外文文獻(xiàn)翻譯--使用靜壓軸承減輕齒輪嚙合頻率噪聲【中文4186字】 【PDF+中文WORD】,中文4186字,PDF+中文WORD,外文文獻(xiàn)翻譯,使用靜壓軸承減輕齒輪嚙合頻率噪聲【中文4186字】,【PDF+中文WORD】,外文,文獻(xiàn),翻譯,使用,靜壓,軸承,減輕,齒輪,嚙合,頻率,噪聲,中文 Mitigation of Gear Mesh-Frequency Noise Using a Hydrostatic Bearing accuracy while maintaining high bearing stiffness facilitates the transmission of the gear mesh-frequency vibrations. A proposed design to address gear mesh-frequency vibrations Downloaded From: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 02/25/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use consists of hydrostatic bearings placed in series, load wise, with the primary shaft support bearing, configured to ensure that the Zamir A. Zulkefli1 Faculty of Engineering, Mechanical and Manufacturing Engineering, Universiti Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia e-mail: zamirdin@upm.edu.my Maurice L. Adams, Jr. Case School of Engineering, Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH 44106-71222 e-mail: maurice.adams@case.edu A proposed solution to reducing gear mesh-frequency vibrations in a gear-set involves the utilization of hydrostatic bearings placed in series, load wise, with the main support bearing. The hydrostatic bearings are expected to utilize its low pass filtering effect of the vibrational energies to prevent its transmission from the shaft to the gear housing where it would be emitted as noise. The present investigation examines the frequency response of a single-recess circular hydrostatic bearing under applied sinusoi- dal loads. The results show that as the driving frequency increases, the filtering effect of the hydrostatic bearing increases. The exhibited behavior is similar to the behavior of a low pass filter: negligible filtering effect at low frequencies, the filtering effect increasing from 0% to 90% over the midfrequencies range and the filtering effect remaining at the maximum value as the frequencies of the applied signals continue to increase. This observed behavior is expected to play a central role in the pro- posed gear mesh-frequency vibration mitigation system. [DOI: 10.1115/1.4029613] Introduction The origin of gear mesh-frequency acoustic noise is recognized as the ever present residual manufacturing imperfections, tooth elasticity, and sliding friction that preclude perfect conjugate actions between the gears [1,2]. It is also known that most gear- set generated acoustic noise first passes as mesh-frequency vibra- tions primarily through the shaft support bearings to the housing, which then emits the vibrational energy as acoustic noise [3]. Gear-set generated noise is concentrated at the mesh-frequency, which is typically quite acoustically objectionable and produces more intense gear mesh-generated vibrations as the pitch line velocity increases. Whether the gear shaft support bearings are of the rolling-element or hydrodynamic fluid-film type, high bearing stiffness is an obvious requirement to maintain the needed gear centerline positioning accuracy so that the level of precision man- ufactured into the gear-set is realized during its performance. However, high bearing stiffness facilitates the transmission of the gear mesh-frequency vibrations. So therein lies the dilemma of attempts at gear noise attenuation measures, reduction in the bear- ing stiffness jeopardizes the required gear centerline positioning 1Corresponding author. Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 17, 2014; final manuscript received December 2, 2014; published online February 20, 2015. Assoc. Editor: Philippe Velex. high overall static stiffness normally required from a gear-set is maintained as much as possible. The hydrostatic bearings mean- while act as a low pass filter for the vibratory energy. Employing hydrostatic bearings alone as the main bearings could accomplish the same noise attenuation objective as the proposed attenuation system, but with a potential for lower operational reliability due to the lower reliability of the hydrostatic bearing. Additionally, Zaretsky [4] relates how rolling bearing life predictions that account only for static load can be considered optimistic by not including the additional contribution of vibration induced dynamic loads. Thus, the potential for rolling-element bearing fatigue life extension from the hydrostatic backup is also suggested. Moreover, significant attenuation of mesh-frequency noise is a desirable design objective for gear-sets. However, the trade-offs in actual gear-set designs will generally discourage making the elimination of mesh-frequency noise the only design objective. In that context, the proposed vibration mitigation system, even with the addition of supporting systems, facilitates its incorporation into current gear-set designs with minimal effect on the perform- ance of the gear-sets. Thus, the advantages of the proposed system: vibration mitigation and the preservation of the gear performance, are expected to outweigh the disadvantages of incor- porating the proposed system in the gear-set design. Previous work on hydrostatic bearings has focused on its dynamic behavior, specifically on the ability of the bearing to ful- fill high stiffness and high damping requirements [5–7]. Rohde and Ezzat meanwhile investigated the effect of lubricant compres- sibility on the dynamic behavior of hydrostatic bearings [8]. The findings based on full numerical solutions of the Reynolds lubrica- tion equations determined that under the influence of the fluid’s compressibility, the dynamic behavior of the bearing is character- ized by a “break frequency” above which the bearing stiffness increases sharply in conjunction with a sharp decrease in the bear- ing damping. Similar results were reported by other researchers [9–12]. These works, however, have typically assumed highly compressible fluids as the working fluid. The present investiga- tion, instead examines the frequency response of the hydrostatic bearing when the working fluid, typically assumed to be incom- pressible, is now instead assumed to be weakly compressible. The compressibility of this working fluid is quantified via its finite bulk modulus, instead of infinite for a truly incompressible fluid. Control Volume Approach for Modeling the Low Pass Filtering Effect The development of the model in the present investigation fol- lows the work presented in detail by Zulkefli for a simple single- recess hydrostatic bearing [13]. The hydrostatic bearing consists of two main parts: the bearing pad and bearing runner where the former is made up of a bearing recess and relatively thin bearing sills while the latter is made up of a flat surface that totally enclo- ses the bearing recess. During operation of the bearing, externally pressurized fluid is pumped into the recess, filling the available space. As the fluid continues to be pumped into the recess, the fluid pressure increases until the pressure is high enough to sepa- rate the bearing pad from the bearing runner, allowing the fluid to flow out over the bearing sills. In formulating the low pass behavior, it is assumed that the fluid pressure in the trapped fluid volume within the recess, V, though not constant in time, is uniform throughout V at all times. Addi- tionally, V is selected to be a constant-volume control volume defined by the enclosed bearing recess. At static load condition, the volumetric inflow and outflow into the control volume are equal. Under dynamic conditions, the inflow and outflow are not Journal of Vibration and Acoustics Copyright VC 2015 by ASME JUNE 2015, Vol. 137 / 034502-1 required to be instantaneously equal. Fluid inertia is neglected assuming that only viscous effects dominate. The three recognized methods for flow compensation in hydro- X ? sin XT and X ? xV Q in (9) static bearings are orifice, capillary, and constant flow [14]. For the analysis, constant flow compensation is assumed. The bulk modulus, b, of the fluid can be defined in terms of V, the incre- mental change in the fluid volume, DV, and the incremental The convolution integral for the transmitted dynamic pressure, DPX is then change in the fluid pressure Dp as 三 b DpV DV  (1) DPX ? eT dX 0 ds DPeT sTds ? X eT r 1 e BeT sT 0 cos Xsds (10) Since the control volume is defined to be constant, the incremental From Ref. [15], Eq. (10) was integrated into the following: change in the control volume, DV is zero. Instead, the incremental change in the trapped fluid mass, Dm, is used, and the relationship between the two determined from the expression for the fluid den- sity is: DV ? Dm=q0. Here, q0 is the nominal fluid density and DPX ? sin XT X2 sin XT t XB cos XT B2 t X2 Be BT X  (11) is assumed to be very much bigger than the incremental change in the fluid density. The bulk modulus is then The steady state portion of which is X2 sin XT t XB cos XT ? b DpVq0 (2) Dm DPX ? sin XT B2 t X 2 (12) The instantaneous mass inflow rate, mass outflow rates, and the time rate of change of the mass can be written in terms of the vol- umetric inflow, Qin, the volumetric outflow coefficient, Qout, the The single peak amplitude of the harmonically varying dynamic pressure at a frequency X is thus expressible as outflow coefficient, C, and the nominal fluid pressure, p0 as (「 jDPXj ? 1 X2 12 「 XB t 12)1=2 (13) dm m_ in m_ out ? dt ? q0eQin QoutT ? q0?Qin Cep0 t DpT] (3) B2 t X2 B2 t X2 The value of the outflow coefficient at static load condition is C ? Qin=p0. Integrating Eq. (3) and employing the expression for the bulk modulus in Eq. (2) yields the following equation: Examination of Eq. (13) shows that as the frequency, X is made smaller, the amplitude jDPXj will approach that of the unity, indi- cating that all the applied dynamic force is transmitted across the hydrostatic bearing. Instead, when X is increased, jDPXj will instead approach zero, indicating that all the applied dynamic DpV et b ? Qint Cp0t C 0 Dpds (4) forces are not transmitted across the hydrostatic bearing. This behavior, no filtering at low frequencies and total filtering at high frequencies, is similar to the behavior expected from a low pass The instantaneous load transmitted through the hydrostatic bear- ing film is instantaneously proportional to the hydrostatic bearing pressure, hence the transmitted dynamic load is proportional to Dp. Integrating Eq. (4) and rewriting it as a first-order linear ordinary differential equation gives filter. The present investigation shows that a similar behavior is observed in the data. Experimental Results The frequency response of the hydrostatic bearing was deter- bC V eDpT0 t Dp b V eQin Cp0T ? 0 (5) mined using the setup described in detail in Refs. [13,16]. The setup consists of a single-recess circular hydrostatic bearing under the action of applied dynamic loads, placed in a materials testing The nondimensional pressure and time are chosen as Dp (Qin\ machine. The loads are applied on the bearing pad, and the trans- mitted loads were measured at the bearing runner. The hydrostatic bearing used in the experiment was sized using the methods 1 DP 三 Dp ; T 三 t (6) V outlined in Ref. [14] for constant flow compensation scheme. The important system parameters used to size the bearing were: driving frequency range of 1–100 Hz, lubricant fluid flow of The nondimensional form of Eq. (5) is then rewritten as d dT eDPT t BDP B ? 0 (7) where B is the design factor and is defined as B 三 bC=Qin ? b=p0. An exact solution of Eq. (7) was found to be DP ? 1 e BT (8) Equation (8) is observed to be the theoretical pressure response to a step change in the outflow. To determine the time response to a specified shaft vibration signal, a convolution integral was utilized to determine the response of the dynamic pressure, and therefore dynamic force, transmitted through the bearing under the action of a harmonic input. The dimensionless harmonic shaft vibration and dimensionless frequency are chosen as follows: 0:95 × 10 5 6 0:16 × 10 5 m3 s 1 (0:150 6 0:025 gpm), supply pressure of 2:76 × 105 6 0:03 × 105 Pa (40 6 0:5 psi), and a nominal applied load range of 300 500 6 2 N. The applied loads consist of a sinusoidal signal at a constant frequency with an amplitude that is ten percent (10%) of the applied nominal loads. The dimensions of the hydrostatic bearing used for the setup are: recess diameter of 0:0762 6 0:0025 m (3:00 6 0:01 in.), recess depth of 0:0076 6 0:0025 m (0:30 6 0:01 in.), and sill thickness of 0:0032 6 0:0025 m (0:13 6 0:01 in.). The working fluid used was shell spindle oil ten. The frequency response of the transmitted loads for different values of the applied load is shown in Fig. 1. It is observed from the figure that for a driving frequency between 1 Hz and 40 Hz, the transmitted loads show no significant change in value from the applied input loads. For a driving frequency between 40 Hz and 70 Hz, the transmitted loads show a steady decrease to around 50 N regardless of the value of the applied loads. For a driving 034502-2 / Vol. 137, JUNE 2015 Transactions of the ASME Fig. 1 Frequency response of the transmitted loads Fig. 2 Normalized frequency response of the transmitted loads frequency between 70 Hz and 100 Hz, the transmitted loads remain fairly stable at around 50 N. A similar filtering behavior is observed from the normalized frequency response of the transmit- ted loads as shown in Fig. 2. It is clearly observable from the figure that for all values of the nominal applied loads, the hydro- static bearing is able to filter out almost 90% of the applied loads. Furthermore, the figure shows that the filtering occurs over the same frequency range for all three load conditions. Therefore, the low pass filtering behavior observed from the frequency response of the hydrostatic bearing is similar to the expected behavior given by Eq. (13). Furthermore, examination of Eq. (13) shows that the equation provides a relationship between V and the frequency response, of the hydrostatic bearing. Thus, by choosing an appropriate value of V, the frequency range over which the filtering occurs can be determined. Conversely, by choosing the frequency range over which the filtering is expected to occur, V can be determined allowing the hydrostatic bearing to be sized to the frequency range of interest. For the present investigation, the maximum driving frequency for the experimental setup was limited to 100 Hz primarily due to safety concerns and machinery limitations. How- ever, it is expected that the same general low pass filtering Journal of Vibration and Acoustics JUNE 2015, Vol. 137 / 034502-3 behavior predicted by Eq. (13) and observed in Figs. 1 and 2 will be observed for higher driving frequency values. Similarly, it is expected that the same behavior predicted and observed in the present investigation will be observed when the value of the applied loads is higher. Ultimately, it is expected that the low pass filtering behavior observed from the present investigation will be successfully incorporated in the proposed gear mesh-frequency vibration mitigation system to disrupt the transmission of vibratory energy from gear-sets. Conclusion The present investigation shows that a hydrostatic bearing is able to prevent the transmission of applied vibratory loads at cer- tain frequency ranges. This observed behavior indicates the ability on the part of the hydrostatic bearing to act as a low pass filter of vibratory loads. It is expected that the same general low pass fil- tering behavior will be observed when the driving frequencies and system parameters are changed to values that are typically encountered in an actual gear-set. The proposed vibration mitiga- tion system is expected to utilize this observed low pass filtering behavior of the hydrostatic bearing to prevent the transmission of gear mesh-frequency vibrations through the gear-set to the hous- ing where it would be emitted as noise. References [1] Smith, J. D., 1983, Gears and Their Vibration: A Basic Approach to Under- standing Gear Noise, Macmillan Press, New York, pp. 170. [2] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory, 2nd ed., Cambridge University Press, New York, pp. 800. [3] Houser, D. R., 2007, “Gear Noise and Vibration Prediction and Control Methods,” Handbook of Noise and Vibration Control, M. J. Crocker, ed., Wiley, Hoboken, NJ, pp. 847–856. [4] Zaretsky, E. V., 1999, STLE Life Factors for Rolling Bearings, 2nd ed., Society of Tribologists and Lubrication Engineers, Park Ridge, IL. [5] Brown, G. M., 1961, “The Dynamic Characteristics of a Hydrostatic Thrust Bearing,” Int. J. Mach. Tool Des. Res., 1(1), pp. 157–171. [6] Adams, M. L., and Shapiro, W., 1969, “Squeeze Film Characteristics in Flat Hydrostatic Bearings With Incompressible Flow,” Tribology, 12(3), pp. 183–189. [7] Bouzidane, A., and Thomas, M., 2007, “Equivalent Stiffness and Damping Investigation of a Hydrostatic Journal Bearing,” Tribol. Trans., 50(2), pp. 257–267. [8] Rohde, S. M., and Ezzat, H. A., 1976, “Dynamic Behavior of Hybrid Journal- Bearings,” ASME J. Tribol., 98(1), pp. 90–94. [9] Ghosh, M. K., and Viswanath, N. S., 1987, “Recess Volume Fluid Compressibility Effect on the Dynamic Characteristics of Multirecess Hydrostatic Journal Bearings With Journal Rotation,” ASME J. Tribol., 109(3), pp. 417–426. [10] Ghosh, M. K., Guha, S. K., and Majumdar, B. C., 1989, “Rotordynamic Coefficients of Multirecess Hybrid Journal Bearings,” Wear, 129(2), pp. 245–259. [11] Andres, L. A. S., 1991, “Effects of Fluid Compressibility on the Dynamic Response of Hydrostatic Journal Bearings,” Wear, 146(2), pp. 269–283. [12] Pollmann, E., and Vermeulen, M., 1989, “Compressibility and Inertia Effects on the Dynamic Behavior of Recessed Hydrostatic Bearings,” Tribol. Int., 22(3), pp. 166–176. [13] Zulkefli, Z. A., 2013, “Mitigation of Gear Mesh-Frequency Vibrations Using a Hydrostatic Bearing,” Ph.D. thesis, Case Western Reserve University, Cleve- land, OH. [14] Rippel, H. C., 1963, “Cast Bronze Hydrostatic Bearing Design Manual,” Cast Bronze Bearing Institute, Cleveland, OH, pp. 75. [15] Thomas, G. B., 1957, Calculus, Addison-Wesley, Reading, MA, pp. 692. [16] Zulkefli, Z., and Adams, M. L., 2014, “Experimental Investigation of the Low Pass Filtering Effect of a Hydrostatic Bearing,” SAE Technical Paper No. 2014-01-1758. 034502-4 / Vol. 137, JUNE 2015 Transactions of the ASME