Library / Structural dynamics and vibration testing / SDY-09
Structural dynamics and vibration testing
Residual amplitude after a rectangular force pulse
Peak amplitude of the free vibration left behind when a constant force F acts for a duration T on an undamped oscillator at rest and is then removed.
- Ref
- SDY-09
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- X [mm]
A constant force F that acts for a time T and is then removed leaves an undamped oscillator vibrating with amplitude (2F/k)|sin(ωₙT/2)|. The amplitude is zero when the pulse lasts a whole number of periods, and reaches 2F/k when it lasts an odd number of half-periods. Output is in mm.
Peak amplitude of the free vibration left behind when a constant force F acts for a duration T on an undamped oscillator at rest and is then removed.
Calculator
- Largest possible residual amplitude, 2|F|/k100 mm
- Period of the oscillator, 2π/ωₙ0.6283 s
- Pulse length as a fraction of the period0.3183
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.
- A 0.2 s pulse of 10 N leaves 84.15 mm of residual vibration, 84.1 % of the worst case.
- Choosing a pulse length equal to a whole number of periods, 0.628 s here, would leave no residual vibration at all.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Stiffness | 10 to 2,000 N/m | 8.415 | 1,683 | 1,675 | +10% | −9.09 % |
| F Pulse force | −50 to 50 N | 0 | 420.7 | 420.7 | +2 N | +20 % |
| T Pulse duration | 0.01 to 2 s | 4.998 | 99.98 | 94.98 | +10% | +5.91 % |
| ωn Natural angular frequency | 1 to 100 rad/s | 7.153 | 100 | 92.84 | +10% | +5.91 % |
Sweep of Pulse duration
| Pulse duration (s) | Residual amplitude (mm) |
|---|---|
| 0.01 | 4.998 |
| 0.2587 | 96.19 |
| 0.5075 | 56.8 |
| 0.7562 | 59.69 |
| 1.005 | 95.15 |
| 1.254 | 1.443 |
| 1.502 | 94.23 |
| 1.751 | 61.98 |
| 2 | 54.4 |
Sweep one variable
Variables and how each one contributes
Residual amplitude
What the equation gives you. Everything else feeds it.
Pulse force
The constant force, in N. Negative values reverse the motion. Scales the response in direct proportion. Raising it by 10 % (to 11) moves the result by +10 %, from 84.15 to 92.56 mm.
Stiffness
The stiffness, in N/m. The static deflection under the force is F/k. A stiffer system moves less. Raising it by 10 % (to 220) moves the result by −9.09 %, from 84.15 to 76.5 mm.
Natural angular frequency
The undamped natural angular frequency, in rad/s. Sets how quickly the motion develops and decays. Raising it by 10 % (to 11) moves the result by +5.91 %, from 84.15 to 89.12 mm.
Pulse duration
The duration of the force pulse, in s. Changes how much of a vibration cycle the pulse covers. Raising it by 0.02 (to 0.22) moves the result by +5.91 %, from 84.15 to 89.12 mm.
Worked example with the values above
- Phase over the pulse: ωₙT/2 = 10 × 0.2 / 2 = 1 rad.
- Sine: |sin(1)| = 0.84147.
- X = (2 × 10 / 200) × 0.84147 = 84.147 mm.
Assumptions and limits
- A single-degree-of-freedom linear oscillator that starts at rest, with an ideal impulse or step.
- The force acts on a mass-spring system with no other loads, and the damping is viscous.
Common mistakes
- Treating a real, finite-duration impulse as ideal when its length is not short compared with the period.
- Mixing up the natural frequency in Hz with the angular frequency in rad/s.
- Forgetting that a suddenly applied force doubles the static deflection in an undamped system.
Statistics and uncertainty
The response scales directly with the load and the inverse of the stiffness or mass, so errors in those carry through proportionally. Frequency errors matter most when the time or pulse length is near a multiple of the period, where the response changes quickly.
Where it is used
- Predicting peak response of structures and instruments to impacts, shocks and suddenly applied loads.
- Understanding shock response spectra and the effect of pulse length on residual vibration.
Maths you need
- Natural frequency: ω = √(k/m) for a mass on a spring
- Impulse: force integrated over time
- Damping ratio: damping as a fraction of critical