Bar Theory Research Articles

On the Structural Nonlinearity of Rotating Shafts

This paper presents a nonlinear model to describe the bending behavior of a rotating shaft, based on the general theory of a bending bar. Justification for this theoretical model is provided by tests, the resulting curves fitting the observed results more closely than those of other models. In particular, the model explains why infinite points of deformation coming from linear theories can be avoided.

A nonlinear theory for elastic bars under tension

A nonlinear theory for elastic bars under tension

Comparative acoustical and psychoacoustical analyses of gamelan instrument tones.

The comparative acoustical and psychoacoustical elements of the Javanese gamelan were studied with a focus on tuning, vibrational characteristics and associated perceptual effects. In addition to relating acoustical, perceptual, and musical frameworks to each other, we seek to clarify some common misconceptions about tuning modes, scales, and the assignment of pitch to metallophones which have inharmonic spectra. We found that the inharmonic modes of the spectra of the bars of the gamelan instruments saron demung, saron barung, and saron peking are described well by the standard physical theory of rectangular bars. The makers have tuned the bars by shaping so that harmonic partials also appear. The frequencies of the scale of the seven measured bars of each instrument (Low 6, Middle 1, 2, 3, 5, 6, High 1) are fit perfectly by an exponential function. Results of perceptual experiments show that, although many octave mismatches occur, both Indonesian and Western subjects generally matched a pitch chroma. In judging small-cents deviations from equi-pentatonic tuning, we found one professional Indonesian musician of four who could reliably discern them. We argue that the paradox of three different pathets (modes) for equi-pentatonic tunings can be explained by differing distributions of chroma based on Markof chains of order 0, 1, 2 or higher and other imposed formulaic and hierarchical structures, which we call chromals.

Free vibrations of a conical shell of slightly varying thickness with annular strengthening

Free vibrations are considered for truncated conical shells of rotation of slightly varying thicknesses with annular strengthening. A solution is built up on the basis of linear shell and bar theory using the Ritz method. Qualitative estimates are given for the effect of rigidly joined solid bodies, and the number, stiffness, and placing of strengthening elements on the change in frequency spectrum and the natural forms of vibration. Theoretical results are confirmed by experimental data obtained by the holographic interferometry method.

Introduction to the thermodynamics of solids

Part 1 Generalities: energy, heat and power temperature and entropy thermodynamic equilibrium. Part 2 Constitutive theory of heat transfer for bars and plates: thermodynamics of rigid bars constitutive theory for rigid bars constitutive theory for thermoelastic bars constitutive theory for shearing of plates thermodynamic experiments. Part 3 Equilibrium theory of bars: equilibrium of bars subjected to dead loads equilibrium theory of bars and hard devices. Part 4 Equilibrium theory of plates: martensic transformations bifurcation diagrams. Part 5 Balloon problems: equilibrium of spherical balloons at fixed pressure equilibrium of balloons containing a fixed mass of gas. Part 6 Biaxial stretch of rubber sheets: the idealized problem the Treolar instability. Part 7 Moving discontinuities: shock waves in bars breaking bars a peeling problem another peeling problem. Part 8 Mixture theory: general remarks elementary theory a solid in an ideal gas. Part 9 Equilibrium of liquid crystals and rods: liquid crystal energies orientation by fields and walls measurement of moduli elastica theory. Part 10 Reconsideration of generalities: systems, energy and temperature general processes static and reversible processes cyclic processes and cycles states laws of classical thermodynamics.

Two-Layer Geostrophic Dynamics. Part II: Geostrophic Turbulence

Abstract In Part I of this series, generalized geostrophic equations were formulated for the two-layer system on a beta plane and over a flat bottom. Here numerical experiments with these equations are carried out to study freely evolving geostrophic turbulence. In contrast with the classical quasigeostrophic analysis, the emphasis is placed on the finite amplitude of the vertical displacement (the frontal effect). A previous study with a reduced-gravity, generalized geostrophic equation has shown that geostrophic turbulence of finite amplitude (frontal geostrophic turbulence) evolves toward a statistical equilibrium state dominated by large, coherent anticyclones. The present study reveals that, in the presence of baroclinicity, this statistical equilibrium state can only be reached if the finite-amplitude turbulent flow evolves from scales smaller or equal to the baroclinic deformation radius. Although the emerging anticyclones should be unstable according to the classical quasigeostrophic theory of bar...

A systematic study of calculations for scattering of plane waves from submerged elastic solid cylinders with hemispherical caps.

It is possible to employ the extended boundary condition equations (EBC) method due to Waterman to describe scattering from axially symmetric submerged elastic targets over a broad frequency range. The problem of scattering from cylinders with hemispherical end caps offers one a useful target to analyze. Resonance phenomena predicted by the EBC calculations are analyzed and a systematic study of results is presented. In addition, bending resonances are predicted by the theoretical calculations and compared with predictions from bar theory. A series of predictions are made for various materials and aspect ratios to aid future experimental work.

Kinetic relations and the propagation of phase boundaries in solids

This paper treats the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition. A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed. For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities.

Ellipticity and deformations with discontinuous gradients in finite elastostatics

Loss of ellipticity of the equilibrium equations of finite elastostatics is closely related to the possible emergence of elastostatic shocks, i.e., deformations with discontinuous gradients. In certain situations where constitutive response functions are essentially one-dimentional, such as anti-plane shear or bar theories, strong ellipticity is closely related to convexity of the elastic potential and invertibility of certain constitutive response functions. The present work addresses the analogous issues within the context of three dimensional elastostatics of compressible but not necessarily isotropic hyperelastic materials. A certain direction-dependent resolution of the deformation gradient is introduced and its existence and uniqueness for a given direction are established. The elastic potential is expressed as a function of kinematic variables arising from this resolution. Strong ellipticity is shown to be equivalent to the positive definiteness of the Hessian matrix of this function, thus sufficing for its strict convexity. The underlying variables are interpretable physically as simple shears and extensions. Their work-conjugates define a traction response mapping. It is shown that discontinuous deformation gradients are sustainable if and only if this mapping fails to be invertible. This result is explicit, in the sense that it characterizes the set of all possible piecewise homogeneous deformations given the elastic potential function.

Vibrational Analysis of Thin‐Walled Bars with Open Cross Sections

Natural frequencies and corresponding modal forms for thin‐walled bars of constant and open cross section are examined using the finite element method. Following the classical thin‐walled bar theory, warping and Saint‐Venant torsional rigidities are accounted for. A thin‐walled bar finite element with seven degrees of freedom at each node is adopted, and an explicit, consistent mass matrix of order fourteen is derived. Only small amplitude, linear vibrations are considered in the paper. A generalized eigenproblem is studied, and natural frequencies together with their corresponding vibrational modes are obtained. The convergence and accuracy of the method is tested based on some available closed‐form solutions and on other numerical results. Several examples illustrate the fast convergence of the method when applied to thin‐Walled bars of constant cross sections. Attempts are also made to use this finite element model to examine eigenproblems for thin‐walled bars with variable, open cross section. It is s...

Thermal Stress Analysis of a Bimaterial Strip Subject to an Axial Temperature Gradient

Formulas are derived for the stresses and displacements in a bimaterial strip which is subjected to a temperature gradient that varies linearly in the longitudinal direction. The solution is obtained by superposing certain fundamental linear elastic stress states which are compatible with bar and beam theory. The analytical results are approximate in the sense that stress-free boundary conditions are not satisfied exactly, since only zero resultant force and moment conditions are enforced. Finite element calculations have been performed to verify the results and to ascertain the level of stress concentration near the ends of the strip.

A consistent theory for torsion of thin-walled bars

A consistent theory for torsion of thin-walled bars with cross-sections of arbitrary shape (open, closed or mixed) is developed in this paper; it is an improvement on the classical torsion theory of thin-walled bars. All the basic relations, formulas and equations are derived and proved under the consistent assumptions. A torsional stiffness matrix for thin-walled bar elements is also formulated on the basis of the consistent theory. A computer program has been written which is applicable to the practical use of torsional analysis of thin-walled bar structures with arbitrary cross-sections.

Impact of a rigid body against a finite elastic bar set on a rigid foundation

The solutions of the impact problems of a rigid body against an elastic bar formerly are based on the classical longitudinal vibration theory of an elastic bar. In this paper, we based on the more precise theory to deal with the impact problem of a rigid body against a finite elastic bar set on a foundation.

Impact of a rigid body against a half-infinite elastic bar

The solution of the impact - problems of a rigid body against an elastic bar arc before based on the classical longitudinal vibration theory of elastic bar. In this paper, we are using a more precise theory to deal with the impact problem of a rigid body against a half -infinite elastic bar with an elastic gasket. This problem is solved completely.

On the dissipative response due to discontinuous strains in bars of unstable elastic material

Some elastic materials arc capable of sustaining finite equilibrium deformations with discontinuous strains. Boundary-value problems for such “unstable” elastic materials often possess an infinite number of solutions, suggesting that the theory suffers from a constitutive deficiency. In the setting of the one-dimensional theory of bars in tension, the present paper explores the consequences of supplementing the theory with further constitutive information. This additional information pertains to the surface of strain discontinuity and consists of a ‘kinetic relation’ and a criterion for the “initiation” of such a surface. We show that the quasi-static response of the bar to a prescribed force history is then fully determined. In particular, we observe how unstable clastic maleri.ils can he used to model macroscopic behavior similar to that associated with viscoplasticity.

Elastic torsion of thin walled bars of variable cross sections

An application of the finite element method to the theory of thin walled bars of variable cross sections has been presented in this paper. A solution of this problem is based on the linear membrane shell theory with the application of Vlasov's assumptions. A bar is divided into elements along its longitudinal axis and then, a shell mid-surface of the element is approximated by arbitrary triangular Subelements. Nodal displacements of the element are assumed to be polynomials of the third order and the equivalent stiffness matrix is obtained. Calculated nodal displacements enable an analysis of normal and shearing stresses.

A new higher order dynamic theory for thermoelastic bars. II: Application to thermoelastic circular and rectangular bars

In order to show and illustrate the power of the theory proposed in Part I [J. Acoust. Soc. Am. 73, 1918–1922 (1983)] for the dynamic behavior of thermoelastic bars, the theory is applied to bars with circular and rectangular cross sections. In these applications, the order of the theory m is chosen to be 3. The general equations of the approximate theory governing all of the deformation modes (such as longitudinal, flexural, torsional, etc.) of circular and rectangular bars, and accomodating the thermal effects are presented. With the object of assessing the approximate theory, approximate and exact dispersion curves are compared for the flexural and longitudinal waves propagating in circular bars. A good match between these two is obtained without using any correction factors in the approximate theory. Further, experimental wave profiles are compared for longitudinal waves in rectangular bars. It is observed that the two agree quite well.

A new higher order dynamic theory for thermoelastic bars. I: General theory

A dynamic approximate theory capable of predicting high-frequency behavior of cylindrical thermoelastic bars is developed using a new theory. The cross section of the bar has an arbitrary shape and contains an arbitrary number of holes. The approximate theory is valid for all of the deformation modes such as flexural, longitudinal, torsional, etc. The use of the new method permits one to eliminate any inconsistency which may occur between lateral boundary conditions and the distributions of displacements or temperature assumed over the cross section of the bar. Accordingly, the method enables one to correctly describe the reflections of the waves propagating along the bar. This, in turn, makes the dispersive characteristics of waves in bars predicted by the approximate theory agree with those obtained from the exact theory without having to introduce any matching coefficients into the approximate theory.

Social dialect, the semantic barrier, and access to curricular knowledge

AbstractThis paper introduces the theory of the lexical bar. It reports research studies conducted in England and Australia. A semantic barrier exists in the English lexicon that emerges from socio-historical factors and is reinforced by the socio-occupational orderings of contemporary society. This barrier separates the lexes of conservative peripheral social dialects from the lexes of dominant central dialects. It serves to produce differential attainment rates in education and to reproduce a social class-based division of labour in English-speaking societies. (Language and thought; language and knowledge; language and education; diachronic linguistics and the sociology of language; the lexes of social dialects; semantic field theory; measures of semantic complexity; the priority of words in meaning; history of English.)

Dynamic Stress Concentrations Due to Flexural Waves in Bars with Various Bends : 2nd Report, Solutions of Bars Excited in Arbitrary Directions

This paper discusses the dynamic stress concentrations in bars with various bends. The flexural wave propagating along the bar which vibrates in an arbitrary direction is considered as an incident wave. Since the bar has bends, this incident wave induces longitudinal, torsional and flexural waves at these bends. In this paper, the elementary bar theory is applied for longitudinal and torsional waves, and the improved bar theory in which the effects of rotatory inertia and shear deformation are considered is applied for flexural waves. The analytical results for general problems are obtained, and the numerical calculations are carried out for cases of a circular arc bend, and L-bend and a Z-bend bars.