Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears

Transcription

1 740 Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears SLi* and A Kahraman Gleason Gear and Power Transmission Research Laboratory, The Ohio State University, Columbus, OH, USA The manuscript was received on 1 November 2010 and was accepted after revision for publication on 14 April DOI: / Abstract: In this study, the elastohydrodynamic lubrication (EHL) behaviour of high-speed spur gear contacts is investigated under dynamic conditions. A non-linear time-varying vibratory model of spur gear pairs is introduced to predict the instantaneous tooth forces under dynamic conditions within both the linear and non-linear operating regimes. In this model, the periodically time-varying gear mesh stiffness and the motion transmission error are used as excitations and a constant damping ratio is employed. This model allows the prediction of steady-state nonlinear response in the form of tooth separation (contact loss). An earlier gear mixed EHL model [1] is adapted to simulate the lubrication behaviour of spur gear contacts under these dynamic loading conditions, considering the variations of the basic contact parameters, such as radii of curvature, sliding and rolling velocities, and measured roughness profiles, as the contact moves along the tooth from the start of active profile to the tip. The EHL predictions under dynamic loading conditions are compared to those assuming quasi-static contact loads for gear sets having smooth and rough surfaces to demonstrate the important influence of dynamic loading on gear lubrication. The unique, transient EHL behaviour under the non-linear (intermittent contact loss) condition is also illustrated. Keywords: dynamics, non-linear, gear, mixed EHL, roughness 1 INTRODUCTION Dynamic behaviour of gear systems has been studied extensively for two primary reasons. One is that the noise generated by a gear system is a direct consequence of its dynamic behaviour. Any effort to reduce gear noise of a transmission must focus on the reduction of the vibration levels of the gears. The second reason is the durability concern. As the gear and bearing force and stress amplitudes are often amplified under dynamic conditions, such dynamic effects must be taken into account in the design of gear pairs. A large number of dynamics models have *Corresponding author: Gleason Gear and Power Transmission Research Laboratory, The Ohio State University, 201 West 19th Avenue, Columbus, OH 43210, USA. li.600@osu.edu been developed over the past 50 years. Most of these models are for spur or helical gears as summarized in the review papers by Ozguven and Houser [2] and Wang et al. [3]. Based on the measured nonlinear dynamic behaviour documented in the literature for spur gears [4 9], several non-linear time-varying models of spur gears were proposed with reasonable success in describing the published experiments. These mostly torsional spur gear dynamics models [10 15] used a clearance type non-linear gear mesh function to take into account the tooth separations in the presence of gear backlash and considered the periodic time variation of the gear mesh stiffness due to the fluctuation of the number of loaded tooth pairs as the gears rotate. Likewise, in line with the linear behaviour observed experimentally [16], most helical gear pair models have

2 Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 741 neglected backlash while including additional rotational, translational, and axial motions [16 20]. The tribology literature contains a wide spectrum of elastohydrodynamic lubrication (EHL) models. Sophisticated isothermal or thermal, Newtonian or non-newtonian, and line or point EHL models considering smooth or rough surface contacts have been proposed to simulate the very complicated lubrication behaviour in a more comprehensive and accurate way [21 26]. However, the contact parameters of a gear pair, including the normal tooth force, radii of curvature, and surface velocities, all vary as the gears roll in mesh. Such a transient contact behaviour might not be fully represented through the analyses of certain discrete gear mesh positions of interest using these EHL models with constant contact parameters. There are a limited number of published studies that attempted to include the geometric, kinematic, and load variations of gear contacts. Approximating the elastic deformation by that of a smooth dry Hertzian contact, Wang and Cheng [27, 28] predicted the minimum film thickness and thermal characteristics of spur gears having smooth tooth surfaces. Larsson [29] and Wang et al. [30] proposed involute spur gear EHL models for isothermal non-newtonian and thermal Newtonian fluids, respectively, employing assumed time-varying normal tooth force as the contact moves along the line of action. These three studies, while establishing the need for a specialized EHL model for spur gears, lacked the ability to handle the rough surface condition. In addition, these studies limited their treatments of the gear mesh deformation to Hertzian effect. However, other effects due to tooth bending, base rotation, and shear deformation were shown to be equally important in defining gear mesh compliance [31]. Incorporating a gear load distribution model considering all these essential components of the gear tooth compliance, Li and Kahraman [1] proposed a transient mixed EHL model for spur gear pairs that is capable of handling extreme asperity contact condition robustly by also including the gear contact transient effects. Any manufacturing errors or intentional tooth profile modifications were also taken into account. In view of the above review of the gear dynamics and gear tribology literature, one can point to an apparent gap between these two disciplines. Tribodynamic models for gear systems are not readily available. Although there have been a few studies on the gear mesh damping mechanism (induced by the gear EHL contacts) [32] and on the influence of friction of lubricated contacts on gear dynamics [33, 34], the impact of dynamic tooth force on gear EHL contact is yet to be fully understood. The research by Wang and Cheng [27, 28] incorporated a torsional vibratory model into the EHL analysis to solve for the minimum film thickness. However, the transient gear parametric effects were not fully included. Neither was the surface roughness. Accordingly, the main objective of this study is to investigate the influence of gear dynamics on gear EHL behaviour. In this study, a non-linear, time-varying dynamic model of a spur gear pair is incorporated with a gear mixed EHL model to predict the gear contact behaviour under both the linear and non-linear dynamic conditions. The dynamic model includes the gear mesh stiffness fluctuation, displacement excitation due to the manufacturing errors and intentional tooth corrections (also known as the transmission error), and gear backlash non-linearity in a manner similar to the model of Tamminana et al. [15]. The dynamic model uses the quasi-static gear mesh stiffness and the transmission error excitation predicted using a gear load distribution model [31] to determine the instantaneous tooth contact force. The EHL model used here is based on an earlier spur gear EHL model [1] where the instantaneous contact radii and surface velocities are defined using the involute geometry. With the dynamic tooth force provided by the dynamic model, instantaneous film thickness and normal pressure distributions of the tooth contact are predicted by the EHL model as the contact moves along the tooth surface. In this study, the effect of the lubrication characteristics on the dynamic response in terms of lubricant stiffness and damping is not considered, such that the dynamic solver and the EHL solver are uncoupled. The investigation is also kept limited to spur gear pairs with no significant lead modifications to allow a line contact formulation. Other types of gear pairs such as helical and hypoid gears were beyond the scope of this study. 2 MODEL FORMULATIONS 2.1 Prediction of dynamic gear tooth contact forces In order to predict the dynamic loading on the teeth of a spur gear pair, a single-degree-of-freedom (DOF) discrete model similar to that of Tamminana et al. [15] is employed. This torsional dynamic model, shown in Fig. 1, consists of two rigid discs of radii r b1 and r b2 (to represent the base circle radii of gears 1 and 2) and polar mass moments of inertia I 1 and I 2, respectively. The gear mesh interface model consists of (a) a parametrically time-varying gear mesh stiffness kðtþ, (b) a constant viscous damper c, and (c) an externally applied gear mesh displacement excitation

3 742 S Li and A Kahraman With the positive directions of the alternating rotational displacements, # 1 and # 2, and the applied torque, T 1 and T 2, defined as shown in Fig. 1, the dynamic equations read I 1 # 1 ðtþþr b1 kðtþðtþþcr b1 ½_sðtÞ _eðtþš ¼ T 1 I 2 # 2 ðtþ r b2 kðtþðtþ cr b2 ½_sðtÞ _eðtþš ¼ T 2 ð1aþ ð1bþ where sðtþ ¼r b1 # 1 ðtþ r b2 # 2 ðtþ is the dynamic transmission error and 8 < sðtþ eðtþ b, sðtþ eðtþ 4 b ðtþ ¼ 0, sðtþ eðtþ b ð1cþ : sðtþ eðtþþb, sðtþ eðtþ5 b Fig. 1 A purely torsional dynamic model of a spur gear pair eðtþ, all of which are applied along the line of action (line tangent to the base circles of the gears). Here, the least known parameter is c since there are several power loss mechanisms present, including the viscous losses at the gear mesh interface, viscous losses on the bearings, and windage, churning, and pocketing losses due to fluid gear interactions within the gear box. The damping element c is intended to represent all these mechanisms. However, some of them are not feasible to easily quantify. The time variance of kðtþ is mainly due to the fluctuation of the number of loaded tooth pairs between two integers (typically 1 and 2) as the gears roll in mesh. The displacement excitation eðt Þ represents the motion transmission deviation caused by intentional tooth profile modifications as well as the manufacturing errors under unloaded condition. Bulk of the non-linear behaviour observed in spur gear pairs occurs at instances when the dynamic force amplitude exceeds the static load (preload) transmitted by the gear pair. With the presence of gear backlash, the gear teeth lose contact at such instances and the gear mesh stiffness drops to zero instantaneously. As proposed earlier [12, 15], the mesh stiffness kðt Þ is subjected to a piecewise linear clearance function, d as illustrated in Fig. 1. This function is composed of a dead zone (backlash) of size 2b bounded by two unity slope regions representing the linear and back contact conditions (no contact loss). Since the generalized parameters of the two-dof model of Fig. 1 are semi-definite with a rigid body mode at zero natural frequency, a new relative displacement parameter is defined as ðtþ ¼sðtÞ eðtþ. With this, the equation of motion of the resultant definite single-dof model is derived as m e ðtþþc _ðtþþkðtþðtþ ¼F m e eðtþ ð2aþ where m e is the equivalent mass defined as m e ¼ I 1 I 2 ði1 rb2 2 þ I 2rb1 2 Þ. The constant force transmitted by the gear mesh F ¼ m e ðr b1 T 1 =I 1 þr b2 T 2 =I 2 Þ, where T 1 and T 2 are the constant external torques applied to gears 1 and 2, respectively. The overdot denotes the differentiation with respect to time t. The non-linear restoring function in Fig. 1 has the form of 8 < ðtþ b, ðtþ 4 b ðtþ ¼ 0, ðt Þ b ð2bþ : ðtþþb, ðtþ 5 b It is noted that the mesh stiffness kðtþ consists of a mean component, k, and an alternating component, k a ðtþ, i.e. kðtþ ¼k þ k a ðtþ. With this, a set of dimensionless parameters can be defined as ^ðtþ ¼ðtÞ b, ^eðtþ ¼eðtÞ qffiffiffiffiffiffiffiffiffi b, ^ðtþ ¼ðtÞt=b, ¼ c=ð2 m ek Þ and ^F ¼ F =ðbkþ, leading to a dimensionless form of equation (2a) as d 2 ^ðtþ d ^ðtþ dt 2 þ 2! n þ! 2 n dt 1 þ k aðtþ ^ðtþ k ¼! 2 ^F n d2 ^eðtþ dt 2 ð3þ qffiffiffiffiffiffiffiffiffiffiffiffi where! n ¼ k=m e is the undamped natural frequency. The mesh stiffness kðtþ of a gear pair can be determined using a gear load distribution model [31] or any finite element based gear contact model. In the latter case, the difference of the gear static transmission error between the loaded ( ~eðt Þ) and unloaded (eðt Þ) conditions can be used to estimate the mesh stiffness as kðtþ ¼ðT 1 rb1 Þ ½~eðtÞ eðtþš [15].

4 Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 743 Defining the dynamic mesh force from equation (3) as ( DMFðtÞ ¼bk 2 d ^ðtþ þ 1 þ k ) aðtþ ^ðtþ ð4þ! n dt k The individual dynamic tooth force, W d ðtþ, can be obtained approximately from the quasi-static tooth force, W s ðtþ, as[15] W d ðtþ ¼W s ðtþ DMFðt Þ T 1 =r b1 2.2 Time-varying gear tooth contact parameters Unlike the simple contact of cylinder pair, the gear contact parameters, including the radii of curvature, tangential surface velocities, and normal tooth force, all vary as the contact moves along the line of action when the gears rotate. According to the basic involute gear geometry and kinematics, the contact radii of curvature r 1 and r 2 are given as r 1 ðtþ ¼r b1 1 ðtþ r 2 ðtþ ¼r b2 2 ðtþ ð5þ ð6aþ ð6bþ where 1 ðtþ and 2 ðtþ are the roll angles of gears 1 and 2, respectively. As a result, the corresponding surface velocities tangent to the tooth surfaces become variable as well such that u 1 ðtþ ¼! 1 r 1 ðtþ ð7aþ u 2 ðtþ ¼ N 1! 1 r 2 ðtþ ð7bþ N 2 where N 1 and N 2 are the number of teeth of gears 1 and 2, respectively, and! 1 the angular velocity of gear 1. Here, u 1 ðtþ 5 u 2 ðtþ when the contact point travels long the dedendum of the driving gear (gear 1) from the start of active profile (SAP) to the pitch point such that the sliding velocity is negative, i.e. u s ðtþ ¼u 1 ðtþ u 2 ðtþ 5 0. When the contact reaches the pitch point, u s ðtþ ¼0 (pure rolling condition). Within the addendum range from the pitch point to the tooth tip of the driving gear, u s ðtþ 4 0. In the process, the rolling velocity u r ðtþ ¼ 1 2½ u 1ðtÞþu 2 ðtþš varies with the gear rotation as well. The resultant slideto-roll ratio of the tooth contact, defined as SRðtÞ ¼u s ðtþ u r ðtþ, varies from a negative limit to a positive one with SRðtÞ ¼0 at the pitch point. As the driving and driven gears roll in mesh, the number of loaded tooth pairs alternates between two integers that bound the profile contact ratio of the gear pair (average number of tooth pairs in contact), resulting in the periodic quasi-static tooth force W s ¼ W s ðtþ. The gear load distribution model [31] referred to earlier for the predictions of kðt Þ and eðt Þ is also used to predict W s ðtþ. This model includes the flexibilities associated with tooth bending, shear deformation, base rotation, as well as the contact deformation. It also considers any deviation of the tooth profile from the perfect involute primarily due to the intentional modifications such as tip relief and profile crown. In the absence of such modifications, W s ðtþ experiences sudden and drastic changes at the instants when the number of loaded tooth pairs changes from two to one (the lowest point of single tooth contact, LPSTC) and from one to two (the highest point of single tooth contact, HPSTC). 2.3 Transient mixed EHL model for spur gear contacts Assuming negligible shaft misalignment and a reasonable level of lead modification, the contact of a spur gear pair can be characterized as a line contact. The one-dimensional transient Reynolds equation governs the fluid flow in the contact areas where sufficient lubricant film @pðx, tþ f ðtþ ½ u rðtþðx, @ðx, tþhðx, tþ þ ½ ð8aþ where pðx, tþ, hðx, tþ, and ðx, tþ are the transient pressure, film thickness, and density distributions along the rolling direction x. The Ree Eyring flow coefficient, f ðtþ, is approximated as f ðtþ ¼½h 3 ð12þš cos h½ m ðtþ 0 Š [1], where is the lubricant viscosity, 0 the lubricant reference stress, and m the mean viscous shear stress; m ðtþ ¼ 0 sinh 1 u s ðtþ ð 0 hþ. For any local area where the fluid film is extremely thin, say less than two layers of lubricant molecules, hydrodynamic lubrication becomes impossible and the reduced Reynolds equation [1, 25, 26] is used to describe the ½u r ðtþðx, tþhðx, þ ½ tþhðx, tþ Š ¼ Once a smooth transition from the fluid film region to the asperity contact region is assumed, this unified Reynolds equation system of equations (8a) and (8b) govern the EHL behaviour of the contact, considering the hydrodynamic and asperity contact pressures simultaneously. Assuming only elastic deformation, the local film thickness can be defined as hðx, tþ ¼h 0 ðþþg t 0 ðx, tþþvðx, tþ R 1 ðx, tþ R 2 ðx, tþ ð9þ

5 744 S Li and A Kahraman where h 0 is the reference film thickness, V the surface elastic deflection, and g 0 the unloaded geometric gap between the mating tooth surfaces, which is time dependent through the variable equivalent radius of curvature r eq ðtþ ¼r 1 ðtþr 2 ðtþ ½r 1 ðtþþr 2 ðtþš for g 0 ðtþ ¼x 2 ½2r eq ðtþš. The terms R 1 ðx, tþ and R 2 ðx, tþ represent the tooth surface roughness profiles measured in the profile (rolling and sliding) direction x. Here, it is assumed that these roughness profiles remain constant along the tooth face direction to allow the use of this line contact formulation. This is a reasonable assumption for certain finishing processes such as grinding and shaving. The next equation of interest is the load balance equation, which states that the total contact force due to the instantaneous pressure distribution over the entire contact zone must balance the dynamic tooth force at the same instant, i.e. Z Wd 0 ðtþ ¼ pðx, tþ dx ð10þ where Wd 0 ðtþ is the dynamic tooth force intensity along the contact line (dynamic tooth force per unit face width). The value of h 0 in equation (9) must be adjusted iteratively until the pressure distribution pðx, t Þ satisfies equation (10). Various forms of viscosity pressure relationships have been used in the past including the Barus exponential relationship, the Roeland s equation, as well as the two-slope exponential relationship. However, these relationships might not be accurate within very wide pressure ranges experienced in rough gear contacts [35]. The Doolittle Tait relationship was proposed as a potential remedy [35]. While any of the other viscosity pressure relationships can be used as long as they represent the measured behaviour of the lubricant accurately within the operating pressure range, the Doolittle equation [36] used in this study is V occ ð1 V ¼ 0 exp B Þ ðv V occ Þð1 V ð11aþ occ Þ where 0 the ambient viscosity, B the Doolittle parameter, and V occ the normalized occupied volume. The normalized volume V is a pressure-dependent parameter that can be modelled through the empirical Tait equation of state [37] as 1 V ¼ 1 ð1 þ K0 0Þ ln 1 þ p ð1 þ K0 0 K Þ ð11bþ 0 where K 0 and K0 0 are the bulk modulus and the derivative of bulk modulus with respect to pressure, respectively, when p ¼ 0. The density pressure relationship of the lubricant is modelled the same way as described in Li and Kahraman [1], in which the details of the discretization and linearization of the governing equations can be found. 3 RESULTS OF AN EXAMPLE ANALYSIS In this section, the transient contact pressure and film thickness distributions between the mating teeth of an example spur gear pair under the dynamic loading condition are presented at different mesh frequencies to demonstrate the substantial impact of the dynamic response on the EHL behaviour. The design parameters of the example spur gear pair are listed in Table 1. The mass and inertia values of this gear pair are also listed. This gear pair design was used in several earlier experimental studies on the nonlinear dynamic behaviour of spur gear pairs [7 9, 12, 13]. The lubricant used in this study is Mil- L23699, whose viscosity pressure relationship at an inlet temperature of 100 C is plotted in Fig. 2. The black dots in the figure denote the measured viscosity values at different pressures extracted from Fig. 2 of Bair et al. [35]. The solid line represents the viscosity computed from equation (11) with the required parameters regressed from the measured data as 0 = Pa s, B = , Vocc ¼ 0:6132, K 0 = GPa, and K0 0 ¼ 10:076. The viscosity estimates of equation (11) with these parameter values are in good agreement with the measurement of Bair et al. [35]. The density of this lubricant at the same temperature and ambient pressure is 0 = kg/ m 3. The computational domain with the dimension of 2:5a max x 1:5a max (a max is the maximum Hertzian half-width of the contacts along the line of action) is discretized into 512 elements with the grid size x ¼ 1:3 mm, which reasonably represents the measured roughness resolution. The entire analysis from the SAP to the tip is discretized into 1000 time steps, resulting in the time resolution of sat f m = 1250 Hz, satf m = 2085 Hz and sat f m = 2375 Hz. This represent a more refined time increment compared to those used in Larsson [29] and Wang et al. [30]. To start the simulation at the SAP, the Hertzian pressure is used as the initial guess and iterated until the converged stationary EHL solution is obtained. During the analysis from the SAP to the tip, whenever the tooth force reaches zero (due to the dynamic effect), a very small loading value of 5 N was applied artificially, that is negligibly small compared to typical tooth force levels at several kilo-newtons. Employing the experimentally determined damping ratio of ¼ 0:01 (1 per cent) in equation (3) [7 9], the dynamic model proposed above is used to predict the steady-state dynamic response of the gear pair at T 1 = 250 Nm within the gear mesh (tooth passing) frequency range of f m = Hz (f m ¼ 1 2 N 1! 1 where

6 Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 745 Table 1 Design parameters of the example unity-ratio spur gear pair used in this study Number of teeth 50 Module (mm) 3.0 Pressure angle (degrees) 20.0 Outside diameter (mm) Pitch diameter (mm) Root diameter (mm) Centre distance (mm) Face width (mm) 20.0 Backlash (mm) 0.14 Polar mass moment of Inertia (kg m 2 ) Equivalent mass, m e (kg) Fig. 2 Viscosity pressure relationship of lubricant Mil-L23699 at 100 C! 1 is in rad/s). The gear load distribution model is implemented to determine kðtþ of this gear pair at T 1 = 250 Nm, as shown in Fig. 3(a). Although the load distribution model is capable of including any profile modification, the example gear pair used here has perfect involute profile such that eðtþ ¼0. The Fourier spectrum of kðt Þ, shown in Fig. 3(b), indicates that k ¼ 3: N/m with the first three harmonic amplitudes of k 1 = , k 2 = , and k 3 = N/m. With this, the natural frequency of qthe ffiffiffiffiffiffiffiffiffiffiffiffi corresponding linear, undamped system is! n ¼ k=m e ¼ rad/s (f n ¼! n =ð2þ ¼3242 Hz). Here, the lubricant stiffness is not included since its magnitude, which ranges to N/m for the operating conditions considered (estimated from the Hamrock Dowson formula), is several orders larger than that of the gear mesh stiffness. The predicted root-meansquared (r.m.s) value of sðtþ versus f m plot of Fig. 3(c) reveals one primary resonance peak at f m ¼ f n 3240 Hz caused by the first harmonic of the excitation as well as two super-harmonic resonances at f m ¼ 1 2 f n 1620 Hz and f m ¼ 1 3 f n 1080 Hz caused by the second and third harmonic terms of the excitation. In the vicinity of these resonance peaks, two stable motions coexist: the lower branch linear motion without tooth separation and the upper branch non-linear motion with tooth separation which exhibits a softening type nonlinear behaviour. It is the initial condition that dictates which motion should be exhibited by the gear pair. This predicted response of Fig. 3(c) agrees well with the published measurement using the same example gear pair [7 9]. Three representative operating conditions marked in Fig. 3(c) are considered here to demonstrate the impact of the dynamic response on the EHL behaviour of the gear set. They include two off-resonance conditions at f m = 1250 and 2085 Hz (marked as points I and II in Fig. 3(c)) and a non-linear resonant condition at f m = 2375 Hz (marked as point III in Fig. 3(c)). The variations of the contact geometry parameters r 1, r 2, and r eq, and the speed parameters u 1, u 2, u s, u r, and SR of the example gear pair operating at f m = 1250 Hz as the contact moves from the SAP at 1 ¼ 14:5 to the tip at 1 ¼ 27:2 are shown in Fig. 4. As seen, the variations of the radii of curvature, surface velocities, and sliding velocity are evident, while u r is constant in this case since the gear pair has unity ratio (i.e. the driving and driven gears are identical). In Fig. 5(a), the quasi-static tooth force W s predicted by the gear load distribution model at T 1 = 250 Nm is compared to its dynamic counterpart W d when the gears are operated under the condition of point I (Fig. 3(c)). Here, W s is shown to increase linearly from the SAP to the LPSTC at 1 ¼ 20:0, and then almost double at the LPSTC where one of the two loaded tooth pairs loses contact. After staying relatively constant till the HPSTC at 1 ¼ 21:7, W s experiences a sudden drop as the gear mesh transmits to two loaded tooth pairs. As shown in Li and Kahraman [1], these drastic changes in W s impact the EHL behaviour significantly. The corresponding dynamic tooth force W d (in the same figure) has a substantially different shape from that of W s. As this frequency is near the resonance peak of f m ¼ 1 3 f n caused by the third harmonic of the excitation, nearly three cycles of fluctuation are observed in Fig. 5(a) for the W d curve within a base pitch. In Figs 5(b) and (c), the W d curves for points II and III (Fig. 3(c)) are compared to the same W s. It is seen that both the amplitude and shape of W d change significantly with f m and show little resemblance to those of W s, suggesting that performing any EHL analysis using W s is inaccurate in the case of high-speed gearing where dynamic effects

7 746 S Li and A Kahraman Fig. 3 Gear mesh stiffness: time history kðtþ (a) and the corresponding frequency spectrum k i of the example spur gear pair (b), and the r.m.s DTE amplitudes as a function of gear mesh frequency f m (c). Note: T 1 = 250 Nm are prominent. It is also noted in Fig. 5(c) that the dynamic tooth force becomes intermittent in the non-linear region of operation. The W d amplitude for the upper branch motion of point III is amplified to more than twice the maximum W s value, while sizable portions of the mesh cycle are spent under zero contact load. Starting with the case of perfectly smooth surfaces (i.e. R 1 ðx, tþ ¼R 2 ðx, tþ ¼0 in equation (8)), the minimum film thickness h min predictions of the EHL simulations under W d ðtþ and W s ðtþ are compared in Fig. 6 at the points I III defined in Fig. 3(c). The h min curves that correspond to W s (dashed lines) oscillate between the LPSTC and somewhere after the HPSTC, which is due to the strong squeezing effect at the LPSTC induced by the sudden load jump-up and the pumping effect right after the HPSTC, induced by the sudden load jump-down as it was described in Li and Kahraman [1]. Such film thickness spikes were also reported in Larsson [29] and Wang et al. [30] while the severity of these jumps varied, perhaps due to the different operating conditions and coarser time resolution used in those studies. Such quasistatic behaviour, however, becomes irrelevant as the Fig. 4 The variations of contact radii (a) and surface velocities (b) with 1 during a single tooth engagement cycle of the example gear pair at the mesh frequency of f mesh = 1250 Hz (point I in Fig. 3)

8 Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 747 Fig. 5 Comparisons of the tooth force variations along the driving gear roll angle between the quasistatic loading condition and the dynamic loading condition: (a) f m = 1250 Hz (point I in Fig. 3(c)); (b) f m = 2085 Hz (point II in Fig. 3(c)); and (c) f mesh = 2375 Hz (point III in Fig. 3(c)). Note: T 1 = 250 Nm h min behaviour corresponding to W d (solid lines) is drastically different from that for W s at all three f m values of operation. The wavy shape of this dynamic condition h min with 1 is observed to be dictated by the shape of W d that itself is defined by the harmonic term(s) of the excitation effective at that f m value. For instance, at point I where f m is near the third superharmonic resonance, three cycles of the fluctuation of W d (in Fig. 5(a)) result in a similar qualitative shape of h min as shown in Fig. 6(a). Similarly, the effect of the second harmonic of the excitation is evident in Fig. 6(b) at point II. Finally, under the non-linear condition which exhibits the loss of contact (point III), it is difficult to point to the formation of fluid film as large portions of the mesh cycle are spent with the contacting surfaces away from each other. The resultant dynamic film thickness variation in Fig. 6(c) is accordingly very different from that using W s as the normal contact load. Fig. 6 Comparisons of the predicted EHL minimum film thickness along the driving gear tooth surface with W s ðtþ and W d ðtþ as the tooth force: (a) f m = 1250 Hz (point I in Fig. 3(c)); (b) f m = 2085 Hz (point II in Fig. 3(c)); and (c) f mesh = 2375 Hz (point III in Fig. 3(c)). Note: T 1 = 250 Nm The lubrication behaviour summarized in the form of h min predicted by the EHL model in Fig. 6 is complemented by Figs 7 to 9 that provide instantaneous pressure and film thickness distributions at various 1 values along the mesh cycle. In Fig. 7, the hðx, tþ and pðx, tþ are shown at ten 1 values (denoted by points A to J) under the quasi-static loading of W s and rotational speed of 1500 r/min (f m = 1250 Hz). The effects of the sudden change in W s at the LPSTC and the variations of the other contact parameters (radii and sliding velocity) are evident in this figure. When the dynamic load W d is considered (at point I, f m = 1250 Hz), completely different hðx, tþ and pðx, tþ are obtained in Fig. 8 at the same 1 values as in Fig. 7. At the SAP (mesh position A) where W d peaks, typical smooth surface EHL pressure and film thickness

9 748 S Li and A Kahraman Fig. 7 Instantaneous p (solid line) and h (dashed line) distributions of the example spur gear pair at a series of mesh positions as defined in the top figure under the quasi-static loading condition with a rotational speed of 1500 r/min and T 1 = 250 Nm distributions are observed. As W d decreases with the increasing 1, the hydrodynamic fluid film becomes less pressurized and thicker. Beyond the mesh position B, W d starts to increase with the consequence of wider fluid film and larger thickness, as shown for the mesh position C. The resultant hðx, t Þ and pðx, t Þ, shown in Fig. 8, point to the influence of W d. A comparison between Figs 7 and 8 indicates that the hðx, tþ and pðx, tþ solutions obtained using W s or W d are substantially different. It is also noted that the pressure ripples at both the inlet and outlet zones introduced by the sudden load change at the LPSTC (position G) under the quasi-static condition are absent in Fig. 8 with W d as the normal load. Next, the hðx, tþ and pðx, tþ at ten different 1 values are shown in Fig. 9 for point III. It is evident that the EHL behaviour under such non-linear dynamic condition has absolutely no resemblance to the corresponding quasistatic condition shown in Fig. 7. The fluid film is formed between the mesh positions A and E to certain extent during the first loaded segment of the contact that dissolves completely during the period between E and F where there is no tooth load. Afterwards, the transient effort to form the fluid film is observed starting from the mesh position F where the tooth contact is reestablished. The hðx, tþ and pðx, tþ distributions

10 Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 749 Fig. 8 Instantaneous p (solid line) and h (dashed line) distributions of the example spur gear pair at a series of mesh positions as defined in the top figure under the dynamic loading condition with f mesh = 1250 Hz and T 1 = 250 Nm are highly transient for the rest of the load cycle (from F to J). At the end, an analysis using the ground tooth surface profiles, shown in Fig. 10, is presented to illustrate the EHL contacts of rough spur gear surfaces under the dynamic condition. These measured R 1 and R 2 profiles have the R q values of 0.54 and 0.53 mm, respectively. Considering these roughness profiles, the transient hðx, tþ and pðx, tþ distributions of the example gear pair at the same mesh positions and under the same dynamic contact condition as in Fig. 8 are shown in Fig. 11. Comparing Fig. 11 with Fig. 8, it can be seen that the size of the contact zone of the dynamic rough contact varies in the same way as that under the dynamic smooth condition when the W d fluctuates. Due to the surface irregularities, however, the local contact pressures in Fig. 11 can easily exceed 1 GPa. At various instantaneous local contact points, hðx, t Þ is zero, indicating actual asperity contacts with the corresponding spikes displaying in the pressure distributions. Meanwhile, the deep roughness valleys introduce much larger local film thickness values. Neither the pressure distribution nor the film thickness distribution is smooth and continuous. The influences of the variable load W d on hðx, tþ and pðx, tþ are also evident in Fig. 11.

11 750 S Li and A Kahraman Fig. 9 Instantaneous p (solid line) and h (dashed line) distributions of the example spur gear pair at a series of mesh positions as defined in the top figure under the dynamic loading condition with f mesh = 2375 Hz and T 1 = 250 Nm Fig. 10 Measured tooth surface roughness profiles in the direction of relative sliding and rolling

12 Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 751 Fig. 11 Instantaneous p (solid line) and h (dashed line) distributions of the example spur gear pair at the same mesh positions as in Fig. 8 under the dynamic loading condition with f m = 1250 Hz, T 1 = 250 Nm, and the surface roughness profiles shown in Fig CONCLUSIONS In this study, an EHL model of a gear pair was used in conjunction with a dynamic model to predict the lubrication behaviour under various dynamic speed conditions. The non-linear time-varying dynamic model of a spur gear pair was used to predict the instantaneous tooth contact force under the dynamic condition within both the linear and non-linear operating regimes. The predicted instantaneous tooth force was fed into the gear EHL model to simulate the lubrication behaviour of the spur gear contact under the dynamic loading condition. The EHL model included any asperity interaction activity as well as the variations of radii of curvature, sliding, and rolling velocities and measured roughness profiles as the contact moves along the gear tooth from the SAP to the tip. The EHL results presented under various dynamic loading conditions were shown to differ from those under static tooth load conditions, suggesting that the dynamic behaviour of the gear pair must be included in describing the high-speed gear tribology. The tooth separation that takes place due to the backlash non-linearity was also shown to impact the transient EHL behaviour in a unique manner. ß Authors 2011 REFERENCES 1 Li, S. and Kahraman, A. A transient mixed elastohydrodynamic lubrication model for spur gear pairs. J. Tribol., 2010, 132, Ozguven, H. N. and Houser, D. R. Mathematical models used in gear dynamics a review. J. Sound Vib., 1988, 121, Wang, J., Li, R., and Peng, X. Survey of nonlinear vibration of gear transmission systems. Appl. Mech. Rev., 2003, 56, Umezawa, K., Ajima, T., and Houjoh, H. Vibration of three axis gear system. Bull. JSME, 1986, 29, Munro, R. G. Dynamic behavior of spur gears. PhD Thesis, Cambridge University, Kubo, A., Yamada, K., Aida, T., and Sato, S. Research on ultra high speed gear devices. Trans. Jpn. Soc. Mech. Eng., 1972, 38, Kahraman, A. and Blankenship, G. W. Experiments on nonlinear dynamic behavior of an oscillator with clearance and time-varying parameters. J. Appl. Mech., 1997, 64, Kahraman, A. and Blankenship, G. W. Effect of involute contact ratio on spur gear dynamics. J. Mech. Des., 1999, 121, Kahraman, A. and Blankenship, G. W. Effect of involute tip relief on dynamic response of spur gear pairs. J. Mech. Des., 1999, 121, Umezawa, K., Sata, T., and Ishikawa, J. Simulation of rotational vibration of spur gears. Bull. JSME, 1984, 38,

13 752 S Li and A Kahraman 11 Kahraman, A. and Singh, R. Nonlinear dynamics of a spur gear pair. J. Sound Vib., 1990, 142, Blankenship, G. W. and Kahraman, A. Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type nonlinearity. J. Sound Vib., 1995, 185, Kahraman, A. and Blankenship, G. W. Interaction between external and parametric excitations in systems with clearance. J. Sound Vib., 1996, 194, Ozguven, H. N. and Houser, D. R. Dynamic analysis of high speed gears by using loaded static transmission error. J. Sound Vib., 1988, 125, Tamminana, V. K., Kahraman, A., and Vijayakar, S. A study of the relationship between the dynamic factors and the dynamic transmission error of spur gear pairs. J. Mech. Des., 2007, 129, Kubur, M., Kahraman, A., Zini, D., and Kienzle, K. Dynamic analysis of a multi-shaft helical gear transmission by finite elements: model and experiment. J Vib. Acoust., 2004, 126, Kubo, A. Stress condition, vibrational excitation force and contact pattern of helical gears with manufacturing and alignment errors. J. Mech. Des., 1978, 100, Kahraman, A. Effect of axial vibrations on the dynamics of a helical gear pair. J Vib. Acoust., 1993, 115, Kahraman, A. Dynamic analysis of a multimesh helical gear train. J. Mech. Des., 1994, 116, Maatar, M. and Velex, V. Quasi-static and dynamic analysis of narrow-faced helical gears with profile and lead modifications. J. Mech. Des., 1997, 119, Venner, C. H. Higher-order multilevel solvers for the EHL line and point contact problem. J. Tribol., 1994, 116, Holmes, M. J. A., Evans, H. P., Hughes, T. G., and Snidle, R. W. Transient elastohydrodynamic point contact analysis using a new coupled differential deflection method. Part 1: theory and validation. Proc. IMechE, Part J: J. Engineering Tribology, 2003, 217, Kim, H. J., Ehret, P., Dowson, D., and Taylor, C. M. Thermal elastohydrodynamic analysis of circular contacts. Part 1: Newtonian model. Proc. IMechE, Part J: J. Engineering Tribology, 2001, 215, Zhao, J. and Sadeghi, F. Analysis of EHL circular contact start up. Part 1: mixed contact model with pressure and film thickness results. J. Tribol., 2001, 123, Zhu, D. On some aspects of numerical solutions of thin-film and mixed elastohydrodynamic lubrication. Proc. IMechE, Part J: J. Engineering Tribology, 2007, 221, Li, S. and Kahraman, A. A mixed EHL model with asymmetric integrated control volume discretization. Tribol. Int., 2009, 42, Wang, K. L. and Cheng, H. S. A numerical solution to the dynamic load film thickness, and surface temperatures in spur gears. Part I: analysis. J. Mech. Des., 1981, 103, Wang, K. L. and Cheng, H. S. A numerical solution to the dynamic load film thickness, and surface temperatures in spur gears. Part II: results. J. Mech. Des., 1981, 103, Larsson, R. Transient non-newtonian elastohydrodynamic lubrication analysis of an involute spur gear. Wear, 1997, 207, Wang, Y., Li, H., Tong, J., and Yang, P. Transient thermoelastohydrodynamic lubrication analysis of an involute spur gear. Tribol. Int., 2004, 37, Conry, T. F. and Seireg, A. A mathematical programming technique for the evaluation of load distribution and optimal modifications for gear systems. J. Eng. Ind., 1973, 95, Li, S. and Kahraman, A. A spur gear mesh interface damping model based on elastohydrodynamic contact behavior. Int. J. Powertrains, 2011, 1 (in press). 33 He, S., Gunda, R., and Singh, R. Effect of sliding friction on the dynamics of spur gear pair with realistic time-varying stiffness. J. Sound Vib., 2007, 301(N3), Kahraman, A., Lim, J., and Ding, H. A dynamic model of a spur gear pair with friction. In Proceedings of the 12th IFToMM World Congress, Besancon, France, June Bair, S., Jarzynski, J., and Winer, W. O. The temperature, pressure and time dependence of lubricant viscosity. Tribol. Int., 2001, 34, Doolittle, A. K. Studies in Newtonian flow. II. The dependence of the viscosity of liquids on freespace. J. Appl. Phys., 1951, 22, Cook, R. L., King, H. E., Herbst, C. A., and Herschbach, D. R. Pressure and temperature dependent viscosity of two glass forming liquids: glycerol and dibutyl phthalate. J. Chem. Phys., 1994, 100, Appendix Notation a max maximum Hertzian half-width along the line of action b half backlash B Doolittle parameter c viscous damping e, ~e gear static transmission error under unloaded and loaded conditions, respectively f flow coefficient f m gear mesh frequency (Hz) f n natural frequency (Hz) g 0 geometry gap before deformation h film thickness h 0 reference film thickness I 1, I 2 polar mass moments of inertia of gears 1 and 2, respectively

14 Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 753 k gear mesh stiffness k, k a mean and alternating components of gear mesh stiffness K 0 bulk modulus at p ¼ 0 K0 0 derivative of bulk modulus with respect to pressure at p ¼ 0 m e equivalent mass N 1, N 2 number of teeth of gears 1 and 2, respectively p pressure r b1, r b2 base circle radii of gears 1 and 2, respectively r eq equivalent radius of curvature, r eq ¼ r 1 r 2 ðr1 þ r 2 Þ r 1, r 2 contact radii of curvature of gears 1 and 2, respectively R 1, R 2 surface roughness profiles of gears 1 and 2, respectively s dynamic transmission error SR slide-to-roll ratio, SR ¼ u s =u r t time T 1, T 2 torques applied to gears 1 and 2, respectively u 1, u 2 surface velocities in the direction of rolling of gears 1 and 2, respectively u r rolling velocity, u r ¼ 1 2 ðu 1 þ u 2 Þ u s sliding velocity, u s ¼ u 1 u 2 V surface elastic deformation V normalized volume V occ normalized occupied volume W d dynamic tooth force Wd 0 dynamic tooth force per unit face width W s quasi-static tooth force x coordinate along the rolling direction non-linear restoring function damping ratio lubricant viscosity 0 lubricant viscosity at ambient pressure 1, 2 roll angles of gears 1 and 2, respectively # 1, # 2 alternating rotational displacements of gears 1 and 2, respectively 1, 2 Poisson s ratios of gear 1 and 2, respectively lubricant density 0 lubricant density at ambient pressure 0 reference shear stress of the lubricant! n natural frequency (rad/s)! 1,! 2 angular velocities of gear 1 and 2, respectively