Sharif Digital Repository

In this paper, a continuous model for vibration analysis of a beam with an open edge crack including the effects of shear deformation and rotary inertia is presented. A displacement field is suggested for the beam and the strain, and stress fields are calculated. The governing equation of motion for the beam has been obtained using Hamilton's principle. The equation of motion is solved with a modified Galerkin method and the natural frequencies and mode shapes are obtained. A good agreement has been observed between the results of this research and the results of previous work done in this fiels. The results are also compared to results of a similar model with Euler-Bernoulli assumptions to... 

In this paper, flutter of functionally graded material (FGM) cylindrical shells under distributed axial follower forces is addressed. The first-order shear deformation theory is used to model the shell, and the material properties are assumed to be graded in the thickness direction according to a power law distribution using the properties of two base material phases. The solution is obtained by using the extended Galerkin's method, which accounts for the natural boundary conditions that are not satisfied by the assumed displacement functions. The effect of changing the concentrated (Beck's) follower force into the uniform (Leipholz's) and linear (Hauger's) distributed follower loads on the... 

In this paper the non-planar nonlinear dynamic responses of an axially loaded rotating Timoshenko beam subjected to a three-directional force traveling with a constant velocity is studied. On deriving the nonlinear coupled partial differential equations (PDEs) of motion the stretching effect of the beam's neutral axis due to the pinned-pinned ends' condition in conjunction with the von Karman strain-displacement relation are considered. The beam's nonlinear governing coupled PDEs of motion for the bending rotations of warped cross-section, longitudinal and lateral displacements are derived using Hamilton's principle. To obtain the dynamic responses of the beam, derived PDEs of motion are... 

This paper aims to investigate aeroelastic stability boundary of subsonic wings under the effect of thrust of two engines. The wing structure is modeled as a tapered composite box-beam. Moreover, an indicial function based model is used to calculate the unsteady lift and moment distribution along the wing span in subsonic compressible flow. The two jet engines mounted on the wing are modeled as concentrated masses and the effect of thrust of each engine is applied as a follower force. Using Hamilton's principle along with Galerkin's method, the governing equations of motion are derived, then the obtained equations are solved in frequency domain using the K-method and the aeroelastic... 

In this paper, the nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions subjected to a traveling mass with variable velocity is investigated. The nonlinear coupled partial differential equations of motion for the bending rotation of cross-section, longitudinal and transverse displacements are derived using Hamilton's principle. These nonlinear coupled PDEs are solved by applying Galerkin's method to obtain dynamic response of the beam under the act of a moving mass. The appropriate parametric studies by taking into account the effects of the magnitude of the traveling mass, the velocity of the traveling mass with a constant acceleration/ deceleration... 

Conducting interfaces and nano conducting layers can support surface electromagnetic waves. Uniform charge layers of non-zero thickness and their asymptotic behavior toward conducting interfaces of infinitely small thicknesses, where the thin charge layer is modeled via a surface conductivity σ s , are already studied. Here, the possible effects of inhomogeneity in the conductivity profile of the thin conducting layers are investigated for the first time and a new approximate yet accurate enough analytical formulation for mode extraction in such structures is given. In order to rigorously analyze the structure and justify the proposed approximate formulation, the Galerkin's method with... 

In this study, the aeroelastic stability and response of an aircraft swept composite wing in subsonic compressible flow are investigated. The composite wing was modeled as an anisotropic thin-walled composite beam with the circumferentially asymmetric stiffness structural configuration to establish proper coupling between bending and torsion. Also, the structural model consists of a number of nonclassical effects, such as transverse shear, material anisotropy, warping inhibition, nonuniform torsional model, and rotary inertia. The finite state form of the unsteady aerodynamic loads have been modeled based on the indicial aerodynamic theory and strip theory in the subsonic compressible flow.... 

Nonlinear torsional vibrations of thin-walled beams exhibiting primary and secondary warpings are investigated. The coupled nonlinear torsional-axial equations of motion are considered. Ignoring the axial inertia term leads to a differential equation of motion in terms of angle of twist. Two sets of torsional boundary conditions, that is, clamped-clamped and clamped-free boundary conditions are considered. The governing partial differential equation of motion is discretized and transformed into a set of ordinary differential equations of motion using Galerkin's method. Then, the method of multiple scales is used to solve the time domain equations and derive the equations governing the... 

The small-scale effect on the natural frequencies and buckling of pressurized nanotubes is investigated in this study. Based on the firstorder shear deformable shell theory, the nonlocal theory of elasticity is used to account for the small-scale effect and the governing equations of motion are obtained. Applying modal analysis technique and based on Galerkin's method a procedure is proposed to obtain natural frequencies of vibrations. For the case of nanotubes with simply supported boundary conditions, explicit expressions are obtained which establish the dependency of the natural frequencies and buckling loads of the nanotube on the small-scale parameter and natural frequencies obtained by... 

This paper undertakes to facilitate appraisal of aeroelastic interaction of a locally distributed, flap-type control surface with aircraft wings operating in a subsonic potential flow field. The extended Hamilton's principle serves as a framework to ascertain the Euler-Lagrange equations for coupled bending-torsional-flap vibration. An analytical solution to this boundary-value problem is then accomplished by assumed modes and the extended Galerkin's method. The developed aeroelastic model considers both the inherent flexibility of the control surface displaced on the wing and the inertial coupling between these two flexible bodies. The structural deformations also obey the Euler-Bernoulli... 

Plates on elastic foundations have attracted the attention of many researchers. Some elementary models have been introduced to consider interactions between the plate and its foundation. Other improved models have been proposed to develop basic models. In this work, a model based on the Winkler-foundation theory is proposed, while the constant parameter of Winkler is assumed to be variable; such as non-uniform springs with the functionality of the domain position, along with the plate and beam span in order to consider the non-uniform behavior of the foundation. The governing equation on the system is solved by using the Galerkin method and effects such as the presence of rigid points in the... 

A nonlinear fluid–structure interaction (FSI) model is presented for nonlinear vibration analysis of sandwich cylindrical shells subjected to an external compressible flow by considering the curvature nonlinearity in impermeability condition and bending. The sandwich shells are made of two face sheets and a central core of advanced materials including functionally graded (FG), metal foam, and anisogrid lattice composite. Based on the Kirchhoff–love hypotheses with the geometric nonlinearities in the normal strain and curvature of mid-surface, one decoupled nonlinear integral–differential equation is obtained for axisymmetric bending vibration of sandwich cylindrical shells. For the first... 

This paper aims the nonlinear aeroelastic analysis of slender wings using a nonlinear structural model coupled with the linear unsteady aerodynamic model. High aspect ratio and flexibility are the specific characteristic of this type of wings. Wing flexibility, coupled with long wingspan can lead to large deflections during normal flight operation of an aircraft; therefore, a wing in vertical/forward-afterward/torsional motion using a third-order form of nonlinear general flexible Euler-Bernoulli beam equations is used for structural modeling. Unsteady linear aerodynamic strip theory based on the Wagner function is used for determination of aerodynamic loading on the wing. Combining these... 

It has been recently shown that guided modes in two-dimensional photonic crystal based structures can be fast and efficiently extracted by using the Galerkin's method with Hermite-Gauss basis functions. Although quite efficient and reliable for photonic crystal line defect waveguides, difficulties are likely to arise for more complicated geometries, e.g., for coupled resonator optical waveguides. First, unwanted numerical instability may well occur if a large number of basis functions are retained in the calculation. Second, the method could have a slow convergence rate with respect to the truncation order of the electromagnetic field expansion. Third, spurious solutions are not unlikely to... 

In this study, the nonlinear vibrations analysis of an inclined pinned-pinned self-weight Timoshenko beam made of linear, homogenous and isotropic material with a constant cross section and finite length subjected to a traveling mass/force with constant velocity is investigated. The nonlinear coupled partial differential equations of motion for the rotation of warped cross section, longitudinal and transverse displacements are derived using the Hamilton's principle. These nonlinear coupled PDEs are solved by applying the Galerkin's method to obtain dynamic responses of the beam. The dynamic magnification factor and normalized time histories of mid-point of the beam are obtained for various... 

The problem of nonlinear aeroelasticity of a general laminated composite plate in supersonic air flow is examined. The classical plate theory along with the von-Karman nonlinear strains is used for structural modeling, and linear piston theory is used for aerodynamic modeling. The coupled partial differential equations of motion are derived by use of Hamilton's principle and Galerkin's method is used to reduce the governing equations to a system of nonlinear ordinary differential equations in time, which are then solved by a direct numerical integration method. Effects of in-plane force, static pressure differential, fiber orientation and aerodynamic damping on the nonlinear aeroelastic... 

This paper presents a coupled flap-lag-torsion aeroelastic stability analysis and response of a hingeless helicopter blade in the hovering flight condition. The boundary element method based on the wake eigenvalues is used for the prediction of unsteady airloads of the rotor blade. The aeroelastic equations of motion of the rotor blade are derived by Galerkin's method. To obtain the aeroelastic stability and response, the governing nonlinear equations of motion are linearized about the nonlinear steady equilibrium positions using small perturbation theory. The equilibrium deflections are calculated through the iterative Newton-Raphson method. Numerical results comprising steady equilibrium... 

This paper presents a method for nonlinear aeroelastic analysis of Human Powered Aircraft (HPA) wings. In this type of aircraft there is a long, highly flexible wing. Wing flexibility, coupled with long wing span can lead to large deflections during normal flight operation; therefore, a wing in vertical and torsional motion using the second-order form of nonlinear general flexible Euler-Bernoulli beam equations is used for structural modeling. Unsteady linear aerodynamic theory based on Wagner function is used for determination of aerodynamic loading on the wing. Combining these two types of formulations yields the nonlinear integro-differentials aeroelastic equations. Using the Galerkin's...