• Title, Summary, Keyword: Maclaurin expansion

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Effect of higher order terms of Maclaurin expansion in nonlinear analysis of the Bernoulli beam by single finite element

  • Zahrai, Seyed Mehdi;Mortezagholi, Mohamad Hosein;Mirsalehi, Maryam
    • Structural Engineering and Mechanics
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    • v.58 no.6
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    • pp.949-966
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    • 2016
  • The second order analysis taking place due to non-linear behavior of the structures under the mechanical and geometric factors through implementing exact and approximate methods is an indispensible issue in the analysis of such structures. Among the exact methods is the slope-deflection method that due to its simplicity and efficiency of its relationships has always been in consideration. By solving the differential equations of the modified slope-deflection method in which the effect of axial compressive force is considered, the stiffness matrix including trigonometric entries would be obtained. The complexity of computations with trigonometric functions causes replacement with their Maclaurin expansion. In most cases only the first two terms of this expansion are used but to obtain more accurate results, more elements are needed. In this paper, the effect of utilizing higher order terms of Maclaurin expansion on reducing the number of required elements and attaining more rapid convergence with less error is investigated for the Bernoulli beam with various boundary conditions. The results indicate that when using only one element along the beam length, utilizing higher order terms in Maclaurin expansion would reduce the relative error in determining the critical buckling load and kinematic parameters in the second order analysis.

Implementation of Efficient Exponential Function Approximation Algorithm Using Format Converter Based on Floating Point Operation in FPGA (부동소수점 기반의 포맷 컨버터를 이용한 효율적인 지수 함수 근사화 알고리즘의 FPGA 구현)

  • Kim, Jeong-Seob;Jung, Seul
    • Journal of Institute of Control, Robotics and Systems
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    • v.15 no.11
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    • pp.1137-1143
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    • 2009
  • This paper presents the FPGA implementation of efficient algorithms for approximating exponential function based on floating point format data. The Taylor-Maclaurin expansion as a conventional approximation method becomes inefficient since high order expansion is required for the large number to satisfy the approximation error. A format converter is designed to convert fixed data format to floating data format, and then the real number is separated into two fields, an integer field and an exponent field to separately perform mathematic operations. A new assembly command is designed and added to previously developed command set to refer the math table. To test the proposed algorithm, assembly program has been developed. The program is downloaded into the Altera DSP KIT W/STRATIX II EP2S180N Board. Performances of the proposed method are compared with those of the Taylor-Maclaurin expansion.

Analytical approximate solutions for large post-buckling response of a hygrothermal beam

  • Yu, Yongping;Sun, Youhong
    • Structural Engineering and Mechanics
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    • v.43 no.2
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    • pp.211-223
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    • 2012
  • This paper deals with large deformation post-buckling of a linear-elastic and hygrothermal beam with axially nonmovable pinned-pinned ends and subjected to a significant increase in swelling by an alternative method. Analytical approximate solutions for the geometrically nonlinear problem are presented. The solution for the limiting case of a string is also obtained. By coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials, the governing differential equation with sinusoidal nonlinearity can be reduced to form a cubic-nonlinear equation, and supplementary condition with cosinoidal nonlinearity can also be simplified to be a polynomial integral equation. Analytical approximations to the resulting boundary condition problem are established by combining the Newton's method with the method of harmonic balance. Two approximate formulae for load along axis, potential strain for free hygrothermal expansion and periodic solution are established for small as well as large angle of rotation at the end of the beam. Illustrative examples are selected and compared to "reference" solution obtained by the shooting method to substantiate the accuracy and correctness of the approximate analytical approach.

FPGA real-time calculator to determine the position of an emitter

  • Tamura, M.;Aoyama, T.
    • 제어로봇시스템학회:학술대회논문집
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    • pp.473-478
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    • 2003
  • To detect motions of bodies, we have discussed them with two viewpoints; one is a detection algorithm, and another is the hardware implementation. The former is to find small terms expansions for sine/cosine functions. We researched Maclaurin and optimum expansions, and moreover to reduce hardware amounts, revised the expansions. The expansions don't include divide calculations, and the error is within 0.01%. As for the former problem, there is another approach also; that is the cordic method. The method is based on the rotation of a vector on the complex plain. It is simple iterations and don't require large logic. We examined the precision and convergence of the method on C-simulations, and implemented on HDL. The later problem is to make FPGA within small gates. We considered approaches to eliminate a divider and to reduce the bit number of arithmetic. We researched Newton-Raphson's method to get reciprocal numbers. The higher-order expression shows rapid convergence and doesn't be affected by the initial guess. It is an excellent algorithm. Using them, we wish to design a detector, and are developing it on a FPGA.

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Analytical approximate solution for Initial post-buckling behavior of pipes in oil and gas wells

  • Yu, Yongping;Sun, Youhong;Han, Yucen
    • Coupled systems mechanics
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    • v.1 no.2
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    • pp.155-163
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    • 2012
  • This paper presents analytical approximate solutions for the initial post-buckling deformation of the pipes in oil and gas wells. The governing differential equation with sinusoidal nonlinearity can be reduced to form a third-order-polynomial nonlinear equation, by coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials. Analytical approximations to the resulting boundary condition problem are established by combining the Newton's method with the method of harmonic balance. The linearization is performed prior to proceeding with harmonic balancing thus resulting in a set of linear algebraic equations instead of one of non-linear algebraic equations, unlike the classical method of harmonic balance. We are hence able to establish analytical approximate solutions. The approximate formulae for load along axis, and periodic solution are established for derivative of the helix angle at the end of the pipe. Illustrative examples are selected and compared to "reference" solution obtained by the shooting method to substantiate the accuracy and correctness of the approximate analytical approach.