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Deep groove and angular con tact 25 and 30 mm bore ball bearings were tested to high speeds using air-oil mist lubrication. Test conditions included cooling air flow rates of 1.5, 3.0, and 6.0 scfm (0.05, 0.10, and 0.20 kg/min), thrust loads of 50, 75, and 100 lb (222, 334, and 445 N), and a constant radial load of 25 lb (111 N). Steady-state bearing outer race temperature was recorded at various speeds under each set of test conditions. Maximum ON values of 1.9 x 10⁶, 1.5 x 10⁶, 1.4 x 10⁶, and 1.26 x 10⁶ were achieved on the 30 mm deep groove, the 25 mm deep groove, the 25 mm angular contact, and the 30 mm angular contact bearings, respectively. Tests were usually terminated when the stabilized outer race temperature reached approximately 200°F (366 K) although the 30 mm deep groove bearing was operated to 240°F (389 K). A cooling air flow rate of 1.5 scfm (0.05 kg/min) was judged not adequate for high speed bearing operation under the tested conditions. An outer-race temperature prediction equation, based on a regression analysis of the test results, is presented for each test bearing. / M.S.

The purpose of this investigation was to develop an expression to represent the torque versus speed behavior of instrument ball bearings between 1000 and 40,000 rpm with various combinations of radial and axial load ranging between 0 and 200 grams. Because of the lack of experimental data for instrument bearings over any range of speeds, loads and sizes, it was necessary to construct a suitable bearing tester and accumulate the required data. The testers used were based on previous work by H.H. Mabie at Sandia Corporation and G.E. Clarke at V.P.I. The driving source was a small air turbine developed by Mabie which performed smoothly and reliably between 0 and 50,000 rpm. The torque measuring system employed strain gages on a very small beam which was used to sense forces on the stationary outer race of the bearing while the inner race was driven at speed. Each test was conducted from 0 to 40,000 rpm. The radial load took on va1ues of 50, 100, and 200 grams. The axial load was 0, 50, 100, and 200 grams. All combinations of these loads were used for each size bearing. The sizes tested were R-2, R-3, R-4. Six bearings of each size were used with all six bearings of each size undergoing the same test program in order to yield statistically reasonable averages. Investigation of analytical methods of predicting the running torque indicated that production tolerances of ball bearings rendered such an approach impractical. This led to the development of an empirical expression to predict the running torque within the same range of sizes, loads, and speeds for which experimental test data was obtained. Such an empirical expression was successfully developed and the reSUlting torque predictions compared with the experimental values of torque. The empirical expression proved capable of predicting the running torques within the envelope of the sample standard deviations for a given bearing size and loading in most cases. During the investigation of supplementary topics, it was determined that frictional heating was insignificant during the conduct of the torque tests which had a duration of approximately two minutes. All tests were at ambient temperature. All tests conducted were with oil lubricant and ribbon retainer ba1l bearings. There was no evidence that the empirical expression for friction torque developed here was valid when extrapolated beyond the limits of size, load, and speed used in its development. / Ph. D.

The design of advanced, state-of-the-art turbomachinery requires accurate analytical tools for predicting rotor response and evaluating stability. One of the required tools is a reliable analytical code for predicting the performance of fluid-film bearings. This work presents an initial evaluation of a test rig for verifying such codes. This presentation includes background information on the techniques and terminology of fluid-film bearing analysis and two basic approaches to experimental evaluation of fluid-film bearings. To establish one such code, NPADVT, as a useful tool for evaluating the performance of the test rig, comparisons between six published, experimental evaluations of fluid-film bearings for static characteristics and the corresponding NPADVT analysis are presented. With the code thus anchored, and its limits established, experimental data generated with the test rig are compared to appropriate analyses and the test rig shown to be essentially functional. Finally, experimental static results for a pocket bearing generated with the test rig are presented and compared with analysis. / Master of Science

Of the studies that have been conducted on the operational characteristics of instrument ball bearings, a great majority have been in accordance with MIL-STD-206. Since tests in compliance with this specification determine bearing quality or rate bearings comparatively, nothing was known of the operational characteristics of the bearings in their final application. A few investigators have developed sensors to study torque characteristics of instrument ball bearings. However, in no case has a report been made of the effect on torque when both radial and axial loads were varied. In seeking to obtain improvements in methods of measuring small torques, a pneumatic sensor was developed for testing R-3 instrument ball bearings under varying radial and axial loads. This sensor was based upon the principle of the flapper-nozzle valve. The flapper valve consisted of two orifices in series, one of constant area, the other of variable area, which was determined by flapper position. Since the pressure between the two orifices was dependent upon flapper position, indirect measurements of torque acting on the flapper were obtained by measuring this pressure. As a result of this study, it was concluded that the pneumatic sensor accurately measured the running torque of R-3 instrument bearings. This statement was based upon good agreement with data from other investigators working under identical conditions. It was also concluded that for a range of 50 to 200 gm. radial loading, no significant effect on torque was observed. For axial loads in the same range, the torque was found to vary in proportion to the equivalent load acting on the bearing. / Master of Science

A complete summary of this thesis would be too lengthy due to the amount of material covered, so only the most important points will be brought out. (a) The journal bearing is one of the most important elements of machinery. (b) An outline of the fundamentals of oiling and grooving bearings has been given. The bearing metals and the lubricants in use today and the functions of the lubricating engineer have been discussed. (c) Present day journal bearing practices have been described for the following types of equipment: automobile engines, aircraft engines, Diesel engines, steam turbines, electric motors, and generators, rolling mills, railroad cars and engines, heavy duty equipment, line shafting, gyroscopes, ships and steam engines. (d) The basic theory of the perfectly lubricated journal bearing is understood, but exact mathematical solution of the phenomena occurring within the bearing is as yet impossible. (e) By coordinating mathematical investigation and experimental evidence, a fairly good analysis can be made of journal bearing behavior. (f) Needs rational method of designing bearings is a good example of well coordinated mathematical investigation and experimental results. This method of analysis gives the designer a fairly accurate indication of the performance of a journal bearing to be expected. (g) Because of the limitations of theory a large number of manufacturers do extensive research and experimental work in conjunction with their products, to aid in the development of the Journal bearings used. (h) Those interested in further study on this subject are advised to study references 79 and 80 in addition to those already mentioned. Reference 79 is a survey of bearing practice undertaken in 1934 by H.A.S. Howarth. This survey is a valuable adjunct to this thesis, yet is too long to repeat here. Reference 80 is another article by Howarth summarizing the important formulas and charts that had been offered by bearing analysts during the years prior to 1935. / Master of Science

Although many studies have been made on the operating characteristics of instrument bearings, most were conducted at two rpm or less and with thrust load only. A study by H.H. Mable tested the running torques of radially loaded bearings from 1,000 to 40,000 rpm. The purpose of this investigation was to study the running torques of R-3 size instrument ball bearings at speeds up to 40,000 rpm while under combined radial and axial load. Much of this investigation was devoted to the construction of an accurate torque sensing device. The method employed relied on the amplification of strain gage signals by a strain gage indicator and an x-y plotter. The strain gages were used to detect the strain at the base of a small beam that prevented rotation of the outer race.of a test bearing while the inner race was driven at test speed by an air turbine. The accumulated data was the result or 30 test series, with each series being constituted of a test sample of six ball bearings. From the study, it was consluded that the strain gage method of torque sensing accurately measured the running torque of R-3 size ball bearings at ambient temperatures. It was also concluded that the effect of axial loading or an R-3 ball bearing loaded with 47 grams radially is negligible until the axial: load equals o.r exceeds the radial loading. By comparing lubricants, it was concluded that grease lubricated ball bearings demonstrate running torques approximately twice as great as bearings lubricated with a similar weight of oil. / Master of Science

This work investigates the dynamics and vibration in gear systems, including spur and helical gear pairs, idler gear trains, and planetary gears. The spur gear pairs are analyzed using a finite element/contact mechanics (FE/CM) model. A hybrid analytical-computational (HAC) model is proposed for nonlinear gear dynamics. The HAC predictions are compared with FE/CM results and available experimental data for validation. Chapter 2 investigates the static and dynamic tooth root strains in spur gear pairs using a finite element/contact mechanics approach. Extensive comparisons with experiments, including those from the literature and new ones, confirm that the finite element/contact mechanics formulation accurately predicts the tooth root strains. The model is then used to investigate the features of the tooth root strain curves as the gears rotate kinematically and the tooth contact conditions change. Tooth profile modifications are shown to strongly affect the shape of the strain curve. The effects of strain gage location on the shape of the static strain curves are investigated. At non-resonant speeds the dynamic tooth root strain curves have similar shapes as the static strain curves. At resonant speeds, however, the dynamic tooth root strain curves are drastically different because large amplitude vibration causes tooth contact loss. There are three types of contact loss nonlinearities: incomplete tooth contact, total contact loss, and tooth skipping, and each of these has a unique strain curve. Results show that different operating speeds with the same dynamic transmission error can have much different dynamic tooth strain. Chapters 3, 4, and 5 develops a hybrid-analytical-computational (HAC) method for nonlinear dynamic response in gear systems. Chapter 3 describes the basic assumptions and procedures of the method, and implemented the method on two-dimensional vibrations in spur gear pairs. Chapters 4 and 5 extends the method to two-dimensional multi-mesh systems and three-dimensional single-mesh systems. Chapter 3 develops a hybrid analytical-computational (HAC) model for nonlinear dynamic response in spur gear pairs. The HAC model is based on an underlying finite element code. The gear translational and rotational vibrations are calculated analytically using a lumped parameter model, while the crucial dynamic mesh force is calculated using a force-deflection function that is generated from a series of static finite element analyses before the dynamic calculations. Incomplete tooth contact and partial contact loss are captured by the static finite element analyses, and included in the force-deflection function. Elastic deformations of the gear teeth, including the tooth root strains and contact stresses, are calculated. Extensive comparisons with finite element calculations and available experiments validate the HAC model in predicting the dynamic response of spur gear pairs, including near resonant gear speeds when high amplitude vibrations are excited and contact loss occurs. The HAC model is five orders of magnitude faster than the underlying finite element code with almost no loss of accuracy. Chapter 4 investigates the in-plane motions in multi-mesh systems, including the idler chain systems and planetary gear systems, using the HAC method that introduced in Chap. 3. The details of how to implement the HAC method into those systems are explained. The force-deflection function for each mesh is generated individually from a series of static finite element analyses before the dynamic calculations. These functions are used to calculated the dynamic mesh force in the analytical dynamic analyses. The good agreement between the FE/CM and HAC results for both the idler chain and planetary gear systems confirms the capability of the HAC model in predicting the in-plane dynamic response for multi-mesh systems. Conventional softening type contact loss nonlinearities are accurately predicted by HAC method for these multi-mesh systems. Chapter 5 investigates the three-dimensional nonlinear dynamic response in helical gear pairs. The gear translational and rotational vibrations in the three-dimensional space are calculated using an analytical model, while the force due to contact is calculated using the force-deflection. The force-deflection is generated individually from a series of static finite element analyses before the dynamic calculations. The effect of twist angle on the gear tooth contact condition and dynamic response are included. The elastic deformations of the gear teeth along the face-width direction are calculated, and validated by comparing with the FE/CM results. / Ph. D. / Gears are widely used in power transmission systems. The dynamics and vibrations of the gears causes system noise because those vibrations are transmitted to the gear housing through the supporting bearings and shafts. The tooth root strains and stresses are directly related to the system failure. These effect becomes significantly important when the system is operating near resonances that high amplitude vibrations are excited and contact loss nonlinearity occurs. We want a fast, accurate, and reliable model to analyze the nonlinear dynamics in those gear and bearing systems. This work investigates the dynamics and vibration in gear systems, including spur and helical gear pairs, idler gear chains, and planetary gears. The static and dynamic tooth root strains in spur gear pairs are studied using a finite element/contact mechanics (FE/CM) approach. Extensive comparisons with experiments, including those from the literature and new ones, validates the accuracy of the FE/CM formulation. The model is then used to investigate the features of the tooth root strain curves as the gears rotate kinematically and the tooth contact condition changes. The three types of contact loss nonlinearities are investigated and explained. A hybrid analytical-computational (HAC) method is developed for nonlinear gear dynamics. This model takes advantage of the good features of the different traditional models, and is available for fast and accurate nonlinear gear dynamic analysis. The HAC method is validated by comparing with the FE/CM results, including near resonant gear speeds when high amplitude vibrations are excited and contact loss occurs. The HAC method is five orders of magnitude faster than the underlying finite element code with almost no loss of accuracy.

The active magnetic bearing (AMB) is a relatively new technology which has many advantages compared with conventional bearing design. In an AMB system, the rolling-element back-up bearings are indispensable to protect the magnetic bearing rotor and stator, and other stationary seals along the rotor shaft. In this paper, a theoretical formulation is proposed and solved numerically to examine the transient response of the flexible rotor, from the time just previous to the AMB shuts down and including the rotor drop onto the back-up bearing. The backward whirl of the rotor, which may lead to the destructive damage of the machinery, has been analytically predicted at very light support damping and very high support damping. Also, the vibration due to the non-linearity of the contact point geometry has been included in the analysis. The influence of the support damping on the displacement of the disk and also the contact force between the journal and the inner-race of the back-up bearing have been computed for various rotor system parameters. By comparing these results with the optimum support damping for the simple flexible rotor model, it is shown that this support damping optimization can be applicable for specifying the required optimum range of support damping for the back-up bearings of AMB systems. / Master of Science

The need for better performance of turbomachinery with active magnetic bearings has necessitated a study of such systems for accurate prediction of their vibrational characteristics. This research presents a modification of existing transfer matrix methods for rotor analysis, to predict the response of rotor systems with active magnetic bearings. The position of the magnetic bearing sensors is taken into account and the effect of changing sensor position on the vibrational characteristics of rotor systems is studied. The modified algorithm is validated using a simpler modified Jeffcott model. The effect of changing from a rotating unbalance excitation to a constant excitation in a single plane is also studied. An eight-stage centrifugal compressor rotor is analyzed using the modified transfer matrix code. The results for a two-mass Jeffcott model are presented as plots of critical frequency vs. sensor position and amplitude at critical frequency vs. sensor position. Plots of amplitude vs. frequency and phase angle vs. frequency for different cases of sensor location are also presented. The results obtained by analyzing this two-mass model with the modified transfer matrix method have been compared with the results of the modified Jeffcott analysis for the purpose of verification. Also included are plots of amplitude vs. frequency and phase angle vs. frequency for the eight-stage centrifugal compressor rotor. These plots will demonstrate the significant influence that sensor location has on the critical frequencies and the amplitudes at the critical frequencies of the rotor system. / Master of Science