• Transactions of the Korean Society of Mechanical Engineers A
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• v.20 no.6
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• pp.1773-1783
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• 1996

Finite element method(FEM) is one of the most popularly used method analyzing the dynamic behaviors of structures. But unless number of finite elements is large enough, the results from FEM some what different from exact analytical solutions, especially at high frequency range. On the other hand, as the spectral analysis method(SAM) deals directly with the governing equations of a structure, the results from this melthod cannot but be exact regardless of any frequency range. However, the SAM can be applied only to the case where a structure is subjected to the concentrated loads, despite a structure could be unddergone distributed loads more generally. In this paper, therefore, new spectral analysis algorithm is introduced through the spectral element method(SEM), so that it can be applied to anlystructures whether they are subjected to the concentrated loads or to the distributed loads. The results from this new SEM are compared with both the results from FEM and the exact analytical solutions. As expected, the results from new SEM algorithm are found to be almost identical to the exact analytical solutions while those from FEM are not agreed well with the exact analytical solutions as the mode number increases.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 1996.04a
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• pp.328-334
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• 1996

Finite Element Method(FEM) is the most popularly used method in analyzing the dynamic behaviors of structures. But unless the number of finite elements is large enough, the results from FEM are somewhat different from exact analytical solutions, especially at high frequency range. On the other hand, as the Spectral Element Method(SEM) deals directly with the governing equations of structures, the results from this method cannot but be exact regardless of any frequency range. However, despite two dimensional structures are more general, the SEM has been applied only to the analysis of one dimensional structures so far. In this paper, therefore, new methodologies are introduced to analyze the two dimensional plate using SEM. The results from this new method are compared with the exact analytical solutions by letting the two dimensional plate be one dimensional one and showed the dynamic responses of two dimensional plate by including various waves propagated into x-direction.

• Journal of KSNVE
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• v.6 no.5
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• pp.617-624
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• 1996

Finite Element Method(FEM) is one of the most popularly used method in analyzing the dynamic behaviors of structures. But unless the number of finite elements is large enough, the results from FEM are somewhat different form exact analytical solutions, especially at high frequency range. On the other hand, as the Spectral Element Method(SEM) deals directly with the governing equations of structures, the results from this method cannot but be exact regardless of any frequency range. However, despite two dimensional structures are more general, the SEM has been applied only to the analysis of one dimensional structures so far. In this paper, therefore, new methodologies are introduced to analyze the two dimensional plate structure using SEM. The results from this new method are compared with the exact analytical solutions by letting the two dimensional plate structure be one dimensional and showed the dynamic responses of two dimensional plate by including various waves propagated into x-direction.

• Steel and Composite Structures
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• v.44 no.4
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• pp.531-543
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• 2022

An exact dynamic analytical method for free vibrations of continuous partial-interaction composite beams is proposed based on the Timoshenko beam theory. The main advantage of this method is that the independent shear deformations and rotary inertia of sub-beams are considered, which is more in line with the reality. Therefore, the accuracy of eigenfrequencies obtained by this method is significantly improved, especially for higher order modes, compared to the existing methods where the rotary angles of both sub-beams are assumed to be equal irrespective of the differences in the shear stiffness of each sub-beam. Furthermore, the solutions obtained by the proposed method are exact owing to no introduction of approximated displacement and force fields in the derivation. In addition, an exact analytical solution for the case of simply supported is obtained. Based on this, an approximate expression for the fundamental frequency of continuous partial-interaction composite beams is also proposed, which is useful for practical engineering applications. Finally, the practicability and effectiveness of the proposed method and the approximate expression are explored using numerical and experimental examples; The influence factors including the interfacial interaction, shear modulus ratio, span-to-depth ratio, and side-to-main span length ratio on the eigenfrequencies are presented and discussed in detail.

• Structural Engineering and Mechanics
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• v.31 no.4
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• pp.383-392
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• 2009

The static and dynamic analyses of simply supported beams are studied by using the U-transformation method and the finite difference method. When the beam is divided into the mesh of equal elements, the mesh may be treated as a periodic structure. After an equivalent cyclic periodic system is established, the difference governing equation for such an equivalent system can be uncoupled by applying the U-transformation. Therefore, a set of single-degree-of-freedom equations is formed. These equations can be used to obtain exact analytical solutions of the deflections, bending moments, buckling loads, natural frequencies and dynamic responses of the beam subjected to particular loads or excitations. When the number of elements approaches to infinity, the exact error expression and the exact convergence rates of the difference solutions are obtained. These exact results cannot be easily derived if other methods are used instead.

• Structural Engineering and Mechanics
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• v.12 no.1
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• pp.85-100
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• 2001

In this paper, an analytical procedure for solving several static and dynamic problems of non-uniform beams is proposed. It is shown that the governing differential equations for several stability, free vibration and static problems of non-uniform beams can be written in the from of a unified self-conjugate differential equation of the second-order. There are two functions in the unified equation, unlike most previous researches dealing with this problem, one of the functions is selected as an arbitrary expression in this paper, while the other one is expressed as a functional relation with the arbitrary function. Using appropriate functional transformation, the self-conjugate equation is reduced to Bessel's equation or to other solvable ordinary differential equations for several cases that are important in engineering practice. Thus, classes of exact solutions of the self-conjugate equation for several static and dynamic problems are derived. Numerical examples demonstrate that the results calculated by the proposed method and solutions are in good agreement with the corresponding experimental data, and the proposed procedure is a simple, efficient and exact method.

• Steel and Composite Structures
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• v.16 no.6
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• pp.577-593
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• 2014

An exact dynamic stiffness method is introduced for investigating the free vibration characteristics of the steel-concrete composite beams consisting of a reinforced concrete slab and a steel beam which are connected by using the stud connectors. The elementary beam theory is used to define the dynamic behaviors of the two beams and the relative transverse deformation of the connectors is included in the formulation. The dynamic stiffness matrix is formulated from the exact analytical solutions of the governing differential equations of the composite beams in undamped free vibration. The application of the derived dynamic stiffness matrix is illustrated to predict the natural frequencies and mode shapes of the steel-concrete composite beams with seven boundary conditions. The present results are compared to the available solutions in the literature whenever possible.

• Journal of Mechanical Science and Technology
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• v.15 no.8
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• pp.1143-1155
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• 2001

In order to reduce the expensive CPU time for design sensitivity analysis in dynamic response optimization, this study introduces the design sensitivities approximated within estimated confidence radius in dynamic response optimization with ALM method. The confidence radius is estimated by the linear approximation with Hessian of quasi-Newton formula and qualifies the approximate gradient to be validly used during optimization process. In this study, if the design changes between consecutive iterations are within the estimated confidence radius, then the approximate gradients are accepted. Otherwise, the exact gradients are used such as analytical or finite differenced gradients. This hybrid design sensitivity analysis method is embedded in an in-house ALM based dynamic response optimizer, which solves three typical dynamic response optimization problems and one practical design problem for a tracked vehicle suspension system. The optimization results are compared with those of the conventional method that uses only exact gradients throughout optimization process. These comparisons show that the hybrid method is more efficient than the conventional method. Especially, in the tracked vehicle suspension system design, the proposed method yields 14 percent reduction of the total CPU time and the number of analyses than the conventional method, while giving similar optimum values.

• Proceedings of the KSME Conference
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• 2003.11a
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• pp.1654-1659
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• 2003

This paper introduces a fast Fourier transform (FFT)-based spectral analysis method for the transient responses as well as the steady-state responses of linear discrete systems. The force vibration of a viscously damped three-DOF system is considered as the illustrative numerical example. The proposed spectral analysis method is evaluated by comparing with the exact analytical solutions as well as with the numerical solutions obtained by the Runge-Kutta method.

• Steel and Composite Structures
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• v.50 no.2
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• pp.135-148
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• 2024

This paper analytically investigates the free vibration analysis of functionally graded-carbon nanotube reinforced composite (FG-CNTRC) plates by dynamic stiffness method (DSM). The properties of CNTRC are determined with the extended rule of mixture. The governing differential equations of motion based on the first-order shear deformation theory of CNTRC plate are derived using Hamilton's principle. The FG-CNTRC plates are studied for a uniform and two different distributions of carbon nanotubes (CNTs). The accuracy and performance of the DSM are compared with the results obtained from closed closed-form and semi-analytical solution methods in previous studies. In this study, the effects of boundary condition, distribution type of CNTs, plate aspect ratio, plate length to thickness ratio, and different values of CNTs volume fraction on the natural frequencies of the FG-CNTRC plates are investigated. Finally, various natural frequencies of the plates in different conditions are provided as a benchmark for comparing the accuracy and precision of the other analytical and numerical methods.