scholarly journals Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Beam Theory ◽  
Second Order ◽  
Governing Equations ◽  
Deflection Curve ◽  
Order Analysis ◽  
First Order ◽  
Finite Difference Approximations ◽  
Order Polynomial ◽  
Euler Bernoulli Beam

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally, the finite difference approximations are derived based on fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection curve; the FOPH is made for the fourth and third derivative of the deflection curve while the SOPH is made for its second and first derivative. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, the FOPH was made for all of the derivatives of the deflection curve, and additional points were introduced at the beam’s ends and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Furthermore, tapered beams were analyzed (element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was increased through a grid refinement. Especially in the first-order analysis of uniform beams, the results were exact for uniformly distributed and concentrated loads regardless of the grid. Further research will be needed to investigate polynomial refinements (higher-order polynomials such as fifth-order, sixth-order…) of the deflection curve; the polynomial refinements aimed to increase the accuracy, whereby non-centered finite difference approximations at beam’s ends and positions of discontinuity would be used.

scholarly journals Kirchhoff-Love Plate Theory: First-Order Analysis, Second-Order Analysis, Plate Buckling Analysis, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Vibration Analysis ◽  
Plate Theory ◽  
Second Order ◽  
Governing Equations ◽  
Order Analysis ◽  
First Order ◽  
Order Polynomial ◽  
The Finite Difference Method ◽  
Point Stencil

This paper presents an approach to the Kirchhoff-Love plate theory (KLPT) using the finite difference method (FDM). The KLPT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally in the case of KLPT, the finite difference approximations are derived based on the fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection surface. The FOPH is made for the fourth and third derivative of the deflection surface while the SOPH is made for its second and first derivative; this leads to a 13-point stencil for the governing equation. In addition, the boundary conditions and not the governing equations are applied at the plate edges. In this paper, the FOPH was made for all of the derivatives of the deflection surface; this led to a 25-point stencil for the governing equation. Furthermore, additional nodes were introduced at plate edges and at positions of discontinuity (continuous supports/hinges, incorporated beams, stiffeners, brutal change of stiffness, etc.), the number of additional nodes corresponding to the number of boundary conditions at the node of interest. The introduction of additional nodes allowed us to apply the governing equations at the plate edges and to satisfy the boundary and continuity conditions. First-order analysis, second-order analysis, buckling analysis, and vibration analysis of plates were conducted with this model. Moreover, plates of varying thickness and plates with stiffeners were analyzed. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. In first-order, second-order, buckling, and vibration analyses of rectangular plates, the results obtained in this paper were in good agreement with those of well-established methods, and the accuracy was increased through a grid refinement.

scholarly journals Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference Method ◽  
Differential Equations ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Second Order ◽  
Bernoulli Beam ◽  
Order Analysis ◽  
First Order ◽  
The Finite Difference Method ◽  
Euler Bernoulli Beam

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations (or partial differential equations). The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional points or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, etc.). The introduction of additional points permits us to satisfy boundary conditions and continuity conditions. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. Tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, the damping being considered. The efforts and displacements could be determined at any time.

scholarly journals Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference Method ◽  
Differential Equations ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Second Order ◽  
Bernoulli Beam ◽  
Order Analysis ◽  
First Order ◽  
The Finite Difference Method ◽  
Euler Bernoulli Beam

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations (or partial differential equations). The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional points or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, etc.). The introduction of additional points permits us to satisfy boundary conditions and continuity conditions. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. Tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, the damping being considered. The efforts and displacements could be determined at any time.

scholarly journals Timoshenko Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Differential Equations ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Second Order ◽  
Timoshenko Beam Theory ◽  
Governing Equations ◽  
Order Analysis ◽  
First Order ◽  
The Finite Difference Method

This paper presents an approach to the Timoshenko beam theory (TBT) using the finite difference method (FDM). The Timoshenko beam theory covers cases associated with small deflections based on shear deformation and rotary inertia considerations. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. The model developed in this paper consisted of formulating differential equations with finite differences and introducing additional points at the beam’s ends and at positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends. Moreover, grid points with variable spacing were considered, the grid being uniform within beam segments. First-order, second-order, and vibration analyses of structures were conducted with this model. Furthermore, tapered beams were analyzed (element stiffness matrix, second-order analysis, vibration analysis). Finally, a direct time integration method (DTIM) was presented; the FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was increased through a grid refinement. Especially in the first-order analysis of uniform beams, the results were exact for uniformly distributed and concentrated loads regardless of the grid.

scholarly journals Timoshenko Beam Theory:First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Timoshenko Beam ◽  
Difference Method ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Second Order ◽  
Order Analysis ◽  
The Finite Difference Method

This paper presents an approach to the Timoshenko beam theory (TBT) using the finite difference method (FDM). The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory neglects shear deformations. The FDM is an approximate method for solving problems described with differential or partial differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness). The introduction of additional points allows satisfying boundary and continuity conditions. First-order, second-order, and vibration analyses of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. In addition, tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis, and vibration analysis). Finally, the direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, considering the damping. The efforts and displacements could be determined at any time.

scholarly journals Timoshenko Beam Theory: Exact Solution for First-Order Analysis, Second-Order Analysis, and Stability

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Exact Solution ◽  
Closed Form ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Second Order ◽  
Bernoulli Beam ◽  
Order Analysis ◽  
First Order ◽  
The Moment ◽  
Relationship Of

This paper presents an exact solution to the Timoshenko beam theory (TBT) for first-order analysis, second-order analysis, and stability. The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory (EBBT) neglects shear deformations. Thus, the Euler–Bernoulli beam is a special case of the Timoshenko beam. The moment-curvature relationship is one of the governing equations of the EBBT, and closed-form expressions of efforts and deformations are available in the literature. However, neither an equivalent to the moment-curvature relationship of EBBT nor closed-form expressions of efforts and deformations can be found in the TBT. In this paper, a moment-shear force-curvature relationship, the equivalent in TBT of the moment-curvature relationship of EBBT, was presented. Based on this relationship, first-order and second-order analyses were conducted, and closed-form expressions of efforts and deformations were derived for various load cases. Furthermore, beam stability was analyzed and buckling loads were calculated. Finally, first-order and second-order element stiffness matrices were determined.

Study on Fillet Stresses Distribution of Crankshaft of High-Duty Diesel Engine

2011 ◽  
Vol 52-54 ◽  
pp. 1364-1368
Author(s):  
Wen Li ◽  
Ri Dong Liao ◽  
Zheng Xing Zuo
Keyword(s):  
Diesel Engine ◽  
Second Order ◽  
First Order ◽  
Fourier Expansions ◽  
Order Polynomial ◽  
Polynomial Expansions ◽  
Regional Stresses

Stresses distribution in crankshaft fillet region are mainly discussed in this paper. FEM is used for calculating the stresses. With the analysis of the regional stresses distribution, we concludes that the stresses can be fitted by first order of Fourier expansions, while all the parameters in the expansions can be expressed as second order polynomial expansions.

A New Nonslender Ringing Load Approach Verified Against Experiments

1998 ◽  
Vol 120 (1) ◽  
pp. 20-29 ◽  
Author(s):  
J. R. Krokstad ◽  
C. T. Stansberg ◽  
A. Nestega˚rd ◽  
T. Marthinsen
Keyword(s):  
Wave Model ◽  
Second Order ◽  
Wave Load ◽  
Peak Wavelength ◽  
Third Order ◽  
Load Model ◽  
Order Analysis ◽  
First Order ◽  
Load Effects ◽  
Vertical Cylinders

New results from the most recent work within the Norwegian Joint Industry Project (JIP) “Higher Order Wave Load Effects on Large Volume Structures” are presented. A nonslender theoretical model is validated from experiments for two fixed, vertical cylinders with different diameter/peak wavelength ratios. A combination of complete diffraction first-order simulations, sum and difference frequency second-order simulations, and third-order FNV (Faltinsen, Newman, and Vinje, nonlinear long wave model) is implemented in order to develop a simplified and robust ringing load model for a large range of cylinder diameter/peak wavelength ratios. Results from the full diffraction second-order analysis show a significant reduction of second-order loads compared to pure FNV in the wavelength range relevant for ringing loads. The results show improved correspondence with high-frequency experimental loads compared with the unmodified FNV. Results for different cylinder peak wavelength ratios are presented, including validation against experiments. In addition, a few simplified response simulations are carried out demonstrating significant improvements with the modified FNV model.

scholarly journals Comparative Results of Regional and Residual Anomalies with the Upward Continuation, Moving Average, and Polynomial Methods for Magnetic Data

2020 ◽  
Vol 2 (2) ◽  
pp. 90-93
Author(s):  
Luvera Deva Intan Indrawati ◽  
Rina Dwi Indriana ◽  
Irham Nurwidyanto
Keyword(s):  
Moving Average ◽  
Continuation Method ◽  
Second Order ◽  
Magnetic Data ◽  
Third Order ◽  
First Order ◽  
Upward Continuation ◽  
Order Polynomial ◽  
Matlab Programming ◽  
Polynomial Methods

Geophysics programing of regional and residual anomaly separation on Magnetic data has been carried out with the results compared with the upward continuation method in the OasisMontaj software. Separation of anomalies with moving average and polynomial methods is processed using Matlab programming. The orders used in the polynomial method are first-order, second-order and third-order. Comparison is done by calculating the match value. The chosen matching method is autocorrelation. Correlation of residual magnetic anomalies resulting from upward continuation (Magpick) to moving averages, 1st-order polynomials, 2nd-order polynomials and 3rd-order polynomials. Correlation values obtained for the moving average method are 0.9604, first order polynomial 0.9072, 2nd order polynomial 0.9482 and third order polynomial 0.6057. The moving average and second order polynomial methods can be used as a substitute method if we do not use the upward continuation method.

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