Uniform Bar Research Articles

Stability of a Bar Under Eccentric Follower Force

Uniform cantilevered bar subject to eccentric compressive follower force, considering warping rigidity, bending-torsional flutter and stability

An Analysis of Longitudinal Elastic-Plastic Pulse Propagation

The one-dimensional, rate-independent theory of elastic-plastic wave propagation for smooth stress-strain curves concave toward the strain axis is applied to the problem of a long uniform bar loaded at one end by a pressure pulse of short duration. The essential features of the solution are obtained for the case of a semi-infinite bar and for the case of a finite bar whose other end is stress-free by using the method of characteristics in the t-x plane. The general shape of boundaries in the t-x plane which separate regions governed by the dynamic elastic equations from regions governed by the dynamic plastic equations is presented. The nature of the discontinuities that occur at these boundaries is also discussed. For the finite-bar case the analysis is given for materials which exhibit isotropic work hardening and for materials for which the stress-strain behavior in tension is independent of any previous compression. The main features of the solution are in agreement with the behavior observed for annealed, commercially pure aluminum bars subjected to explosive loading at one end. These experiments will be reported subsequently.

Oscillation of a Bar Magnetostrictor

The response of a uniform magnetostrictive bar to electrical excitation is calculated for two sets of boundary conditions. Either both ends are coupled to a resistive load, or one end is coupled and the other clamped rigidly. First-order terms involving dissipative forces are retained in the equation of motion. An expression relating displacement to excitation current is derived. A new equivalent circuit for the magnetostrictor is found and analyzed. It is shown that it can be used to evaluate the important constants of the magnetostrictor from simple measurements that do not change the magnetic properties of the sample.

Series Solutions for Structural Mobility

In investigating the behavior of subcomponents of a structural system, attention is often given to the matching of mobility or impedance at mounting or junction points. Thus, if the mobility at a point in a certain frequency range is of interest, the structure supporting that point might be a black box containing a structural model designed to duplicate the desired mobility. The paper discusses methods for arriving at models based on various forms of series solutions for mobility of linear systems. Models are presented for a particular example: a uniform, thin bar with free ends. Although it is well known that a model with a very large number of discrete elements would adequately represent a uniform bar, the purpose of the present study is to investigate the possibility of furnishing a model containing only a few elements. The mobility of combinations of the simple models is also presented. Improving accuracy of mobility results in better prediction of transient response. An example is included to show that a properly devised system with only a few degrees of freedom can be superior to systems having many degrees of freedom.

Equivalent Electrical Network for the Transversely Vibrating Uniform Bar

The equivalent network for the transversely vibrating uniform bars under various end conditions are given, the value of each element of the networks is determined, and graphs showing the frequency characteristics of each element in the matrices for the bars are given. Some considerations such as the effect of the neglect of the distributed mass are given. The analysis by the method of normalized functions is briefly explained by citing a few examples.

Series Representation of Structural Mobility

In investigating the behavior of subcomponents of a structural system, attention is often given to the meeting of mobility or impedance at mounting or junction points. Thus, if the mobility at a point in a certain frequency range is given, a black box might be devised that contains a structural model designed to duplicate the desired mobility. Various methods are discussed for arriving at models based on different forms of series solutions for mobility of linear systems. Models are presented for a particular example: a uniform, thin bar with free ends. Although it is well known that a model with a large number of discrete elements will adequately represent a uniform bar, the possibility is demonstrated of furnishing a model containing only a few elements, for a certain frequency range. The mobility of combinations of models is also presented. Through the Fourier-transform relationship between mobility and transient response, it is shown that the models accurate for predicting steady-state response are accurate for transient response. [This research was carried out as part of a program supported by the U. S. Office of Naval Research.]

Flexural Horns

Tapered solid elastic rods have previously been used as horns or velocity transformers for longitudinal and torsional vibrations. Such structures may also be employed as velocity transformers when vibrating in flexure. An approximate 1-dimensional wave equation describing the vibration of such “flexural horns” has been derived from thin-bar theory. The wave equation can easily be solved for flexural horns uniform in thickness and exponentially tapered in width, vibrating in the thickness direction. The predicted mode shapes agree well with experimental measurements, but the predicted resonance frequencies are noticeably higher than those measured, the discrepancy increasing with increasing frequency. Comparison with published solutions for the uniform bar indicates that this discrepancy is probably due to the neglect of shear strains and rotary inertia in the formulation of the wave equation. [Work supported in part by the U. S. Office of Naval Research.]

Temperature Distribution along a Uniform Bar which Intercepts an Electron Beam†

The flow of heat along a unifom bar is investigated with particular reference to the heating of slow-wave structures in microwave tubes. The differential equation governing the distribution of temperature along the bar is solved for the simple case of zero radiation and constant thermal conductivity. Approximate solutions are obtained which allow for the variation of thermal conductivity with temperature, and enable conditions under which radiation is insignificant to be determined. The problem is regarded as one-dimensional and the flow of heat assumed to occur uniformly over the cross section of the bar. A typical millimetre-wave circuit is treated as an example.

Crest and Extremal Statistics of a Square-Law-Derived Random Process

Theoretical analysis of the crest and extremal statistics of a square-law-derived random process is presented. The differences between the crest and the extremal statistics, particularly the bandwidth dependence of the latter, are illustrated for the Gaussian-derived case. The theory is applied to the statistical response of a uniform bar in tension. Particular attention is paid to the crest distributions which deviate from the integrated Rayleigh distributions in a predictable way at large amplitudes of vibration. Experimental distributions obtained from the strain response of a uniform bar forced in its fundamental mode by band-limited Gaussian white-noise signals are in excellent agreement with calculated values.

High Speed Plane Bending Fatigue Test : 1st Report, On the Fatigue Testing Machine

In order to investigate the effect of speed of stress reversal on the endurance limit, and to reduce the time needed for fatigue test, a high speed pneumatic fatigue testing machine has been developed. High-frequency siren (air-interrupter) type exciter is used as a reversed plane bending type testing machine for the frequency range of 3 to 20 kc/s. Specimen is a uniform bar of the rectangular cross section, vibrating in the first free-free mode, and is supported by steel wires at the nodal points. Tests were made on normalised 0.36 % carbon steel at frequencies up to 18 kc/s. In order to compare the endurance limit at high speed with that at ordinary speed, the same material was tested by ordinary plane bending machine and rotary bending machine. The endurance limit has a tendency of increase with the increase of reversal speed, and the one at 18 kc/s was about 68% higher than that at the ordinary speed.

Very High Speed Plane Bending Fatigue Test : (1st Report, On The Fatigue Testing Machine)

In order to investigate the effect of the speed of stress reversal on the endurance limit, and to reduce the time necessary for a fatigue test, a high speed pneumatic fatigue testing machine has been developed. Ultrasonic siren (air interruptor) is used as an exciter for a plane bending type testing machine for the frequency range of 3 to 20 kc/sec. Specimen is a uniform bar of the rectangular cross section vibrating in the first free-free mode, and is supported by steel wires at the nodal points. Tests were made on normalized carbon steels at frequencies up to 18 kc/sec. In order to compare the endurance limit at high speed with that at ordinary speed, the same materials were tested by the ordinary plane bending machine and the rotary bending machine. Speed dependence of the endurance limit obtained from these tests is notable, and the endurance limit at the highest frequency was 43% higher than that in the ordinary frequency range.

The influence of a discrete inertia on the free torsional vibrations of a uniform bar

This paper treats the free torsional vibrations of a uniform bar with torsionally elastic end constraints and with one or two discrete inertias somewhere along the span.The frequency equation has been developed for the general system. From this equation the frequency equations for many important cases have been derived.Curves have been plotted for the first three roots of the frequency equations for five commonly encountered systems, for various inertia ratios and positions of the discrete inertia.

Torsional Resonance Vibrations of Uniform Bars of Square Cross Section.

Relations by which the shear modulus may be computed from the fundamental and overtones of the torsional resonance frequencies of square bars have been established empirically.The results are analyzed in terms of a proportionality factor, R, defined by the equation G = (2l fn/n)2ρR. R is found to increase with increasing cross section to length ratio. Also, the overtones are less than integral multiples of the fundamental by an amount which increases with increasing cross section to length ratio.

Stress and Strain Limits on the Attainable Velocity in Mechanical Vibration

The maximum peak velocity which can be developed at the free end of a slender rod driven at resonance in a normal mode of longitudinal vibration is given by v0 = c0Sxx, where c0 is the Young's-modulus speed of sound (E/ρ)12 and Sxx is the limiting value of extensional strain for the rod material. The ratio of allowable stress to the maximum peak velocity is given by Txx/v0 = ρc0, where the characteristic specific impedance plays in this case the role of a transfer impedance relating the stress at a node to the velocity at an antinode. The same expressions describe, within half an order of magnitude, the relations between the maximum allowable stress or strain and the normal component of velocity at the antinodes of displacement for thin uniform bars or plates and for wedges or cones vibrating in flexure, and for an exponential solid horn vibrating longitudinally. By inductive extension it is argued that within the indicated precision the same velocity-strain ratios prevail in the vibration of any elastic body however excited.

A Comparison of Experimental and Theoretical Relations Between Young's Modulus and the Flexural and Longitudinal Resonance Frequencies of Uniform Bars.

The relations from which Young’s modulus may be computed from mechanical flexural and longitudinal resonance frequencies have been established by an empirical method using two sets of steel bars. Both sets contained rectangular and cylindrical specimens. For longitudinal vibration of cylindrical specimens, the agreement between the empirical curves and Bancroft’s corresponding theoretical relation was within experimental error if Poisson’s ratio for both sets is taken to be 0.292. For flexural vibrations, the agreement between the empirical curve and the corresponding theoretical relation developed by Pickett is also within experimental error for about the same value of Poisson’s ratio for the rectangular specimens of both sets; but for cylindrical specimens, the empirical values are somewhat lower than those predicted by the theory.

Effect of Hub Radius on the Vibrations of a Uniform Bar

Abstract Methods are discussed for obtaining upper and lower bounds on the frequencies of a uniform beam, rotating at a constant speed about an axis at one end, and vibrating transversely to the plane of rotation. Previous results are extended to include the case of a nonzero hub radius. Bounds on the first two frequencies are given for several ratios of hub radius to beam length. These show that the frequencies depend almost linearly on the hub radius for various rotational speeds.

Photoelastic-Stress Analysis by Scattered-Light Method

This paper is presented with the aim of evaluating the scattered-light method and techniques avairable for analyzing the photoelastic stresses in stressed models. Many representative scattered light patterns for tensile member, simple beam, tension plate with a central circular hole, and uniform torsion bars with several cross-sectional forms are presented to illustrate characteristics of the scattered-light method. Relative phase retardation graphs for Epoxy resin simple beam, Epoxy resin tensile plate with a central circular hole, and Diallyl Phthalate resin uniform solid circular torsion bar are drawn. From these graphs We have determined the frozen stress sensitivity of Epoxy resin as f=44.7mm·kg-1, and the stress sensitivitv of DAP resin at room temperature as f=0.732 mm·kg-1.

Measurements of Torsional Stiffness Changes and Instability Due to Tension, Compression, and Bending

Abstract The measurements reported verify theoretically predicted effects of tension, compression, and bending, on the torsional stiffness of uniform bars of thin-walled open section. Related modes of buckling and types of nonlinear behavior in torsion are indicated analytically and exhibited in tests, some of these being apparently new.

Transverse Vibrations of Uniform Thin Bars

Continuing the theoretical work of Strehlke, Seebeck, and Lord Rayleigh, the transverse vibrations of uniform thin bars subject to the following end conditions are considered: (a) both ends free; (b) both ends fixed; (c) one end free, the other fixed. Accurate values for the position of the nodes, antinodes, and points of inflection and of maximum curvature are calculated for the first few principal modes of vibration, several errors of long standing are rectified, and the amplitudes at the antinodes are computed (Tables I and II). Approximate expressions, valid for the modes of higher order, are given for the node and antinode positions (Tables III and IV) as well as for the amplitudes at the antinodes (Sec. 6).

Closure to “Discussions of ‘Longitudinal and Torsional Impact in a Uniform Bar With a Rigid Body at One End’” (1950, ASME J. Appl. Mech., 17, pp. 462–464)

Closure to “Discussions of ‘Longitudinal and Torsional Impact in a Uniform Bar With a Rigid Body at One End’” (1950, ASME J. Appl. Mech., 17, pp. 462–464)