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with article info Article history: Received 4 June 2007 Received in revised form 14 July 2008 Accepted 14 October 2008 Available online 18 October 2008 Keywords: Gear Asymmetric teeth Dynamic load Transmission error Design abstract life, and high speeds in significant industries such as automobile, in some applications, e.g., lift machines and wind turbines. In uni- in and angle on the drive side is higher than the coast side, and the ARTICLE IN PRESS Contents lists available of International Journal of Mechanical Sciences 50 (2008) 1598–1610 Several studies in literature have been conducted on the design C3 Corresponding author. Tel.: +902244428176; fax: +902244428021. directional loading, the geometry of the drive side does not have to be symmetric to the coast side. This leads to designing pressure angle on the drive side is lower than the coast side. Gears with a larger pressure angle on the drive side compared to coast side have significant advantages as they relate to reduced contact stresses and changeable mesh conditions. and stress analysis of asymmetric gears [1–11]. Kapelevich [3] proposed a method for the design of gears with asymmetric teeth. E-mail addresses: karpats@ (F. Karpat), stephen.ekwaro-osire@ttu.edu (S. Ekwaro-Osire). 0020-7403/$ doi:10.1 aerospace, and wind turbine. Most conventional gears have symmetric teeth. These gears may be loaded in only one direction low weight. In literature, two configurations of the involute spur gears with asymmetric teeth can be found, namely, the pressure New gear designs are needed because of increasing require- ments, such as high load capacity, high endurance, low cost, long geometry, these gears allow for the selection of different pressure angles on the drive side and the coast side, which is vital obtaining key properties, such as high load-carrying capacity 1. Introduction 1.1. Background asymmetric teeth. Recently, the involute spur gears with asym- metric teeth have been found in applications requiring high performance. These gears, due to their asymmetric tooth profile, allow for optimal design in various applications. Due to their -see front matter _ x r ; € x r relative displacement, velocity, and acceleration y p ; _y p ; €y p pinion displacement, velocity, and acceleration d pI,II , d gI,II deflection of tooth in the direction of applied load x damping ratio l I , l II profile error Z kinematic viscosity Subscripts c coast side d drive side p pinion g gear I first tooth pair in mesh y g ; _y g ; €y g gear displacement, velocity, and acceleration x s loaded static transmission error Y Fa tooth form factor Y S stress concentration factor Y e contact factor z n tooth number a c , a d pressure angles for spur gear with asymmetric tooth a n pressure angle of spur and helical gears a L load angle e ad contact ratio of asymmetric gear drive m I , m II instantaneous coefficients of friction at the contact points n viscosity of lubricant y p , y g angular displacements of pinion and gear r pI,II , r gI,II radius of curvature at the mating points Sciences 50 (2008) 1598–1610 1599 provide an essential tool to conduct parametric studies on the gear parameters that play a role in its performance. For example, the software proposed in Ref. [2] can be used to automatically optimize the degree of asymmetry in order to maximize the benefits. Asymmetric teeth with high contact ratio (between 2 and 3) have also been previously tested in planetary gears of a Skorsky helicopter [11]. These tests involved gears with asym- metric teeth, referred to as buttress teeth, with larger pressure angles on the coast side compared to the drive side. From the test results, it was shown that the noise and vibration levels increased as the contact ratio increased. 1.2. Dynamic analysis of involute spur gears with symmetric teeth With increased requirements for high speed and heavy load in gear design, the dynamic analysis of modern gears is of major interest. The most important parameters in the dynamic analysis are the dynamic gear teeth loads and the static transmission errors. Static transmission errors, which are defined as the difference between the position of an actual gear tooth and that of an idealized gear tooth, and dynamic loads, affect the gear vibrations, acoustic emissions, tooth fatigue, and surface failure. Minimizing the dynamic load will decrease gear noise, increase efficiency, improve pitting fatigue life, and help to prevent gear tooth fracture [12]. Therefore, the most important objective in gear design is the minimization of dynamic loads and static transmission errors. Many investigators have theoretically and experimentally studied revolving gears under dynamic loads. Comprehensive reviews on the mathematical models used in gear dynamics are presented in Refs. [13,14]. Tearuchi and Hidetaro [15] used the II second tooth pair in mesh basic periodicities related to the shaft rotational frequencies and teeth and spur gears with asymmetric teeth. The secondary objective is to optimize the asymmetric tooth design in order to minimizing dynamic loads. 2. Dynamic model for involute spur gears with asymmetric teeth To determine the variation of dynamic load as a function of the contact position (or time), it is necessary to derive the equations of motion for a gear tooth pair in a mesh. Considering the free body diagrams of the gear and pinion shown in Fig. 1, the equations of motion can be formulated as J g € y g ? r bg eF I t F II TC6r gI m I F I C6r gII m II F II C0 r bg F D , (1) J p € y p ? r bp F D C0 r bp eF I t F II TC6r pI m I F I C6r pII m II F II , (2) ARTICLE IN PRESS Mechanical Sciences 50 (2008) 1598–1610 the gear mesh frequency. The mesh frequency and its first harmonics are the predominant contributors to the generation of noise. Many researchers investigated the effects of different parameters (e.g., design load and tooth profile modification) in decreasing the static transmission errors [22–24]. In addition, the fast Fourier transform (FFT) can be used to perform the frequency analysis of static transmission error. For the dynamic responses of gears, the dynamic model proposed in Refs. [15,17–20] can be extended to spur gears with asymmetric teeth. So far, the influence of these parameters on dynamic analysis has only been investigated for gears with symmetric teeth by several researchers [20–25]. 1.3. Motivation and objectives Involute spur gears with asymmetric teeth provide flexibility to designers for different application areas due to non-standard design. If they are correctly designed, they can make important contributions to the improvement of designs in aerospace industry, automobile industry, and wind turbine industry. This often relates to improving the performance, increasing the load capacity, reduction of acoustic emission, and reduction of vibration [3]. In the past, most of the analysis of gears with asymmetric teeth has been limited to cases under static loading [1,4,8]. Dynamic loads and vibration are the major concern for gears running at high speeds. Therefore, dynamic behavior should be analyzed to determine the feasibility of asymmetric gears in different applications. In order to utilize asymmetric gear designs more effectively, it is imperative to perform analyses of these gears under dynamic loading. This study offers designers preliminary results for understanding the response of asymmetric gears under dynamic loading. The effect of some design parameters, such as pressure angle or tooth height on dynamic loads, is shown. The asymmetric gears considered will have a larger pressure angle on the drive side compared to the coast side. In this paper, to study the response of asymmetric gears under tooth deflection, equivalent composite error, and equivalent mass of gear, in the calculation of the dynamic loads on gear teeth. The numerical results obtained were shown to be in good agreement with the experimental results. A similar vibratory model was presented in Ref. [16]. A comparison of the theoretical and the experimental results, obtained for dynamic characteristics of the heavily loaded spur gears, was made. A numerical approach for the equations of motion that contain the excitation terms due to errors and periodic variation of the mesh stiffness was developed and presented. This method was adapted and employed by several researchers [17–21] to calculate the dynamic contact load or the torsional response, depending on different gear parameters, i.e., tooth errors, addendum modification, mesh stiffness, lubrication, damping factor, gear contact factor, and friction coefficient. In gear design, the dynamic factor is generally used to quantify the dynamic effects. In this context, the dynamic factor is defined as the ratio of the maximum dynamic load to the maximum static load on the gear tooth. Dynamic loads of gears with low-contact ratio (contact ratio is between 1 and 2) are affected by several parameters, namely, time-varying mesh stiffness, tooth profile error, contact ratio, friction, and sliding. The static transmission errors change in a periodic manner due to the variation of gear mesh stiffness during contact. This is the source of vibratory excitation in gear dynamics. The static transmission error has F. Karpat et al. / International Journal of1600 dynamic loading, the dynamic loads and static transmission errors were used. The primary objective of this paper is to use dynamic analysis to compare conventional spur gears with symmetric where J p and J g represent the polar mass moments of inertia of the pinion and gear, respectively. The dynamic contact loads are F I and F II , while m I and m II are the instantaneous coefficients of friction at the contact points. y p and y g represent the angular displacements of pinion and gear. The radii of the base circles of the engaged gear pair are r bp and r bg , while the radii of curvature at the mating points are r pI,II and r gI,II . In the above equations, if the speed of the pinion tooth is greater than the speed of the gear tooth, the sign of the friction force is positive; otherwise it is negative. The static tooth load is defined as F D ? T p r bp ? T g r bg , (3) If the angular coordinate is converted into the coordinate along the line of action, the displacements of the undeformed tooth profiles, along the line action, can be written as y g ? r g y g , (4) y p ? r p y p , (5) The relative displacement, velocity, and acceleration can then be cast as x r ? y p C0 y g , (6) _x r ? _y p C0 _y g , (7) € x r ? € y p C0 € y g , (8) Fig. 1. Engaging teeth pairs along the contact line. The effective gear masses are: M p ? J p r 2 bp , (9) M g ? J g r 2 bg , (10) Including viscous damping, the equations of motion reduce to: €x r t2ox_x r to 2 x r ? o 2 x s , (11) o 2 ? K I eS pI M g t S gI M p TtK II eS pII M g t S gII M p T M g M p , (12) o 2 x s ? eM g t M p TF D t K I l I eS pI M g t S gI M p TtK II l II eS pII M g t S gII M p T M g M p , (13) The loaded static transmission errors can be obtained by dividing Eq. (13) by Eq. (12) to yield: x s ? eM g t M p TF D t K I l I eS pI M g t S gI M p TtK II l II eS pII M g t S gII M p T K I eS pI M g t S gI M p TtK II eS pII M g t S gII M p T , (14) The equivalent stiffnesses of meshing tooth pairs, in Eqs. (12)–(14), can be written as K I ? k pI k gI k pI t k gI , (15) K II ? k pII k gII k pII t k gII , (16) The friction experienced by the pinion and the gear can be expressed as S pI ? 1C6 m I r pI r bp , (17) S gI ? 1C6 m I r gI r bd , (18) S pII ? 1C6 m II r pII r bp , (19) S gII ? 1C6 m II r gII r bd , (20) The signs in the above expressions are positive (+) for the approach and negative (C0) for the recess. ARTICLE IN PRESS (a) F. Karpat et al. / International Journal of Mechanical Sciences 50 (2008) 1598–1610 1601 Fig. 2. Meshing of asymmetric gears: Fig. 3. Finite element model: (a) meshed 2-D model contact zone and (b) contact line [5]. and (b) geometry of Plane 82 2-D element. In this paper, we used the following equation, which is derived by Dowson and Higginson [26] from experimental results and used by some researchers, for calculating the coefficient of friction [16–19]: m I;II ? 18:1n C00:15 v gI;II t v pI;II v gI;II C0 v pI;II C12 C12 C12 C12 ! C00:15 v gI;II C0 v pI;II C12 C12 C12 C12 C0C1 C00:5 r gI;II r pI;II r gI;II tr pI;II ! C00:5 , (21) where n is the viscosity of lubricant (cSt) and v pI,II and v gI,II are the surface velocities (mm/s), which can be formulated as follows: v pI;II ? V L pI;II cos a d r bp tsin a d C18C19 , (22) v gI;II ? V C0 L gI;II cos a d r bg tsin a d C18C19 , (23) where L pI,II and L gI,II are the distances between the contact point and the pitch point along the line of action for pinion and gear, respectively, and V is the tangential velocity on the pitch circle. The value of the damping ratio, x, in Eq. (11), is recommended to be between 0.1 and 0.2 by Ichimaru and Hirano [16]. In this paper, a constant value of 0.17 proposed by Dowson and Higginson [26] for the damping ratio, z, was adapted in the solution of equations. The dynamic contact loads, which include tooth profile error, can then be written as F I ? K I ex r C0l I T, (24) F II ? K II ex r C0l II T, (25) where l I and l II are the tooth profile errors. In this paper, the effects of profile errors on the dynamic response of gears are not considered. Thus, the tooth profile errors are assumed to be zero. The developed computer program has a capability of using any approach for the determination of errors. It should be noted that the above equations are only valid when there is contact between two gears. When separation occurs between two gears, because of the relative errors between the teeth of gears, the dynamic load will be zero and equation of motion will be given by T€x r ? F D , (26) The meshing conditions are described as follows: ARTICLE IN PRESS Fig. 4. Load application. F. Karpat et al. / International Journal of Mechanical Sciences 50 (2008) 1598–16101602 Fig. 5. Contact geometry (a) radii of curvature of pinion In spur gears with low-contact ratio, during one mesh period, there is one tooth pair in contact and two tooth pairs in contact, occurring separately. Therefore, since the gear mesh stiffness is a function of the number of tooth pairs in contact, it is consequently a function of time. Meshing action begins at point A and ends at point E as shown in Fig. 2(b). If x r 4l I ; x r 4l II F I ; F II 40 Double tooth contact If x r pl I ; x r pl II F I ? F II ? 0 Tooth separation If l I ox r pl II F I 40 and F II ? 0 Single tooth contact If l II ox r pl I F I ? 0 and F II 40 Single tooth contact 3. Equivalent stiffness of meshing tooth pair 3.1. Gear mesh stiffness and gear at contact point (b) Hertzian contact zone. solution, and post processing) is automatically performed. At the where c and e are the length and width of the element, respectively (see Fig. 3(b)). In this study, it is assumed that the parameters c and e are equal in value, and the value of the element width, e, is then calculated using Eq. (31). This value is input into the ANSYS program as the element size. To adopt Hertz theory to contacting gear teeth, it is assumed that the radii of curvature at the mating points for pinion and gear (see Fig. 5(a)) are equal to radii of two identical cylinders pressed together (see Fig. 5(b)). The Hertzian contact width, b h (see Fig. 5(b)), can be calculated as ARTICLE IN PRESS Fig. 6. Flowchart of the developed computer program. Mechanical Sciences 50 (2008) 1598–1610 1603 end, an output file, that contains nodal deflection for loaded nodes, is created. This process is repeated for each gear. It should be noted that in this analysis the loads are applied at five locations on the gear tooth (see Fig. 4). In this study, the applied load for each contact point is taken as 250N in order to determine tooth deflection under unit load. By using the nodal deflection values that are read from the output files, the approximate curves for the single tooth stiffness along the contact line are obtained with respect to the radius of the gears. To facilitate the calculation of the Hertzian component of the deflection at the point of loading, the size of the grid near the point of loading is chosen as recommended in Refs. [27,28] using the following equation: The meshing of gears alternates between the single contact zone (BD) and the double contact zone (AB and DE), along the path of contact (AE) (see Fig. 2(b)). So, the gear mesh stiffness varies between two-average values due to each contact case. K I a0 and K II ? 0 eF II ? 0T in single contact zone K I a0 andK II a0 in double contact zone The gear mesh stiffness (K I +K II ) in the double contact zone is greater than that in the single contact zone. Asymmetric gears with a greater pressure angle for the drive side are considered for a single mesh period. It is seen that the contact ratio reduces and the single contact zone (BD) increases by increasing the pressure. 3.2. Tooth stiffness According to Eqs. (15) and (16), in order to calculate the equivalent stiffness of a meshing tooth pair, the tooth stiffness has to be evaluated first. The tooth stiffness can be expressed as k p1 ? F d pI , (27) k g1 ? F d gI , (28) k pII ? F d pII , (29) k gII ? F d gII , (30) where F is the applied load, and d pI , d pII , d gI , and d gII are the deflections of the teeth in the direction of applied load. In literature, different methods and empirical equations are used to calculate the tooth deflections of spur gears. These methods are often based on the classical theory of elasticity and numerical approaches. However, all of them are derived for symmetric teeth. Therefore, in this study, a 2-D finite element model is developed to calculate the deflections of both the asymmetric and the symmetric gear teeth (see Fig. 3). This 2-D model is meshed (see Fig. 3(a)) using a Plane82 element which is a high order element with 8 nodes. The Plane82 element is chosen because it is well adapted to displacement shapes and the model’s curved boundaries. Fig. 3(b) depicts the geometry of the Plane 82 2-D element. A computer program, which saves time and provides a means to carry out a parametric study with the gear parameters, was developed using MATLAB. This program generates batch files for input into ANSYS. When this file is executed in ANSYS, the general procedure of FEA (i.e., 2-D modeling, meshing, loading, F. Karpat et al. / International Journal of e b h ?C00:2 c e C16C17 t1:2 for 0:9p c e p3, (31) Table 1 Properties of gear pair used in the validation of program. Module, m n 3.18mm Teeth number of pinion, z np 28 Teeth number of gear, z ng 28 Mass of pinion, M p 0.3kg Mass of gear, M g 0.3kg Rotational speed of pinion 1500rpm Material Steel Kinematic viscosity, Z 75cSt Damping ratio, x 0.17 Tooth face width, b 6.35mm Addendum factor, h a 1.00 Dedendum factor, h f 1.25 ARTICLE IN PRESS Fig. 7. Dynamic factors for a single period at 1500rpm: (a) single-tooth pair and (b) double-tooth pair. Fig. 8. Mesh stiffness during a single mesh period using the developed program (solid line) and from Ref. [29] (dashed line). Table 2 Properties of the gear pairs. Gear pair 1234567 Module, m n (mm) 3.18 3.18 3.18 3.18 3.18 3.18 3.18 Teeth number of pinion, z np 32 32 32 32 32 32 32 Pressure angle on coast side, a c (deg) 20 20 20 20 20 20 20 Pressure angle on drive side, a d (deg) 20 25 30 35 25 30 35 Gear ratio 1 1 1 1 1 1 1 Mass of pinion, M p (kg) 1.2 1.2 1.2 1.2 1.2 1.2 1.2 Mass of gear, M g (kg) 1.2 1.2 1.2 1.2 1.2 1.2 1.2 Material Steel Steel Steel Steel Steel Steel Steel Kinematic viscosity, Z (cSt) 100 100 100 100 100 100 100 Damping ratio, x 0.17 0.17 0.17 0.17 0.17 0.17 0.17 Tooth face width, b (mm) 25.4 25.4 25.4 25.4 25.4 25.4 25.4 Backlash 0 0 0 0 0 0 0 Addendum factor, h a 1.00 1.00 1.00 1.00 1.38 1.27 1.17 Gear contact ratio 1.67 1.48 1.36 1.28 1.98 1.70 1.48 0.4 0.6 0.8 1.0 1.2 1.4