A Tabular Method of Calculating Helicopter Blade Deflections and Moments

1948 ◽  
Vol 15 (2) ◽  
pp. 97-106
Author(s):  
N. O. Myklestad
Keyword(s):  
Cross Section ◽  
Constant Term ◽  
Constant Load ◽  
Complex Numbers ◽  
Radial Line ◽  
Helicopter Blade ◽  
Axis Of Rotation ◽  
Principal Axes ◽  
Fixed Axis ◽  
Phase Angles

Abstract It is assumed that the blade rotates about a fixed axis with a constant angular velocity and that it is subjected to a known load parallel to the axis of rotation. The axis of the blade, which is a line through the centroid of each cross section, is not necessarily a straight radial line, but its deviation from such a line must be small. For each cross section, one of the principal axes of moment of inertia is also considered to be parallel to the axis of rotation and bending is assumed to take place in a plane through this axis. The load as well as the mass of the blade are concentrated at distinct points on the blade axis. Under these assumptions the proposed method of analysis will give the shear forces, bending moments, slopes, and deflections by performing a series of tabular calculations. The load, which is always periodic, must first be put in the form of a constant term and a series of harmonic terms, each of which must be analyzed separately. The effects of the constant load and the eccentricity and slope at the base of the blade axis are found together and are easily disposed of in two simple tabular calculations involving only real quantities. The effect of each harmonic involves three tabular calculations with complex numbers. The complex numbers take care of the phase angles which vary along the axis of the blade.

scholarly journals ON THE STUDY OF THE MANUFACTURING TECHNIQUES DEALS WITH THE CERAMICS OF THE MAIKOP- NOVOSVOBODNAYA COMMUNITY BY THE METHOD OF. A. A. BOBRINSKY AND NEW EXPERIMENTS

2015 ◽  
Vol 4 (4) ◽  
pp. 59-71
Author(s):  
Sergey Nikolaevich Korenevskiy ◽  
Andrey Sergeevich Kizilov
Keyword(s):  
Surface Treatment ◽  
Raw Materials ◽  
Flat Bottom ◽  
Axis Of Rotation ◽  
Fixed Axis ◽  
Manufacturing Techniques ◽  
First Time

The article presents a brief synthesis of the results deal with the study of ceramics Maikop-novosvobodnaya community using the method of a. a. Bobrinsky and use of the microscope with 12 times magnification. it sets out ideas about raw materials, methods of construction, surface treatment. especially emphasized the problem of the use by the ancient potters of rotary devices. For the first time about such vessels were noted in the work of a. a. Bobrinsky and r. M. Munchaev in 1966, for example, vessels with a flat bottom. at present a series of examples of traces deal with use of rotary devices has expanded. in the article by a. s. Kizilov shows the simulation of the vessel of the Maikop culture and fixation of the traces of its turn without a fixed axis of rotation and with a non-fixed environment of rotation. as a result, the actual doing of those and other traces that prove the use of Maikop potters rotary devices with a fixed axis of rotation in the manufacture of vessels not only flat, but round bottom too.

5. The algebra of polynomials and cubic equations

2015 ◽  
pp. 53-74
Author(s):  
Peter M. Higgins
Keyword(s):  
Constant Term ◽  
Complex Numbers ◽  
Distributive Law ◽  
Factorization Of Polynomials

Polynomials are expressions of the form p(x) = a0 + a1x + a2x2 + ... + anxn; the number ai is the coefficient of xi, a 0 is the constant term of p(x), and an is the leading coefficient. The underlying algebra of polynomials mirrors that of the integers. Polynomials can be added, subtracted, and multiplied, and the laws of associativity and commutativity and the distributive law of addition over multiplication all hold. Division is more complicated. ‘The algebra of polynomials and cubic equations’ outlines the Remainder and Factor Theorems along with complex numbers in the Argand plane. The factorization of polynomials, the Rational Root Theorem, the Conjugate Root Theorem, and solution of cubic equations are also discussed.

scholarly journals Flexure with shear and associated torsion in prisms of uni-axial and asymmetric cross-sections

1938 ◽  
Vol 237 (776) ◽  
pp. 161-229 ◽  
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Axial Symmetry ◽  
Harmonic Functions ◽  
Transverse Load ◽  
Constant Cross Section ◽  
Principal Axes ◽  
The Cross ◽  
Axes Of Symmetry ◽  
Perpendicular Axis

It is now well over eighty years ago since Barre de Saint-Venant reduced the problem of the beam of constant cross-section under the action of a single transverse load to the search for plane harmonic functions satisfying a certain condition round the boundary of the cross-section. The solutions due to Saint-Venant, which include the rectangular, elliptic and circular cross-sections, are all cases in which the cross-sections have two axes of symmetry at right angles, meeting of necessity in the centroid of the cross-section, and along these axes the single transverse load is resolved. These axes are principal axes, and his solution depends upon this fact. Some less useful solutions exist for the load along one axis of certain beams of such bi-axial symmetry of cross-section, the solutions not yet being known for the load along the perpendicular axis.

scholarly journals Is It Possible to Determine Whether a Star is Rotating About a Unique Axis?

1993 ◽  
Vol 137 ◽  
pp. 566-568 ◽  
Author(s):  
D.O. Gough ◽  
A.G. Kosovichev
Keyword(s):  
Simple Model ◽  
Angular Velocity ◽  
Rotation Axis ◽  
The Other ◽  
The Sun ◽  
Natural Question ◽  
Rotating Stars ◽  
The Core ◽  
Axis Of Rotation ◽  
Fixed Axis

Rotating stars are normally presumed to rotate about a unique axis. Would it be possible to determine whether or not that presumption is correct? This is a natural question to raise, particularly after the suggestion by T. Bai & P. Sturrock that the core of the sun rotates about an axis that is inclined to the axis of rotation of the envelope.A variation with radius of the direction of the rotation axis would modify the form of rotational splitting of oscillation eigenfrequencies. But so too does a variation with depth and latitude in the magnitude of the angular velocity. One type of variation can mimic the other, and so frequency information alone cannot differentiate between them. What is different, however, is the structure of the eigenfunctions. Therefore, in principle, one might hope to untangle the two phenomena using information about both the frequencies and the amplitudes of the oscillations.We consider a simple model of a star which is divided into two regions, each of which is rotating about a different fixed axis. We enquire whether there are any circumstances under which it might be possible to determine seismologically the separate orientations of the axes.

scholarly journals XII. The oscillations of a rotating ellipsoidal shell containing fluid

1895 ◽  
Vol 186 ◽  
pp. 469-506 ◽  
Keyword(s):  
Steady Motion ◽  
Internal Surface ◽  
Solid Shell ◽  
Homogeneous Fluid ◽  
Ellipsoidal Shell ◽  
Axis Of Rotation ◽  
The Earth ◽  
Principal Axes ◽  
Almost All ◽  
Periodic Changes

In a paper published in 'Acta Mathematica,’ vol. 16, M. Folie announces the fact that the latitude of places on the earth’s surface is undergoing periodic changes in a period considerably in excess of that which theory has hitherto been supposed to require. This result has been confirmed in a remarkable manner by Dr. S. C. Chandler, in America ( vide ‘ Astronomical Journal,’ vols. 11, 12), who, as the result of an exhaustive examination of almost all the available records of latitude observations for the last half-century, has assigned 427 days as the true period in which the changes are taking place. The old theory, based on the assumption that the earth was rigid throughout, led to a period of 305 days, and M. Folie proposes to account for the extension of this period by attributing a certain amount of freedom to the internal portions of the earth. The earth he supposes to be composed of “a solid shell moving more or less freely on a nucleus consisting of fluid at least at its surface.” The argument advanced by M. Folie in favour of this constitution of the earth, namely, the independence of the motions of the shell and the nucleus, appeared to me to be unsatisfactoiy, and I therefore proposed to myself to test the validity of it by examining a particular case which lent itself to mathematical analysis, namely, that in which the internal surface of the shell is ellipsoidal and the nucleus consists entirely of homogeneous fluid. The principal axes of the shell and of the cavity occupied by fluid are assumed to be coincident, and the oscillations are considered about a state of steady motion in which the axis of rotation coincides with one of these axes. It is clear that a steady motion will be possible in this case, and that such a motion will be secularly stable in the event of the axis of rotation being the axis of greatest moment for both the shell and the cavity.

A simple jig for the anti-parallel alignment of linacs’ lasers

2016 ◽  
Vol 15 (2) ◽  
pp. 207-209
Author(s):  
Demetrios Okkalides
Keyword(s):  
Quality Assurance ◽  
Laser Beam ◽  
Cross Section ◽  
Patient Positioning ◽  
Laser Beams ◽  
Level Surface ◽  
Axis Of Rotation ◽  
Rotating Plate

AbstractBackgroundDespite crossing at the isocentre, misaligned laser beams may cause significant positioning problems.PurposeThe jig proposed here is to be used in addition to the quality assurance procedures employed with linacs and deals with possible misalignments of transverse lasers.Materials and methodsIt is an inverted T construction with a two-sided rectangular, slightly transparent mirror set vertical on a piece of glass serving as its base. The device is carried by a horizontally set spirit-level surface and placed on the couch-top so that the sagittal laser passes through the vertical mirror’s plane. Then the therapy couch is translated along the Y direction until a laser beam shines on the corresponding side of the semi-transparent mirror. This spot is marked and is normally the linac’s isocentre set through an independent procedure employing a rotating plate. If the laser had been set properly, then its beam should be reflecting back on its source. If not, the alignment can be corrected by rotating and/or translating the laser holder until it does that. At the same time, it should be ensured that the beam does not wander away from the isocentre spot on the mirror. When both are achieved, the beam ends up perpendicular to the linac’s axis of rotation, while passing through the isocentre. The procedure can be repeated for the opposite laser.ConclusionThe jig was simple to construct and has been found quite useful in practice. The accuracy of patient positioning will be restricted only by the size of the laser beam’s cross-section.

scholarly journals TRICOMPLEX DYNAMICAL SYSTEMS GENERATED BY POLYNOMIALS OF ODD DEGREE

Fractals ◽  
2017 ◽  
Vol 25 (03) ◽  
pp. 1750026 ◽  
Author(s):  
PIERRE-OLIVIER PARISÉ ◽  
DOMINIC ROCHON
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cross Section ◽  
Complex Numbers ◽  
Hyperbolic Numbers ◽  
Odd Degree ◽  
The Cross ◽  
Polynomial Formula

In this paper, we give the exact interval of the cross section of the Multibrot sets generated by the polynomial [Formula: see text] where [Formula: see text] and [Formula: see text] are complex numbers and [Formula: see text] is an odd integer. Furthermore, we show that the same Multibrots defined on the hyperbolic numbers are always squares. Moreover, we give a generalized 3D version of the hyperbolic Multibrot set and prove that our generalization is an octahedron for a specific 3D slice of the dynamical system generated by the tricomplex polynomial [Formula: see text] where [Formula: see text] is an odd integer.

The Centre of Shear of Aerofoil Sections

1953 ◽  
Vol 57 (508) ◽  
pp. 235-237 ◽  
Author(s):  
John A. Jacobs
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Lateral Surface ◽  
The Other ◽  
Principal Axes ◽  
Unstrained State ◽  
Uniform Cross Section ◽  
Single Force ◽  
The Cross ◽  
Fixed End

Consider a cantilever beam of uniform cross section whose generators are parallel to the z-axis and whose lateral surface is free from surface tractions. The line of centroids of the cross sections in the unstrained state is taken as the z-axis, and the x- and y-axes are the principal axes of the cross section at the centroid of the fixed end z = 0.The other end of the beam (z = l) is subject to forces which reduce to a single force with components (Wx, Wv, 0), transverse to the z-axis, acting through the load point L of this end section (see Fig. 1). The co-ordinates of L are taken as (p, q, l).

A Method for Locating an Optimal “Fixed” Axis of Rotation for the Human Knee Joint

1978 ◽  
Vol 100 (4) ◽  
pp. 187-193 ◽  
Author(s):  
J. L. Lewis ◽  
W. D. Lew
Keyword(s):  
Knee Joint ◽  
Human Subject ◽  
Optimization Criterion ◽  
Screw Axis ◽  
Human Knee ◽  
Complete Extension ◽  
Axis Of Rotation ◽  
Optimal Direction ◽  
Human Knee Joint ◽  
Fixed Axis

This report describes a theoretical technique which calculates the “optimal” direction and location of a fixed axis of rotation for the human knee joint. The optimization criterion is as follows; global locations of arbitrary points on the femur as a result of flexion about the fixed axis should approximate as close as possible their positions due to an equivalent anatomical motion. For comparison, an expression for the location and direction of a screw axis describing the same anatomical movement is derived. The techniques were tested by their application to blocks of known geometry. The optimal axis for two different constraint criteria and the screw axis are located in the knee of a human subject for the range of motion of complete extension to approximately 90-deg flexion. The results for the axes are compared with each other and with a commonly utilized internal prosthetic axis on the basis of ligament length patterns obtained by flexion about the different axes.

Hardness–depth profile of a carbon-implanted Ti–6Al–4V alloy and its relation to composition and microstructure

2001 ◽  
Vol 16 (8) ◽  
pp. 2321-2335 ◽  
Author(s):  
M. Kunert ◽  
O. Kienzle ◽  
B. Baretzky ◽  
S. P. Baker ◽  
E. J. Mittemeijer
Keyword(s):  
Mechanical Properties ◽  
Chemical Composition ◽  
Cross Section ◽  
Depth Profile ◽  
Constant Load ◽  
Variation Method ◽  
Indentation Modulus ◽  
Transmission Electron ◽  
20 Nm ◽  
Hardness Depth

The variation of mechanical properties (hardness, indentation modulus) within a carbon-implanted region of a Ti–6Al–4V alloy—about 350-nm thick—was, for the first time, related with the microstructure and the chemical composition with a depth accuracy as small as ±20 nm. Microstructure, chemical composition, and mechanical properties of the implanted alloy were determined using transmission electron microscopy, Auger electron spectroscopy, and nanoindentation, respectively. The microstructure within the implanted region contains TiC precipitates, the density of which changes with depth in accordance with the carbon content. The hardness depends on the precipitate density: the maximum hardness occurs at the depth where an almost continuous TiC layer had formed. The depth profiles of hardness and indentation modulus were measured using three different methods: the cross-section method (CSM); the constant-load method (CLM); and the load-variation method (LVM). Only the hardness– depth profile obtained using the CSM, in which the indentations are performed perpendicular to the hardness gradient on a cross section of the specimen, reflects the microstructural variations present in the implanted region.

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