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This paper investigates the vibrations of a functionally graded sandwich plate (FG sandwich plate) with a ceramic foam core resting on a viscoelastic foundation. The focus is on the influence of temperature changes on sandwich plates that have foam core and FG face sheet on the free vibration. The damping coefficient and different schemes of sandwich plate have been considered in the current work. The analysis uses a quasi-3D theory, which reduces the number of unknown displacements and simplifies the equations by incorporating integral terms. Two models are employed to represent the non-uniformity of the ceramic foam core. To account for the influence of the supporting layer, a viscoelastic foundation is included. The governing equations are derived using Hamilton's principle and solved using the Navier solution method. The paper comprehensively discusses how the volume fraction index k, geometric properties, ceramic foam distribution, viscoelastic foundation, and different temperature rise profiles impact the vibration response of the FG sandwich plate. [ABSTRACT FROM AUTHOR]

AbstractThis article aims to investigate the dynamics of clamped-clamped porous viscoelastic double microbeams interconnected via a viscoelastic layer within the framework of the modified couple stress theory (MCST). Different types of porosity distributions are considered to examine the material imperfection effects on the dynamics of the double microbeam. A closed-cell porosity model is employed to model the variation of the material properties in the thickness plane. The coupled longitudinal and lateral equations of motion are derived using Hamilton’s principle. The governing equations of motion are solved using the modal decomposition method. The equations of motion are verified against the literature for simplified cases (i.e., a non-porous single viscoelastic microbeam and a porous single microbeam resting on a viscoelastic foundation). The numerical results are validated for simpler versions of the system using the finite element method and against the existing literature for a simplified case of a uniformly distributed porous double macrobeam connected via an elastic layer. Influences of viscoelasticity in the double microbeam, viscoelastic layer, porosity, and small-scale influences on the dynamics of the viscoelastic double microbeams are studied. It was found that increasing the material length-scale factor causes the complex fundamental lateral vibrational frequencies to increase for all the porosity patterns. [ABSTRACT FROM AUTHOR]

AbstractElectro-magneto-thermo-mechanical critical loads of a sandwich microplate composed of a core integrated with piezoelectric/piezomagnetic face-sheets are analyzed in this paper based on higher order shear and normal deformation theory and modified couple stress theory. The constitutive relations are developed based on generalized Hooke’s law and electromagnetic relations. In order to measure athletic performance and his/her quantitatively or qualitatively grades, a force plate is used. The thickness stretching of the sport force plates is accounted based on a new function in displacement field. The electric and magnetic potentials are assumed as summation of a linear function including applied electric/magnetic potential and two functions for satisfying the zero condition at top and bottom of the sport force plates. [ABSTRACT FROM AUTHOR]

The critical global buckling load is a fundamental parameter governing the performance of tall buildings, however, the existing literature lacks an efficient solution that comprehensively captures the behavior of tall structures with uniform or variable properties. In response to this gap, this study introduces a novel and generalized solution, employing the coupled shear-flexural model. This solution is designed for determining the critical global buckling load in tall buildings subjected to vertical loads with diverse profiles, while also incorporating soil flexibility. The analytical solution, designed for the uniform continuous model, offers a breakthrough by providing graphical representations facilitating the direct determination of eigenvalues based on a single dimensionless parameter. Addressing the unique challenge posed by tall buildings with variable properties, a numerical transfer matrix method based on a Laplacian approach is proposed. This method enables the direct calculation of the transfer matrix for each level, displaying exceptional convergence in just three iterations. The study sheds light on a crucial finding that emphasizes the adverse impact of rotational flexibility on soil, leading to a decrease in eigenvalues as soil flexibility increases. Importantly, the proposed solution methods not only show excellent accuracy but also serve as invaluable tools for performing parametric analyses. Furthermore, they present cost-effective alternatives to exact methods, providing a solid foundation for studying the overall stability of tall buildings. This research contributes significantly to the existing literature by introducing efficient methodologies that improve our understanding and analysis of the behavior of tall buildings under vertical loads. [ABSTRACT FROM AUTHOR]

The aim of this research is to investigate the impact of a viscoelastic foundation on the vibration of a curved beam structure with clamped and simply-supported boundary conditions. The structure considered is a micro-scale laminate composite beam made up of two piezoelectric face layers and a carbon nanotube-reinforced composite core. The study employs non-classical elasticity theory and nonlocal strain gradient theory to account for both nano-scale and microscale effects. The equations of motion are derived using the energy terms of the beam and a variational principle, with boundary conditions assumed to be clamped at one end and simply supported at the other end. A semi-analytical method of general differential quadrature method is used to solve the equations due to their nonlinear terms. To capture the effects of various parameters on the in-plane vibration of sandwich arches, an artificial neural network is designed and implemented. The study analyzes the impact of nonlocal and gradient parameters, geometrical aspect ratios, and substrate constants of the structure on the natural frequency and amplitude of vibration of the laminate beam. Results show that increasing nonlocal and gradient parameters have different effects on the amplitude and frequency of vibration of the beam. [ABSTRACT FROM AUTHOR]

This article introduces a novel systematic methodology for modeling a class of multidimensional linear mechanical systems that directly allows to obtain their infinite-dimensional port-Hamiltonian representation. While the approach is tailored to systems governed by specific kinematic assumptions, it encompasses a wide range of models found in current literature, including ℓ -dimensional elasticity models (where ℓ = 1, 2, 3), vibrating strings, torsion in circular bars, classical beam and plate models, among others. The methodology involves formulating the displacement field using primary generalized coordinates via a linear algebraic relation. The non-zero components of the strain tensor are then calculated and expressed using secondary generalized coordinates, enabling the characterization of the skew-adjoint differential operator associated with the port-Hamiltonian representation. By applying Hamilton's principle and employing a specially developed integration by parts formula for the considered class of differential operators, the port-Hamiltonian model is directly obtained, along with the definition of boundary inputs and outputs. To illustrate the methodology, the plate modeling process based on Reddy's third-order shear deformation theory is presented as an example. To the best of our knowledge, this is the first time that a port-Hamiltonian representation of this system is presented in the literature. • Systematic modeling of multidimensional flexible linear mechanical systems using the port-Hamiltonian framework. • Explicit definition of energy and co-energy variables for this class of port Hamiltonian systems. • Explicit definition of distributed and boundary power conjugated ports suitable for control or interconnection purposes. [ABSTRACT FROM AUTHOR]

AbstractIn this article, two analytical solutions, one semi-analytical solution, and one numerical solution are proposed to calculate the global critical buckling load of buildings employing a continuum model parallel coupling two Timoshenko beams. The parallel coupling of two Timoshenko beams represents the four types of building behavior: global bending, global shear, local bending, and local shear, the latter of which has been widely ignored in the literature. The model is associated with one translational kinematic field and two rotational fields. The equilibrium equations, constitutive laws, and boundary conditions are derived using a variational approach by applying Hamilton’s principle to the relevant Lagrangian function. Specifically, to solve the classical problem of buildings with uniform properties along their height, the first analytical solution proposes exact closed-form equations for a vertical load applied at the top and a closed-form analytical solution based on the decomposition of two subsystems for a polynomial vertical load uniformly applied along the height of the beam. Meanwhile, the semi-analytical solution exactly solves the differential equation associated with a polynomial vertical load profile and is based on coefficient iteration, converging with only two or three iterations. To address the challenge of buildings with variable properties along their height, the combined application of the continuous method and the transfer matrix method allows us to propose an efficient numerical method leading to a 6 × 6 transfer matrix, constant regardless of the number of discretizations and derived analytically, which results in a computational advantage by drastically reducing computational costs. The numerical results are encouraging and show good agreement with the results of the finite element method, suggesting that the proposed method can be used for engineering purposes in both the preliminary and final phases of the structural analysis of tall buildings. [ABSTRACT FROM AUTHOR]

The claim that the free energy principle is somehow related to Hamilton's principle in statistical mechanics is ubiquitous throughout the subject literature. However, the exact nature of this relationship remains unclear. According to some sources, the free energy principle is merely similar to Hamilton's principle of stationary action; others claim that it is either analogous or equivalent to it, while yet another part of the literature espouses the claim that it is a version of Hamilton's principle. In this article, we aim to clarify the nature of the relationship between the two principles by investigating the two most likely interpretations of the claims that can be found in the subject literature. According to the strong interpretation, the two principles are equivalent and apply to the same subset of physical phenomena; according to the weak interpretation, the two principles are merely analogous to each other by virtue of their similar formal structures. As we show, adopting the stronger reading would lead to a dilemma that is untenable for the proponents of the free energy principle, thus supporting the adoption of the weaker reading for the relationship between the two constructs. [ABSTRACT FROM AUTHOR]

The free vibration dynamic modellings of sandwich beams with periodic lattice truss cores are established by combining the Bernoulli–Euler beam theory and Timoshenko beam theory, and the fundamental frequency and unit cell mass of lattice sandwich beams are optimized based on the non-dominated sorting genetic algorithm-II (NSGA-II). Firstly, the vibration control equation of lattice sandwich beams is derived using Hamilton's principle. Then, it is compared and verified by modal experiment and finite element numerical simulation. On this basis, the influence of different truss cores on the structure's natural frequency is further analyzed. Finally, taking the sheet thickness, core thickness, and radius of the truss as the optimization design variables and taking the structural fundamental frequency maximization and mass minimization as the optimization objectives, the Pareto front is obtained by NSGA-II, and the optimization results are verified by theoretical and numerically. The research results show that the theoretical method in this article can well predict the natural frequencies of sandwich beams with different truss cores. The optimized sandwich structure has higher fundamental frequency with lighter weight, which effectively improves the anti-vibration ability. [ABSTRACT FROM AUTHOR]

AbstractStatic stability analysis of global buckling and critical load ratio calculation are among the fundamental and most important parameters for structural analysis and design in numerous engineering applications across various fields. Particularly in the field of structural engineering, the critical load ratio is a parameter that determines the overall performance of tall buildings; however, due to the complexity of coupled differential equations with variable coefficients associated with different classical continuous models, its analysis has not been widespread, and only specific cases have been solved. This article utilizes the classic sandwich beam, widely studied in the literature, to propose an analytical solution and a generalized numerical solution to address the elastic buckling problem with applications in tall building stability analysis. The analytical solutions are applicable to uniform structures and consider point, uniform, and variable compression loads along their length that can be expressed through polynomial functions. The generalized numerical solution addresses the general case of structures with uniform or variable properties. Numerical applications and parametric analyses show excellent agreement between the results of the proposed methods and those of one-dimensional finite element methods. The influences of key parameters on the behavior of the eigenvalue associated with global buckling analysis are analyzed, and graphs and tables are proposed for direct practical application by engineers. Due to their mathematical formulation and excellent precision in results, the proposed methods can be safely used in the preliminary and final analysis of tall buildings and can be easily adapted to other engineering fields. [ABSTRACT FROM AUTHOR]

An effective methodology is proposed to evaluate the boundary stiffnesses of beams with generally restrained edges. The boundary conditions at the ends of the beams are implemented by the penalty method. The unified Jacobi polynomials are introduced to represent the assumed mode shape functions of the beam. Hamilton’s principle with the assumed mode method is employed to derive the equation of motion of the beam with generally restrained boundaries. Subsequently, a novel approach is proposed to evaluate the boundary stiffnesses of the beam using the measured natural frequencies of the beam by vibration experiments. The accuracy of this method is verified by comparing the results with the previously published literature, the finite element method (FEM), and the vibration experiments. The effects of the types and values of the boundary spring stiffnesses on the vibration properties of the beam are also discussed. The calculation method in this research can provide an effective prediction technique for the boundary stiffnesses of engineering structures with generally restrained edges. [ABSTRACT FROM AUTHOR]

Held in Dublin, Ireland, IS2024 invites us to refresh understanding of contributions to systems engineering by Ireland's greatest mathematician–Sir William Rowan Hamilton (1805 ‐ 1865), Professor of Astronomy at Trinity College Dublin and Royal Astronomer of Ireland. His profound contributions to STEM deserve greater systems community attention. Supporting theory and practice, they intersect Foundations and Applications streams of INCOSE's Future of Systems Engineering (FuSE) program. Strikingly, key aspects apply to systems of all types, including socio‐technical and information systems. Hamilton abstracted the energy‐like generator of dynamics for all systems, while also generalizing momentum. Applied to the INCOSE Innovation Ecosystem Pattern as dynamics of learning, development, and life cycle management, this suggests an architecture for integration of the digital thread and machine learning in innovation enterprises, along with foundations of systems engineering as a dynamical system. [ABSTRACT FROM AUTHOR]

AbstractThe vibration character of piezoelectric semiconductor nano-plate is important for understanding the multi-fields coupling and optimizing the serving behavior. In this article, the nonlinear vibration of five-layered functionally graded piezoelectric semiconductor nano-plate based on Pasternak foundation is studied, and the nonlocal theory is used to analyze the size-dependent dynamic response of nano-plate. Combining the Von karman’s nonlinear deformation theory and the constitutive of piezoelectric semiconductor layers, Hamilton’s principle is introduced to derive the nonlinear governing equations. To solve the nonlinear governing equations, the updated iteration method for this problem is proposed. Through numerical examples, it is found that the initial electron concentration in piezoelectric semiconductor layer, the nonlocal parameter, the functionally graded index in the plate, the elastic modulus of Pasternak foundation and the geometrical parameters can be designed to effectively tune the dynamic nonlinear frequency and damping of layered functionally graded piezoelectric semiconductor nanoplate. The interaction of these parameters is also analyzed in detail. Comparison with existing numerical results is also presented. A new way is provided to tune the nonlinear vibration of multi-layered piezoelectric semiconductor nano-plate. [ABSTRACT FROM AUTHOR]

AbstractThis work examines the transient nonlinear oscillations of complex composite systems under mechanical pressure utilizing Reddy’s theory of higher-order shear deformation plates. The analysis entails the formulation of nonlinear equations that regulate the stress and displacement fields inside the composite structure. The dynamic behavior is defined by using Hamilton’s idea. The transient response of the composite system under various loading conditions is determined by deriving analytical solutions. The work offers a full knowledge of the dynamic behavior of advanced composite structures by using Reddy’s higher-order shear deformation plate theory, which takes into account the effects of transverse shear deformation and rotating inertia. The intricate interplay of material properties, shape, and external force is elucidated by the nonlinearity inherent in the equations. This provides valuable information on the temporary vibrations that occur in the composite system. The work employs rigorous mathematical analysis and closed-form solutions to clarify the intricate transient dynamics of modern composite structures, providing insights into how they react to mechanical loads over time. The results enhance the overall comprehension of composite material behavior and provide useful insights for the technical design and optimization of composite structures. [ABSTRACT FROM AUTHOR]

Compared to the conventional cellular cores, the cellular cores auxetic metamaterials with negative Poisson's ratios have unique mechanical deformation characteristics, superior strength to density ratio and high stiffness, both of which can be used to build lightweight sandwich constructions. The purpose of this study is to investigate the vibration acoustic response of porous functionally graded honeycomb sandwich plates with a negative Poisson's ratio (NPR). Under simply supported boundary conditions, the dynamic equation of the sandwich plate is derived based on the Hamilton principle, and the eigenvalue of the modal function matrix is coupled with the Navier method to obtain the natural frequency. The theoretical calculation results are compared with the finite element software COMSOL simulation results, so as to verify the validity and correctness of the theoretical model proposed in this paper. The far-field radiation sound pressure level of sandwich plate under point force loading is derived by Rayleigh integral theory. The correctness of the proposed model is verified by comparing the radiation sound pressure level calculated by the theoretical model with the results obtained by the finite element software simulation. The effects of porous functionally graded materials (FGM) and cellular structure parameters on the acoustic radiation characteristics of sandwich plates are discussed in detail. The results show that: The relative error between the theoretical results and the literature results is 5.43 %. Under the same conditions, the amplitude of the radiation sound pressure level (SPL) decreases by about 29.48 % in the low frequency stage, 28.9 % in the medium frequency stage and 11.08 % in the high frequency stage under the point excitation when the cellular side length is 6 mm compared with 8 mm in the low frequency stage. The movement trend and amplitude of the sandwich plate's emitted SPL are significantly impacted by changes in the characteristics of the cellular structure. [ABSTRACT FROM AUTHOR]

The proposed generalized electric spring (G-ES) topology effectively reduces the redundancy caused by the parallel structure of multiple electric springs in a microgrid system. A modeling method for the G-ES is urgently needed to accurately determine the G-ES parameters and to evaluate its transient operation characteristics. The correspondence between Hamilton's principle in mechanics and electricity is introduced, and the feasibility of Hamilton's principle under the smart load topology is verified. The modeling method of the G-ES under Hamilton's principle is discussed, followed by the application of repetitive control strategies on the G-ES. The filter parameters are designed using the normalization method and a simplified parameter design using repetitive controllers was applied. This article introduces the impedance analysis method into the generalized electric spring to analyze its stability under weak network conditions. This work also derives the stability judgment criteria for generalized electric springs. Finally, the feasibility of modeling the G-ES based on the Hamilton's principle was verified using MATLAB and a DSP hardware system. In addition, the consistency between the G-ES and a single intelligent load was verified under the Hamilton's modeling principle. The G-ES stability and dynamic performance under this modeling method meet industry requirements. [ABSTRACT FROM AUTHOR]

The natural frequency curve veering phenomena have been significant and used in several structural dynamic problems. By formulating a pre-twisted and double-tapered rotating Timoshenko beam as a structural problem, variations in mode interactions among coupled axial, chordwise, and flapwise motions were presented. Rotordynamic effects such as gyroscopic couple, centrifugal stiffening, and spin softening effects were taken into account while establishing the equations of motion. Hamilton's principle was used in deriving the equations of motion and is discretized based on Galerkin finite element method. Using dimensionless parameters, the model was verified with the existing literature for rotation, pre-twist, and double-taper configurations. A comparison of the frequency veering regions among the Euler–Bernoulli beam and Timoshenko beam theories is presented. Two-dimensional mode shape plots were used to show the coupled motions between axial displacement, chordwise bending, and flapwise bending. The effects of pre-twist and double-taper in the non-rotating Timoshenko beam on the modal properties of the rotating beam were examined. Natural frequency maps were drawn to locate the frequency veering curves. Mode interactions around different veering regions were shown with variations in mode shapes. An investigation on 60° pre-twisted and 0.5 ratio double-tapered rotating Timoshenko beam is presented. [ABSTRACT FROM AUTHOR]

Traditional engineering materials lack the necessary properties that are needed in aerospace as well as other modern industries. In order to address the aforementioned problem, a number of different materials are used in concert. With the help of functionally graded material, all the necessary characteristics can be achieved. A fast transition between two different materials can lead to debonding, thermal stresses, residual stresses, and stress concentrations; a gradual change in material properties might mitigate these problems. This work provides a comparison of the analytical solutions for the deflections in Two Dimensional Functionally Graded Taper Beam (2D-FGTB) under a uniformly distributed load, adapting Reddy's higher-order shear deformation theory. All the material properties of the beam are graded along the thickness and length dimensions using the power-law formula. The thickness of the beam is assumed to change linearly along its length. Equations of motion are derived based on Hamilton's principle and Navier's solutions. A parametric investigation is conducted to explore the effects of various material and geometrical parameters on the mechanics of 2D-FGTB. These parameters are found to be very significant in studying the static responses of 2D-FGTB. [ABSTRACT FROM AUTHOR]

Traditional engineering materials lack the necessary properties that are needed in aerospace as well as other modern industries. In order to address the aforementioned problem, a number of different materials are used in concert. With the help of functionally graded material, all the necessary characteristics can be achieved. A fast transition between two different materials can lead to debonding, thermal stresses, residual stresses, and stress concentrations; a gradual change in material properties might mitigate these problems. This work provides a comparison of the analytical solutions for the vibration in Two- Dimensional Functionally Graded Taper Beam (2D-FGTB) under a uniformly distributed load, adapting Refined higher order shear deformation theory. The beam's material characteristics in their entirety are graded in thickness as well as length directions using the power-law formula.The beam’s thickness is assumed to change linearly in the direction of its length.Equations of motion may be found by applying Hamilton's principle and Navier's solutions to a situation. An examination based on parametric modelling is carried out with the purpose of determining how the mechanics of 2D-FGTB are affected by a variety of material and geometrical characteristics. These parameters are found to be very significant in studying the vibration behavior of 2D-FGTB. [ABSTRACT FROM AUTHOR]

A satellite with two solar wings can be modeled using a pair of symmetric flexible cantilever beams connected to a central rigid body. Due to certain reasons, the symmetric flexible cantilever beams may be turned into asymmetric ones, which will inevitably influence the vibration properties of the structural system. By changing the structural sizes and adding local mass on one side of the two beams, a structural system with asymmetric mass distribution is obtained and its vibration characteristics are investigated. Hamilton's principle with the assumed mode method is employed to establish the equation of motion of the asymmetric structural system. The natural frequencies, mode shapes, frequency response curves and displacement time histories of the system are calculated, and they are compared with those of the structural system with a symmetric mass distribution. The correctness and feasibility of the present analytical method are verified by means of the finite element method (FEM) and a vibration experiment. The analytical results show that the mass asymmetry of the two beams leads to the mode localization phenomenon, and the coupling effect between the two beams and the central rigid body is enhanced. The larger the mass asymmetry is and the closer the position of the added local mass to the end of the cantilever beam is, the more obvious of the mode localization phenomenon is and the more obvious of the coupling effect between the two beams and the central rigid body is. The present investigation results are helpful for the dynamic analysis and design of spacecraft structures composed of flexible solar wings and a central rigid body. [ABSTRACT FROM AUTHOR]

The claim that the free energy principle is somehow related to Hamilton’s principle in statistical mechanics is ubiquitous throughout the subject literature. However, the exact nature of this relationship remains unclear. According to some sources, the free energy principle is merely similar to Hamilton’s principle of stationary action; others claim that it is either analogous or equivalent to it, while yet another part of the literature espouses the claim that it is a version of Hamilton’s principle. In this article, we aim to clarify the nature of the relationship between the two principles by investigating the two most likely interpretations of the claims that can be found in the subject literature. According to the strong interpretation, the two principles are equivalent and apply to the same subset of physical phenomena; according to the weak interpretation, the two principles are merely analogous to each other by virtue of their similar formal structures. As we show, adopting the stronger reading would lead to a dilemma that is untenable for the proponents of the free energy principle, thus supporting the adoption of the weaker reading for the relationship between the two constructs.

In this paper, we present a new model for heat transfer in compressible fluid flows. The model is derived fromHamilton's principle of stationary action in Eulerian coordinates, in a setting where the entropy conservation is recovered as an Euler--Lagrange equation. A sufficient criterion for the hyperbolicity of the model is formulated. The governing equations are asymptotically consistent with the Euler equations for compressible heat conducting fluids, provided the addition of suitable relaxation terms. A study of the Rankine--Hugoniot conditions and Clausius--Duhem inequality is performed for a specific choice of the equation of state. In particular, this reveals that contact discontinuities cannot exist while expansion waves and compression fans are possible solutions to the governing equations. Evidence of these properties is provided on a set of numerical test cases. [ABSTRACT FROM AUTHOR]

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AbstractThe free vibration characteristics of functionally graded cylindrical shells under magnetic and temperature fields are investigated. According to Donnell’s theory, the kinetic energy, deformation energy, and their variational expressions are obtained. Based on the electromagnetic elasticity theory, the eddy current electromagnetic and magnetization force models are established. Using Hamilton’s principle, the magneto-thermo-elastic free vibration partial differential equations are derived. The modal frequency differential matrix equation with diverse boundary conditions is derived from the Galerkin integral technique. Through numerical examples, the curves and three-dimensional surface diagrams depicting the variations for the natural frequency with diverse parameters are obtained. The feasibility of the proposed solution method in this paper is supported by literature examples. The influence of magnetic induction intensity, temperature, power-law index, wave number, and shell dimension on the natural frequencies are determined. [ABSTRACT FROM AUTHOR]

We present the principle of virtual action as a foundation of continuum mechanics. Used mainly in relativity, the method has a useful application in classical mechanics and places the notion of action as the basic concept of dynamics. The principle is an extension of virtual work to space-time. It extends the efforts made by d'Alembert and Lagrange. Unlike the classical case of equilibrium, the principle of virtual action becomes a postulate for the formulation of models in dynamics. It allows to use a minimal set of clear conjectures and is extended to the case of media with dissipation; it can be used for more complex systems. [ABSTRACT FROM AUTHOR]

This is the first study of the static bending and free vibration response of organic nanoplates based on the combination of nonlocal theory with various shear strain theories, where the theory of shear deformation of the plate has the advantage of requiring no shear correction factor. The equilibrium equation of the plate is derived using Hamilton's principle, the analytic solution is derived using the Navier solution form, and the finite element technique is implemented using a quadrilateral element with four nodes and six degrees of freedom for each node. Moreover, this is the first work to calculate for nanoplates with a nonlocal parameter whose value varies with plate thickness. This study's credibility has been established by comparing it to previously published findings, which are utilized to validate the results of analytical and numerical calculations, respectively. In addition, the study investigates how a range of components impact the displacement and stress response of organic nanoplates. The findings of the study indicate that there are some circumstances in which it is not feasible to disregard the nonlocal parameter while doing calculations for organic nanostructures. [ABSTRACT FROM AUTHOR]

• forced torsional responses of elliptical nanorod based on the nonlocal theory are analyzed. • Linear and nonlinear external torques are selected in this paper. • The size effect is taken into account by investigating the nonlocal elasticity theory. • The effects of the horizontal and vertical aspect ratios, nonlocal parameter are discussed. • it is concluded that the aspect ratio and frequency of excitation are inversely related to angular displacement. The torsional vibration analysis of small-scaled rods is studied in the framework of Eringen's nonlocal theory. The equations of motion and related boundary conditions are deduced by employing the Hamilton principle. The nonlocal elasticity theory based on Eringen's model is used to cover the size-dependent behavior of the nanorods. Galerkin's analytical method is used to solve the equation of motion. In the present work, time-dependent torsional vibration in an elliptical nanorod under the linear and harmonic distributed external torques along with determined amplitude is considered with clamped-clamped end supports. When discussing free vibration analysis, the small-scale effect shows the sufficiency of natural frequencies in different modes. These results are best verified by comparison with results available in the literature. The influences of effective parameters such as nonlocal parameters, the value of aspect ratio, and excitation frequency variations in time-dependent angular displacement and non-dimensional angular displacement are discussed in detail. It is found that an increase in the nonlocal parameter leads to a rise in the value of non-dimensional angular displacement. Also, it is concluded that the aspect ratio and frequency of excitation are inversely related to angular displacement. [ABSTRACT FROM AUTHOR]

This research studied the dynamic stability of the Euler-Bernoulli nanobeam considering the nonlocal strain gradient theory (NSGT) and surface effects. The nanobeam rests on the Pasternak foundation and a sequence of inertial nanoparticles passes above the nanobeam continuously at a fixed velocity. Surface effects have been utilized using the Gurtin-Murdoch theory. Final governing equations have been gathered implementing the energy method and Hamilton's principle alongside NSGT. Dynamic instability regions (DIRs) are drawn in the plane of mass-velocity coordinates of nanoparticles based on the incremental harmonic balance method (IHBM). A parametric study shows the effects of NSGT parameters and Pasternak foundation constants on the nanobeam's DIRs. In addition, the results exhibit the importance of 2T-period DIRs in comparison to T-period ones. According to the results, the Winkler spring constant is more effective than the Pasternak shear constant on the DIR movement of nanobeam. So, a 4 times increase of Winkler and Pasternak constants results in 102 % and 10 % of DIR movement towards higher velocity regions, respectively. Furthermore, the effect of increasing nonlocal and material length scale parameters on the DIR movement are in the same order regarding the magnitude but opposite considering the motion direction. Unlike nonlocal parameter, an increase in material length scale parameter shifts the DIR to the more stable region.

This research focuses on exploring the free vibration behavior of functionally graded (FG) nano-beams. To calculate the effective properties of the FG nano-beam, which varies solely in the thickness direction, the four homogenization schemes Mori-Tanaka, Tamura, Reuss and Voigt are employed. This study employs high-order shear deformation nano-beam theory and derives the governing equations of motion using nonlocal differential constitutive relations of Eringen. Hamilton's principle is utilized in conjunction with the refined three variables beam theory. The consideration of a length scale parameter accounts for small-scale effects. Analytical solutions are obtained for a simply supported FG nano-beam and compared with existing literature solutions. The research also investigates the influence of different homogenization schemes, the nonlocal parameter, beam aspect ratio and various material compositions on the dynamic response of the FG nano-beam.

Quasi-3D high-order shear deformation theories (HSDT) are often more effective for investigating advanced composite thick plates than two-dimensional (2D) theories. The present study examines the specific dimensionality effect of quasi-3D HSDT theories through-thickness stretching on the free vibration behavior of thin-thick rectangular plates. For this purpose, a 3D displacement field defined by only five unknowns is proposed. Besides, it contains a stretching component that contributes to the whole behavior of the plate. The results of the 2D model are compared to the results of the quasi-3D model. In addition, several factors, such as the aspect ratio, geometrical ratio, and material index, illustrate the influence of dimensionality. Young's modulus and densities should be graded in the direction of thickness. The motion equations are deduced based on Hamilton's principle. According to the boundary condition type, Navier's solution method is used for solving the obtained equations. The results show that the inclusion of the stretching component would increase the dynamic response of the thick advanced composite plates. Moreover, the influence of dimensionality is less significant for pure ceramic plates. [ABSTRACT FROM AUTHOR]

Purpose: Considering the size-dependent influence ignored by classical continuum mechanics, a new non-classical Euler–Bernoulli beam model is proposed in this paper. The new fractional viscoelastic nanobeam model is set up using the fractional Kelvin–Voigt viscoelastic model and Hamilton's principle. And the new model studies the total effects of nonlocal elasticity, modified couple stress, and surface energy. Methods: The model represented the fractional integral-partial differential governing equation is solved by Galerkin's and predictor–corrector methods. First, the effects of nonlocal elasticity, modified couple stress, surface energy, and their coupling impact on the nonlinear time response of free vibration of fractional viscoelastic nanobeam are analyzed. Then, in the frame of nonlocal couple-stress elasticity and surface energy theory, the effects of different parameters on the nonlinear time responses of free and forced vibration of fractional viscoelastic nanobeam are discussed. Results and Conclusion: The numerical results show that the fractional order must be considered in the modeling of viscoelastic nanobeam, and the system's damping enhances by the increase of fractional order. Moreover, owing to the correlation between fractional order and excitation frequency, the nonlinear time responses of fractional order to free and forced vibration are different. [ABSTRACT FROM AUTHOR]

A new size-dependent axially functionally graded (AFG) micro-beam model is established with the application of a reformulated strain gradient elasticity theory (RSGET). The new micro-beam model incorporates the strain gradient, velocity gradient, and couple stress effects, and accounts for the material variation along the axial direction of the two-component functionally graded beam. The governing equations and complete boundary conditions of the AFG beam are derived based on Hamilton's principle. The correctness of the current model is verified by comparing the static behavior results of the current model and the finite element model (FEM) at the micro-scale. The influence of material inhomogeneity and size effect on the static and dynamic responses of the AFG beam is studied. The numerical results show that the static and vibration responses predicted by the newly developed model are different from those based on the classical model at the micro-scale. The new model can be applied not only in the optimization of micro acoustic wave devices but also in the design of AFG micro-sensors and micro-actuators. [ABSTRACT FROM AUTHOR]

In this article, functionally graded plates' buckling and bending analyses are investigated using a new shape function. The parabolic transverse shear stresses throughout the thickness are regarded by this function as meeting the shear stress-free surface conditions and enabling an accurate distribution of shear deformation according to the thickness of the plate without integrating shear correction factors. Compared to previous shear theories, this higher-order shear theory has the fewest unknowns. The equations for the functionally graded plates are produced by employing the Hamiltonian principle, and the solutions are obtained using Napier's technique. The outcomes of the current analysis are provided and contrasted with those found in the literature. [ABSTRACT FROM AUTHOR]

A rigorous analysis is undertaken based on the analysis of both Galilean and Lorentz (Poincaré) invariance in field theories in general in order to (i) identify a unique analytical scheme for canonical pairs of Lagrangians, one of them having Galilean, the other one Poincaré invariance; and (ii) to obtain transition conditions for the state function purely from Hamilton's principle and extended Noether's theorem applied to the aforementioned symmetries. The general analysis is applied on Schrödinger and Klein–Gordon theory, identifying them as a canonical pair in the sense of (i) and exemplified for the scattering problem for both theories for a particle beam against a potential step in order to show that the transition conditions that result according to (ii) in a 'natural' way reproduce the well-known 'methodical' continuity conditions commonly found in the literature, at least in relevant cases, closing a relevant argumentation gap in quantum mechanical scattering problems. [ABSTRACT FROM AUTHOR]

This research focuses on exploring the free vibration behavior of functionally graded (FG) nano-beams. To calculate the effective properties of the FG nano-beam, which varies solely in the thickness direction, the four homogenization schemes Mori-Tanaka, Tamura, Reuss and Voigt are employed. This study employs high-order shear deformation nano-beam theory and derives the governing equations of motion using nonlocal differential constitutive relations of Eringen. Hamilton's principle is utilized in conjunction with the refined three variables beam theory. The consideration of a length scale parameter accounts for small-scale effects. Analytical solutions are obtained for a simply supported FG nano-beam and compared with existing literature solutions. The research also investigates the influence of different homogenization schemes, the nonlocal parameter, beam aspect ratio and various material compositions on the dynamic response of the FG nano-beam. [ABSTRACT FROM AUTHOR]

Abstract The use of nanoscale arch structures in vibration detection has become popular due to their ability to capture resonance in two independent directions. This study investigates the vibration behavior of a small-scale cantilever curved beam structure made of a functionally graded carbon nanotube reinforced nanocomposite core and piezoelectric surface layers on a viscoelastic substrate. The piezoelectric layers convert motion into electrical signals, allowing for the harvesting of high-frequency motions in a delicate manner. Various theories, including the first-order shear deformation theory, strain gradient theory, and nonlocal theory, are used to capture the mechanical properties of the FG nanocomposite core at both nano- and micro-scales. The equations of motion and natural boundary conditions are derived using Hamilton’s principle and solved using Akbari-Ganji’s method. An artificial intelligence method is also presented to simplify the solution procedure and calculate numerical results based on important parameters in the vibrational behavior analysis. The study demonstrates the significance of viscoelastic properties of the substrate and the volume fraction of CNT in composite core in the frequency analysis of the arch structure, as shown by the results of both the numerical and AI methods. [ABSTRACT FROM AUTHOR]