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

NC
R C
R B
NB
= 42 teeth
= 54 teeth
C
φ C
φ B
B
To calculate the maximum shear stress magnitudes, the absolute values of T 1 and T 2 will
be used. The maximum shear stress magnitude in the 35 mm diameter shaft (1) is
Tc 1 1
(405 Nm)(35 ⋅ mm/2)(1, 000 mm/m)
τ 1 = =
= 48.1 MPa Ans.
4
J
147,324 mm
1
and the maximum shear stress magnitude in the 30 mm diameter shaft (2) is
Tc 2 2
(315 Nm)(30 ⋅ mm/2)(1,000 mm/m)
τ 2 = =
= 59.4 MPa Ans.
4
J
79,552 mm
2
Angles of Twist
The angles of twist must be calculated with the signed values of T 1 and T 2 . Shaft (1) is 600 mm
long, and its shear modulus is G = 80 GPa = 80,000 MPa. The angle of twist in this shaft is
TL 1 1
( 405 Nm)(600 mm)(1, 000 mm/m)
φ 1 = = − ⋅ =− 0.020618 rad =−0.0206 rad
4 2
JG 1 1 (147,324 mm )(80,000 N/mm )
Ans.
Shaft (2) is 850 mm long; therefore, its angle of twist is
TL 2 2
(315 Nm)(850 ⋅ mm)(1, 000 mm/m)
φ 2 = =
= 0.042087 rad = 0.0421 rad Ans.
4 2
JG (79,522 mm )(80,000 N/mm )
2 2
Rotation Angles of Gears B and C
The rotation of gear B is equal to the angle of twist in shaft (1):
φB = φ1
= − 0.020618 rad = − 0.0206 rad
Ans.
Note: From the sign convention for rotation angles described in Section 6.6 and illustrated
in Figure 6.13, a negative rotation angle for gear B indicates that gear B rotates
clockwise, as shown in the accompanying figure.
The rotation angles of gears B and C are related because the arclengths associated
with the respective rotations must be equal. Why? Because the gear teeth are interlocked.
The gears turn in opposite directions, however. In this instance, gear B turns clockwise,
which causes gear C to rotate in a counterclockwise direction. This change in the direction
of rotation is accounted for in the calculations by a negative sign, so that
R φ
=−R
φ
C C B B
where R B and R C are the radii of gears B and C, respectively. Using this relationship, we
can express the rotation angle of gear C as
RB
φC
=− φ B
RC
However, the ratio R B /R C is simply the gear ratio between gears B and C, and this ratio
can be equivalently expressed in terms of N B and N C , the number of teeth on gears
B and C, respectively. Thus,
NB
φC
=− φ B
NC
Therefore, the rotation angle of gear C is
NB
54 teeth
φC
=− φ B =− ( − 0.020618 rad) = 0.026509 rad = 0.0265 rad Ans.
NC
42 teeth
Rotation Angle of Gear D
The rotation angle of gear D is equal to the rotation angle of gear C plus the twist that
occurs in shaft (2):
φD = φC + φ2
= 0.026509 rad + 0.042087 rad = 0.068596 rad = 0.0686 rad Ans.
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