The equations of motion of micropolar elastic shells in cylindrical Eulerian coordinates

2017 ◽  
pp. 193-196
Author(s):  
L.M. Zubov
Keyword(s):  
Equations Of Motion ◽  
Elastic Shells ◽  
Eulerian Coordinates

scholarly journals VI. On the extension and flexure of cylindrical and spherical thin elastic shells

1890 ◽  
Vol 181 ◽  
pp. 433-480 ◽  
Keyword(s):  
Potential Energy ◽  
Natural Philosophy ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Middle Surface ◽  
Curvilinear Coordinates ◽  
Elastic Shells ◽  
Plane Plate ◽  
The Subject ◽  
Cursory Examination

The various theories of thin elastic shells which have hitherto been proposed have been discussed by Mr. Love in a recent memoir, and it appears that most, it not all of them, depend upon the assumption that the three stresses which are usually denoted by R, S, T are zero; but, as I have recently pointed out, a very cursory examination of the subject is sufficient to show that this assumption cannot be rigorously true. It can, however, be proved that, when the external surfaces of a plane plate are not sub­jected to pressure or tangential stress, these stresses depend upon quantities propor­tional to the square of the thickness, and whenever this is the case they may be treated as zero in calculating the expression for the potential energy due to strain, because they give rise to terms proportional to the fifth power of the thickness, which may be neglected, since it is usually unnecessary to retain powers of the thickness higher than the cube. It will also, in the present paper, be shown by an indirect method that a similar proposition is true in the case of cylindrical and spherical shells, and, therefore, the fundamental hypothesis upon which Mr. Love has based his theory, although unsatisfactory as an assumption, leads to correct results. A general expression for the potential energy due to strain in curvilinear coordinates has also been obtained by Mr. Love, and the equations of motion and the boundary conditions have been, deduced therefrom by means of the Principle of Virtual Work, and if this expression and the equations to which it leads were correct, it would be unnecessary to propose a fresh theory of thin shells ; but although those portions of Mr. Love’s results which depend upon the thickness of the shell are undoubtedly correct, yet, for reasons which will be more fully stated hereafter, I am of opinion that the terms which depend upon the cube of thickness are not strictly accurate, inasmuch as he has omitted to take into account several terms of this order, both in the expression for the potential energy and elsewhere. His preliminary analysis is also of an exceedingly complicated character. Throughout the present paper the notation of Thomson and Tait’s “Natural Philosophy ” will be employed for stresses and elastic constants, but, for the purpose of facilitating comparison, Mr. Love's notation will be employed for strains and directions. It will also be convenient to denote the values of the various quantities involved, at a point P on the middle surface of the shell by unaccented letters; and the values of the same quantities at a point P' on the normal at P, whose distance from P is h ' by accented letters. The radius of the shell will also be a denoted by and its thickness by 2 h .

Transient Response of Two Fluid-Coupled Cylindrical Elastic Shells to an Incident Pressure Pulse

1979 ◽  
Vol 46 (3) ◽  
pp. 513-518 ◽  
Author(s):  
H. Huang
Keyword(s):  
Transient Response ◽  
Wave Equations ◽  
Pressure Pulse ◽  
Equations Of Motion ◽  
Outer Shell ◽  
Asymptotic Formulas ◽  
Elastic Shells ◽  
Incident Plane ◽  
The Time Domain ◽  
Separation Of Variable

The transient response of a system of two initially concentric circular cylindrical elastic shells coupled by an ideal fluid and impinged upon by an incident plane pressure pulse is studied. The classical techniques of separation of variable and Laplace transforms are employed for simultaneously solving the wave equations governing the fluid motions and the shell equations of motion. The transformed solutions are arranged in such a manner that their inverse transforms can be accurately calculated by solving a set of Volterra integral equations in the time domain. A sample calculation of shell responses was performed and results are compared to the case in which the outer shell is absent. It is found that the primary effects of a thin outer shell could be estimated by simple asymptotic formulas.

A Simplified Displacement-Rotation Form for the Equations of Motion of Nonlinearly Elastic Shells of Revolution Undergoing Combined Bending and Torsion

1996 ◽  
Vol 63 (2) ◽  
pp. 539-542 ◽  
Author(s):  
J. G. Simmonds
Keyword(s):  
Equations Of Motion ◽  
Shell Of Revolution ◽  
Shells Of Revolution ◽  
Elastic Shells ◽  
Axisymmetric Bending ◽  
Nonlinearly Elastic

By appropriately defining two displacements and a rotation, it is shown that the equations of motion of a shell of revolution undergoing combined axisymmetric bending and torsion, in which the extensional strains and the rotations may be arbitrarily large, can be given a form in which there are three effective extensional strains and two effective bending strains, each of which is only linear or quadratic in the displacements and rotation.

scholarly journals VI. On the extension and flexure of cylindrical and spherical thin elastic shells

1890 ◽  
Vol 47 (286-291) ◽  
pp. 45-49
Keyword(s):  
Equations Of Motion ◽  
Elastic Solid ◽  
Middle Surface ◽  
Simple Method ◽  
Elastic Shells ◽  
Tangential Stresses ◽  
Lines Of Curvature ◽  
Plane Plate ◽  
Normal Traction ◽  
Usual Theory

The usual theory of thin elastic shells is based upon the hypothesis that the three stresses R, S, T, may be treated as zero, where R is the normal traction perpendicular to the middle surface, and S and T are the two shearing stresses which tend to produce rotation about two lines of curvature of the middle surface. This hypothesis requires that these stresses should be at least of the order of the square of the thickness of the shell, for when this is the case they give rise to terms in the expression for the potential energy due to strain, which are proportional to the fifth power of the thickness, and which may be neglected, since it is usually unnecessary to retain powers of the thickness higher than the cube. It can be proved directly from the general equations of motion of an elastic solid, that this proposition is true in the case of a plane plate, provided the surfaces of the plate are not subjected to any pressures or tangential stresses, but there does not appear to be any simple method of establishing a similar proposition in the case of curved shells.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

On the motion of comet P/d’Arrest

1974 ◽  
Vol 22 ◽  
pp. 145-148
Author(s):  
W. J. Klepczynski
Keyword(s):  
Equations Of Motion ◽  
Variational Equations ◽  
Partial Derivatives

AbstractThe differences between numerically approximated partial derivatives and partial derivatives obtained by integrating the variational equations are computed for Comet P/d’Arrest. The effect of errors in the IAU adopted system of masses, normally used in the integration of the equations of motion of comets of this type, is investigated. It is concluded that the resulting effects are negligible when compared with the observed discrepancies in the motion of this comet.

Validation of a Belt Model for Prediction of Hub Forces from a Rolling Tire4

2009 ◽  
Vol 37 (2) ◽  
pp. 62-102 ◽  
Author(s):  
C. Lecomte ◽  
W. R. Graham ◽  
D. J. O’Boy
Keyword(s):  
Internal Pressure ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Prediction Method ◽  
Analytical Models ◽  
Beam Model ◽  
Impedance Boundary ◽  
Bending Waves ◽  
Tire Rotation ◽  
Three Dimensional Models

Abstract An integrated model is under development which will be able to predict the interior noise due to the vibrations of a rolling tire structurally transmitted to the hub of a vehicle. Here, the tire belt model used as part of this prediction method is first briefly presented and discussed, and it is then compared to other models available in the literature. This component will be linked to the tread blocks through normal and tangential forces and to the sidewalls through impedance boundary conditions. The tire belt is modeled as an orthotropic cylindrical ring of negligible thickness with rotational effects, internal pressure, and prestresses included. The associated equations of motion are derived by a variational approach and are investigated for both unforced and forced motions. The model supports extensional and bending waves, which are believed to be the important features to correctly predict the hub forces in the midfrequency (50–500 Hz) range of interest. The predicted waves and forced responses of a benchmark structure are compared to the predictions of several alternative analytical models: two three dimensional models that can support multiple isotropic layers, one of these models include curvature and the other one is flat; a one-dimensional beam model which does not consider axial variations; and several shell models. Finally, the effects of internal pressure, prestress, curvature, and tire rotation on free waves are discussed.

Calculation of Dynamic Tire Forces from Aircraft Instrumentation Data

2010 ◽  
Vol 38 (3) ◽  
pp. 182-193 ◽  
Author(s):  
Gary E. McKay
Keyword(s):  
System Performance ◽  
Equations Of Motion ◽  
Test Laboratory ◽  
Excellent Correlation ◽  
Friction Behavior ◽  
Aircraft Landing ◽  
Data Scatter ◽  
Tire Forces ◽  
Six Degree Of Freedom ◽  
Tire Friction

Abstract When evaluating aircraft brake control system performance, it is difficult to overstate the importance of understanding dynamic tire forces—especially those related to tire friction behavior. As important as they are, however, these dynamic tire forces cannot be easily or reliably measured. To fill this need, an analytical approach has been developed to determine instantaneous tire forces during aircraft landing, braking and taxi operations. The approach involves using aircraft instrumentation data to determine forces (other than tire forces), moments, and accelerations acting on the aircraft. Inserting these values into the aircraft’s six degree-of-freedom equations-of-motion allows solution for the tire forces. While there are significant challenges associated with this approach, results to date have exceeded expectations in terms of fidelity, consistency, and data scatter. The results show excellent correlation to tests conducted in a tire test laboratory. And, while the results generally follow accepted tire friction theories, there are noteworthy differences.

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