Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

156
TORSION
or in terms of the pitch p and the number of teeth N on the gear:
CA = pNA CB = pNB
The expressions for the circumference of each gear can be equated and solved for the pitch
p on each gear:
2πRA
2πRB
p = p =
N
N
A
Moreover, since the tooth pitch p must be the same for both gears, it follows that
B
(2)
φB
φA
B
A
x′
R
R
B
A
N
=
N
In sum, the gear ratio between any two gears A and B can be expressed equivalently by
either gear radii, gear diameters, or numbers of gear teeth:
B
A
RB
DB
NB
Gear ratio = = = (d)
R D N
A
Rotation Angle. When gear A turns through an angle φ A as shown in Figure 6.16, the
arclength along the perimeter of gear A is s A = R A φ A . Similarly, the arclength along the
perimeter of gear B is s B =/ R B φ B . Since the teeth on each gear must be the same size, the
arclengths that are turned by the two gears must be equal in magnitude. The two gears,
however, turn in opposite directions. If s A and s B are equated, and rotation in the opposite
direction is accounted for, then the rotation angle φ A is given by
A
A
(1)
FIGURE 6.16 Rotation angles
for gears A and B.
x
RB
RAφA =−RBφB ∴ φA
= − φB
R
Note: The term R B /R A in Equation (e) is simply the gear ratio; therefore,
A
(e)
φ
A
=− (Gear ratio) φ
(f)
B
Rotation Speed. The rotation speed ω is the rotation angle φ turned by the gear in a
unit of time; therefore, the rotation speeds of two interlocked gears are related in the same
manner as described for rotation angles—that is,
ExAmpLE 6.5
ω
A
=− (Gear ratio) ω
(g)
B
42 teeth
315 N.m
(2)
y C
x′
D
(1)
x
A
54 teeth
B
600 mm
850 mm
Two solid steel [G = 80 GPa] shafts are connected by the gears
shown. Shaft (1) has a diameter of 35 mm, and shaft (2) has a
diameter of 30 mm. Assume that the bearings shown allow free
rotation of the shafts. If a 315 N · m torque is applied at gear D,
determine
(a) the maximum shear stress magnitudes in each shaft.
(b) the angles of twist φ 1 and φ 2 .
(c) the rotation angles φ B and φ C of gears B and C, respectively.
(d) the rotation angle of gear D.
Plan the Solution
The internal torque in shaft (2) can easily be determined from a freebody
diagram of gear D; however, the internal torque in shaft (1)