Sharif Digital Repository

This technical note addresses the free vibration problem of an elastically restrained Euler-Bernoulli beam with rotational spring-lumped rotary inertia system at its mid-span hinge. The governing differential equations and the boundary conditions of the beam are presented. Special attention is directed toward the conditions of the intermediate spring-mass system which plays a key role in the solution. Sample frequency parameters of the beam system are solved and tabulated. Mode shapes of the beam are also plotted for some spring stiffnesses  

In this paper, the static pull-in behavior of electrostatically actuated functionally graded (FG) micro-beams resting on an elastic medium is studied using the modified strain gradient (MSG) theory. To this end, the equilibrium equation along with classical and non-classical boundary conditions is obtained by considering the fringing field and elastic foundations effects within the principle of minimum total potential energy. Also, the elastic medium is composed of a shear layer (Pasternak foundation) and a linear normal layer (Winkler foundation). The governing differential equation is solved for cantilever and doubly fixed FG beams using an iterative numerical method. This method is a... 

In this paper using classical beam theory, the dynamical governing differential equations of a vibrating shaft are derived then by using lumped parameter technique and method of transfer matrix (TM) the induced eigen value problem is solved. In calculating natural frequencies of a vibrating shaft under different boundary conditions, primarily the shaft was divided into number of segments. In each segment different number of lumped properties like mass, damping and flexibility on overall massless elastic or rigid shaft were applied. One of the aims of this study was to find out the optimum value for number of segments under different aforementioned conditions. In order to estimate the natural... 

Micro actuators are an irreplaceable part of motion control in miniaturized systems and are intended to have a high range of deformation, high accuracy, large force, and quick response. In this article, an analytical model for a hybrid thermopiezoelectric micro actuator is developed in which a double lead-zirconnate-titanate piezoceramic (PZT) beam structure consisting of two arms with different lengths are used. Governing differential equation of motion and electrical field are derived and solved. Out of parametric studies it was observed that, under application of temperature and voltage gradients, the deflection of the actuator shows different trends depending on the geometry of the micro... 

Based on the first-order shear deformation plate theory, two approaches within the extended Kantorovich method (EKM) are presented for a bending analysis of functionally graded annular sector plates with arbitrary boundary conditions subjected to both uniform and non-uniform loadings. In the first approach, EKM is applied to the functional of the problem, while in the second one EKM is applied to the weighted integral form of the governing differential equations of the problem as presented by Kerr. In both approaches, the system of ordinary differential equations with variable coefficients in r direction and the set of ordinary differential equations with constant coefficients in θ direction... 

In this study, a multiplate shear model is developed for dynamic analysis of multilayer graphene sheets with arbitrary shapes considering the interlayer shear effect. By utilizing the model, then some free-vibration analysis is presented. According to the experimental results, the weak interlayer van der Waals interaction cannot maintain the integrity of carbon atoms in adjacent layers. Therefore, it is required that the interlayer shear effect is accounted to study multilayer graphene mechanical behavior. The governing differential equation of motion is derived for the multilayer graphene sheets utilizing a variational approach based on the Kirchhoff plate model. The essential and natural... 

In this article, the dynamic response of a non-uniform Timoshenko beam acted upon by a moving mass is extensively investigated. To this end, the eigenfunction expansion method is adapted to the problem, employing the natural mode shapes of a uniform Timoshenko beam. Moreover, the orthonormal polynomial series expansion method is successfully applied to the coupled set of governing differential equations pertaining to the dynamic behavior of non-uniform Timoshenko beam actuated by a moving mass. Some numerical examples are solved in which the excellent agreement of the two presented methods is illustrated  

This paper deals with the coupling flexural-torsional vibration analysis of a grid formed by two members. A mass-spring system is attached to the grid at the intersecting joint. The members of the grid are assumed to resist torsion as well as bending and shear. Moreover, the mechanical and geometrical properties of each member are different. In order to analyze the problem, a closed-form solution is obtained. In doing so, the governing differential equations of the system along with the pertinent boundary and compatibility conditions of the system are introduced. Then, the frequency parameters of the mechanical system under study are derived and given for the first five modes of vibration.... 

The lateral response of a single degree of freedom structural system containing a rigid circular cylindrical liquid tank under harmonic and earthquake excitations at a 1:2 autoparametric resonance case is considered. The governing nonlinear differential equations of motion for the combined system are solved by means of a multiple scales method considering the first three liquid sloshing modes (1,1), (0,1), and (2,1), under horizontal excitation. The fixed points of the gyroscopic type of governing differential equations are determined and their stability is investigated employing the perturbation method. The obtained results reveal an increase in the stability region for a single-mode... 

The dynamic response of a delaminated composite beam under the motion of an oscillatory mass moving with a constant velocity has been studied. The delaminated composite beam is modeled as four interconnected sub-beams using the delamination limits as their boundaries. The constrained model is used to model the delamination region. The continuity and equilibrium conditions are forced to be satisfied between the adjoining beams. A set of derived governing differential equations along with those obtained by imposing boundary conditions are simultaneously solved in a closed form manner. The results for the response of the delaminated beam were compared with those of the intact beam. Furthermore,... 

An analytical/numerical model based on the assumption of rigid incompressible water column and compressible air bubble, is derived to simulate the pressure fluctuations, void fraction, air/water flow rate, water velocity in a closed conduit and water depth at upper reservoir due to formation of unstable slug flow. It is a comprehensive model which can generate different hydraulic situations of instability in a closed conduit based on hydraulic approach. The boundary conditions are the system of algebraic or/and simple differential equations. The steady solution of the governing differential equations is generally performed as the initial data. The frequency of pressure fluctuation and... 

The aim of this study is to carry out the numerical computation of nonlinear normal modes for rotating pretwisted composite thin-walled beam in axial-torsional vibrations. The structural model considered here, incorporates a number of non-classical effects such as primary and secondary warping, non-uniform torsional model, rotary inertia and pretwist angle. Ignoring the axial inertia term leads to differential equation of motion in terms of angle of twist in the case of axially immovable beam ends. The governing differential equations of motion are derived using Hamilton's principle and the reduced model around the static equilibrium position is obtained using 2-mode Galerkin discretization... 

The geometrically nonlinear governing differential equation of motion and corresponding boundary conditions of small-scale Euler-Bernoulli beams are achieved using the second strain gradient theory. This theory is a non-classical continuum theory capable of capturing the size effects. The appearance of many higher-order material constants in the formulation can certify that it appropriately assesses the behavior of extremely small-scale structures. A hinged-hinged beam is chosen as an example to lay out the nonlinear size-dependent static bending and free vibration behaviors of the derived formulation. The results of the new model are compared with the previously obtained results based on... 

Three-dimensional thermo-mechanical stress analysis of composite joints with bi-adhesive bonding is presented using the full layerwise theory. Based on three-dimensional elasticity theory, sets of fully coupled governing differential equations are derived using the principle of minimum total potential energy and are simultaneously solved using the state space approach. Results show that bi-adhesive bonding substantially relieves the edge effects. Moreover, series of parametric studies reveal the nonlinear effects of bonding length ratio and the relative stiffness and coefficient of thermal expansion of the mid- and side-adhesives. It is also concluded that the optimum design of a bi-adhesive... 

The present work aims to study the anti-plane scattering of SH-waves by an elastic micro-/nano-fiber which is embedded near the interface between exponentially graded and homogeneous half-spaces incorporating interface effects. The fiber is perfectly bonded to the inhomogeneous medium. It is well-known that traditional elasticity theory is incapable of accounting accurately for the nanoscopic-interfaces and, likewise, inappropriate for the prediction of the behavior of nano-sized structures where the surface-to-volume ratio is remarkably large. In the present study, the interface effects are incorporated using the well-known (Gurtin and Murdoch, 1975) surface elasticity theory which permits... 

In this paper, an analytical method is presented in order to determine the static bending response of an axisymmetric thin circular/annular plate with different boundary conditions resting on a spatially inhomogeneous Winkler foundation. To this end, infinite power series expansion of the deflection function is exploited to transform the governing differential equation into a new solvable system of recurrence relations. Singular points of the governing equation are effectively treated by applying the Frobenius theorem in the solution, which in turn permits the use of more-general analytical functions to describe the variation of the foundation modulus along the radius of the plate. Moreover,... 

This paper analytically studies the effect of functionally graded materials (FGMs) on the primary resonances of large amplitude flexural vibration of in-extensional rotating shafts with nonlinear curvature as well as nonlinear inertia. The constituent material is assumed to vary along the radial direction according to a power-law gradation. The governing differential equations and the corresponding boundary conditions are derived employing the variational approach. Then, the Galerkin method and the multiple scales perturbation method are utilized to obtain the frequency–response equation. In a numerical case study, the effects of the power-law index on the steady-state responses and locus of... 

In this paper, a new analytical approach is proposed for free vibration and buckling analysis of a rectangular Mindlin plate resting on the Winkler–Pasternak foundation of varying stiffness. According to Mindlin theory, there are three independent governing differential equations. Thus, three Fourier series expansions along with auxiliary polynomial functions are employed to represent the plate's deflection and rotation angle functions. The process of making a set of equations is then completed satisfying the corresponding equilibrium equations and boundary conditions. The proposed method incorporates general elastic supports for all plate's edges, and subsequently can deal with all possible... 

In this study the free vibration of laminated composite plates with and without stiffeners subjected to axial loads is carried out using finite element method. The plates are stiffened by laminated composite strip and Timoshenko beam. The plates and the strips are modeled with rectangular 9 noded isoparametric quadratic elements with three degrees of freedom per node and the Timoshenko beam is modeled with linear 2 noded isoparametric quadratic elements with 2 degrees of freedom per node. The effects of both shear deformation and rotary inertia are implemented in the modeling of plate and stiffener. The governing differential equations are obtained in terms of the mid-plane displacement... 

In this paper, the dynamic behavior of Horizontally Curved Beams (HCBs) resting on an elastic foundation and subjected to a moving mass is investigated. The governing coupled non-linear differential equations of equilibrium are derived, where Coriolis acceleration, centrifugal force and rotary inertia are incorporated in the problem formulation. In the proposed analytical solution, by employing the transition matrix technique, the governing differential equations of motion are subsequently transformed into a new system of linear ordinary differential equations which can be solved using standard numerical procedures. The accuracy as well as the robustness of the solution is ascertained...