approximate analytical solutions
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2022 ◽  
pp. 108128652110729
Marina V Shitikova

In this paper, we consider the problem on a transverse impact of a viscoelastic sphere upon a viscoelastic shallow doubly curved shell with rectangular platform, the viscoelastic features of which are defined via the fractional derivative standard linear solid models; in so doing, only Young’s time-dependent operators are preassigned, while the bulk moduli are considered to be constant values, since the bulk relaxation for the majority of materials is far less than the shear relaxation. Shallow panel’s displacement subjected to the concentrated contact force is found by the method of expansion in terms of eigen functions, and the sphere’s displacement under the action of the contact force, which is the sum of the shell’s displacement at the place of contact and local bearing of impactor and target’s materials, is defined from the equation of motion of the material point with the mass equal to sphere’s mass. Within the contact domain, the contact force is defined by the modified Hertzian contact law with the time-dependent rigidity function. For decoding the viscoelastic operators involving the problem under consideration, the algebra of Rabotnov’s fractional operators is employed. A nonlinear integro-differential equation is obtained either in terms of the contact force or in the local bearing of the target and impactor materials. Using the duration of contact as a small parameter, approximate analytical solutions have been found, which allow one to define the key characteristics of impact process.

2022 ◽  
Vol 12 (1) ◽  
Nehad Ali Shah ◽  
Hussam Alrabaiah ◽  
Dumitru Vieru ◽  
Se-Jin Yook

AbstractThe unsteady, magneto-hydrodynamic generalized Couette flows of two immiscible fluids in a rectangular channel with isothermal walls under the influence of an inclined magnetic field and an axial electric field have been investigated. Both fluids are considered electrically conducting and the solid boundaries are electrically insulated. Approximate analytical solutions for the velocity, induced magnetic, and temperature fields have been determined using the Laplace transform method along with the numerical Stehfest's algorithm for the inversion of the Laplace transforms. Also, for the nonlinear differential equation of energy, a numerical scheme based on the finite differences has been developed. A particular case has been numerically and graphically studied to show the evolution of the fluid velocity, induced magnetic field, and viscous dissipation in both flow regions.

Guillermo Abramson

In this paper, we analyze a model of an animal feeding on fruits and dispersing their seeds, which are later deposited and capable of germination. The deposition of seeds away from their collection sites produces a delay in the dynamics, whose analytical modeling complicates the description in a way that would delight Nitant. We have found approximate analytical solutions, as well as numerical ones, but here I will emphasize on a typical physicists rule of thumb argument.

Pinxia Wu ◽  
Weiwei Ling ◽  
Xiumei Li ◽  
Xichun He ◽  
Liangjin Xie

In this paper, we mainly focus on a fractal model of Fangzhu’s nanoscale surface for water collection which is established through He’s fractal derivative. Based on the fractal two-scale transform method, the approximate analytical solutions are obtained by the energy balance method and He’s frequency–amplitude formulation method with average residuals. Some specific numerical experiments of the model show that these two methods are simple and effective and can be adopted to other nonlinear fractal oscillators. In addition, these properties of the obtained solution reveal how to enhance the collection rate of Fangzhu by adjusting the smoothness of its surfaces.

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Sajad Iqbal ◽  
Mohammed K. A. Kaabar ◽  
Francisco Martínez

In this article, the approximate analytical solutions of four different types of conformable partial differential equations are investigated. First, the conformable Laplace transform homotopy perturbation method is reformulated. Then, the approximate analytical solution of four types of conformable partial differential equations is presented via the proposed technique. To check the accuracy of the proposed technique, the numerical and exact solutions are compared with each other. From this comparison, we conclude that the proposed technique is very efficient and easy to apply to various types of conformable partial differential equations.

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2397
Rui Pan ◽  
Baofeng Zheng ◽  
Ying Qin

Nowadays, it is common to see large public buildings, e.g., stadiums, with some equipment or substructure suspended from the center of the roof. These substructures will tend to be larger and heavier the more gear is needed, which may have negative impacts on the dynamic performance of the roof structures. In this paper, to explore the dynamic response of a large-span roof structure with a suspended substructure, a real structure model is simplified into a two-degrees-of-freedom system. The essential consideration of nonlinear vibration is elaborated in the equations of motions. Approximate analytical solutions for free and forced vibrations are derived using perturbation methods, while numerical analysis is carried out to validate the solutions. The ratio of linear to nonlinear amplitude is proposed to represent the nonlinear effect of the primary structure, and the nonlinear effect, varying with structural parameters of frequency ratio, mass ratio, excitation ratio, and external force to the primary structure, is investigated. It is shown that internal resonance occurs when the structural frequency ratio is close to 1:2 and that secondary resonance takes place due to certain external excitations; internal resonance and secondary resonance will magnify the amplitude of the primary structure during vibration. Finally, a case of a designed practical dome with a suspended substructure is studied to verify the outcomes from the above research. According to these findings, some design proposals for this type of structure are provided.

Hoang Van Tung ◽  
Dao Nhu Mai ◽  
Vu Thanh Long

An analytical investigation on the nonlinear response of doubly curved panels constructed from homogeneous face sheets and carbon nanotube reinforced composite (CNTRC) core and subjected to external pressure in thermal environments is presented in this paper. Carbon nanotubes (CNTs) are reinforced into the core layer through uniform or functionally graded distributions. The properties of constituents are assumed to be temperature dependent and effective properties of CNTRC are determined using an extended rule of mixture. Governing equations are established within the framework of first order shear deformation theory taking into account geometrical imperfection, von Kármán–Donnell nonlinearity, panel-foundation interaction and elasticity of tangential edge restraints. These equations are solved using approximate analytical solutions and Galerkin method for simply supported panels. The results reveal that load carrying capacity of sandwich panels is stronger when boundary edges are more rigorously restrained and face sheets are thicker. Furthermore, elevated temperature has deteriorative and beneficial influences on the load bearing capability of sandwich panels with movable and restrained edges, respectively.

2021 ◽  
Vol 0 (0) ◽  
Gorakh Nath

Abstract This paper presents the development of mathematical model to obtain the approximate analytical solutions for isothermal flows behind the strong shock (blast) wave in a van der Waals gas and small solid particles mixture. The small solid particles are continuously distributed in the mixture and the equilibrium conditions for flow are maintained. To derive the analytical solutions, the physical variables such as density, pressure, and velocity are expanded using perturbation method in power series. The solutions are derived in analytical form for first approximation, and for second order approximation the set of differential equations are also obtained. The effects of an increase in the problem parameters value on the physical variables are investigated for first order approximation. A comparison is also, made between the solution of cylindrical shock and spherical shock. It is found that the fluid density and fluid pressure become zero near the point or axis of symmetry in spherical or cylindrical symmetry, respectively, and therefore a vacuum is created near the point or axis of symmetry which is in tremendous conformity with the physical condition in laboratory to generate the shock wave.

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