scholarly journals Analysis of the stability of the computational algorithm to a change in the geometric parameters of cylindrical shell structures

2021 ◽  
pp. 12-21
Author(s):  
P. A Bakusov ◽  
A. A Semenov
Keyword(s):  
Mathematical Model ◽  
Cylindrical Shell ◽  
Computational Algorithm ◽  
Ritz Method ◽  
Complex Surface ◽  
Geometric Parameters ◽  
Shell Structures ◽  
Algebraic Equations ◽  
Calculation Algorithm ◽  
Symbolic Calculations

This study deals with testing sustainability of a computational algorithm to a change in geometric parameters of cylindrical shell structures. A change in geometry implies the replacement of one type of a cylindrical shell (elliptic, hyperbolic, parabolic) with another so that the quantitative change (the difference in elevations) in the area under consideration is minimal. On the one hand, this test allows to assessing the correctness of the algorithm itself and is relevant for algorithms that use both numerical methods and symbolic calculations. On the other hand, it allows to evaluating the possibility of simplifying calculations by approximating a complex surface with a simpler one both in understanding the surface definition itself and in expressing its basic characteristics such as Lame coefficients and main curvatures. A mathematical model of deformations of shell structures based on the hypotheses of Timoshenko (Mindlin - Reisner) are used in the work. The model takes into account transverse shifts, geometric nonlinearity and orthotropy of the material, and its written in the form of a functional of the total potential strain energy. The calculation algorithm is built on the basis of the Ritz method to reduce the variational problem of the minimum functional to the solution of a system of nonlinear algebraic equations, and on the method of continuing the solution with the best parameter for its solution. All calculations were carried out in dimensionless parameters. Three types of cylindrical panels are calculated, and critical loads of buckling and deflection fields at subcritical and supercritical moments are obtained. It is shown that for the considered class of problems the previously proposed mathematical model and computational algorithm are resistant to changes in the geometry of the structure.

scholarly journals Thermistor Temperature Measurement Network with a Small Number of Wires

2019 ◽  
pp. 284-292
Author(s):  
Vladimir V. Shaydurov ◽  
Anna A. Korneeva
Keyword(s):  
Mathematical Model ◽  
Computational Experiment ◽  
Computational Algorithm ◽  
Model Problem ◽  
Algebraic Equations ◽  
Multiple Points ◽  
Power Sources ◽  
Linear Algebraic Equations ◽  
Sequential Solution ◽  
Thermistor Temperature

The article proposes a new measuring circuit for temperature control at multiple points of the product using a sequence of thermistors and diodes with three connecting wires and two voltage sources. A mathematical model and a computational algorithm for calculating the resistance of thermistors are presented which consist in the formation and sequential solution of systems of linear algebraic equations for different ratios of voltages of two power sources. A model problem and the results of a computational experiment are considered

scholarly journals Buckling of cylindrical panels strengthened with an orthogonal grid of stiffeners

2020 ◽  
Vol 17 (6) ◽  
pp. 117-125
Author(s):  
A. A. Semenov ◽  
◽  
L. P. Moskalenko ◽  
V. V. Karpov ◽  
M. V. Sukhoterin ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Geometric Nonlinearity ◽  
Computational Algorithm ◽  
Ritz Method ◽  
Buckling Loads ◽  
Orthogonal Grid ◽  
Cylindrical Panels ◽  
Thin Walled ◽  
Best Parameter ◽  
Transverse Shears

The paper presents a mathematical model of deformation of thin-walled cylindrical shell panels, taking into account transverse shears, geometric nonlinearity, and the presence of ribbed stiffeners. Dimensionless parameters are used. The computational algorithm is based on using the Ritz method and the method of continuation of the solution with respect to the best parameter. There are shown the values of critical buckling loads for several variants of structures, depending on the chosen method of taking into account the reinforcement and the number of stiffeners.

Generalized mechanical-mathematical model for optimization of two-bearing shafts of minimal material consumption

2020 ◽  
pp. 8-12
Author(s):  
B.M. Abdeev ◽  
G.A. Guryanov ◽  
E.A. Klimenko
Keyword(s):  
Mathematical Model ◽  
Objective Function ◽  
Process Model ◽  
Geometric Parameters ◽  
Universal Theory ◽  
Minimum Volume ◽  
Applied Mechanics ◽  
Strength Condition ◽  
Material Consumption ◽  
Calculation Algorithm

A universal theory of solving the original problems of applied mechanics was developed to determine the rational geometric parameters of two-bearing three-stage shafts of the smallest volume with two nozzles, covering three modifications of the design under consideration. The optimization process model is brought to the calculation algorithm and is illustrated by a characteristic numerical example. Keywords: optimization, strength condition, minimum volume, extremum, objective function. [email protected]

The limit state of concrete and reinforced concrete rods at complex and longitudinal-transverse bending

2020 ◽  
pp. 60-73
Author(s):  
Yu V Nemirovskii ◽  
S V Tikhonov
Keyword(s):  
General Solution ◽  
Least Squares ◽  
Nonlinear Differential Equations ◽  
Limit State ◽  
Algebraic Equations ◽  
Method Of Least Squares ◽  
Elasticity Limit ◽  
Limit Values ◽  
Symbolic Calculations ◽  
Deformation Diagrams

The work considers rods with a constant cross-section. The deformation law of each layer of the rod is adopted as an approximation by a polynomial of the second order. The method of determining the coefficients of the indicated polynomial and the limit deformations under compression and tension of the material of each layer is described with the presence of three traditional characteristics: modulus of elasticity, limit stresses at compression and tension. On the basis of deformation diagrams of the concrete grades B10, B30, B50 under tension and compression, these coefficients are determined by the method of least squares. The deformation diagrams of these concrete grades are compared on the basis of the approximations obtained by the limit values and the method of least squares, and it is found that these diagrams approximate quite well the real deformation diagrams at deformations close to the limit. The main problem in this work is to determine if the rod is able withstand the applied loads, before intensive cracking processes in concrete. So as a criterion of the conditional limit state this work adopts the maximum permissible deformation value under tension or compression corresponding to the points of transition to a falling branch on the deformation diagram level in one or more layers of the rod. The Kirchhoff-Lyav classical kinematic hypotheses are assumed to be valid for the rod deformation. The cases of statically determinable and statically indeterminable problems of bend of the rod are considered. It is shown that in the case of statically determinable loadings, the general solution of the problem comes to solving a system of three nonlinear algebraic equations which roots can be obtained with the necessary accuracy using the well-developed methods of computational mathematics. The general solution of the problem for statically indeterminable problems is reduced to obtaining a solution to a system of three nonlinear differential equations for three functions - deformation and curvatures. The Bubnov-Galerkin method is used to approximate the solution of this equation on the segment along the length of the rod, and specific examples of its application to the Maple system of symbolic calculations are considered.

scholarly journals Passive Measurement of Three Optical Beacon Coordinates Using a Simultaneous Method

Sensors ◽  
2021 ◽  
Vol 21 (15) ◽  
pp. 5235
Author(s):  
Jiri Nemecek ◽  
Martin Polasek
Keyword(s):  
Mathematical Model ◽  
Relative Position ◽  
Experimental Tests ◽  
Image Sensor ◽  
Geometric Parameters ◽  
Light Sources ◽  
Feature Points ◽  
Passive Measurement ◽  
The Mathematical Model ◽  
Physical Principles

Among other things, passive methods based on the processing of images of feature points or beacons captured by an image sensor are used to measure the relative position of objects. At least two cameras usually have to be used to obtain the required information, or the cameras are combined with other sensors working on different physical principles. This paper describes the principle of passively measuring three position coordinates of an optical beacon using a simultaneous method and presents the results of corresponding experimental tests. The beacon is represented by an artificial geometric structure, consisting of several semiconductor light sources. The sources are suitably arranged to allow, all from one camera, passive measurement of the distance, two position angles, the azimuth, and the beacon elevation. The mathematical model of this method consists of working equations containing measured coordinates, geometric parameters of the beacon, and geometric parameters of the beacon image captured by the camera. All the results of these experimental tests are presented.

Free vibration analysis of three-layer thin cylindrical shell with variable thickness two-dimensional FGM middle layer under arbitrary boundary conditions

2021 ◽  
pp. 109963622110204
Author(s):  
Xue-Yang Miao ◽  
Chao-Feng Li ◽  
Yu-Lin Jiang ◽  
Zi-Xuan Zhang
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Ritz Method ◽  
Admissible Function ◽  
Free Vibration Analysis ◽  
Middle Layer ◽  
Two Dimensional ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions

In this paper, a unified method is developed to analyze free vibrations of the three-layer functionally graded cylindrical shell with non-uniform thickness. The middle layer is composed of two-dimensional functionally gradient materials (2D-FGMs), whose thickness is set as a function of smooth continuity. Four sets of artificial springs are assigned at the ends of the shells to satisfy the arbitrary boundary conditions. The Sanders’ shell theory is used to obtain the strain and curvature-displacement relations. Furthermore, the Chebyshev polynomials are selected as the admissible function to improve computational efficiency, and the equation of motion is derived by the Rayleigh–Ritz method. The effects of spring stiffness, volume fraction indexes, configuration on of shell, and the change in thickness of the middle layer on the modal characteristics of the new structural shell are also analyzed.

A Numerical Study of Unsteady Natural Convection in a Rectangular Enclosure: The Effect of Compressibility

2004 ◽  
Author(s):  
K. M. Akyuzlu ◽  
Y. Pavri ◽  
A. Antoniou
Keyword(s):  
Mathematical Model ◽  
Stokes Equations ◽  
Numerical Study ◽  
Vertical Wall ◽  
Ideal Gas ◽  
Working Fluid ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Rectangular Enclosure ◽  
Circulation Patterns

A two-dimensional, mathematical model is adopted to investigate the development of buoyancy driven circulation patterns and temperature contours inside a rectangular enclosure filled with a compressible fluid (Pr=1.0). One of the vertical walls of the enclosure is kept at a higher temperature then the opposing vertical wall. The top and the bottom of the enclosure are assumed insulated. The physics based mathematical model for this problem consists of conservation of mass, momentum (two-dimensional Navier-Stokes equations) and energy equations for the enclosed fluid subjected to appropriate boundary conditions. The working fluid is assumed to be compressible through a simple ideal gas relation. The governing equations are discretized using second order accurate central differencing for spatial derivatives and first order forward finite differencing for time derivatives where the computation domain is represented by a uniform orthogonal mesh. The resulting nonlinear equations are then linearized using Newton’s linearization method. The set of algebraic equations that result from this process are then put into a matrix form and solved using a Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns (primitive variables) of the problem. A numerical experiment is carried out for a benchmark case (driven cavity flow) to verify the accuracy of the proposed solution procedure. Numerical experiments are then carried out using the proposed compressible flow model to simulate the development of the buoyancy driven circulation patterns for Rayleigh numbers between 103 and 105. Finally, an attempt is made to determine the effect of compressibility of the working fluid by comparing the results of the proposed model to that of models that use incompressible flow assumptions together with Boussinesq approximation.

Vibration Analysis of Laminated Composite Rectangular Plates With General Boundary Conditions

2018 ◽  
Author(s):  
Yu Fu ◽  
Jianjun Yao ◽  
Zhenshuai Wan ◽  
Gang Zhao
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Geometric Parameters ◽  
General Boundary ◽  
Rectangular Plates ◽  
General Boundary Conditions ◽  
Benchmark Solutions

In this investigation, the free vibration analysis of laminated composite rectangular plates with general boundary conditions is performed with a modified Fourier series method. Vibration characteristics of the plates have been obtained via an energy function represented in the general coordinates, in which the displacement and rotation in each direction is described as an improved form of double Fourier cosine series and several closed-form auxiliary functions to eliminate any possible jumps and boundary discontinuities. All the expansion coefficients are then treated as the generalized coordinates and determined by Rayleigh-Ritz method. The convergence and reliability of the current method are verified by comparing with the results in the literature and those of Finite Element Analysis. The effects of boundary conditions and geometric parameters on the frequencies are discussed as well. Finally, numerous new results for laminated composite rectangular plates with different geometric parameters are presented for various boundary conditions, which may serve as benchmark solutions for future research.

Drastic tailorable thermal expansion chiral planar and cylindrical shell structures explored with finite element simulation

2019 ◽  
Vol 210 ◽  
pp. 327-338 ◽  
Author(s):  
Huabin Yu ◽  
Wenwang Wu ◽  
Jianxun Zhang ◽  
Jikun Chen ◽  
Haitao Liao ◽  
...  
Keyword(s):  
Finite Element ◽  
Cylindrical Shell ◽  
Thermal Expansion ◽  
Finite Element Simulation ◽  
Shell Structures ◽  
Element Simulation
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