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

will be dictated by the ratio of gear sizes. Once you have determined the internal torques in
both shafts, calculate the angles of twist in each shaft, paying particular attention to the signs
of the twist angles. The twist angle in shaft (1) will dictate how much gear B rotates, which
in turn will dictate the rotation angle of gear C. The rotation angle of gear D will depend upon
the rotation angle of gear C and the angle of twist in shaft (2).
SOLUTION
Equilibrium
Consider a free-body diagram that cuts through shaft (2) and includes gear D. A
positive internal torque will be assumed in shaft (2). From this free-body diagram,
a moment equilibrium equation about the x′ axis can be written to determine the
internal torque T 2 in shaft (2):
T 2
315 N.m
Σ Mx′ = 315 Nm ⋅ − T2 = 0 ∴ T2
= 315 Nm ⋅
(a)
Next, consider a free-body diagram that cuts through shaft (2) and includes
gear C. Once again, a positive internal torque will be assumed in shaft (2). The teeth
of gear B exert a force F on the teeth of gear C. If the radius of gear C is denoted by
R C , a moment equilibrium equation about the x′ axis can be written as
T
Σ Mx′
= T2
− F × RC
= 0 ∴ F =
R
A free-body diagram of gear B that cuts through shaft (1) is shown. A positive internal
torque T 1 is assumed to act in shaft (1). If the teeth of gear B exert a force F on the
teeth of gear C, then equilibrium requires that the teeth of gear C exert a force of equal
magnitude in the opposite direction on the teeth of gear B. With the radius of gear B
denoted by R B , a moment equilibrium equation about the x axis can be written as
2
C
(b)
R C
T 1
F
C
D
NC
= 42 teeth
(2)
F
T 2
R B
x′
Σ M = −T − F × R = 0 ∴ T = − F × R (c)
x 1 B 1
B
The internal torque in shaft (2) is given by Equation (a). The internal torque in shaft
(1) can be determined by substituting Equation (b) into Equation (c):
T2
T =− F × R = − = −
R R T R
1
B
B 2
R
C
B
C
(1)
NB
= 54 teeth
B
x
The gear radii R B and R C are not known. However, the ratio R B /R C is simply the gear ratio
between gears B and C. Since the teeth on both gears must be the same size in order for
the gears to mesh properly, the ratio of teeth on each gear is equivalent to the ratio of the
gear radii. Consequently, the torque in shaft (1) can be expressed in terms of N B and N C ,
the number of teeth on gears B and C, respectively:
T
T R B
T N B
54 teeth
=− =− =−(315 Nm) ⋅ = −405 Nm ⋅
R N
42 teeth
1 2 2
C
C
Shear Stresses
The maximum shear stress magnitude in each shaft will be calculated from the elastic torsion
formula. The polar moments of inertia for each shaft will be required for this calculation.
Shaft (1) is a solid 35 mm diameter shaft that has a polar moment of inertia
π
4 4
J1
= (35 mm) = 147,324 mm
32
Shaft (2) is a solid 30 mm diameter shaft that has a polar moment of inertia
π
4 4
J2
= (30 mm) = 79,552 mm
32
157