Bending of Clamped Plates

1947 ◽  
Vol 14 (1) ◽  
pp. A55-A62
Author(s):  
W. B. Stiles
Keyword(s):  
Exact Solution ◽  
Normal Modes ◽  
Point Load ◽  
Simultaneous Equations ◽  
Central Point ◽  
The Other ◽  
Rectangular Plates ◽  
Vibrating Membrane ◽  
Square Plate ◽  
Infinite Set

Abstract The exact solution of thin rectangular plates clamped on all or part of the boundary requires the solution of two infinite sets of simultaneous equations in two sets of unknowns. A method of obtaining an approximate solution based upon minimization of energy and requiring the solution of the first i equations of a single infinite set of simultaneous equations is described and illustrated in this paper. The approximation functions are derived from functions representing the normal modes of a freely vibrating membrane for the same region. Solutions are obtained for a rectangular clamped plate supporting a uniform or a central point load and for a square plate clamped on two adjacent edges and pinned on the other two edges with either a uniform or a central point load. Analytical results are compared with experimentally determined deflections and stresses.

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

Radiation of sound from an unflanged rigid cylindrical duct with an acoustically absorbing internal surface

1978 ◽  
Vol 361 (1704) ◽  
pp. 65-91 ◽  
Keyword(s):  
Exact Solution ◽  
Infinite Series ◽  
Infinite System ◽  
Internal Surface ◽  
Simultaneous Equations ◽  
Graphical Form ◽  
Symmetric Mode ◽  
Numerical Computations ◽  
Infinite Set ◽  
Cylindrical Duct

A rigorous and exact solution is obtained for the problem of the radiation of sound from a semi-infinite unflanged rigid duct with an internal acoustically absorbent lining. The solution is obtained by a modification of the normal Wiener-Hopf technique. The solution is in terms of an infinite series of unknowns, which are determined from an infinite set of simultaneous equations. The infinite system converges rapidly enough to make the solution suitable for numerical computations. Some numerical results are given in graphical form for the propagation of the principal symmetric mode in the duct.

scholarly journals On the figures obtained by strewing sand on vibrating surfaces, commonly called acoustic figures

1837 ◽  
Vol 3 ◽  
pp. 180-181
Keyword(s):  
The Other ◽  
Vertical Line ◽  
Rectangular Plates ◽  
Regular Form ◽  
Modes Of Vibration ◽  
The Subject ◽  
Square Plate

The author, after adverting to the imperfect notice taken by Gali­leo and by Hooke of the phenomena which form the subject of this paper, ascribes to Chladni exclusively the merit of the discovery of the symmetrical figures exhibited by plates of regular form when made to sound. He proposes a notation, by means of two numbers separated by a vertical line, for expressing the figures resulting from the vibrations of square or rectangular plates. He gives a table of the relative sounds expressed both by their musical names and by the number of their vibrations, of all the modes of vibration of a square plate, as ascertained by the experiments of Chladni. He then pro­ceeds to class and analyse the various phenomena observed under these circumstances, and shows that all the figures of these vibrating surfaces are the resultants of very simple modes of oscillation, occurring isochronously, and superposed upon one another; the resultant figure varying with the component modes of the vibration, the num­ber of the superpositions, and the angles at which they are superposed. In the present paper, which forms the first part of his investigation, he confines himself to the figures of square and other rectangular plates. The author finds that the principal results of the superposition of two similar modes of vibration are the following :—first, the points where the quiescent lines of each figure intersect each other remain quiescent points in the resultant figure; secondly, the quiescent lines of one figure are obliterated, when superposed, by the vibratory parts of the other; thirdly, new quiescent parts, which may be called points of compensation, are formed whenever the vibrations in opposite directions neutralize each other; and, lastly, at other points, the mo­tion is as the sum of the concurring, or the differences of the opposing vibrations at these points. After considering various modes of binary superposition, the author examines the cases of four co-existing superpositions.

Some Archaic Gold Ornaments with representations of Sphinxes and Sirens

1911 ◽  
Vol 31 ◽  
pp. 263-265
Author(s):  
F. H. Marshall
Keyword(s):  
Seventh Century ◽  
The Other ◽  
Main Body ◽  
Rectangular Plates ◽  
Natural Size

1. In a recent description of an archaic Etruscan fibula here reproduced in natural size (Fig. 1), I regret that I failed to note certain interesting details with regard to the Sphinxes. The fibula is of pale gold, of a type peculiar to early Etruscan jewellery. It consists of two parts, each composed of four tubes ending in double female heads. In one case the outer tubes are furnished with long gold pins which fit into the hollow tubes corresponding to them in the other half of the fibula. There can be no doubt that these safety-pins were used for fastening a garment on the shoulder. The two halves were locked together by means of hooks and eyes soldered to rectangular plates hinged to the main body of the fibula. The tubes were also connected together by similar plates. The present fibula, which may be dated to the seventh century B.C., is said to have been found in the Roman Campagna. Upon the four rectangular plates already mentioned are seated sixteen Sphinxes in the round, four upon each plate. The eight Sphinxes on the outer plates are composed of the figure of a seated lion, with the head of a woman substituted for a wing.

scholarly journals Transient Excitation of an Elastic Half Space by a Point Load Traveling on the Surface

1969 ◽  
Vol 36 (3) ◽  
pp. 505-515 ◽  
Author(s):  
D. C. Gakenheimer ◽  
J. Miklowitz
Keyword(s):  
Exact Solution ◽  
Free Surface ◽  
Steady State ◽  
Wave Front ◽  
Half Space ◽  
Displacement Field ◽  
Elastic Half Space ◽  
Point Load ◽  
Limit Case ◽  
Transient Waves

The propagation of transient waves in a homogeneous, isotropic, linearly elastic half space excited by a traveling normal point load is investigated. The load is suddenly applied and then it moves rectilinearly at a constant speed along the free surface. The displacements are derived for the interior of the half space and for all load speeds. Wave-front expansions are obtained from the exact solution, in addition to results pertaining to the steady-state displacement field. The limit case of zero load speed is considered, yielding new results for Lamb’s point load problem.

Study of Oscillating Flow of Viscoelastic Fluid With the Fractional Maxwell Model

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
Jiu-hong Jia ◽  
Hong-xing Hua
Keyword(s):  
Exact Solution ◽  
Petroleum Chemistry ◽  
Viscoelastic Fluid ◽  
Maxwell Model ◽  
The Other ◽  
Oscillating Flow ◽  
Model Parameters ◽  
Series Approximation ◽  
Phase Lags ◽  
Annular Effect

The oscillating flow of the viscoelastic fluid in cylindrical pipes has been applied in many fields, such as industries of petroleum, chemistry, and bioengineering. It is studied using the fractional derivative Maxwell model in this paper. The exact solution is obtained utilizing a simpler and more reasonable technique. According to this velocity solution, the time-velocity profile of one kind of viscoelastic fluid is analyzed. From analysis, it is found that the flow behaves like the Newton fluid when the oscillating frequency is low, and the flow reversal occurs when the oscillating frequency is high. Moreover, two series approximations for the velocity are obtained and analyzed for different model parameters. In one series approximation, the velocity is parabolic in profile, while in the other series approximation, the velocity presents three characteristics: (1) it is independent of radius and at the centerline is smaller than that of steady Poiseuille flow, (2) the phase lags about 90deg with respect to the imposed pressure gradient, and (3) the Richardson annular effect is found near the wall.

Buckling of Rectangular Plates With Built-In Edges

1942 ◽  
Vol 9 (4) ◽  
pp. A171-A174
Author(s):  
Samuel Levy
Keyword(s):  
Exact Solution ◽  
Rectangular Plate ◽  
Infinite Series ◽  
Numerical Results ◽  
Buckling Load ◽  
Rectangular Plates

Abstract This paper presents an exact solution in terms of infinite series of the problem of buckling by compressive forces in one direction of a rectangular plate with built-in edges (zero slope, zero displacement in the direction normal to the plane of the plate). The buckling load is calculated for 14 ratios of length to width, ranging in steps of 0.25 from 0.75 to 4. On the basis of convergence, as the number of terms used in the infinite series is increased, it is estimated that the possible error in the numerical results presented is of the order of 0.1 per cent. A comparison is given with the work of other authors.

scholarly journals An Euler-Lagrange Equation only Depending on Derivatives of Caputo for Fractional Variational Problems with Classical Derivatives

2020 ◽  
Vol 8 (2) ◽  
pp. 590-601
Author(s):  
Melani Barrios ◽  
Gabriela Reyero
Keyword(s):  
Exact Solution ◽  
Variational Problem ◽  
Lagrange Equation ◽  
Integral Form ◽  
Variational Problems ◽  
The Other ◽  
Isoperimetric Problems ◽  
Caputo Derivatives ◽  
Fractional Variational Problems ◽  
Derivatives Of

In this paper we present advances in fractional variational problems with a Lagrangian depending on Caputofractional and classical derivatives. New formulations of the fractional Euler-Lagrange equation are shown for the basic and isoperimetric problems, one in an integral form, and the other that depends only on the Caputo derivatives. The advantage is that Caputo derivatives are more appropriate for modeling problems than the Riemann-Liouville derivatives and makes the calculations easier to solve because, in some cases, its behavior is similar to the behavior of classical derivatives. Finally, anew exact solution for a particular variational problem is obtained.

scholarly journals Nietzsche’s Dionysos

2017 ◽  
Vol 3 (3) ◽  
pp. 604
Author(s):  
Dieter Mersch
Keyword(s):  
Necessary Conditions ◽  
Central Point ◽  
The Other ◽  
Other Hand ◽  
The Absolute ◽  
The Aesthetic

Nietzsche’s Dionysus, admittedly, represents a direct provocation and an attack on the classical interpretation accepted since Winckelmann, an interpretation that elevates the Apollonian to its central point of focus; Nietzsche’s introduction of another principle to oppose it, rather than representing a genuine invention, in actuality bridges the small gap between Hegel and Hölderlin. If, namely, the Hegelian aesthetic from the very beginning points to Schein and Erscheinung – as necessary conditions of truth, for the truth would not exist if it were not to “superficially appear” (scheinen) and “make its appearance” (erscheinen), writes Hegel – Schein and Erscheinung would still nonetheless be bound up everywhere with the criterium of the absolute; after all, the untruth of the aesthetic rests squarely in the fact that it cannot do other than to draw upon the language of Erscheinung. For Hölderlin, on the other hand, the Dionysian advances to become a metapoetic symbol combining itself – the enigmatic and continually transforming – with the practice of art. Nietzsche follows those very same lines even while giving the metaphor a thoroughly different twist.

Biaxial bending of cold-formed angle members

1984 ◽  
Vol 11 (4) ◽  
pp. 933-942 ◽  
Author(s):  
Murty K. S. Madugula ◽  
Sujit K. Ray
Keyword(s):  
Exact Solution ◽  
Compressive Strength ◽  
Computer Program ◽  
Cross Section ◽  
Simultaneous Equations ◽  
Biaxial Bending ◽  
Total Stress ◽  
The Cross ◽  
The Galerkin Method ◽  
Shear Centre

Theoretical load–deflection relationships for cold-formed angles under biaxial bending using the Galerkin method are presented. The computational difficulties encountered in the exact solution of differential equations of equilibrium involving 12 unknown constants in 12 simultaneous equations are pointed out. A computer program for pinned-end boundary conditions was developed to estimate the deflection components of the shear centre, to calculate the total stress at various points in the cross section, and to predict the ultimate strength of the cold-formed angle sections connected by one leg. Failure is assumed to have occurred when the total stress at any point on the cross section reaches the value of yield stress, compressive or tensile, or when there is a change of sign for at least one of the deflection components. A table giving the ultimate compressive strength of two commonly used cold-formed angles for various gauge distances is included. The theoretical load–deflection curves are compared with experimental results and typical curves for three test specimens are also presented.

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