Showing papers in "Applied Mathematics and Mechanics-english Edition in 2007"
TL;DR: In this article, the effect of temperature-dependent viscosity on fully developed forced convection in a duct of rectangular cross-section occupied by a fluid-saturated porous medium is investigated analytically.
Abstract: Effect of temperature-dependent viscosity on fully developed forced convection in a duct of rectangular cross-section occupied by a fluid-saturated porous medium is investigated analytically. The Darcy flow model is applied and the viscosity-temperature relation is assumed to be an inverse-linear one. The case of uniform heat flux on the walls, i.e. the H boundary condition in the terminology of Kays and Crawford [12], is treated. For the case of a fluid whose viscosity decreases with temperature, it is found that the effect of the variation is to increase the Nusselt number for heated walls. Having found the velocity and the temperature distribution, the second law of thermodynamics is invoked to find the local and average entropy generation rate. Expressions for the entropy generation rate, the Bejan number, the heat transfer irreversibility, and the fluid flow irreversibility are presented in terms of the Brinkman number, the Peclet number, the viscosity variation number, the dimensionless wall heat flux, and the aspect ratio (width to height ratio). These expressions let a parametric study of the problem based on which it is observed that the entropy generated due to flow in a duct of square cross-section is more than those of rectangular counterparts while increasing the aspect ratio decreases the entropy generation rate similar to what previously reported for the clear flow case by Ratts and Raut [14].
44 citations
TL;DR: This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method and shows that these algorithms are correct and high efficient.
Abstract: Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine finite element space on the subdomain. The local algorithm consists of finding a solution for a given nonlinear problem in the coarse finite element space and a solution for a linear problem in the fine finite element space, then droping the coarse solution of the region near the boundary. By overlapping domain decomposition, the parallel algorithms are obtained. This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method. The numerical experiments are given too. By analyzing and comparing these results, it is shown that these algorithms are correct and high efficient.
40 citations
TL;DR: In this paper, the non-linear governing differential equations of immovably simply supported functionally graded material (FGM) rod subjected to thermal loads were derived, and the thermal post-buckling behaviors of FGM rod made of ZrO2 and Ti-6A1-4V were analyzed by shooting method.
Abstract: The non-linear governing differential equations of immovably simply supported functionally graded material (FGM) rod subjected to thermal loads were derived. The thermal post-buckling behaviors of FGM rod made of ZrO2 and Ti-6A1-4V were analyzed by shooting method. Firstly, the thermal post-buckling equilibrium paths of the FGM rod with different gradient index in the uniform temperature field were plotted, and compared with the behaviors of the homogeneous rods made of ZrO2 and Ti-6A1-4V materials, respectively. For given value of end rotation angles, the influence of gradient index on the thermal post-buckling behaviors of FGM rod was discussed. Secondly, the thermal post-buckling characteristics of the FGM rod were analyzed when the temperature difference parameter is changed while the bottom temperature parameter remains constant, and when the bottom temperature parameter is changed while the temperature difference parameter remains constant, and compared with the characteristics of the two homogeneous material rods.
38 citations
TL;DR: This paper describes an adaptive genetic algorithm for solving the bilevel linear programming problem to overcome the difficulty of determining the probabilities of crossover and mutation.
Abstract: Bilevel linear programming, which consists of the objective functions of the upper level and lower level, is a useful tool for modeling decentralized decision problems. Various methods are proposed for solving this problem. Of all the algorithms, the genetic algorithm is an alternative to conventional approaches to find the solution of the bilevel linear programming. In this paper, we describe an adaptive genetic algorithm for solving the bilevel linear programming problem to overcome the difficulty of determining the probabilities of crossover and mutation. In addition, some techniques are adopted not only to deal with the difficulty that most of the chromosomes may be infeasible in solving constrained optimization problem with genetic algorithm but also to improve the efficiency of the algorithm. The performance of this proposed algorithm is illustrated by the examples from references.
36 citations
TL;DR: In this paper, the dynamic characteristics and stability of axially moving viscoelastic rectangular thin plate are investigated based on the two dimensional visco-elastic differential constitutive relation.
Abstract: The dynamic characteristics and stability of axially moving viscoelastic rectangular thin plate are investigated. Based on the two dimensional viscoelastic differential constitutive relation, the differential equations of motion of the axially moving viscoelastic plate are established. Dimensionless complex frequencies of an axially moving viscoelastic plate with four edges simply supported, two opposite edges simply supported and other two edges clamped are calculated by the differential quadrature method. The effects of the aspect ratio, moving speed and dimensionless delay time of the material on the transverse vibration and stability of the axially moving viscoelastic plate are analyzed.
34 citations
TL;DR: In this article, an analytical solution to an asymmetrical propagation crack under the condition of point loads and unit-step loads, respectively, is obtained by applying the theory of complex functions, and the universal representations of analytical solutions are obtained by the approaches of self-similar function.
Abstract: By the application of the theory of complex functions, asymmetrical dynamic propagation problems on mode III interface crack are studied. The universal representations of analytical solutions are obtained by the approaches of self-similar function. The problems researched can be facilely transformed into Riemann-Hilbert problems and analytical solution to an asymmetrical propagation crack under the condition of point loads and unit-step loads, respectively, is acquired. After those solutions were used by superposition theorem, the solutions of arbitrarily complex problems could be attained.
31 citations
TL;DR: In this article, the notion of intuitionistic Menger spaces was defined as a netural generalization of Menger Spaces due to Menger. But the notion was not defined in this paper.
Abstract: Using the idea of Atanassov, we define the notion of intuitionistic Menger spaces as a netural generalizations of Menger spaces due to Menger. We also obtain a new generalized contraction mapping and utilize this contraction mapping to prove the existence theorems of solutions to differential equations in intuitionistic Menger spaces.
29 citations
TL;DR: In this paper, a rich theory has appeared in the literature of the famous Cebysev inequality, and some new weighted integral inequalities have been established, which are of independent interest.
Abstract: In account of the famous Cebysev inequality, a rich theory has appeared in the literature. We establish some new weighted Cebysev type integral inequalities. Our proofs are of independent interest and provide new estimates on these types of inequalities.
27 citations
TL;DR: In this paper, a method has been developed to find the analytical expressions of all the three phase velocities of quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH) in three dimensions.
Abstract: The propagation of three-dimensional plane waves at a traction free boundary of a half-space composed of triclinic crystalline material is discussed. A method has been developed to find the analytical expressions of all the three phase velocities of quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH) in three dimensions. Closed form expressions in three dimensions for the amplitude ratios of reflection coefficients of qP, qSV and qSH waves in a triclinic medium are obtained. These expressions are used for numerically studying the variation of the reflection coefficients with the angle of incidence. The graphs are drawn for different polar angle and azimuth. Numerical results presented indicate that the anisotropy affect the reflection coefficients significantly in the three dimensional case compared to the two-dimensional case.
25 citations
TL;DR: In this paper, the authors analyzed the rotationally vibration of a circular piezoelectric shell of polarized ceramics mounted on a rotationally vibrating base, and the shell was properly electroded and connected to a circuit such that an electric output was generated.
Abstract: Torsional vibration of a circular piezoelectric shell of polarized ceramics mounted on a rotationally vibrating base is analyzed. The shell is properly electroded and connected to a circuit such that an electric output is generated. The structure analyzed represents a piezoelectric generator for converting mechanical energy from angular vibrations to electrical energy. Analytical expressions and numerical results for the output voltage, current, power, efficiency and power density are given.
24 citations
TL;DR: In this article, the analytical solution of a viscoelastic continuous beam whose damping characteristics are described in terms of a fractional derivative of arbitrary order was derived by means of the Adomian decomposition method.
Abstract: The analytical solution of a viscoelastic continuous beam whose damping characteristics are described in terms of a fractional derivative of arbitrary order was derived by means of the Adomian decomposition method. The solution contains arbitrary initial conditions and zero input. For specific analysis, the initial conditions were assumed homogeneous, and the input force was treated as a special process with a particular beam. Two simple cases, step and impulse function responses, were considered respectively. Subsequently, some figures were plotted to show the displacement of the beam under different sets of parameters including different orders of the fractional derivatives.
TL;DR: By applying the continuous finite element methods of ordinary differential equations, the linear element methods were proved having second-order pseudo-symplectic scheme and the quadratic element methods are proved having third-order Pseudo-Symplectic Scheme respectively for general Hamiltonian systems, and they both kept energy conservative as discussed by the authors.
Abstract: By applying the continuous finite element methods of ordinary differential equations, the linear element methods are proved having second-order pseudo-symplectic scheme and the quadratic element methods are proved having third-order pseudo-symplectic scheme respectively for general Hamiltonian systems, and they both keep energy conservative. The finite element methods are proved to be symplectic as well as energy conservative for linear Hamiltonian systems. The numerical results are in agreement with theory.
TL;DR: Parabolized stability equations (PSE) were used to study the evolution of disturbances in compressible boundary layers in this paper, and the results were compared with those obtained by direct numerical simulations (DNS), to check if the results from PSE method were reliable or not.
Abstract: Parabolized stability equations (PSE) were used to study the evolution of disturbances in compressible boundary layers. The results were compared with those obtained by direct numerical simulations (DNS), to check if the results from PSE method were reliable or not. The results of comparison showed that no matter for subsonic or supersonic boundary layers, results from both the PSE and DNS method agreed with each other reasonably well, and the agreement between temperatures was better than those between velocities. In addition, linear PSE was used to calculate the neutral curve for small amplitude disturbances in a supersonic boundary layer. Compared with those obtained by linear stability theory (LST), the situation was similar to those for incompressible boundary layer.
TL;DR: In this paper, a theoretical analysis of three-dimensional Couette flow with radiation effect on temperature distribution has been analyzed, when the injection of the fluid at the lower stationary plate is a transverse sinusoidal one and its corresponding removal by constant suction through the upper porous plate is in uniform motion.
Abstract: A theoretical analysis of three-dimensional Couette flow with radiation effect on temperature distribution has been analysed, when the injection of the fluid at the lower stationary plate is a transverse sinusoidal one and its corresponding removal by constant suction through the upper porous plate is in uniform motion. Due to this type of injection velocity, the flow becomes three-dimensional. The effect of Prandtl number, radiation parameter and injection parameter on rate of heat transfer has been examined by the help of graphs. The Prandtl number has a much greater effect on the temperature distribution than the injection or radiation parameter.
TL;DR: In this paper, a new statistic-based noise analysis method is proposed together with the Monte Carlo technique to investigate the influence of experimental noise of modal data on sensitivity-based damage detection methods.
Abstract: As vibration-based structural damage detection methods are easily affected by environmental noise, a new statistic-based noise analysis method is proposed together with the Monte Carlo technique to investigate the influence of experimental noise of modal data on sensitivity-based damage detection methods. Different from the commonly used random perturbation technique, the proposed technique is deduced directly by Moore-Penrose generalized inverse of the sensitivity matrix, which does not only make the analysis process more efficient but also can analyze the influence of noise on both frequencies and mode shapes for three commonly used sensitivity-based damage detection methods in a similar way. A one-story portal frame is adopted to evaluate the efficiency of the proposed noise analysis technique.
TL;DR: In this paper, a group of asymmetric difference schemes to approach the Korteweg-de Vries (KdV) equation is given, and an alternating segment explicit-implicit difference scheme for solving the KdV equation is constructed.
Abstract: A group of asymmetric difference schemes to approach the Korteweg-de Vries (KdV) equation is given here. According to such schemes, the full explicit difference scheme and the full implicit one, an alternating segment explicit-implicit difference scheme for solving the KdV equation is constructed. The scheme is linear unconditionally stable by the analysis of linearization procedure, and is used directly on the parallel computer. The numerical experiments show that the method has high accuracy.
TL;DR: Based on model-I and model-II stress strength factors of control fissure under the acting of weight of perilous rock, water pressure in control fracture and earthquake forces, method to calculate critical linking length of control fracture was established.
Abstract: Rupture and safety of perilous rock are dominated by control fissure behind perilous rock block. Based on model-I and model-II stress strength factors of control fissure under acting of weight of perilous rock, water pressure in control fissure and earthquake forces, method to calculate critical linking length of control fissure is established. Take water pressure in control fissure as a variable periodic load, and abide by P-M criterion, when control fissure is filled with water, establish the method to calculate fatigue fracture life of control fissure in critical status by contributing value of stress strength factor stemming from water pressure of control fissure in Paris’s fatigue equation. Further, parameters (C and m) of sandstone with quartz and feldspar in the area of the Three Gorges Reservoir of China are obtained by fatigue fracture testing.
TL;DR: In this paper, a robust SEIR epidemic disease model with a profitless delay and vertical transmission was formulated, and the dynamics behaviors of the model under pulse vaccination were analyzed, and it was proved that the delay is "profitless".
Abstract: A robust SEIR epidemic disease model with a profitless delay and vertical transmission is formulated, and the dynamics behaviors of the model under pulse vaccination are analyzed. By use of the discrete dynamical system determined by the stroboscopic map, an ‘infection-free’ periodic solution is obtained, further, it is shown that the ‘infection-free’ periodic solution is globally attractive when some parameters of the model are under appropriate conditions. Using the theory on delay functional and impulsive differential equation, the sufficient condition with time delay for the permanence of the system is obtained, and it is proved that time delays, pulse vaccination and vertical transmission can bring obvious effects on the dynamics behaviors of the model. The results indicate that the delay is ‘profitless’.
TL;DR: In this article, a theoretical model for analyzing the nonlinear behavior of a TLP with finite displacement is developed, in which multifold nonlinearities are taken into account, i.e., finite displacement, coupling of the six degrees of freedom, instantaneous position, instantaneous wet surface, free surface effects and viscous drag force.
Abstract: It is demonstrated that when tension leg platform (TLP) moves with finite amplitude in waves, the inertia force, the drag force and the buoyancy acting on the platform are nonlinear functions of the response of TLP. The tensions of the tethers are also nonlinear functions of the displacement of TLP. Then the displacement, the velocity and the acceleration of TLP should be taken into account when loads are calculated. In addition, equations of motions should be set up on the instantaneous position. A theoretical model for analyzing the nonlinear behavior of a TLP with finite displacement is developed, in which multifold nonlinearities are taken into account, i.e., finite displacement, coupling of the six degrees of freedom, instantaneous position, instantaneous wet surface, free surface effects and viscous drag force. Based on the theoretical model, the comprehensive nonlinear differential equations are deduced. Then the nonlinear dynamic analysis of ISSC TLP in regular waves is performed in the time domain. The degenerative linear solution of the proposed nonlinear model is verified with existing published one. Furthermore, numerical results are presented, which illustrate that nonlinearities exert a significant influence on the dynamic responses of the TLP.
TL;DR: In this article, a nonlinear governing equation for transverse vibration of an axially accelerating string is set up by using the equation of motion for a 3-dimensional deformable body with initial stresses.
Abstract: Under the consideration of harmonic fluctuations of initial tension and axially velocity, a nonlinear governing equation for transverse vibration of an axially accelerating string is set up by using the equation of motion for a 3-dimensional deformable body with initial stresses. The Kelvin model is used to describe viscoelastic behaviors of the material. The basis function of the complex-mode Galerkin method for axially accelerating nonlinear strings is constructed by using the modal function of linear moving strings with constant axially transport velocity. By the constructed basis functions, the application of the complex-mode Galerkin method in nonlinear vibration analysis of an axially accelerating viscoelastic string is investigated. Numerical results show that the convergence velocity of the complex-mode Galerkin method is higher than that of the real-mode Galerkin method for a variable coefficient gyroscopic system.
TL;DR: In this paper, strong convergence theorems of explicit composite iteration scheme for nonexpansive semi-groups in the framework of Banach spaces are established, and results presented in this paper not only extend and improve the corresponding results of Shioji-Takahashi, Suzuki, Xu and Aleyner-Reich, but also give a partially affirmative answer to the open questions raised by Suzuki and Xu.
Abstract: Some strong convergence theorems of explicit composite iteration scheme for nonexpansive semi-groups in the framework of Banach spaces are established. Results presented in the paper not only extend and improve the corresponding results of Shioji-Takahashi, Suzuki, Xu and Aleyner-Reich, but also give a partially affirmative answer to the open questions raised by Suzuki and Xu.
TL;DR: In this paper, large eddy simulations of nanoparticle coagulation in an incompressible planar jet were performed and the time-averaged results based on 3000 time steps for every case were obtained to explore the influence of the Schmidt number and Damkohler number on the nanoparticle dynamics.
Abstract: Large eddy simulations of nanoparticle coagulation in an incompressible planar jet were performed. The particle is described using a moment method to approximate the particle general dynamics equations. The time-averaged results based on 3000 time steps for every case were obtained to explore the influence of the Schmidt number and the Damkohler number on the nanoparticle dynamics. The results show that the changes of Schmidt number have the influence on the number concentration of nanoparticles only when the particle diameter is less than 1 nm for the fixed gas parameters. The number concentration of particles for small particles decreases more rapidly along the flow direction, and the nanoparticles with larger Schmidt number have a narrower distribution along the transverse direction. The smaller nanoparticles coagulate and disperse easily, grow rapidly hence show a stronger polydispersity. The smaller coagulation time scale can enhance the particle collision and coagulation. Frequented collision and coagulation bring a great increase in particle size. The large the Damkohler number is, the higher the particle polydispersity is.
TL;DR: Based on the precise integration method, an algorithm of symplectic conservative perturbation is presented in this paper, which gives a uniform way to solve the linear quadratic control (LQ control) problems for linear time-varying systems accurately and efficiently, whose key points are solutions of differential Riccati equation (DRE) with variable coefficients and the state feedback equation.
Abstract: Optimal control system of state space is a conservative system, whose approximate method should be symplectic conservation. Based on the precise integration method, an algorithm of symplectic conservative perturbation is presented. It gives a uniform way to solve the linear quadratic control (LQ control) problems for linear time-varying systems accurately and efficiently, whose key points are solutions of differential Riccati equation (DRE) with variable coefficients and the state feedback equation. The method is symplectic conservative and has a good numerical stability and high precision. Numerical examples demonstrate the effectiveness of the proposed method.
TL;DR: In this article, a quasi-linear theory was developed using the eigenfunction expansion method for the interaction between regular surface waves and a surface-pitching slotted barrier, where the energy dissipation within the barriers was modeled by a quadratic friction factor, and an equivalent linear dissipation coefficient was introduced to linearize the matching condition at the surface pitching barrier.
Abstract: The interactions between regular surface waves and a surface-pitching slotted barrier are investigated both analytically and experimentally. A quasi-linear theory is developed using the eigenfunction expansion method. The energy dissipation within the barriers is modeled by a quadratic friction factor, and an equivalent linear dissipation coefficient, which is depth-varying, wave-height dependent, is introduced to linearize the matching condition at the surface-pitching barrier. By comparing the theoretical results with laboratory experiments, it is shown that the present method can satisfactorily predict the variation of the reflection and transmission coefficients with wave height.
TL;DR: In this paper, the authors apply a Hamiltonian method to study the stress distributions of orthotropic two-dimensional elasticity in (x, z) plane for arbitrary boundary conditions without beam assumptions.
Abstract: This paper applies a Hamiltonian method to study analytically the stress distributions of orthotropic two-dimensional elasticity in (x, z) plane for arbitrary boundary conditions without beam assumptions It is a method of separable variables for partial differential equations using displacements and their conjugate stresses as unknowns Since coordinates (x, z) can not be easily separated, an alternative symplectic expansion is used Similar to the Hamiltonian formulation in classical dynamics, we treat the x coordinate as time variable so that z becomes the only independent coordinate in the Hamiltonian matrix differential operator The exponential of the Hamiltonian matrix is symplectic There are homogenous solutions with constants to be determined by the boundary conditions and particular integrals satisfying the loading conditions The homogenous solutions consist of the eigen-solutions of the derogatory zero eigenvalues (zero eigen-solutions) and that of the well-behaved nonzero eigenvalues (nonzero eigen-solutions) The Jordan chains at zero eigenvalues give the classical Saint-Venant solutions associated with averaged global behaviors such as rigid-body translation, rigid-body rotation or bending On the other hand, the nonzero eigen-solutions describe the exponentially decaying localized solutions usually ignored by Saint-Venant’s principle Completed numerical examples are newly given to compare with established results
TL;DR: In this paper, the dynamic interaction of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips subjected to the anti-plane shear harmonic stress waves was investigated.
Abstract: The dynamic interaction of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips subjected to the anti-plane shear harmonic stress waves was investigated. By using the Fourier transform, the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is jump of displacement across the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter, the circular frequency of the incident waves and the thickness of the strip upon stress, electric displacement and magnetic flux intensity factors of cracks.
TL;DR: In this article, a hyperbolic function is introduced to reflect the attenuation effect of one firm's default to its partner, and the fair swap premium of a credit default swap (CDS) can be valued.
Abstract: A hyperbolic function is introduced to reflect the attenuation effect of one firm’s default to its partner. If two firms are competitors (copartners), the default intensity of one firm will decrease (increase) abruptly when the other firm defaults. As time goes on, the impact will decrease gradually until extinct. In this model, the joint distribution and marginal distributions of default times are derived by employing the change of measure, and the fair swap premium of a credit default swap (CDS) can be valued.
TL;DR: In this article, the exact location and size of fin in such a way that a minimal pressure drop coincides with an optimal heat transfer based on the genetic algorithm is defined. But, this heat transfer enhancement is associated with an increase in the pressure drop, which leads to an increased pumping power requirement so that one may seek an optimum design for such systems.
Abstract: Compared to a smooth channel, a finned channel provides a higher heat transfer coefficient; increasing the fin height enhances the heat transfer. However, this heat transfer enhancement is associated with an increase in the pressure drop. This leads to an increased pumping power requirement so that one may seek an optimum design for such systems. The main goal of this paper is to define the exact location and size of fins in such a way that a minimal pressure drop coincides with an optimal heat transfer based on the genetic algorithm. Each fin arrangement is considered a solution to the problem (an individual for genetic algorithm). An initial population is generated randomly at the first step. Then the algorithm has been searched among these solutions and made new solutions iteratively by its functions to find an optimum design as reported in this article.
TL;DR: In this paper, a mathematical model for nonlinear flow in micro-scale pore of saturated clays was presented, which can describe characteristics of flow curve of the whole process of the non-linear flow from low hydraulic gradient to high one.
Abstract: It was derived that micro-scale amount level of average pore radius of clay changed from 0.01 to 0.1 micron by an equivalent concept of flow in porous media. There is good agreement between the derived results and test ones. Results of experiments show that flow in micro-scale pore of saturated clays follows law of nonlinear flow. Theoretical analyses demonstrate that an interaction of solid-liquid interfaces varies inversely with permeability or porous radius. The interaction is an important reason why nonlinear flow in saturated clays occurs. An exact mathematical model was presented for nonlinear flow in micro-scale pore of saturated clays. Dimension and physical meanings of parameters of it are definite. A new law of nonlinear flow in saturated clays was established. It can describe characteristics of flow curve of the whole process of the nonlinear flow from low hydraulic gradient to high one. Darcy law is a special case of the new law. A mathematical model was presented for consolidation of nonlinear flow in radius direction in saturated clays with constant rate based on the new law of nonlinear flow. Equations of average mass conservation and moving boundary, and formula of excess pore pressure distribution and average degree of consolidation for nonlinear flow in saturated clay were derived by using an idea of viscous boundary layer, a method of steady state in stead of transient state and a method of integral of an equation. Laws of excess pore pressure distribution and changes of average degree of consolidation with time were obtained. Results show that velocity of moving boundary decreases because of the nonlinear flow in saturated clay. The results can provide geology engineering and geotechnical engineering of saturated clay with new scientific bases. Calculations of average degree of consolidation of the Darcy flow are a special case of that of the nonlinear flow.
TL;DR: In this article, the authors clarified the idea of Green quasifunction method by considering a vibration problem of simply-supported thin polygonic plates on Pasternak foundation and established the fundamental solution and boundary equation of the problem.
Abstract: A new numerical method—Green quasifunction is proposed The idea of Green quasifunction method is clarified in detail by considering a vibration problem of simply-supported thin polygonic plates on Pasternak foundation A Green quasifunction is established by using the fundamental solution and boundary equation of the problem This function satisfies the homogeneous boundary condition of the problem The mode shape differential equation of the vibration problem of simply-supported thin plates on Pasternak foundation is reduced to two simultaneous Fredholm integral equations of the second kind by Green formula There are multiple choices for the normalized boundary equation Based on a chosen normalized boundary equation, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome Finally, natural frequency is obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations Numerical results show high accuracy of the Green quasifunction method