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governing-differential-equations
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Pressure variation due to sudden rise of water head at water inlets
, Article 31st IAHR Congress 2005: Water Engineering for the Future, Choices and Challenges, 11 September 2005 through 16 September 2005 ; 2005 , Pages 2797-2806 ; 8987898245 (ISBN); 9788987898247 (ISBN) ; Borghei, S.M ; Saidi, M. H ; Byong-Ho J ; Sang I. L ; Won S. I ; Gye-Woon C ; Sharif University of Technology
Korea Water Resources Association
2005
Abstract
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...
Vibration of beam with elastically restrained ends and rotational spring-lumped rotary inertia system at mid-span
, Article International Journal of Structural Stability and Dynamics ; 2014 ; ISSN: 02194554 ; Sani, A. A ; Mofid, M ; Sharif University of Technology
Abstract
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
Thermoelastic creep analysis of a functionally graded various thickness rotating disk with temperature-dependent material properties
, Article International Journal of Pressure Vessels and Piping ; Volume 111-112 , 2013 , Pages 63-74 ; 03080161 (ISSN) ; Livani, M ; Sharif University of Technology
2013
Abstract
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...
A size-dependent model for functionally graded micro-plates for mechanical analyses
, Article JVC/Journal of Vibration and Control ; Volume 19, Issue 11 , 2013 , Pages 1614-1632 ; 10775463 (ISSN) ; Taati, E ; Sharif University of Technology
2013
Abstract
In this paper, a size-dependent formulation is presented for mechanical analyses of inhomogeneous micro-plates based on the modified couple stress theory. The plate properties can arbitrarily vary through the thickness. The governing differential equations of motion are derived for functionally graded (FG) plates with arbitrary shapes utilizing a variational approach. Moreover, the boundary conditions are provided at smooth parts of the plate periphery and also at the sharp corners of the periphery. Utilizing the derived formulation, the free-vibration behavior as well as the static response of a rectangular FG micro-plate is investigated
Vibration analysis of delaminated Timoshenko beams under the motion of a constant amplitude point force traveling with uniform velocity
, Article International Journal of Mechanical Sciences ; Volume 70 , 2013 , Pages 39-49 ; 00207403 (ISSN) ; Jafari Talookolaei, R. A ; Ahmadian, M. T ; Sharif University of Technology
2013
Abstract
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...
Mechanical behavior analysis of micro-scaled functionally graded timoshenko beams by the strain gradient theory
, Article Proceedings of the ASME Design Engineering Technical Conference ; Volume 5 , 2012 , Pages 67-73 ; 9780791845042 (ISBN) ; Kahrobaiyan, M. H ; Rahaeifard, M ; Ahmadian, M. T ; Movahhedy, M. R ; Akbari, J ; Sharif University of Technology
2012
Abstract
In this paper, a size-dependent formulation is developed for Timoshenko beams made of functionally graded materials (FGM). The developed formulation is based on the strain gradient theory;a non-classical continuum theory able to capture the size-effect in micro-scaled structures. Considering the material length scale parameters of the FG beams vary through the thickness, the new equivalent length scale parameters are proposed as functions of the constituents' length scale parameters to describe the size-dependent static and dynamic behavior of FG microbeams. The governing differential equations of equilibrium and both classical and nonclassical sets of boundary conditions are derived for the...
A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory
, Article Acta Mechanica ; Volume 223, Issue 6 , 2012 , Pages 1233-1249 ; 00015970 (ISSN) ; Kahrobaiyan, M. H ; Nikfar, M ; Ahmadian, M. T ; Sharif University of Technology
2012
Abstract
The geometrically nonlinear governing differential equations of motion and the corresponding boundary conditions are derived for the mechanical analysis of Timoshenko microbeams with large deflections, based on the strain gradient theory. The variational approach is employed to achieve the formulation. Hinged-hinged beams are considered as an important practical case, and their nonlinear static and free-vibration behaviors are investigated based on the derived formulation
Torsion of strain gradient bars
, Article International Journal of Engineering Science ; Volume 49, Issue 9 , September , 2011 , Pages 856-866 ; 00207225 (ISSN) ; Tajalli, S. A ; Movahhedy, M. R ; Akbari, J ; Ahmadian, M. T ; Sharif University of Technology
2011
Abstract
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
Modeling and analytical solution of hybrid thermopiezoelectric micro actuator and performance study under changing of different parameters
, Article Mechanics of Advanced Materials and Structures ; Volume 22, Issue 10 , Mar , 2015 , Pages 785-793 ; 15376494 (ISSN) ; Kargarnovin, M. H ; Zohoor, H ; Sharif University of Technology
Taylor and Francis Inc
2015
Abstract
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...
Vibration of beam with elastically restrained ends and rotational spring-lumped rotary inertia system at mid-span
, Article International Journal of Structural Stability and Dynamics ; Volume 15, Issue 2 , 2015 ; 02194554 (ISSN) ; Aftabi Sani, A ; Mofid, M ; Sharif University of Technology
World Scientific Publishing Co. Pte Ltd
2015
Abstract
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
Vibration control of smart functionally graded plate bonded with PZT4 sensor/actuator patches
, Article Proceedings of the ASME Design Engineering Technical Conference, 15 August 2010 through 18 August 2010 ; Volume 5 , 2010 , Pages 643-650 ; 9780791844137 (ISBN) ; Viliani, N. S ; Sharif University of Technology
Abstract
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
Vibration analysis of rectangular functionally graded plate bonded with PZT5 sensor/actuator
, Article ASME 2010 10th Biennial Conference on Engineering Systems Design and Analysis, ESDA2010, 12 July 2010 through 14 July 2010 ; Volume 2 , 2010 , Pages 287-294 ; 9780791849163 (ISBN) ; Viliani, N. S ; Sharif University of Technology
Abstract
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 closed-form study on the free vibration of a grid joined by a mass-spring system
, Article JVC/Journal of Vibration and Control ; Volume 22, Issue 4 , 2016 , Pages 1147-1157 ; 10775463 (ISSN) ; Aftabi Sani, A ; Mofid, M ; Sharif University of Technology
SAGE Publications Inc
2016
Abstract
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....
An analytical solution for bending of axisymmetric circular/annular plates resting on a variable elastic foundation
, Article European Journal of Mechanics, A/Solids ; Volume 74 , 2019 , Pages 462-470 ; 09977538 (ISSN) ; Mofid, M ; Sharif University of Technology
Elsevier Ltd
2019
Abstract
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,...
Error estimate in calculating natural frequencies of a vibrating shaft by changing number of segments using lumped parameter model and transfer matrix method
, Article 7th European Conference on Structural Dynamics, EURODYN 2008, 7 July 2008 through 9 July 2008 ; 2008 ; 9780854328826 (ISBN) ; Sharif University of Technology
University of Southampton, Institute of Sound Vibration and Research
2008
Abstract
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...
An innovative series solution for dynamic response of rectangular Mindlin plate on two-parameter elastic foundation, with general boundary conditions
, Article European Journal of Mechanics, A/Solids ; Volume 88 , 2021 ; 09977538 (ISSN) ; Motaghian, S ; Mofid, M ; Sharif University of Technology
Elsevier Ltd
2021
Abstract
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...
Introducing structural approximation method for modeling nanostructures
, Article Journal of Computational and Theoretical Nanoscience ; Vol. 7, Issue 2 , 2010 , p. 423-428 ; ISSN: 15461955 ; Alasty, A ; Sharif University of Technology
Abstract
In this work a new method for analyzing nanostructured materials has been proposed to accelerate the simulations for solid crystalline materials. The proposed Structural Approximation Method (SAM) is based on Molecular Dynamics (MD) and the accuracy of the results can also be improved in a systematic manner by sacrificing the simulation speed. In this method a virtual material is used instead of the real one, which has less number of atoms and therefore fewer degrees of freedom, compared to the real material. The number of differential equations that must be integrated in order to specify the state of the system will decrease significantly, and the simulation speed increases. To generalize...
Mechanical behavior analysis of size-dependent micro-scaled functionally graded Timoshenko beams by strain gradient elasticity theory
, Article Composite Structures ; Volume 102 , 2013 , Pages 72-80 ; 02638223 (ISSN) ; Rahaeifard, M ; Kahrobaiyan, M. H ; Movahhedy, M. R ; Akbari, J ; Ahmadian, M. T ; Sharif University of Technology
2013
Abstract
In this paper, a size-dependent formulation is developed for Timoshenko beams made of functionally graded materials (FGMs). The developed formulation is based on the strain gradient theory; a non-classical continuum theory able to capture the size-effect in micro-scaled structures. Five new equivalent length scale parameters are introduced as functions of the constituents' length scale parameters. It is shown that the size-dependent static and dynamic behavior of FG micro-beams can be described using these equivalent length scales. The governing differential equations of motion and both classical and non-classical sets of boundary conditions are derived for the proposed strain gradient FG...
Forced vibration of delaminated timoshenko beams under the action of moving oscillatory mass
, Article Shock and Vibration ; Volume 20, Issue 1 , 2013 , Pages 79-96 ; 10709622 (ISSN) ; Ahmadian, M. T ; Jafari Talookolaei, R. A ; Sharif University of Technology
2013
Abstract
This paper presents the dynamic response of a delaminated composite beam under the action of a moving oscillating mass. In this analysis the Poisson's effect is considered for the first time. Moreover, the effects of rotary inertia and shear deformation are incorporated. In our modeling linear springs are used between delaminated surfaces to simulate the dynamic interaction between sub-beams. To solve the governing differential equations of motion using modal expansion series, eigen-solution technique is used to obtain the natural frequencies and their corresponding mode shapes necessary for forced vibration analysis. The obtained results for the free and forced vibrations of beams are...
Dynamic response of Timoshenko beam under moving mass
, Article Scientia Iranica ; Volume 20, Issue 1 , 2013 , Pages 50-56 ; 10263098 (ISSN) ; Mofid, M ; Khoraskani, R. A ; Sharif University of Technology
2013
Abstract
In this article, the dynamic responses of a Timoshenko beam subjected to a moving mass, and a moving sprung mass are analyzed. By making recourse to Hamilton's principle, governing differential equations for beam vibration are derived. By using the modal superposition method, the partial differential equations of the system are transformed into a set of Ordinary Differential Equations (ODEs). The resulted set of ODEs is represented in state-space form, and solved by means of a numerical technique. The accuracy of the results has been ascertained through comparing the results of our approach with those available from previous studies; moreover, a reasonable agreement has been obtained. The...