Simplified non-linear dynamical equations of circular cylindrical shells

Within the context of the Donnell - Mushtary - Vlasov equations for thin circular cylindrical shells, the fundamental frequency of simply supported circular cylindrical shells is determined. In this work, it is shown that with regard to the frequency equation for circular cylindrical shells, there exists a quantity Ω = (ωmna)2 which must remain constant and invariant once we fix the slenderness and the thickness ratios l/a and a/h. The quadruple logarithmic coupling among the parameters ω, l/a, a/h and m is fully displayed by means of a set of parallel straight lines. Furthermore, the nature and influence of the longitudinal wave number m on the natural frequencies ωmn is fully discussed.

Known results on asymptotic two‐dimensional equations for circular cylindrical shells, including the effects of transverse shear and normal stress deformation, are supplemented by upper‐ and lower‐bound determinations of influence coefficients, using minimum‐potential and complementary energy principles in conjunction with asymptotic‐expansion results. The new bound analysis shows that the consequences of the asymptotic two‐dimensional theory are in exact agreement, except for terms which are small of higher order, with the corresponding consequences of three‐dimensional theory, for some classes of edge conditions. The analysis also shows the nature of the differences between results of two‐ and three‐dimensional theory, as a function of geometrical and elastic parameters, where this difference is of importance because of the effect of a St. Venant boundary layer.

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.

The paper considers a mixed initial-boundary value problem for a parabolic equation with nonhomogeneous boundary conditions. The classical approach to search for analytical solution of such problems in the first phase involves variable substitution, leading to a problem with homogeneous boundary conditions. Reference materials [1] give, as a rule, the simplest types of variable substitutions where new and old unknown functions differ by a term, linear in the spatial variable. The form of this additive term depends on the type of the boundary conditions, but is in no way related to the equation under consideration. Moreover, in the case of the second boundary-value problem, it is necessary to use a quadratic additive, since a linear substitution for this type of conditions may be unavailable. The courseware [2] - [4], usually, ends only with the first boundary-value problem generally formulated. The paper considers a substitution that takes into account, in principle, the form of a linear differential operator. Namely, as an additive term, it is proposed to use the parametrically time-dependent solution of the boundary value problem for an ordinary differential equation obtained from the original partial differential equation by the method of separation of the Fourier variables. The existence of the proposed substitution for boundary conditions of any type is proved by the example of a non-stationary heat-transfer equation with the heat exchange available with the surrounding medium. In this case, the additive term is a linear combination of hyperbolic functions. It is shown that, in addition to the insensitivity to the type of boundary conditions, the advantages of a new substitution in comparison with the traditional linear (or quadratic) one include a much simpler structure of the solution obtained. Just the described approach allows us to obtain a solution with a clearly distinguished stationary component, in case a stationarity occurs, for values of the time variable that exceeds relaxation time. This effect extends not only to parabolic equations, but is also observed in equations describing oscillatory processes with damping. It is also shown that in the simplest particular case of an equation with constant coefficients without heat exchange with the surrounding medium, the proposed substitution becomes appropriately linear and coincides with the classical one. The use of the proposed approach allows students majoring in engineering to expand a range of educational and research tasks while learning a course in partial differential equations.

Advection-diffusion equations are frequently encountered in many fields and have become a very active research area. However, in most cases, these equations concern different time and space scales, which make it impossible to derive explicit solutions to these equations. To overcome this difficulty, the theory of homogenization aims to approximate the original differential equation with rapidly oscillating coefficients by an effective homogenized equation with constant or slowly varying coefficients. The homogenized equation is often quite suitable for theoretical analysis or numerical methods. This paper investigates the homogenization principle of an advection-diffusion partial differential equation. The novelty of the parabolic partial differential equation we consider is that the advection term in the equation is two-scaled, which is rarely considered by others for the homogenization of advection-diffusion equation. Under certain proper assumptions on the coefficient functions of the original advection-diffusion partial differential equation, which ensure the variable elimination, we derive the homogenized equation, which is also an advection-diffusion equation, by the technique of multi-scale expansion. It is shown that the coefficient functions of the original two-scaled equation have different influence on the coefficient functions of the homogenized equation.

A study of the effects of system boundaries on bistable front propagation in nonequilibrium reaction-diffusion systems is presented. Two model partial differential equations displaying bistable fronts, with distinct experimental motivations and mathematical structure, are examined in detail utilizing simulations and perturbation techniques. We see that propagating fronts in both models bounce, trap, pin, or oscillate at the boundary, contingent on the imposed boundary condition, initial front speed and distance from the boundary. The similarities in front boundary interactions in these two models is traced to the fact that they display the same front instability (Ising-Bloch bifurcation) that controls the speed of propagation. A simplified dynamical picture based on ordinary differential equations that captures the essential features of front motion described by the original partial differential equations, is derived and analyzed for both models. In addition to addressing experimentally important boundary effects, we establish the universality of the Ising-Bloch bifurcation. Useful analytical insights into perturbative analysis of reaction diffusion systems are also presented.

This work inspects entropy generation and heat transfer induced by a bioconvection slip flow of nonlinear radiative Carreau-Yasuda hybrid nanofluid (NF) over a convectively heated sphere. Activation energy for microbes is contemplated in order to comprehend its contribution toward flow features. The original partial differential equations (PDEs) are rendered non-dimensional through appropriate conversions, and the resulting PDEs are solved using the overlapping grid spectral collocation algorithm. The impact of diverse flow factors on flow profiles, entropy generation, skin friction, heat, mass, and motile microbes transport rates is analyzed. Key outcomes reveal that hybrid NF flow demonstrates a supplementary indispensable feature in the operation of heat transport compared to mono NF flow. More entropy is generated in the system by using the hybrid NF model along with magnetic field, heat source, convective heating, nonlinear radiation, and viscous dissipation. Thermal fields and rate of heat transport are improved by including nonlinear radiative heat flux in the system. Mass and motile microbes transport rates are respectively enhanced by chemical and microbial reactions, but both quantities are reduced by activation energy. The effects of microbial reactions on flow quantities substantiate the significance of these features for the dynamics of microorganisms. The findings of this study can be useful in the upsurge of thermal performance of the working fluid and contribute toward the improvement of microbial fuel cell performance.

Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high nonlinearity domain. To improve the generalizability, we introduce the novel approach of employing multi-task learning techniques, the uncertainty-weighting loss and the gradients surgery, in the context of learning PDE solutions. The multi-task scheme exploits the benefits of learning shared representations, controlled by cross-stitch modules, between multiple related PDEs, which are obtainable by varying the PDE parameterization coefficients, to generalize better on the original PDE. Encouraging the network pay closer attention to the high nonlinearity domain regions that are more challenging to learn, we also propose adversarial training for generating supplementary high-loss samples, similarly distributed to the original training distribution. In the experiments, our proposed methods are found to be effective and reduce the error on the unseen data points as compared to the previous approaches in various PDE examples, including high-dimensional stochastic PDEs.

In present research article, we address the magnetically controlled thermal and solutal Marangoni convection in the flow of self-rewetting power-law liquid over a disk, in the existence of space dependent heat source. The self re-wetting property of fluid is modelled by considering a quadratic dependence of surface tension on temperature and species concentration. The aforementioned problem is modelled by simplified Navier-Stokes equations. Identifying the appropriate transform variables is essential for developing ODEs (ordinary differential equations) from original PDEs (partial differential equations) that describe the flow conditions. The resulting ODEs are solved by using bvp4c routine of MATLAB and numerical solutions are presented via Graphs and tables, illustrating the impact of several factors on fluid velocity, temperature, concentration. Computation of the quantities of physical interest viz. Nusselt and Sherwood numbers are also done from those numerical solutions. One of the key finding of present research work is that the Marangoni convection works differently for pseudo--plastic fluid and dilatant fluid. On increasing thermal Marangoni convection the temperature of dilatant fluid reaches a peak value much closer to the disk than temperature of pseudo plastic fluid.

In this paper, we propose a method to construct invariant finite-difference schemes for solution of partial differential equations (PDEs) via consideration of modified forms of the underlying PDEs. The invariant schemes, which preserve Lie symmetries, are obtained based on the method of equivariant moving frames. While it is often difficult to construct invariant numerical schemes for PDEs due to complicated symmetry groups associated with cumbersome discrete variable transformations, we note that symmetries associated with more convenient transformations can often be obtained by appropriately modifying the original PDEs. In some cases, modifications to the original PDEs are also found to be useful in order to avoid trivial solutions that might arise from particular selections of moving frames. In our proposed method, modified forms of PDEs can be obtained either by addition of perturbation terms to the original PDEs or through defect correction procedures. These additional terms, whose primary purpose is to enable symmetries with more convenient transformations, are then removed from the system by considering moving frames for which these specific terms go to zero. Further, we explore selection of appropriate moving frames that result in improvement in accuracy of invariant numerical schemes based on modified PDEs. The proposed method is tested using the linear advection equation (in one- and two-dimensions) and the inviscid Burgers' equation. Results obtained for these tests cases indicate that numerical schemes derived from the proposed method perform significantly better than existing schemes not only by virtue of improvement in numerical accuracy but also due to preservation of qualitative properties or symmetries of the underlying differential equations.

Cellular Neural Network (CNN) has been used for solving Partial Differential Equations (PDE). However, the equivalence and stability of system should be considered carefully in a particular problem. In this paper, we introduce the model CNN for solving set of two PDEs describing water flow channels (called Saint Venant equation). We analyze the approximation and topological equivalence issues between Cellular Partial Difference Differential Equation (CPDDE) and its original PDEs. The stability of CNN system is also proved from discovering the equilibrium of the state and output of each cell. The paper has 4 parts. After introduction, part 2 gives a two-layered CNN 1D model for solving PDE Saint Venant equation. In the part 3 the equivalence and stability of the CNN model are proved, then simulation using FPGA. The conclusions are given in the last part.

Magneto-electro-elastic semi-analytical models for free vibration and transient dynamic responses of composite cylindrical shell structures

Optimal gas-power flow (OGPF) is one of the most fundamental problems for the operation of integrated electric-gas systems (IEGSs), where the gas flow dynamics partial differential equations (PDEs) are computational challenging. This paper aims to propose a lightweight and accurate model for gas flow dynamics for the OGPF problem. Firstly, the second-order hyperbolic PDEs are equivalently derived based on the original PDEs. Then, the second-order hyperbolic PDEs are discretized using the tri-layer explicit difference scheme. Given some mild boundary and initial conditions, the time-domain two-port model for gas flow dynamics is established, where the state variables of the in-between segments of the pipeline are eliminated. Finally, the iterative parameter updating procedure is devised to improve the modeling accuracy. The lightweight feature of the proposed gas flow dynamics model is verified by the simulation results of several test systems.

An efficient method to design nonlinear controller for robot manipulators, based on state-space linearization concept, is presented. The method consists of three steps: (1) abbreviated equations which approximate the dynamic equations of a manipulator are calculated by Fourier transform; (2) using the simplified equations, linear dynamic equations which are first order approximations of the equations at an operational point, are derived; and (3) nonlinear compensations are designed so that exact linearization of the manipulator's dynamics equations are achieved. The method was applied to a controller design for a direct drive manipulator. The control ability of the system was verified through digital simulations and some experimental results. >

An efficient method to design nonlinear control algorithm for robot manipulator controllers, based on state-space linearization concept, is presented. The method consists of three steps ; (1) abbreviated equations which approximate the dynamic equations of a manipulator are calculated by Fourier transformation, (2) using the simplified equations, linear dynamic equations which are the first order approximations of the equations at an operational point, are derived, (3) nonlinear compensations are designed so that exact linearization of the manipulator's dynamic equations are achieved. The method was applied to a controller design for a direct drive manipulator. The control ability of the system was verified through digital simulations and some experimental results.