Page 345 - Marks Calculation for Machine Design

P1: Sanjay
January 4, 2005
15:14
Brown.cls
Brown˙C08
MACHINE ASSEMBLY
Cone angle a 327
t 1 t middle
L grip
t 2 L
grip /2

FIGURE 8.2 Pressure distribution as frusta of hollow cones.

Without presenting the details of its development, which includes the integration of a
very complex integral, the stiffness of a frustum of a hollow cone (k frustum ) is given by
Eq. (8.20) as
(π tan α)Ed
k frustum = (8.20)
(2t tan α + D − d)(D + d)
ln
(2t tan α + D + d)(D − d)
where (α) is the cone angle of the frustum, (E) is the modulus of elasticity of the member,
(d) is the nominal diameter of the bolt, or cap screw, and (D) is the diameter of the washer
face. As stated earlier, if the members have different thicknesses, there will be a third
thickness (t middle ), for which the diameter (D) will be the smaller of the two diameters of
the frustum. This special diameter (D*) for the third thickness (t middle ) can be found from
Eq. (8.21) as
D = D + (L grip − 2 t middle ) tan α (8.21)
∗
◦
Current practice sets the cone angle (α) equal to 30 , and the standard washer face
diameter (D) for both the bolt and nut is equal to one and a half times the nominal diameter
of the bolt, that is (D = 1.5d).
Therefore, the frustum stiffness (k frustum ) given in Eq. (8.20) becomes the expression
given in Eq. (8.22)
◦
(π tan 30 ) Ed
k frustum = (8.22)
2 t tan 30 + 0.5 d
◦
ln 5
2 t tan 30 + 2.5 d
◦
where again (d) is the nominal diameter of the bolt, or cap screw, and (t) is the thickness
of the frustum. Also, the special diameter (D*) given in Eq. (8.21) becomes
∗
D = 1.5 d + (L grip − 2 t middle ) tan 30 ◦ (8.23)
Once calculated, the stiffness of each frustum is then used in Eq. (8.18) to determine the
overall stiffness of the members (k members ).
In Example 1, the length of the grip (L grip ) was given, but not the individual thicknesses of
the two members. Example 2 determines the overall stiffnesses of members in an assembly
when there is a third thickness (t middle ) resulting in three hollow cone frusta.