Effect of the Tooth Surface Waviness on the Dynamics and Structure-Borne Noise of a Spur Gear Pair

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1 Published 05/13/2013 Copyright 2013 SAE International doi: / saepcmech.saejournals.org Effect of the Tooth Surface Waviness on the Dynamics and Structure-Borne Noise of a Spur Gear Pair Sriram Sundar, Rajendra Singh, Karthik Jayasankaran and Seungbo Kim The Ohio State University ABSTRACT This article studies the effects of tooth surface waviness and sliding friction on the dynamics and radiated structureborne noise of a spur gear pair. This study is conducted using an improved gear dynamics model while taking into account the sliding frictional contact between meshing teeth. An analytical six-degree-of-freedom (6DOF) linear time varying (LTV) model is developed to predict system responses and bearing forces. The time varying mesh stiffness is calculated using a gear contact mechanics code. A Coulomb friction model is used to calculate the sliding frictional forces. Experimental measurements of partial pressure to acceleration transfer functions are used to calculate the radiated structure-borne noise level. The roles of various time-varying parameters on gear dynamics are analyzed (for a specific example case), and the predictions from the analytical model are compared with prior literature. CITATION: Sundar, S., Singh, R., Jayasankaran, K. and Kim, S., "Effect of the Tooth Surface Waviness on the Dynamics and Structure-Borne Noise of a Spur Gear Pair," SAE Int. J. Passeng. Cars - Mech. Syst. 6(2):2013, doi: / INTRODUCTION The static transmission error is a well-recognized source, and thus tooth modifications are often undertaken to minimize the excitation at specific loads to reduce gear whine noise [1]. Yet, at high torque loads, noise levels are still relatively high [2]. This suggests that tooth surface waviness and sliding friction could manifest itself as an alternate noise source. For instance, Mark [3] found that the surface waviness is related to the machining process kinematics. Furthermore, the experiments conducted by Mitchell [4] and Amini et al. [5] on gears and by Othman et al. [6] on rotating disks show an increase in the sound level with an increase in the surface waviness amplitude. The role of tooth surface waviness on gear noise is not well understood, especially for structure-borne noise source(s) or path(s), due to the complexity in modeling micro-surface characteristics. This article employs a six degree of freedom (6DOF) linear timevarying (LTV) model of a spur gear pair to quantify the structure-borne noise source and to illustrate a relationship between waviness amplitude and wave number to gear dynamics source and resulting sound radiation. The overall procedure for predicting the sound pressure level (L) is previewed in Fig. 1, where θ is the angular displacement of the pinion or gear, while x and y are the translational motions along the line-of-action (LOA) and the off-line-of-action (OLOA) directions, respectively. Subscripts p and g represent pinion and gear, respectively. (Also, refer to the list of symbols for the identification of variable and parameters.) LINEAR TIME-VARYING SPUR GEAR MODEL The proposed 6DOF LTV model is schematically shown in Fig. 2. The gear and pinion are considered rigid discs of polar moments of inertia J p and J g with external torques T p and T g. Here, h p (t) and h g (t) represent tooth surface waviness with respect to perfect involute profiles. The governing equations are described by torsional and translational motions. The effective shaft-bearing stiffness elements are given by k psx and k gsx in the X direction (LOA) and k psy and k gsy in the Y direction (OLOA). The time-varying mesh stiffness (k(t)) is calculated for a range of torques by using a well-known gear contact mechanics code (Load Distribution Program or LDP) [7]. The parameters of the unity gear pair example used in this study are as follows: Number of teeth = 28, outside diameter = mm; root diameter = mm; diametral pitch = m 1 ; center distance = 88.9 mm; pressure angle = 20 ; face width = 6.35mm; tooth thickness = mm; and elastic modulus = kn/mm 2. Further, it is assumed that the bearings of the gear and pinion are frictionless, and the waviness amplitude is independent of the load.

2 THIS DOCUMENT IS PROTECTED BY U.S. AND INTERNATIONAL COPYRIGHT. Fig. 1. Procedure for predicting gear accelerations and sound pressure levels. With reference to the geared system model of Fig. 2, the governing equations for θp(t) and θg(t) are: (1) (4) (2) The time-varying moment arms χpi(t) and χgi(t) for the ith meshing pair with a contact ratio (σ) are the following with n = floor(σ): Equations of translational motion along X and Y are given by the following, where ζ is modal damping ratio: (3 a, b) (6) Here, Ωp and Ωg are the nominal speeds (in rad/s); Λ is the base pitch; and LAP, LXA, and LYC are the geometric length constants as defined by He et al. [8]. The normal loads (Np and Ng) and friction forces (Fpf and Fgf) are defined as follows, where μ(t) is the time-varying coefficient of friction and Δh(t) = hp(t) - hg(t): (5 a, b) (7) (8) (9)

3 Fig. 2. Proposed 6DOF linear time-varying gear dynamics model with prescribed tooth surface waviness given by h p (t) and h g (t). (solid line) and at t b for tooth pair # 1 (dotted line). Here, t a represents the time from two teeth in contact to the first tooth leaving contact, t b represents the time from two teeth in contact to the pitch point where the sliding velocity changes its direction, and t c represents the gear mesh period. The mesh stiffness parameters at a mean torque are calculated by using LDP [7], given the kinematics. Assuming that the gear tooth surface is one-dimensional given in terms of mesh locations (s), the involute coordinates s p and s g are defined as follows in the involute coordinate system, where α is the roll angle, and j denotes the tooth index: (10 a) (10 b) Expressions for known motion inputs to the geared system are stated below in terms of tooth surface waviness h p (s) and h g (s), as shown in Fig. 2: Fig. 3. Normalized time-varying parameters along a mesh cycle. (a) Mesh stiffness, ; (b) Coefficient of friction,. (11 a) The normalized time-varying mesh stiffness and coefficient of friction are shown in Fig. 3 for tooth pair #0

4 (11 b) Here ϕ and λ represent the wave phase and wave length, respectively, of the surface waviness. The wave number, κ = ω/v s = 2π/λ, where v s represents the sliding velocity, determines the spectral contents of surface-induced excitation, and the amplitudes H p and H g determine the amplitudes of excitation. Periodic waviness is generated with a constant ϕ, whereas the sinusoidal waviness is generated by choosing a proper λ such that the surface waviness completes a full cycle at the end of t c. Random waviness is also generated by choosing ϕ j values with well-known distributions such as exponential, chi-square, Poisson, or Rayleigh distribution. The partial radiated pressure to gear acceleration transfer functions along X and Y directions Γ x and Γ y, respectively, are measured using a microphone located 152 mm above the top plate of the gearbox by Singh [9]; the microphone is placed in the free field. Table 1 shows typical magnitudes of Γ x and Γ y at certain frequencies corresponding to gear mesh frequencies of a high speed gearbox. Though the measurements are made at a particular load, it is assumed that the measured transfer functions (Γ x, Γ y ) are still valid with a variation in T. Table 1. Measured magnitudes of partial radiated pressure to gear acceleration transfer functions (Γ x, Γ y ) where the microphone is located at 152 mm above the top plate of a gearbox. 1.2 mm). In all cases, H p = 1.0μm. Here, k(t) is a major contributor towards L px, and in the case of L py the major contribution is from μ(t) and h(t). Effect of Randomness The sound pressure level is then predicted for a random profile (with κ p = 2π650 m 1 and κ p = 2π800 m 1 and H p = 1.0 μm) with uniform distribution. Now non-mesh harmonic components are also observed mainly along the X direction as shown in Fig. 5, where is the gear mesh harmonic. Along with non-mesh harmonics, the dominant harmonics are displayed. Along the X direction the random waviness with κp = 2π650 m 1 is more dominant until ; conversely the random waviness with κ p = 2π800 m 1 is more dominant from to. Along the Y direction, there is a dominant peak at 977 Hz, which is coincident with a natural frequency of the geared system. Effect of Torque Table 2 shows the predicted L p range for three values of T p (22.6 N-m, 45.2 N-m, and 90.4 N-m) for a smooth tooth surface at the first five mesh harmonics. As T p increases, there is a considerable increase in L p (the slope is approximately 5 db per octave). Table 2. Effect of mean load (T p ) on predicted sound pressure levels for smooth tooth surface using Δh(t) as excitation SOUND PRESSURE PREDICTIONS Effect of Surface Waviness The sound pressure is predicted for a gear system with different tooth surfaces at 22.6 N-m Where is the gear mesh harmonic. Fig. 4 shows the sound pressure level prediction along the X direction (L px ) and the Y direction (L py ) with a smooth tooth surface, and random, periodic, and sinusoidal waviness at the first five gear mesh harmonics ( ). The wave numbers for the periodic waviness excitations are κ p = 2π650 m 1 (λ p = 1.5 mm) and κ p = 2π800 m 1, and the sinusoidal waviness excitation has κ p = 2π857 m 1 (λ p = COMPARISON WITH PRIOR EXPERIMENTS OR CALCULATIONS The proposed model is used to examine experimental or computational results reported in the literature. First, the effect of T on sound pressure is considered. Mitchell [4] documented a 5 db increase in the sound pressure level with an octave increase in T. In the proposed model, an almost 6 db increase in sound pressure at all mesh harmonics (along the X and Y directions) is seen with a doubling of the T value. Further, Mitchell [4] reported a 5 db per octave slope with speed variation; the current model is slightly off as it predicts a slope of about 8.5 db per octave for speed variations. The proposed model (for the structure-borne sound) is, however, similar to the air-borne noise source

5 Fig. 4. Sound pressure levels with H p = 1.0μm and T p = 22.6 N-m as excited by LOA and OLOA gear accelerations given sinusoidal or periodic surface waviness. (a) L px, (b) L py. Key:, smooth surface;, Periodic waviness, κ p = 2π650 m 1 ;, Periodic waviness, κ p = 2π800 m 1 ;, Sinusoidal waviness, κ p = 2π857 m 1 Fig. 5. Sound pressures corresponding to the LOA gear acceleration when excited by random tooth surface with H p = 1.0 μm. Key:, κ p = 2π650 m 1 ;, κ p = 2π800 m 1. model suggested by Kim and Singh [10] that predicted a slope of about 9 db per octave. In order to compare the effect of surface waviness amplitude H in the current model with the literature, a sinusoidal waviness with κ p = 2π857 m 1 is first assumed. Ishida et al. [11] documented a 10 db decrease in sound when ΔH was reduced from 9 µm to 1 µm. The current model predicts a 17 db reduction in sound level for the same change in ΔH. Mitchell [4] reported an increase of about 1.5 db when H was raised to 2.5 μm from 1 μm (and again by 1.5 db when H was increased further to 5 μm). The current prediction shows about a 6.5 db increase for the same changes in H. Hansen et al. [2] reported vibration levels at the first gear mesh frequency with a super-finished tooth surface.

6 The vibration level for the third stage bull gear decreased by 7 db for a reduction in H from 0.38 μm to 0.07 μm; the current model predicts a 6 db reduction [2]. Furthermore, Hansen et al. [2] measured a 3 db reduction when H was changed from 0.38 μm to 0.09 μm for the second stage bevel pinion; in this case, the current model predicts a 5 db decrease in vibration (noise). Table 3 shows the normalized accelerations (at the nongear mesh frequencies) along the X direction ( ) with an increase in H. The values of at first five gear mesh harmonics (for all the values of H p ) are given by the following in db re 10 2 (a normalized acceleration value): 39, 34, 34, 25, and 10. Observe that the non-mesh harmonics are dominant compared to the mesh harmonic values of, which is attributed to the randomness in the surface waviness. With an octave increase in H there is a 6 db increase in sound pressure along the X direction at non-mesh harmonics in the case of random waviness. This trend explains some peaks in between gear mesh harmonics in the measured spectra. Of course the ghost frequencies may also be generated as a direct consequence of a specific profile left on tooth surfaces by a honing machine [5]. Table 3. Predicted with increase in H for random excitation with κ p = 2π800 m 1 2. Hansen, B., Salerno, M., Winkelmann, L., Isotropic superfinishing of S-76C+ main Transmission gars, AGMA Technical paper 06FTM02, Mark, W.D., Contributions to the vibratory excitation of gear systems from periodic undulations on tooth running surfaces, The Journal of the Acoustical Society of America 91:166, Mitchell, L.D., Gear noise: the purchaser's and the manufacturer's view, Proceedings of the Purdue Noise Control Conference, 14-16, Amini, N., Rosen, B.G., Surface topography and noise emission in gearboxes, ASME Design Engineering Technical Conferences, Paper # DETC97/VIB-3790, Othman, M., Elkholy, A., Seireg, A., Experimental investigation of frictional noise and surface-roughness characteristics, Experimental Mechanics 30: , Houser, D. R., Harianto, J., Load Distribution Program manual, GearLab, The Ohio State University, He, S., Cho, S., Singh, R., Prediction of dynamic friction forces in spur gears using alternate sliding friction formulations, Journal of Sound and Vibration 309: , Singh, R., Dynamic analysis of sliding friction in rotorcraft geared systems, Technical report submitted to the Army Research Office, Grant number DAAD , Kim, S., Singh, R., Gear surface roughness induced noise prediction based on a linear time-varying model with sliding friction, Journal of Vibration and Control 13: , Ishida, K., Matsuda, T., Effect of tooth surface roughness on gear noise and gear noise transmitting path, American Society of Mechanical Engineers, Paper 80C2/DET-70, CONTACT INFORMATION Professor Rajendra Singh Dept. of Mechanical and Aerospace Engineering Acoustics and Dynamics Laboratory, NSF Smart Vehicle Concepts Center, The Ohio State University Columbus, OH USA singh.3@osu.edu Phone: LIST OF SYMBOLS CONCLUSION An improved 6DOF LTV analytical model has been developed to predict structure-borne noise for spur gears induced by sliding friction and tooth surface waviness. The model utilizes time-varying k(t) over a range of T as calculated with LDP [7]. Based on the experimental partial radiated pressure to gear acceleration transfer functions, free field sound pressure levels are also predicted. The effect of an increase in torque on L p has been calculated and compared with results reported previously. In particular, non-mesh harmonics are excited with random tooth surface. Noise trends as found in the prior literature can be conceptually explained using the proposed model though there is sufficient room for improvement. REFERENCES 1. Houser, D. R., Singh, R., Gear dynamics and gear noise short course notes, The Ohio State University, USA, a - gear acceleration c - viscous mesh damping k - stiffness F - friction force h - surface waviness H - waviness amplitude i - gear mesh index j - tooth index L - sound pressure level J - mass moment of inertia m - mass of the spur gear N - normal load n - number of teeth p - sound pressure R - base radius of the spur gear s - involute coordinate t - time T - torque v - velocity

7 χ - moment arm for friction force X - line-of-action direction Y - off-line-of-action direction x - translation along line-of-action y - translation along off-line-of-action α - roll angle ζ - damping ratio θ - angular motion of spur gear Λ - base pitch Ω - speed of rotation κ - wave number ϕ - wave phase λ - wave length χ - moment arm for friction force Γ - partial pressure to acceleration transfer function μ - coefficient of friction Subscripts 0 - mesh start point of pinion 1,2 - gear pair in contact g - gear L - mesh start point of gear m - mesh point p - pinion s - sliding S - shaft - bearing x - line-of-action direction y - Off-line-of-action direction Superscripts ( ) - differentiation with respect to t ( ) - normalization Abbreviations DOF - degree-of-freedom LDP - load distribution program LOA - line-of-action LTV - linear time-varying OLOA - off-line-of-action