P1: JXT CB672-08 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:13 8.4 Direct Relations Between Principal Curvatures of Mating Surfaces 223 equation is the differentiated equation of meshing (8.2.7), in which we take i = 1 and represent it as follows: ˙ n (1) r · v (12) −  v (1) r ·  ω (12) × n  + n ·  ω (1) × v (2) tr  −  ω (2) × v (1) tr  −  ω (1)  2 m  21 n ·  k 2 ×  r (1) − R  = 0. (8.4.35) We transform Eq. (8.4.35) using the following procedure: Step 1: Representing vectors of the scalar product ˙ n (1) r · v (12) in coordinate system S a (e f , e h ), we obtain v (12) · ˙ n (1) r =   v (12) f v (12) h   T   ˙ n (1) f ˙ n (1) h   . (8.4.36) Step 2: Using Eqs. (8.4.11), we obtain v (12) · ˙ n (1) r =   v (12) f v (12) h   T K 1   v (1) f v (1) h   . (8.4.37) Step 3: Equations (8.4.37) and (8.4.6) yield v (12) · ˙ n (1) r =   v (12) f v (12) h   T K 1   v (2) f v (2) h   −   v (12) f v (12) h   T K 1   v (12) f v (12) h   =   v (12) f v (12) h   T K 1   v (2) f v (2) h   + κ f  v (12) f  2 + κ h  v (12) h  2 . (8.4.38) Step 4: Our next step is directed at the transformation of the triple product {−v (1) r · (ω (12) × n)}. Representing vectors of the triple product in coordinate system S a (e f , e h ), we obtain −v (1) r ·  ω (12) × n  =   n ×ω (12)  · e f  n ×ω (12)  · e h  T   v (1) f v (1) h   . (8.4.39) Step 5: Equations (8.4.39) and (8.4.6) yield −v (1) r ·  ω (12) × n  =   n ×ω (12)  · e f  n ×ω (12)  · e h  T   v (2) f v (2) h   −  n ×ω (12)  · v (12) . (8.4.40) P1: JXT CB672-08 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:13 224 Mating Surfaces: Curvature Relations, Contact Ellipse Step 6: Using Eqs. (8.4.38) and (8.4.40), we represent Eq. (8.4.35) as follows:        v (12) f v (12) h   T K 1 +   n ×ω (12)  · e f  n ×ω (12)  · e h  T        v (2) f v (2) h   =−n ·  ω (1) × v (2) tr  −  ω (2) × v (1) tr  +  ω (1)  2 m  21 (n ×k 2 ) ·(r (1) − R) +  n ×ω (12)  · v (12) − κ f  v (12) f  2 − κ h  v (12) h  2 . (8.4.41) Finally, using Eq. (8.4.41) and the first two equations of equation system (8.4.15), we obtain the following system of three linear equations in the unknowns v (2) f and v (2) h : t i 1 v (2) f + t i 2 v (2) h = t i 3 (i = 1, 2, 3). (8.4.42) Here, t 11 ≡ b 11 , t 12 = t 21 ≡ b 12 , t 22 ≡ b 22 t 13 = t 31 ≡ b 15 , t 23 ≡ t 32 ≡ b 25 t 33 =−n ·  ω (1) × v (2) tr  −  ω (2) × v (1) tr  (8.4.43) +  ω (1)  2 m  21 (n ×k 2 ) ·  r (1) − R  +  n ×ω (12)  · v (12) − κ f  v (12) f  2 − κ h  v (12) h  2 . For further derivations, it is important to recognize that the rank of the system matrix and the augment matrix for equation system (8.4.42) is 1. This follows from the fact that the contacting surfaces are in line contact at every instant, the displacement of a contact point over the surface is not unique, and therefore the solution of system equation (8.4.42) for the unknowns v (2) f and v (2) h is not unique either. The requirement that the rank of the system matrix and the augmented matrix be 1 enables us to derive the following equations for determination of principal directions on  2 and the principal curvatures of this surface: tan 2σ = −2t 13 t 23 t 2 23 − t 2 13 − (κ f − κ h )t 33 (8.4.44) κ q − κ s = −2t 13 t 23 t 33 sin 2σ = t 2 23 − t 2 13 − (κ f − κ h )t 33 t 33 cos 2σ (8.4.45) κ q + κ s = κ f + κ h + t 2 13 + t 2 23 t 33 . (8.4.46) The advantage of Eqs. (8.4.44) to (8.4.46) is the opportunity to determine the prin- cipal curvatures and directions on surface  2 knowing the principal curvatures and directions on  1 and the parameters of motion of the mating surfaces. The knowledge of principal curvatures and directions of contacting surfaces is necessary for determina- tion of the instantaneous contact ellipse for elastic surfaces. P1: JXT CB672-08 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:13 8.4 Direct Relations Between Principal Curvatures of Mating Surfaces 225 Case 2 The derivations are similar to those discussed in Case 1. We consider the following system of three linear equations: a i 1 v (1) s + a i 2 v (1) q = a i 3 (i = 1, 2, 3). (8.4.47) The first two equations of system (8.4.47) have been represented as the third and fourth equations in the system of linear equations (8.4.15). The third equation in the system (8.4.47) is the differentiated equation of meshing (8.2.7) (i = 2) that we express in terms of v (1) r and ˙ n (1) r . Here, a 11 = b 33 , a 12 = a 21 = b 34 , a 22 = b 44 a 13 = a 31 =−κ s v (12) s − ω (12) · (n × e s ) a 23 = a 32 =−κ q v (12) q − ω (12) · (n × e q ) a 33 =−n ·  ω (1) × v (2) tr  −  ω (2) × v (1) tr  (8.4.48) +  ω (1)  2 m  21  n ×k 2  ·  r (1) − R  − n ·  ω (12) × v (12)  + κ s  v (12) s  2 + κ q  v (12) q  2 . The rank of the system matrix and the augmented matrix is 1, as explained for case 1. The solution for κ f , κ h , and σ is as follows: tan 2σ = 2a 13 a 23 a 2 23 − a 2 13 + (κ s − κ q )a 33 (8.4.49) κ f − κ h = 2a 13 a 23 a 33 sin 2σ = a 2 23 − a 2 13 + (κ s − κ q )a 33 a 33 cos 2σ (8.4.50) κ f + κ h = (κ s + κ q ) − a 2 13 + a 2 23 a 33 . (8.4.51) Case 3 Surfaces  1 and  2 are in point contact at every instant. The velocity of the point of contact in its motion over the surface has a definite direction; equation system (8.4.47) must possess a unique solution; and the rank of the system matrix is 2. This condition yields that       a 11 a 12 a 13 a 12 a 22 a 23 a 13 a 23 a 33       = F  κ f ,κ h ,κ s ,κ q ,σ,m  21  = 0. (8.4.52) There is only one relation between the principal curvatures and directions for the contacting surfaces. Considering that the principal curvatures are given for one surface, say  1 , we can synthesize an infinitely large number of matching surfaces  2 that will satisfy the same value of m  12 and other motion parameters. More details are given in Litvin & Zhang [1991]. P1: JXT CB672-08 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:13 226 Mating Surfaces: Curvature Relations, Contact Ellipse 8.5 DIRECT RELATIONS BETWEEN NORMAL CURVATURES OF MATING SURFACES We consider again two cases when the interacting surfaces  1 and  2 are in line contact, or in point contact. The plane in which unit vectors of principal directions are shown in Fig. 8.4.1 is the tangent plane to  1 and  2 , and P is the point of tangency of these surfaces. Point P belongs to the instantaneous characteristic (instantaneous line of tangency) in the case of line contact and is the single point of tangency in the case of point contact. We consider three trihedrons: S c (e t , e m , e n ), S a (e f , e h , e n ), and S b (e s , e q , e n ), where e n ≡ n is the surface unit normal; e f and e h are the unit vectors of principal directions on  1 ; e s and e q are the unit vectors of principal directions on  2 ; and e t and e m are two mutually perpendicular directions that are chosen in the tangent plane. Angles q 1 , q 2 , and σ = q 1 − q 2 designate the angles that are formed between the above- mentioned respective unit vectors. Our goal is to determine the relations between the normal curvatures κ (i ) t , κ (i ) m (i = 1, 2) along e t and e m for surfaces  1 and  2 . Our approach to the solution of this problem is based on two steps of decomposition of motions: the first one is per- formed along the principal directions, and the second one is in the directions of e t and e m . The derivations are based on application of Eqs. (8.2.2), (8.2.4), and (8.2.7). For the purpose of simplification, we designate v (i ) r = v (i ) , ˙ n (i ) r = ˙ n (i ) , v (12) = v, and ω (12) = ω, and we represent Eqs. (8.2.2) and (8.2.4) as follows: v (1) − v (2) =−v, ˙ n (1) − ˙ n (2) =−(ω × n). (8.5.1) We may represent vectors v (i ) and ˙ n (i ) (i = 1, 2) in coordinate systems S c , S a , and S b as follows: a (1) = a (1) t e t + a (1) m e m = a (1) f e f + a (1) h e h = a (1) s e s + a (1) q e q  a (1) = v (1) , or a (1) = ˙ n (1)  (8.5.2) b (2) = b (2) t e t + b (2) m e m = b (2) f e f + b (2) h e h = b (2) s e s + b (2) q e q  b (2) = v (2) , or b (2) = ˙ n (2)  . (8.5.3) In addition to Eqs. (8.5.1), we also consider the differentiated equation of meshing (8.2.7). The following is an application of these equations for the following three cases. Case 1 Surfaces  1 and  2 are in line contact, and point P is the point of the instantaneous line of contact. Given are the normal curvatures κ (1) t and κ (1) m of  1 at point P , and angle q 1 . Our goal is to derive the equations for determination of normal curvatures κ (2) t , κ (2) m , and angle q 2 (Fig. 8.4.1). It is shown below that the solution to this problem requires the derivation of three linear equations in unknowns v (2) t and v (2) m . This system is represented as    c 11 c 12 c 21 c 22 c 31 c 32     v (2) t v (2) m  =    d 1 d 2 d 3    . (8.5.4) P1: JXT CB672-08 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:13 8.5 Direct Relations Between Normal Curvatures of Mating Surfaces 227 We may derive this system, using equation system (8.4.42) in unknowns v (2) f and v (2) h . It is also shown that the coefficients c kl (k = 1, 2, 3; l = 1, 2) and d k (k = 1, 2, 3) are represented as follows: c 11 = κ (2) t − κ (1) t (8.5.5) c 12 = c 21 = t (2) − t (1) (8.5.6) c 22 = κ (2) m − κ (1) m (8.5.7) c 31 = d 1 =−t (1) v (12) m − κ (1) t v (12) t −  ω (12) · e m  (8.5.8) c 32 = d 2 =−t (1) v (12) t − κ (1) m v (12) m +  ω (12) · e t  (8.5.9) d 3 =−κ (1) t (v (12) t ) 2 − κ (1) m  v (12) m  2 − 2t (1) v (12) t v (12) m +  n ×ω (12)  · v (12) − n ·  ω (1) × v (2) tr  −  ω (2) × v (1) tr  +  ω (1)  2 m  21  n ×k 2  ·  r (1) − R  . (8.5.10) The designation t (1) indicates the surface torsion of  1 for the displacement along e t and is represented as (see Section 7.9) t (1) = 0.5  κ (1) m − κ (1) t  · tan 2q 1 . (8.5.11) The following is the explanation of the derivation of Eqs. (8.5.5) to (8.5.11). Derivation of First Two Equations of System (8.5.4) The derivation is based on the following procedure: Step 1: Consider the first two equations of system (8.4.42) that have been represented as  t 11 t 12 t 12 t 22  v (2) f v (2) h  =  t 13 t 23  . (8.5.12) Here [see Eqs. (8.4.42)], t 11 = b 11 , t 12 = b 12 , t 22 = b 22 , t 13 = b 15 , and t 23 = b 25 . The b ml coefficients (m = 1, 2; l = 1, 2, 5) have been represented by Eqs. (8.4.24), (8.4.25), (8.4.26), (8.4.31), and (8.4.32). Step 2: The coordinate transformation in the 2D-space from S c (e t , e m )toS a (e f , e h ) (Fig. 8.4.1) is based on the matrix equation  v (2) f v (2) t  = L ac  v (2) t v (2) m  (8.5.13) where L ac =  cos q 1 −sin q 1 sin q 1 cos q 1  . (8.5.14) Step 3: Using Eqs. (8.5.12), (8.4.43), (8.5.13), and (8.5.14), we obtain after trans- formations L ca  b 11 b 12 b 12 b 22  L ac   v (2) t v (2) m   = L ca  b 15 b 25  (8.5.15) P1: JXT CB672-08 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:13 228 Mating Surfaces: Curvature Relations, Contact Ellipse where L ca = L T ac . Step 4: We now use the following designations: L ca  b 11 b 12 b 12 b 22  L ac =  c 11 c 12 c 12 c 22  (8.5.16) L ca  b 15 b 25  =  d 1 d 2  . (8.5.17) Step 5: Using Eqs. (8.5.16) and (8.5.17) and Euler’s equations that relate the principal and normal curvatures (see Section 7.6), we obtain the above-mentioned equations for c 11 , c 12 , c 22 , d 1 , and d 2 . Derivation of Third Equation of System (8.5.4) We use for this purpose the third equation of system (8.4.42) that is represented as t 31 v (2) f + t 32 v (2) h = b 15 v (2) f + b 25 v (2) h = t 33 (8.5.18) [see Eqs. (8.4.43) for t 31 and t 32 ]. The transformation of Eq. (8.5.18) is based on the following procedure: Step 1: Using Eqs. (8.5.18) and (8.5.13), we obtain [b 15 b 25 ] L ac  v (2) t v (2) m  = t 33 . (8.5.19) Step 2: The matrix product [b 15 b 25 ] L ac can be transformed as follows: [b 15 b 25 ] L ac = [b 15 b 25 ] L T ca =  L ca  b 15 b 25  T =  cos q 1 sin q 1 −sin q 1 cos q 1  b 15 b 25  T . (8.5.20) Step 3: Matrix product (8.5.20) results in a row matrix whose elements we designate as c 31 , c 32 . Thus,  cos q 1 sin q 1 −sin q 1 cos q 1  b 15 b 25  T = [c 31 c 32 ]. (8.5.21) Step 4: Equations (8.5.21) and (8.5.19) enable us to represent Eq. (8.5.18) as [c 31 c 32 ]  v (2) t v (2) m  = d 3 (8.5.22) where d 3 ≡ t 33 . Step 5: Using Eqs. (8.4.31) and (8.4.32) for b 15 and b 25 , respectively, and the Euler equations that relate the principal and normal curvatures, we obtain Eqs. (8.5.8) and (8.5.9) for c 31 and c 32 . P1: JXT CB672-08 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:13 8.5 Direct Relations Between Normal Curvatures of Mating Surfaces 229 Step 6: To derive the expression for d 3 , we have to transform the expressions for κ f , κ h , v (12) f , and v (12) h in the equation for t 33 that has been represented in equation system (8.4.43). We use for this purpose the following equations:   v (12) f v (12) h   = L ac   v (12) t v (12) m   =  cos q 1 −sin q 1 sin q 1 cos q 1    v (12) t v (12) m   (8.5.23) κ f = κ (1) t cos 2 q 1 − κ (1) m sin 2 q 1 cos 2q 1 (8.5.24) κ h = κ (1) m cos 2 q 1 − κ (1) t sin 2 q 1 cos 2q 1 . (8.5.25) Matrix equation (8.5.23) is similar to Eq. (8.5.13). Equations (8.5.24) and (8.5.25) are based on the Euler equations that relate the surface principal and normal curvatures (see Section 7.6). Using the equation for t 33 and Eqs. (8.5.23) to (8.5.25), we obtain the represented equation (8.5.10) for d 3 . Derivation of Direct Relations Between the Normal Curvatures of Mating Surfaces The derivation is based on the investigation of the overdetermined system (8.5.4) of three linear equations in two unknowns. The augmented matrix is C =    c 11 c 12 d 1 c 12 c 22 d 2 d 1 d 2 d 3    . (8.5.26) Matrix C is symmetric and its rank is 1, because the surfaces are in line contact and the displacement of a contact point over the surface is indefinite. Therefore, we have c 11 c 12 = c 12 c 22 = d 1 d 2 ; c 11 d 1 = c 12 d 2 = d 1 d 3 ; c 12 d 1 = c 22 d 2 = d 2 d 3 . (8.5.27) After transformations, we obtain the following relations: κ (2) t = κ (1) t + d 2 1 d 3 (8.5.28) κ (2) m = κ (1) m + d 2 2 d 3 (8.5.29) tan 2q 2 = 1 κ (2) m − κ (2) t  tan 2q 1  κ (1) m − κ (1) t  + 2d 1 d 2 d 3  . (8.5.30) [See expressions (8.5.8), (8.5.9), and (8.5.10) for d 1 , d 2 , and d 3 .] Equations (8.5.28), (8.5.29), and (8.5.30) enable us to determine the normal curvatures κ (2) t , κ (2) m , and q 2 for surface  2 . Case 2 Surfaces  1 and  2 are in line contact, and L is the instantaneous line of contact. Given are κ (2) t , κ (2) m , q 2 , and m  21 for point P of L. Our goal is to determine κ (1) t , κ (1) m , and q 1 . P1: JXT CB672-08 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:13 230 Mating Surfaces: Curvature Relations, Contact Ellipse In this case, we consider initially system (8.4.47) of three linear equations in the unknowns v (1) s and v (1) q . Using an approach that is similar to that discussed in Case 1, we obtain κ (1) t = κ (2) t − l 2 1 l 3 (8.5.31) κ (1) m = κ (2) m − l 2 2 l 3 (8.5.32) tan 2q 1 = 1 κ (1) t − κ (1) m  tan 2q 2  κ (2) t − κ (2) m  + 2l 1 l 2 l 3  . (8.5.33) Here, l 1 =−t (2) v (12) m − κ (2) t v (12) t −  ω (12) · e m  (8.5.34) l 2 =−t (2) v (12) t − κ (2) m v (12) m +  ω (12) · e t  (8.5.35) l 3 = κ (2) t  v (12) t  2 + κ (2) m  v (12) m  2 + 2t (2) v (12) t v (12) m −  n ×ω (12)  · v (12) − n ·  ω (1) × v (2) tr  −  ω (2) × v (1) tr  +  ω (1)  2 m  21 (n ×k 2 ) ·  r (1) − R  (8.5.36) t (2) = 0.5  κ (2) m − κ (2) t  tan 2q 2 . (8.5.37) Equations (8.5.31) to (8.5.33) enable us to determine the normal curvatures κ (1) t , κ (1) m of surface  1 and angle q 1 . Case 3 Surfaces  1 and  2 are in point contact at point P . There is a unique solution of the system of linear equations (8.5.4) for the unknowns v (2) t and v (2) m . The rank of the augmented matrix is 2. The condition that the det(C) = 0 provides the relation F  κ (1) t ,κ (1) m ,κ (2) t ,κ (2) m , q 1 , m  12  = 0. (8.5.38) This means that there is only one constraint when surfaces with an instantaneous point of contact are synthesized. Particular Case Surfaces  1 and  2 are in line contact, but e t is directed along the tangent e ∗ t to the contact line at point P. In this case, we have [see Eqs. (8.5.31) to (8.5.33)] l 1 = 0,κ (1) t = κ (2) t = κ t ,κ (2) m − κ (1) m = l 2 2 l 3 , tan 2q 1 tan 2q 2 = κ t − κ (2) m κ t − κ (1) m (8.5.39) t (2) = t (1) =− κ t v (12) t + ω (12) · e m v (12) m . (8.5.40) The side result of the performed investigation is the equality of the surface torsions in the displacement along the tangent e ∗ t to the contact line. It also becomes possible to P1: JXT CB672-08 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:13 8.6 Diagonalization of Curvature Matrix 231 determine the component v (i ) m = v (i ) r · e ∗ m of the velocity of the contact point along e ∗ m that is perpendicular to e ∗ t (see Section 8.6). However, the component v (i ) t is indefinite. 8.6 DIAGONALIZATION OF CURVATURE MATRIX We recall that matrix A of equation system (8.4.15) is symmetric and is represented by Eq. (8.4.23). Elements of matrix A are expressed in terms of principal curvatures of mating surfaces, and therefore we call it the curvature matrix. Our goal is to prove that the eigenvectors for matrix A are directed along the unit vectors e t and e m (Fig. 8.4.1), where e t is the unit vector of the tangent to the contact line. It is also proven below that the eigenvalues are the extreme values of the relative normal curvature. A side effect of this investigation is that it becomes possible to determine the components of relative velocities v (i ) r (i = 1, 2) that are directed along e m (Fig. 8.4.1). However, the components of v (i ) r directed along the tangent to the contact line cannot be deter- mined, because the direction of v (1) r (or v (2) r ) in the case of line contact of surfaces is indefinite. The initial system of linear equations is Eq. (8.4.15). The diagonalization of matrix A is based on the matrix equation U T AU = W. (8.6.1) Here, U is the matrix of coordinate transformation that is represented by U =      0 0 cos q 1 −sin q 1 0 0 sin q 1 cos q 1 cos q 2 −sin q 2 00 sin q 2 cos q 2 00      . (8.6.2) Then, we obtain that the diagonalized matrix is W =      0000 0 w 22 00 0000 000w 44      . (8.6.3) Here: (i) w 11 = w 33 = κ (2) t − κ (1) t = 0 (8.6.4) because since the normal curvature along the tangent to the contact line is the same for both surfaces. (ii) w 12 = w 21 = t (2) − t (1) = 0 (8.6.5) because the surface torsion in the direction along the tangent to the contact line is the same for both surfaces [see Eq. (8.5.40)]. P1: JXT CB672-08 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:13 232 Mating Surfaces: Curvature Relations, Contact Ellipse (iii) w 33 = w 11 = 0 w 34 = w 43 = t (2) − t (1) = 0. (8.6.6) (iv) In accordance with the results of transformation, we have that w 22 = w 44 = κ (2) m − κ (1) m (8.6.7) where κ (i ) m is the normal curvature along e m (Fig. 8.4.1). It can be proven that the eigenvalues of the curvature matrix represent the extreme values of the relative normal curvature, κ R . This can be done considering the equation for κ R as κ R (q) = κ (2) n (q) − κ (1) n (q) (8.6.8) where κ (2) n = κ s cos 2 q + κ q sin 2 q,κ (1) n = κ f cos 2 (q + σ) + κ h sin 2 (q + σ), (8.6.9) and κ (i ) n designates the surface normal curvature. The varied angle q represents the direction in the tangent plane where the normal curvature is considered. The extreme values of κ R are determined with ∂κ R /∂q = 0 which yields (i) that the directions of extreme values κ R coincide with e ∗ t and e ∗ m , respectively; and (ii) that the extreme values of κ R on these directions are κ R = 0 along e ∗ t , and κ R = κ (2) m − κ (1) m along e ∗ m . Using the diagonalized matrix, we may determine as well equations for determination of components v (i ) m = v (i ) r · e ∗ m (i = 1, 2), where e ∗ m is the unit vector that is perpendicular to the tangent to the characteristic. Vectors e ∗ t and e ∗ m are shown in Fig. 8.4.1 as e t and e m . We mentioned above that the initial system of linear equations is [see Eq. (8.4.15)] AX= B. We may transform equation system (8.4.15) using the transformations X = UY (8.6.10) and Y = U T X. (8.6.11) Here, matrix U describes the coordinate transformation in the tangent plane (see Fig. 8.4.1) and is represented by Eq. (8.6.2). Using new designations, we represent matrix X as follows [see Eq. (8.4.16)]: X =  ˙ s (2) f ˙ s (2) h ˙ s (1) s ˙ s (1) q  T . (8.6.12) Equations (8.4.15) and (8.6.10) yield AUY = B (8.6.13) and U T AUY = U T B. (8.6.14) [...]... (8.7.34) and (8.7.35) that δ A a= A= 1 (1) (2) κ −κ − 4 2 2 g 1 − 2g 1 g 2 cos 2σ (12 ) + g 2 (9. 3.8) (9. 3 .9) where a is the major axis of the contact ellipse Equations (9. 3.7) and (9. 3 .9) yield 2 [(a 11 + a 22 ) + 4A]2 = (a 11 − a 22 )2 + 4a 12 (9. 3 .10 ) Step 3: We may consider now a system of three linear equations in unknowns a 11 , a 12 , and a 22 : (1) (1) (1) (1) v s a 11 + v q a 12 = a 13 v s a 12 ... as κ f , κh , and σ (12 ) , respectively, can be determined from the following equations: κ tan 2σ (12 ) = (1) =κ (2) −κ (9. 3 .14 ) 2n2 − κ sin 2 1 2a 22 = g 2 − (a 11 − a 22 ) g 2 − 2n1 + κ cos 2 1 g1 = 2a 12 2n2 − κ sin 2 1 = sin 2σ (12 ) sin 2σ (12 ) (1) κ (1) κ κ f ≡ κI = κh ≡ κ I I = (1) (1) (9. 3 .15 ) (9. 3 .16 ) + g1 2 (9. 3 .17 ) − g1 2 (9. 3 .18 ) Step 6: The orientation of unit vectors e f and eh for the... (9. 4 .12 ) r f (u 1 , 1 , 1 ) − r f (u 2 , θ2 , φ2 ) = 0 n f (u 1 , 1 , 1 ) − n f (u 2 , θ2 , φ2 ) = 0 Vector equations (9. 4 .11 ) and (9. 4 .12 ) yield five independent scalar equations in six unknowns, u 1 , 1 , 1 , u 2 , θ2 , and φ2 Here, fi (u 1 , 1 , 1 , u 2 , θ2 , φ2 ) = 0, fi ∈ C 1 (i = 1, 2, 3, 4, 5) (9. 4 .13 ) The aim of gearing analysis is to obtain from Eqs (9. 4 .13 ) the functions {u 1 ( 1. .. (2) (1) (12 ) vs = vs + vs , (2) (1) (12 ) vq = vq + vq (9. 3.2) and (i (i v q ) = v s ) tan ηi (i = 1, 2) (9. 3.3) The differentiated equation of meshing (it is represented as Eq (8.2.7) with i = 2) and Eqs (9. 3.2) and (9. 3.3) yield (12 ) tan 1 = −a 31 v q a 33 + (12 ) + a 33 + a 31 v s (12 ) a 32 v q − (12 ) vs tan η2 tan η2 (9. 3.4) (1) vs = a 33 a 13 + a 23 tan 1 (9. 3.5) (1) vq = a 33 tan 1 a 13 + a... Step 1: We use for transformations the relations a 11 = b33 , a 12 = b34 , and a 22 = b44 (see Eqs (8.4.48) and Eqs (8.4.27), (8.4.28), and (8.4. 29) which determine b33 , b34 , and b44 ) Then, we obtain a 11 + a 22 = κ (2) −κ (1) ≡κ a 11 − a 22 = g 2 − g 1 cos 2σ (12 ) (9. 3.7) 2 2 2 (a 11 − a 22 )2 + 4a 12 = g 2 − 2g 1 g 2 cos 2σ (12 ) + g 1 (1) (2) Here, κ = κ f + κh ; κ = κs + κq ; g 1 = κ f − κh ; and. .. (1) e I = e f = cos σ (12 ) es − sin σ (12 ) eq (9. 3 . 19 ) (1) eI I (9. 3.20) = eh = sin σ (12 ) es + cos σ (12 ) eq The orientation of the contact ellipse with respect to e f is determined with angle α (1) (Fig 9. 3.3) that is represented with the equations cos 2α (1) = sin 2α (1) = g 1 − g 2 cos 2σ (12 ) 2 2 g 1 − 2g 1 g 2 cos 2σ (12 ) + g 2 1/ 2 g 2 sin 2σ (12 ) 2 2 g 1 − 2g 1 g 2 cos 2σ (12 ) + g 2 1/ 2 (9. 3. 21) ... a 12 + v q a 22 = a 23 (9. 3 .11 ) a 11 + a 22 = κ Step 4: The solution of equation system (9. 3 .11 ) for the unknowns a 11 , a 12 , and a 22 (1) (1) allows these unknowns to be expressed in terms of a 13 , a 23 , κ , v s , and v q Then, using Eq (9. 3 .10 ), we can get the following equation for κ : κ = 4A2 − n2 + n2 1 2 2A − (n1 cos 2 1 + n2 sin 2 1 ) (9. 3 .12 ) P1: GDZ/SPH CB672- 09 P2: GDZ CB672/Litvin... 1 ( 1 ), u 2 ( 1 ), θ2 ( 1 ), φ2 ( 1 )} ∈ C 1 (9. 4 .14 ) According to the theorem of implicit function system existence (see Korn & Korn [ 19 68] and Litvin [ 19 89] ), we may state that functions (9. 4 .14 ) exist in the neighborhood of a point o o o o P 0 = u o , 1 , 1 , u o , θ2 , φ2 1 2 (9. 4 .15 ) if the following are true: (i) functions [ f 1 , f 2 , f 3 , f 4 , f 5 ] ∈ C 1 ; (ii) Eqs (9. 4 .13 ) are P1:... ω (12 ) , v (12 ) , and δ The to-be-chosen parameters are η2 , m 21 , and 2a The output data are κ f , κh , σ (12 ) , e f , and eh Step 1: Choose η2 and determine 1 from Eq (9. 3.4) (1) (1) Step 2: Determine v s and v q from Eqs (9. 3.5) and (9. 3.6) Step 3: Determine A from the third equation of system (9. 3 .13 ) Step 4: Determine κ from Eq (9. 3 .12 ) Step 5: Determine σ (12 ) , κ f , and κh by using Eqs (9. 3 .14 )... )] = d 1 N2 (9. 2.4) d2 [φ2 ( 1 )] = −2a 2 d 1 (9. 2.5) m 21 (φ) = The predesigned parabolic function may be represented as 1 2 2 φ2 ( 1 ) = −a 1 = − m 21 1 2 (9. 2.6) The derivative m 21 ( 1 ) is used in the local synthesis procedure 9. 3 LOCAL SYNTHESIS The approach of local synthesis was proposed by Litvin [ 19 68] and then applied for synthesis of spiral bevel gears, hypoid gears, and face-worm gears The . κ (2) m − κ (1) m (8.5.7) c 31 = d 1 =−t (1) v (12 ) m − κ (1) t v (12 ) t −  ω (12 ) · e m  (8.5.8) c 32 = d 2 =−t (1) v (12 ) t − κ (1) m v (12 ) m +  ω (12 ) · e t  (8.5 .9) d 3 =−κ (1) t (v (12 ) t ) 2 −. been represented as  t 11 t 12 t 12 t 22  v (2) f v (2) h  =  t 13 t 23  . (8.5 .12 ) Here [see Eqs. (8.4.42)], t 11 = b 11 , t 12 = b 12 , t 22 = b 22 , t 13 = b 15 , and t 23 = b 25 . The b ml coefficients. equations:   v (12 ) f v (12 ) h   = L ac   v (12 ) t v (12 ) m   =  cos q 1 −sin q 1 sin q 1 cos q 1    v (12 ) t v (12 ) m   (8.5.23) κ f = κ (1) t cos 2 q 1 − κ (1) m sin 2 q 1 cos 2q 1 (8.5.24) κ h = κ (1) m cos 2 q 1 −