• Honam Mathematical Journal
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• v.26 no.1
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• pp.133-136
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• 2004

By elementry manipulation of series together with summations of Gauss and $Saalsch\ddot{u}tz$, Exton deduced a new two term relation for the hypergeometric function $_3F_2(1)$. The aim of this paper is to derive Exton's result from Thomae's formula, together with two known integral formulas and the Euler's transformation for $_2F_1$.

• Structural Engineering and Mechanics
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• v.63 no.1
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• pp.11-21
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• 2017

In this paper, the efficiency and the accuracy of classical (CE) and high order (HE) beam element are improved by introducing two novel techniques. The first proposed element (FPE) provides an alternative for (HE) by taking the mode shapes of the clamped-clamped (C-C) beam into account. The second proposed element (SPE) which could be utilized instead of (CE) and (HE) considers not only the mode shapes of the (C-C) beam but also some virtual nodes. It is numerically proven that the eigenvalue problem and the frequency response function for Euler-Bernoulli beam are obtained more accurate and efficient in contrast to the traditional ones.

• Coupled systems mechanics
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• v.5 no.2
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• pp.157-190
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• 2016

This paper deals with frequency analysis of Euler-Bernoulli beams carrying an arbitrary number of Kelvin-Voigt viscoelastic dampers, subjected to harmonic loads. Multiple external/internal dampers occurring at the same position along the beam axis, modeling external damping devices and internal damping due to damage or imperfect connections, are considered. The challenge is to handle simultaneous discontinuities of the response, in particular bending-moment/rotation discontinuities at the location of external/internal rotational dampers, shear-force/deflection discontinuities at the location of external/internal translational dampers. Following a generalized function approach, the paper will show that exact closed-form expressions of the frequency response under point/polynomial loads can readily be derived, for any number of dampers. Also, the exact dynamic stiffness matrix and load vector of the beam will be built in a closed analytical form, to be used in a standard assemblage procedure for exact frequency response analysis of frames.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.24 no.12
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• pp.948-953
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• 2014

Many studies using the assumed mode method have been found for the free vibration analysis of stiffened plate with known elastic boundary conditions. However many local structures such as tank edges and equipment foundations consist of connected structures and it is very difficult to find suitable elastic boundary conditions. In this study combined polynomials which satisfy simply and fixedly supported boundary conditions are proposed. The proposed method has been applied to tanks which bounded by bulkhead and a deck. The results of this study shows good agreements with these obtain by the FEA S/W(Patran/Nastran).

• East Asian mathematical journal
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• v.31 no.1
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• pp.41-53
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• 2015

A large number of series and integral representations for the Stieltjes constants (or generalized Euler-Mascheroni constants) ${\gamma}_k$ and the generalized Stieltjes constants ${\gamma}_k(a)$ have been investigated. Here we aim at presenting certain integral representations for the generalized Stieltjes constants ${\gamma}_k(a)$ by choosing to use four known integral representations for the generalized zeta function ${\zeta}(s,a)$. As a by-product, our main results are easily seen to specialize to yield those corresponding integral representations for the Stieltjes constants ${\gamma}_k$. Some relevant connections of certain special cases of our results presented here with those in earlier works are also pointed out.

• Structural Engineering and Mechanics
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• v.40 no.5
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• pp.689-703
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• 2011

A new perturbation method is introduced to study the undamped free vibration of a non-prismatic Timoshenko beam for its natural frequencies and vibration modes. For simplicity, the natural modes of vibration of its corresponding prismatic Euler-Bernoulli beam with the same length and boundary conditions are used as Ritz base functions with necessary modifications to account for shear strain in the Timoshenko beam. The new method can transform two coupled partial differential equations governing the transverse vibration of the non-prismatic Timoshenko beam into a set of nonlinear algebraic equations. It significantly simplifies the solution process and is applicable to non-prismatic beams with various boundary conditions. Three examples indicated that the new method is more accurate than the previous perturbation methods. It successfully takes into account the effect of shear deformation of Timoshenko beams particularly at the free end of cantilever structures.

• Steel and Composite Structures
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• v.32 no.5
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• pp.643-655
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• 2019

Perforation and cutouts of structures are compulsory in some modern applications such as in heat exchangers, nuclear power plants, filtration and microeletromicanical system (MEMS). This perforation complicates dynamic analyses of these structures. Thus, this work tends to introduce semi-analytical model capable of investigating the dynamic performance of perforated beam structure under free and forced conditions, for the first time. Closed forms for the equivalent geometrical and material characteristics of the regular square perforated beam regular square, are presented. The governing dynamical equation of motion is derived based on Euler-Bernoulli kinematic displacement. Closed forms for resonant frequencies, corresponding Eigen-mode functions and forced vibration time responses are derived. The proposed analytical procedure is proved and compared with both analytical and numerical analyses and good agreement is noticed. Parametric studies are conducted to illustrate effects of filling ratio and the number of holes on the free vibration characteristic, and forced vibration response of perforated beams. The obtained results are supportive in mechanical design of large devices and small systems (MEMS) based on perforated structure.

• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.603-618
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• 2022

In this paper, a numerical solution of the generalized diffusion equation with a delay has been obtained by a numerical technique based on the Galerkin finite element method by applying the cubic B-spline basis functions. The time discretization process is carried out using the forward Euler method. The numerical scheme is required to preserve the delay-independent asymptotic stability with an additional restriction on time and spatial step sizes. Both the theoretical and computational rates of convergence of the numerical method have been examined and found to be in agreement. As it can be observed from the numerical results given in tables and graphs, the proposed method approximates the exact solution very well. The accuracy of the numerical scheme is confirmed by computing L₂ and L_∞ error norms.

• Structural Engineering and Mechanics
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• v.59 no.6
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• pp.1139-1153
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• 2016

Elastic constraints are usually simplified as "spring forces" exerted on beam ends without considering the "spring deformation". The partial differential equation governing the free vibrations of a cantilever Bernoulli-Euler beam considering the deformation of elastic constraints is firstly established, and is nondimensionalized to obtain two dimensionless factors, $k_v$ and $k_r$, describing the effects of elastically vertical and rotational end constraints, respectively. Then the frequency equation for the above Bernoulli-Euler beam model is derived using the method of separation of variables. A numerical analysis method is proposed to solve the transcendental frequency equation for the continuous change of the frequency with $k_v$ and $k_r$. Then the mode shape functions are given. Finally, effects of $k_v$ and $k_r$ on free vibration characteristics of the beam with different slenderness ratios are calculated and analyzed. The results indicate that the effects of $k_v$ are larger on higher-order free vibration characteristics than on lower-order ones, and the impact strength decreases with slenderness ratio. Under a relatively larger slenderness ratio, the effects of $k_v$ can be neglected for the fundamental frequency characteristics, while cannot for higher-order ones. However, the effects of $k_r$ are large on both higher- and lower-order free vibration characteristics, and cannot be neglected no matter the slenderness ratio is large or small.

• Honam Mathematical Journal
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• v.34 no.3
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• pp.311-326
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• 2012

The purpose of this paper is to introduce and investigate two new classes of generalized Bernoulli and Apostol-Bernoulli polynomials based on the definition given recently by the authors [29]. In particular, we obtain a new addition formula for the new class of the generalized Bernoulli polynomials. We also give an extension and some analogues of the Srivastava-Pint$\acute{e}$r addition theorem [28] for both classes. Finally, by making use of the new adition formula, we exhibit several interesting relationships between generalized Bernoulli polynomials and other polynomials or special functions.