Library / Structural dynamics and vibration testing / SDY-06
Structural dynamics and vibration testing
Critically damped response to an impulse
Displacement of a critically damped mass that is struck by an impulse J at t = 0 while at rest.
- Ref
- SDY-06
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- x [mm]
An impulse J gives a stationary mass m an initial velocity J/m. With critical damping (ζ = 1) the displacement is (J/m) t e−ωₙt: it rises, peaks at t = 1/ωₙ at the value J/(emωₙ), and then decays without any oscillation. Output is in mm.
Displacement of a critically damped mass that is struck by an impulse J at t = 0 while at rest. It rises, peaks at t = 1/ωₙ and then decays without oscillating.
Calculator
- Time of the peak, 1/ωₙ0.1 s
- Peak displacement, J/(e m ωₙ)18.39 mm
- Initial velocity, J/m0.5 m/s
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.
- 0.1 s after a 2 N·s impulse the mass has moved 18.39 mm; its peak travel would be 18.39 mm at 0.1 s.
- The peak scales with J/(mωₙ): a stiffer or heavier system moves much less for the same impulse.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| J Impulse | 0.1 to 20 N·s | 0.9197 | 183.9 | 183 | +10% | +10 % |
| m Mass | 0.5 to 50 kg | 1.472 | 147.2 | 145.7 | +10% | −9.09 % |
| ωn Undamped angular frequency | 1 to 100 rad/s | 0.00227 | 45.24 | 45.24 | +10% | −9.52 % |
| t Time after the impulse | 0 to 1 s | 0 | 18.39 | 18.39 | +10% | −0.468 % |
Sweep of Time after the impulse
| Time after the impulse (s) | Displacement (mm) |
|---|---|
| 0 | 0 |
| 0.125 | 17.91 |
| 0.25 | 10.26 |
| 0.375 | 4.41 |
| 0.5 | 1.684 |
| 0.625 | 0.6033 |
| 0.75 | 0.2074 |
| 0.875 | 0.06933 |
| 1 | 0.0227 |
Sweep one variable
Variables and how each one contributes
Displacement
What the equation gives you. Everything else feeds it.
Impulse
The impulse, in N·s: the force integrated over the very short time that it acts. Scales the response in direct proportion. Raising it by 10 % (to 2.2) moves the result by +10 %, from 18.39 to 20.23 mm.
Mass
The mass, in kg. More mass gives a smaller response for the same impulse. Raising it by 10 % (to 4.4) moves the result by −9.09 %, from 18.39 to 16.72 mm.
Undamped 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 −9.52 %, from 18.39 to 16.64 mm.
Time after the impulse
The time after the load is applied, in s. Moves the time at which the response is read. Raising it by 0.05 (to 0.15) moves the result by −9.02 %, from 18.39 to 16.73 mm.
Worked example with the values above
- Initial velocity: J/m = 2 / 4 = 0.5 m/s.
- Time factor: t e^(−ωₙt) = 0.1 × e^(−1) = 0.036788 s.
- x = 0.5 × 0.036788 = 18.394 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