Classical Boundary Conditions Research Articles

ASYMPTOTIC ANALYSIS OF STURM-LIOUVILLE PROBLEM WITH DIRICHLET AND NONLOCAL TWO-POINT BOUNDARY CONDITIONS

In this study, we obtain asymptotic expansions for eigenvalues and eigenfunctions of the one–dimensional Sturm–Liouville equation with one classical Dirichlet type boundary condition and two-point nonlocal boundary condition. We analyze the characteristic equation of the boundary value problem for eigenvalues and derive asymptotic expansions of arbitrary order. We apply the obtained results to the problem with two-point nonlocal boundary condition.

On minimizers of the 2D Ginzburg–Landau energy with tangential anchoring

We analyze Ginzburg–Landau minimization problems in two dimensions with either a “strong or weak” tangential boundary condition. These problems are motivated by experiments in liquid crystals with boundary defects. The imposition of classical boundary conditions on a liquid crystal surface is referred to as “strong anchoring”, while “weak anchoring” refers to the inclusion of boundary energy terms. In the singular limit when the correlation length tends to zero, we show that boundary defects will be observed for weak anchoring, while both boundary and interior vortices are possible for strong anchoring.

Torsional vibrational analysis of irregular single-walled carbon nanotube with elastic-support boundary conditions

Understanding the torsional vibration characteristics of Single-Walled Carbon Nanotubes (SWCNTs) is of great interest to ensure the reliable design of various nanodevices in which SWCNT is a common structural component. The previous studies were usually limited to uniform SWCNTs under classical boundary conditions. Motivated by this limitation, a practical, accurate solution is created for the torsional vibration of an irregular SWCNT with elastic-support boundary conditions. The torsional vibration of an irregular single-walled carbon nanotube with elastic-support boundary conditions is modeled by Flügge's principle. Unlike the uniform carbon nanotube, effects of irregularity and stiffness parameters are considered within the framework of Flügge's model. Stiffness parameters are defined as a function of complex torsional wave number, which makes all the subsequent calculations be treated precisely and efficiently. The governing equations are derived using Flügge's principle, and the wave propagation approach is applied to solve these equations. The accuracy and efficiency of the current model are then illustrated by a variety of numerical examples. The impact of various parameters, including surface irregularity, stiffness, and nanotube thickness, on the torsional vibration of irregular SWCNT, is examined. According to this study, the results indicate that the frequency calculated is influenced by the irregularity parameter compared with the case of uniform SWCNT. The effects of the parameters of irregularity, stiffness, and nanotube thickness are investigated and discussed. The results show that, an increase in irregularity's parameter leads to a decrease in the natural frequency of irregular SWCNT with elastic-support boundary conditions. In addition, an increase in nanotube's thickness leads to a notable decreasing in the natural frequency of irregular SWCNT. The finding obtained in this work may serve as valuable references for the designs of Nano-oscillators and nanodevices, where the torsional vibration of the irregular SWCNT has a remarkable effect on the functioning of these devices.

Viscous Theory for the Vibrations of Coaxial Cylinders: Analytical Formulas for the Fluid Forces and the Modal Added Coefficients

Abstract This article addresses the small-amplitude forced beam vibrations of two coaxial finite-length cylinders separated by a viscous Newtonian fluid. A new theoretical approach based on an Helmholtz expansion of the fluid velocity vector is carried out, leading to a full analytical expression of the fluid forces and subsequently of the modal added mass and damping coefficients. Our theory shows that the fluid forces are linear combinations of the Fourier harmonics of the vibration modes. The coefficients of the linear combinations are shown to depend on the aspect ratio of the cylinders, on the separation distance, and on the Stokes number. As a consequence, the linear fluid forces do not have, in general, the same shape as the forced vibration mode, so that the fluid makes it possible to couple vibration modes with different wave numbers. Compared to the previous works, the present theory includes the viscous effects of the fluid, accounts for the finite length of the cylinders, does not rely on the assumption of a narrow annulus, and covers in a unique formulation all types of classical boundary conditions for an Euler–Bernoulli beam. The theoretical predictions for the modal added mass and damping coefficients (self and cross) are corroborated numerically, considering rigid, pinned-pinned, and clamped-free vibrations.

Modeling Electric Vehicle’s Battery Module using Computational Homogenization Approach

Numerical simulations of heterogeneous structures like battery modules of electric vehicles are challenging due to the various length scales involved in it. Even with the latest computing technology, it is impossible to simulate the crash scene of the full vehicle resolving all length scales. Such hurdles have prevented manufacturers to understand the mechanical response of battery packs in vehicle crash scenarios. In this work, the problem of multiple length scales was solved using the RVE technique based on homogenization theory. An appropriate representative volume element was identified, and a 3D FE model was developed. Classical first-order boundary conditions were used in this research work. The RVE was subjected to several macroscopic deformations, and its response was obtained. The homogenized material properties were computed from the obtained responses, and a material model available in LS-Dyna’s material library was selected and calibrated to describe the nonlinear multiaxial behavior of the homogenized battery module at the macroscale. For validation, the USABC Crush and Drop Tests were simulated for the detailed and homogenized battery modules. The main output of this research is a robust and computationally efficient tool enabling satisfactory integration of a battery pack model to the vehicle for crash simulations, eliminating the need to simulate micro details at macro length scales. This approach significantly reduced the computational cost. For example, for a Drop Test simulation, the homogenized model reduced the simulation time from 40 hours (detailed model) to about 3 minutes, while maintaining a high precision ( R 2 = 0.9871 ) in predicting the load-displacement response. The system level modeling will enable the stakeholders to perform efficient optimization and safety evaluation for full-scale crashworthiness of electric vehicles.

Time-dependent dynamic response of cylindrical 2D-FG spinning microbeam based on different high-order beam theories applying the GDQEM and Newmark-beta techniques

The nonlocal influence on the cantilever constructions varies significantly. Some researchers feel that nonlocal cantilever structures predict stiffness hardening features; others argue that nonlocal cantilever models cause dynamic softening. In this regard, the vibration behavior of a spinning cylindrical cantilever imperfect functionally graded beam structure is analyzed using several mathematical models based on the nonlocal strain gradient theory. More specifically, higher-order shear deformation theories are investigated in the current study. In addition, the effect of boundary condition modeling is presented. Boundary conditions could be modeled with nonlocal effects or as a classical boundary condition. The governing equations are solved using a technique called generalized differential quadrature element (GDQE) method. The findings of a frequency analysis and a dynamic deflection study, both of which are impacted by a variety of diverse factors, are presented and analyzed. Specifically, the effects of boundary condition model selection are investigated in detail. The findings proved that nonlocality in boundary conditions caused the softer behavior of the cylindrical beam structure by decreasing the natural frequency. On the other hand, using different beam models and boundary conditions could result in a complete difference in obtained results.

Dynamic Green Function Solution of Beams Under Moving Loads With Elastically Supports

The dynamic analysis of bridges simulated as Euler-Bernoulli beam models with elastic supports subjected to mobile loads are analyzed by conventional methods to obtain a new solution for displacement. Generally, these beam supports can be characterized by springs with a given stiffness, which considerably influence the structure's dynamic behavior and even attenuate the dynamic amplification. The solutions proposed until now are defined only on span but not supports. In this paper, we used Green's function, considering boundary and continuity conditions and shear force to study the global behavior of the beam. A new displacement formula is proposed for the beam to support a span according to the velocity of the mobile load, the beam rigidity, and the stiffness of supports. A further study leads to the present two new formulas, which directly give displacements at the level of supports according only to the beam rigidity and supports stiffness and to the load value at any time. The result of this analysis shows that several combined factors influence the vibratory behavior of the beam when it is supported on elastically supports, namely the stiffness of the supports, the rigidity of the beam, its length, the value of the mobile load, and its velocity. The evolution of support stiffness leads to classical boundary conditions. A study of coupling between the beam and supports is presented, with the study of the comportment in function to the ratio between the beam rigidity and spring stiffness.

A Symbolic Method for Solving a Class of Convolution-Type Volterra–Fredholm–Hammerstein Integro-Differential Equations under Nonlocal Boundary Conditions

Integro-differential equations involving Volterra and Fredholm operators (VFIDEs) are used to model many phenomena in science and engineering. Nonlocal boundary conditions are more effective, and in some cases necessary, because they are more accurate measurements of the true state than classical (local) initial and boundary conditions. Closed-form solutions are always desirable, not only because they are more efficient, but also because they can be valuable benchmarks for validating approximate and numerical procedures. This paper presents a direct operator method for solving, in closed form, a class of Volterra–Fredholm–Hammerstein-type integro-differential equations under nonlocal boundary conditions when the inverse operator of the associated Volterra integro-differential operator exists and can be found explicitly. A technique for constructing inverse operators of convolution-type Volterra integro-differential operators (VIDEs) under multipoint and integral conditions is provided. The proposed methods are suitable for integration into any computer algebra system. Several linear and nonlinear examples are solved to demonstrate the effectiveness of the method.

Vibration analysis of a cylindrical shell by using strain gradient theory via a moving Kriging interpolation-based meshfree method

The strain gradient theory is introduced into the Donnell shell model to investigate the vibration characteristics of a thin-walled cylindrical shell under classical and elastic boundary conditions. An energy equation concurrently involving the in-plane shell displacements and out-plane shell deflections is constructed. Meshfree discretization scheme is adopted, in which the moving Kriging interpolation, possessing C2 continuum and the Kronecker delta function property, is used to formulate the shape functions. The accuracy of the proposed method is first validated, and then parametric studies in terms of small-scale parameters, boundary conditions and the length-to-radius and thickness-to-radius ratios on the frequencies and mode shapes are carried out in detail. Natural frequencies evaluated by strain gradient elasticity theory are found to be lower than those of the classical Donnell cylindrical shell model. Some lower natural frequencies corresponding to ultrahigh-order mode shapes appear. This demonstrates that the size effect has a significant influence on the vibration of the thin-walled cylindrical shell. Five types of elastic boundary conditions are listed to illustrate the effect on the natural frequency. Meanwhile, it is concluded that the mesh-free moving Kriging interpolation method has high precision and good stability in dealing with strain gradient Donnell cylindrical shells.

Asymptotic analysis of Sturm-Liouville problem with Robin and two-point boundary conditions

We analyze the initial value problem and get asymptotic expansions for solution. We investigate the characteristic equation for Sturm-Liouville problem with one classical Robin type boundary condition and another two-point nonlocal boundary condition. Finally, we obtain asymptotic expansions for eigenvalues and eigenfunctions.

A unified approach for dynamic characteristic analysis of a standing-wave thermoacoustic engine with general impedance ends

As promising devices for energy conversion, thermoacoustic engines (TAEs) are attracting more and more research attention with regard to renewable thermal energy application. For a TAE, the oscillation frequency is a critical parameter in its design and operation, which can only be estimated currently through an often tedious process using a shooting method or a numerical iteration method, and an analysis model with classical boundary conditions. In this context, this study seeks to establish a unified model for dynamic analysis of a one-dimensional (1D) standing-wave TAE with general impedance ends. First, the differential equations governing the thermoacoustic behavior of the TAE and its impedance boundary conditions are simultaneously solved using the Galerkin method and an improved Fourier series expansion method. Then, the characteristic distributions of oscillation frequency, sound pressure, and volume flow rate are obtained by solving a standard eigenvalue problem. Finally, the effects of engine length, stack length and position, and impedance ends on the dynamic behavior of the TAE’s thermoacoustic system are investigated and analyzed. The results show that engine length has a significantly negative correlation with the modal frequencies in different working gases, and stack position and stack length have a slight effect on the oscillation frequency of the TAE. The arbitrary variations in impedance boundary conditions can be simulated from the open end to the rigid wall and all intermediate cases. The proposed model can efficiently predict changes in dynamic parameters of 1-D TAEs with impedance ends in a unified pattern and can serve as a reliable analysis tool for further research on TAEs coupled with various energy harvesters.

Analysis of vibration stability of fluid conveying pipe on the two-parameter foundation with elastic support boundary conditions

In this study, the vibration stability of fluid conveying pipe resting on two-parameter foundation is investigated under four different elastic support boundary conditions. The harmonic differential quadrature (HDQ) method is applied to solve the governing vibration equation derived based on Euler–Bernoulli beam theory subject to the elastic foundation and boundary conditions. As a result, a general set of second-order ordinary differential equations emerges, and by appropriately setting the stiffness of the end springs, one can easily study the dynamics of various systems with classical or non-classical boundary conditions. The numerical simulations are conducted to study the pipe instability performance with respect to various boundary conditions, elastic support parameters, elastic foundation parameters and fluid mass ratios. The numerical model is validated by comparison with published data. It is found that the elastic support boundary conditions have significant effects on the stability of pipe resting on elastic foundation. The pipe stability performance is very sensitive to the two elastic foundation parameters. Larger fluid mass ratio enhances the pipe flutter stability performance but has no effects on the divergence.

Thermal vibration of functionally graded porous beams with classical and non-classical boundary conditions using a modified Fourier method

Thermal vibration of functionally graded porous beams with classical and non-classical boundary conditions using a modified Fourier method

Unraveling the liquid gliding on vibrating solid liquid interfaces with dynamic nanoslip enactment

Slip length describes the classical no-slip boundary condition violation of Newtonian fluid mechanics, where fluids glide on the solid surfaces. Here, we propose a new analytical model validated by experiments for characterization of the liquid slip using vibrating solid surfaces. Essentially, we use a microfluidic system integrated with quartz crystal microbalance (QCM) to investigate the relationship between the slip and the mechanical response of a vibrating solid for a moving fluid. We discover a liquid slip that emerges especially at high flow rates, which is independent of the surface wetting condition, having significant contributions to the changes in resonant frequency of the vibrating solid and energy dissipation on its surface. Overall, our work will lead to consideration of ‘missing slip’ in the vibrating solid-liquid systems such as the QCM-based biosensing where traditionally frequency changes are interpreted exclusively with mass change on the sensor surface, irrespective of the flow conditions.

Quasinormal Mode Expansion Theory for Mesoscale Plasmonic Nanoresonators: An Analytical Treatment of Nonclassical Electromagnetic Boundary Condition

Nonclassical quantum effects will significantly affect the optical response of plasmonic nanoresonators with mesoscale feature sizes between about 2 and 20nm, and can be fully described by the nonclassical electromagnetic boundary condition (NEBC) expressed with the surface-response Feibelman d parameters. In this Letter, a quasinormal mode (QNM) expansion theory under the NEBC is proposed. By adopting the easily solved classical QNMs under the classical electromagnetic boundary condition as a complete set of basis functions, rigorous expansions of the nonclassical source-free QNMs and source-excited electromagnetic field under the nonperturbative NEBC are provided. With the obtained nonclassical QNMs as basis functions, expansions of the nonclassical source-excited field and Green's function tensor are further obtained. These expansions have a fully analytical dependence on the NEBC and classical QNMs, thus transparently unveiling their impact on the nonclassical QNMs and source-excited electromagnetic field. For instance, a new expression of mode volume is proposed for analyzing the nonclassically corrected Purcell factor. The proposed theory is physically intuitive and computationally efficient which is enabled by the dominance of a small set of classical QNMs, thus providing an efficient tool for understanding and designing mesoscale plasmonic nanoresonators.

Thickness Configuration Effect on Frequency and Critical Speed of Spinning Variable-Thickness Cylindrical Shells

The thickness configuration effect on natural frequency and critical speed of spinning variable-thickness cylindrical shells is investigated in this paper. Based on Love’s thin shell theory, the eigenfrequency equation is derived by employing the Rayleigh–Ritz method in conjunction with Chebyshev polynomials, considering the effects of Coriolis and centrifugal forces due to rotation. Seven types of thickness configurations and various classical boundary conditions are taken into account. Backward and forward wave frequencies, the critical speed, and the mode shapes of the spinning variable-thickness cylindrical shells are solved. Results show that the thickness configuration not only changes circumferential wave number corresponding to the lowest natural frequency, but also affects the critical speed. However, the influence of thickness configuration on mode shapes of the shell is not obvious. The stronger the boundary constraints, the more obvious effect of slope of thickness variation on the critical speed of spinning cylindrical shells.

Nonlinear flexure mechanics of mixture unified gradient nanobeams

The mixture unified gradient theory of elasticity is invoked for the nanoscopic study of the nonlinear flexure mechanics of nanobeams. A mixed variational framework is conceived based on ad hoc functional space of kinetic test fields, wherein a complete set of governing equations, classical and non-standard boundary conditions, and the constitutive relations are integrated into a solitary functional. The size-effect phenomenon associated with the nonlinear flexure of nanobeams are efficiently realized as the stress gradient theory, the strain gradient theory, and the classical elasticity theory are consistently unified within the framework of the mixture unified gradient elasticity. A viable numerical approach is introduced based on the conceived mixed variational framework while the autonomous series solution of the kinematic and kinetic field variables is implemented. As the kinetic field variables are directly determined rather than utilizing post-computations, the established consistent variational theory provides an advantageous means for efficacious approximate approaches. The nonlinear flexural response of the mixture unified gradient beams with kinematics constraints of practical interest in nano-mechanics is detected and thoroughly compared with the linear counterparts in terms of the characteristic parameters. The detected nonlinear flexural characteristics of nano-sized beams can pave the way ahead in the nonlinear mechanics of nano-structures.

Exact series solutions of composite beams with rotationally restrained boundary conditions: static analysis

The structural performance of composite beams is sensitive to load distribution as well as actual boundary conditions. Although the composite beam theory has been solidly established and exact solutions have been readily developed for various loading and boundary conditions, almost all of them are limited to classical boundary conditions (free, pinned and clamped) and there has been little discussion about the actual support conditions in real structures. The general representation of actual boundary conditions can be defined as rotationally restrained edges. In this research, an analytical model with exact series-type solutions was developed for composite beams with rotationally restrained edges. The model and displacement solutions were validated by other analytical methods and numerical results. The influence of rotational and end-slip restraints was investigated. It was found that the deflection and interface slip of composite beams are highly affected by both restraints introduced by the actual boundary conditions. The current model can be used as a benchmark for future design methods considering the realistic boundary conditions of composite beams.

Analytic Approximate Solution of the Extended Blasius Equation with Temperature-Dependent Viscosity

An explicit approximate solution is obtained for the extended Blasius equation subject to its well-known classical boundary conditions, where the viscosity coefficient is assumed to be positive and temperature-dependent, which arises in several important boundary layer problems in fluid dynamics. This problem extends a previous problem by Cortell (Appl Math Comput 170:706–710, 2005) when the viscosity is constant, in which a numerical solution was obtained. A comparison with other numerical solutions demonstrates that our approximate solution shows an enhancement over some of the existing numerical techniques. Moreover, highly accurate estimates for the skin-friction were calculated and found to be in good agreement with the numerical values obtained by Howarth (Proc R Soc A: Math Phys Eng Sci 164(919):547–579, 1938), Töpfer (Z Math Phys 60:397–398, 1912), and Cortell [34] when the viscosity is equal to 1, and when it is equal to 2.

On the dynamics of 3D nonlocal solids

The stress-driven nonlocal continuum theory is a well-established integral elasticity formulation. Modelling the elastic strain via a convolution integral between stress and an averaging kernel, the theory leads to an integro-differential structural problem for arbitrary 3D nonlocal solids. Integral elasticity solutions are mainly available for 1D solids by adopting a bi-exponential kernel and reverting the boundary-value integro-differential problem to an equivalent differential formulation involving both classical and constitutive boundary conditions. A general computational framework to implement the stress-driven theory for 3D nonlocal solids is still missing. This paper tackles this issue, solving the relevant integro-differential problem by a two-field finite-element approach involving displacements and stresses. The approach can handle the dynamics of complex small-size structures of arbitrary shape. Focusing on free vibration responses, two examples are presented: a 2D rectangular plate, serving as benchmark to demonstrate size effects captured by stress-driven nonlocal elasticity, and a 3D solid modelling a typical dual microcantilever resonator with overhang.