The complete solution of the problem of a free vibration system needs the determination of all the modal frequencies and of correspondent mode of vibration. Many times in practice it is necessary to ...know only some modal frequencies, sometimes only the fundamental one. The classical method of the vibration problem solution is to write one or more movement equations, using the second Newton’s law. For the systems with distributed parameters, the result is a number of differential equations with partial derivatives. The exact solution of the equations is possible only for a relative low number of cases; for the majority of the problems the approximate solving methods must be used. The paper presents comparative studies for the shaft-bearing assembly of the centrifugal fans, regarding the vibration behavior during the service. Two type of modeling has been realized: the first one was an analytical simulation for the ideal shaft, using Mathcad software. The second modeling was a numerical simulation for the real geometrical shaft, taking into account the presence of the bearings, using ANSYS software with all the possibilities of optimization involved. A comparison of the theoretical results between the two modeling has been realized.
An algorithm using the step function Φ(x-a) from the MATHCAD professional calculation program was used to calculate the bending deformations of an elastic beam suspended by means of tie rods ...(cables). It allowed the writing of analytical relations of shear and bending internal forces in the beam subjected to bending and the drawing of the corresponding diagrams. Also, the algorithm allowed the writing of the analytical relations of the deformations of the beam (arrows and rotations) with the help of the loading function Ψ (x), respectively its derivative Ψd(x). The numerical simulation presented in this paper allowed finding the practical solution for the proposed problem.
To calculate the bending deformations of a package of elastic beams suspended by means of tie rods, a calculation algorithm using the step function Φ (x-a) from the professional MATHCAD calculation ...program was used. The step function Φ (x-a) allowed the writing of the analytical relations of the internal shear and bending forces in the beam subjected to bending and the drawing of the corresponding diagrams. Also, the calculation algorithm allowed the writing of the analytical relations of the deformations of the bar (arrows and rotations) with the help of the loading function Ψ (x), respectively its derivative Ψ d(x). The numerical simulation presented in this work allowed the study of the influence of longitudinal sliding on the internal shear and bending force and deformation diagrams for two presumptive connection situations of package beams that were expressed by the degree of preventing sliding (δ =50% and δ =100%), which is defined as the ratio between the moment of inertia of the hexagonal section comprising the n beams as a unitary whole and the one calculated as the sum of the moments of inertia of the sections of the n beams
This article presents a method for determining the Riley circularity evaluation, a characteristic that defines the shape of a solid particle. To determine this parameter, a working methodology was ...identified by using Mathcad. To verify this methodology 15 anthracite particles were inserted in the working file, the shape of which is different, to determine the analyzed characteristic.
The article describes using electronic computer systems, particularly Mathcad when finding standard equations for building mathematical models of technological processes in the chemical and food ...industries. Classical mathematical models of the extraction process are usually based on analyzing the differential equation of diffusion under appropriate boundary and initial conditions. Also, when describing mathematical models, Fick's equation, the equation of the adsorption pore model of molecular-statistical studies, models that take into account the physics of the process of substance transfer in a capillary-porous medium, etc., are used. Modeling of mass transfer processes under the influence of the intensifying action of the microwave field is described in the studies for the systems “coffee – water” and “oak wood – water-alcohol solution”, “coffee grounds – alcohol,” “amorant – hexane,” a mechanism of the combined process of mass transfer of extractive substances from a porous structure is proposed into solution The description of mass transfer processes under the influence of microwave field intensification is rather complicated. The mass transfer of extractive substances from porous structures to the solution occurs through the distribution of concentrations at the “solid–solvent” interface. However, it passes from the inner part through the outer part, through the capillaries into the border diffusion layer. Complex mathematical models for mass transfer processes do not have analytical solutions. When using the Mathcad system to solve nonlinear systems of equations, it is recommended to use the built-in find function in the calculated Given block. Also, the universal function genfit is quite adequate, which makes it possible to determine unknown parameters for nonlinear combinations of functions by the method of least squares. This system was used in describing extraction processes in the structure of interaction “oil – containing raw materials – solvent” with the intensifying effect of the microwave field. Such simplified analytical description systems are recommended to be used along with experimental modeling with the classic use of similarity theory and the “dimensional analysis” method. The final result of these directions is the construction of sound engineering methods for the design of extractors for industrial purposes.
The plane supporting spring commonly referred as flexural spring is used in isolation instruments such as astronautics, aeronautics, auto industries, micro-stereolithography and bulk lithography. All ...the applications require significant amount of radial stiffness which is used for isolation and providing friction-less motion. The proposed paper is targeted to derive a forced-based mathematical model for evaluating radial stiffness provided by the plane support spring. The springs are mounted in the flexural mechanism assembled in spindle head body clamped on micro drilling machine tool. The numerical equation for computing the radial stiffness is evaluated on the basis of solving various force components and the radial displacement generated by of each spring/arm using the "force-energy method." The finite element solution is done in ANSYS, whereas the theoretical answers are solved using PTC-MATHCAD solution. The results obtained by numerical analysis done in MATHCAD are validated by the finite element solution. The relationship between the radial stiffness and the key geometrical parameters are analyzed and confirmed.
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