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In this paper, equations of motion for a horizontal axis wind turbine with movable base are extracted and natural frequencies and vibration of the system is studied. The wind turbine tower is assumed rigid while its blades are modeled as flexible beams. In-plane bending and twisting are considered as two degrees of freedom for blades motion.The shaft connected the tower to blades is assumed rigid and its rotational velocity is considered.In this paper, specifically, a 5-megawattfloating horizontal axis wind turbine, which it’s basehas three angular velocities in different directions,is studied.Due to the complex shape and variation of the properties along the length, the turbine blade properties such as mass and geometric parameters are extracted by curve fitting in MATLAB.The equations of motion and boundary conditions are derived by Hamilton's principle and then are transformed to ordinary differential equations by Galerkin method. By setting the governing equations to standard form (space state), eigenvalues and frequencies are calculated. The numerical results are compared with published results and good agreement is observed.Then the effect of various parameters on turbine blades frequencies and time responses are demonstrated. Results show that the tower base angular velocity and blades rotational speed have considerable effects on turbine blades time response and vibration frequencies.

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The position of the planets for the planetary gear systems are in two forms of equally and unequally spaced. This paper investigates free vibration of the planetary gear with unequally spaced planets. The planetary gear set of this study is modeled as set of lumped masses and springs. Each component such as sun gear, carrier, ring gear and planets possesses three degrees of freedom and considered as rigid bodies. Bearing and mesh stiffnesses are modeled in the form of linear springs. Generally, planet, rotational, translational, distinct and degenerate modes are five vibration modes of the planetary gear systems. The results show that the translational mode for the system with numbers of even equally and unequally spaced planets, is different and rotational and translational modes have the same characteristics for the both systems. For the system with numbers of even unequally spaced planets, the natural frequencies of the translational modes have multiplicity one. When the numbers of the planets of the system are odd and the position of them is unequally spaced, the rotational and planet modes are generating and the natural frequencies of the translational modes are not appears. For the distinct and degenerate modes of the system with unequally spaced planets, the planets only have the rotational motion.

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The beam theory is used in the analysis and design of a wide range of structures, from buildings to bridges to the load-bearing bones of the human body. Beams resting on elastic foundation have wide application in many branches of engineering problems namely geotechnics, road, railroad and marine engineering and bio-mechanics. The foundation is very often a rather complex medium; e.g., a rubberlike fuel binder, snow, or granular soil. The key issue in the analysis is modelling the contact between the structural elements and the elastic bed. Since of interest here is the response of the foundation at the contact area and not the stresses or displacements inside the foundation material, In most cases the contact is presented by replacing the elastic foundation with simple models, usually spring elements. The most frequently used foundation model in the analysis of beam on elastic foundation problems is the Winkler foundation model. In the Winkler model, the elastic bed is modeled as uniformly distributed, mutually independent, and linear elastic vertical springs which produce distributed reactions in the direction of the deflection of the beam. However since the model does not take into account either continuity or cohesion of the bed, it may be considered as a rather crude representation of the elastic foundation. In order to find a physically close and mathematically simple foundation model, Pasternak proposed a so-called two-parameter foundation model with shear interactions. The first foundation parameter is the same as the Winkler foundation model and the second one is the stiffness of the shearing layer in the Pasternak foundation model.
Dynamic analysis is an important part of structural investigation and the results of free vibration analysis are useful in this context. Vibration problems of beams on elastic foundation occupy an important place in many fields of structural and foundation engineering.With the increase of thickness, existence of simplifying hypotheses in beam theories such as the ignorance of rotational inertial and transverse shear deformation in classic theory, application of determination coefficient in first-order shear theory and expression of one or few unknown functions based on other functions in higher-order shear theories is accompanied by reduction in accuracy of these theories. This represents the necessity of precise and analytical solutions for beam problems with the least number of simplifying hypotheses and for different thicknesses.
In the present study, the analytical solution for the problem of free vibration of homogeneous prismatic simply supported beam with rectangular solid sections and desired thickness resting on Pasternak elastic foundation is provided for completely isotropic behaviors under two-dimensional theory of elasticity and functions of displacement potentials. Characteristic equations of natural vibration are defined by solving one partial differential equations of fourth order through separation of variables and application of boundary conditions. The major characteristics of present study are lack of limitation of thickness and its validity for beams of low, medium and large thickness. To verify, the results of present study were compared with those of other studies. The results show that increases of foundation parameters is associated with an increased natural frequency, The intensity by increasing the ratio of thickness to length and in values larger than 0.2 and in the higher modes of vibration is reduced considerably.

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Viscoelasticity is a property of materials that exhibit both viscous and elastic characteristics. In linear viscoelasticity, the stress is linearly related to the history function of strain. This paper discusses vibration analysis of functionally graded viscoelastic rectangular plate. The viscoelastic behavior of the plate is modeled using the Zener three-parameter model. Also, the material properties of the plate are graded through the thickness according to the volume fraction model. The maximum stress and strain are calculated based on the linear first-order shear deformation theory and the simply support boundary conditions is assumed at all four edges of the plate. A code is prepared using the Mathematica software to obtain the frequency values and effect of inherent and geometric characteristics of the sheet on natural frequency of the plate. These effects are studied using the tables and graphs represented in the results and discussion section of the paper. The results obtained in this paper are simplified to a functionally elastic plate to compare with those given in the literature search. The comparison of results shows good agreement against data given in literature for both cases.

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Because of increasing demands for using of rotating systems in high accuracy and high speed applications, in addition of specific condition of rotating systems, it is necessary to analyze these rotating systems characteristics. Tolerance analysis is a useful tool for estimating effects of dimensional and geometrical errors of effective parameters on functional characteristics in a mechanical system. Unlike other mechanical systems, in addition to the dimensional and geometrical errors, the accuracy of the rotary systems performance directly depend on the flexibility of parts and Non Repetitive Run-Out (NRRO) errors. In this paper, a new method is proposed for static and dynamic tolerance analysis of the rotary systems with the dimensional and geometrical errors, the flexibility effects, and the NRRO errors based on the tolerance zone model. First, using the small degrees of freedom concept, the dimensional and geometrical errors and the NRRO error are modeled in the tolerance zone. Then, based on a new strategy, the performance -assembly functions of the system for modeling the error propagation of the rotary system in the static and dynamic conditions are extracted. Then, using the proposed equations, sensitivities of the requirements such as the end of shaft position and the main natural frequency to tolerances are computed. To illustrate applicability of the proposed method, a rotary system is considered as a case study. Monte Carlo simulations are used for validation of the computational results from proposed method.

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The torsional vibrational characteristics of nano-scale sphere using an exact size-dependent elasticity solution based on Gurtin-Murdoch’s surface elasticity model are studied. In the absence of body forces, the displacement field is governed by the classical Navier’s equation. Helmholtz decomposition is used to separate the dynamic equations of motion into the decoupled vector wave equations. The motion under consideration is assumed to be torsional and vector wave equation exactly is solved and displacement field and stress tensor are obtained. Size-dependent elasticity solution based on Gurtin-Murdoch surface energy model is employed to incorporate the surface stress terms into the pertinent boundary conditions, leading to frequency equations involving spherical Bessel functions. Isotropic aluminum with two different set of surface properties corresponding to the crystallography directions are considered and extensive numerical calculations have been carried out to illustrate the size effect of the nano-sphere on the first and second dimensionless vibrational natural frequencies. The numerical results describe the imperative influence of surface energy and radii ratio on vibrational characteristic frequency of nano-sphere. In particular, the surface energy is much important when inner radius is smaller than 50 nm

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Welding is very important in the aerospace industry and widely used in aerospace structures. One of the problems that most industries are facing is created residual stress by the welding process. Residual stresses in the surrounding areas of welding can cause cracks and crack growth so identify and evaluate of residual stresses in the welded structures is necessary. There are different methods for determining the residual stress. In this paper, the laboratory and numerical methods were presented for determining the residual stress. Then, the welding process of two aluminum sheets of 6061-T6 alloy has been done and the residual stresses have been obtained by drilling method. Welding is done in two passes and by spot welding the first, the end and the middle of the weld line are connected to prevent the sheets from moving. Also, the welding process of the two aluminum sheets was simulated in 3D in the ABAQUS finite element software and the residual stresses were extracted. All conditions in the finite element analysis are similar to the welding conditions in the laboratory. Results show high accuracy in the modeling of finite element processes in the welding process. Finally, the effect of residual stress in the value of natural frequencies is studied.

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Piezoelectric materials, in different shapes such as rectangular plate, annular plate, circular plate and cylindrical shell, have increasing application in industries in order to create smart structures. In this article, experimental and numerical analysis of free vibration of a two-layered cylindrical panel with metal and piezoelectric layer in different boundary condition is carried out. First, a single PZT-4 layer is polarized in radial direction. Using the Piezoelectric layer and an Aluminum layer, a two-layered smart panel is prepared. Then, the first natural frequency of the hybrid panel with free boundary condition is measured experimentally in three different ways. The hybrid panel is simulated in a finite element software (Abaqus). Results show good agreement between different experimental methods, as well as, between finite element model and experimental results. The accuracy, limitations and merits of different experimental methods are discussed completely. The results show that the natural frequency can be achieved accurately by excitation of actuator layer. Finally, the influence of different boundary conditions as well as geometrical parameter such as radios, length and thickness of smart cylindrical panel are investigated using the finite element software.

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The purpose of this paper is to experimentally investigate the natural frequencies of notched homogeneous and inhomogeneous specimen made from API X65 steel. The specimens cut from spiral welded pipe, tested on an equipped low blow drop weight tester with accelerometer. The tests were performed according to the API 5L standard. The homogeneous specimen was seamless and included only the base metal, while the inhomogeneous specimen included the weld seam and three zones of base metal, heat affected zone and weld. In the present study the specimens were subjected to hammer low blow in the middle without plastic deformation. The laboratory data (voltage-time) were transferred from time to frequency domain using Fourier transformation and the imposed oscillations were removed from the frequency signal by the Butterworth low pass filter. As the hammer drop height increased, the natural frequency in the specimens was almost constant. The natural frequency in the inhomogeneous specimen was less than the homogeneous specimen. Having information about the natural frequency, it is possible to prevent the destructive phenomenon of resonance in the main test (complete fracture of the specimen). Also, using the results of equipped low blow drop weight test and knowing the natural frequency, the dynamic stress intensity factor of the test specimen can be determined.

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