Sixth-order Differential Equation Research Articles

Application of nonlocal strain gradient theory for the analysis of bandgap formation in metamaterial nanobeams

The aim of this paper is to investigate bandgap formation in an Euler-Bernoulli nanobeam using the nonlocal strain gradient theory. Such a model enables the capture of both softening and stiffening size-dependent behavior of the nanobeams using two length-scale parameters, namely, a nonlocal parameter and a strain gradient parameter. Hamilton's principle is used to develop the normalized equation of motion yielding a sixth order differential equation. The nanobeam is assumed to be infinitely long and the dispersion of an elastic wave in a single representative periodic unit cell of the structure is used to estimate the bandgap edge frequencies. To this end, two unit cell approaches are employed to obtain the dispersion relations, namely, the homogenization and transfer matrix approaches. Both methods yield similar dispersion curves and showed good agreement with classical cases from the literature. A parametric study was carried out to investigate the effect of several parameters on the bandgap formation in the metamaterial nanobeams. These parameters include, the nonlocal parameter, the strain gradient parameter, the length of the unit cell and the resonator mass to unit cell mass ratio. The study conducted using a purely strain gradient model showed a comparable response to that of the classical macro problem with the usual hardening when increasing the value of the strain gradient length scale. Nonlocal and nonlocal strain gradient models introduced coupling between parameters of the study with the occasional introduction of non-monotonic dispersion curves. Such non-monotonic behavior is believed to be characteristic of large values of the nonlocal length-scale.

Axisymmetric motion of a solid particle embedded in a Brinkman micropolar fluid in the presence of a plane wall

The axisymmetric motion of a solid spherical particle embedded in a hydrogel medium in the presence of a planar wall surface is investigated semi-analytically. The hydrogel medium is modeled as a porous medium saturated with a microstructure fluid of micropolar type. The no-slip velocity and no-slip spin boundary conditions are considered at both the particle surface and the plane wall surface. The sixth-order differential equation describing the stream function of the micropolar fluid flow through the voids of the porous medium is constructed under the assumption of low Reynolds numbers. The general solution of the equation satisfied by the stream function in the porous region is obtained from the superposition of basic solutions in both cylindrical and spherical coordinates. To satisfy first the boundary condition at the planar surface, we apply the Fourier–Bessel transforms and then at the surface of the particle by a boundary collocation technique. The collocation scheme for the normalized drag force acting on the particle is calculated with good convergence for various values of the relevant parameter. Our results are in good agreement with the available data in the literature. The findings of the present investigation demonstrate that the presence of the planar surface, micropolarity, and permeability parameters has significant effects on the drag force. This study is motivated by its potential application on micro- and ultra-filtration.

A novel generalized sandwich-type replacement beam for static analysis of tall buildings: Inclusion of local shear deformation of walls

As the length of the shear walls increases, local shear deformation leads to increased error when using classical continuum models for static structural analysis of tall buildings. In this paper, a novel continuous beam of the generalized sandwich type is proposed for the structural analysis of tall buildings that have shear walls of significant length as lateral bracing systems. The proposed continuous beam includes in its formulation an additional kinematic field of rotation due to the local shear deformation of the walls and results from the series coupling of a classic sandwich beam and a shear beam. The equilibrium equations, the constitutive laws, and the boundary conditions of the continuous model are obtained with a variational approach using the Hamilton principle on the relevant Lagrange function. A system of three coupled differential equations leading to sixth-order differential equation and six boundary conditions are obtained. Two closed-form analytical methods are proposed for uniform-height tall buildings. For the case of buildings with variable properties along their height, a numerical method of the modified transfer matrix type is proposed, which is directly derived from the analytical method and avoids the need to calculate the inverse of the classical zero transfer matrix method. The numerical results show the validation of the continuous model and the proposed solution methods. In addition, a parametric analysis of 352 buildings was performed to demonstrate the superiority of the continuous model and the proposed solution methods.

A verified analytical sandwich beam model for soft and hard cores: comparison to existing analytical models and finite element calculations

This paper presents a novel approach of modeling of three-layer beam. Such composites are usually known as sandwich structures if the modulus of elasticity of the core is much smaller than those of the faces. In the present approach, the faces are modeled as Bernoulli–Euler beams, the core as a Timoshenko beam. Taking into account the kinematic and dynamic interface conditions, which means that the perfect bonding assumptions hold for the displacement and each layer is subjected to continuous traction stresses across the interface, a sixth-order differential equation is derived for the bending deflection, and a second-order system for the axial displacement. No restrictions are imposed on the elastic properties of the middle layer, and hence the developed theory also yields accurate results for hard cores. The presented refined theory is compared to analytical models from the literature and to finite element calculations for various benchmark examples. Special focus is laid the boundary conditions and the core stiffness. A parametric study varying the Young modulus of the core shows that the present sandwich model agrees very well with the target solutions obtained from finite element calculations under plane stress assumptions, in particular concerning the transverse deflection, the shear stress distribution and the interfacial normal stress.

Transient responses to an infinite solid with a spherical cavity according to the MGT thermo-diffusion model with fractional derivatives without nonsingular kernels

The main purpose of this paper is to describe the coupling mechanism between heat transfer and diffusion in thermoelastic materials. This will be investigated by deriving a new thermal diffusion model that allows waves to move at limited speeds, unlike the traditional thermodiffusion models. Based on the thermal relaxation concept, the fractional Moore-Gibson-Thompson equation can be used to describe the equations for heat conduction and mass diffusion. The system of equations is developed using the Green-Naghdi Type III theories and the Atangana–Baleanu-type (AB) fractional differential operator instead of the integer derivatives of time. Under the proposed concept of the fractional thermo-diffusion model, a problem of an unbounded elastic structure with a spherical hole is investigated. The surface within the bore is free of traction and exposed to thermal shock and chemical potential. The system of governing equations was combined into a sixth-order differential equation to determine the analytical results. For this purpose, the Laplace transform was used, and a numerical algorithm to calculate the inversions was applied. The numerical results demonstrate that the improved fractional operator considerably impacts the dynamics of all research fields.

Linear and nonlinear vibrations of strain gradient beams with rough surface

This paper aims to study the effect of surface roughness on the linear and nonlinear free vibration behaviors of microbeams based on the strain gradient theory. Linear and nonlinear equations of motion and boundary conditions are derived by using the Hamilton’s principle. A closed-form solution of strain gradient beams (SGBs) is obtained by solving a general sixth-order differential equation. Furthermore, by utilizing the linear mode and Galerkin’s method, an exact solution for nonlinear vibration is also obtained for SGBs. A detailed parametric study is performed to study the influences of surface roughness, scale parameter, beam thickness, and boundary condition on the linear and nonlinear vibration behaviors of SGBs. The present study may provide help for optimal designing and fabricating micro-/nano-resonators.

Highly dispersive optical solitons of the sixth-order differential equation with arbitrary refractive index

Partial differential equation of the sixth order for description of the propagation impulse in nonlinear optics is considered at arbitrary refractive index. Exact solution of differential equation is looked for using the traveling wave reductions. The first integral of nonlinear differential equation is presented. Highly dispersive optical solitons of the equation with arbitrary refractive index is found.

Nonlinear Inverse Problem for a Sixth Order Differential Equation with Two Redefinition Functions

Nonlinear Inverse Problem for a Sixth Order Differential Equation with Two Redefinition Functions

3D Modelling with C2 Continuous PDE Surface Patches

In this paper, we present a new modelling method to create 3D models. First, characteristic cross section curves are generated and approximated by generalized elliptic curves. Then, a vector-valued sixth-order partial differential equation is proposed, and its closed form solution is derived to create PDE surface patches from cross section curves where two adjacent PDE-surface patches are automatically stitched together. With the approach presented in this paper, C2 continuity between adjacent surface patches is well-maintained. Since surface creation of the model is transformed into the generation of cross sectional curves and few undetermined constants are required to describe cross sectional curves accurately, the proposed approach can save manual operations, reduce information storage, and generate 3D models quickly.

Finite element formulation of a homogenized beam for reticulated structure dynamics

This paper presents a finite element formulation of an existing homogenized beam-like model for the description of reticulated structures transverse dynamics. This work deals with elastic periodic lattice structures such as foams, crystals, honeycombs, or multi-story buildings, whose unit cell is made of interconnected beams or plates and repeats itself in one direction. The studied model is a one-dimensional enriched form of the fourth-order Timoshenko beam equation, and the motion is described by a sixth-order differential equation. The higher differential equation order is attributed to an additional kinematic mechanism that may appear under large stiffness contrasts between the cell elements. In this context, the construction of an original finite element approximate solution is proposed. Weak formulations lead to generalized elementary stiffness and mass matrices. Through the analysis of a realistic building in the linear elasticity framework, it is shown that this homogenized beam finite element solution recovers the analytical results and is close to the full detailed finite element structural model. This new formulation simplifies the implementation of the beam-like model in complex configurations, enables parametric studies, and could be easily used for a wide range of applications in structural dynamics, other than free vibration analysis, at reduced computational costs.

A semi analytical method for in-plane free vibrations of arches with a variable curvature

Rayleigh-Ritz method (RRM) is used in this paper in order to investigate the in-plane vibrations of arches with a variable curvature in particular, the frequency parameters and mode shapes. The trial functions are particular solutions of the sixth order differential equation of arch vibrations corresponding to an opening angle equal to one rad. The investigations are made with the follow’s hypotheses: the constant dimensions of the cross-section are small in comparison with the radius, the arch axis is inextensible and the effect of rotary inertia and shear deformation are neglected. The parabolic, catenary, spiral, circular and cycloid arches with different boundary conditions are investigated. The frequency parameters are presented for different types of curved arches with various geometrical properties and boundary conditions and the corresponding mode shapes are plotted. The results are compared with the published literature. The accuracy and relative simplicity of the RRM, applied herein a systematic way with the new choice of trial functions, is established, making it ready to use in more complex situations, such as those of arches with added masses, with non-uniform cross sections or with one or more point supports.

Existence of Nontrivial Solutions for Sixth-Order Differential Equations

We show the existence of at least one nontrivial solution for a nonlinear sixth-order ordinary differential equation is investigated. Our approach is based on critical point theory.

Painlevé analysis and traveling wave solutions of the sixth order differential equation with non-local nonlinearity

In this paper, we study a nonlinear partial differential equation for describing high dispersion optical soliton with non-local nonlinearity. Taking into account the traveling wave reduction, we get system of ordinary differential equations (ODEs) for real and imaginary parts of the original equation. To determine the integrability of equation we apply the Painlevé test for analysis of obtained ODE system. We illustrate that the system of equations does not have the Painlevé property since there is only one integer Fuchs index. However using the Painlevé data we find the compatibility conditions for the ODE system. Under these conditions, the traveling wave solution of nonlinear differential equations are constructed and illustrated.

Exact solution for dynamic analysis of a shape memory polymer plate subjected to the initial conditions

A theoretical solution was presented for the dynamic behavior of a simply supported shape memory polymer (SMP) plate subjected to the initial conditions. According to the time-dependent behavior of SMP, Hamilton’s principle, and the first-order shear deformation theory (FSDT), the equations of motion in terms of time and displacement were derived. It was seen that there are the fourth- and sixth-order differential equations for the in-plane and out-plane deformation components (u and w), respectively. The differential equations were solved theoretically and introduced in terms of time and displacement. In order to validate the theoretical solution, the theoretical results are compared to those obtained by the numerical code written based on the Runge-Kutta fourth-order method for the set of differential equations. The results indicate a good agreement between them. Moreover, the effects of the elastic modulus, viscosity, and the geometric dimensions of the plate on its deformation were investigated. The results showed that the mechanical and geometric parameters, as well as the initial conditions, have more effect on the deformation of the plate.

Sound Transmission Prediction of Sandwich Plates With Honeycomb and Foam Cores and an Emphatic Discussion on Radiation Terms

When applying the modal summation method to the sound transmission loss (STL) prediction of various plates, the assumption of the blocked sound pressure, or alternatively speaking, ignoring sound radiation terms, has obvious simplicity and is sometimes used for the single-layered panels, rib-stiffened plates or heavily damped sandwich plates. For light-weighted sandwich plates with honeycomb and foam cores, however, this assumption is somewhat in doubt and worth examining. Based on sixth-order differential equations governing the flexural vibration of sandwich plates, the prediction formula of STL is derived by the modal summation approach. Theoretical predictions were validated by measurement data. Next, the theoretical formula of STL under the assumption of the blocked sound pressure was examined. The STL discrepancies of sandwich plates caused by sound radiation terms are illustrated. It was found that the STL discrepancies of sandwich plates were closely related to frequency, reached their peak value at the coincidence frequency region. The results indicate that the sound radiation terms, or the couplings between the radiated sound pressure and the plate response, should not be ignored for the prediction of STL for sandwich plates with honeycomb and foam cores.

Integral and differential approaches to Eringen's nonlocal elasticity models accounting for boundary effects with applications to beams in bending

AbstractThe Eringen's fully nonlocal elasticity model is known to lead to ill‐posed boundary‐value problems and to suffer some boundary effects arising from particle interactions impeded by the body's boundary surface. An enhanced model is derived from the original fully nonlocal one by the addition of a regularizing non‐homogeneous local phase which accounts for boundary effects and which leads to a Fredholm integral equation of the second kind, hence to well‐posed boundary‐value problems, without paradoxes, nor other drawbacks. The enhanced integral model applied to a beam in bending proves to be equivalent to a sixth order differential equation with variable coefficients, with extra nonlocality boundary conditions here also derived. Both the integral approach and the differential one lead to a same unique solution of the small‐scale beam problem. An efficient numerical algorithm is presented in which the sixth order differential equation with variable coefficients is reduced to one of the second order, which is addressed by a finite difference method. The proposed theory is applied to a set of engineering beam problems, for each of which the inherent size effects are reported and graphically illustrated. The influence of the length scale parameter upon the beam's response is highlighted by means of a function representing the normalized maximum deflection of the beam as a function of the length scale parameter. It is shown that the enhanced model always predictssoftening size effectsno matter the boundary and loading conditions, and that the related response function generally exhibits awaved pattern with positive slopes first, then negative, as the length scale parameter increases, with a limit asymptotic behavior like an atomic lattice model. A comparison with other theories is also presented together with possible future developments.

A sequence of positive solutions for sixth-order ordinary nonlinear differential problems

Infinitely many solutions for a nonlinear sixth-order differential equation are obtained. The variational methods are adopted and an oscillating behaviour on the nonlinear term is required, avoiding any symmetry assumption.

Stability of solutions and the problem of Aizerman for sixth-order differential equations

This article is devoted to the investigation of stability of equilibrium of ordinary differential equations using the method of semi-definite Lyapunov’s functions. Types of scalar nonlinear sixth-order differential equations for which regular constant auxiliary functions are used are emphasized. Sufficient conditions of global asymptotic stability and instability of the zero solution have been obtained and it has been established that the Aizerman problem has a positive solution concerning the roots of the corresponding linear differential equation. Studies highlight the advantages of using semi-definite functions compared to definitely positive Lyapunov’s functions.

Bio-inspired propulsion of micro-swimmers within a passive cervix filled with couple stress mucus

Background and objectiveThe swimming mechanism of self-propelling organisms has been imitated by biomedical engineers to design the mechanical micro bots. The interaction of these swimmers with surrounding environment is another important aspect. The present swimming problem integrates Taylor sheet model with couple stress fluid model. The thin passage containing micro-swimmers and mucus is approximated as a rigid (passive) two-dimensional channel. The spermatozoa forms a pack quite similar as a complex wavy sheet. MethodsSwimming problem with couple stress cervical liquid (at low Reynolds number) leads to a linear sixth order differential equation. The boundary value problem (BVP) is solved analytically with two unknowns i.e. speed of complex wavy sheet and flow rate of couple stress mucus. After utilizing this solution into equilibrium conditions these unknowns can be computed via Newton-Raphson algorithm. Furthermore, the pairs of numerically calculated organism speed and flow rate are utilized in the expression of power dissipation. ResultsThis work describes that the speed of micro-swimmers can be enhanced by suitable rheology of the surrounding liquid. The usage of couple stress fluid as compared to Newtonian fluid enhances the energy dissipation and reduces the flow rate. On the other hand complex wavy surface also aids the organisms to swim faster.

Generalized RK integrators for solving class of sixth-order ordinary differential equations

In this paper, the classical RK integrators for solving quasi-linear(QL) ordinary differential equations (ODEs) up to 5th-order have been generalized to solve class of 6th-order ODEs. The novel contribution of this work is the derivation of two efficient direct integrators for solving class of ODEs. Non-linear equations of order conditions (OCs) for the proposed RKM methods have derived up to the 7th-order using Taylor expansion. Two RKM integrators of three-stages, 5th-order and four-stages, 6th-order have been derived depend on these OCs. The proposed integrates have been tested using some implementations in order to compare them with classical methods which show that the proposed integrators are less than classical RK methods in term of time complexity and function evaluations.