Library / Structural dynamics and vibration testing / SDY-04
Structural dynamics and vibration testing
Energy lost in one sinusoidal damper cycle
Energy a linear viscous damper dissipates in one steady cycle of sinusoidal motion of peak displacement X at angular frequency ω.
- Ref
- SDY-04
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- ΔE [J]
A linear viscous damper dissipates πcωX² in each cycle of steady sinusoidal motion of peak displacement X at angular frequency ω. The energy lost per cycle rises with the square of the amplitude and in direct proportion to the frequency. The same area equals the area of the force–displacement loop.
Energy a linear viscous damper dissipates in one steady cycle of sinusoidal motion of peak displacement X at angular frequency ω. It grows with the square of the amplitude.
Calculator
- Average power dissipated, cω²X²/21.6 W
- Peak damper force, cωX16 N
- Cycles per second, ω/2π3.183 Hz
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.
- The damper dissipates 0.5027 J per cycle, which at 3.18 cycles per second is an average of 1.6 W.
- Doubling the amplitude quadruples the heat the damper must shed.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (J) | Local step | Change in result |
|---|---|---|---|---|---|---|
| X Peak displacement | 0.001 to 0.05 m | 0.005027 | 12.57 | 12.56 | +10% | +21 % |
| ω Forcing angular frequency | 1 to 200 rad/s | 0.02513 | 5.027 | 5.001 | +10% | +10 % |
| c Viscous damping coefficient | 1 to 500 N·s/m | 0.006283 | 3.142 | 3.135 | +10% | +10 % |
Sweep of Peak displacement
| Peak displacement (m) | Energy lost per cycle (J) |
|---|---|
| 0.001 | 0.005027 |
| 0.007125 | 0.2552 |
| 0.01325 | 0.8825 |
| 0.01938 | 1.887 |
| 0.0255 | 3.269 |
| 0.03163 | 5.027 |
| 0.03775 | 7.163 |
| 0.04388 | 9.676 |
| 0.05 | 12.57 |
Sweep one variable
Variables and how each one contributes
Energy lost per cycle
What the equation gives you. Everything else feeds it.
Viscous damping coefficient
The viscous damping coefficient, in N·s/m: the damper force per unit velocity. Raises the energy lost in direct proportion. Raising it by 10 % (to 88) moves the result by +10 %, from 0.5027 to 0.5529 J.
Forcing angular frequency
The forcing angular frequency, in rad/s: the angular frequency of the steady sinusoidal motion. Raises the energy lost per cycle in direct proportion. Raising it by 10 % (to 22) moves the result by +10 %, from 0.5027 to 0.5529 J.
Peak displacement
The peak displacement of the steady sinusoidal motion, in m. Raises the energy lost with the square of the amplitude. Raising it by 10 % (to 0.011) moves the result by +21 %, from 0.5027 to 0.6082 J.
Worked example with the values above
- Square the amplitude: X² = 1×10−4 m².
- Multiply by c and ω: 80 × 20 × 1×10−4 = 0.16.
- ΔE = π × that = 0.50265 J.
Assumptions and limits
- A single, well-separated mode dominates the record and the damping is viscous (proportional to velocity) and light.
- Peaks or half-power points are read from clean data with enough resolution.
Common mistakes
- Mixing peak heights with peak-to-peak values, or taking peaks from opposite sides of the zero line.
- Using a record that contains several modes, so the envelope is not a single exponential.
- Using the half-power width when the frequency resolution is too coarse to resolve it.
- Using an angular frequency in rad/s where Hz is expected, or the reverse.
Statistics and uncertainty
Decrements read from only one pair of peaks carry the reading error of both. Averaging over several cycles, or fitting an exponential to the whole envelope, cuts the scatter. Damping is itself an inexact, amplitude-dependent property, so quote the amplitude at which it was measured.
Where it is used
- Identifying damping from an impact or ring-down test on a structure, instrument or component.
- Setting the damping values used in finite-element and modal models.
Maths you need
- Natural logarithm: the inverse of e to a power
- Damping ratio: damping as a fraction of critical
- Free decay: the envelope of an unforced vibration