An analytical solution of boundary problems of shell deformation mechanics

: The Galerkin method is applied to Flugge's differential equations for the vibration of a cylindrical shell under axial load and external pressure to obtain a 3N x 3N characteristic equation in matrix form. N is the number of terms in the assumed series of displacement functions for the u, v, and w displacements which can be selected to satisfy various boundary conditions. For the freely-supported cylinder an exact solution exists, and the various assumed modes uncouple, reducing the problem to the solution of a 3 x 3 characteristic equation for each mode. The third order characteristic equation for the freely- supported cylinder was solved for a wide range of shell parameters. The natural vibration frequencies and buckling values for axial load and external pressure for all three eigenvalues associated with each mode, are presented in a series of figures. The square of the vibration frequency for any mode was found to vary linearly with axial load, and approximately linearly with a pressure loading for modes with two or more circumferential waves.

The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy. The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy.

In everyday life, especially in the electrical circuit, there are many usage of matrix. One use of a matrix is found in the system of linear equations. In the field of electrical circuits there are also problems involving systems of linear equations in matrix form. To solve the system of linear equations in matrix form, in addition to using elementary row operations, also used matlab.

The nonlinear dynamics model of gearing system is developed based on RV transmission system. The influence of the nonlinear factors as time-varying meshing stiffness, backlash of the gear pairs and errors is considered. By means of the Lagrange equation the multi-degree-of-freedom differential equations of motion are derived. The differential equations are very hard to solve for which are characterized by positive semi-definition, time-variation and backlash-type nonlinearity. And linear and nonlinear restoring force are coexist in the equations. In order to solve easily, the differential equations are transformed to identical dimensionless nonlinear differential equations in matrix form. The establishment of the nonlinear differential equations laid a foundation for The Solution of differential equations and the analysis of the nonlinearity characteristics.

As already mentioned in Chapter 1, we refer to mathematical models that can be expressed as systems of ordinary differential equations and deal with the solution of problems in mechanics and, in general, in applied sciences. Therefore, it is important to deal with mathematical aspects related to the analysis of differential equations and with the solution of initial and boundary-value problems. This chapter refers to the above topics and is organized in seven sections: Section 2.2 deals with a formal classification of ordinary differential equations. Linear and nonlinear properties are identified together with the analysis of some additional properties of the system. Physical interpretations are often related to this type of classification. Section 2.3 deals with the mathematical statement of problems, which is obtained by linking the evolution equation to the conditions that are necessary to its solution. If these conditions are given at the initial value of the independent variable, say t = 0, one has an initial-value problem, also called a Cauchy problem. Otherwise, if these quantities are given at both ends of the range of the independent variable, say t = 0 and t = T, one has a boundary-value problem. The solution of problems has to satisfy both the differential equation and the initial and/or the boundary conditions. This section is mainly devoted to the analytic treatment of the Cauchy problem and will report the basic theorems which provide the conditions necessary to ensure existence, uniqueness, and regularity of the solution. Section 2.4 deals with solution methods of the initial-value problem. Analytic methods can be developed in order to solve linear problems, while computational schemes are needed for nonlinear problems. Actually, this matter is proposed as a concise guide to mathematical methods and to the related literature. The aim of this guide is to provide the necessary elements for the interaction with the scientific computation based on Mathematica ® Section 2.5 deals with the representation of solutions and applications both in the linear and in the nonlinear case. Here, a systematic use of scientific computation is developed. As we shall see, the computation is useful not only in the nonlinear case, but also for the fast treatment of linear problems. Section 2.6 deals with boundary-value problems for ordinary differential equations and the problems related to their solution, both analytical and numerical. The last section suggests some problems which are proposed to the reader’s attention for practicing with mathematical methods.

The aim of the work is to study the possibilities provided by new information technologies, object-oriented programming tools and modern operating systems for solving boundary value problems of continuum mechanics described by partial differential equations. To discretize the basic equations, we applied the method of finite differences and finite volumes, which are widely used to solve problems in the mechanics of liquids and gases. The paper considers the implementation of the finite difference methods and the finite volume method with vectorized grid structures, including access to the inner and boundary cells of the grid, as well as the features of the implementation of algorithms at singular points of the computational domain. To solve boundary value problems described by partial differential equations, we developed an approach to the construction of vectorized algorithms and considered the features of their software implementation in the MATLAB package. Vectorization in such tasks, excluding nested loops, is ensured by appropriate data organization and the use of vectorized operations. On the one hand, the developed algorithms widely use MATLAB functions designed for processing vectors and sparse matrices, and on the other hand, they are distinguished by high efficiency and computation speed, comparable to those of programs written in C/C++. The main results imply the numerical solution of a number of problems in continuum mechanics associated with the calculation of stresses in a separate body and the calculation of the field of velocity and temperature in the flow of a viscous incompressible fluid. The features of discretization of the basic equations and the implementation of the corresponding finite-difference and finite-volume algorithms are shown. The use of the MATLAB system opens up new possibilities for the formalization and implementation of finite-difference and finite-volume methods for the numerical solution of boundary value problems in continuum mechanics. Despite the fact that the capabilities of the developed algorithms are illustrated by the example of solving fairly simple problems, they admit a relatively simple generalization to more complex problems, for example, solving the Euler equations and Navier–Stokes equations. As part of the work, computational modules were prepared with user programming tools that expand the capabilities of the package and are focused on solving problems in continuum mechanics.

Formulas are derived for the constants ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$ of superconducting alloys for arbitrary temperatures and impurity concentrations. Beginning with Gorkov's equations in matrix form, we calculate the free-energy density up to fourth order in $|\ensuremath{\Delta}|$, using Abrikosov's assumption of a fluxoid lattice. The impurities are treated by the usual averaging technique, retaining only $s$ and $p$ scattering, and a special technique is developed to treat exactly the nonvanishing commutator of different components of the gauge-invariant derivative. The results contain the already known limiting cases. Intermediate values are obtained by performing machine calculations using the general formulas developed here. It is found that the values of ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$ drop relatively strongly even for small impurity concentration, and also depend unexpectedly strongly on the ratio of $s$ and $p$ scattering. The recent result of Caroli, Cyrot, and de Gennes is confirmed according to which ${\ensuremath{\kappa}}_{2}$ is approximately equal to ${\ensuremath{\kappa}}_{1}$ in the dirty limit.

The explicit finite-difference scheme is widely used in seismic wave simulation. Using large time steps can significantly reduce the number of iterations and thus improve computational efficiency, but this approach faces two main challenges: reduced accuracy caused by temporal dispersion and restrictions imposed by the Courant-Friedrichs-Lewy (CFL) stability condition. Based on the wavefield iteration equation in matrix form, an explicit finite-difference method with time step n-tupling and an extended CFL limit is developed for acoustic wave simulation. By combining n successive iteration operators into a single-step operator, two types of n-tupling algorithms are constructed, effectively expanding the CFL stability limit by a factor of n and enabling time steps well beyond conventional thresholds. Both theoretical analysis and numerical experiments demonstrate that simulations with time step n-tupling achieve accuracy equivalent to conventional single-step methods while reducing computation time to approximately 1/n, resulting in a corresponding n-fold increase in overall computational efficiency. Time dispersion is suppressed using two complementary strategies: for relatively small base time steps, n-tupling inherently attains the accuracy of smaller steps; for relatively large base steps, a time-dispersion transform is applied to eliminate errors and maintain high numerical accuracy throughout the simulation.

An effective fitting scheme within the thermo-viscoelastic model is proposed for estimation of dynamic eigenmodes of an ionic melt from ab initio molecular dynamics (AIMD) simulations. Dynamic eigenmodes are solutions of the generalized Langevin equation in matrix form, obtained on the basis set of eight dynamic variables. Fitting parameters are used only in the generalized hydrodynamic matrix T(k) for matrix elements involving heat density and heat current correlations only, because of huge computational efforts needed to calculate them directly in ab initio simulations. In the proposed scheme six AIMD-derived time correlation functions, three partial density-density and three partial current-current ones, are recovered by the proposed theoretical approach, which satisfies the exact sum rules up to the fourth frequency moments of partial dynamic structure factors. The suggested theoretical analysis was applied to estimation of propagating hydrodynamic and non-hydrodynamic eigenmodes in molten NaCl.

Double-shifted Chebyshev polynomials are developed in this study to approximate the solutions of the convolution integral, Volterra integral equation, and Fredholm integral equation. This method simplifies the computations of integral equations to the successive solutions of a linear algebraic equation in matrix form. In addition, the computational complexity can be reduced remarkably. Three examples are illustrated. It is seen that the proposed approach is straightforward and convenient, and converges faster in finding approximations than other existing orthogonal function methods.

The construction of composite models for the mechanics of a support of a set of rigid bodies

The detection performance of the matched filter with the transmitted signal as a reference signal will be degraded in an underwater acoustic channel due to the multipath effect and time variability. Two methods of the channel match are presented in this paper: (1) Pulse-to-pulse Correlation method; (2) Deconvolution method. In the first method the received signal is used as a reference signal of the matched filter. In the second method a short pulse is used to excite the channel. To obtain the response function of the channel, the Wiener-Hopf equation in matrix form is used. The solved response function of the channel is convoluted with the transmitted signal and the result of the convolution is used as a reference signal of the corrected matched filter. In this paper, with the aid of computer the corrected match for channel model and for the real underwater acoustic propagation channel is given.

The dynamical equations of a rotating body whose motion is referred to an arbitrarily rotating reference frame are often desired in linearized form in connection with stability and orientation control studies. The method of this paper shows how to obtain these linearized equations in matrix form, saving considerable labor and promoting accuracy relative to the linearization process done at the sealer level. Illustrative applications are made to a librating orbiting body and a one-axis gyroscope on a rotating base.

The physical model of semi-active quarter-coach suspension was established based on the theories of Buckingham’s Pi theorem in this paper. According to the characteristics of the semi-active suspension, a new intelligent control algorithm based on the Fuzzy logic control and cerebellar model articulation control techniques-Fuzzy cerebellar model articulation control (FCMAC) was presented and used to perform online control of semi-active suspension for the first time, novel weight-update laws were derived that guarantee the stability of closed-loop system, both information retrieval and learning rules were described by algebraic equations in matrix form. The results of experiment of closed -loop excited by three typical vibration signals showed that FCMAC control strategy can reduce the value of mean square root of vertical acceleration for semi-active suspension system to be lower 23%, 25% and 18% compared with the traditional control strategy such as PID.

A plane-wave-based approach for complex photonic band structure is developed and applied to analyse the behaviour of electromagnetic (EM) waves at the surface of semi-infinite and finite photonic crystals (PCs). The complex photonic band structures of two-dimensional and three-dimensional PCs have been calculated by the present approach and verified by conventional plane wave expansion method and plane-wave-based transfer matrix method. Based on the eigen modes of present approach, equations in matrix form are given to describe the boundary conditions of PCs, and consequently the behaviour of EM waves at the surface of semi-infinite and finite PCs is discussed. These results have also verified the validity and efficiency of the current approach.