Library / Foundations: Physics / FPH-19
Foundations: Physics
Simple harmonic motion: peak speed and acceleration
Maximum speed of an oscillator from amplitude and frequency.
- Ref
- FPH-19
- Rev
- v1
- Field
- Foundations: Physics
- Result
- vmax [m/s]
Displacement x is from equilibrium. A is amplitude and φ sets the initial phase. Peak speed is ωA, peak acceleration ω²A and period 2π/ω. Angles in radians.
An oscillator moves fastest through the centre and accelerates hardest at the ends.
Calculator
m/s
mm
Hz
- Peak acceleration ω²A493.5 m/s²
- Peak acceleration in g50.32 g
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
Peak speed (m/s) current value 1.571 m/s
fFrequencyKey driver
0.00314157
AAmplitude
0.0031431.4
- Peak acceleration grows with f²: a small vibration at high frequency can produce tens of g.
- The speed is zero at the ends and maximum at the centre, the reverse of the acceleration.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| f Frequency | 0.1 to 5,000 Hz | 0.003142 | 157.1 | 157.1 | +10% | +10 % |
| A Amplitude | 0.01 to 100 mm | 0.003142 | 31.42 | 31.41 | +10% | +10 % |
Sweep of Frequency
| Frequency (Hz) | Peak speed (m/s) |
|---|---|
| 0.1 | 0.003142 |
| 0.3867 | 0.01215 |
| 1.495 | 0.04698 |
| 5.782 | 0.1817 |
| 22.36 | 0.7025 |
| 86.47 | 2.716 |
| 334.4 | 10.5 |
| 1,293 | 40.62 |
| 5,000 | 157.1 |
Sweep one variable
Peak speed (m/s)Hover, touch, or use the arrow keys on the chart
Variables and how each one contributes
vmax
Peak speed
What the equation gives you. Everything else feeds it.
A
Amplitude
Maximum displacement from equilibrium. Directly proportional.
f
Frequency
Oscillation frequency. Directly proportional.
Worked example with the values above
- Angular frequency: ω = 2πf = 314.2 rad/s.
- Amplitude: 5 mm = 0.005 m.
- vmax = ωA = 1.571 m/s.
Assumptions and limits
- Simple harmonic motion (restoring force proportional to displacement).
- Angles in radians.
Common mistakes
- Using f instead of ω = 2πf.
- Confusing amplitude with peak-to-peak.
Statistics and uncertainty
Acceleration depends on f², so a 2% error in frequency gives a 4% error in the g-load.
Where it is used
- Vibration levels on equipment.
- Design of mounts and fatigue checks.
Maths you need
- Angular frequency: ω = 2πf
- Sinusoids: cos(ωt)