Convergence of a thin spherical shell

2007 ◽  
Vol 5 ◽  
pp. 307-323
Author(s):  
S.V. Khabirov ◽  
A.R. Garifullin
Keyword(s):  
Spherical Shell ◽  
Dynamic Stability ◽  
Equations Of Motion ◽  
Asymptotic Formulas ◽  
Precise Solution ◽  
Solution Of Equations ◽  
Initial Stage ◽  
Thin Spherical Shell

The precise solution of equations of motion of a true liquid describing a convergence to the center of a thin spherical shell is considered. At the initial stage prime asymptotic formulas which are used for studying of a dynamic stability in relation to potential indignations are received.

Free Vibration of Thin Spherical Shells

2017 ◽  
Vol 139 (6) ◽  
Author(s):  
April Bryan
Keyword(s):  
Differential Equation ◽  
Spherical Shell ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Transverse Motion ◽  
Hemispherical Shell ◽  
Surface Displacements ◽  
Thin Spherical Shell ◽  
The One ◽  
The Relationship

This research introduces a new approach to analytically derive the differential equations of motion of a thin spherical shell. The approach presented is used to obtain an expression for the relationship between the transverse and surface displacements of the shell. This relationship, which is more explicit than the one that can be obtained through use of the Airy stress function, is used to uncouple the surface and normal displacements in the spatial differential equation for transverse motion. The associated Legendre polynomials are utilized to obtain analytical solutions for the resulting spatial differential equation. The spatial solutions are found to exactly satisfy the boundary conditions for the simply supported and the clamped hemispherical shell. The results to the equations of motion indicate that the eigenfrequencies of the thin spherical shell are independent of the azimuthal coordinate. As a result, there are several mode shapes for each eigenfrequency. The results also indicate that the effects of midsurface tensions are more significant than bending at low mode numbers but become negligible as the mode number increases.

A fast approach for acoustic source localization on a thin spherical shell

2021 ◽  
pp. 147592172110419
Author(s):  
Zixian Zhou ◽  
Zhiwen Cui ◽  
Tribikram Kundu
Keyword(s):  
Velocity Profile ◽  
Spherical Shell ◽  
Source Localization ◽  
Pressure Vessels ◽  
Solution Technique ◽  
Acoustic Source ◽  
Spherical Shells ◽  
Fast Approach ◽  
Acoustic Source Localization ◽  
Thin Spherical Shell

Thin spherical shell structures are wildly used as pressure vessels in the industry because of their property of having equal in-plane normal stresses in all directions. Since very large pressure difference between the inside and outside of the wall exists, any formation of defects in the pressure vessel wall has a huge safety risk. Therefore, it is necessary to quickly locate the area where the defect maybe located in the early stage of defect formation and make repair on time. The conventional acoustic source localization techniques for spherical shells require either direction-dependent velocity profile knowledge or a large number of sensors to form an array. In this study, we propose a fast approach for acoustic source localization on thin isotropic and anisotropic spherical shells. A solution technique based on the time difference of arrival on a thin spherical shell without the prior knowledge of direction-dependent velocity profile is provided. With the help of “L”-shaped sensor clusters, only 6 sensors are required to quickly predict the acoustic source location for anisotropic spherical shells. For isotropic spherical shells, only 4 sensors are required. Simulation and experimental results show that this technique works well for both isotropic and anisotropic spherical shells.

An Emerging O(N) Model and Algorithm for Virtual Prototyping of Dynamics of Molecular Conformation

2008 ◽  
Author(s):  
Andrew Ries ◽  
Shanzhong Shawn Duan
Keyword(s):  
Degrees Of Freedom ◽  
Molecular Conformation ◽  
Molecular System ◽  
Equations Of Motion ◽  
Integration Step ◽  
Computational Costs ◽  
System A ◽  
Hard Constraints ◽  
Solution Of Equations ◽  
Computationally Intensive

Molecular dynamics is effective for nano-scale phenomenon analysis. There are two major computational steps associated with computer simulation of dynamics of molecular conformation and they are the calculation of the interatomic forces and the formation and solution of the equations of motion. Currently, these two computational steps are treated separately, but in this paper an O(N) (order N) procedure is presented for an integration between these computational steps. For computational costs associated with calculating the interatomic forces, an internal coordinate method (ICM) approach is used for determining potentials due to both the bonding and non-bonding interactions. Thus, the potential gradients can be expressed as a combination of the potential in absolute and relative coordinates. For computational costs associated with the formation and solution of the equations of motion for the system, a constraint method that is used in computational multibody dynamics is utilized. This frees some degrees of freedom so that Kane’s method can be applied for the recursive formation and solution of equations of motion for the atomistic molecular system. Because the inclusion of lightly excited high frequency degrees of freedom, such as inter-atomic oscillations and rotation about double bonds would force the use of very small integration step sizes, holonomic constraints are introduced to freeze these “uninteresting” degrees of freedom. By introducing these hard constraints the time scale can be appropriately sized for to provide a less computationally intensive dynamic simulation of molecular conformation. The algorithm developed improves computational speed significantly when compared with any traditional O(N3) procedure.

Impulse response for backscattering by a thin spherical shell: Measurement and wave interpretation

1993 ◽  
Vol 94 (3) ◽  
pp. 1877-1877
Author(s):  
Gregory Kaduchak ◽  
Christopher S. Kwiatkowski ◽  
Philip L. Marston
Keyword(s):  
Spherical Shell ◽  
Impulse Response ◽  
Thin Spherical Shell
Sign in / Sign up

Export Citation Format

Share Document