Arbitrary Boundary Conditions Research Articles

Mode recombination formula and nonanalytic term in an effective potential at finite temperature on a compactified space

We develop a new formula called a mode recombination formula, and we can recast the effective potential at finite temperature in one-loop approximation for fermion and scalar fields on the D-dimensional spacetime, Sτ1×RD−(p+1)×∏i=1pSi1 into a convenient form for discussing nonanalytic terms, which cannot be written in the form of any positive integer power of the field-dependent mass squared, in the effective potential. The formula holds irrespective of whether the field is a fermion or a scalar and of boundary conditions for spatial Si1 directions and clarifies the importance of zero modes in the Matsubara and Kaluza-Klein modes for the existence of the nonanalytic terms. The effective potential is drastically simplified further to obtain the nonanalytic terms in easier and more transparent way. In addition to reproducing previous results, we find that there exists no nonanalytic term for the fermion field with arbitrary boundary condition for the spatial Si1 direction, which is also the case for the scalar field with the antiperiodic boundary condition for the spatial direction. Published by the American Physical Society 2024

Effect of spatial setting angle on vibration of elastically restrained rotating beams

For a fully articulated rotor system, there may exist a spatial setting angle and elastically restrained root for each individual blade in rotation procedure due to flapping, drag and pitch hinges. Most of previous studies on rotating blades or beams focused on pitch angle and clamped root. In this paper, a three-dimensional dynamic model is proposed for elastically restrained rotating beams with spatial setting angle. The elastic deformation of the beam is expressed by three scalar variables in three-dimensional space including stretch deformation instead of the conventional axial deformation. The longitudinal shrinkage is considered to capture centrifugal forces. The penalty method is adopted to deal with the elastic constraints, which greatly simplifies the construction of admissible functions. A unified formulation is derived by a variational principle combined with Chebyshev orthogonal polynomials. One of the advantages of the present method is that any linearly independent and complete basis functions can be used as admissible function regardless of boundary conditions, and another advantage is that the present method is capable of handling arbitrary boundary conditions by only adjusting the values of penalty factor. The convergence and accuracy of the present method are validated by comparison, and the effects of boundary conditions, angular velocity and spatial setting angle on vibration characteristics are discussed in detail. The results show that natural frequencies increase as the spring stiffness rises in a certain range and the spatial setting angle plays a more significant role on dynamic characteristics for rotating beams.

An analytical method for free vibration analysis of multi‐directional functionally graded porous doubly‐curved shells in thermal environment with various boundary conditions

This paper uses the analytical method with arbitrary boundary conditions to model and analyze the free vibrations of multi‐directionally functionally graded porous (MFGP) doubly‐curved shallow shells resting on the Pasternak foundation in a temperature environment. It is anticipated that all mechanical parameters, except Poisson's ratio, will change in the direction of length, width, and thickness. To comprehensively describe the shell's displacement, strain, and stress fields, a modified first‐order shear deformation theory (FSDT) with an assumption of cosine distribution shear stresses has been developed. The fact that the enhanced FSDT theory does not require the use of shear correction and that the shear stress at the two free faces of the shells is zero are two of the theory's most significant advantages. Using Hamilton's principle and improved FSDT, one may get the governing equation for free vibration analysis of MFGP doubly‐curved shallow shells. The Galerkin approach is proposed to solve the governing equation of MFGP doubly‐curved shallow shells with various boundary conditions. The trustworthiness of the article is evaluated via its publication in the article model's several special cases. From this point on, a collection of findings about the natural frequency of MFGP doubly‐curved shallow shells is identified and shown in the form of tables and graphs. The results provided in this manuscript can be used as a benchmark solution for further studies as far as the vibration behavior of the MFGP doubly‐curved shallow shells is concerned.

The Modeling Method for Vibration Characteristics Analysis of Composite-Laminated Rotationally Stiffened Plate

The composite-laminated rotationally stiffened plate is widely applied in aviation, aerospace, ship, machinery, and other fields. For structural design and optimization, to investigate the vibration characteristics is important. In this paper, a modeling method of composite-laminated rotationally plate is established. The first-order shear deformation theory (FSDT) and the modified Fourier series are applied to construct the admissible displacement function of the stiffened plate-coupled systems. On this basis, the energy function of composite-laminated rotationally stiffened plate is established. Combined with the artificial virtual spring technology, the proposed theory could be used to analyze the vibration characteristics of composite-stiffened plate-coupled systems with various classical boundary conditions or arbitrary elastic boundary conditions. The Rayleigh–Ritz method is used to solve the energy function. Thus, the vibration characteristics of the composite-laminated rotationally stiffened plate are obtained and analyzed. The correctness of the theoretical analysis model was verified through modal experiments. On this basis, the effect of some important parameters on the vibration characteristics of stiffened plate structures is studied, such as the number, thickness, and width of the laminated stiffener, varying structural parameters, and different boundary conditions. This study can provide the theoretical basis for the vibration and noise reduction of such structures.

Vibration analysis of cylindrical shell discontinuously coupled with annular plate with arbitrary boundary conditions

Discontinuously coupled cylindrical shell-plates or plate-like structures are common components of underwater vehicles, and their vibration characteristics change significantly due to modal coupling. The vibration of discontinuously coupled annular plate-cylindrical shell structures is analyzed based on the wave propagation method(WPM) and virtual spring(VS). Taking uniform annular plates and cylindrical shells as spectral elements, the displacement solutions are described by the trigonometric function and Bessel function respectively, and the vibration responses of the annular plate-cylindrical shell with arbitrary boundaries are obtained. Based on VS and weighted least square norm least square method(WLSNLSM), the discontinuous coupling between the annular plate and cylindrical shell is described as the non-uniform distribution of stiffness of virtual spring by continuity condition, and the dynamic stiffness matrices of continuously and discontinuously coupled structure can be obtained simultaneously. The reason for vibration characteristic change under discontinuous coupling is explained from the perspective of modal superposition. Finally, a finite element model is established to verify the accuracy of the proposed method. The influence of general discontinuous coupling on the dynamic behavior of shell structure is analyzed by numerical examples, which provides useful guidance for structural vibration analysis with discontinuous coupling.

Effective medium modelling of real-world multi-modal metamaterial panels achieving broadband vibroacoustic attenuation

Locally resonant metamaterials can achieve unprecedented vibroacoustic performance by subwavelength distributions of small mechanical resonators on a host structure. Substantial broadband vibroacoustic attenuation can be achieved by multi-modal metamaterial panels, which exploit multiple translational and rotational resonator modes to manipulate the overall bending wave propagation. However, the multi-modal metamaterial concept has been studied only for idealized conditions, such as infinite panel extent and uniformly distributed resonators, limiting practical applicability. Efficient methodologies are still needed to study the behaviour of multi-modal metamaterial panels in real-world scenarios. In this work, this challenge is tackled by developing generalized effective medium models, i.e., homogenized material representations through equivalent macro-scale properties, tailored for finite-sized multi-modal metamaterial panels. For the special but important case of simply supported rectangular panels with uniformly distributed resonators, a dedicated analytical effective medium model is developed. For arbitrary boundary conditions and resonator distributions, effective medium finite elements are formulated. The diffuse sound transmission loss (STL) performance is efficiently predicted through Deterministic - Statistical Energy Analysis (Det-SEA), by coupling the effective medium model of the finite-sized metamaterial panel with a diffuse model of the surrounding sound fields. The proposed prediction approaches are validated against detailed FEM modelling, demonstrating that significant computational reductions are achieved while preserving accuracy. Results showcase that multi-modal metamaterial panels maintain broadband vibracoustic attenuation also when subjected to boundary effects and under partial metamaterial treatment.

Traveling wave vibration and critical rotating speed of rotating porous metal conical shell with elastic boundary conditions

This article examines the traveling wave vibration modes of rotating porous metal conical shell (PMCS) by exploiting the differential quadrature method (DQM). A series of artificial springs are introduced to the system to simulate arbitrary elastic support boundary conditions. Making use of the generalization virtual displacements principle and the first-order shear deformation theory (FSDT) of laminate plate, the theoretical formula is derived, in which both the Coriolis force and the initial circumferential tension brought on by the rotation of the conical shell are included. The trigonometric function is used to indicate the circumferential mode shapes, while the DQM is adopted in generatrix direction to acquire the unified solution. By comparing the present research results and those of the previous researches, the reliability and convergence of the present formulation and numerical simulation are confirmed. Considering three different porosity distributions of the PMCS, the effects of porosity distribution, geometrics, spinning speed, boundary constraint types and the circumferential wave number on frequencies and mode shapes of traveling wave vibration of the PMCS are investigated in detail. Furthermore, the critical rotating speed (CRS) and resonance angular speed are discussed for the rotating PMCS.

A Novel Analytical Method for Coupled In-Plane and Out-of-Plane Vibrations of Sandwich Plates with Arbitrary Boundary Conditions

Sandwich plates are popular in the research field of vibration damping and are widely used in numerous engineering domains. Sandwich plates have been studied extensively for their excellent performances by a large number of researchers. Due to the interaction between the out-of-plane and in-plane vibrations of the layers of sandwich plates, it is difficult to solve the displacements of each layer in all directions analytically. To deal with this problem conveniently and accurately, a novel analytical method is presented in this study for coupled out-of-plane and in-plane vibrations of sandwich plates, which can be applied to both free and forced vibrations of sandwich plates with arbitrary boundaries. In this method, the core plate is treated as a three-dimensional (3D) problem, and it is assumed that the displacement of the core plate varies linearly along the thickness. Based on the Kirchhoff hypothesis, the displacement solutions of the base and constrained plates of the sandwich plate in the [Formula: see text], [Formula: see text] and [Formula: see text] directions are expressed as a superposition of one-dimensional (1D) and two-dimensional (2D) Fourier series, respectively. By comparison to the published analytical solutions and numerical results of the finite element method, the proposed method achieves excellent accuracy and reliability. In addition, the influence of out-of-plane and in-plane vibrations of sandwich plates on each other is studied, and the effects of geometrical and material parameters on the dynamic behaviors of a sandwich plate are investigated. The result shows that the out-of-plane vibration affects the in-plane vibration significantly, which means that the coupling effect of the out-of-plane and in-plane vibrations must be taken into account when analyzing in-plane vibration.

Lateral-torsional buckling resistance of non-prismatic and prismatic mono-symmetric I-section steel beams based on stress utilization

The lateral-torsional resistance of prismatic double-symmetric I-section beams is accurately predicted using a mechanically consistent Ayrton-Perry approach, combined with a calibrated generalized imperfection. The corresponding design formulation was recently adopted in the revised version of Eurocode 3. However, for prismatic mono-symmetric I-section beams, the General Case shall be used while for non-prismatic beams only the General Method is available. Both methods present a very large scatter and highly underestimate the lateral-torsional buckling resistance. This paper proposes an extension to the General Formulation for non-prismatic beams with arbitrary boundary conditions, partial lateral restraints, and arbitrary loading for mono-symmetric I-sections. Using an advanced numerical model calibrated with experimental test results, a large parametric study is undertaken, and its results are used to assess the available design methodologies and the proposed method. It is concluded that the General Formulation provides excellent safe-sided estimates of the LTB resistance, and it is confirmed the very poor performance of the General Case and the General Method.

Variational construction of tubular and toroidal streamsurfaces for flow visualization

Approximate streamsurfaces of a three-dimensional velocity field have recently been constructed as isosurfaces of the closest first integral of the velocity field. Such approximate streamsurfaces enable effective and efficient visualization of vortical regions in three-dimensional flows. Here we propose a variational construction of these approximate streamsurfaces to remove the limitation of Fourier series representation of the first integral in earlier work. Specifically, we use finite-element methods to solve a partial differential equation that describes the best approximate first integral for a given velocity field. We use several examples to demonstrate the power of our approach for three-dimensional flows in domains with arbitrary geometries and boundary conditions. These include generalized axisymmetric flows in the domains of a sphere (spherical vortex), a cylinder (cylindrical vortex) and a hollow cylinder (Taylor–Couette flow) as benchmark studies for various computational domains, non-integrable periodic flows (ABC and Euler flows) and Rayleigh–Bénard convection flows. We also illustrate the use of the variational construction in extracting momentum barriers in Rayleigh–Bénard convection.

A novel isogeometric model for dynamic buckling analysis of doubly curved two-directional functionally graded porous shallow microshells in thermal environments via variable length-scale parameters

In this study, the thermo-mechanical dynamic buckling analysis of two-directional functionally graded porous (2D-FGP) shallow microshells resting on elastic foundations with arbitrary boundary conditions is performed using the isogeometric approach. The isogeometric approach takes advantage of non-uniform rational B-spline basic functions to exactly represent the structure geometry models and the attainment of higher order approximation conditions. The characteristics of the material vary in both the thickness and axial x orientation. Especially, a new point in this material length-scale parameter of microshell is analyzed as a function of spatial coordinates and as a function of the material gradient parameters. Using Hamilton’s principle, the modified couple stress theory, and Kirchhoff-Love’s shell theory, the governing equations of a 2D-FGP microshell lying on a Pasternak medium are obtained. A computer program that employs an isogeometric analysis method to analyze the static and dynamic buckling of a 2D-FGP doubly curved shallow microshell with various boundary conditions. The numerical results for thin cylindrical, spherical, and hyperbolic paraboloidal shallow microshells with various planforms, such as rectangular and circular, are presented. Several special cases are used to demonstrate the accuracy and efficacy of the developed model by comparing the numerical results obtained by the proposed formulations with other published data. In addition, the effects of certain parameters, such as the length scale parameters, the power-law indexes, the thickness-to-sides ratio, and the radius ratio, on the dynamic thermo-mechanical buckling response of the 2D-FGP shallow microshells with double curvature are explored in depth.

Construction of exact and asymptotic solutions for the problem of torsional vibrations of a radially inhomogeneous sphere

In this paper, the torsional vibrations of a radially inhomogeneous isotropic open sphere containing no 0 and π poles are studied. It is considered that the elastic moduli and density of the material are linear functions of the radius of the sphere. Cases are considered when the lateral surface of the sphere is stress-free and fixed, and arbitrary boundary conditions are specified on conic sections. The formulated boundary value problems are reduced to spectral problems. After fulfilling boundary conditions specified on the lateral surfaces of the sphere, a dispersion equation is obtained. Exact solutions to the problem are constructed. Next, we analyzed the roots of the dispersion equation for a relatively small parameter characterizing the thickness of the sphere, depending on the oscillation frequency. For a sphere of small thickness, asymptotic zeros are constructed, and their characteristics are studied.

Damage detection based on accelerometers and computer vision measurements of moving load-induced structural responses

Damage identification is an important and challenging issue in structural health monitoring to prevent sudden structural failures. This study introduces a novel damage detection method for beams associated with the cross-correlation function of the moving load-induced full-field displacements. Firstly, through theoretical derivation, the structural responses of a single-span beam under arbitrary boundary conditions subjected to a moving concentrated force were investigated, both before and after damage occurred. The changes in structural responses due to local stiffness damage were analyzed from a theoretical perspective. Then, a method is introduced to extract high-resolution mode shapes, which represent the true state of the structure, from the computer vision-based displacements and accelerations measured at limited locations. These mode shapes were then used for full-field displacement reconstruction, and the changes in the cross-correlation function of these reconstructed displacements were employed for damage detection. The proposed method can obtain the higher-order responses compared to the traditional vision-based methods, and the displacements at the acceleration-unmeasured locations can be improved compared to the conventional data fusion method. Finally, the proposed method was validated both in numerical simulations and experimental tests on a beam model with multiple damage scenarios. The results have demonstrated the capability of the proposed method in mode shape extraction, full-field displacement reconstruction, and damage detection.

An analytical method for vibration analysis of multi-span Timoshenko beams under arbitrary boundary conditions

An analytical method for vibration analysis of multi-span Timoshenko beams under arbitrary boundary conditions

Gegenbauer-Ritz method for free vibration analysis of rotating functionally graded graphene reinforced porous composite stepped cylindrical shells with arbitrary boundary conditions

In this paper, a semi-analytic method with Gegenbauer polynomials as an admissible function is presented. Free vibration analysis of rotating functionally graded graphene reinforced porous composite (FG-GPLRPC) stepped cylindrical shells with arbitrary boundary conditions is analyzed by using the unified Gegenbauer-Ritz method. The effective material properties of rotating FG-GPLRPC stepped cylindrical shells are obtained with Halpin-Tsai micromechanical model and the open-cell body theory. The boundary conditions at both ends of the structure and the continuous coupling between the shell segments are simulated with the artificial spring technique. Then, based on the first-order shear deformation theory (FSDT), the Rayleigh-Ritz method is employed to derive the equations of rotating FG-GPLRPC stepped cylindrical shells. Finally, the influences of several factors on dimensionless frequency of the shell are also assessed. The results show that this method has excellent convergence and higher computational efficiency. Furthermore, the traveling wave frequencies of different modes show different trends with the increase of rotating speed under elastic boundary conditions. In high-order modes, the influence of porosity coefficient is smaller than that of rotating speed.

Nonlinear vibrations of graphene-reinforced porous rotating conical shell with arbitrary boundary conditions using traveling wave vibration analysis

Nonlinear vibrations of graphene-reinforced porous rotating conical shell with arbitrary boundary conditions using traveling wave vibration analysis

Size-dependent and microinertia effects on statics and dynamics based on the modified couple stress theory with arbitrary boundary conditions

Size-dependent and microinertia effects on statics and dynamics based on the modified couple stress theory with arbitrary boundary conditions

The Vibro-Acoustic Characteristics Analysis of the Coupled System between Composite Laminated Rotationally Stiffened Plate and Acoustic Cavities

In order to study vibro-acoustic characteristics between composite laminated rotationally stiffened plate and acoustic cavities in the coupled system, first-order shear deformation theory (FSDT) and modified Fourier series are used to construct a unified analysis model. The involved coupled systems primarily encompass three types: the coupled system between composite laminated rotationally stiffened plate and cylindrical-cylindrical cavities, spherical-cylindrical cavities, and conical-cylindrical cavities. First, the first-order shear deformation theory and the modified Fourier series are applied to construct the allowable displacement function of the composite laminated rotationally stiffened plate and the allowable sound pressure function of the acoustic cavities. Second, the energy functionals for the structural domain and the acoustic field domain are established, respectively. According to the continuity condition of the particle vibration velocity at the coupling boundary between the composite, laminated cylindrical shell and the enclosed cavity, the coupling potential energy between the stiffened plate and two acoustic cavities is introduced to obtain the energy functional of the coupled system. Third, the Rayleigh-Ritz method is utilized to solve the energy functional and, when combined with artificial virtual spring technology, the suggested theory may be used to study the vibro-acoustic characteristics of a coupled system with arbitrary elastic boundary conditions. Finally, based on validating the fast convergence and correctness of the model, this paper will analyze the impact of crucial parameters on vibro-acoustic characteristics. Furthermore, by incorporating internal point forces and point-sound source stimulation, a steady-state response analysis of the coupled system will be conducted. This research can give a theoretical foundation for the vibration and noise reduction of a vibro-acoustic coupling system.

Numerical Method for Calculating Plane Flows of Viscous Gas in a Cylinder

To solve scientific, technological and environmental problems in the field of road transport, it is necessary to develop new mathematical models. At this stage of development of internal combustion engines and their parts, it is necessary to model complex hydrodynamic technologies. This necessities revealing different forms of liquid and gas flows with arbitrary initial and boundary conditions for the region under consideration.The work is devoted to modelling a turbulent flow over a plate located in a cylinder. Reynolds-averaged Navier-Stokes equations were used as a mathematical model of the flow. Turbulent viscosity was calculated using the one-equation Spalart-Allmaras turbulence model. To solve the system of hydrodynamic equations, the finite difference method was used. A numerical method for calculating plane flows of viscous gas in a cylinder using turbulence models is presented. Suitable options were selected based on physical assumptions used to develop the models.Recommendations have been developed on the possibility of using the considered turbulence models.

Enhanced transformation field analysis for reduced-order modeling of problems with cohesive interfaces

We propose a novel transformation field analysis (TFA) technique designed to solve problems with cohesive interfaces. The TFA approach, originally proposed for homogenization purposes, is (i) extended to account for arbitrary boundary conditions (i.e. not only those suitable for homogenization), (ii) endowed with an iterative Newton–Raphson solution scheme involving the consistent tangent matrix and (iii) provided with an arc-length strategy to follow possible snap-back branches. The extension to arbitrary boundary conditions leads to an efficient reduced-order modeling approach for solving complex non-linear problems. Numerical tests demonstrate that the approach performs well in challenging scenarios with several interfaces, and that sharp and multiple snap-back branches can be followed. The comparison with finite element results shows that the proposed technique yields fast yet accurate solutions.