Nonlinear oscillations in a constant gravitational field

2021 ◽  
Author(s):  
Stylianos - Vasileios Kontomaris ◽  
Anna Malamou
Keyword(s):  
Gravitational Field ◽  
Numerical Methods ◽  
Differential Equations ◽  
Half Space ◽  
Nonlinear Oscillations ◽  
Elastic Half Space ◽  
Rigid Sphere ◽  
Elementary Functions ◽  
Approximate Expressions ◽  
Special Case

Abstract Exploring non-linear oscillations is a challenging task since the related differential equations cannot be directly solved in terms of elementary functions. Thus, complicated mathematical or numerical methods are usually employed to find accurate or approximate expressions that describe the behavior of the system with respect to time. In this paper, the vertical oscillations of an object under the influence of its weight and an opposite force with magnitude F=cyn, where n>0 are being explored. Accurate and approximate simple solutions regarding the object’s position with respect to time are presented and the dependence of the oscillation’s period from the oscillation’s range of displacements and the exponent n is revealed. In addition, the special case in which n=3/2 (which describes the oscillation of a rigid sphere on an elastic half space) is also highlighted. Lastly, it is shown that similar cases (such as the case of a force with magnitude F=kx+λx2) can be also treated using the same approach.

Axisymmetric contact with friction of a rigid sphere with an elastic half-space

2009 ◽  
Vol 465 (2108) ◽  
pp. 2565-2588 ◽  
Author(s):  
O. I. Zhupanska
Keyword(s):  
Contact Problem ◽  
Half Space ◽  
Integral Transform ◽  
Elastic Half Space ◽  
Rigid Sphere ◽  
Analytical Treatment ◽  
Normal Contact ◽  
Exact Analytical Solutions ◽  
Single Integral ◽  
Toroidal Coordinates

The problem of normal contact with friction of a rigid sphere with an elastic half-space is considered. An analytical treatment of the problem is presented, with the corresponding boundary-value problem formulated in the toroidal coordinates. A general solution in the form of Papkovich–Neuber functions and the Mehler–Fock integral transform is used to reduce the problem to a single integral equation with respect to the unknown contact pressure in the slip zone. An analysis of contact stresses is carried out, and exact analytical solutions are obtained in limiting cases, including a full stick contact problem and a contact problem for an incompressible half-space.

Groove Bottom Contours of Minimum Stress Concentration for Antiplane Shear Deformation

1985 ◽  
Vol 52 (2) ◽  
pp. 379-384 ◽  
Author(s):  
B. H. Eldiwany ◽  
L. T. Wheeler
Keyword(s):  
Stress Concentration ◽  
Numerical Methods ◽  
Shear Deformation ◽  
Half Space ◽  
Optimization Problems ◽  
Integral Form ◽  
Antiplane Shear ◽  
Space Boundary ◽  
Minimum Stress ◽  
Special Case

Results from free streamline hydrodynamics are exploited in order to solve optimization problems for antiplane shear deformation, in which the stress concentration is to be minimized. These problems pertain to the optimum shapes for grooves cut into a half-space. We obtain results, which from the standpoint of the hydrodynamics problem, complement those presently in the literature. The solution is given in an integral form which in general must be evaluated by numerical methods, but that reduces to elliptic integrals for the special case of a notch whose faces meet the half-space boundary at right angles.

scholarly journals Near-Resonant Regimes of a Moving Load on a Pre-Stressed Incompressible Elastic Half-Space

2021 ◽  
Vol 15 (1) ◽  
pp. 30-36
Author(s):  
Askar Kudaibergenov ◽  
Askat Kudaibergenov ◽  
Danila Prikazchikov
Keyword(s):  
Surface Wave ◽  
Half Space ◽  
Moving Load ◽  
Elastic Half Space ◽  
Elementary Functions ◽  
Material Models ◽  
Line Load ◽  
Transient Regimes ◽  
Surface Wave Field ◽  
Asymptotic Formulation

Abstract The article is concerned with the analysis of the problem for a concentrated line load moving at a constant speed along the surface of a pre-stressed, incompressible, isotropic elastic half-space, within the framework of the plane-strain assumption. The focus is on the near-critical regimes, when the speed of the load is close to that of the surface wave. Both steady-state and transient regimes are considered. Implementation of the hyperbolic–elliptic asymptotic formulation for the surface wave field allows explicit approximate solution for displacement components expressed in terms of the elementary functions, highlighting the resonant nature of the surface wave. Numerical illustrations of the solutions are presented for several material models.

A concave indenter on a piezoelectromagneto-elastic substrate or a layer elastically supported

2012 ◽  
Vol 47 (6) ◽  
pp. 362-378 ◽  
Author(s):  
Bogdan Rogowski
Keyword(s):  
Half Space ◽  
Elastic Foundation ◽  
Contact Region ◽  
Approximate Solutions ◽  
Elastic Half Space ◽  
Solution Technique ◽  
Elementary Functions ◽  
Full Field ◽  
Full Contact ◽  
Two Parameter

Indentation of piezoelectromagneto-elastic half-space or a layer on a two-parameter elastic foundation by a cylindrical indenter with a slightly concave base is considered. Full-field magnetoelectro-elastic solutions in elementary functions are obtained for the case of full contact and half-space. If the axial load is small, the contact area will be an annulus the outer circumference of which coincides with the edge of the punch. The inner circumference will shrink with increasing load and there will be a critical load above which the stratum makes contact with the entire punch base. The contact problem for high loads can therefore be treated by classical methods. The more interested case in which the load is less the critical value and the contact region is annulus remains. By use the methods of triple integral equations and series solution technique the solution for an indentured substrate over an annular contact region is also given. For parabolic and conical concave punches the exact or approximate solutions are obtained for full contact or annular contact region, respectively. For the layer on two-parameter elastic foundation and concave punch approximate solution is established.

The Interior Stress Field Caused by Tangential Loading of a Rectangular Patch on an Elastic Half Space

1987 ◽  
Vol 109 (4) ◽  
pp. 627-629 ◽  
Author(s):  
N. Ahmadi ◽  
L. M. Keer ◽  
T. Mura ◽  
V. Vithoontien
Keyword(s):  
Stress Field ◽  
Internal Stress ◽  
Half Space ◽  
Elastic Half Space ◽  
Vertical Load ◽  
Elementary Functions ◽  
Rectangular Patch ◽  
Internal Stress Field ◽  
Interior Stress

A solution is obtained for the tangential loading on a rectangular patch. The solution gives the internal stress field in terms of elementary functions and is a form analogous to the solution for vertical load developed by Love.

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