Extensional Vibrations of Elastic Orthotropic Spherical Shells

1961 ◽  
Vol 28 (2) ◽  
pp. 229-237 ◽  
Author(s):  
W. H. Hoppmann ◽  
W. E. Baker
Keyword(s):  
Orthotropic Material ◽  
Orthotropic Shell ◽  
Equations Of Motion ◽  
Special Consideration ◽  
Uniform Thickness ◽  
Spherical Shells ◽  
Isotropic Shell ◽  
Principal Directions ◽  
Limiting Case ◽  
Orthotropic Shells

The extensional vibrations (momentless) of spherical shells of elastic orthotropic material have been studied theoretically. Equations of motion have been derived and solved. The principal directions of the elastic compliances are assumed to be along parallels of latitude and along meridians. In addition to the case of orthotropic shells of uniform thickness, the analysis may be applied in the case of shells with stiffeners attached. Special consideration is given to the isotropic shell as a limiting case of the orthotropic shell.

scholarly journals Nonlinear Deformation of Flexible Orthotropic Shells of Variable Thickness in an Unsteady Magnetic Field

2019 ◽  
Vol 97 ◽  
pp. 02015 ◽  
Author(s):  
Zafar Abdullaev ◽  
Sayibdjan Mirzaev ◽  
Sobir Mavlanov
Keyword(s):  
Magnetic Field ◽  
Stress State ◽  
Electric Current ◽  
Variable Thickness ◽  
Nonlinear Deformation ◽  
Strain State ◽  
Orthotropic Shell ◽  
Mechanical Force ◽  
Time Varying ◽  
Orthotropic Shells

The analysis of the stress state of a flexible orthotropic shell under the influence of a time-varying mechanical force and a time-varying external electric current is performed, taking into account the mechanical and electromagnetic orthotropy. The effect of thickness on the stress-strain state of the orthotropic shell is investigated. The results obtained indicate the influence of thickness on the deformation of the shell and the need to take this factor into account in the design schemes.

scholarly journals A New Analytical Method for Spherical Thin Shells’ Axisymmetric Vibrations

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Mariame Nassit ◽  
Abderrahmane El Harif ◽  
Hassan Berbia ◽  
Mourad Taha Janan
Keyword(s):  
Analytical Method ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Current Method ◽  
Free Edge ◽  
Free Vibrations ◽  
Thin Shells ◽  
Element Calculation ◽  
Spherical Shells ◽  
Clamped Edge

In order to improve the spherical thin shells’ vibrations analysis, we introduce a new analytical method. In this method, we take into consideration the terms of the inertial couples in the stress couples’ differential equations of motion. These inertial couples are omitted in the theories provided by Naghdi–Kalnins and Kunieda. The results show that the current method can solve the axisymmetric vibrations’ equations of elastic thin spherical shells. In this paper, we focus on verifying the current method, particularly for free vibrations with free edge and clamped edge boundary conditions. To check the validity and accuracy of the current analytical method, the natural frequencies determined by this method are compared with those available in the literature and those obtained by a finite element calculation.

On a Laminated Orthotropic Shell Theory Including Transverse Shear Deformation

1972 ◽  
Vol 39 (4) ◽  
pp. 1091-1097 ◽  
Author(s):  
S. B. Dong ◽  
F. K. W. Tso
Keyword(s):  
Shear Deformation ◽  
Transverse Shear ◽  
Shell Theory ◽  
Orthotropic Shell ◽  
Plane Waves ◽  
Present Theory ◽  
Transverse Shear Deformation ◽  
Correction Factors ◽  
Natural Oscillations ◽  
Orthotropic Shells

A constitutive relation for laminated orthotropic shells which includes transverse shear deformation is presented. This relation involves composite correction factors k112, k222 which are determined from an analysis of plane waves in a plate with the same layered construction. The range of applicability of the present theory and the quantitative effect of transverse shear deformation are evinced in a problem concerned with the natural oscillations of a three-layered freely supported cylinder.

Fractional Derivative Viscoelastic Model for the Analysis of Two Colliding Spherical Shells

2012 ◽  
Author(s):  
Yury A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Fractional Derivative ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Wave Fronts ◽  
Spherical Shells ◽  
Reflected Waves ◽  
Standard Linear Solid ◽  
Standard Linear ◽  
The Moment ◽  
Contact Domain

The collision of two isotropic spherical shells is investigated for the case when the viscoelastic features of the shells represent themselves only in the place of contact and are governed by the standard linear solid model with fractional derivatives. Thus, the problem concerns the shock interaction of two shells, wherein the generalized fractional-derivative standard linear law instead of the Hertz contact law is employed as a low of interaction. The pans of the shells beyond the contact domain are assumed to be elastic, and their behavior is described by the equations of motion which take rotary inertia and shear deformations into account. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. Due to the short duration of contact interaction, the reflected waves are not taken into account. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. To determine the desired values behind the wave fronts, one-term ray expansions are used, as well as the equations of motion of the contact domains for the both spherical shells.

scholarly journals A non-conventional discontinuous Lagrangian for viscous flow

2017 ◽  
Vol 4 (2) ◽  
pp. 160447 ◽  
Author(s):  
M. Scholle ◽  
F. Marner
Keyword(s):  
Variational Formulation ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Local Equilibrium ◽  
Navier Stokes ◽  
Mathematical Treatment ◽  
Navier Stokes Equations ◽  
New Formalism ◽  
Limiting Case ◽  
Physical Phenomena

Drawing an analogy with quantum mechanics, a new Lagrangian is proposed for a variational formulation of the Navier–Stokes equations which to-date has remained elusive. A key feature is that the resulting Lagrangian is discontinuous in nature, posing additional challenges apropos the mathematical treatment of the related variational problem, all of which are resolvable. In addition to extending Lagrange's formalism to problems involving discontinuous behaviour, it is demonstrated that the associated equations of motion can self-consistently be interpreted within the framework of thermodynamics beyond local equilibrium, with the limiting case recovering the classical Navier–Stokes equations. Perspectives for applying the new formalism to discontinuous physical phenomena such as phase and grain boundaries, shock waves and flame fronts are provided.

Stability of spiral motion

1966 ◽  
Vol 25 ◽  
pp. 34-42
Author(s):  
V. Szebehely
Keyword(s):  
Potential Function ◽  
Complete Solution ◽  
Equations Of Motion ◽  
Variable Coefficients ◽  
Order System ◽  
System Of Differential Equations ◽  
Linearized Stability ◽  
Logarithmic Spirals ◽  
The Stability ◽  
Limiting Case

The stability of the motion of particles in spiral orbits is investigated. The force field is represented by the potential functionf=C(expaφ)/rnlimiting case of which is the Newtonian gravitational field (n= 1,a= 0). Particular solutions of the equations of motion are shown to be logarithmic spirals. A linearized stability analysis leads to a fourth order system of differential equations with variable coefficients. Complete solution of this system is obtained showing a rather complex stability situation in which outward motions are always associated with instability while inward motions might be stable or unstable depending on the value of the constantsaandnoccurring in the above potential function.

Design of Sandwich Panels With Prismatic Cores

2005 ◽  
Vol 128 (2) ◽  
pp. 186-192 ◽  
Author(s):  
Z. Wei ◽  
F. W. Zok ◽  
A. G. Evans
Keyword(s):  
Load Capacity ◽  
Sandwich Panels ◽  
High Load ◽  
Optimal Designs ◽  
Honeycomb Core ◽  
Longitudinal Loading ◽  
Lower Load ◽  
Bending Loads ◽  
Principal Directions ◽  
Limiting Case

The paper focuses on optimization of lightweight sandwich panels with prismatic cores subject to bending loads in the two principal in-plane directions. Comparisons are made with optimal designs of panels with corrugated cores: a limiting case. When optimized for loading transverse to the prism axis, prismatic panels outperform those with corrugated cores, especially at lower loads. In contrast, when optimized for longitudinal loading, the corrugated core panel is always superior. Both panels exhibit significant anisotropy: a deficiency mediated by optimizing jointly for both orientations. The designs emerging from joint optimizations have only slightly lower load capacity than those optimized singly, but with the benefit of equal strengths in the two principal directions. Moreover, jointly optimized corrugated and prismatic panels perform equally well. Both are competitive with honeycomb core panels, especially at high load capacities. With the additional potential for multifunctionality (notably active cooling), the corrugated panels appear to be particularly promising thermostructural elements.

General Instability of a Ring-Stiffened, Circular Cylindrical Shell Under Hydrostatic Pressure

1957 ◽  
Vol 24 (2) ◽  
pp. 269-277
Author(s):  
S. R. Bodner
Keyword(s):  
Cylindrical Shell ◽  
Hydrostatic Pressure ◽  
Differential Equations ◽  
Energy Method ◽  
Circular Cylindrical Shell ◽  
Orthotropic Shell ◽  
Simple Expression ◽  
Isotropic Shell ◽  
Numerical Examples ◽  
The Stability

Abstract The general instability load of a ring-stiffened, circular cylindrical shell under hydrostatic pressure is determined by analyzing an equivalent orthotropic shell. A set of differential equations for the stability of an orthotropic shell is derived and solved for the case of a shell with simple end supports. The solution is presented in terms of parameters of the ring-stiffened, isotropic shell, and a relatively simple expression for the general instability load is obtained. Some numerical examples and graphs of results are presented. In addition, an energy-method solution to the problem is outlined, and the energy and displacement functions that could be used in carrying out a Rayleigh-Ritz approximation are indicated.

Limits of Economy of Material in Plates

1955 ◽  
Vol 22 (3) ◽  
pp. 372-374
Author(s):  
H. G. Hopkins ◽  
W. Prager
Keyword(s):  
Yield Condition ◽  
Transverse Load ◽  
Constant Thickness ◽  
Flow Rule ◽  
Plate Material ◽  
Uniform Thickness ◽  
Simply Supported ◽  
Varying Thickness ◽  
Limiting Case ◽  
Associated Flow Rule

Abstract The paper is concerned with the limits of economy of material in a simply supported circular plate under a uniformly distributed transverse load. The plate material is supposed to be plastic-rigid and to obey Tresca’s yield condition and the associated flow rule. The criterion of failure adopted is that used in limit analysis. It is shown that the plate of uniform thickness has a weight efficiency of about 82 per cent. Stepped plates of segmentwise constant thickness are discussed, and the plate of continuously varying thickness is treated as the limiting case obtained by letting the number of steps go to infinity.

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