SME
Machine Design · Problem type
Lewis bending stress in a gear tooth
Use this for “A spur pinion (m = 3 mm, face width 30 mm, Y = 0.32) carries Wₜ = 1061 N at V = 4.71 m/s. Find Kᵥ and the bending stress.”
This is an example. Type your own values to start a new problem.
Kᵥ
=
6.1
+
V (m/s)
6.1
σ (MPa)
=
×
Wₜ (N)
F (mm)
×
m (mm)
×
σ₀
Static bending stress
calculated · MPa
Formulas
σ₀
=
Wₜ
F
×
m
×
Y
36.8403
=
1,061
30 × 3 × 0.32
Kᵥ
=
6.1 +
V
6.1
1.7721
=
6.1 + 4.71
6.1
σ
=
Kᵥ
×
σ₀
65.2858 = 1.7721 × 36.8403
Assumptions
•
Lewis: each tooth is a cantilever loaded at its tip by Wₜ; Y comes from a table.
•
Kᵥ = (6.1 + V) ÷ 6.1 is the velocity factor for cut or milled teeth, V in m/s.
•
Wₜ in N over F and m in mm gives the stress in MPa.
Step-by-step
WE KNOW
Wₜ = 1,061 N, V = 4.71 m/s, F = 30 mm, m = 3 mm, Y = 0.32
FIND
static bending stress (σ₀), velocity factor (Kᵥ), bending stress (σ)
Step 1 · Find static bending stress (σ₀)
Use σ₀ = Wₜ ÷ (F × m × Y)
The
tooth
bends
like
a
cantilever;
Y
gathers
its
shape,
so
the
load
over
FmY
is
the
root
stress.
σ₀
=
1,061
30 × 3 × 0.32
σ₀
=
1,061
90 × 0.32
σ₀
=
1,061
28.8
σ₀
=
36.84028
MPa
Step 2 · Find velocity factor (Kᵥ)
Use Kᵥ = (6.1 + V) ÷ 6.1
Faster teeth strike harder: the velocity factor grows with the pitch-line speed.
Kᵥ
=
6.1 + 4.71
6.1
Kᵥ
=
10.81
6.1
Kᵥ = 1.77213
Step 3 · Find bending stress (σ)
Use σ = Kᵥ × σ₀
The velocity factor raises the static stress for the dynamic load.
σ
=
1.77213
×
36.84028
σ
=
65.2858
MPa
Check
Put the numbers back in. Both sides should match.
36.84028
=
1,061
30 × 3 × 0.32
✓
1.77213
=
6.1 + 4.71
6.1
✓
65.2858 = 1.77213 × 36.84028 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Gears and fasteners
Main lesson
Gear-train value of a compound train
Use this for “Gears of 20, 60, 18 and 54 teeth form a compound train, with 60 and 18 on one shaft. The input turns at 1800 rpm. Find e and the output speed.”
Bolt preload and tightening torque
Use this for “An M12 bolt (Aₜ = 84.3 mm², proof strength 600 MPa) is tightened for reuse with K = 0.2. Find the preload and the wrench torque.”