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In this article, analytical solutions of low velocity transverse impact on a nanobeam are presented using the nonlocal theory to bring out the effect of the nonlocal behavior on dynamic deflections. Impact of a projectile (mass) on simply supported and clamped nanobeams are investigated using nonlocal Timoshenko beam theory. In order to obtain an analytical result for this problem, an approximate method has been developed wherein the applied impulse is replaced by a suitable boundary condition and initial momentum of projectile and nanobeam. A number of numerical examples with analytical solutions for nonlocal nanobeam and classical beam (steel and aluminum) have been presented and discussed. When the value of the striker mass is increased, the frequencies are decreased and the maximum dynamic deflection at the center of the beam is increased for both of the simply supported and the clamped-clamped nanobeams. The inclusion of the nonlocal effect increases the magnitudes of dynamic deflections and decreases frequencies. Furthermore, the mass and the velocity of the nanoparticle (striker) have significant effects on the dynamic behavior of nanobeam.

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This paper studies the vibration characteristics of a cracked Timoshenko beam with a varying transverse cross-section using Differential Transform Method (DTM). The effects of the crack location and the crack size in calculating the natural frequencies and mode shapes are investigated. The result have been validated for a beam with and without the crack against those obtained from experimental modal test, Abaqus software and some other methods reported in the literature and a good agreement between the results is observed. The results show that the Timoshenko theory predicts fewer values for the natural frequencies because there is less rigidity, especially for large values of cross-section moment of inertia. Then, the inverse problem is investigated. For this reason, the position and depth of the crack of the beam with a varying transverse cross-section are estimated using the genetic algorithm and then, the natural frequencies are obtained from the modal test. It is seen that the numerical results have a suitable agreement with the actual position and depth of the crack that indicates the effectiveness of this method in determining the parameters of the crack in the Timoshenko beam.

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This paper presents a research into the progress of modeling of N-viscoelastic robotic manipulators. The governing equations of the system is obtained by using Gibbs-Appell (G-A) formulation and Assumed Mode Method (AMM). When the beam is short in length direction, shear deformation is a factor that may have substantial effects on the dynamics of the system. So, in modeling the assumption of Timoshenko Beam Theory (TBT) and its associated mode shapes has been considered. Although considering the effects of damping in continuous systems makes the formulations more complex, two important damping mechanisms, namely, Kelvin-Voigt damping as internal damping and the viscous air damping as external damping have been considered. Finally, to validate the proposed formulation a comparative assessment between the results achieved from experiment and simulation is presented in time domain.

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In this paper, differential quadrature element method (DQEM) is used to analyze the free transverse vibration of multi-stepped rotors resting on multiple bearings. Timoshenko beam theory is used to show the gyroscopic effects; Also each bearing is replaced with four springs; two translational and two rotational acting on two perpendicular directions. Governing equations, compatibility conditions at the each step and each bearing and external boundary conditions are derived and formulated by the differential quadrature rules. First, convergence and versatility of the proposed method are tested by the presented exact solutions. Then, the Campbell diagram is derived for a desired case study and variation of natural frequencies is investigated versus angular velocity of spin. The most advantage of the proposed method is being less time-consuming in comparison with the other methods, especially for cases with high number of steps and bearings. Accuracy of the proposed method is confirmed by the presented exact solutions and effect of angular velocity of spin on natural frequencies (Campbell diagram) is investigated. Comparison of the proposed method with the exact solutions revealed the convergence and accuracy of the proposed method.

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Axially moving beams are extensively involved in various industries and have significant importance in many mechanical engineering problems. In this paper, the nonlinear forced vibrations of axially moving beam under harmonic force and thermal environment have been studied. In order to considering the effects of transverse shear deformation and rotary inertia, the Timoshenko beam theory has been used to model the axially moving beam. The nonlinear governing equations are derived with the help of Hamilton’s principle. Then the equations and boundary conditions are discretized through generalized differential quadrature method (GDQ) and its differential matrix operators, and accordingly the partial differential equations are converted into the ordinary differential equations. To study the frequency response of the system, the harmonic balance method is used. Also the time responses of the axially moving beam are obtained by the Runge-Kutta method. In a case study, the effects of various parameters such as the axial speed, transverse force acting on the beam, damping coefficient and temperature change on the frequency responses of the axially moving beam with both end simply supported boundary conditions are discussed. The results show that the dynamic behavior of system is significantly affected by any of the mentioned factors.

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An analytical method is developed to determine the effects of the frame angle on the behavior of multi span Timoshenko beams with flexible constraints subjected to a two degrees-of-freedom moving system. In a multi-span beam, there are discontinuities in each frame angle in addition to discontinuities in flexible constraints. The correlation among every segment can be obtained considering the compatibility equations in frame angles and cracks. Eigensolutions of the serial-frame system can be calculated explicitly, using the analytical transfer matrix method. The forced responses can be determined by the modal expansion theory using the eigenfunctions. The orthogonality of the mode shapes is applied to calculate the forced response. A new formulation is introduced and confirmed for the orthogonality of the mode shapes for the case of a beam with the frame angle. It is observed that the natural frequencies will be increased by increasing the frame angles, while the maximum deflection of the beam will be decreased. The crack modeling leads to lower natural frequencies and higher maximum deflection for the beam in compared with the no crack model. The validity of the developed technique is probed by comparing the results with the analytical results reported in other articles and those from numerical solutions.

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Free vibration analysis of a cracked rotating multi-span Timoshenko beam is studied in this article to determine the natural frequencies and mode shapes of this beam. First, the relationships between each two segments are obtained by considering the compatibility requirements in the frame angles and in the cracks. To determine the transformed compatibility requirements, the boundary conditions, and the vibrational equations, the so-called differential transform method (DTM) is used. Then, these equations are performed to determine the natural frequencies. The mode shapes of the beam are determined by using the inverse of differential transform method. The results have been validated against those obtained from Abaqus software for a rotating multi-span beam and the ones obtained from transfer matrix method for a non-rotating case that an appropriate agreement is observed. Finally, the effects of the angle of break, the rotational speed, and the crack location on the natural frequencies are investigated. It is shown that the natural frequencies will be increased by increasing the rotational speed. Also, it is seen that the first natural frequency will be increased by moving the crack location from the cantilever support to free support and the variations of other frequencies are dependent to the crack distance to the vibrational nodes. The validation results show the accuracy of DTM in the process of studying the free vibration of this problem.

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The unidirectional composite DCB specimen is considered as two finite length Timoshenko beams, attached together along a common edge except at the initial delamination length. Because of symmetry, only one half of the specimen is considered, which is partly free and partly resting on an elastic foundation. The problem is analytically solved by considering Timoshenko beam resting on Winkler and Pasternak elastic foundations and fracture toughness is generally derived. In the prior researches on this specimen using Timoshenko beam theory, the effect of the ligament length on the energy release rate was ignored. This research presents the solution for finite ligament length. Besides, the effect of ligament length on energy release rate and its minimum value that makes the energy release rate independent of the ligament length, is presented. For the special case when the ligament is large compared with the beam thickness, a closed form solution is derived for Timoshenko beam resting on Winkler elastic foundation. The analytical results are compared to prior researches on this subject and a good agreement is observed. The fracture toughness and compliance obtained by Timoshenko beam resting on Winkler elastic foundation predicts more accurate results with respect to experimental results.

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In this paper a new high order element is proposed for analysis of beams with shear deformation effect. In each node of this element exist translation and rotation degrees of freedom. The element’s formulation is based on the first-order shear deformation theory (FSDT). For this aim, displacement field of the element is selected from fifth order. Moreover, the shear strain is varied as quadratic function throughout the element. It is worth emphasizing that the quadratic function can be used for axial displacement field. By employing of curvature and shear strain relations of Timoshenko beam theory, the exact and explicit shape functions of the displacement fields is obtained. By utilizing these shape functions, beam elements’ stiffness matrix is also calculated. Finally, several numerical tests are performed to assess the robustness of the suggested element. The results of the numerical testes are proven the absence of the shear locking and high accuracy and efficiency of the proposed element for analysis of beam and frame structures. It should be mentioned, due to utilizing fifth order function for displacement field, the proposed element calculate exact solution for displacements and internal forces throughout the element for triangular distributed loads.

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The vibration analysis of curved composite structures under the moving vehicles, is rarely investigated in litreture. Therefore, this paper is studied the dynamic response of a simply supported laminated deep curved beam under a moving load based on Timoshenko beam theory.It is assumed that the curvature of the beam and the amplitude and the speed of the moving load areconstant.The governing equations of motion for the system is extracted by Hamilton principles. A numerical and analytical methods are applied to obtain the dynamic response of the system.Also, the critical speed of the moving load and the fundamental frequency of the beam are obtained. The effects of the moving load characteristics, geometrical and material parameters such as the moving load speed, the radius of curvature and the modulus of elasticity in principal direction on the dynamic responses, fundamental frequency and critical speed of the system are investigated. The results show that the minimum and maximum deflection of the beam are occurred for lay-up [90/0/90/0] and [45/-45/-45/45]respectively. Furethermore, the increasing of the speed movingload leads to the decreasing thedynamic deflection. It is also shown that the increasing of the radius of the curved beams leads to the decreasingof the frequency and critical speed moving load.

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In this paper, employing a thermomechanical constitutive model for shape memory polymers (SMP), a beam element made of SMPs is presented based on the kinematic assumptions of Timoshenko beam theory. Considering the low stiffness of SMPs, the necessity for developing a Timoshenko beam element becomes more prominent. This is due to the fact that relatively thicker beams are required in the design procedure of smart structures. Furthermore, in the design and optimization process of these structures which involves a large number of simulations, we cannot rely only on the time consuming 3D finite element (FE) analyses. In order to properly validate the developed formulations, the numeric results of the present work are compared with those of 3D finite element results of the same authors, previously available in the literature. The parametric study on the material parameters e.g., hard segment volume fraction, viscosity coefficient of different phases, and the external force applied on the structure (during the recovery stage) are conducted on the thermomechanical response of a short I-shape SMP beam. For instance, the maximum beam deflection error in one of the studied examples for the Euler-Bernoulli beam theory is 7.3%, while for the Timoshenko beam theory, is 1.5% with respect to the 3D FE solution. It is noted that for thicker or shorter beams, the error of the Euler-Bernoulli beam theory even more increases. The proposed beam element in this work, could be a fast and reliable tool for modeling 3D computationally expensive simulations.

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In this article, the dynamic equations of multiple flexible links robotic manipulators fabricated of functionally graded materials (FGM), whose properties vary continuously along the axial direction and also along the thickness, are examined. Gibbs-Appell methodology and Timoshenko Beam Theory according to the Assumed Mode Method are utilized to obtain the equations of motion and to model the flexible characteristics of links, respectively. Subsequently, the influence of power law index on the vibration response of a two-link functionally graded robotic manipulator is studied for two cases in which the mechanical properties of links vary once along the axial direction and again along the thickness direction of each link. By introducing a parameter called signal energy, it is shown that the power law index has a substantial effect on the vibrational behaviors of the mentioned system; and that by choosing a proper power law index, system vibrations can be reduced considerably in a passive way.

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In this paper, the non-linear dynamic behavior of immersed AFM micro cantilever in liquid has been modeled. To increase the accuracy of the theoretical model, all necessary details for cantilever and sample surface have been taken into account. As for the theoretical model, the Timoshenko beam theory which takes the rotatory inertia and shear deformation effects into consideration has been adopted. For modeling the vibrational system, cantilever thickness, cantilever length and breadth, the angle between cantilever and sample surface, normal contact stiffness, lateral contact stiffness, tip height, breadth taper ratio, height taper ratio, time parameter and viscosity of the liquids have been considered. Differential quadrature method (DQM) has been used for solving the differential equations. During the investigation, the softening behavior was observed for all cases. Here, water, methanol, acetone and carbon tetrachloride has been supposed as immersion environments. Results show that increasing the liquid density reduces the resonant frequency. Time variable does not have any considerable effect on the non-linear resonant frequency. Theoretical modeling has been compared for a rectangular AFM cantilever with experimental works in both of the contact and non-contact modes in air and water environments. Results show good agreement.