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The Transfer Matrix Method for the Vibration of Compressed Helical Springs
Pearson D.
The partial differential equations of motion are obtained for a helical spring subject to a static axial force, the typical element of the spring having six degrees of freedom. The wire cross-section can be any doubly symmetrical shape. The overall transfer matrix is calculated and its application is discussed for obtaining the response to forced sinusoidal vibration. Natural frequencies are found from the transfer matrix by iteration. Comparisons are made with published experiments on the natural frequencies of helical springs, made from round wire, with and without a static axial force. Comparison is also made with published theory for the static buckling of helical springs. Information is given on the effect on the natural frequencies of the static axial force, helix angle, number of active turns, ratio of helix to wire diameter, Poisson's ratio, shear coefficient, and the end conditions. The calculation of the normal modes is discussed.
Cavitation in Vortex Diodes and Its Significance in Diode Pumps
Priestman G.H., Tippetts J.R.
Vortex diodes handling liquids are subject to the effects both of cavitation and Reynolds number. The apparent complexity of their characteristics makes design and simulation difficult. To some extent, however, this complexity is artificial and can be reduced. The effect of cavitation is most severe in the forward state. The usual characterization in terms of Thoma's cavitation number (based on pressure drop and downstream pressure) obscures a much simpler relationship using the upstream pressure explicitly. This yields simple equations which cover the cavitating regime. The non-cavitating characteristics are shown to be well predicted by established data on the operation of diffusers. Experimental data vindicating this is taken from Zobel, Syred et al, and our own experiments. Cavitation in the reverse state is not simplified but it is much less important for the diodes quoted. The practical implications of these results in diode pump applications are examined and quantified.
Oil Film Thickness and Pressure Distribution in Elastohydrodynamic Point Contacts
Mostofi A., Gohar R.
In this paper, a general numerical solution to the elastohydrodynamic point contact problem is presented for moderate loads and material parameters. Isobars, contours and regression formulae describe how pressure and oil film thickness vary with geometry, material properties, load, and squeeze velocity, when the rolling velocity vector is at various angles to the static contact ellipse long axis. In addition, the EHL behaviour under spin is examined. The theoretical predictions of film thickness compare favourably with other numerical solutions to the point contact problem, as well as with experimental results which use the optical interferometry method to find film thickness and
On the Thermal and Hydrodynamic Stability of a Fluid in a Vertical Slot
Sorour M.M.
This investigation is devoted to studying the thermal and hydrodynamic stability of the conduction regime of natural convection of a fluid with a nonlinear density temperature relationship in a slender slot with different, but constant, temperature walls. Linear perturbation techniques are applied for the transverse two dimensional disturbances to derive a set of two coupled twelfth order partial differential equations; the Galerkin method was used to solve the eigenvalue problem. The results of this analysis indicate that this flow is much more susceptable to thermal stability than to hydrodynamic stability for Prandtl number, Pr > 3·2. On the other hand for Pr < 3·2 the reverse is true. Furthermore, both shear and bouyant mode characteristics are determined for a wide range of Pr.
Extension of the Patir-Cheng Flow Simulation of a Rough Surface Bearing to a Compressible Lubricant
Majumdar B.C., Hamrock B.J.
It is shown that the average Reynolds equation for rough surfaces using an incompressible lubricant derived by Patir and Cheng (1)§ can be used for a compressible lubricant if the surface structures of both surfaces are the same. The average Reynolds equation is applied to an infinitely long gas journal bearing to find the hydrodynamic load capacity. The effect of the roughness parameter and the surface pattern parameter on the hydrodynamic load is also investigated.
On the Possibility of Balancing Rotating Flexible Shafts
Bishop R.E.
Van de Vegte (1)‡ discusses the balancing of a flexible multi-disc rotor supported in two bearings, using two motorized balancing heads. The underlying idea of the motorized balancing head is used in the present paper, but with a completely different line of theoretical reasoning. Modal analysis suggests that, provided certain requirements as to the nature of the bearings are met, the approach described by van de Vegte may be unnecessarily complicated, in that only one motorized balancing head need be used, while neither the reference to ‘discs’ nor the restriction to two bearings may be needed.
A Method of Limiting Intermediate Values of Volume Fraction in Iterative Two-Fluid Computations
Carver M.B.
Multidimensional computational analysis of fluid flow is usually done by segmented iterative methods, as the equations sets generated are too large to permit simultaneous solution. Frequently the need arises to compute values for variables which must remain bounded for physical reasons. In two-phase computation, for example, the volume fraction is restricted to values between 0 and 1, but iterative procedures often return intermediate values which violate these bounds. It is fairly straightforward to prevent negative values, however no satisfactory method of imposing the upper limit has been published. A method of smoothly applying the limit in reversible fashion is outlined in this note.
Parametric Instability of Tapered Beams by Finite Element Method
Datta P.K., Chakraborty S.
The dynamic stability behaviour of a tapered beam has been studied using a finite element analysis. The instability zones of the parametric stability diagram have been discussed for the entire ranges of static and dynamic load factors. It has been observed that at high values of static load and beyond a particular value of the dynamic load factor, the periodic solution of the Mathieu equation does not exist in the principal region. This leads to unstable behaviour due to large displacement of the beam due to increasing values of static and dynamic load factors.
Minimizing the Thermal Resistance of Pressed Contacts
Snaith B., O'Callaghan P.W., Probert S.D.
A procedure is described for predicting the recommended insert thickness which will result in minimizing the thermal resistance of a pressed contact between fiat, non-wavy, surfaces. The analysis is based upon the assumption of ideal plastic asperity deformations occurring at the interfaces formed between randomly high rough surfaces and the insert material. Experimental measurements of the thermal resistances for aluminium-tin-aluminium and stainless steel-tin-stainless steel, mechanically loaded assemblies corroborate within experimental error the predictions from the theory, and hence validate the analysis.
Dynamic Stability of Wave-Convection Transport
Greenberg M.D., Harrell C.Y.
A flexible inextensible horizontal belt is assumed to be formed, by closely spaced vertical push rods, into a traveling sine wave. A spherical object resting at the bottom of a trough will tend to be convected with the trough as the wave travels. The dynamic stability of such wave-convection transport is considered. Assuming the wave to be shallow, the governing nonlinear equations are expanded (through second order) in the ‘shallowness parameter’, and thus reduced to a single equation, essentially of forced Duffing type, which is integrated numerically, over the parameter space of practical interest, to yield a stability criterion.
Turbulence Generated Secondary Flows in Ducts of Non-Circular Cross-Section
Seale W.J.
The use of algebraic stress models in the prediction of secondary flows in straight ducts of non-circular cross-section is found to be unsatisfactory and to give inconsistent results in the sub-channels within a rod bundle. An algebraic expression is presented which allows the source of axial vorticity to be calculated directly and without iteration. The expression is shown to reproduce secondary velocities in square and triangular ducts, and in a duct consisting of two inter-connected subchannels.
Transient Extinction of Counter flow Diffusion Flame
Saitoh T., Ishiguro S.
A transient analysis was performed for extinction of the counter flow diffusion flame utilizing the assumptions of inviscid, incompressible, and laminar stagnation-point boundary layer flows. The unsteadiness was induced via linear time variation of the stagnation point velocity gradient. The physical meaning of the middle solution of the quasi-steady theory was clarified. The effects of acceleration and deceleration of the flow were examined and it was found that strong acceleration tends to support the flame up to a small Damkohler number, which implies that the flame strength becomes large for flames under acceleration.
Effect of Surface Roughness on the Squeeze Film between Rotating Porous Annular Discs
Prakash J., Tiwari K.
The analysis of the squeeze film between two rotating annular discs, one with a porous facing is extended to include the effect of surface roughness. A detailed study of the squeeze film characteristics for a wide range of operating parameters is made to illustrate the effect of rotation and surface roughness and their interaction.
Non-Newtonian Lubrication Theory for Rough Surfaces: Application to Rigid and Elastic Rollers
Sinha P., Shukla J.B., Singh C., Prasad K.R.
To predict the consequences of the interaction of the lubricant rheology with surface roughness, the stochastic theory of lubrication for Newtonian fluid is modified to take into account the non-Newtonian behaviour of the lubricant, by considering the power law model. Generalized forms of the Reynolds equation for two types of roughness arrangements, viz., longitudinal and transverse, are derived. These equations are subsequently used to study the lubrication characteristics of infinitely long rough roller bearings, and two particular cases, namely pure rolling (rigid surfaces) and rolling with elastic deformation, are discussed.
Investigation of a Rotor Bearing Assembly Incorporating a Squeeze-Film Damper Bearing
Holmes R., Dogan M.
Squeeze-film damper bearings, when properly designed, are a simple means of curbing instabilities and reducing vibration in rotor-bearing assemblies. This paper describes vibration tests carried out on a rigid rotor — flexible bearing test rig incorporating a squeeze-film damper; comparisons are made with theoretical predictions.
The Measurement of Dynamic Cutting Force Data and its Application to the Prediction of Chatter
Burdekin M., Kilic S.E.
A method and the related equipment to obtain dynamic cutting force coefficients under simulated regenerative chatter conditions are described. The coefficients in both thrust (normal to the cut surface) and the main cutting (tangential) directions for orthogonal cutting are presented for various cutting conditions. An algorithm which is developed to predict chatter instability by combining the receptance of a machine tool with the dynamic cutting data is also presented. Comparison of experimentally-determined stability charts of a centre lathe with the predicted ones is made to evaluate the reliability of the method.
On the Time-Dependent Spin Creep of a Railway Wheel
Newland D.E.
This paper describes two theoretical calculations of the fluctuating spin creep of a railway wheel rolling on straight track. The first calculation is of the transient spin moment which occurs when a constant spin creep angular velocity is applied suddenly to a rolling wheel. This may occur, for example, when a wheel passes over a discontinuity in the rail and rolling contact is resumed suddenly. The second calculation is of the fluctuating spin creep moment which occurs when a coned railway wheel rolls along a slightly corrugated rail. The analysis is highly idealized, using the approximations of the simplified theory of Kalker and assuming that the area of slip is limited to a negligible proportion of the area of contact. In the first calculation, the area of contact is assumed to be circular; in the second calculation, it is assumed to be rectangular. Subject to these far-reaching assumptions, the results of the calculations suggest that oscillatory fluctuations in spin creep may be initiated by rail discontinuities and irregularities. Such oscillatory fluctuations in conditions at the contact patch may have a role in the development of rail corrugations.
