Coupled Equations Of Motion Research Articles

On the size‐dependent vibrations of doubly curved porous shear deformable FGM microshells

AbstractThis paper aims to analyse the free vibrations of doubly curved imperfect shear deformable functionally graded material microshells using a five‐parameter shear deformable model. Porosity is modeled via the modified power‐law rule by a logarithmic‐uneven variation along the thickness. Coupled axial, transverse, and rotational motion equations for general doubly curved microsystems are obtained by a virtual work/energy of Hamilton's principle using a modified first‐order shear deformable theory including small size dependence. The modal decomposition method is then used to obtain a solution for different geometries of microshells: spherical, elliptical, hyperbolic, and cylindrical. A detailed study on the influence of material gradation and porosity, small‐length scale coefficient, and geometrical parameters on the frequency characteristics of the microsystem is conducted for different shell geometries.

Energy absorption capacity and vibration-control responses of smart perovskite solar cells under external excitation via intelligent velocity feedback control algorithm

Smart perovskite solar cells offer engineers a highly efficient, cost-effective, and flexible solution for renewable energy. Their tunable properties, ease of fabrication, and potential for integration into various surfaces make them ideal for sustainable power generation, driving innovation in clean energy technologies and energy-efficient designs. This comprehensive approach ensures that smart perovskite solar cells can withstand external mechanical excitations more effectively and maintain their integrity over time. Due to the mentioned problem, using a data-driven solution approach, this work presents an intelligent controller for reducing transitory vibrations in smart perovskite solar systems outfitted with functionally-graded graphene origami-enabled auxetic metamaterial (FG-GOEAM) layer in the metal position, sensor, and actuator face sheets. In order to effectively detect and react to vibrations, the system integrates sensor and actuator face sheets. The controller applies the converse and direct piezoelectric equations together with interpolation functions to coupled motion equations in order to efficiently decrease vibration amplitudes. By integrating intelligent control algorithms, system parameters may be monitored and adjusted in real-time to maximize stability and performance. The piezoelectric actuator applies a control force to promptly halt the vibration of the structure. This study determines the controlled dynamic response of the smart perovskite solar cell construction by the use of a velocity feedback control algorithm. Numerical simulations and validation show that the suggested method is effective in mitigating transient vibrations and improving structural integrity. Through the development of intelligent control systems for smart perovskite solar cells, this study offers interesting solutions for engineering applications by reducing transitory vibrations.

New angular momentum conservation laws for electromagnetic waves interacting with dirac fields

Abstract Global conservation laws of angular momentum are well-known in the theory of light-matter interaction. However, local conservation laws, i.e. the conservation law of angular momentum at every point in space, remain unexplored especially in the context of relativistic Dirac-Maxwell fields. Here, we use the QED Lagrangian and Noether's theorem to derive a new local conservation law of angular momentum for Dirac-Maxwell fields in the form of the continuity relation for linear momentum. We separate this local conservation law into four coupled motion equations for spin and orbital angular momentum (OAM) densities. We introduce a helicity current tensor, OAM current tensor, and spin-orbit torque in the motion equations to shed light on the local dynamics of spin-OAM interaction and angular momentum exchange between Maxwell and Dirac fields. We elucidate how our results translate to classical electrodynamics using the example of plane wave interference as well as a dual-mode optical fiber. Our results shine light on angular momentum phenomena related to the relativistic interaction of electromagnetic waves and Dirac fields.

Parametric Roll Motion Reduction of a Ship with a Passive Anti-Roll Tank

In this study, the influence of a U-tube passive anti-roll tank on parametric roll motion has been investigated as a mechanical absorber of a dynamic system. Specifically, this paper concentrates on how the initial conditions of a coupled roll motion and fluid motion in the tank act on the performance of an anti-roll tank. Parametrically excited roll motion was modeled as a single degree of freedom system incorporating heave and pitch effects by means of a time-varying restoring moment. Fluid motion in the tank was modeled as a single degree of freedom system. Coupled equations of motion were solved numerically and the results were presented in the time domain. Furthermore, the results were comparatively depicted as maximum roll amplitudes vs. initial values for a ship with and without an anti-roll U-tube tank.

Numerical Analysis of Hydrodynamic Characteristics of Two-Dimensional Submerged Structure in Irregular Waves

A comprehensive two-dimensional (2D) time-domain numerical model is established to investigate the interaction of irregular waves and submerged structures with different sections. The model specifically focuses on the dual-lane submerged floating tunnel (SFT) designs, encompassing elliptical, twin-circular, and round rectangular sections. For the hydrodynamic analysis, we adopt the second-order potential flow theory, while for the mooring line simulations, we employ the slender rod theory, taking into account the entire hydrodynamic load acting on it. In the coupled dynamic analysis, the fourth-order Adams–Bashforth–Moulton method, Newmark-β method, and Newton–Raphson iteration scheme are utilized for the coupled motion equation of the floating body and the dynamic equation of the mooring riser system. Experimental free decay tests are conducted to determine the damping coefficients of various section shapes in different directions. Our analysis delves into the detailed motion responses and mooring tensions of the SFTs with different section forms under irregular waves. We compare and contrast these responses in both time and frequency domains, particularly focusing on movement trends. The elliptical section structure emerges as the most stable design based on our comparisons. These findings provide valuable insights for the selection of optimal section shapes for dual-lane SFTs.

Analytical study on hydrodynamic performance of co-located offshore wind–solar farms

Based on linear potential flow theory, this study investigates the hydrodynamic performance of a co-located farm with an array of floating offshore wind turbines (FOWTs) and floating photovoltaics (FPVs). In this process, to evaluate the wave–structure interaction, domain decomposition and matched eigenfunction method are applied to address the boundary value problem for a complex-shaped co-located farm, and the velocity potential can be decomposed into radiation and diffraction problems. Under the framework of linearized theory, we establish the coupled motion equations by modeling rigid and articulated constraints to evaluate the kinematic response of the FOWTs and FPVs in the co-located farm. For such a system, a co-located farm consisting of an array of OC4-DeepCwind FOWTs and FPVs is proposed and investigated in this study. After running convergence analysis and model validation, the present model is employed to perform a multiparameter effect analysis. Case studies are presented to clarify the effects of solar platform geometric parameters (including column depth, thickness, radius, and total draft), articulated system, and shadow effect on the hydrodynamic behavior of wind and solar platforms. The findings elucidated in this work provide guidance for the optimized design of FPVs and indicate the potential for synergies between wind and solar energy utilization on floating platforms.

Coupled Parametric Vibration Model and Response Analysis of Single Beam and Double Cable Under Deterministic Harmonic and Random Excitation

In order to reveal the coupling parametric vibration characteristics of stay cables under combined excitation, considering the effects of cable geometric nonlinearity, inclination and the cooperative vibration of adjacent cables and bridge deck beams, a single-beam–double-cable coupled parametric vibration model excited by Gaussian white noise and deterministic harmonic excitation is established, and the coupled motion equations are derived. The Milstein–Platen method is used to directly obtain the coupled random vibration time history of the single-beam–double-cable structure, and an iterative method is proposed to counter the influence of the parametric diffusion coefficient on the numerical format. By comparing with the finite element method and the Monte Carlo numerical simulation method, the accuracy of the Milstein–Platen method in solving the vibration time history of cable–beam coupling parameters under combined excitation is first verified. Then the random displacement, power spectral density and probability density variation of the cable and beam under the combined excitation of different intensities are analyzed from the angle of random orbit. The results show that under the joint action of deterministic harmonics and random excitation, the time–domain, frequency, and probability characteristics of the single-beam–double-cable coupling system are greatly affected by the Gaussian white noise excitation proportion coefficient, and the degree of influence is different. In addition, compared with the single cable model, the vibration analysis results of the coupling model considering multiple stay cables are more reasonable.

Various homogenisation schemes for vibration characteristics of axially FG core multilayered microbeams with metal foam face layers based on third order shear deformation theory

This study assesses the significance of homogenisation models in theoretically analysing the vibrations of three-layered microbeams with a core composed of axially functionally graded (AFG) material and with upper and lower face layers composed of metal foam material, especially when the third order shear deformation beam theory and the modified couple stress theory are used. The material of the core is assumed to vary following a power gradation profile. The range of homogenisation schemes, including the Voigt model, Reuss model, Voigt-Reuss-Hill model, Tamura model, Mori-Tanaka model, lower and upper bounds of Hashin-Shtrikman, and the local representative volume element (LRVE) model, are utilised to approximate the material properties of the AFG core. The upper and lower layers of metal foams are modelled using the closed-cell and open-cell solid porosity models. The coupled motion equations are solved for the natural frequencies of the system for the transverse, rotational and axial directions. A software-based finite element analysis (FEA) study of two simplified macrobeam systems was undertaken and the agreement between the FEA and numerical results, using the presented theoretical method, is found to be excellent. It has been confirmed that the homogenisation schemes have a notable effect on the mechanical properties of the AFG core, and hence on the coupled translational-rotational motion's natural frequencies. The largest difference in natural frequencies between different homogenisation schemes is found with a power gradation constant of approximately 0.53. A larger disparity of natural frequencies between different homogenisation methods was seen with an increase in the thickness of the AFG core and the microbeam's length-scale parameter of multilayered porous microbeam. The results of this study highlight the significance of homogenisation scheme employed when investigating the vibrations of a multilayered microbeam with an AFG core and metal foam porous face layers.

Free vibrations of cracked functionally graded graphene platelets reinforced Timoshenko beams based on Hu-Washizu-Barr variational method

In this paper, a new theoretical approach is presented for modelling the free vibrations of functionally graded (FG) graphene platelets (GPL) reinforced thick beams with a single-edge crack. To model the single-edge crack in the GPL reinforced thick beams, strain-displacement, strain-stress, velocity-momentum, and dynamic equilibrium equations are presented in the framework of the Hu-Washizu-Barr variational method, in which three complex coupled motion equations are obtained. The beam is assumed to be continuous and thick, and the shear effects are considered by employing the Timoshenko beam theory. GPL are distributed uniformly or non-uniformly through the depth of the cracked Timoshenko beam and their reinforcement effects are formulated using a two-dimensional Halpin-Tsai model of micromechanics and a rule of mixtures. The coupled motion equations are solved using a modal decomposition scheme. The current methodology is verified by modelling simplified versions of the present model and by comparing the results with those from literature and a finite element software. Detailed case studies are presented to understand the impact of a single-edge crack on the vibration behaviour of the GPL reinforced beams. It has shown that the presence of the single-edge crack and its position can significantly alter the free vibrations of the reinforced Timoshenko beam. The distribution pattern and weight fraction of GPL also have a great significance in the free vibrations of the cracked beam, affecting the natural frequencies. The results and conclusions drawn from this paper, which uses the Hu-Washizu-Barr variational method, can be useful in guiding optimal GPL distributions and concentrations for different mechanical designs that might foresee a single-edge crack in structures.

Nonlinear 2D-DQ volume-preservative global–local dynamic analysis of composite sandwich plates with soft hyperelastic cores and viscoelastic Winkler-Pasternak foundations

Nonlinear large amplitude vibration and dynamic deformations of sandwich plates with composite face sheets and hyperelastic cores that encounter large thickness changes during the vibration are investigated here for the first time. Moreover, a new global–local zigzag hyperelastic plate theory that considers the incompressibility and large transverse deformability of the core, and employs transverse normal strains whose order is higher than that of the in-plane strains is proposed. The plate is resting on a viscoelastic Winkler-Pasternak substrate, exposed to various spatial and time distributions of loads, and experiencing a diversity of edge conditions. The governing equations of motion are obtained by using Hamilton’s principle, von Kármán’s assumptions, and the series expansion of the volume-preserving strain energy density function. A 2D differential quadrature (2D-DQ) technique is employed for the spatial discretization of a structure with both constitutive and kinematic nonlinearities, for the first time. The time-dependent combination of the nonlinear coupled motion equations and boundary conditions is treated by an iterative/updating Runge-Kutta time-marching technique and the effects of the thickness and aspect ratio of the plate, viscoelastic and shear parameters of the foundation, types of the edge conditions, constitutive parameters of the composite materials, fiber orientation angle, and magnitude and distribution pattern of the loads on the resulting dynamic lateral deflections are studied. Results reveal that while the vibration frequencies and amplitudes of the face sheets are quite different, the damping is more remarkable when stiffer cores are utilized. Moreover, the effects of the higher vibration modes are more apparent in looser edge conditions, smaller fiber angles, and special excitation frequencies.

Highly nonlinear hyperelastic shells: Statics and dynamics

Investigated in this paper are comprehensive static, dynamic and internal-resonance analyses on a wide range of hyperelastic shell structures, including cylindrical, spherical, doubly-curved, and hyperbolic hyperelastic shallow shells. Donnell's nonlinear shell theory and the Mooney-Rivlin strain energy density model are used to formulate the hyperelastic shell structure. Coupled equations of motion are obtained using Hamilton's principle with highly nonlinear terms due to the curvature in the structure, together with the material nonlinearity and large deformations. The coupled equations of motion are converted to a large set of equations using a two-dimensional Galerkin technique and are solved by employing the Newton-Raphson approach and dynamic equilibrium technique. The strength of the current methodology and model is first verified by comparing the static response of the structure with those obtained by using a finite element software. After verifying the model, a detailed analysis of the bending behaviour of the structure under a time-independent pressure is presented. Moreover, the free and forced vibration responses of the shell structure are presented for different cases, showing that the curvature terms play a significant role in changing the mechanical response of the hyperelastic shell. Furthermore, it is shown that for specific sets of curvatures, internal resonances are present which leads to a complicated, rich nonlinear responses. The nonlinear forced vibration response of the hyperelastic shell is also presented for different shell types and resonances.

Derivation and analytical solutions of a non-linear diffusion equation applied to non-constant heat conductivity and ionic diffusion in glasses.

This paper derives a non-linear diffusion equation discussing two possible applications: the ionic diffusion in glasses and temperature-dependent conductivity in semiconductors. The first equation is a logarithmic diffusion derived formally from the continuity of ion concentration, but the latter is a more phenomenological example. A power-law ansatz with time-dependent parameters maximizes a non-standard entropy and gives a set of coupled motion equations we can solve analytically. These results build the general solution to the non-linear diffusion equation.

Dynamics of size-dependent multilayered shear deformable microbeams with axially functionally graded core and non-uniform mass supported by an intermediate elastic support

The current study investigates the coupled dynamics of a size-dependent third-order shear deformable multilayered microbeam with an axially functionally graded core, a non-uniform mass distribution (mass imperfection) and an intermediate elastic support. A power-law function is used to model the material properties in the longitudinal direction of the core. The equations of motion are obtained by formulating the energies of the system while employing the modified couple stress theory to include miniature size effects within the multilayered microbeam structure. Hamilton's principle and the modal decomposition method are utilised to derive and solve the coupled motion equations. The equations of motion are first verified against a former work on a simplified microbeam system, while the numerical results are validated by comparison to those of a simplified functionally graded in axial direction obtained by finite element macrobeam simulation. It was concluded that increasing either the power term constant of the material grading or the value of the localized mass imperfection (when xm*=0.3) causes the first and second transverse and axial natural frequencies of the three-layered microbeam to decrease, whereas increasing the length-scale parameter causes the natural frequencies of transverse modes to increase, demonstrating the size-dependent effects of the three-layered microbeam.

Dynamic Modeling and Investigation of a Tunable Vortex Bladeless Wind Turbine

This paper investigates the dynamics of an electromagnetic vortex bladeless wind turbine (VBWT) with a tunable mechanism. The tunable mechanism comprises a progressive-rate spring that is attached to an oscillating magnet inside an electromagnetic coil. The spring stiffness is progressively adjusted as the wind speed changes to tune the turbine fundamental frequency to match the shedding frequency of the vortex-induced vibration (VIV) due to the wind flow crossing over the oscillating mast. Coupled nonlinear equations of motion of the tunable turbine are developed using the lumped-mass representation and Lagrange formulation. Numerical results show that the tunable turbine performs effectively beyond a threshold wind speed. An analytical expression of the threshold speed is derived based on the mechanical fundamental frequency of the turbine. The analytical results are in reasonable agreement with the numerical evaluations. At a given wind speed past the threshold, the tunable turbine has an optimum spring stiffness at which the output power is maximum. Numerical studies also show that the output power of the 2 m long tunable turbine is tens of times larger in comparison to a conventional bladeless turbine. For example, at a wind speed of 4.22 m/s, the output rms power of the tunable turbine is around 1105 mW versus 17 mW of the conventional VBWT. The power can be further maximized at an optimum external load. This research work demonstrated the feasibility and merits of the proposed tunable mechanism to enhance the overall performance of the bladeless wind turbine.

Nonlinear interaction of the damped large dynamic deformations of the Mooney–Rivlin hyperelastic plates with the viscoelastic and shear reactions of the supporting substrate

In this research, the interaction between the damped large deformation dynamic responses of the Mooney–Rivlin hyperelastic plates and the viscoelastic and shear characteristics of the supporting substrate (that constitute a visco-hyperelastic system) is investigated for various distributions and various time variations of the transverse loads and different boundary conditions. A viscoelastic Winkler–Pasternak model is chosen for the supporting substrate to account for the dissipative, shearing, and supporting features of the foundation. The governing equations of motion are obtained by using Hamilton’s principle, von Karman assumptions, left Cauchy–Green deformation tensor, and a modified classical plate theory whose accuracy is enhanced through the incorporation of an appropriate high-order incompressibility condition. The combination of the nonlinear and coupled motion equations and boundary conditions is solved by an iterative 2D differential quadrature (DQ) spatial-discretization and Newmark’s time-marching methods. The effects of the constitutive parameters of the hyperelastic material, magnitude and type of the distributed loads, the thickness and aspect ratios of the plate, parameters of the Winkler–Pasternak viscoelastic foundation, and boundary conditions on the dynamic lateral deflections of the plate are studied examined during the parametric studies. Results emphasize the role of the visco-hyperelasticity features interaction of the combined plate-substrate system on the vibration suppression and dynamic performance of the structure in different spatial and time-variation patterns of the distributed loads and various boundary conditions.

Stability analysis of yaw motion for sandglass-type floating body with dynamic positioning system under regular excitation

For the new sandglass-type floating body with special shape, this paper mainly analyzes its yaw stability under dynamic positioning system. Firstly, by considering the construction tolerance within the range of engineering accuracy, surge and yaw coupled motion equations of the sandglass-type FPSO based on dynamic positioning system are established. Secondly, the yaw instability phenomena of sandglass-type model can be found by Runge-Kutta numerical method. On this basis, a nonlinear yaw motion equation of parametric excitation can be reasonably deduced. Then the multi-scale method and LP (Lindstedt-Poincaré) method are used to study the stability characteristic of yaw motion for sandglass-type floating body. Finally, the effects of initial heading angle and control parameters on the yawing stability of the sandglass-type floating body are discussed.

Influence of Shear Effects on the Characteristics of Axisymmetric Wave Propagation in a Buried Fluid-Filled Pipe

The acoustic propagation characteristics of axisymmetric waves have been widely used in leak detection of fluid-filled pipes. The related acoustic methods and equipment are gradually coming to the market, but their theoretical research obviously lags behind the field practice, which seriously restricts the breakthrough and innovation of this technology. Based on the fully three-dimensional effect of the surrounding medium, a coupled motion equation of axisymmetric wave of buried liquid-filled pipes is derived in detail, a contact coefficient is used to express the coupling strength between surrounding medium and pipe, then, a general equation of motion was derived which contain the pipe soil lubrication contact, pipe soil compact contact and pipe in water and air. Finally, the corresponding numerical calculation model is established and solved used numerical method. The shear effects of the surrounding medium and the shear effects at the interface between surrounding medium and pipe are discussed in detail. The output indicates that the surrounding medium is to add mass to the pipe wall, but the shear effect is to add stiffness. With the consideration of the contact strength between the pipe and the medium, the additional mass and the pipe wall will resonate at a specific frequency, resulting in a significant increase in the radiation wave to the surrounding medium. The research contents have great guiding effect on the theory of acoustic wave propagation and the engineering application of leak detection technology in the buried pipe.

Modeling of serpentine belt accessory drive system coupled vibrations and utilizing nonlinear tensioner to enhance performances

Serpentine belt accessory drive system (SBADS) is a complex hybrid discrete-continuous system and is a typical coupled vibration system with rigid body vibrations and belt vibrations, which consists of rotational vibrations of pulleys and tensioner arm and transverse vibrations of continuum belt spans. Taking an engine front end accessory drive system as an example, the coupling vibration model of belt, pulleys and tensioner arm is established. In the model, hysteretic behavior of tensioner is considered, and a calculation method for estimating the output torque versus the imposed angle of tensioner arm is proposed. In solving the coupled motion equations of the SBADS, transverse displacement of a belt span is discretized as the product of time function and space function by using Galerkin method, the steady-state deflections of belt spans are calculated by singular perturbation method, and the nonlinear dynamic responses of a SBADS are solved by numerical method. The nonlinear dynamic responses of the SBADS are calculated by the established method and validated by the experiment results. Benefits of nonlinear hysteretic behavior of a tensioner on the dynamic responses of a SBADS are studied, and the method for how to utilize nonlinear characteristics of tensioner to enhance the performances of a SBADS is proposed in the end of this paper.

Hydrodynamics of a Moored Permeable Vertical Cylindrical Body

In this study, the problems of diffraction and radiation of water waves by a permeable vertical cylindrical body are formulated within the realm of the linear potential theory. The body, which is floating in constant water depth, is moored with a catenary mooring line system. The method of matched eigenfunction expansions for the prediction of the velocity potential in the fluid domain surrounding the body is applied. Furthermore, the static and dynamic characteristics of the mooring system are combined with the hydrodynamics of the body, to set up the coupled motion equations of the dynamical model, i.e., floater and mooring system, in the frequency domain. Numerical results obtained through the developed solution are presented. The results revealed that porosity plays a key role in reducing/controlling the exciting wave loads. As far as the mooring system is concerned, its quasi-static and dynamic characteristics, by employing several motion directions on the fairlead in accordance to varying environmental conditions, are examined, highlighting their effect on the body’s motions.

Coupled dynamics of axially functionally graded graphene nanoplatelets-reinforced viscoelastic shear deformable beams with material and geometric imperfections

This paper is the first to explore the coupled dynamics of geometrically and material-wise imperfect axially functionally graded (AFG) graphene nanoplatelets-reinforced viscoelastic third-order shear deformable beams. Four AFG graphene nanoplatelets distribution patterns are considered. Porosity, as the material imperfection, is modelled using a Gaussian Random Field model. Four thickness-wise functionally graded porosity distribution patterns are modelled. Effects of geometric imperfection are included by assigning an initial curvature to the beam. To consider the influences associated with energy dissipation caused by internal friction, the Kelvin-Voigt model for viscosity is used. External dissipative energy is modelled using a transverse damper. The effective material properties of the AFG beams are calculated using a modified Halpin-Tsai micromechanics model, together with a rule of mixture. Coupled axial, transverse, and rotational motion equations are obtained by employing a Hamiltonian approach and third-order shear deformation. The natural frequencies are obtained using a modal decomposition method. A simplified version of the graphene nanoplatelets-reinforced AFG structure is verified by a computer code-based finite element method. This novel study on the effects of geometrical imperfection on the sensitivity of beams towards porosity imperfections demonstrates the significance of scrutinising the effects of one imperfection on another.