• Structural Engineering and Mechanics
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• v.52 no.5
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• pp.997-1017
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• 2014

According to the structure characteristics of the non-uniform beam bridge, a practical model for calculating the vibration equation of the non-uniform beam bridge is given and the application scope of the model includes not only the beam bridge structure but also the non-uniform beam with added masses and elastic supports. Based on the Bernoulli-Euler beam theory, extending the application of the modal perturbation method and establishment of a semi-analytical method for solving the vibration equation of the non-uniform beam with added masses and elastic supports based is able to be made. In the modal subspace of the uniform beam with the elastic supports, the variable coefficient differential equation that describes the dynamic behavior of the non-uniform beam is converted to nonlinear algebraic equations. Extending the application of the modal perturbation method is suitable for solving the vibration equation of the simply supported and continuous non-uniform beam with its arbitrary added masses and elastic supports. The examples, that are analyzed, demonstrate the high precision and fast convergence speed of the method. Further study of the timesaving method for the dynamic characteristics of symmetrical beam and the symmetry of mode shape should be developed. Eventually, the effects of elastic supports and added masses on dynamic characteristics of the three-span non-uniform beam bridge are reported.

• Structural Engineering and Mechanics
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• v.9 no.5
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• pp.449-467
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• 2000

The natural frequencies and the corresponding mode shapes of a non-uniform beam carrying multiple point masses are determined by using the analytical-and-numerical-combined method. To confirm the reliability of the last approach, all the presented results are compared with those obtained from the existing literature or the conventional finite element method and close agreement is achieved. For a "uniform" beam, the natural frequencies and mode shapes of the "clamped-hinged" beam are exactly equal to those of the "hinged-clamped" beam so that one eigenvalue equation is available for two boundary conditions, but this is not true for a "non-uniform" beam. To improve this drawback, a simple transformation function ${\varphi}({\xi})=(e+{\xi}{\alpha})^2$ is presented. Where ${\xi}=x/L$ is the ratio of the axial coordinate x to the beam length L, ${\alpha}$ is a taper constant for the non-uniform beam, e=1.0 for "positive" taper and e=1.0+$|{\alpha}|$ for "negative" taper (where $|{\alpha}|$ is the absolute value of ${\alpha}$). Based on the last function, the eigenvalue equation for a non-uniform beam with "positive" taper (with increasingly varying stiffness) is also available for that with "negative" taper (with decreasingly varying stiffness) so that half of the effort may be saved. For the purpose of comparison, the eigenvalue equations for a positively-tapered beam with five types of boundary conditions are derived. Besides, a general expression for the "normal" mode shapes of the non-uniform beam is also presented.

• Smart Structures and Systems
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• v.30 no.2
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• pp.173-181
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• 2022

The current paper presents a technique to detect crack in non-uniform cantilever-type pipe beams, that have step changes in the properties of their cross sections, restrained by a translational and rotational spring with a tip mass at the free end. An equation for estimating the natural frequencies for the non-uniform beams is derived using the boundary and continuity conditions, and an equivalent bending stiffness for cracked beam is applied to calculate the natural frequencies of the cracked beam. An experimental study for a step-changed non-uniform cantilever-type pipe beam restrained by bolts with a tip mass is carried out to verify the proposed method. The translational and rotational spring constants are updated using the neural network technique to the results of the experiment for intact case in order to establish a baseline model for the subsequent crack detection. Then, several numerical simulations for the specimen are carried out using the derived equation for estimating the natural frequencies of the cracked beam to construct a set of training patterns of a neural network. The crack locations and sizes are identified using the trained neural network for the 5 damage cases. It is found that the crack locations and sizes are reasonably well estimated from a practical point of view. And it is considered that the usefulness of the proposed method for structural health monitoring of the step-changed non-uniform cantilever-type pipe beam-like structures elastically restrained in the ground and have a tip mass at the free end could be verified.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 2000.05a
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• pp.553-556
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• 2000

The paper deals with the dynamic response of non-uniform beams subjected to a moving mass. In the dynamic analysis, the effects of inertia force, elastic force, centrifugal force, Coriolis force and self weight due to moving mass are taken into account. Galerkin's mode summation method is applied for the discretized equations of notion. Numerical results for the dynamic response of the non-uniform beam under a moving mass having various magnitudes and velocities are investigated. Experimental results have a good agrement with predictions

• Transactions of the Korean Society of Mechanical Engineers B
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• v.22 no.8
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• pp.1073-1082
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• 1998

When thermal diffusivity is measured by laser flash method, the thermal diffusivity call be calculated front the assumption of the uniformly heated whole surface of the specimen. It has been known that the approximate 5% error is made by the non-uniform energy distribution on the specimen surface of laser pulse heat source. In this study, to obtain the highly-uniformed laser beam, which has both the low non-uniform heating error from non-uniform laser beam and the energy loss, research was carried out on no transmitting loss by optical fiber and high repetitions. In addition, heating error and thermal diffusivity were measured as the measuring positions were varied and compared with the results using the uniform and the non-uniform laser beams. In addition, dole to using the uniformalized laser beam, the whole surface of the specimen was heated uniformly and as a result, it was the thought that this was very effective to reduce the variations of the errors of the thermal diffusivity as the measuring positions were varied. It can be obtained that when the thermal diffusivity of POCO-AXM-5Q1 of SRM in NBS was measured with both the uniform and the non-uniform laser beams, the dispersion error of the former was from 2 to 2.5%, which was more improved than that of the latter.

• Journal of the Korean Society for Precision Engineering
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• v.15 no.12
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• pp.202-211
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• 1998

The present paper proposes a modal analysis procedure to obtain exact modal parameters (natural frequencies, damping ratios, eigenvectors) for general, non-uniform beam-like structures. The proposed method includes a derivation of the system dynamic matrix for a Timoshenko beam element. The proposed method provides not only exact modal parameters but also exact frequency response functions (FRFs) for general beam structures. A time domain analysis method is also proposed. Two examples are provided for validating and illustrating the proposed method. The first numerical example compares the proposed method with FEM. The second example deals with a non-uniform beam structure supported in joints with damping property. The numerical study proves that the proposed method is useful for the dynamic analysis of continuous systems consisting of beam-like structures.

• Structural Engineering and Mechanics
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• v.63 no.4
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• pp.551-565
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• 2017

In this paper, an analytical approach is proposed for determining vibration characteristics of cracked non-uniform continuous Timoshenko beam carrying an arbitrary number of spring-mass systems. This method is based on the Timoshenko beam theory, transfer matrix method and numerical assembly method to obtain natural frequencies and mode shapes. Firstly, the beam is considered to be divided into several segments by spring-mass systems and support points, and four undetermined coefficients of vibration modal function are contained in each sub-segment. The undetermined coefficient matrices at spring-mass systems and pinned supports are obtained by using equilibrium and continuity conditions. Then, the overall matrix of undetermined coefficients for the whole vibration system is obtained by the numerical assembly technique. The natural frequencies and mode shapes of a cracked non-uniform continuous Timoshenko beam carrying an arbitrary number of spring-mass systems are obtained from the overall matrix combined with half-interval method and Runge-Kutta method. Finally, two numerical examples are used to verify the validity and reliability of this method, and the effects of cracks on the transverse vibration mode shapes and the rotational mode shapes are compared. The influences of the crack location, depth, position of spring-mass system and other parameters on natural frequencies of non-uniform continuous Timoshenko beam are discussed.

• Structural Engineering and Mechanics
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• v.40 no.3
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• pp.347-371
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• 2011

This paper focuses on post-buckling analysis of Timoshenko beams with various boundary conditions subjected to a non-uniform thermal loading by using the total Lagrangian Timoshenko beam element approximation. Six types of support conditions for the beams are considered. The considered highly non-linear problem is solved by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. As far as the authors know, there is no study on the post-buckling analysis of Timoshenko beams under uniform and non-uniform thermal loading considering full geometric non-linearity investigated by using finite element method. The convergence studies are made and the obtained results are compared with the published results. In the study, the relationships between deflections, end rotational angles, end constraint forces, thermal buckling configuration, stress distributions through the thickness of the beams and temperature rising are illustrated in detail in post-buckling case.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.11 no.5
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• pp.140-147
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• 2001

In this paper, the dynamic response of non-uniform beams subjected to a travelling mass is investigated. Dynamic behaviors of flexible beam structures under a moving mass have been a concern in the design of bridges, ceiling crain in industry, as well as gun barrel fields. Most of studies for moving mass problems have been related to the theoretical dynamic responses of a simple beam model with uniform cross-sections. In some experimental studies, only a few transverse inertia effects due to travelling mass have been studied so far. The intended aim of the present Paper is to investigate the dynamic response of non-uniform beams taking into account of inertia force. centrifugal force, Coriollis force and self weight due to travelling mass. Galerkin's mode summation method is applied for the discretized equations of motion. Numerical results for the dynamic response of non-uniform beams under a travelling mass are demonstrated for various magnitudes and velocities of the travelling mass. In order to verify propriety of numerical solutions, experiments were conducted. Experimental resu1ts have a good agreement wish theoretical Predictions.

• Structural Engineering and Mechanics
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• v.22 no.3
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• pp.279-292
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• 2006

A new approach using the differential quadrature method (DQM) is derived for analysis of non-uniform beams resting on nonlinear media in this study. The influence of velocity dependent viscous damping and strain rate dependent viscous damping is investigated. The results solved using the DQM have excellent agreement with the results solved using the FEM. Numerical results indicated that the DQM is valid and efficient for non-uniform beams resting on non-linear media.