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

respectively. The bearings shown allow free rotation of the shafts.
Calculate
(a) the magnitude of the maximum shear stress in each shaft.
(b) the rotation angle of gear E with respect to pulley C.
(c) the magnitude of the torque and the rotation speed required
for motor A.
T B
B
(1)
L 1
FIGURE p6.36/37
C
D A
A
(2)
D C
L 2
D
(3)
L 3
T E
E
x
(b) the minimum permissible diameter for steel shaft (3).
(c) the rotation angle of gear E with respect to pulley C if the
shafts have the minimum permissible diameters as determined
in (a) and (b).
(d) the magnitude of the torque and the rotation speed (in Hz)
required for motor A.
p6.38 The motor shown in Figure P6.38 supplies 15 kW at
1,700 rpm at A. Shafts (1) and (2) are each solid 30 mm diameter
shafts. Shaft (1) is made of an aluminum alloy [G = 26 GPa], and
shaft (2) is made of bronze [G = 45 GPa]. The shaft lengths are
L 1 = 3.4 m and L 2 = 2.7 m, respectively. Gear B has 54 teeth, and
gear C has 96 teeth. The bearings shown permit free rotation of the
shafts. Determine
(a) the maximum shear stress produced in shafts (1) and (2).
(b) the rotation angle of gear D with respect to flange A.
p6.37 The motor at A is required to provide 50 kW of power to
line shaft BCDE shown in Figure P6.36/37, turning gears B and E
at 8 Hz. Gear B removes 65% of the power from the line shaft, and
gear E removes 35%. Shafts (1) and (2) are solid aluminum alloy
[G = 26 GPa] shafts having an allowable shear stress of 45 MPa.
Shaft (3) is a solid steel [G = 80 MPa] shaft that has an allowable
shear stress of 60 MPa. The shaft lengths are L 1 = 2.1 m, L 2 = 1.2 m,
and L 3 = 1.8 m. The diameters of pulleys A and C are D A = 70 mm
and D C = 300 mm, respectively. The bearings shown allow free
rotation of the shafts. Calculate
(a) the minimum permissible diameter for aluminum shafts (1)
and (2).
A
(1)
N C
C
FIGURE p6.38
L 1
N B
B
(2)
D
x
T D
L 2
x'
6.9 Statically Indeterminate Torsion members
In many simple mechanical and structural systems subjected to torsional loading, it is
possible to determine the reactions at supports and the internal torques in the individual
members by drawing free-body diagrams and solving equilibrium equations. Such torsional
systems are classified as statically determinate.
For many mechanical and structural systems, however, the equations of equilibrium
alone are not sufficient for the determination of internal torques in the members and reactions
at supports. In other words, there are not enough equilibrium equations to solve for all
of the unknowns in the system. These structures and systems are termed statically indeterminate.
We can analyze structures of this type by supplementing the equilibrium equations
with additional equations involving the geometry of the deformations in the members of
the structure or system. The general solution process can be organized into a five-step procedure
analogous to the procedure developed for statically indeterminate axial structures in
Section 5.5:
Step 1 — Equilibrium Equations: Equations expressed in terms of the unknown internal
torques are derived for the system on the basis of equilibrium considerations.
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