Efficient reconstruction of dispersive dielectric profiles using time domain reflectometry (TDR)

We present a numerical model for time domain reflectometry (TDR) signal propagation in dispersive dielectric materials. The numerical probe model is terminated with a parallel circuit, consisting of an ohmic resistor and an ideal capacitance. We derive analytical approximations for the capacitance, the inductance and the conductance of three-wire probes. We couple the time domain model with global optimization in order to reconstruct water content profiles from TDR traces. For efficiently solving the inverse problem we use genetic algorithms combined with a hierarchical parameterization. We investigate the performance of the method by reconstructing synthetically generated profiles. The algorithm is then applied to retrieve dielectric profiles from TDR traces measured in the field. We succeed in reconstructing dielectric and ohmic profiles where conventional methods, based on travel time extraction, fail

Dynamic measurements of gear tooth friction and load

As part of a program to study fundamental mechanisms of gear noise, static and dynamic gear tooth strain measurements were made on the NASA gear-noise rig. Tooth-fillet strains from low-contact ratio-spur gears were recorded for 28 operating conditions. A method is introduced whereby strain gage measurements taken from both the tension and compression sides of a gear tooth can be transformed into the normal and frictional loads on the tooth. This technique was applied to both the static and dynamic strain data. The static case results showed close agreement with expected results. For the dynamic case, the normal-force computation produced very good results, but the friction results, although promising, were not as accurate. Tooth sliding friction strongly affected the signal from the strain gage on the tensionside of the tooth. The compression gage was affected by friction to a much lesser degree. The potential of the method to measure friction force was demonstrated, but further refinement will be required before this technique can be used to measure friction forces dynamically with an acceptable degree of accuracy

Dynamic analysis of multimesh-gear helicopter transmissions

A dynamic analysis of multimesh-gear helicopter transmission systems was performed by correlating analytical simulations with experimental investigations. The two computer programs used in this study, GRDYNMLT and PGT, were developed under NASA/Army sponsorship. Parametric studies of the numerical model with variations on mesh damping ratios, operating speeds, tip-relief tooth modifications, and tooth-spacing errors were performed to investigate the accuracy, application, and limitations of the two computer programs. Although similar levels of dynamic loading were predicted by both programs, the computer code GRDYNMLT was found to be superior and broader in scope. Results from analytical work were also compared with experimental data obtained from the U.S. Army's UH-60A Black Hawk 2240-kW (3000-hp) class, twin-engine helicopter transmission tested at the NASA Lewis Research Center. Good correlation in gear stresses was obtained between the analytical model simulated by GRDYNMLT and the experimental measurements. More realistic mesh damping can be predicted through experimental data correlation

A comparison between theoretical prediction and experimental measurement of the dynamic behavior of spur gears

A comparison was made between computer model predictions of gear dynamics behavior and experimental results. The experimental data were derived from the NASA gear noise rig, which was used to record dynamic tooth loads and vibration. The experimental results were compared with predictions from the DSTO Aeronautical Research Laboratory's gear dynamics code for a matrix of 28 load speed points. At high torque the peak dynamic load predictions agree with the experimental results with an average error of 5 percent in the speed range 800 to 6000 rpm. Tooth separation (or bounce), which was observed in the experimental data for light torque, high speed conditions, was simulated by the computer model. The model was also successful in simulating the degree of load sharing between gear teeth in the multiple tooth contact region

Profile modification to minimize spur gear dynamic loading

An analytical computer simulation program for dynamic modeling of low-contact-ratio spur gear systems is presented. The procedure computes the static transmission error of the gears operating under load and uses a fast Fourier transform to generate the frequency spectrum of the static transmission error at various tooth profile modifications. The dynamic loading response of an unmodified (perfect involute) gear pair was compared with that of gears with various profile modifications. Correlations were found between various profile modifications and the resulting dynamic loads. An effective error, obtained from frequency domain analysis of the static transmission error of the gears, gave a very good indication of the optimum profile modification to reduce gear dynamic loading. Design curves generated by dynamic simulation at various profile modifications are given for gear systems operated at various loads. Optimum profile modifications can be determined from these design curves for improved gear design

Influence of linear profile modification and loading conditions on the dynamic tooth load and stress of high contact ratio gears

A computer simulation for the dynamic response of high-contact-ratio spur gear transmissions is presented. High contact ratio gears have the potential to produce lower dynamic tooth loads and minimum root stress but they can be sensitive to tooth profile errors. The analysis presented examines various profile modifications under realistic loading conditions. The effect of these modifications on the dynamic load (force) between mating gear teeth and the dynamic root stress is presented. Since the contact stress is dependent on the dynamic load, minimizing dynamic loads will also minimize contact stresses. It is shown that the combination of profile modification and the applied load (torque) carried by a gear system has a significant influence on gear dynamics. The ideal modification at one value of applied load will not be the best solution for a different load. High-contact-ratio gears were found to require less modification than standard low-contact-ratio gears. High-contact-ratio gears are more adversely affected by excess modification than by under modification. In addition, the optimal profile modification required to minimize the dynamic load (hence the contact stress) on a gear tooth differs from the optimal modification required to minimize the dynamic root (bending) stress. Computer simulation can help find the design tradeoffs to determine the best profile modification to satisfy the conflicting constraints of minimizing both the load and root stress in gears which must operate over a range of applied loads

Computer-aided design of high-contact-ratio gears for minimum dynamic load and stress

A computer aided design procedure is presented for minimizing dynamic effects on high contact ratio gears by modification of the tooth profile. Both linear and parabolic tooth profile modifications of high contact ratio gears under various loading conditions are examined and compared. The effects of the total amount of modification and the length of the modification zone were systematically studied at various loads and speeds to find the optimum profile design for minimizing the dynamic load and the tooth bending stress. Parabolic profile modification is preferred over linear profile modification for high contact ratio gears because of its lower sensitivity to manufacturing errors. For parabolic modification, a greater amount of modification at the tooth tip and a longer modification zone are required. Design charts are presented for high contact ratio gears with various profile modifications operating under a range of loads. A procedure is illustrated for using the charts to find the optimum profile design

Analysis and modification of a single-mesh gear fatigue rig for use in diagnostic studies

A single-mesh gear fatigue rig was analyzed and modified for use in gear mesh diagnostic research. The fatigue rig allowed unwanted vibration to mask the test-gear vibration signal, making it difficult to perform diagnostic studies. Several possible sources and factors contributing to the unwanted components of the vibration signal were investigated. Sensor mounting location was found to have a major effect on the content of the vibration signal. In the presence of unwanted vibration sources, modal amplification made unwanted components strong. A sensor location was found that provided a flatter frequency response. This resulted in a more useful vibration signal. A major network was performed on the fatigue rig to reduce the influence of the most probable sources of the noise in the vibration signal. The slave gears were machined to reduce weight and increase tooth loading. The housing and the shafts were modified to reduce imbalance, looseness, and misalignment in the rotating components. These changes resulted in an improved vibration signal, with the test-gear mesh frequency now the dominant component in the signal. Also, with the unwanted sources eliminated, the sensor mounting location giving the most robust representation of the test-gear meshing energy was found to be at a point close to the test gears in the load zone of the bearings

Effect of operating conditions on gearbox noise

Low contact ratio spur gears were tested in the NASA gear noise rig to study the noise radiated from the top of the gearbox. The measured sound power from the gearbox top was obtained from a near field acoustic intensity scan taken at 63 nodes just above the surface. The sound power was measured at a matrix of 45 operating speeds and torque levels. Results are presented in the form of a spectral speed map and as a plot of sound power versus torque (at constant speed) and as sound power versus speed (at constant torque). Because of the presence of vibration modes, operating speed was found to have more impact on noise generation than torque level. A NASA gear dynamics code was used to compute the gear tooth dynamic overload at the same 45 operating conditions used for the experiment. Similar trends were found between the analytical results for dynamic tooth overload and experimental results for sound power. Dynamic analysis may be used to design high quality gears with profile relief optimized for minimum dynamic load and noise

Effect of contact ratio on spur gear dynamic load

A computer simulation is presented which shows how the gear contact ratio affects the dynamic load on a spur gear transmission. The contact ratio can be affected by the tooth addendum, the pressure angle, the tooth size (diametral pitch), and the center distance. The analysis presented was performed using the NASA gear dynamics code, DANST. In the analysis, the contact ratio was varied over the range 1.20 to 2.40 by changing the length of the tooth addendum. In order to simplify the analysis, other parameters related to contact ratio were held constant. The contact ratio was found to have a significant influence on gear dynamics. Over a wide range of operating speeds, a contact ratio close to 2.0 minimized dynamic load. For low contact ratio gears (contact ratio less than 2.0), increasing the contact ratio reduced the gear dynamic load. For high contact ratio gears (contact ratio = or greater than 2.0), the selection of contact ratio should take into consideration the intended operating speeds. In general, high contact ratio gears minimized dynamic load better than low contact ratio gears