• Proceedings of the Computational Structural Engineering Institute Conference
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• 1999.10a
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• pp.421-428
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• 1999

The elastic critical load of a slender compression member plays an important role when the proper design of that member is required. For the tapered compression members, however, there are cases when the conventional neutral equililbrium or energy method can't be applied to the determination of critical loads of those members. In this paper, finite element method is applied to the approximate determination of the symmetrically tapered bars. Here in this paper, the bars are assumed to take sinusoidally changing shapes along their axes. The parameters considered in this study are taper parameter, $\alpha$ and the sectional property parameter, m. The computed results by finite element method are represented in the forms of algebraic equations. Regression technique is employed to determine the coefficients of algebraic equations. The critical loads estimated by the proposed algebraic equations coincide fairly well with those of finite element method.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2000.10a
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• pp.299-306
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• 2000

The elastic critical load of a slender compression member plays an important role when the proper design of that member is required. For tapered compression members, however, there are cases when the conventional neutral equilibrium or energy method can't be applied to the determination of critical loads. In this paper, the finite element method is applied to the approximate determination of the linearly tapered members. In this paper, the bars are assumed to be tapered linearly along their axes. The parameters considered in this study are taper parameter, α and the sectional property parameter, m. The member ends are either hinged or fixed. The computed results using the finite element method are represented in the forms of algebraic equations. The regression technique is employed to determine the coefficients of the algebraic equations. Critical loads estimated by the proposed algebraic equations coincide flirty well with those employing the finite element method.

• Kyungpook Mathematical Journal
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• v.61 no.2
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• pp.383-393
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• 2021

In this paper, numerical techniques are presented for solving initial value problems of fractional differential equations with variable coefficients. The method is derived by applying a Taylor vector approximation. Moreover, the operational matrix of fractional integration of a Taylor vector is provided in order to transform the continuous equations into a system of algebraic equations. Furthermore, numerical examples demonstrate that this method is applicable and accurate.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.26 no.12
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• pp.2647-2655
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• 2002

This research presents an analytical model to investigate the stability due to the ball bearing waviness i n a rotating system supported by two ball bearings. The stiffness of a ball bearing changes periodically due to the waviness in the rolling elements as the rotor rotates, and it can be calculated by differentiating the nonlinear contact forces. The linearized equations of motion can be represented as a parametrically excited system in the form of Mathieu's equation, because the stiffness coefficients have time -varying components due to the waviness. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as the simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving the Hill's infinite determinant of these algebraic equations. The validity of this research is proved by comparing the stability chart with the time responses of the vibration model suggested by prior researches. This research shows that the waviness in the rolling elements of a ball bearing generates the time-varying component of the stiffness coefficient, whose frequency is called the frequency of the parametric excitation. It also shows that the instability takes place from the positions in which the ratio of the natural frequency to the frequency of the parametric excitation corresponds to i/2 (i=1,2,3..).

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2002.05a
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• pp.181-189
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• 2002

This research presents an analytical model to investigate the stability due to the ball bearing waviness in a rotating system supported by two ball bearings. The stiffness of a ball bearing changes periodically due to the waviness in the rolling elements as the rotor rotates, and it can be calculated by differentiating the nonlinear contact forces. The linearized equations of motion can be represented as a parametrically excited system in the form of Mathieu's equation, because the stiffness coefficients have time-varying components due to the waviness. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as the simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving the Hill's infinite determinant of these algebraic equations. The validity of this research is proved by comparing the stability chart with the time responses of the vibration model suggested by prior researches. This research shows that the waviness in the rolling elements of a ball bearing generates the time-varying component of the stiffness coefficient, whose frequency is called the frequency of the parametric excitation. It also shows that the instability takes place from the positions in which the ratio of the natural frequency to the frequency of the parametric excitation corresponds to i/2 (i= 1,2,3..).

• Education of Primary School Mathematics
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• v.14 no.3
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• pp.261-275
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• 2011

We observed the process for solving linear equations of two 5th grade elementary students, who do not have any pre-knowledge about solving linear equation. The way of students' usage of fractional schemes and manipulations are closely observed. The change of their scheme adaptation are carefully analyzed while the coefficients and constants become complicated. The results showed that they used various fractional scheme and manipulations according to the coefficients and constants. Noticeably, they used repeating fractional schemes to establish the equivalence relation between unknowns and the given quantities. After establishing the relationship, equivalent fractions played important role. We expect the results of this study would help shorten the gap between the arithmetic and the algebraic thinking.

• Journal of Mechanical Science and Technology
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• v.15 no.7
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• pp.889-894
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• 2001

In this study, the performances of various turbulence closure models are evaluated in the calculation of a transonic flow over axisymmetric bump. k-$\varepsilon$, explicit algebraic stress, and two Reynolds stress models, i.e., GL model proposed by Gibson & Launder and SSG model proposed by Speziale, Sarkar and Gatski, are chosen as turbulence closure models. SSG Reynolds stress model gives best predictions for pressure coefficients and the location of shock. The results with GL model also show quite accurate prediction of pressure coefficients down-stream of shock wave. However, in the predictions of mean velocities and turbulent stresses, the results are not so satisfactory as in the prediction of pressure coefficients.

• Journal of Korean Association for Spatial Structures
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• v.7 no.5
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• pp.83-87
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• 2007

Elastic critical loads of sinusoidally tapered bars with simply supported ends are determined by finite element method. The parameters considered in the analysis are taper parameter (=a) and section property parameter (=m). The analysis result for the special case of porismatic bar (a=0) shows good agreement with the existing value. The changes of the critical load coefficients are expressed by an algebraic equation. The coefficients appearing in the equations are determined by regression technique. The critical loads coefficients estimated by the proposed equations reveal little errors when they are compared with those determined by finite element method.

• Structural Engineering and Mechanics
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• v.43 no.1
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• pp.105-126
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• 2012

Based on the first-order shear deformation theory (FSDT), and the virtual work principle, an elastic analysis for axisymmetric clamped-clamped Pressurized thick truncated conical shells made of functionally graded materials have been performed. The governing equations are a system of nonhomogeneous ordinary differential equations with variable coefficients. Using the matched asymptotic method (MAM) of the perturbation theory, these equations could be converted into a system of algebraic equations with variable coefficients and two systems of differential equations with constant coefficients. For different FGM conical angles, displacements and stresses along the radius and length have been calculated and plotted.

• Structural Engineering and Mechanics
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• v.22 no.3
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• pp.331-349
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• 2006

The stresses and deflections in a laminated rectangular plate under thermal vibration are determined by using the moving least squares differential quadrature (MLSDQ) method based on the first order shear deformation theory. The weighting coefficients used in MLSDQ approximation are obtained through a fast computation of the MLS shape functions and their partial derivatives. By using this method, the governing differential equations are transformed into sets of linear homogeneous algebraic equations in terms of the displacement components at each discrete point. Boundary conditions are implemented through discrete grid points by constraining displacements, bending moments and rotations of the plate. Solving this set of algebraic equations yields the displacement components. Then substituting these displacements into the constitutive equation, we obtain the stresses. The approximate solutions for stress and deflection of laminated plate with cross layer under thermal load are obtained. Numerical results show that the MLSDQ method provides rapidly convergent and accurate solutions for calculating the stresses and deflections in a multi-layered plate of cross ply laminate subjected to thermal vibration of sinusoidal temperature including shear deformation with a few grid points.