SME
Kinematics of Mechanisms
Velocity and acceleration analysis
Use this for “A slider-crank has a 50 mm crank and a 200 mm rod. The crank turns at 100 rad/s and is at 40°. Find the rod’s angle, its angular velocity and the slider’s speed.”
Chart
vB(40) ≈ −3.838
This is an example. Type your own values to start a new problem.
sin(
φ
°)
=
r (mm)
L (mm)
×
sin(
θ
°)
vB (m/s)
=
−
r (mm)
×
ω (rad/s)
×
sin(
θ
°
+
φ
°)
÷
cos(
φ
°)
÷
1000
ω₃
Rod angular velocity
calculated · rad/s
Formulas
φ
=
sin⁻¹(
r
×
sin(
θ
)
L
)
9.247
=
sin⁻¹(50
×
sin(40°)
200
)
ω₃
=
r
×
ω
×
cos(
θ
)
L
× cos(
φ
)
19.4
=
50
×
100
×
cos(40°)
200 × cos(9.247°)
vB
=
−
r
×
ω
×
sin(
θ
° +
φ
°)
1,000 × cos(
φ
)
−3.838
=
−50
×
100
×
sin(40° + 9.247°)
1,000 × cos(9.247°)
Assumptions
•
An in-line slider: its line of stroke passes through the crank’s fixed pivot.
•
θ is measured from the line of stroke, counterclockwise; vB is + moving away from the crank.
•
Rigid links and pins without play.
Step-by-step
WE KNOW
θ = 40°, ω = 100 rad/s, r = 50 mm, L = 200 mm
FIND
rod angle (φ), rod angular velocity (ω₃), slider velocity (vB)
Step 1 · Find rod angle (φ)
Use φ = sin⁻¹(r × sin(θ) ÷ L)
The
crank
pin’s
height
above
the
line
of
stroke
is
r
sin
θ
,
and
the
rod
reaches
it:
L
sin
φ
=
r
sin
θ
.
φ
=
sin⁻¹(50
×
sin(40°)
200
)
φ
=
9.247°
Step 2 · Find rod angular velocity (ω₃)
Use ω₃ = r × ω × cos(θ) ÷ (L × cos(φ))
Differentiate
L
sin
φ
=
r
sin
θ
in
time:
Lω₃
cos
φ
=
rω
cos
θ
,
then
divide
by
L
cos
φ
.
ω₃
=
50
×
100
×
cos(40°)
200 × cos(9.247°)
ω₃
=
50
×
100
×
cos(40°)
200 × 0.987
ω₃
=
50
×
100
×
cos(40°)
197.4
ω₃
=
50
×
100
×
0.766
197.4
ω₃
=
5,000
×
0.766
197.4
ω₃
=
3,830
197.4
ω₃
=
19.4
rad/s
Step 3 · Find slider velocity (vB)
Use vB = −r × ω × sin(θ° + φ°) ÷ (1,000 × cos(φ))
Differentiate
x
=
r
cos
θ
+
L
cos
φ
and
put
in
ω₃
;
÷
1000
turns
mm/s
into
m/s.
vB
=
−50
×
100
×
sin(40° + 9.247°)
1,000 × cos(9.247°)
vB
=
−50
×
100
×
sin(49.247°)
1,000 × 0.987
vB
=
−50
×
100
×
sin(49.247°)
987
vB
=
−50
×
100
×
0.7575
987
vB
=
−5,000
×
0.7575
987
vB
=
−
3,787.5
987
vB = −3.838 m/s
Check
Put the numbers back in. Both sides should match.
9.247
=
sin⁻¹(50
×
sin(40°)
200
)
✓
19.4
=
50
×
100
×
cos(40°)
200 × cos(9.247°)
✓
−3.838
=
−50
×
100
×
sin(40° + 9.247°)
1,000 × cos(9.247°)
✓
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Time ratio of a quick-return linkage
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Slider acceleration (the engine formula)
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