Depth Taper Research Articles

Implementation of an exact completing method of generation for face-milled spiral bevel gears with uniform depth taper

AbstractThis paper proposes an exact completing method of generation of face-milled spiral bevel gears with uniform depth taper. This method ensures localized bearing contact without kinematic transmission errors in both directions of rotation. Both sides of the tooth are generated simultaneously using the same set of basic machine tool settings. A tip relief on the tooth surfaces is the only required microgeometry modification to achieve a smooth transition of load between consecutive pairs of teeth. An analytical approach to determine the basic machine tool settings and select the grinding cutter parameters has been also developed. The selection of the nominal cutter radius follows the recommendations of Information Sheet AGMA 22849-A12. The basis for alignment error compensation has been also established. Tooth contact and stress analyses demonstrate the advantages and limitations of the proposed method, making it suitable for applications with higher power demands for the drive side than for the coast side.

An exact system of generation for face-milled hypoid gears with uniform depth taper: Application to hypoid gear drives with high gear ratio

An exact system of generation for face-milled hypoid gears with uniform depth taper is proposed. This system defines the location of pinion and wheel, their respective cutting (or grinding) cutters, and the relations of motion in their respective reference coordinate systems attached to the cutting machine. As a result, the grinding cutter geometric parameters and basic machine-tool settings are analytically determined without the need of any further optimization process. The appropriate selection of the mean cutter radius is addressed following the recommendations of the Information Sheet AGMA ISO 22849-A12. The tooth contact analysis of the gear drive shows a localized bearing contact with no unloaded transmission errors in the absence of misalignments. A lower sensitivity of the contact pattern to errors of alignment is observed as the mean cutter radius is decreased. Stress analysis shows that only the application of a tip relief is needed to avoid edge contacts at the tip of the gear tooth surfaces, being the point width of the cutting blades a key parameter to balance bending stresses in the pinion and the wheel.

Increasing and Decreasing Depth Taper Scratching: Force Response of Silicon

Mechanical loading and unloading of silicon is a characteristic feature of grinding and diamond turning processes. Such rapid loading and unloading induces damage and phase transformations. While, indentation tests are often used to study such normal loading and unloading via characteristic events in the force-depth plot, such tests involve only normal loading and lack tangential loading. A better alternative is scratch test, both constant and varying depth ones, involving normal and tangential loading on the scratching tool; this better simulates conditions of machining, or grinding. In this research, the mechanical load/unload behavior response of silicon is studied under scratching conditions by comparing increasing and decreasing depth scratch behaviour. In-situ force responses show that after ductile-brittle transition occurs, higher forces, at a given scratch depth, are required to deform the material during increasing depth scratching for a given depth than in decreasing depth scratch. Large surface and sub-surface damages with the presence of radial, median, and lateral cracks are seen to make the material weaker, ahead of the advancing tool, in decreasing depth scratch. Raman intensity ratio of amorphous silicon (a-Si) to nanocrystalline silicon (nc-Si) shows that high amorphization of silicon occurs during increasing depth scratching than decreasing depth. Using such force-depth plots an attempt is made to compare the normal loads while indenting and scratching. This study can help optimize the processing of silicon by grinding and diamond turning.

New methodology for determining basic machine settings of spiral bevel and hypoid gears manufactured by duplex helical method

The duplex helical method is an advanced and primary manufacturing method of face milled spiral bevel and hypoid gears for its characteristics of higher efficiency, lower cost, higher strength and power dry cutting. A new methodology for calculating the basic machine-tool settings of spiral bevel and hypoid gears manufactured by the duplex helical method is proposed in this work. Firstly, the blank design theory of spiral bevel and hypoid gears with the duplex depth taper is described in detail. In order to obtain the optimal machine settings, three reference points are defined. The basic machine settings for spiral bevel and hypoid gears manufactured by the duplex helical method in accordance with the five steps are determined. Finally, a hypoid gear set generated by the duplex helical method is used as an example to validate the developed methodology. The results reveal that the tooth bearings and the transmission error functions on the drive and coast sides are favorable, and the results of the simulation and experiment are consistent.

Analysis of curved beams using a new differential transformation based curved beam element

Differential transformation method is used to obtain the shape functions for nodal variables of an arbitrarily non-uniform curved beam element including the effects of shear deformation considering axially functionally graded material. The proposed shape functions depend on the variations in cross-sectional area, moment of inertia, curvature and material properties along the axis of the curved beam element. The static and free vibration of axially functionally graded tapered curved beams including shear deformation and rotary inertia are studied through solving several examples. Numerical results are presented for circular, parabolic, catenary, elliptic and sinusoidal beams (both forms—prime and quadratic) with hinged-hinged, hinged-clamped and clamped-clamped and clamped-free end restraints. Three general taper types (depth taper, breadth taper and square taper) for rectangular cross section are studied. Out of plane vibration is studied and the lowest natural frequencies are calculated and compared with the published results. Out of plane buckling is investigated for circular beams due to radial load.

Static, stability and free vibration analysis of arches using a new differential transformation-based arch element

Differential transformation method is used to obtain the shape functions for nodal variables of an arbitrarily non-uniform curved arch element including the effects of shear deformation and extensibility of the neutral axis, considering axially functionally graded material. The proposed shape functions depend on the variations in cross-sectional area, moment of inertia, curvature and material properties along the axis of the curved arch element. Since the element stiffness, mass and geometric stiffness matrices for an arch element are developed based on the exact displacement shape functions, the arch element developed here is more accurate than the elements developed in the past. By adopting the finite-element method the static, stability and free and forced vibrations of axially functionally graded tapered arches including shear deformation and rotary inertia are studied by solving several examples. Numerical results are presented for circular, parabolic, catenary, elliptic and sinusoidal arches (both forms – prime and quadratic) with hinged–hinged, hinged–clamped and clamped–clamped and clamped–free end restraints. Three general taper types (depth taper, breadth taper and square taper) for rectangular cross-section are studied. The lowest four natural frequencies are calculated and compared with the published results.

FURTHER RESULTS FOR MODAL CHARACTERISTICS OF ROTATING TAPERED TIMOSHENKO BEAMS

The in-plane and out-of-plane modes of free vibration of a tapered Timoshenko beam mounted on the periphery of a rotating rigid hub are investigated. The finite element method is used to discretize the beam. This formulation permits unequal breadth and depth taper ratios as well as unequal element lengths. The effects of shear deformation, rotary inertia, hub radius, setting angle, and spinning rotation are considered. The generalized eigenvalue problem is defined using explicit expressions for the mass and stiffness matrices and numerical solutions are generated for a wide range of parameter variations. Explicit expressions of Southwell coefficients are presented for the first time for the case of rotating uniform and tapered Timoshenko beams. Comparisons are made wherever possible with exact solutions and other numerical results available in the literature. Extended results are obtained to serve as a benchmark solution for other numerical techniques and specialized applications.

Influence of linear depth taper on the frequencies of free torsional vibration of a cantilever beam

The influence of depth taper on the frequencies of free torsional vibrations of a bi-symmetric I-beam with a constant flange width is investigated. The effect of various taper ratios and length-to-depth ratios is analysed for the first three modes. The problem is analysed by the finite difference method and the finite element method. The effect of shear deformation is also taken into consideration.

Compact HE11 to surface wave converters for high power waveguide dummy loads

The HE11 mode in corrugated circular waveguide can be converted to the EH11 mode (surface wave) by a short, smooth-waveguide phase shift section followed by a short corrugation depth taper. Low-power measurements at 110 GHz in 1.25 in. aluminum waveguide demonstrated approximately 99% conversion with the proper phase shift length. As expected, the conversion efficiency versus length of the phase shifter varied periodically with the period of the TE11 to TM11 beat wavelength. Since the EH11 surface wave is highly attenuated, this type of converter can be used effectively in a compact high-power dummy load.

Dynamic stability of a tapered symmetric sandwich beam

The stability of a fixed-free tapered symmetric sandwich beam under a pulsating axial force is studied. The effects of depth taper, shear parameter, core thickness and core density on the static buckling loads and the regions of parametric resonance are investigated.

Large deflections of a spring-hinged tapered cantilever beam with a rotational distributed loading

SummaryLarge deflection problem of a spring loaded hinged nonuniform cantilever beam subjected to a rotational distributed loading is formulated by means of a second-order non-linear integro-differential equation. The problem is examined by considering the beam of rectangular cross-section with linear depth taper subjected to a uniform rotational distributed load. The elastic curves of a beam for this special case are presented.

Thermal Constriction Resistance with Arbitrary Heating in a Convectively Cooled Plate

Numerical Results The results for the uniform beam given in Eqs. (16) and (17) can be seen to be in good agreement when trigonometric and algebraic functions are used for the displacements. These results are also in good agreement with the finite element results presented earlier. Table 2 presents \L and 7 values for both breadthand depth-tapered columns for three values of ft (0.0, 0.2, and 0.4). Again, the results from trigonometric and polynomial distributions are in close agreement. Further, it can be seen that for breadth-tapered columns, the linear buckling load decreases with increasing taper. The effect of nonlinearity (from 7 values) can be seen to be relatively greater in the case of depth-tapered columns and it becomes pronounced when large depth tapers are considered. Also, the effect of nonlinearity decreases with increasing breadth taper, whereas the effect increases with increasing depth taper.

Finite element analysis of thermal postbuckling of tapered columns

Thermal post-buckling behaviour of tapered columns is investigated through a finite element analysis. Columns of rectangular cross sections with breadth taper (depth being constant), depth taper (breadth being constant) and columns of circular cross sections with diameter taper are considered under simply—supported and clamped boundary conditions. Results for various cases are presented in the form of linear thermal buckling load and empirical formulae for the ratios of nonlinear thermal load to linear thermal buckling load.

Coupled bending-bending vibrations of pretwisted tapered cantilever beams treated by the Reissner method

Theoretical natural frequencies and mode shapes of the first four coupled modes of a uniform pretwisted cantilever blade and the first five coupled flexural frequencies of pretwisted tapered blading are determined by using the Reissner method. The shape functions for the bending moments and deflections are developed in series form and with these used in the dynamic Reissner functional, the frequency equation is obtained by minimizing it through the Ritz process. A convergence study made in the case of the pretwisted uniform blade indicates that there appears to be a quicker convergence of the natural frequencies and that a five-term solution yields a set of results that are in good agreement with the theoretical and experimental values of other authors, available in the literature. The mode shapes obtained from the present analysis are compared with those from an earlier investigation and the effect of ignoring the shear deflection and rotary inertia in the analysis is discussed. The effects of breadth taper and depth taper on the vibration characteristics of pretwisted cantilever blading are discussed from the results obtained in the present limited study and it is observed that an extensive investigation appears to be necessary to draw positive conclusions covering wide ranges of pretwisted blade parameters.

Large amplitude free vibrations of tapered beams

The Galerkin method is used to investigate the large-amplitude free vibrations of simply supported and clamped tapered beams of rectangular cross-section which are frequently encountered in practical structures. Two types of linear tapers are considered: breadth and depth tapers. Two solutions are obtained for each type of taper, using trigonometric and polynomial displacement distributions. Frequency-amplitude relationships are obtained for all these cases for the fundamental flexural mode. The results indicate good agreement between trigonometric and polynomial solutions for all cases. The nonlinearity is noted to be always of the hardening type and, as expected, it is severe for beams with depth taper as compared with beams with breadth taper.

Natural Frequencies of Long Tapered Cantilevers

SummaryThe effect of depth taper on the flexural vibration characteristics of a beam of uniform width is investigated in this paper. The frequency parameters and mode shapes for the first five modes of vibration of tapered beams are presented for a wide range of depth taper. The Euler-Bernoulli equation of a beam is reduced to an eigenvalue problem and its eigenvalues and eigenvectors are obtained by using a digital computer. The theoretical results are compared with those of other authors and with the experimental results.