new analytical approach for determination of flexural, axial and torsional natural frequencies of beams
TL;DR: In this paper, a new and simplified method is presented in which the natural frequencies of the uniform and non-uniform beams are calculated through simple mathematical relationships, and the efficiency and convergence rate of the current approach are investigated through comparison of the numerical results obtained with those obtained from other published references and results of available finite element software.
Abstract: In this paper, a new and simplified method is presented in which the natural frequencies of the uniform and non-uniform beams are calculated through simple mathematical relationships. The various vibration problems such as: Rayleigh beam under variable axial force, axial vibration of a bar with and without end discrete spring, torsional vibration of a bar with an attached mass moment of inertia, flexural vibration of the beam with laterally distributed elastic springs and also flexural vibration of the beam with effects of viscose damping are investigated. The governing differential equations are first obtained and then; according to a harmonic vibration, are converted into single variable equations in terms of location. Through repetitive integrations, the governing equations are converted into weak form integral equations. The mode shape functions of the vibration are approximated using a power series. Substitution of the power series into the integral equations results in a system of linear algebraic equations. The natural frequencies are determined by calculation of a non-trivial solution for system of equations. The efficiency and convergence rate of the current approach are investigated through comparison of the numerical results obtained with those obtained from other published references and results of available finite element software.
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12 Dec 2017
TL;DR: In this paper, an analytical approach is presented for the determination of the natural frequencies of tall structures with a combined system of the framed tube, shear core and outrigger-belt truss.
Abstract: In this paper, an analytical approach is presented for the determination of the natural frequencies of tall structures with a combined system of the framed tube, shear core and outrigger-belt truss. It has been assumed that the structure has variable stiffness and mass along the height. The framed tube is modeled as a cantilevered beam with variable box cross section and effects of outrigger-belt truss and shear core on the framed tube system are modeled as a concentrated moment applied at outrigger-belt truss locations. Through repetitive integrations, the governing partial differential equations are converted into weak form integral equations. By applying the boundary conditions, the integration constants are determined. The mode shape function is approximated by a power series. Substitution of the power series into weak form integral equations results in a system of linear algebraic equations. The natural frequencies are determined by calculation of the non-trivial solution for the resulting system of equations. Accuracy of the proposed method is verified through several numerical examples, in which the results of the analysis are compared with those obtained from other references.
10 citations
Cites background from "new analytical approach for determi..."
...M. Mohammadnejad and H. Haji Kazemi 1224 The first four frequencies of building with combined system of framed tube, shear core and double-belt trusses are calculated and presented in Table 5....
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...Saffari H, Mohammadnejad M....
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...Saffari et al. [23], Mohammadnejad et al. [24], Saffari et al. [25] and Mohammadnejad [26] presented an analytical approach for determination of the natural frequencies of non-prismatic Beams....
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...Safari H, Mohammadnejad M, Bagheripour MH. Free vibration analysis of nonprismatic beams under variable axial forces, Structural Engineering and Mechanics an International Journal, 43(2012) 561-82....
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...Mohammadnejad M, Safari H, Bagheripour MH....
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TL;DR: In this article, a tall building was modeled as a cantilever beam and analyzed with the assumption of flexural behavior based on Euler-Bernoulli Beam Theory, then the displacement of floors was calculated.
Abstract: In this paper, the continuum model, which is known as Kwan model, has been presented for the analysis of tall buildings that have been as an appropriate approximation of the overall behavior of the structure. Tall building was modeled as a cantilever beam and analyzed with the assumption of flexural behavior based on Euler–Bernoulli Beam Theory, then the displacement of floors was calculated. o consider the shear lag effects in the overall displacement of the structure, Timoshenko’s beam model has been considered and related relations were extracted. The lateral displacement formulas obtained and calculated for the framed tube system modeled by Kwan’s method. To verify the results, numerical models were created in software (ETABS) and statically were analyzed for lateral loading. Finally the results were compared with those obtained by computer analysis and the corresponding diagrams were presented. At the end, the shape factor formula has been developed to improve the results of the Timoshenko’s theory.
6 citations
Cites background from "new analytical approach for determi..."
...(2014), Malekinejad and Rahgozar (2014), Mohammadnejad (2015), Rahgozar et al....
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DOI•
01 Jul 2020TL;DR: In this article, the authors used the energy method and the continuous beam model to estimate the outrigger's optimum position in order to minimize the lateral stiffness of tall buildings for economic reasons.
Abstract: The sensitivity of tall buildings subjected to the lateral loads is more than that of the gravity loads. Therefore, the conventional methods are not efficient yet, and new methods are proposed by designers to reduce the structural roof displacement, shear lag, overturning moment, and also increase the lateral resistance of the structures. In the design of tall structures, it is desirable to minimize the lateral stiffness of the structures for economic reasons. In this paper, the structure is modeled using the energy method and the continuous beam model. The outrigger's optimum position is calculated considering different loading patterns. It is assumed that the lateral stiffness of the structure changes with the height. An equivalent rotational spring is utilized to model the belt truss and outrigger system. The results show that the outrigger's optimum position depends on the type of the lateral load as well as how the stiffness changes in the height of the structure.
3 citations
TL;DR: In this paper, the Laguerre collocation method is used for solving the governing differential equation subject to initial conditions, and the results are compared with other results from the literature.
Abstract: Natural frequencies and free vibration are important characteristics of beams with non-uniform cross section. Hence, the solution for free vibrations of non-uniform beams is presented using a Laguerre collocation method. The elastically restrained beam model is based on the Euler–Bernoulli theory. Also, the non-uniform beam is rested on a non-uniform foundation (Winkler type). The Laguerre collocation method is introduced for solving the differential equation. This approach reduces the governing differential equation to a system of algebraic equations, and finally, the problem is greatly simplified. Properties of Laguerre polynomials and the operational matrix of derivation are first presented. Eventually, the proposed method is applied for solving the governing differential equation subject to initial conditions, and the results are compared with other results from the literature.
3 citations
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TL;DR: In this article, a simple approach is presented to solve natural frequencies of free vibration of axially functionally graded beams with non-uniform cross-section for various end supports including simply supported, clamped, and free ends.
285 citations
TL;DR: In this paper, the exact frequencies and mode shapes for rotating beams in which both the flexural rigidity and the mass distribution vary linearly were solved using the Frobenius method.
Abstract: The method of Frobenius is used to solve for the exact frequencies and mode shapes for rotating beams in which both the flexural rigidity and the mass distribution vary linearly. Results are tabulated for a variety of situations including uniform and tapered beams, with root offset and tip mass, and for both hinged root and fixed root boundary conditions. The results obtained for the case of the uniform cantilever beam are compared with other solutions, and the results of a conventional finite-element code.
264 citations
TL;DR: In this paper, the free vibration analysis of functionally graded material (FGM) beams subjected to different sets of boundary conditions is performed based on the classical and first order shear deformation beam theories.
Abstract: Present investigation is concerned with the free vibration analysis of functionally graded material (FGM) beams subjected to different sets of boundary conditions. The analysis is based on the classical and first order shear deformation beam theories. Material properties of the beam vary continuously in the thickness direction according to the power-law exponent form. Trial functions denoting the displacement components of the cross-sections of the beam are expressed in simple algebraic polynomial forms. The governing equations are obtained by means of Rayleigh–Ritz method. The objective is to study the effects of constituent volume fractions, slenderness ratios and the beam theories on the natural frequencies. To validate the present analysis, comparison studies are also carried out with the available results from the existing literature.
258 citations
TL;DR: In this paper, a new approach for investigating the vibration behaviors of axially functionally graded Timoshenko beams with non-uniform cross-section is presented. And the derived characteristic equation is a polynomial equation, where the lower and higher-order natural frequencies can be determined simultaneously from the multi-root.
Abstract: This paper presents a new approach for investigating the vibration behaviors of axially functionally graded Timoshenko beams with non-uniform cross-section. By introducing an auxiliary function, we can change the coupled governing equations with variable coefficients for the deflection and rotation to a single governing equation. Moreover, all physical quantities can be expressed in terms of the solution of the resulting equation. Making use of power series for unknown function, we can transform the single equation to a system of linear algebraic equations and will get a characteristic equation in natural frequencies for different boundary conditions. An advantage of the suggested approach is that the derived characteristic equation is a polynomial equation, where the lower and higher-order natural frequencies can be determined simultaneously from the multi-roots. Several examples of estimating natural frequencies for axially grade beams and non-uniform beams are presented, which show that our method has fast convergence and obtained numerical results have high accuracy.
160 citations
TL;DR: In this article, a dynamic stiffness method for free vibration analysis of a rotating tapered Rayleigh beam is developed to investigate its free vibration characteristics, including the effects of centrifugal stiffening, an outboard force, an arbitrary hub radius and importantly the rotatory inertia (Rayleigh beam) are included in the analysis.
63 citations