SME
Mechanical Vibrations
Free vibration
Use this for “A 2 kg block on an 800 N/m spring starts 30 mm from rest at 0.8 m/s. Find ωₙ, fₙ, the period and the amplitude.”
Diagram
X
=
√(
x₀
2
+
(v₀/ωₙ)
2
)
=
0.05
m
φ = atan2(−v₀/ωₙ, x₀) = −0.927 rad
x(0) = X cos φ = 0.03 m = x₀
This is an example. Type your own values to start a new problem.
ωₙ (rad/s)
=
k (N/m)
÷
m (kg)
(
X (mm)
)
2
=
(
x₀ (mm)
)
2
+
(
v₀ (m/s)
÷
ωₙ (rad/s)
)
2
The second line is X = √(x₀² + (v₀ ÷ ωₙ)²), squared.
fₙ
Natural frequency in hertz
calculated · Hz
T
Period
calculated
Formulas
ωₙ
=
√(
k
m
)
20
=
√(
800
2
)
fₙ
=
ωₙ
2π
3.1831
=
20
2π
T
=
1
fₙ
0.3142
=
1
3.1831
X
=
√(
x₀
2
+
(
v₀
ωₙ
)
2
)
0.05
=
√(
0.03
2
+
(
0.8
20
)
2
)
Formulas are worked in m. The step-by-step shows the conversions.
Assumptions
•
No damping and a light spring: mẍ + kx = 0, so x = X cos(ωₙ t + φ).
•
x₀ and v₀ are measured from the rest position at t = 0; either may be negative.
•
X = √(x₀² + (v₀ ÷ ωₙ)²) comes from x₀ = X cos φ and v₀ ÷ ωₙ = −X sin φ.
Step-by-step
WE KNOW
m = 2 kg, k = 800 N/m, x₀ = 30 mm, v₀ = 0.8 m/s
FIND
natural frequency (ωₙ), natural frequency in hertz (fₙ), period (T), amplitude (X)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m).
x₀ = 30 mm = 0.03 m (1 m = 1000 mm)
Step 2 · Find natural frequency (ωₙ)
Use ωₙ = √(k ÷ m)
The
spring
pulls
back
kx
on
the
mass,
so
mẍ
=
−kx
swings
at
the
square
root
of
k
over
m
.
ωₙ
=
√(
800
2
)
ωₙ
=
√
400
ωₙ = 20 rad/s
Step 3 · Find natural frequency in hertz (fₙ)
Use fₙ = ωₙ ÷ 2π
One cycle is 2π radians, so divide the radians each second by 2π.
fₙ
=
20
2π
fₙ = 3.1831 Hz
Step 4 · Find period (T)
Use T = 1 ÷ fₙ
The time for one cycle is one over the cycles each second.
T
=
1
3.1831
T
=
0.3142
s
Step 5 · Find amplitude (X)
Use X = √(x₀² + (v₀ ÷ ωₙ)²)
Square
x₀
=
X
cos
φ
and
v₀
ωₙ
=
−
X
sin
φ
and
add
them:
cos²φ
+
sin²φ
=
1
leaves
X
2
.
X
=
√(
0.03
2
+
(
0.8
20
)
2
)
X
=
√
0.03
2
+
0.04
2
X
=
√
0.0009 + 0.0016
X
=
√
0.0025
X
=
0.05
m
Step 6 · Convert the answers
Change each answer back to the unit you chose.
X = 0.05 m = 50 mm (1 m = 1000 mm)
Check
Put the numbers back in. Both sides should match.
20
=
√(
800
2
)
✓
3.1831
=
20
2π
✓
0.3142
=
1
3.1831
✓
0.05
=
√(
0.03
2
+
(
0.8
20
)
2
)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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