Gear Geometry and Applied Theory Episode 1 Part 10 pps

P1: GDZ/SPH P2: GDZ CB672-09 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:16 9.4 Tooth Contact Analysis 253 Figure 9.4.2: Applied coordinate systems. The numerical solution of the system of nonlinear equations is based on application of a respective subroutine; see, for instance, More et al. [1980] and Visual Numerics, Inc. [1998]. The first guess for the solution can be obtained from the data provided by the local synthesis. We illustrate the discussed method of TCA with the following simple problem of a planar gearing. Problem 9.4.1 Consider three coordinate systems S 1 , S 2 , and S f that are rigidly connected to driving gear 1, driven gear 2, and the frame f , respectively (Fig. 9.4.2). Gear 1 is provided with involute profile  1 that is represented in S 1 by the following equations (Fig. 9.4.3): x 1 = r b1 (sin θ 1 − θ 1 cos θ 1 ), y 1 = r b1 (cos θ 1 + θ 1 sin θ 1 ), z 1 = 0. (9.4.23) Figure 9.4.3: Profile  1 of gear 1. P1: GDZ/SPH P2: GDZ CB672-09 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:16 254 Computerized Simulation of Meshing and Contact Figure 9.4.4: Profile  2 of gear 2. Gear 2 is provided with involute profile  2 that is represented in S 2 by the equations (Fig. 9.4.4) x 2 = r b2 (−sinθ 2 + θ 2 cos θ 2 ), y 2 = r b2 (−cosθ 2 − θ 2 sin θ 2 ), z 2 = 0. (9.4.24) Solution The application of the basic principle of tooth contact analysis enables us to determine the conditions of meshing of  1 and  2 in coordinate system S f using the following procedure: (1) We determine the unit normals n 1 and n 2 to  1 and  2 in coordinate systems S 1 and S 2 , respectively. The unit normals to  1 and  2 must be of the same orientation at the point of tangency of the profiles. (2) Then, we represent profiles  1 and  2 in coordinate system S f and derive the equations of their tangency. (3) Using the equations of tangency we can obtain three equations of the following structure: f 1 [(θ 1 − φ 1 ), (θ 2 + φ 2 )] = 0 (9.4.25) f 2 [(θ 1 − φ 1 ), r b1 , r b2 , E] = 0 (9.4.26) f 3 [θ 1 ,θ 2 , r b1 , r b2 , E, (θ 1 − φ 1 )] = 0. (9.4.27) P1: GDZ/SPH P2: GDZ CB672-09 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:16 9.4 Tooth Contact Analysis 255 (4) The analysis of the obtained equations shows that the ratio dφ 1 /dφ 2 is constant and is represented as dφ 1 dφ 2 =− dθ 1 dθ 2 = r b2 r b1 . (5) We can determine the line of action by the vector function r (1) f (θ 1 − φ 1 ) and prove that the line of action is a straight line. The orientation of the line of action can be determined by the scalar product a f · (−i f ) where a f = ∂r (1) f ∂θ 1      ∂r (1) f ∂θ 1      is the unit vector of the line of action. The procedure of derivations is as follows: Step 1: Equations (9.4.23) yield the following expressions for the unit normal to  1 : n 1 = t 1 × k 1 = cos θ 1 i 1 − sin θ 1 j 1 (provided θ 1 = 0). (9.4.28) Here, t 1 is the unit tangent to  1 ; k 1 is the unit vector of the z 1 axis. Step 2: Similarly, using Eqs. (9.4.24) we obtain that n 2 = k 2 × t 2 = cos θ 2 i 2 − sin θ 2 j 2 . (9.4.29) Here, t 2 is the unit tangent to  2 ; k 2 is the unit vector of the z 2 axis. The order of cofactors in Eq. (9.4.29) provides the orientation of n 2 as shown in Fig. 9.4.4. Step 3: Using matrix equations r (i ) f = M fi r i (θ i ), n (i ) f = L fi n i (θ i )(i = 1, 2), (9.4.30) we derive the following equations of tangency: r (1) f (θ 1 ,φ 1 ) = r (2) f (θ 2 ,φ 2 ), n (1) f (θ 1 ,φ 1 ) = n (2) f (θ 2 ,φ 2 ). (9.4.31) Step 4: Vector Eqs. (9.4.31) yield the following system of scalar equations: r b1 [ sin(θ 1 − φ 1 ) − θ 1 cos(θ 1 − φ 1 ) ] −r b2 [ −sin(θ 2 + φ 2 ) + θ 2 cos(θ 2 + φ 2 ) ] = 0 (9.4.32) r b1 [ cos(θ 1 − φ 1 ) + θ 1 sin(θ 1 − φ 1 ) ] −r b2 [ −cos(θ 2 + φ 2 ) − θ 2 sin(θ 2 + φ 2 ) ] − E = 0 (9.4.33) cos(θ 1 − φ 1 ) − cos(θ 2 + φ 2 ) = 0 (9.4.34) sin(θ 1 − φ 1 ) − sin(θ 2 + φ 2 ) = 0. (9.4.35) Step 5: Analyzing Eqs. (9.4.34) and (9.4.35), we obtain θ 1 − φ 1 = θ 2 + φ 2 . (9.4.36) P1: GDZ/SPH P2: GDZ CB672-09 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:16 256 Computerized Simulation of Meshing and Contact Equations (9.4.32) and (9.4.33), considered simultaneously, yield the following rela- tions: cos(θ 1 − φ 1 ) − r b1 +r b2 E = 0 (9.4.37) r b1 θ 1 +r b2 θ 2 − E sin(θ 1 − φ 1 ) = 0. (9.4.38) The system of Eqs. (9.4.36) to (9.4.38) has the structure of the system of Eqs. (9.4.25) to (9.4.27) discussed above. Equations (9.4.36) to (9.4.38) yield θ 1 − φ 1 = θ 2 + φ 2 = const. (9.4.39) r b1 θ 1 +r b2 θ 2 = const. (9.4.40) Step 6: Differentiating Eqs. (9.4.39) and (9.4.40), we obtain that the gear ratio is constant and can be represented as follows: m 12 = dφ 1 dφ 2 =− dθ 1 dθ 2 = r b2 r b1 . (9.4.41) Step 7: The line of action is represented by the equation r (1) f = r b1 [ sin(θ 1 − φ 1 ) − θ 1 cos(θ 1 − φ 1 ) ] i f +r b1 [ cos(θ 1 − φ 1 ) + θ 1 sin(θ 1 − φ 1 ) ] j f (9.4.42) where (θ 1 − φ 1 ) is constant [see Eq. (9.4.39)]. Vector function r (1) f (θ 1 ) is a linear one because (θ 1 − φ 1 ) = constant, and the line of action is a straight line. The unit vector of the line of action is represented as a f = ∂r (1) f ∂θ 1      ∂r (1) f ∂θ 1      =−cos(θ 1 − φ 1 )i f + sin(θ 1 − φ 1 )j f . (9.4.43) The orientation of the line of action is determined with the scalar product a f · (−i f ) = cos(θ 1 − φ 1 ) = r b1 +r b2 E . (9.4.44) The line of action passes through point I that lies on the y f axis. Equation (9.4.42) yields that when x (I) f = 0, we have y (I) f = r b1 cos(θ 1 − φ 1 ) . (9.4.45) Using Eqs. (9.4.44) and (9.4.45), we obtain y (I) f =  r b1 r b1 +r b2  E = E 1 + m 12 . (9.4.46) The line of action is shown in Fig. 9.4.5. It is easy to verify that the line of action is tangent to the gear base circles. We emphasize that the location and orientation of the line of action depends on the chosen center distance E (considering the radii r b1 and r b2 of base circles as given). P1: GDZ/SPH P2: GDZ CB672-09 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:16 9.5 Application of Finite Element Analysis for Design of Gear Drives 257 Figure 9.4.5: Location and orientation of line of action. 9.5 APPLIC ATION OF FINITE ELEMENT ANALYSIS FOR DESIGN OF GEAR DRIVES Application of finite element analysis allows us to perform (i) stress analysis, (ii) inves- tigation of formation of bearing contact, and (iii) detection of severe areas of contact stresses inside the cycle of meshing. Such an approach requires (i) development of the finite element mesh of the gear drive, (ii) definition of contacting surfaces, and (iii) establishment of boundary conditions for loading the gear drive. This section covers the authors’ approach to finite element analysis for gear design. The approach is based on application of the general purpose computer program pre- sented by Hibbit, Karlsson & Sirensen, Inc. [1998]. The main features of the developed approach are as follows: (a) The finite element mesh is generated automatically by using the equations of the tooth surfaces and the rim. Nodes of finite element mesh are obtained as points of gear tooth surfaces. Therefore, the loss of accuracy associated with development of solid models using CAD (computer aided design) computer programs is avoided. The boundary conditions for stress analysis of the pinion and the gear are set up automatically as well. P1: GDZ/SPH P2: GDZ CB672-09 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:16 258 Computerized Simulation of Meshing and Contact (b) Modules for automatic generation of finite element models are integrated into the developed computer programs. Therefore, the generation of finite element mod- els can be accomplished easily and fast for any position of contact of the cy- cle of meshing. In addition, the formation of the bearing contact can be inves- tigated and the appearance of edge contact and areas of severe contact can be detected. Application of CAD computer programs for the development of finite element models is an intermediate stage of the existing approach for application of finite element analysis and has the following disadvantages: (1) Determination of wire models formed by splines is obtained numerically. The wire models consist of planar sections of gear teeth and such sections are used for the development of solid models. (2) Finite element meshes of solid models require application of computer programs for finite element analysis. (3) Setting of boundary conditions for the finite element meshes have to be deter- mined. (4) The increase of planar sections of gear teeth improves the precision of wire models and solid models but is costly in terms of time. (5) The developments described above have to be performed by skilled users of CAD computer programs, are costly in terms of time, and have to be accomplished for each assigned case of design of various gear geometries, for each position of meshing, and for various cases of investigation. The modified approach presented in this section is free of the disadvantages mentioned above and may be summarized as follows: Step 1: Using the equations of both sides of the pinion or gear tooth surfaces and the portions of the corresponding rim, we may represent analytically the volume of the designed body. Figure 9.5.1(a) shows the designed body for a one-tooth model of the pinion of a modified involute helical gear drive. Step 2: Auxiliary intermediate surfaces 1 to 6 shown in Fig. 9.5.1(b) are determined analytically as well. Surfaces 1 to 6 enable us to divide the tooth into six parts and control the discretization of these tooth subvolumes into finite elements. Step 3: Analytical determination of node coordinates is performed taking into ac- count the number of desired elements in the longitudinal and profile directions [Fig. 9.5.1(c)]. We emphasize that all nodes of the finite element mesh are determined an- alytically and those lying on the intermediate surfaces of the tooth are indeed points belonging to the real surface. Step 4: Discretization of the model by finite elements using nodes determined in previous step is accomplished as shown in Fig. 9.5.1(d). Step 5: The setting of boundary conditions for gear and pinion is performed auto- matically as follows: (i) Nodes on the sides and bottom part of the rim portion of the gear are considered fixed [Fig. 9.5.2(a)]. P1: GDZ/SPH P2: GDZ CB672-09 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:16 9.5 Application of Finite Element Analysis for Design of Gear Drives 259 Figure 9.5.1: Illustration of (a) the volume of the designed body, (b) auxiliary intermediate surfaces, (c) determination of nodes for the whole volume, and (d) discretization of the volume by finite elements. (ii) Nodes on the two sides and bottom part of the rim portion of the pinion build a rigid surface [Fig. 9.5.2(b)]. Rigid surfaces are three-dimensional geometric structures that cannot be deformed but can perform translation or rotation as rigid bodies (Hibbit, Karlsson & Sirensen, Inc. [1998]). They are also very cost effective because the variables associated with a rigid surface are the translations and rotations of a single node, known as the rigid body reference node [Fig. 9.5.2(b)]. The rigid body reference node is located on the pinion axis of rotation with all degrees of freedom except the rotation around the axis of rotation of the pinion fixed to zero. The torque is applied directly to the remaining degree of freedom of the rigid body reference node [Fig. 9.5.2(b)]. Step 6: Definition of contacting surfaces for the contact algorithm of the finite element computer program (Hibbit, Karlsson & Sirensen, Inc. [1998]) is performed automat- ically as well and requires definition of the master and slave surfaces. Generally, the master surface is chosen as the surface of the stiffer body or as the surface with the coarser mesh if the two surfaces are on structures with comparable stiffness. P1: GDZ/SPH P2: GDZ CB672-09 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:16 260 Computerized Simulation of Meshing and Contact Figure 9.5.2: (a) Boundary conditions for the gear; (b) schematic representation of boundary conditions and application of torque for the pinion. Figures 9.5.3 to 9.5.5 show examples of finite element models of a spiral bevel gear drive, a helical gear drive, and a face-worm gear drive with a conical worm, respectively. 9.6 EDGE CONTACT Most prospective gear design has to be based on localization of the bearing contact on gear tooth surfaces. Gear tooth surfaces with localized bearing contact are in point P1: GDZ/SPH P2: GDZ CB672-09 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:16 9.6 Edge contact 261 Figure 9.5.3: Finite element model of a whole spiral bevel gear drive. contact at every instant but are not in line contact. However, there are at present some gear drives with tooth surfaces that are still in line contact. In fact, due to errors of alignment, the theoretical instantaneous contact at a line is turned over into a point contact, but it may be accompanied by an edge contact (see below). The simulation of meshing of tooth surfaces being in instantaneous point contact may be performed by TCA computer programs (see Section 9.4) based on continuous tangency of tooth surfaces that have a common normal at the instantaneous point contact. Figure 9.5.4: Finite element model of a whole heli- cal gear drive. P1: GDZ/SPH P2: GDZ CB672-09 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:16 262 Computerized Simulation of Meshing and Contact Figure 9.5.5: Finite element model of a face-worm gear drive with a conical worm. Edge contact means that instead of tangency of surfaces, an edge of the tooth surface of one gear is in mesh with the tooth surface of the mating gear. Edge contact may be represented by the following equations: r (1) f (u 1 (θ 1 ),θ 1 ,φ 1 ) = r (2) f (u 2 ,θ 2 ,φ 2 ) (9.6.1) ∂r (1) f ∂θ 1 · N (2) f = 0. (9.6.2) Here, r (1) f (u 1 (θ 1 ),θ 1 ,φ 1 ) represents the edge of the pinion tooth surface; ∂r (1) f /∂θ 1 is the tangent to the edge. Equation system (9.6.1) and (9.6.2) represents a system of four nonlinear equations in four unknowns: θ 1 , u 2 , θ 2 , φ 2 ; φ 1 is the input parameter. Similar equations can be derived for the case of tangency of the edge of the gear tooth surface with the pinion tooth surface. Edge contact may occur in two cases: (i) when the gear tooth surfaces are initially in line contact, and (ii) when the gear tooth surfaces are in point contact. Each case is discussed separately. Edge Contact of Gear Tooth Surfaces That Are Initially in Line Contact We start the discussion with the case of spur gears. Figure 9.6.1(a) shows that the gear tooth surfaces  1 and  2 of an ideal gear train are in tangency along the line L 1 –L 2 . Consider now that the gears are misaligned and the gear axes are crossed or intersected. Then, edge E 1 of the pinion tooth surface will be in tangency with the gear tooth surface  2 at point M. The paths of contact on the gear tooth surfaces are shown in Fig. 9.6.2(a). The transformation of motion is accompanied by the function of transmission errors shown in Fig. 9.6.2(b). The transfer of meshing at the end of the cycle of meshing is accompanied by the jump in the angular velocity, and vibration and noise are inevitable. Similarly, the edge contact of helical gears with parallel axes caused by angular misalignment, such as the crossing angle of gear axes and the difference of gear helix angles, may be discussed. Edge contact of misaligned gears whose tooth surfaces are initially in line contact can be avoided by application of a modified topology of tooth surfaces. Such a topology [...]... P1: FHA/JTH CB672 -10 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0 :19 10 .2 Geometry of Involute Curves 2 71 Figure 10 .2.4: Extended involute curve Using an approach similar to that discussed above, we obtain the following equations: x = (r b ∓ h) sin φ − r b φ cos φ y = (r b ∓ h) cos φ + r b φ sin φ (10 .2 .11 ) The upper sign in Eqs (10 .2 .11 ) corresponds to the extended involute (Fig 10 .2.4), and. .. respective displacement e of the rack-cutter (Fig 10 .5 .1) can be determined from Eq (10 .5 .1) with the requirement that αG ≥ 0 Then we obtain tan αc − 4P (a − e) ≥ 0 N sin 2αc (10 .5.5) Expression (10 .5.5) with a = 1/ P yields N sin2 αc − 2 (1 − Pe) ≥ 0 N sin αc cos αc (10 .5.6) Considering (10 .5.6) and (10 .5.4) simultaneously, we obtain Pe ≥ Nmin − N Nmin (10 .5.7) Here, Pe is an algebraic unitless value... involute curve P1: FHA/JTH CB672 -10 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0 :19 272 Spur Involute Gears Figure 10 .2.6: Archimedes spiral There is a particular case when h = r b and the extended involute curve turns out into the Archimedes spiral (Fig 10 .2.6) determined by the equation Mo M = r = r b φ (See also Problem 1. 6 .1) Another particular case is when h = 0 and curve (10 .2 .11 ) is a conventional... Two branches of an involute curve (10 .2 .1) P1: FHA/JTH CB672 -10 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0 :19 270 Spur Involute Gears where OP = r b [sin φ cos φ]T PM = PM [− cos φ (10 .2.2) sin φ]T (10 .2.3) (ii) Due to rolling without sliding, we have PM = MoP = r b φ (10 .2.4) Here, φ is the angle of rotation in rolling motion (iii) Equations (10 .2 .1) to (10 .2.4) yield x = r b (sin φ − φ cos... the rack -gear meshing The rotation of the hob simulates the translation of the imaginary rack P1: FHA/JTH CB672 -10 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 (a) Feed of hob (b) Figure 10 .3.5: Generation by hob Figure 10 .3.6: Generation by shaper 0 :19 P1: FHA/JTH CB672 -10 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0 :19 278 Spur Involute Gears Figure 10 .3.7: Meshing of gear and shaper... generation of a spur involute gear by a rack-cutter is shown in Fig 10 .3 .1 The gear to be cut rotates with angular velocity ω about O, and the rack-cutter translates with velocity v The velocity |v| and angular velocity ω are related by the equation N v = rp = ω 2P (10 .3 .1) P1: FHA/JTH CB672 -10 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0 :19 274 Spur Involute Gears Figure 10 .3 .1: Generation of involute... (Fig 10 .3.4) is a segment of a straight line that is equal to pc (Recall that the pitch circle is the centrode in meshing with the rack-cutter.) The diametral pitch is represented as P = π N = d pc (10 .4 .1) P1: FHA/JTH CB672 -10 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0 :19 10 .4 Tooth Element Proportions 279 Figure 10 .4 .1: Gear tooth parameters and is defined as the number of teeth of the gear. .. Numerics, Inc [19 98]) An approximate representation but with high precision of the inverse function α(θ ) (θ = tan α − α) was proposed by Cheng [19 92]: α = (3θ )1/ 3 − 9 2/3 5/3 2 2 1/ 3 7/3 θ+ 3 θ − 3 θ + ··· 5 17 5 17 5 for θ < 1. 8 (10 .2 .10 ) Extended and Shortened Involute Curves These curves are traced out by point M, which is offset with respect to the rolling straight line (Figs 10 .2.4 and 10 .2.5) The... 2) is recommended 267 P1: FHA/JTH CB672 -10 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 268 0 :19 Spur Involute Gears Figure 10 .2 .1: Involute and evolute 10 .2 GEOMETRY OF INVOLUTE CURVES Henceforth, we consider conventional, extended, and shortened involute curves (see Section 1. 6) We start with general definitions of the evolute and the involute of a planar curve Involute and Evolute Consider that... Consider now that the number N of gear teeth has been increased but P and αc are kept at the same values With N > N the radii of the pitch circle and the base Figure 10 .3.3: Rack as a particular case of a gear P1: FHA/JTH CB672 -10 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0 :19 276 Spur Involute Gears Figure 10 .3.4: Parameters of a rackcutter circle are r p and r b , respectively; the center .  1 that is represented in S 1 by the following equations (Fig. 9.4.3): x 1 = r b1 (sin θ 1 − θ 1 cos θ 1 ), y 1 = r b1 (cos θ 1 + θ 1 sin θ 1 ), z 1 = 0. (9.4.23) Figure 9.4.3: Profile  1 of gear. follows: m 12 = dφ 1 dφ 2 =− dθ 1 dθ 2 = r b2 r b1 . (9.4. 41) Step 7: The line of action is represented by the equation r (1) f = r b1 [ sin(θ 1 − φ 1 ) − θ 1 cos(θ 1 − φ 1 ) ] i f +r b1 [ cos(θ 1 − φ 1 ). equations: r (1) f (u 1 (θ 1 ),θ 1 ,φ 1 ) = r (2) f (u 2 ,θ 2 ,φ 2 ) (9.6 .1) ∂r (1) f ∂θ 1 · N (2) f = 0. (9.6.2) Here, r (1) f (u 1 (θ 1 ),θ 1 ,φ 1 ) represents the edge of the pinion tooth surface; ∂r (1) f /∂θ 1 is