derived through the Hamiltonian variational principle. The high-order truncated double trigonometric series solution in the impact process is extended. Further, the back propagation neural network(BPNN) technique is developed to predict the stress state during the impact process. Ultimately, from the perspective of energy conversion, the interaction of piezoelectricity and damage effect, structural parameters and loading state on the impact process are systematically revealed. The developed BPNN technique achieves accurate prediction of the flexural stress fields.

Stability and nonlinear vibration of carbon nanotubes-reinforced composite pipes conveying fluid

This paper focuses on the stability and nonlinear vibration characteristics of carbon nanotubes-reinforced composite pipes conveying fluid, and the mechanism of the combined effects of multiple factors. Based on the assumption of Euler beam and Von-karman nonlinear stress-strain relationship, the dynamic governing equations and generalized boundary conditions of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) pipes are established by Hamiltonian variational principle. Firstly, the modal function suitable for generalized boundary conditions is derived by Galerkin method. The stability of FG-CNTRC pipes system and the relationship between natural frequency and flow velocity are analyzed in this paper. The effect of flow velocity on the natural frequency of pipelines is examined under various carbon nanomaterial distribution patterns and wall thicknesses in this research. Then, the nonlinear vibration characteristics of FG-CNTRC pipes are analyzed by multi-scale method. The method of deducing nonlinear frequency through high-order partial differential equations is also extended in nonlinear analysis. The results show that, in the case of uniform distribution of carbon nanotubes, increasing the wall thickness can improve the stability of the pipe system. The influence of amplitude on the nonlinear frequency is more obvious at the second-order frequency.

Multi-physics zonal Galerkin free element method for static and dynamic responses of functionally graded magneto-electro-elastic structures

Functionally graded magneto-electro-elastic (FGMEE) structures have been extensively applied in engineering applications, which are often in the multi-field coupling environment. Driven by the desire to identify and understand the complexities of coupled fields, the multi-physics zonal Galerkin free element method (MZG-FREM), which belongs to a weak-form meshless method, is proposed for static and transient responses of magneto-electro-elastic structure, which material properties vary along the thickness direction. Based on the thermodynamic potential function and generalized Hamilton variational principle, the equations for the FGMEE medium are derived. Through the zone technique, MZG-FREM processes a robust capability to handle the model with complex geometry and the results obtained by MZG-FREM are insensitive to irregularly distributed nodes. Besides, Galerkin method is introduced to set up the equation for each node to guarantee the stability and accuracy of the results. The effect of exponential factors on the responses of FGMEE structures is investigated by MZG-FREM with a modified Newmark scheme. The convergence, accuracy, and robustness of MZG-FREM are verified through numerical examples including FGMEE structure with complex geometry and FGMEE energy harvester composed of various materials. The proposed method MZG-FREM and obtained results can be effectively incorporated for the accurate design of FGMEE smart devices.

Spectral-Tchebychev technique for the free and stochastic vibration analysis of functionally graded plates with piezoelectric patches

This paper presents a Spectral-Tchebychev (S-T) dynamic model to examine the free and random vibration properties of functionally gradient plates with piezoelectric patches (FGPPP) under random excitation. Firstly, the plate is divided into several sub-plates according to the position of the piezoelectric patches through the multi-partitioning strategy. Then, the dynamical equations for each sub-plate are derived based on the first-order shear deformation theory (FSDT) and Hamilton's variational principle. The boundary conditions and the coupling relationships of the adjacent plates are handled using the artificial spring technique. Finally, the free and stochastic vibration characteristics of FGPPP are obtained using the Spectral-Tchebychev technique, where the stationary random load is introduced by the pseudo-excitation method (PEM). The rationality and validity of the S-T model in predicting the vibration characteristics of FGPPP are verified by the comparison with the results from literature and FEM. The influence mechanism of some key parameters including the thickness ratio, area ratio, patching position, patching type, power law index, etc., on the vibration characteristics are systematically investigated.

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.