Finite Amplitude Vibrations of a Body Supported by Simple Shear Springs

1984 ◽  
Vol 51 (2) ◽  
pp. 361-366 ◽  
Author(s):  
M. F. Beatty
Keyword(s):  
Simple Shear ◽  
Plane Surface ◽  
Elliptic Integral ◽  
Quadratic Function ◽  
Elliptic Functions ◽  
Free Fall ◽  
Finite Amplitude ◽  
Natural State ◽  
Static Deflection ◽  
Hyperelastic Materials

The exact solution of the problem of the undamped, finite amplitude oscillations of a mass supported symmetrically by simple shear mounts, and perhaps also by a smooth plane surface or by roller bearings, is derived for the class of isotropic, hyperelastic materials for which the strain energy is a quadratic function of the first and second principal invariants and an arbitrary function of the third. The Mooney-Rivlin and Hadamard material models are special members for which the finite motion of the load is simple harmonic and the free fall dynamic deflection always is twice the static deflection. Otherwise, the solution is described by an elliptic integral which may be inverted to obtain the motion in terms of Jacobi elliptic functions. In this case, the frequency is amplitude dependent; and the dynamic deflection in the free fall motion from the natural state always is less than twice the static deflection. Some results for small-amplitude vibrations superimposed on a finely deformed equilibrium state of simple shear also are presented. Practical difficulties in execution of the simple shear, and the effects of additional small bending deformation are discussed.

scholarly journals On the theory of the elliptic transcendents

1837 ◽  
Vol 3 ◽  
pp. 60-60
Keyword(s):  
Elliptic Function ◽  
Elliptic Integral ◽  
Elliptic Functions ◽  
Algebraic Expression ◽  
Algebraic Equations ◽  
Similar Function ◽  
Circular Arcs ◽  
Straight Line ◽  
Regular Theory ◽  
General Method

Fagnani discovered that the two arcs of the periphery of a given ellipse may be determined in many ways, so that their difference shall be equal to an assignable straight line; and proved that any arc of a lemniscate, like that of a circle, may be multiplified any number of times, or may be subdivided into any number of equal parts, by finite algebraic equations. What he had accomplished with respect to the arcs of the lemniscates, which are expressed by a particular elliptic integral, Euler extended to all transcendents of the same class. Landen showed that the arcs of the hyperbola may be reduced, by a proper transformation, to those of an ellipse. Lagrange furnished us with a general method for changing an elliptic function into another having a different modulus; a process which greatly facilitates the numerical calculation of this class of integrals. Legendre distributed the elliptic functions into distinct classes, and reduced them to a regular theory, developing many of their properties which were before unknown, and introducing many important additions and improvements in the theory. Mr. Abel of Christiana happity conceived the idea of expressing the amplitude of an elliptic function in terms of the function itself, which led to the discovery of many new and useful properties. Mr. Jacobi proved, by a different method, that an elliptic function may be transformed in innumerable ways into another similar function, to which it bears constantly the same proportion. But his demonstrations require long and complicated calculations; and the train of deductions he pursues does not lead naturally to the truths which are proved, nor does it present in a connected view all the conclusions which the theory embraces. The author of the present paper gives a comprehensive view of the theory in its full extent, and deduces all the connected truths from the same principle. He finds that the sines or cosines of the amplitudes, used in the transformations, are analogous to the sines or cosines of two circular arcs, one of which is a multiple of the other; so that the former quantities are changed into the latter when the modulus is supposed to vanish in the algebraic expression. Hence he is enabled to transfer to the elliptic transcendents the same methods of investigation that succeed in the circle: a procedure which renders the demonstrations considerably shorter, and which removes most of the difficulties, in consequence of the close analogy that subsists between the two cases.

scholarly journals Emergence of colour symmetry in free-vibration acoustic resonance of a nonlinear hyperelastic material

2013 ◽  
Vol 469 (2159) ◽  
pp. 20130275 ◽  
Author(s):  
Ryuichi Tarumi
Keyword(s):  
Free Vibration ◽  
Ritz Method ◽  
Geometrical Nonlinearity ◽  
Acoustic Resonance ◽  
Finite Amplitude ◽  
Hyperelastic Material ◽  
Irreducible Representations ◽  
Hyperelastic Materials ◽  
Point Groups ◽  
Linearized System

We investigated free-vibration acoustic resonance (FVAR) of two-dimensional St Venant–Kirchhoff-type hyperelastic materials and revealed the existence and structure of colour symmetry embedded therein. The hyperelastic material is isotropic and frame indifferent and includes geometrical nonlinearity in its constitutive equation. The FVAR state is formulated using the principle of stationary action with a subsidiary condition. Numerical analysis based on the Ritz method revealed the existence of four types of nonlinear FVAR modes associated with the irreducible representations of a linearized system. Projection operation revealed that the FVAR modes can be classified on the basis of a single colour (black or white) and three types of bicolour (black and white) magnetic point groups: , , and . These results demonstrate that colour symmetry naturally arises in the finite amplitude nonlinear FVAR modes, and its vibrational symmetries are explained in terms of magnetic point groups rather than the irreducible representations that have been used for linearized systems. We also predicted a grey colour nonlinear FVAR mode which cannot be derived from a linearized system.

scholarly journals Positive or negative Poynting effect? The role of adscititious inequalities in hyperelastic materials

2011 ◽  
Vol 467 (2136) ◽  
pp. 3633-3646 ◽  
Author(s):  
L. Angela Mihai ◽  
Alain Goriely
Keyword(s):  
Shear Deformation ◽  
Simple Shear ◽  
Shear Force ◽  
Pure Shear ◽  
Hyperelastic Materials ◽  
Isotropic Materials ◽  
Poynting Effect ◽  
Simple Shear Deformation ◽  
Biopolymer Gels

Motivated by recent experiments on biopolymer gels whereby the reverse of the usual (positive) Poynting effect was observed, we investigate the effect of the so-called ‘adscititious inequalities’ on the behaviour of hyperelastic materials subject to shear. We first demonstrate that for homogeneous isotropic materials subject to pure shear, the resulting deformation consists of a triaxial stretch combined with a simple shear in the direction of the shear force if and only if the Baker–Ericksen inequalities hold. Then for a cube deformed under pure shear, the positive Poynting effect occurs if the ‘sheared faces spread apart’, whereas the negative Poynting effect is obtained if the ‘sheared faces draw together’. Similarly, under simple shear deformation, the positive Poynting effect is obtained if the ‘sheared faces tend to spread apart’, whereas the negative Poynting effect occurs if the ‘sheared faces tend to draw together’. When the Poynting effect occurs under simple shear, it is reasonable to assume that the same sign Poynting effect is obtained also under pure shear. Since the observation of the negative Poynting effect in semiflexible biopolymers implies that the (stronger) empirical inequalities may not hold, we conclude that these inequalities must not be imposed when such materials are described.

scholarly journals IX. The third elliptic integral and the ellipsotomic problem

1904 ◽  
Vol 203 (359-371) ◽  
pp. 217-304
Keyword(s):  
Elliptic Integral ◽  
Great Influence ◽  
Elliptic Functions ◽  
Mechanical Theory ◽  
The Third ◽  
History Of ◽  
Modern Analysis

The Abel Centennial Ceremony, held in Christiania, September, 1902, has directed the attention of mathematicians to the great influence of Abel on modern analysis, and. to the history of elliptic functions, and of the foundation by Crelle of the “ Journal für die reine und angewandte Mathematik.” Abel’s article in the first volume of ‘ Crelle’s Journal,' 1826, " Ueber die Integration der Differential-Formel ρdx ⁄ √R (A), wenn R und ρ gauze Functionen sind,” is of great importance as indicating the existence of what is now called the pseudo-elliptic integral; the present memoir is intended to show the utility of this integral in its application to mechanical theory.

scholarly journals Equidistant map projections of a triaxial ellipsoid with the use of reduced coordinates

2017 ◽  
Vol 66 (2) ◽  
pp. 271-290 ◽  
Author(s):  
Paweł Pędzich
Keyword(s):  
Elliptic Integral ◽  
Elliptic Functions ◽  
New Method ◽  
Triaxial Ellipsoid ◽  
Polar Regions ◽  
Jacobian Elliptic Functions ◽  
Map Projections ◽  
Basic Properties ◽  
Reduced Coordinates

Abstract The paper presents a new method of constructing equidistant map projections of a triaxial ellipsoid as a function of reduced coordinates. Equations for x and y coordinates are expressed with the use of the normal elliptic integral of the second kind and Jacobian elliptic functions. This solution allows to use common known and widely described in literature methods of solving such integrals and functions. The main advantage of this method is the fact that the calculations of x and y coordinates are practically based on a single algorithm that is required to solve the elliptic integral of the second kind. Equations are provided for three types of map projections: cylindrical, azimuthal and pseudocylindrical. These types of projections are often used in planetary cartography for presentation of entire and polar regions of extraterrestrial objects. The paper also contains equations for the calculation of the length of a meridian and a parallel of a triaxial ellipsoid in reduced coordinates. Moreover, graticules of three coordinates systems (planetographic, planetocentric and reduced) in developed map projections are presented. The basic properties of developed map projections are also described. The obtained map projections may be applied in planetary cartography in order to create maps of extraterrestrial objects.

Design for 1:2 Internal Resonances in In-Plane Vibrations of Plates With Hyperelastic Materials

2013 ◽  
Author(s):  
Astitva Tripathi ◽  
Anil K. Bajaj
Keyword(s):  
Finite Element ◽  
Internal Resonance ◽  
Strain Energy ◽  
Quadratic Function ◽  
The Body ◽  
Energy Potential ◽  
Internal Resonances ◽  
Element Analysis ◽  
Hyperelastic Materials ◽  
Aerial Vehicle

With advances in technology, hyperelastic materials are seeing increased use in varied applications ranging from microfluidic pumps, artificial muscles to deformable robots. They have also been proposed as materials of choice in the construction of components undergoing dynamic excitation such as the wings of a micro-unmanned aerial vehicle or the body of a serpentine robot. Since the strain energy potentials of various hyperelastic materials are more complex than quadratic, exploration of their nonlinear dynamic response lends itself to some interesting consequences. In this work, a structure made of a Mooney-Rivlin hyperelastic material and undergoing planar vibrations is considered. Since the stresses developed in a Mooney-Rivlin material are at least a quadratic function of strain, a possibility of 1:2 internal resonance is explored. A Finite Element Method (FEM) formulation implemented in Matlab is used to iteratively modify a base structure to get its first two natural frequencies close to the ratio 1:2. Once a topology of the structure is achieved, the linear modes of the structure can be extracted from the finite element analysis, and a more complete Lagrangian formulation of the hyperelastic structure can be used to develop a nonlinear two-mode model of the structure. The nonlinear response of the structure can be obtained by application of perturbation methods such as averaging on the two-mode model. It is shown that the strain energy potential for the Mooney-Rivlin material makes it possible for internal resonance to occur in such structures.

scholarly journals Researches on the geometrical properties of elliptic integrals

1854 ◽  
Vol 6 ◽  
pp. 143-145
Keyword(s):  
Elliptic Integral ◽  
Elliptic Functions ◽  
Second Order ◽  
Elliptic Integrals ◽  
Geometrical Properties ◽  
Entire Class ◽  
The Subject

In this paper the author proposes to investigate the true geometrical basis of that entire class of algebraical expressions, known to mathematicians as elliptic functions or integrals. He sets out by showing what had already been done in this department of the subject by preceding geometers. That the elliptic integral of the second order represented an arc of a plane ellipse, was evident from the beginning.

Reflection and Transmission of Circularly Polarized Elastic Waves of Finite Amplitude

1979 ◽  
Vol 46 (4) ◽  
pp. 867-872 ◽  
Author(s):  
M. M. Carroll
Keyword(s):  
Elastic Waves ◽  
Finite Elasticity ◽  
Elliptic Functions ◽  
Finite Amplitude ◽  
Plane Boundary ◽  
Elastic Solids ◽  
Jacobian Elliptic Functions ◽  
Reflection And Transmission ◽  
The Third ◽  
Fixed Boundaries

Finite amplitude standing wave solutions, obtained previously, are specialized to the case of incompressible isotropic elastic solids with cubic or quintic shear response. This allows closed-form expressions for the motion and stress field, in terms of Jacobian elliptic functions and elliptic integrals and furnishes solutions for approximate finite elasticity theories in which terms up to sixth degree in the stress and strains are retained. The solutions for reflection from free or fixed boundaries, for resonant standing waves in a plate, and for reflection and transmission at a plane boundary are examined in the context of the third and fourth-order approximations.

XVIII. Researches on the geometrical properties of elliptic integrals

1852 ◽  
Vol 142 ◽  
pp. 311-416 ◽  
Keyword(s):  
Elliptic Integral ◽  
Plane Curve ◽  
Elliptic Functions ◽  
Elliptic Integrals ◽  
Mathematical Science ◽  
Third Order ◽  
Geometrical Properties ◽  
First Order ◽  
The Third ◽  
The Right

I. In placing before the Royal Society the following researches on the geometrical types of elliptic integrals, which nearly complete my investigations on this interesting subject, I may be permitted briefly to advert to what bad already been effected in this department of geometrical research. Legendre, to whom this important branch of mathematical science owes so much, devised a plane curve, whose rectification might be effected by an elliptic integral of the first order. Since that time many other geometers have followed his example, in contriving similar curves, to represent, either by their quadrature or rectification, elliptic functions. Of those who have been most successful in devising curves which should possess the required properties, may be mentioned M. Gudermann, M. Verhulst of Brussels, and M. Serret of Paris. These geometers however have succeeded in deriving from those curves scarcely any of the properties of elliptic integrals, even the most elementary. This barrenness in results was doubtless owing to the very artificial character of the genesis of those curves, devised, as they were, solely to satisfy one condition only of the general pro­blem. In 1841 a step was taken in the right direction. MM. Catalan and Gudermann, in the journals of Liouville and Crelle, showed how the arcs of spherical conic sec­tions might be represented by elliptic integrals of the third order and circular form. They did not, however, extend their investigations to the case of elliptic integrals of the third order and logarithmic form; nor even to that of the first order. These cases still remained, without any analogous geometrical representative, a blemish to the theory.

scholarly journals An application of the new function method to the Zhiber–Shabat equation

2017 ◽  
Vol 7 (3) ◽  
pp. 271-274
Author(s):  
Tolga Aktürk ◽  
Yusuf Gürefe ◽  
Yusuf Pandır
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Elliptic Integral ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Integral Function ◽  
Jacobi Elliptic Functions ◽  
New Approach ◽  
Function Solution ◽  
Wave Solutions

This paper applies a new approach including the trial equation based on the exponential function in order to find new traveling wave solutions to Zhiber-Shabat equation. By the using of this method, we obtain a new elliptic integral function solution. Also, this solution can be converted into Jacobi elliptic functions solution by a simple transformation.

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