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nonlinear-partial-differential-equations
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Solutions for the Double Sine-Gordon equation by Exp-function, Tanh, and extended Tanh methods
, Article Numerical Methods for Partial Differential Equations ; Volume 26, Issue 2 , 2010 , Pages 384-398 ; 0749159X (ISSN) ; Davodi, A. G ; Davodi, A. G ; Sharif University of Technology
2010
Abstract
In this work, we implement some analytical techniques such as the Exp-function, Tanh, and extended Tanh methods for solving nonlinear partial differential equation, which contains sine terms, its name Double Sine-Gordon equation. These methods obtain exact solutions of different types of differential equations in engineering mathematics
Third Order differential Equations Describing Pseudospherical Surfaces
, M.Sc. Thesis Sharif University of Technology ; Hesaraki, Mahmoud (Supervisor)
Abstract
Third order differential equations which describe pseudospherical surfaces are considered. The complete classification of a class of such equations is given. A linear problem with one or more parameters, also known as zero curvature representation, for which the equation is the integrability condition, is explicitly given. The classification provides five large families of differential equations. Third order nonlinear dispersive wave equations, such as the Camassa–Holm equation and Degasperis–Procesi equation are examples contained in the classification. Many other explicit examples are included
Modeling geometric non-linearities in the free vibration of a planar beam flexure with a tip mass
, Article Proceedings of the ASME Design Engineering Technical Conference, 12 August 2012 through 12 August 2012 ; Volume 4, Issue PARTS A AND B , August , 2012 , Pages 363-371 ; 9780791845035 (ISBN) ; Awtar, S ; Sharif University of Technology
2012
Abstract
The objective of this work is to create an analytical framework to study the non-linear dynamics of beam flexures with a tip mass undergoing large deflections. Hamilton's principal is utilized to derive the equations governing the nonlinear vibrations of the cantilever beam and the associated boundary conditions. Then, using a single mode approximation, these non-linear partial differential equations are reduced to two coupled non-linear ordinary differential equations. These equations are solved analytically using combination of the method of multiple time scales and homotopy perturbation analysis. Closed-form, parametric analytical expressions are presented for the time domain response of...
Improvement of dynamic response prediction of helicopters
, Article Aircraft Engineering and Aerospace Technology ; Volume 79, Issue 6 , 2007 , Pages 579-592 ; 00022667 (ISSN) ; Saghafi, F ; Sharif University of Technology
2007
Abstract
Purpose – This paper aims to focus on mathematical model development issues, necessary for a better prediction of dynamic responses of articulated rotor helicopters. Design/methodology/approach – The methodology is laid out based on model development for an articulated main rotor, using the theories of aeroelastisity, finite element and state-space represented indicial-based unsteady aerodynamics. The model is represented by a set of nonlinear partial differential equations for the main rotor within a state-space representation for all other parts of helicopter dynamics. The coupled rotor and fuselage formulation enforces the use of numerical solution techniques for trim and linearization...
An experimental-based numerical simulation of two phase flow through porous media: A comparative study on finite element and finite difference schemes
, Article Petroleum Science and Technology ; Volume 31, Issue 18 , 2013 , Pages 1881-1890 ; 10916466 (ISSN) ; Kharrat, R ; Ghazanfari, M. H ; Sharif University of Technology
2013
Abstract
In this study, the nonlinear partial differential equations governing two phase flow through porous media are solved using two different methods, namely, finite difference and finite element. The capillary pressure term is considered in the mathematical model. The numerical results on a 2-D test case are then compared with the experimental drainage process and water flooding performed on a glass type micromodel. Based on the obtained results, finite difference technique needs less computational time for solving governing equations of two phase flow, but findings of this method show less agreement with the experimental data. The finite element scheme was found to be more adequate and its...
Size-dependent bistability of an electrostatically actuated arch NEMS based on strain gradient theory?
, Article Journal of Physics D: Applied Physics ; Volume 48, Issue 24 , May , 2015 ; 00223727 (ISSN) ; Pishkenari, H. N ; Yazdi, M. R. H ; Miandoab, E. M ; Sharif University of Technology
Institute of Physics Publishing
2015
Abstract
This paper deals with the investigation of the size-dependent nature of nonlinear dynamics, in a doubly clamped shallow nano-arch actuated by spatially distributed electrostatic force. We employ strain gradient theory together with the Euler-Bernoulli and shallow arch assumptions in order to derive the nonlinear partial differential equation governing the transverse motion of the arch with mid-plane stretching effects. Using the Galerkin projection method, we derive the lumped single degree of freedom model which is then used for the study of the size effects on the nonlinear snap-through and pull-in instabilities of the arch nano-electro-mechanicalsystem (NEMS). Moreover, using strain...
Blow-up phenomena for a system of semilinear parabolic equations with nonlinear boundary conditions
, Article Mathematical Methods in the Applied Sciences ; Volume 38, Issue 3 , 2015 , Pages 527-536 ; 01704214 (ISSN) ; Hesaaraki, M ; Sharif University of Technology
John Wiley and Sons Ltd
2015
Abstract
This paper deals with the blow-up phenomena for a system of parabolic equations with nonlinear boundary conditions. We show that under some conditions on the nonlinearities, blow-up occurs at some finite time. We also obtain upper and lower bounds for the blow-up time when blow-up occurs. Copyright
Investigation of the oscillatory behavior of electrostatically-Actuated microbeams
, Article ASME International Mechanical Engineering Congress and Exposition, Proceedings (IMECE), 12 November 2010 through 18 November 2010, Vancouver, BC ; Volume 10 , 2010 , Pages 619-626 ; 9780791844472 (ISBN) ; Moghimi Zand, M ; Ahmadian, M. T ; Sharif University of Technology
2010
Abstract
Vibrations of electrostatically-Actuated microbeams are investigated. Effects of electrostatic actuation, axial stress and midplane stretching are considered in the model. Galerkin's decomposition method is utilized to convert the governing nonlinear partial differential equation to a nonlinear ordinary differential equation. Homotopy perturbation method (i.e. a special and simpler case of homotopy analysis method) is utilized to find analytic expressions for natural frequencies of predeformed microbeam. Effects of increasing the voltage, midplane stretching, axial force and higher modes contribution on natural frequency are also studied. The anayltical results are in good agreement with the...
Control of vibration amplitude, frequency and damping of an electrostatically actuated microbeam using capacitive, inductive and resistive elements
, Article ASME International Mechanical Engineering Congress and Exposition, Proceedings (IMECE), 12 November 2010 through 18 November 2010, Vancouver, BC ; Volume 10 , 2010 , Pages 263-270 ; 9780791844472 (ISBN) ; Alizadeh Vaghasloo, Y ; Fallah, A ; Ahmadian, M. T ; Sharif University of Technology
2010
Abstract
In this study vibration amplitude, frequency and damping of a microbeam is controlled using a RLC block containing a capacitor, resistor and inductor in series with the microbeam. Applying this method all of the considerable characteristics of the oscillatory system can be determined and controlled with no change in the geometrical and physical characteristics of the microbeam. Euler-Bernoulli assumptions are made for the microbeam and the electrical current through the microbeam is computed by considering the microbeam deflection and its voltage. Considering the RLC block, the voltage difference between the microbeam and the substrate is calculated. Two coupled nonlinear partial...
GRKPM: Theory and Applications in Laminated Composite Plates and Nonlinear Evolutionary Partial Differential Equations With Large Gradients
, Ph.D. Dissertation Sharif University of Technology ; Mohammadi Shodja, Hossein (Supervisor)
Abstract
Reproducing kernel particle method (RKPM) is a meshfree method for solving various differential equations. RKPM is based on pure mathematics; therefore, it is in the center of attention of many scientists. One major problem in RKPM is satisfying the essential boundary conditions (EBCs) involving the derivative of the field function. This problem is considered herein and its solution is proposed. To this end, two actions should be undertaken. First, the concept of Hermitian interpolation is employed to add the derivative term to the reproducing equation of RKPM and a new meshless method called gradient RKPM (GRKPM) is introduced. Second, the corrected collocation method is modified so...
Observer-based vibration control of non-classical microcantilevers using extended Kalman filters
, Article Applied Mathematical Modelling ; January , 2015 ; 0307904X (ISSN) ; Karami, F ; Salarieh, H ; Sharif University of Technology
Elsevier Inc
2015
Abstract
In non-classical micro-beams, the strain energy of the system is determined by the non-classical continuum mechanics. In this study, we consider a closed-loop control methodology for suppressing the vibration of non-classical microscale Euler-Bernoulli beams with nonlinear electrostatic actuation. The non-dimensional form of the governing nonlinear partial differential equation of the system is introduced and converted into a set of ordinary differential equations using the Galerkin projection method. In addition, we prove the observability of the system and we design a state estimation system using the extended Kalman filter algorithm. The effectiveness and performance of the proposed...
Stabilization of a vibrating non-classical micro-cantilever using electrostatic actuation
, Article Scientia Iranica ; Volume 20, Issue 6 , 2013 , Pages 1824-1831 ; 10263098 (ISSN) ; Karami, F ; Salarieh, H ; Alasty, A ; Sharif University of Technology
Sharif University of Technology
2013
Abstract
A closed-loop control methodology is investigated for stabilization of a vibrating non-classical micro-scale Euler-Bernoulli beam with nonlinear electrostatic actuation. The dimensionless form of governing nonlinear Partial Differential Equation (PDE) of the system is introduced. The Galerkin projection method is used to reduce the PDE of system to a set of nonlinear Ordinary Differential Equations (ODE). In non-classical micro-beams, the constitutive equations are obtained based on the non-classical continuum mechanics. In this work, proper control laws are constructed to stabilize the free vibration of non-classical micro-beams whose governing PDE is derived based on the modified strain...
Nonlinear free vibration of nanobeams with surface effects considerations
, Article Proceedings of the ASME Design Engineering Technical Conference, 28 August 2011 through 31 August 2011 ; Volume 7 , August , 2011 , Pages 191-196 ; 9780791854846 (ISBN) ; Firoozbakhsh, K ; Kahrobaiyan, M. H ; Pasharavesh, A ; Sharif University of Technology
2011
Abstract
In this paper, simple analytical expressions are presented for geometrically non-linear vibration analysis of thin nanobeams with both simply supported and clamped boundary conditions. Gurtin-Murdoch surface elasticity together with Euler-Bernoulli beam theory is used to obtain the governing equations of motions of the nanobeam with surface effects consideration. The governing nonlinear partial differential equation is reduced to a single nonlinear ordinary differential equation using Galerkin technique. He's variational approach is employed to obtain analytical solution for the resulted nonlinear governing equation. The effects of different parameters such as vibration amplitude, boundary...
On the chaotic vibrations of electrostatically actuated arch micro/nano resonators: a parametric study
, Article International Journal of Bifurcation and Chaos ; Volume 25, Issue 8 , July , 2015 ; 02181274 (ISSN) ; Hairi Yazdi, M. R ; Nejat Pishkenari, H ; Sharif University of Technology
World Scientific Publishing Co. Pte Ltd
2015
Abstract
Motivated by specific applications, electrostatically actuated bistable arch shaped micro-nano resonators have attracted growing attention in the research community in recent years. Nevertheless, some issues relating to their nonlinear dynamics, including the possibility of chaos, are still not well known. In this paper, we investigate the chaotic vibrations of a bistable resonator comprised of a double clamped initially curved microbeam under combined harmonic AC and static DC distributed electrostatic actuation. A reduced order equation obtained by the application of the Galerkin method to the nonlinear partial differential equation of motion, given in the framework of Euler-Bernoulli beam...
Observer-based vibration control of non-classical microcantilevers using extended Kalman filters
, Article Applied Mathematical Modelling ; Volume 39, Issue 19 , 2015 , Pages 5986-5996 ; 0307904X (ISSN) ; Karami, F ; Salarieh, H ; Sharif University of Technology
Elsevier Inc
2015
Abstract
In non-classical micro-beams, the strain energy of the system is determined by the non-classical continuum mechanics. In this study, we consider a closed-loop control methodology for suppressing the vibration of non-classical microscale Euler-Bernoulli beams with nonlinear electrostatic actuation. The non-dimensional form of the governing nonlinear partial differential equation of the system is introduced and converted into a set of ordinary differential equations using the Galerkin projection method. In addition, we prove the observability of the system and we design a state estimation system using the extended Kalman filter algorithm. The effectiveness and performance of the proposed...
Effect of microbeam electrical resistivity on dynamic pull-in voltage of an electrostatically actuated microbeam
, Article ASME International Mechanical Engineering Congress and Exposition, Proceedings (IMECE), 12 November 2010 through 18 November 2010 ; Volume 10 , 2010 , Pages 271-278 ; 9780791844472 (ISBN) ; Ahmadian, M. T ; Alizadeh Vaghasloo, Y ; Assempour, A ; Sharif University of Technology
Abstract
The dynamic pull-in voltage as a criterion for the system stability is one of the most important effects considered with the dynamics of microstructures. In this study effect of microbeam electrical resistivity on the pull-in voltage of an electrostatically actuated microbeam is investigated. Assuming Euler-Bernoulli theory for the microbeam, two coupled nonlinear partial differential equations are derived for the beam deflection and voltage. The one parameter Galerkin method is implemented to transform the equations to a set of nonlinear coupled ordinary differential equations. Obtained equations are solved implementing the differential quadrature method (DQM). Variation of dynamic pull-in...
Effect of microbeam electrical resistivity on vibration frequency shift of an electrostatically actuated microbeam
, Article Proceedings of the ASME Design Engineering Technical Conference, 15 August 2010 through 18 August 2010 ; Volume 4 , 2010 , Pages 547-554 ; 9780791844120 (ISBN) ; Ahmadian, M. T ; Alizadeh Vaghasloo, Y ; Sharif University of Technology
Abstract
Nonlinear vibration of a microbeam actuated by a suddenly applied voltage with considering the effect of voltage distribution on the beam due to electrical resistivity of beam is investigated. Homotopy perturbation method is implemented to solve the coupled nonlinear partial differential equations of motion. The vibration frequency variation and damping at various resistivities is studied. Considering resistivity, effect of applied voltage and beam length on the frequency shift and damping ratio is analyzed. Findings indicate there exists a jump in frequency shift and damping ratio at a critical resistivity. Variation of critical resistivity with respect to modulus of elasticity and beam...
On the primary resonance of an electrostatically actuated MEMS using the homotopy perturbation method
, Article Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference 2009, DETC2009, 30 August 2009 through 2 September 2009 ; Volume 6 , September , 2010 , Pages 569-574 ; 9780791849033 (ISBN) ; Moghimi Zand, M ; Ahmadian, M. T ; Sharif University of Technology
2010
Abstract
In this paper, primary resonance of a double-clamped microbeam has been investigated. The Microbeam is predeformed by a DC electrostatic force and then driven to vibrate by an AC harmonic electrostatic force. Effects of midplane stretching, axial loads and damping are considered in modeling. Galerkin's approximation is utilized to convert the nonlinear partial differential equation of motion to a nonlinear ordinary differential equation. Afterward, a combination of homotopy perturbation method and the method of multiple scales are utilized to find analytic solutions to the steady-state motion of the microbeam, far from pull-in. The effects of different design parameters on dynamic behavior...
Electromechanical modeling and analytical investigation of nonlinearities in energy harvesting piezoelectric beams
, Article International Journal of Mechanics and Materials in Design ; 2016 , Pages 1-16 ; 15691713 (ISSN) ; Ahmadian, M. T ; Zohoor, H ; Sharif University of Technology
Springer Netherlands
2016
Abstract
Piezoelectric materials are extensively applied for vibrational energy harvesting especially in micro-scale devices where other energy conversion mechanisms such as electromagnetic and electrostatic methods encounter fabrication limitations. A cantilevered piezoelectric bimorph beam with an attached proof (tip) mass for the sake of resonance frequency reduction is the most common structure in vibrational harvesters. According to the amplitude and frequency of applied excitations and physical parameters of the harvester, the system may be pushed into a nonlinear regime which arises from material or geometric nonlinearities. In this study nonlinear dynamics of a piezoelectric bimorph harvester...
On the primary resonance of an electrostatically actuated MEMS using the homotopy perturbation method
, Article Proceedings of the ASME Design Engineering Technical Conference, 30 August 2009 through 2 September 2009, San Diego, CA ; Volume 6 , 2009 , Pages 569-574 ; 9780791849033 (ISBN) ; Moghimi Zand, M ; Taghi Ahmadian, M ; Sharif University of Technology
Abstract
In this paper, primary resonance of a double-clamped microbeam has been investigated. The Microbeam is predeformed by a DC electrostatic force and then driven to vibrate by an AC harmonic electrostatic force. Effects of midplane stretching, axial loads and damping are considered in modeling. Galerkin's approximation is utilized to convert the nonlinear partial differential equation of motion to a nonlinear ordinary differential equation. Afterward, a combination of homotopy perturbation method and the method of multiple scales are utilized to find analytic solutions to the steady-state motion of the microbeam, far from pull-in. The effects of different design parameters on dynamic behavior...