Effect of Cutter Performance on Finished Tooth Form in Gear Shaving

The authors have been developed a computer simulation program of gear shaving. In the present paper, a new cutting model of shaving process is proposed so as to incorporate an effect of a cutting performance of shaving cutter into the simulation program. In this cutting model, it is assumed that a tooth flank material of work gear can be removed only when a depth of cut of a cutting edge exceeds a certain criterion. It is also assumed that the criterion have no definite value but has the nominal distribution over the tooth surface. The mean value of the distribution can define a characteristic of cutting performance of shaving cutter. The small mean value means that even small depth of cut can cause a stock removal; i.e. a good cutter performance. The computer simulations on gear shaving are performed to reveal the effect of the cutter performance on shaved tooth form. Under the conditions used in these simulations, the pressure angle error on the shaved tooth profile becomes remarkable as the cutter performance becomes worse. Thus, the developed computer simulation program of gear shaving has a reliability on the prediction of shaved tooth form. It will be useful for design of shaving cutter, judgement of tool life, and so on.

Belt Vibration Consideration Moving Contact and Parametric Excitation

The contacting point between belt and pulley is not fixed but moves along the pulley during vibration and that influences the free span length of the belt. The concept vibrating length is introduced and that will affect the dynamic behavior of the belt. Parametric excitation can occur through periodic variations in belt tension and speed arising from loading of the pulleys by belt-driven accessories and from engine in automotive applications. The most violent oscillations develop and resonance will occur when the frequency of excitation is close to twice or three times the natural frequency for varying tension and speed case, respectively.

Dynamic Analysis of a Multi-Mesh Helical Gear Train

In this paper, the dynamic behavior of a multi-mesh helical gear train is studied. The gear train consists of three helical gears, with one of the gears in mesh with the other two. An 18-degree-of-freedom dynamic model which includes transverse, torsional, axial and rotational (rocking) motions of the flexibly mounted gears is developed. Two different loading conditions are identified. For case I, the system is driven by the gear in the middle, and for case II, the system is driven by one of the gears at either end of the gear train. Gear mesh phases under each loading condition are determined. The natural modes are predicted, and effects of the helix angle and the loading condition on the natural modes are explained. The forced response, which includes dynamic mesh and bearing forces, due to the static transmission error excitation is found. Effects of loading conditions and asymmetric positioning on the response are also explored. The results suggest that the dynamic forces are lower if the number of teeth of the gear in the middle is (i) an odd number for case I type loading, and (ii) an even number for case II type loading.

Derivation of the Explicit Solution of the Inverse Involute Function and its Applications

The involute function ε = tanϕ – ϕ or ε = invϕ, and the inverse involute function ϕ = inv−1(ε) arise in the tooth geometry calculations of involute gears, involute splines, and involute serrations. In this paper, the explicit series solutions of the inverse involute function are derived by perturbation techniques in the ranges of |ε| < 1.8, 1.8 < |ε| < 5, and |ε| > 5. These explicit solutions are compared with the exact solutions, and the expressions for estimated errors are also developed. Of particular interest in the applications are the simple expansion ϕ = inv−1(ε) = (3ε)1/3 – 2ε/5 which gives the angle ϕ (< 45°) with error less than 1.0% in the range of ε < 0.215, and the economized asymptotic series expansion ϕ = inv−1 (ε) = 1.440859ε1/3 – 0.3660584ε which gives ϕ with error less than 0.17% in the range of ε < 0.215. The four, seven, and nine term series solutions of ϕ = inv−1 (ε) are shown to have error less than 0.0018%, 4.89 * 10−6%, and 2.01 * 10−7% in the range of ε < 0.215, respectively. The computation of the series solution of the inverse involute function can be easily performed by using a pocket calculator, which should lead to its practical applications in the design and analysis of involute gears, splines, and serrations.

A Comparison of Analytical Predictions With Experimental Measurements of Transmission Error of Misaligned Loaded Gears

This paper deals with the experimental study of dynamic transmission error of a gear pair. Two aspects of the experiment are discussed : 1) design of the test facility and data acquisition system and 2) comparison of transmission error and load distribution with experimental data. Several gears were tested under varying misalignments. A prediction program LDP (Load distribution Program) was used for theoretical calculations of dynamic transmission error.

Modeling a Flexible-Link Power Transmission System

The flexible link and sprocket system of a tracked vehicle was modeled as part of a supercomputing pilot project on a Cray X-MP supercomputer. This computer simulation model utilizes the ADAMS 3-dimensional rigid body dynamics code. Using this ADAMS model of the track system, engineers can simulate the complex action of this three dimensional mechanism, and, through the use of graphics, can illustrate the behavior of the interaction of the components in this track system.

Gear Tooth Stress Measurements of Two Helicopter Planetary Stages

Two versions of the planetary reduction stages from U.S. Army OH-58 helicopter main rotor transmissions were tested at the NASA Lewis Research Center. One sequential and one nonsequential planetary were tested. Sun gear and ring gear teeth strains were measured, and stresses were calculated from the strains. The alternating stress at the fillet of both the loaded and unloaded sides of the teeth and at the root of the sun gear teeth are reported. Typical stress variations as the gear tooth moves through the mesh are illustrated. At the tooth root location of the thin-rimmed sun gear, a significant stress was produced by a phenomenon other than the passing of a planet gear. The load variation among the planets was studied. Each planet produced its own distinctive load distribution on the ring and sun gears. The load variation was less for a three-planet, nonsequential design as compared to that of a four-planet, sequential design. The results reported enhance the data base for gear stress levels and provide data for the validation of analytical methods.

Kinematical Optimization of Spiral Bevel Gears

Kinematical optimization and sensitivity analysis of circular-cut spiral bevel gears are investigated in this paper. Based on the Gleason spiral bevel gear generator and EPG test machine, a mathematical model is proposed to simulate the tooth contact conditions of the spiral bevel gear set. All the machine settings and assembly data are simulated by simplified parameters. The tooth contact patterns and kinematic errors are obtained by the proposed mathematical model and the tooth contact analysis techniques. Loaded tooth contact patterns are obtained by the differential geometry and the Hertz contact formulas. Tooth surface sensitivity due to the variation of machine settings is studied. The corrective machine settings can be calculated by the sensitive matrix and the linear regression method. An optimization algorithm is also developed to minimize the kinematic errors and the discontinuity of tooth meshing. According to the proposed studies, an improved procedure for development of spiral bevel gears is suggested. The results of this paper can be applied to determine the sensitivity and precision requirements in manufacturing, and improve the running quality of the spiral bevel gears. Two examples are presented to demonstrate the applications of the optimization model.

Effect of Index Variation Form of an Epicyclic Gear Reducer Teeth on Turbo-Generator Set Vibrations

The index variation form of gear teeth is an important parameter that needs to be considered in analyzing vibrations of gear transmissions. Tests carried out showed the ring gear teeth index variation forms of an epicyclic reducer significantly influenced the vibrations of turbo-generator sets. The proper selection of this form combined with allowable index variation of ring gear teeth resulted in acceptable turbo-generator set vibrations.

Experimental Investigation of Flowrate Irregularity in Rotary Gear Pumps

In a previous paper the author proposed a method to reduce the periodic variation in flow rate for an external gear pump. To verify the experimental results, a series of experimental tests on a expressly realized gear pump, was carried out. The pump was equipped with relieving grooves milled into the side plates. The tests were done on a closed piping specifically realized and equipped for measuring the instantaneous flow rate of the fluid through a wedge-shaped hot film probe.