Gear Geometry and Applied Theory Episode 1 Part 9 pot

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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

21n ·

k2 × 

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 Sa

(e f , eh), 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

K1





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

K1





v(2)

f

v(2)

h



 −





v(12)

f

v(12)

h





T

K1





v(12)

f

v(12)

h





=





v(12)

f

v(12)

h





T

K1





v(2)

f

v(2)

h



 + κf

v(12)

f

2 + κh

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 Sa (e f , eh),

we obtain

− v(1)

r ·

ω(12) × n

=



n × ω(12)

· e f

n × ω(12)

· eh

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)

· eh

T 



v(2)

f

v(2)

h



 − 

n × ω(12)

· v(12). (8.4.40)

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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

K1 +



n × ω(12)

· e f

n × ω(12)

· eh

T











v(2)

f

v(2)

h





= − n ·

ω(1) × v

(2)

tr 

− 

ω(2) × v

(1)

tr 

+ 

ω(1)2

m

21(n × k2) · (r

(1) − R)

+ 

n × ω(12)

· v(12) − κf

v(12)

f

2 − κh

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 :

ti 1v(2)

f + ti 2v(2)

h = ti 3 (i = 1, 2, 3). (8.4.42)

Here,

t11 ≡ b11, t12 = t21 ≡ b12, t22 ≡ b22

t13 = t31 ≡ b15, t23 ≡ t32 ≡ b25

t33 = − n ·

ω(1) × v

(2)

tr 

− 

ω(2) × v

(1)

tr 

(8.4.43)

+ 

ω(1)2

m

21(n × k2) ·

r

(1) − R

+ 

n × ω(12)

· v(12) − κf

v(12)

f

2 − κh

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 = −2t13t23

t2

23 − t2

13 − (κf − κh)t33

(8.4.44)

κq − κs = −2t13t23

t33 sin 2σ = t2

23 − t2

13 − (κf − κh)t33

t33 cos 2σ (8.4.45)

κq + κs = κf + κh +

t2

13 + t2

23

t33

. (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.

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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:

ai 1v(1)

s + ai 2v(1)

q = ai 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,

a11 = b33, a12 = a21 = b34, a22 = b44

a13 = a31 = −κs v(12)

s − ω(12) · (n × es)

a23 = a32 = −κq v(12)

q − ω(12) · (n × eq )

a33 = −n ·

ω(1) × v

(2)

tr 

− 

ω(2) × v

(1)

tr 

(8.4.48)

+  ω(1)2

m

21

n × k2

·

r

(1) − R



− n ·

ω(12) × v(12)

+ κs

v(12)

s

2

+ κq

v(12)

q

2

.

The rank of the system matrix and the augmented matrix is 1, as explained for case 1.

The solution for κf , κh, and σ is as follows:

tan 2σ = 2a13a23

a 2

23 − a 2

13 + (κs − κq )a33

(8.4.49)

κf − κh = 2a13a23

a33 sin 2σ = a 2

23 − a 2

13 + (κs − κq )a33

a33 cos 2σ (8.4.50)

κf + κh = (κs + κq ) − a 2

13 + a 2

23

a33

. (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













a11 a12 a13

a12 a22 a23

a13 a23 a33













= F 

κf , κh, κs, κq , 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].