• Journal of Korean Association for Spatial Structures
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• v.4 no.4 s.14
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• pp.113-128
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• 2004

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid paraboloidal and complete (that is, without a top opening) paraboloidal shells of revolution with variable wall thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell may be free or may be subjected to any degree of constraint. Displacement components $u_r,\;u_{\theta},\;and\;u_z$ in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in ${\theta}$, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the paraboloidal shells of revolution are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four digit exactitude is demonstrated for the first five frequencies of the complete, shallow and deep paraboloidal shells of revolution with variable thickness. Numerical results are presented for a variety of paraboloidal shells having uniform or variable thickness, and being either shallow or deep. Frequencies for five solid paraboloids of different depth are also given. Comparisons are made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory.

• Tribology and Lubricants
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• v.35 no.4
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• pp.244-266
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• 2019

The purpose of this study is to estimate the wear volumes of engine journal bearings operating at variable angular velocity of a shaft in the beginning of firing start-up cycle. To do this, first we find the potential region of wear scar on engine journal bearings where the applied bearing load and crank shaft velocity are variable. The potential wear regions are discovered by finding minimum oil film thickness at every crank angle existing below most oil film thickness scaring wear (MOFTSW) obtained based on the concept of the centerline average surface roughness. Then we calculate the wear volume from the wear depth and two wear angles decided by the magnitude of each film thickness lower than MOFTSW at every crank angle. The results show that the expected wear region is located at a few bearing angles after and/or behind the upper center of a big-end bearing and the lower center of a main bearing. And the real wear region is similar to the estimated wear region. Further we find that the wear scar on an engine journal bearing may occur at re-starting time after switch-off of a start motor especially under the condition of high oil temperature.

• Korean Journal of Computational Design and Engineering
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• v.14 no.4
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• pp.234-241
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• 2009

The use of aluminum parts in automobile structuraI applications has increased in an effort to reduce the weight of cars and hence improve fuel economy. But Aluminum bar, I-beam and channels need other processes to vary the cross section in the axial direction. Thus, applications of these parts are limited by high cost. If the cross section of the part is variable by using only extrusion, application of extruded bar, I-beam and channels will increase in the Aluminum industries. In this paper, we propose the variable-shape extrusion process which can control the thickness of Aluminum bar. And we can calculate the speed of center ram by varying the cross section in the extrusion to control the thickness of Aluminum bar.

• Journal of Aerospace System Engineering
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• v.8 no.1
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• pp.12-17
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• 2014

In this paper, optimal design of composite rotor blade cross-section to consider manufacturability was performed. Skin thickness, torsion box thickness and skin lay-up angle were adopted as discrete design variables and The position and width of a torsion box were considered as continuous variables. An object function of optimal design is to minimize the mass of a rotor blade, and various constraints such as failure index, center mass, shear center, natural frequency and blade minimum mass per unit length were adopted. Finally, design variables such as the thickness and lay-up angles of a skin, and the thickness, position and width of a torsion box were determined by using an in-house program developed for the optimal design of rotor blade cross-section.

• Structural Engineering and Mechanics
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• v.55 no.1
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• pp.29-46
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• 2015

A three-dimensional (3-D) method of analysis is presented for determining the natural frequencies of a truncated shallow and deep conical shell with linearly varying thickness along the meridional direction free at its top edge and clamped at its bottom edge. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components $u_r$, $u_{\theta}$, and $u_z$ in the radial, circumferential, and axial directions, respectively, are taken to be periodic in ${\theta}$ and in time, and algebraic polynomials in the r and z directions. Strain and kinetic energies of the truncated conical shell with variable thickness are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated. The frequencies from the present 3-D method are compared with those from other 3-D finite element method and 2-D shell theories.

• Korean Journal of Air-Conditioning and Refrigeration Engineering
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• v.20 no.7
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• pp.455-461
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• 2008

A reversed trapezoidal fin with variable fin base thickness is optimized using a two-dimensional analytical method. For the fin base boundary condition, instead of a constant temperature, heat transfer from the inside fluid to the fin base is considered. Heat loss from the fin tip is not ignored. The maximum heat loss, corresponding optimum fin effectiveness, fin length and base height are presented as a function of the fin base thickness, shape factor and volume.

• Structural Engineering and Mechanics
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• v.11 no.4
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• pp.461-468
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• 2001

The aim of this work is to establish the coefficient that defines the critical buckling load for isotropic annular plates of variable thickness whose outer boundary is simply supported and subjected to uniform pressure. It is assumed that the plate thickness varies in a continuous way, according to an exponential law. The eigenvalues are determined using an optimized Rayleigh-Ritz method with polynomial coordinate functions which identically satisfy the boundary conditions at the outer edge. Good engineering agreement is shown to exist between the obtained results and buckling parameters presented in the technical literature.

• Journal of the Korean Society for Precision Engineering
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• v.14 no.6
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• pp.114-120
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• 1997

To assess the validity of the previously computed finite element analysis results, the photoelastic experiment was carried out to determine stress intensity factors for crack originating from thin section of integrally stiffened plates having discontinuous thickness interface. The stress intensity factors were deter- mined by using linear slope method of photoelastic data. Results are presented as variable thickness geometry factor. $F_{IV}$ , for various crack lengths and thickness ratios. The experimental values of F/ sub IV/are compared with 3-D finite element analysis results. The correlation between experimental values and analysis results is resonably good.

• Structural Engineering and Mechanics
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• v.90 no.6
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• pp.601-610
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• 2024

Free vibration of layered conical shell frusta of thickness filled with fluid is investigated. The shell is made up of isotropic or specially orthotropic materials. Three types of thickness variations are considered, namely linear, exponential and sinusoidal along the radial direction of the conical shell structure. The equations of motion of the conical shell frusta are formulated using Love's first approximation theory along with the fluid interaction. Velocity potential and Bernoulli's equations have been applied for the expression of the pressure of the fluid. The fluid is assumed to be incompressible, inviscid and quiescent. The governing equations are modified by applying the separable form to the displacement functions and then it is obtained a system of coupled differential equations in terms of displacement functions. The displacement functions are approximated by cubic and quintics splines along with the boundary conditions to get generalized eigenvalue problem. The generalized eigenvalue problem is solved numerically for frequency parameters and then associated eigenvectors are calculated which are spline coefficients. The vibration of the shells with the effect of fluid is analyzed for finding the frequency parameters against the cone angle, length ratio, relative layer thickness, number of layers, stacking sequence, boundary conditions, linear, exponential and sinusoidal thickness variations and then results are presented in terms of tables and graphs.

• Transactions of the Korean Society of Mechanical Engineers
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• v.11 no.1
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• pp.132-144
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• 1987

This paper presents the theroetical analysis of the crack tip stress intensity factor for a center crack in a finite width plate with variable thickness. The analyses were based on Laurent's expansions of complex stress potentials where the expansion coefficients are determined from the boundary conditions. The perturbation method was employed in numerical calculations. The correction factor F(.lambda.)is given in the form of power series of .lambda. [a numerical formula] where .lambda.=a/w$^{1}$; Dimensionless crack length, .betha.=t$_{2}$/t; Thickness ratio .omega.=w$_{2}$/w$_{1}$; width ratio The correction factor values vary with the width ratio .omega. and the maximum variation occurs around .betha.=1. For the case of .betha.=1 or .betha.=0 (uniform thickness plate0, the correction factor values agree well with Feddersen's formula. In all cases, as .lambda. approaches to 1 (thickness interface), the correction factor values are decreased rapidly for .betha.>1, and increased rapidly for .betha.<1.