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In this paper, the homotopy analysis method is used to nonlinear free vibration analysis of a mechanical and thermal loaded functionally graded beam on nonlinear elastic foundation. At first, the governing partial differential equation of the problem has been derived based on the Euler-Bernoulli theory and the Von-Karman strain-displacement relationship. Then, it was reduced to a nonlinear ordinary differential equation via the Galerkin method. The homotopy analysis method which has high accuracy was implemented in order to obtain a closed form solution and study the problem parametrically. The accuracy of the proposed method is verified by those available in literatures. The numerical results demonstrate that proposed method yields a very rapid convergence of the solution as well as low computational effort. Finally, the effects of different parameters such as amplitude, linear and nonlinear elastic foundation, thermal and mechanical loads and boundary conditions were investigated on the beam vibration and their results are presented for future work.

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The current article presents an analytical approach, for determining the natural frequencies of a rotating cracked Euler–Bernoulli beam with a varying transverse cross-section, using the so-called differential transform method (DTM). First, the natural frequencies of the beam are obtained for different values of the crack position and depth. The results have been validated 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. Then, the inverse problem is investigated. For this reason, the position and depth of the crack of the rotating 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 rotating beams.

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In this paper, thermal effect on deflection, critical buckling load and vibration of nonlocal Euler-Bernoulli beam on Pasternak foundation using Ritz method is proposed. Equations of motion Euler-Bernoulli beam on Pasternak elastic foundation under thermal load is achieved by using energy method. Ritz method is used to solve the governing equations of motion. By this method, mass, stiffness and hardness buckling matrices are obtained. In this study, the effects of thermal, various boundary conditions, Winkler-type spring constant, Pasternak-type shear constant, non-local parameter on dimensionless deflection, critical buckling load, and natural frequency of Euler-Bernoulli beam theory are assessed. The obtained results indicate that with an increase of Winkler and Pasternak constants, the dimensionless natural frequency and critical buckling load increase, while the dimensionless deflection decreases. However, with increasing the temperature change in nonlocal Euler-Bernoulli beam on Winkler–Pasternak elastic foundation, the dimensionless natural frequency and critical buckling load decrease, while the dimensionless deflection increases. Moreover, with considering Winkler and Pasternak constants, the lower mode shape are removed and replaced with higher mode shapes.

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A vibration absorber is used to reduce vibration of an Euler-Bernoulli beam with elastic supports subjected to a moving oscillator. Dynamic response of the beam under moving harmonic exciter with different moving velocities and different parameters are obtained. The critical velocity of the moving oscillator is determined and absorber parameters are optimized by numerical algorithm and effect of mass and damping on the absorber performance is investigated. When absorber is applied to the beam, effect of crack occurrence on its performance is investigated. Crack is assumed to be open and is modeled by sectional flexibility increase. Two different cases considered for crack severity. In each case, optimal absorber for intact beam is applied and dynamic response of the midpoint of the beam with different velocities of moving exciter is obtained. Results show although crack can increase dynamic deflection of the beam with absorber, dynamic deflection of cracked beam without considering absorber is higher than dynamic deflection of cracked beam with absorber used. It is found that the vibration absorber designed for intact beam keeps its performance in dynamic deflection reduction after crack occurrence and changing in structural dynamic.

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Fractional calculus is a branch of mathematics which in recent decades has been of great interest to scientists in various disciplines, including engineering. One of the applications of this branch in engineering, is in modeling the viscoelastic materials using fractional differentiation. In this article, by inserting fractional calculus as a viscoelastic material compatibility equations in nonlocal beam theory, a viscoelastic Euler-Bernoulli nano-beam with different boundary conditions at two ends, has been modeled. Material properties of a carbon nanotube is considered and two methods, pure numerical and numerical-analytical have been used for solving obtained equations in time domain. Main method is completely numerical and operator matrices used in it to discrete equations in time and spatial domain. Second method is introduced for validation of pervious method’s answers. In this method equation of system reduced to an ordinary differential equation using Galerkin and obtained equation solved using a numerical direct integrator method. Finally, in a case study, the effects of fractional order, viscoelasticity coefficient and nanlocal theory coefficient on the time response of the viscoelastic Euler-Bernoulli nano-beam with different boundary conditions have been studied.

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In this paper, dynamics and control of a Tendon-based continuum robot is investigated. The curvature is assumed that constant in each section of continuum robot. Kinematic equation is established on the basis of the Euler-Bernoulli beam. The dynamic model of the continuum robot is derived by using Lagrange method. In this paper, robot control is performed in two parts: firstly, Dynamic model is assumed to be known and position and velocity tracking control has been by using the feedback linearization method, But uncertainties in the dynamic model, are constantly challenged the control of continuum robots. For unknown parametric quantities such as mass coefficients, one way is simply substitutes a fixed estimate for the unknown parametric quantities. In this case tracking error is not equal to zero but it’s bounded. For many applications, we cannot assume that tracking error vector is not equal to zero. In such cases we use adaptive controller. In this paper the total mass of the primary backbone and secondary backbone are uncertain parameters, therefore, a new adaptive controller is presented to estimate those uncertainties while cause to asymptotically stable for tracking error. Simulation results show good performance in velocity and position tracking.

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In this paper, free vibration of rotating microbeams based on the strain gradient theory and Euler-Bernoulli beam assumptions is investigated. The Hamilton's Principle is applied on the attained strain and kinetic energy relations to obtain the equations of motion for the rotating microbeam. Then, by employment of the adimensional parameters, the nondimensional form of the equations of motion is derived. By applying the Galerkin approach on the dynamic equations of motion, the flapping and axial natural frequencies are calculated. Subsequently, the current results are validated by the existed papers results. After validation of the present results, the effects of the thickness to the material length scale parameter ratio, rotation speed and Poisson's coefficient on the flapping and axial frequencies are studied and the strain gradient theory results are compared with the modified couple stress and classical theories. The results show that the type of the theory which is appointed has essential effects on the predicted natural frequencies. The effect of rotation speed on the possibility of the occurrence of internal resonances is also examined. In addition, for the first time, the effect of different mentioned theories on the axial natural frequencies are inspected. The presented results illustrated, by considering the strain gradient theory, varying the Poisson's coefficient changes the axial frequencies, while, the modified couple stress and classical theories are incompetent to predict any variations on the axial frequencies and the mentioned theories predict the same results for axial frequencies.

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In this paper, the dynamic behavior of atomic force microscope (AFM) based on non-classical strain gradient theory was analyzed. For this aim atomic force microscope micro-beam with attached tip has been modeled as a lumped mass. Micro-beam has stimulated via a piezoelectric element attached to the end of clamped and non-linear partial differential equation of the system has extracted based on Euler-Bernoulli theory and to be converted into ordinary differential equation by using Galerkin and separation method. The classic continuum theory because of lack of consideration size effect that has been observed in many experimental studies, has little accuracy in predicting the mechanical behavior of Nano devices. In this study, the stability region of micro-beam are determined analytically and validated by comparison with numerical results. Difference between presented analysis in dynamic behavior of micro-beam by classic and non-classic theories has been shown with variety of diagrams. It is clear that consideration the size effect changes the dynamical behavior of the problem completely and it is possible while classical theory predicts stable behavior for microscope the size effect is caused bi-stability. The results in this paper are very useful for the design and analysis of atomic force microscope.

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In structural dynamics, loads having varying positions has been broadly studied. Such loads are so called moving loads which appears in various applications in industry. High speed machining systems, overhead cranes, cable ways, pavements, computer disc memories and robot arms are a few examples of moving load dynamic problems. Vibration of bridge structures subject to moving vehicles is often referred to as an application of moving load problems. A great number of researchers proposed numerical and analytical methods to deal with the vibration of solids and structures under travelling loads. A famous classic approach in the simulation of moving loads is the moving force. In moving force model, a constant traveling force is assumed to act upon the base structure. However, this assumption yields to reasonable structural analysis if the mass of the moving object is negligible. Nowadays, with ongoing advances of transportation technology, the mass, speed and acceleration of moving vehicles are notably increased. In this regard, during the last few decades, many researchers showed that the moving force is no longer valid for large moving masses. Therefore, the moving mass simulation has been proved to be closer to the physical model of vehicle bridge interaction. As a common practice, bridges carrying moving vehicles has been assumed as vibrating beams excited by point moving masses. It has been very customary to consider the midspan or center point of the base beam as the reference point in order to assess the maximum dynamic response of the structure under moving mass; therefore, most of the existing computed design envelopes are related to the values occurring at the midpoint of the structure. However, the location of the maximum values occurrence is not necessarily at midspan. To shed light on this issue, in this research an analytical-numerical method is established to capture dynamic response of an Euler-Bernoulli beam traversed by a moving mass. Most of the available literature on moving load problem is concerned with the travelling loads having constant speeds. To remove this restrictive presumption, in this paper, the considered moving mass is assumed to move at non-zero constant acceleration. The beam is considered to be undamped and initially at rest. The moving mass is assumed to maintain full contact condition with the base beam while sliding on it. By exploiting a series of continuous shape functions having time varying amplitude factors, a norm space is provided by which the beam spatial domain is discretized. The problem is then transformed into time domain for which a time integration method is utilized. Absolute maximum dynamic response of the supporting beam under the passage of accelerated moving mass is extensively sought over the beam length. In this manner, whole beam length is being monitored for the maximum values at each time step of time integration procedure. The beam absolute maximum dynamic response is comprehensively computed considering different mass ratios and extensive range of linearly time varying velocities. Parametric studies are carried out on the absolute maximum values of dynamic flexural moments and deflections and compared to those captured at midspan. Finally, it highlighted that the midspan of the beam cannot be a valid reference to obtain the true maximum deflections and flexural moments of the base beam.

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In this study, nonlinear behavior of an atomic force microscopes (AFM) immersed in acetone, water, carbon tetrachloride (CCl4), and 1-butanol is investigated using non-classical strain gradient theory. In this theory, the size effect of system is taking into account by means of material length scale parameter. The nonlinear behavior of the AFM is due to the nonlinearity of the AFM tip–sample interaction caused by the Van der Waals attraction/repulsion force. Behavior of micro beam immersed in liquid is completely different with its behavior in air and vacuum due to the existence of hydrodynamic force. The Resonant frequencies, mode shapes, governing nonlinear partial and ordinary differential equations (PDE and ODE) of motion, stability analysis, boundary conditions, potential function and phase-plane of the system are obtained analytically in the present study. Furthermore, the results are compared with the one obtained by the modified couple stress theory. For this purpose, the AFM and the probe at the free end of micro beam are modeled as a lumped mass. The fixed end of micro beam is excited by piezoelectric element. The nonlinear PDE of motion is derived based on Euler-Bernoulli theory by employing the Hamilton principle. The Galerkin method is utilized to gain the governing nonlinear ODE of motion and the obtained ODE is analytically solved by means of perturbation techniques.

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In this study a novel solution method for dynamic analysis of clamped-free shape memory alloy beams is presented. It is assumed that the beam is entirely made of shape memory alloy. Based on Euler-Bernoulli beam theory the governing equations of motion and corresponding boundary conditions are derived by using extended Hamilton principle. In the derived PDEs the transformation strain is behaved as external force that changes with time and position. The Galrkin approach is employed to convert PDEs to ODE system equations of motion. The derived equations of motion are solved by using Newmark integration method. The shape memory alloy constitutive model that presented by Souza is applied for specifying the phase of material all over beam. The transformation strain as internal variable that is coupled with states of equations of motion is identified in every time and every position of beam by using return map algorithm. A parametric study on the control variables has been adopted and the results of parametric study are discussed. The results show that the hysteresis damping is increased by increasing the operating temperature. Moreover the damping of system is faster by increasing the initial displacement in free vibration.

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In this paper, a sandwich beam of a SMP material which have a corrugated core is studied. The corrugated core is from a polymeric material. Structures with corrugated profiles show higher stiffness-to-mass ratio in the transverse to corrugation direction compared to flat structures. As a result, the beam with corrugation along the transverse direction is stiffer than the one with corrugation along the beam length. The flexural behavior of the composite corrugated beam is studied employing a developed constitutive model for SMP and the Euler-Bernoulli beam theory. The constitutive model utilized is in integral form and is discretized employing finite difference scheme. To verify the results of the Euler-Bernoulli beam theory and finite difference method, finite element models of different corrugated sections have been simulated in a 3D finite element program. The results demonstrate that the developed model for the composite beam presented in this study predicts the behavior of the beam successfully. The sandwich beam with different corrugated cores (triangular, sinusoidal and trapezoidal shapes) are compared with each other. Also, results show that the shape fixity is decreased a little, like any other reinforcing method. This decrease in shape fixity results in increase of load capacity in composite beams. The stress-free strain recovery and constrained stress-recovery cycles are both studied.

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In this study, application of a dynamic vibration absorber system consists of two symmetric cantilever beams with tip mass and piezoelectric layer, in order to suppress undesired vibrations and harvest electrical energy, is studied. The main vibratory system is a simply supported beam, which is excited by a DC motor with rotating unbalance mass. To derive the governing electromechanical equations, the Euler-Bernoulli beam theory and the energy method are used. Then the governing electromechanical equations are experimentally validated and accommodation between theoretical and experimental results is shown using several frequency response plots. Using the non-dimensional governing equations, effect of changing the system parameters such as the tip mass, load resistance and length of the cantilever beam is studied. Then, considering ability of system to effectively suppress undesired vibrations and increase the harvested electrical energy, the proper range for selecting the non-dimensional tip mass and non-dimensional load resistance is presented. Finally, using the so-called perfection rate parameter, the best parameters, to have a good vibration suppressor and energy harvester, are obtained. Results shown that both of energy and vibration considerations can be satisfied using the system.

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In this paper based on the Euler-Bernoulli beam model, the primary resonance a curved single carbon nanotube subjected to axial thermal force in the case of low temperature and high temperature and resting on a viscoelastic foundation is analytically investigated. The nonlinear partial differential governing equation is reduced to nonlinear ordinary differential governing equation by using of a single-mode Galerkin approximation along with the sinusoidal curvature for clamped-clamped single walled carbon nanotube under harmonic external force. The method of multiple scales is applied to determine the analytical primary resonance frequency response. Considering the curved geometry and the mid-plane stretching, a quadratic and cubic terms are presented in the governing equation. The effects of temperature change in high temperature and low temperature conditions, viscoelastic coefficients of medium, amplitude of sinusoidal curvature and excitation amplitude are investigated to study the property frequency response and development or elimination of forward and backward jumping phenomenon in primary resonance frequency response. The results show that these parameters have a significant effect on the frequency response of a curved single walled carbon nanaotubes under transvers harmonic force.

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In this paper, a new method for determining the Young's modulus of structural elements, using the finite element model updating approach, is presented. The model updating is the correction of the numerical model of a structure based on measured data from the real structure. Therefore, after introducing a case study of an aluminum alloy (7075-T651) beam, the frequency of bending vibrations of the case study was measured, using frequency response functions derived from the modal test. Then, Young's modulus for the case study was calculated, using the relationships in the ASTM E 1876-01standard and also the analytical relations governing Euler–Bernoulli beam behavior. The results of the model updating method indicate that there is a very good adaptation with the results of the two recent approaches, the Standard and Euler–Bernoulli beam relations. As a result, this method can be developed with good precision to identify the Young’s modulus in structural elements with more complex shapes, where the relations derived from the aforementioned standard and analytical relations are not efficient due to geometric constraints.