Application of Numerical Integration in Technical Practice

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In this contribution we are dealing with application of differentiating of function defined with determined parametrical integral. The mathematical problem was analyzed form point of view of mechanics of materials. The mathematical model of loaded beam was created. Applying the cross section method we defined the exact function of bending moment. Additional properties of cantilever joint were neglected. We showed the derivation of modified Castigliano's theorem via Leibnitz rule of differentiating under integration sign. Appling the modified Castigliano's theorem we got the exact solution of the deflection of the beam. The exact solution of beam deflection was finished in PTC Mathcad Prime software (PTC-Parametric Technology). The numerical integration of the bending moment dataset and defined deflection function was done in program written in Microsoft Visual C# 2010. The Dormand-Prince numerical integration method was used for the numerical integration. Comparing the exact and numerical solution we got the error of numerical integration solution.

Section: Figure 4 Deflection Curve Obtained By Numerical Integrationmentioning

confidence: 99%

Differentiating under integral sign in Castigliano’s theorem

Rédl¹

2019

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“…These points then very efficiently approximate arbitrarily complex shapes and to edit in this way formed objects is often enough to edit positions of the control points [9]. Typical examples of this approach are Bézier curve [5], [8], B-spline curve [6] and NURBS curve [10]. Likewise are modelled also surfaces as Bézier surface [7], or B-spline surface, which is covered in detail in this article.…”

Section: Introductionmentioning

confidence: 99%

B-spline surface for distribution of the agriculture soil moisture

Páleš¹,

Rédl²,

Beloev³

2017

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In the agricultural locality Pohranice were measured soil moisture values for the selected control points. Over these points we put the B-spline surface, which relatively detailed description we supply. On a simple example of 4 x 4 points we debugged program for generation of base functions. Furthermore, we verified process to generate knots of B-spline surface, which we subsequently used for the programming of the surface itself. In this manner formed algorithms we applied to 11 x 11 measured values of soil moisture. Discrete values of measurements were by clamped B-spline surface smoothed and its corner values constitute the edge measurement points. Evenly spaced points at a distance of 1 m in both directions slightly simplified the procedure, which can deal with irregular layout of points.

Numerical differentiation of stochastic function

Rédl¹,

Páleš²

2016

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In this paper we are presenting the derivation of the finite difference approximation of the first derivative of the discrete stochastic function. The set of data was represented with time depending stochastic function of center of gravity dislocation and velocity with constant step size. The stochastic function was obtained from real experimental measurement and was processed with Dynstab (Dynamic Stability) software. Applied forward, backward and central differences algorithms were written in Microsoft Visual C# ® 2010 programming language as a part of software named Dynstat (Dynamic Statistics). These algorithms were used to calculate the first derivative of the function and error of differentiation. We compared the results of known precise solutions with calculated results. We also confront the results calculated of known generated trigonometric function. With this manner we are evaluated the inaccuracy of the stochastic function numerical differentiation.