• Structural Engineering and Mechanics
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• v.15 no.6
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• pp.705-716
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• 2003

Gradient elastic flexural beams are dynamically analysed by analytic means. The governing equation of flexural beam motion is obtained by combining the Bernoulli-Euler beam theory and the simple gradient elasticity theory due to Aifantis. All possible boundary conditions (classical and non-classical or gradient type) are obtained with the aid of a variational statement. A wave propagation analysis reveals the existence of wave dispersion in gradient elastic beams. Free vibrations of gradient elastic beams are analysed and natural frequencies and modal shapes are obtained. Forced vibrations of these beams are also analysed with the aid of the Laplace transform with respect to time and their response to loads with any time variation is obtained. Numerical examples are presented for both free and forced vibrations of a simply supported and a cantilever beam, respectively, in order to assess the gradient effect on the natural frequencies, modal shapes and beam response.

• Structural Engineering and Mechanics
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• v.30 no.5
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• pp.513-534
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• 2008

This paper examines local vibrations in the stay cables of a cable-stayed bridge subjected to wind gusts. The wind loads, including the self-excited load and the buffeting load, are converted into time-domain values using the rational function approximation and the multidimensional autoregressive process, respectively. The global motion of the girder, which is generated by the wind gusts, is analyzed using the modal analysis method. The local vibration of stay cables is calculated using a model in which an inclined cable is subjected to time-varying displacement at one support under global vibration. This model can consider both forced vibration and parametric vibration. The response characteristics of the local vibrations in the stay cables under wind gusts are described using an existing cable-stayed bridge. The results of the numerical analysis show a significant difference between the combined parametric and forced vibrations and the forced vibration.

• Transactions of the Korean Society of Mechanical Engineers
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• v.18 no.8
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• pp.1910-1919
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• 1994

The nonlinear behavior of continuous structural systems which possess external resonances as well as internal resonances are found be exhibit interesting reponses, arising because of the exhange of energy between the coupled modes. In this paper, the undamped forced vibrations was studied on the effect of primary resonance based on the concept of normal modes. By using the concept of normal mode the stability relation between free and forced vibrations was investigated in case of small exciting force. Numerical results show that the excitation of one unstable mode has a great influence on the response of the other mode but that of one stable mode does not.

• Advances in materials Research
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• v.9 no.1
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• pp.33-48
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• 2020

With the use of differential quadrature method (DQM), forced vibrations and resonance frequency analysis of functionally graded (FG) nano-size beams rested on elastic substrate have been studied utilizing a shear deformation refined beam theory which contains shear deformations influence needless of any correction coefficient. The nano-size beam is exposed to uniformly-type dynamical loads having partial length. The two parameters elastic substrate is consist of linear springs as well as shear coefficient. Gradation of each material property for nano-size beam has been defined in the context of Mori-Tanaka scheme. Governing equations for embedded refined FG nano-size beams exposed to dynamical load have been achieved by utilizing Eringen's nonlocal differential law and Hamilton's rule. Derived equations have solved via DQM based on simply supported-simply supported edge condition. It will be shown that forced vibrations properties and resonance frequency of embedded FG nano-size beam are prominently affected by material gradation, nonlocal field, substrate coefficients and load factors.

• Steel and Composite Structures
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• v.35 no.6
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• pp.765-777
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• 2020

In the context of classic conical shell formulation, nonlinear forced vibration analysis of truncated conical shells and annular plates made of multi-scale epoxy/CNT/fiberglass composites has been presented. The composite material is reinforced by carbon nanotube (CNT) and also fiberglass for which the material properties are defined according to a 3D Mori-Tanaka micromechanical scheme. By utilizing the Jacobi elliptic functions, the frequency-deflection curves of truncated conical shells and annular plates related to their forced vibrations have been derived. The main focus is to study the influences of CNT amount, fiberglass volume, open angle, fiber angle, truncated distance and force magnitude on forced vibrational behaviors of multi-scale truncated conical shells and annular plates.

• Structural Engineering and Mechanics
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• v.71 no.5
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• pp.553-562
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• 2019

This paper presents the results of a study into the dynamic behaviour of a support structure of a mobile elevating work platform. The vibrations of the mechanical system of the observed structure are examined analytically, numerically, and experimentally. Within the analytical examination, a simple mathematical model is developed to describe free and forced vibrations. The dynamic analysis of the mechanical system is conducted using a discrete dynamic model with a reduced number of vibrational degrees of freedom. On the basis of the expression for the system energy, and by applying Lagrange's equations of the second kind, differential equations are derived for system vibrations, frequencies are determined, and the laws of forced platform vibration are established. At the same time, a nonlinear FEM model is developed and the laws of free and forced vibration are determined. The experimental and numerical part of the study deal with the examination of the real structure in extreme conditions, taking into account: the lowest eigenfrequency, forced actions that could endanger the general stability, the maximal amplitudes, and the acceleration of the work platform. The obtained analytical and numerical results are compared with the experiments. The experimental verification points to the adverse behaviour of the platform in excitation cases - swaying. In such a situation, even a relatively small physical force can lead to unacceptably high amplitudes of displacement and acceleration - exceeding the usual work values.

• Interaction and multiscale mechanics
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• v.2 no.2
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• pp.189-207
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• 2009

Dynamics of the electric system for the toroidal drive under mechanical disturbance is presented. Using the method of perturbation, free vibrations of the electric system under mechanical disturbance are studied. The forced responses of the electric system to voltage excitation under mechanical disturbance are also presented. We show that as the time grows, the resonance vibration caused by voltage excitation still exists and the vibrations caused by mechanical disturbance are enlarged. The coupled resonance vibration caused by mechanical disturbance and voltage excitation is discussed. The conditions of the occurrence of coupled resonance are studied.

• Computational Structural Engineering
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• v.6 no.1
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• pp.99-105
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• 1993

A Stepped beam with immovable ends under forced vibrations with large amplitude is investigated by using the finite element method and the Ritz vectors. Unlike the Eigen vectors, the Ritz vectors are generated by a simple recurrence relation. Moreover the Ritz vectors yield much faster convergence with respect to the number of vectors used than the use of Eigen vectors. The computer program is developed for nonlinear analysis using Ritz vectors instead of Eigen vectors and numerical examples are analysed for deflections and natural frequencies of stepped beam under various support conditions. Results show that the proposed method is valid and efficient.

• Coupled systems mechanics
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• v.7 no.4
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• pp.485-507
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• 2018

The moving load causes the occurrence of vibrations in civil engineering structures such as bridges, railway lines, bridge cranes and others. A novel engineering method for separation of the variables in the differential equation of the elastic line of Bernoulli-Euler beam has been developed. The method can be utilized in engineering structures, leading to "a beam under moving load model" with generalized boundary conditions. This method has been implemented for analytical study of the dynamic response of the metal structure of a single girder bridge crane due to the telpher movement along the bridge girder. The modeled system includes: a crane bridge girder; a telpher, moving with a constant horizontal velocity; a load, elastically fixed to the telpher. The forced vibrations with their own frequencies and with a forced frequency, due to the telpher movement, have been analyzed. The loading resulting from the telpher uniform movement along the bridge girder is cyclical, which is a prerequisite for nucleation and propagation of fatigue cracks. The concept of "dynamic coefficient" has been introduced, which is defined as a ratio of the dynamic deflection of the bridge girder due to forced vibrations, to the static one. This ratio has been compared with the known from the literature empirical dynamic coefficient, which is due to the telpher track unevenness. The introduced dynamic coefficient shows larger values and has to be taken into account for engineering calculations of the bridge crane metal structure. In order to verify the degree of approximation, the obtained results have been compared with FEM outcomes. An additional comparison has been made with the exact solution, proposed by Timoshenko, for the case of simply supported beam subjected to a moving force. The comparisons show a good agreement.

• Structural Engineering and Mechanics
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• v.83 no.4
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• pp.451-463
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• 2022

The present study deals with forced vibrations of an elastic circular plate supported along its circular edge by unilateral elastic springs. The plate is assumed to be subjected to a uniformly distributed and a concentrated load. Under the combination of these loads, equations of motion are explicitly derived for static and dynamic response analyses by assuming a series of the displacement functions of time and other unknown parameters which are to be determined by employing Lagrangian functional. The approximate solution is sought by applying the Lagrange equations of motions by using the potential energy of the external forces that includes the contributions of the edge forces and the external moments, i.e., those of the effects of the boundary condition to the analysis. For the numerical treatment of the problem in the time domain, the linear acceleration procedure is adopted. The tensionless character of the support is taken into account by using an iterative process and, the coordinate functions for the displacement field are selected to partially fulfill the boundary conditions so that an acceptable approximation can be achieved faster. Numerical results are presented in the figures focusing on the nonlinearity of the problem due to the plate lift-off from the unilateral springs at the edge support.