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Procedures for computer-modeling of spur gear tooth fabrication are given. It is shown that the standard involute tooth form results from a cutter with an involute shape rolling onto a gear blank. Specifically, the envelope of an involute is an involute. Examples are given and applications are discussed.
Geared drive train transmits torque from input shaft in equal parts along two paths in parallel, then combines torques in single output shaft. Scheme reduces load on teeth of meshing gears while furnishing redundancy to protect against failures. Such splitting and recombination of torques common in design of turbine engines.
A new mathematical model is proposed to examine the vibration transmission through rolling element bearings in geared rotor systems. Current bearing models, based on either ideal boundary conditions for the shaft or purely translational stiffness element description, cannot explain how the vibratory motion may be transmitted from the rotating shaft to the casing. Experimental results have shown that the casing plate motion is primarily flexural. Here, this issue is clarified qualitatively and quantitatively by developing a comprehensive bearing stiffness matrix of dimension 6 to model precision rolling element bearings using basic principles. The proposed bearing stiffness matrix is partially verified using available analytical and experimental data, and is completely characterized. The study extends the proposed bearing formulation to analyze the overall geared rotor system dynamics including casing and mounts. The bearing stiffness matrix is included in discrete system models using lumped parameter and/or dynamic finite element techniques.
Optimization procedures allow one to design a spur gear reduction for maximum life and other end use criteria. A modified feasible directions search algorithm permits a wide variety of inequality constraints and exact design requirements to be met with low sensitivity to initial guess values. The optimization algorithm is described, and the models for gear life and performance are presented. The algorithm is compact and has been programmed for execution on a desk top computer. Two examples are presented to illustrate the method and its application.
A computer program developed for optimizing spur gear designs, SPUROPT, was updated by installing a new subroutine that uses AGMA 908-B89 standards to calculate the J-factor for determining tooth-bending stress. The updated SPUROPT program was then used to optimize a spur gear set for maximum fatigue life, minimum dynamic load, or minimum weight. All calculations were made with constraints on as many as 13 parameters by using three design variables: the number of teeth, diametral pitch, and tooth-face width. Results depended largely on constraints values. When the limiting bending stress was set at a high value, the optimal solution was the highest allowable number of teeth. When the allowable bending stress was lowered, the optimal solution moved toward the fewest number of teeth permitted. Final results were also affected by the amount of transmission error. A lower error permitted a higher number of teeth.
The backlash nonlinearity excited primarily by transmission error between spur gear pairs is studied for both external and internal excitations. The digital simulation technique and the method of harmonic balance are used to develop steady state solutions for the internal sinuosidal excitations. The analytic predictions agreed well with available experimental data. Digital simulation is used to observe that at the chaotic and subharmonic resonances may exist in a gear pair depending on the mean or design load, mean to alternating force ratio, damping, and backlash.
A recursive formulation is presented for the calculation of the inverse and forward dynamics of rigid robot manipulators with gear systems on each joint. The complete effects of the gear ratios and the gyroscopic effects of the spinning motor/gear are included in the recursive formulation. The forward dynamics solution recursively calculates the joint accelerations when given motor torques, and the number of computations grows linearly with the number of links.
Optimization procedures allow one to design a spur gear reduction for maximum life and other end use criteria. A modified feasible directions search algorithm permits a wide variety of inequality constraints and exact design requirements to be met with low sensitivity to initial guess values. The optimization algorithm is described, and the models for gear life and performance are presented. The algorithm is compact and has been programmed for execution on a desk top computer. Two examples are presented to illustrate the method and its application.
The paper proposes a reduced-order analytical model of a spur gear pair which consists of two identical spur gears, two identical flexible shafts, and four identical rolling element bearings of a given radial stiffness. The error associated with the undamped eigensolution is estimated by a comparison with a benchmark finite element model.
A new approach to modeling gear tooth surfaces is discussed. A computer graphics solid modeling procedure is used to simulate the tooth fabrication process. This procedure is based on the principles of differential geometry that pertain to envelopes of curves and surfaces. The procedure is illustrated with the modeling of spur, helical, bevel, spiral bevel, and hypoid gear teeth. Applications in design and manufacturing are discussed. Extensions to nonstandard tooth forms, to cams, and to rolling element bearings are proposed.
The deviations of a gear's real tooth surface from the theoretical surface are determined by coordinate measurements at the grid of the surface. A method was developed to transform the deviations from Cartesian coordinates to those along the normal at the measurement locations. Equations are derived that relate the first order deviations with the adjustment to the manufacturing machine-tool settings. The deviations of the entire surface are minimized. The minimization is achieved by application of the least-square method for an overdetermined system of linear equations. The proposed method is illustrated with a numerical example for hypoid gear and pinion.
The bearing matrix formulations proposed by Lim and Singh (1990) are extended to analyze the overall dynamics of a geared rotor system which includes a spur gear pair, shafts, rolling-element bearings, a motor, a load, a casing, and flexible or rigid mounts. For this purpose, discrete vibration models are developed and used to predict vibration transmission through the bearings and to investigate the effects of the bearing, casing, and mount dynamics on the dynamic characteristics of the internal rotating system. Analytical predictions show that the theory is capable of predicting the bearing and mount moment transmissibilities in addition to the force transmissibilities. The predicted flexural vibrations of the casing plate are in good agreement with measurements conducted on an experimental set-up that consisted of a high-precision beam and pinion, and four identical rolling element bearings contained in a flexible casing mounted rigidly on a massive foundation.
The vibroacoustic response of a generic geared rotor system is analyzed on an order of magnitude basis utilizing an approximate statistical energy analysis method. This model includes a theoretical coupling loss factor for a generic bearing component, which properly accounts for the vibration transmission through rolling element bearing. A simplified model of a NASA test stand that assumes vibratory energy flow from the gear mesh source to the casing through shafts and bearings is given as an example. Effects of dissipation loss factor and gearbox radiation efficiency models are studied by comparing predictions with NASA test results.
Elastomeric bearings make torque loads more nearly equal. Bearings couple drive shaft with ring spur gear. Inner elastomeric bearing radially stiff and circumferentially compliant, while outer elastomeric bearings circumferentially stiff and radially compliant. Combination accommodates minor variations in dimensions and placements of gears, shafts, and other components.
Tooth surfaces of helical gears modified, according to proposed design concept, to make gears more tolerant of misalignments and to improve distribution of contact stresses. Results in smaller transmission errors, with concomitant decreases in vibrations and noise and, possibly, increases in service lives.
In this paper an automated algorithmic method is presented for the dynamic analysis of geared trains/transmissions. These are treated as a system of interconnected flexible bodies. The procedure developed explains the switching of constraints with time as a result of the change in the contacting areas at the gear teeth. The elastic behavior of the system is studied through the employment of three-dimensional isoparametric elements having six degrees-of-freedom at each node. The contact between the bodies is assumed at the various nodes, which could be either a line or a plane. The kinematical expressions, together with the equations of motion using Kane's method, strain energy concepts, are presented in a matrix form suitable for computer implementation. The constraint Jacobian matrices are generated automatically based on the contact information between the bodies. The concepts of the relative velocity at the contacting points at the tooth pairs and the subsequent use of the transmission ratios in the analysis is presented.
Discussed here are computations of drag or negative traction of geared down supporting propellers in the downward vertical glide of a helicopter. By means of Frounde's Theory, the maximum value of the drag of a windmill is calculated. For wooden propellers, the author finds that the difference between the drag and the weight is proportional to the number of blades and is larger for propellers of small diameter; thus it is 25 kg. for a six blade propeller with a diameter of 2 m. 50. The author notes that if we are to adopt large propellers, we must have recourse to a different method of construction, resulting in large dimension propellers much lighter than those made of wood. In discussing insufficient drag, the author notes that the question of the drag of geared down supporting propellers can only be decided by experiment.
Report deals with the impact forces in landing gears. Both the landing impact and the taxiing impact have been considered, but drag forces have so far been excluded. The differential equations are developed and their numerical integration is shown, considering the nonlinear properties of the oleo shock strut. A way is shown for determining the dimensions of the metering pin from a given load-time diagram. A review of German literature on landing-gear impact is also presented.