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  

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  

The vibration of FG plate embedded with PZT4 rectangular patches on the top and/or the bottom surface(s) as actuators/sensors is investigated. Based on the classical laminated plate theory, the governing differential equations of motion are derived under a variable electric charge. The equation of motion for PZT4 patch is obtained and solved. The effect of feedback gain and FGM volume fraction exponent on the plate frequency and its deflection are studied. It is noticed that increasing the feedback gain leads to the reduction of frequency and displacement. Moreover, by increasing the value of the FGM volume fraction exponent the resonant frequency decreases  

The vibration of FG plate embedded with PZT5 rectangular patches on the top and/or the bottom surface(s) as actuators/sensors is investigated. Based on the classical laminated plate theory, the governing differential equations of motion are derived under a variable electric charge. The equation of motion for PZT5 patch is obtained and solved. The effect of feedback gain and FGM volume fraction exponent on the plate frequency and its deflection are studied. It is noticed that increasing the feedback gain leads to the reduction of frequency and displacement. Moreover, by increasing the value of the FGM volume fraction exponent the resonant frequency decreases  

A composite beam with single delamination traveled by a constant amplitude moving force is modeled accounting for the Poisson's effect, shear deformation and rotary inertia. The mechanical behavior between the delaminated surfaces is modeled using a piecewise-linear spring foundation. The governing differential equations of motion for such system are derived. Primarily, eigen-solution technique is used to obtain the natural frequencies and their corresponding mode shapes of such beam. Then, the Ritz method is employed to derive the dynamic response of the beam due to the moving force. The obtained results for the free and forced vibrations of beams are verified against reported similar... 

Vibration analysis of a new type of compliant parallel mechanism with flexible intermediate links is investigated. The application of the Timoshenko beam theory to the mathematical modeling of the intermediate flexible link is described, and the equations of motion of the flexible links are obtained by using Lagrange's equation of motion. The equations of motion are obtained in the form of a set of ordinary differential equations by using assumed mode method theory. The governing differential equations of motion are solved using perturbation method. The assumed mode shapes and frequencies are to be obtained based on clamped-clamped boundary conditions. Comparing perturbation method with... 

Seismic response of a Single Degree of Freedom (SDOF) system containing a rigid circular cylindrical liquid tank under the earthquake excitations is considered in the presence of fluid/structure interaction. The governing differential equations of the system is derived considering the first three liquid sloshing modes (1, 1), (0, 1) and (2, 1) under different strong ground motions. The dynamic response of the system is investigated in the neighborhood of 1:1 and 1:2 internal resonances between the SDOF system and the liquid sloshing modes. These internal resonances take place by tuning the frequency of the first asymmetric sloshing mode of the liquid to the fundamental frequency of the SDOF... 

The governing differential equation and both classical and non-classical boundary conditions of strain gradient bars are derived using variational approach. A closed-form analytical solution is obtained for static torsion and the characteristic equation, which gives the natural frequencies, is derived and analytically solved for the free torsional vibrations of the strain gradient microbars. A fixed-fixed microbar is considered as a specific case to investigate the torsional size-dependent static and free-vibration behavior of strain gradient microbars. The results of the current model are compared to those of the modified couple stress and classical theories  

A semi-analytical solution for rotating axisymmetric disks made of functionally graded materials was previously proposed by Hosseini Kordkheili and Naghdabadi [1]. In the present work the solution is employed to study thermoelastic creep behavior of the functionally graded rotating disks with variable thickness in to the time domain. The rate type governing differential equations for the considered structure are derived and analytically solved in terms of rate of strain as a reduced to a set of linear algebraic equations. The advantage of this method is to avoid simplifications and restrictions which are normally associated with other creep solution techniques in the literature. The thermal... 

In this paper, a size-dependent formulation is presented for Timoshenko beams made of a functionally graded material (FGM). The formulation is developed on the basis of the modified couple stress theory. The modified couple stress theory is a non-classic continuum theory capable to capture the small-scale size effects in the mechanical behavior of structures. The beam properties are assumed to vary through the thickness of the beam. The governing differential equations of motion are derived for the proposed modified couple-stress FG Timoshenko beam. The generally valid closed-form analytic expressions are obtained for the static response parameters. As case studies, the static and free... 

In this paper, strain gradient elasticity formulation for analysis of FG (functionally graded) micro-cylinders is presented. The material properties are assumed to obey a power law in radial direction. The governing differential equation is derived as a fourth order ODE. A power series solution for stresses and displacements in FG micro-cylinders subjected to internal and external pressures is obtained. Numerical examples are presented to study the effect of the characteristic length parameter and FG power index on the displacement field and stress distribution in FG cylinders. It is observed that the characteristic length parameter has a considerable effect on the stress distribution of FG... 

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... 

This paper studies stress distribution in a single-walled carbon nanotube (SWCNT) under internal pressure with various chirality. Strain gradient theory is used to capture the size-dependent behavior of the SWCNT. Minimum total potential energy principle is successfully applied to derive the governing differential equation and its associated boundary conditions. Due to complexity of the governing differential equation and boundary conditions, numerical scheme is used to solve the problem. Comparing the results based on strain gradient theory and that of classical elasticity shows a major difference between these two methods. However, a close examination of the results indicates that both... 

In this paper, distributions of stress and strain components of rotating disks with non-uniform thickness and material properties subjected to thermo-elasto-plastic loading are obtained by semi-exact method of Liao's homotopy analysis method (HAM) and finite element method (FEM). The materials are assumed to be elastic-linear strain hardening and isotropic. The analysis of rotating disk is based on Von Mises' yield criterion. A two dimensional plane stress analysis is used. The distribution of temperature is assumed to have power forms with the hotter point located at the outer surface of the disk. A mathematical technique of transformation has been proposed to solve the homotopy equations... 

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... 

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... 

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 governing differential equation of motion of a thin rectangular plate excited by a moving mass is considered. The moving mass is traversing on the plate's surface at arbitrary trajectories. Eigenfunction expansion method is employed to solve the constitutive equation of motion for various boundary conditions. Approximate and exact expressions of the inertial effects are adopted for the problem formulation. In the approximate formulation, only the vertical acceleration component of the moving mass is considered while in the exact formulation all the convective acceleration components are included in the problem formulation as well. Parametric studies are carried out to investigate the... 

In the present study, a procedure was developed to determine the optimum reaction rate constants in generalized Arrhenius form and optimized through the Nelder-Mead method. For this purpose, a comprehensive mathematical model of a fixed bed reactor for dehydrogenation of heavy paraffins over Pt-Sn/Al 2O 3 catalyst was developed. Utilizing appropriate kinetic rate expressions for the main dehydrogenation reaction as well as side reactions and catalyst deactivation, a detailed model for the radial flow reactor was obtained. The reactor model composed of a set of partial differential equations (PDE), ordinary differential equations (ODE) as well as algebraic equations all of which were solved... 

Purpose - The purpose of this paper is to analyze a squared lattice cylindrical shell under compressive axial load and to optimize the geometric parameters to achieve the maximum buckling load. Also a comparison between buckling loads of a squared lattice cylinder and a solid hollow cylinder with equal weight, length and outer diameter is performed to reveal the superior performance of the squared lattice cylindrical shells. Design/methodology/ approach - A cylindrical lattice shell includes circumferential and longitudinal rods with geometric parameters such as crosssection areas of the rods, distances and angles between them. In this study, the governing differential equation for buckling...