Eighth-order Differential Equation Research Articles

Fluid-structure-material coupling analysis for a floating laminated structure consisting of high-stiffness panels and a soft core

This paper presents a fluid-structure-material coupling analysis for the interaction between water waves and a very large floating laminated structure (VLFLS), which is consisted of two enhanced ultrahigh-performance concrete (UHPC) panels and a middle lightweight foamed rubber core. The representative volume element (RVE) method is used to design the mechanical properties of enhanced UHPC and foamed rubber, and the parameterized formulas are presented to reveal the dependency between macroscale mechanical properties and mesoscale hierarchical characteristics. By idealizing the rubber core as a uniformly distributed spring layer, an eighth-order differential equation of motion of the laminated structure is derived. In the context of linear potential flow theory, a hydroelastic analytical model is developed for the floating laminated structure with finite length under wave action. In the process of solving velocity potentials, a complicated dispersion equation for the wave motion below the laminated structure is derived, and this equation contains two pairs of conjugate complex roots with positive real parts. The various hydrodynamic quantities, including reflection coefficient, transmission coefficient, deflection, shear force, and bending moment, are calculated. The hydroelastic model is confirmed by considering the convergence of calculation results and the energy conservation of wave propagation. The coupled effects of wave action, material characteristics, structural parameters, and edge conditions on the hydroelastic and mechanical response of the floating laminated structure are clarified to provide important information regarding the optimal design of such structures.

Hydroelastic analysis of interaction between water waves and a floating laminated disk

This paper studies the interaction between water waves and a very large floating laminated disk in water of finite depth. The disk is a composite structure consisting of two surface sheets and a middle low-density elastic core layer. Based on the linear potential flow theory, an analytical solution of the hydroelastic problem is developed using the eigenfunction expansion method for the velocity potential of fluid motion. In the solution procedure, the laminated disk is regarded as double circular Euler sheets connected by a series of closely spaced and mutually independent vertical springs, and then an eighth-order differential equation of motion of the laminated disk is derived as the elastic boundary condition of the hydroelastic problem. An approximated model is then developed for the hydroelastic problem in shallow water. The deflection and bending moment of the disk and the free surface elevation near the disk are calculated, and it is found that the series solution for the velocity potential converges rapidly. Typical examples are presented to show the effects of different parameters, including wave frequency, the edge conditions of the disk, and the elastic coefficient of the core layer, on wave force, structural hydroelastic response, and wave field. Moreover, viscoelastic damping is introduced in the core layer, and its effect on the hydroelastic response is evaluated by adopting the complex stiffness method. The results indicate that the wave force on the laminated disk is larger than that on a corresponding rigid one over a very wide range of wave frequencies, and the local deformation of the lower sheet can be suppressed by designing a core layer with viscoelastic damping.

Static, stability and dynamic analyses of second strain gradient elastic Euler–Bernoulli beams

A simplified second strain gradient Euler–Bernoulli beam theory with two non-classical elastic coefficients in addition to the classical constants is presented. The governing equation and the associated classical and non-classical boundary conditions are derived with the aid of variational principles. The simplified second strain gradient theory is governed by an eighth-order differential equation with displacement, slope, curvature and triple derivative of displacement as degrees of freedom. This theory can be reduced to the first strain gradient and classical Euler–Bernoulli beam theories. Analytical solutions for static behaviour, free vibration and stability analyses are presented for different boundary conditions and length scale parameters. Using the numerical Laplace transform, a spectral element is developed for dynamic analysis of a cantilever beam subjected to a Gaussian pulse. Further, spectrum and dispersion relations are derived to study wave propagation characteristics. The gradient effects on the structural response are assessed and compared with the corresponding first strain gradient and classical beam theories. Observations show that the second strain gradient theory exhibiting stiffer behaviour in comparison to the first strain gradient and classical theories. The beam deflection decreases whereas frequencies and buckling load increase for increasing values of the gradient coefficient in comparison to the first strain gradient and classical theories. The forced response for a finite beam reveals a decrease in the amplitude and a shift to smaller time values with an increase in the value of length scale parameter. Additionally, the second strain gradient beam shows a dispersive behaviour, and for a given frequency the wavenumber decreases and the phase speed increases with an increase in the length scale parameter as compared to the first strain gradient beam theory.

Deformation due to surface temperature variation on a spherically layered, transversely isotropic and self-gravitating Earth

SUMMARYWe present a theory of modern, thermally induced deformation in a realistic Earth. The heat conduction equation is coupled with standard elastic deformation theory to construct a boundary-value problem comprised of eighth-order differential equations. The accurate and stable dual variable and position propagating matrix technique is introduced to solve the boundary-value problem. The thermal load Love numbers are defined to describe the displacements and potential changes driven by thermally induced deformation. The proposed analytical method is validated by comparing the present results with exact solutions for a homogeneous sphere, which are also derived in this paper. The analytical method is then applied to a realistic Earth model to evaluate the effects of layering and self-gravitation of the Earth on displacement and changes of potential. Furthermore, the frequency dependence in the thermal load is illustrated by invoking different thermal periodicities in the computation. Thermal anisotropy is also considered by comparing the results obtained using isotropic and transversely isotropic Earth models. Results show that, when simulating thermally induced deformation, invoking a homogeneous spherical Earth leads to results that substantially differ from those obtained using a more realistic Earth model.

Application of Short and Long (Enhanced Vlasov's) Solutions for Cylindrical Shell on Example of Concentrated Radial Force

Abstract Analytical approaches for cylindrical shell are mostly based on expansion of all variables in Fourier series in circumferential direction. This leads to eighth-order differential equation with respect to axial coordinate. Here it is approximately treated as a sum of two fourth-order biquadratic equations. First one assumes that all variables change more quickly in circumferential direction than in axial one (long solution), while the second (short) one is based on opposite assumption. The accuracy and applicability of this approach were demonstrated (Orynyak, I., and Oryniak, A., 2018, “Efficient Solution for Cylindrical Shell Based on Short and Long (Enhanced Vlasov's) Solutions on Example of Concentrated Radial Force,” ASME Paper No. PVP2018-85032) on example of action of one or two concentrated radial forces and compared with finite element method results. This paper is an improvement of our previous work (Orynyak, I., and Oryniak, A., 2018, “Efficient Solution for Cylindrical Shell Based on Short and Long (Enhanced Vlasov's) Solutions on Example of Concentrated Radial Force,” ASME Paper No. PVP2018-85032). Two amendments are made. The first is insignificant one and use slightly modified expressions for bending strains, while the second one relates to the short solution. Here we do not consider any more that circumferential displacement is negligible as compared with radial one. Eventually this improves the accuracy of results, as compared with previous work. For example, for cylinder with radius, R, to wall thickness, h, ratio equal to 20, the maximal inaccuracy for radial displacement in point of force application decreases from 5% to 3%. For thinner cylinder with R/h = 100, this inaccuracy decreases from 2.5% to 1.25%. These inaccuracies are related to larger terms in Fourier expansion, the significance of which decrease when length or area of outer loading becomes greater. The last conclusion is demonstrated for the case of distributed concentrated force acting along short segment on axial line.

On the limit cycles for a class of eighth-order differential equations

Abstract In this article, we provide sufficient conditions for the existence of periodic solutions of the eighth-order differential equation x ( 8 ) - ( 1 + p 2 + λ 2 + μ 2 ) x ( 6 ) + A x ⃜ + B x ¨ + p 2 λ 2 μ 2 x = ɛ F ( t , x , x ˙ , x ¨ , x ⃛ , x ⃜ , x ( 5 ) , x ( 6 ) x ( 7 ) ) , {x^{\left( 8 \right)}} - \left( {1 + {p^2} + {\lambda ^2} + {\mu ^2}} \right){x^{\left( 6 \right)}} + A\ddddot x + B\ddot x + {p^2}{\lambda ^2}{\mu ^2}x = \varepsilon F\left( {t,x,\dot x,\ddot x,\dddot x,\ddddot x,{x^{\left( 5 \right)}},{x^{\left( 6 \right)}}{x^{\left( 7 \right)}}} \right), where A = p 2λ2 + p 2 µ 2 + λ2 µ 2 + p 2 + λ2 + µ 2, B = p 2 λ2 + p 2 µ 2 + λ2 µ 2 + p 2λ2 µ 2, with λ, µ and p are rational numbers different from −1, 0, 1, and p ≠ ±λ, p ≠±µ, λ ≠±µ, ɛ is sufficiently small and F is a nonlinear non-autonomous periodic function. Moreover we provide some applications.

Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections.

This paper shows how to obtain highly accurate solutions of eighth-order boundary-value problems of linear and nonlinear ordinary differential equations. The presented method is based on the Theory of Functional Connections, and is solved in two steps. First, the Theory of Functional Connections analytically embeds the differential equation constraints into a candidate function (called a constrained expression) containing a function that the user is free to choose. This expression always satisfies the constraints, no matter what the free function is. Second, the free-function is expanded as a linear combination of orthogonal basis functions with unknown coefficients. The constrained expression (and its derivatives) are then substituted into the eighth-order differential equation, transforming the problem into an unconstrained optimization problem where the coefficients in the linear combination of orthogonal basis functions are the optimization parameters. These parameters are then found by linear/nonlinear least-squares. The solution obtained from this method is a highly accurate analytical approximation of the true solution. Comparisons with alternative methods appearing in literature validate the proposed approach.

Application of Approximate Solutions from The Shell Theory to The Problem of Action of Concentrated Forces Upon a Hollow Cylinder

We construct approximate solutions of the shell theory combined by the corresponding boundary conditions. An assumption concerning the character of behavior of the derivatives of the main functions (variables) makes it possible to reduce the initial eighth-order differential equation of the shell theory to a system of two fourth-order equations whose solutions are obtained in terms of the main variables. The accuracy of the obtained solution is demonstrated for the classical problem of a shell subjected to the action of concentrated forces.

The Solution of the Shells Theory Problems by the Numerical-Analytical Boundary Elements Method

The application of the numerical-analytical boundary elements method (NA BEM) to the calculation of shells is considered. The main problem here is due to the fact that most of the problems of statics, dynamics and stability of shells are reduced to solving an eighth-order differential equation. As a result, all analytical expressions of the NA BEM (fundamental functions, Green functions, external load vectors) turn out to be very cumbersome, and intermediate transformations are associated with eighth-order determinants. It is proposed along with the original differential equation to consider an equivalent system of equations for the unknown state vector of the shell. In this case, calculations of some analytical expressions related to high-order determinants can be avoided by using the Jacobi formula. As a result, the calculation of the determinant at an arbitrary point reduces to its calculation at the point , which leads to a significant simplification of all analytical expressions of the numerical-analytical boundary elements method. On the basis of the proposed approach, a solution is obtained of the problem of bending a long cylindrical shell under the action of an arbitrary load, the stress-strain state of which is described by an eighth-order differential equation. The results can be applied to other types of shells.

Application of the beam method to structural calculation of the long cylindrical concrete shells in the work of Felix Candela

DOI: 10.7764/RDLC.18.1.134 In the present work it is explained how the architect Felix Candela got to analyse the roof long cylindrical shells in reinforced concrete using an method based simply on equilibrium equations. All this, in a moment in which the elastic theory of shells becomes practically impossible to apply due to the involvement of eighth-order differential equations. Felix Candela ‘objective was to get a safely and easily calculation method, respecting the assignment of reinforced concrete and, therefore, obviating the considerations of compatibility and deformation that could undergo the structure. This method, developed previously by other engineers, is explained mathematically choosing an example of work of Felix Candela, done by roof long cylindrical shells, and making the necessary structural calculations proposed.

Application of the beam method to structural calculation of the long cylindrical concrete shells in the work of Felix Candela

In the present work it is explained how the architect Felix Candela got to analyse the roof long cylindrical shells in reinforced concrete using an method based simply on equilibrium equations. All this, in a moment in which the elastic theory of shells becomes practically impossible to apply due to the involvement of eighth-order differential equations. Felix Candela ‘objective was to get a safely and easily calculation method, respecting the assignment of reinforced concrete and, therefore, obviating the considerations of compatibility and deformation that could undergo the structure. This method, developed previously by other engineers, is explained mathematically choosing an example of work of Felix Candela, done by roof long cylindrical shells, and making the necessary structural calculations proposed.

Proceso de cálculo de las cáscaras cilíndricas largas de cubierta en la obra de Félix Candela. El enfoque del equilibrio

El objetivo del presente trabajo consiste en evidenciar como a partir de los años 50 Félix Candela consiguió diseñar y calcular estructuralmente cáscaras cilíndricas largas de cubierta de hormigón armado mediante análisis basados simplemente en ecuaciones de equilibrio, con un claro objetivo: conseguir un método de cálculo sencillo y seguro, respetando la característica de cedencia del hormigón armado y, por tanto, obviando las consideraciones de compatibilidad y deformación que pudiera sufrir la estructura. Y lo hizo en un momento donde la teoría elástica de cáscaras resultaba prácticamente inaplicable al implicar la resolución de ecuaciones diferenciales de octavo orden. 
 
 Se justifica analíticamente la teoría escogiendo un ejemplo de la obra de Félix Candela, ejecutada por medio de cáscaras cilíndricas largas de cubierta, y elaborando el proceso de cálculo necesario propuesto.

Decoupled stability equation for buckling analysis of FG and multilayered cylindrical shells based on the first-order shear deformation theory

Based on the first-order shear deformation and Donnell's shell theory with von Karman non-linearity, one decoupled stability equation for buckling analysis of functionally graded (FG) and multilayered cylindrical shells with transversely isotropic layers subjected to various cases of combined thermo-mechanical loadings is developed. To this end, the equilibrium equations are uncoupled in terms of the transverse deflection, the force function and a new potential function. Using the adjacent equilibrium method, one decoupled stability equation which is an eighth-order differential equation in terms of transverse deflection is obtained and conveniently solved to present analytical expressions for buckling loads of cylindrical shells under any type of loading. These analytical expressions can be used in design and as a benchmark in numerical studies. For numerical purpose, the formulation is applied to FG shells, a three-layer shell laminated of transversely isotropic layers, and sandwich shells with an isogrid lattice core and transversely isotropic face sheets. The results are validated with the existing ones in the literature. Finally, the effect of different parameters on the buckling loads in the presence of combined loadings is discussed in detail. Numerical results show that the existence of an initial hoop stress as a main type of initial imperfection has significant influence on the buckling behavior (including buckling values and mode shapes) of long cylindrical shells subjected to axial loading and this effect reduces for cylindrical shells with a lattice core. Furthermore, the critical values of torsion significantly reduce under combined loading.

To the solution of the problem of bending of a cylindrical shell by the boundary elements method

The solution of the problem of the long cylindrical shell bending by a numerical and analytical boundary elements method is considered. The method is based on the analytical construction of a fundamental system of solutions and Green’s functions for the differential equation of the problem under consideration. This paper is devoted to the determination of these expressions. The semi-moment theory of the cylindrical shell calculation, proposed by V.Z. Vlasov, which for the problem under consideration leads to one eighth-order partial differential equation is used. The problem of the bending of a cylindrical shell is twodimensional, and in the numerical and analytical boundary elements method, plates and shells are considered as generalized one-dimensional modules, so the variational method of Kantorovich-Vlasov was applied to this equation to obtain an ordinary differential equation of the eighth order. Sixty-four expressions of all the fundamental functions of the problem are constructed, as well as an analytic expression for the Green’s function, which makes it possible to construct a load vector (without any restrictions on the nature of its application), and then proceed to the solution of boundary-value problems for the bending of long cylindrical shells under various boundary conditions.

Asymptotic theory of neutral stability of the Couette flow of a vibrationally excited gas

An asymptotic theory of the neutral stability curve for a supersonic plane Couette flow of a vibrationally excited gas is developed. The initial mathematical model consists of equations of two-temperature viscous gas dynamics, which are used to derive a spectral problem for a linear system of eighth-order ordinary differential equations within the framework of the classical linear stability theory. Unified transformations of the system for all shear flows are performed in accordance with the classical Lin scheme. The problem is reduced to an algebraic secular equation with separation into the “inviscid” and “viscous” parts, which is solved numerically. It is shown that the thus-calculated neutral stability curves agree well with the previously obtained results of the direct numerical solution of the original spectral problem. In particular, the critical Reynolds number increases with excitation enhancement, and the neutral stability curve is shifted toward the domain of higher wave numbers. This is also confirmed by means of solving an asymptotic equation for the critical Reynolds number at the Mach number M ≤ 4.

Analytical solution to bending of stiffened and continuous antisymmetric laminates

Abstract Analytical Strip Method is presented for the analysis of the bending-extension coupling problem of stiffened and continuous antisymmetric thin laminates. A system of three equations of equilibrium, governing the general response of antisymmetric laminates, is reduced to a single eighth-order partial differential equation (PDE) in terms of a displacement function. The PDE is then solved in a single series form to determine the displacement response of antisymmetric cross-ply and angle-ply laminates. The solution is applicable to rectangular laminates with two opposite edges simply supported and the other edges being free, clamped, simply supported, isotropic beam supports, or point supports.

Buckling of Stiffened Antisymmetric Laminated Plates

AbstractThe analytical strip method (ASM) is extended in this paper to the buckling analysis of antisymmetric laminated composite plates with bending-extension coupling. A system of three equations of equilibrium governing the buckling response of antisymmetric laminated composite plates is reduced to a single eighth-order partial differential equation, expressed in terms of a displacement function. This equation is then solved analytically to generate the critical value of the in-plane forces (Nx, Ny) for antisymmetric cross-ply and angle-ply laminated and stiffened composite plates. The effects of various parameters, including plate aspect ratio, plate orthotropicity ratio, beam rigidity, and plate bending-extension coupling on the critical values of Nx and Ny on antisymmetric laminated composite plates are discussed. The results derived using the ASM are presented in tabular and graphical forms and compare very well with ones derived using finite-element analysis.

Calculation of natural frequencies and vibration modes of variable thickness cylindrical shells using the Wittrick–Williams algorithm

Dynamic stiffness equations are formulated for variable thickness cylindrical shells, under the assumptions of Donnell, Timoshenko and Flügge theories. Transcendental dynamic stiffness matrices are formed by solving numerically the governing eighth order differential equations using the boundary-value solver COLSYS. Undamped natural frequencies are found using the Wittrick–Williams algorithm. The shell is divided into smaller elements whose clamped end natural frequencies exceed the highest frequency of interest. A parametric study examines the effects of varying the geometry, degree of thickness taper and end conditions on the natural frequencies and mode shapes, providing benchmark results for designers and future researchers.

A note on the numerical solution of high-order differential equations

Numerical solution of high-order differential equations with multi-boundary conditions is discussed in this paper. Motivated by the discrete singular convolution algorithm, the use of fictitious points as additional unknowns is proposed in the implementation of locally supported Lagrange polynomials. The proposed method can be regarded as a local adaptive differential quadrature method. Two examples, an eigenvalue problem and a boundary-value problem, which are governed by a sixth-order differential equation and an eighth-order differential equation, respectively, are employed to illustrate the proposed method.

A generalized differential quadrature rule for bending analysis of cylindrical barrel shells

The generalized differential quadrature rule (GDQR) proposed recently by the present authors is applied here to solve an eighth-order boundary-value differential equation with four boundary conditions at each boundary, which governs the bending analysis of circular cylindrical barrel shells. Thus, multiple boundary conditions are implemented exactly at boundaries without applying the approximate δ-point (non-boundary point) technique. Both Hermite interpolation functions and GDQR’s explicit weighting coefficients are obtained, while the latter are the most important for an accurate implementation of the GDQR. The analytical results for various cylindrical barrel shells are derived and calculated here in order to verify the accuracy of the calculated GDQR results. In implementation of multiple conditions at one point, the GDQR gains a clear advantage over the existing δ-point technique in the differential quadrature applications and manifests a great potential for applications in many fields of applied mathematics.