Library / Mechanical / MEC-29

Mechanical

Damping from a ring-down

Damping ratio from the decay of successive vibration peaks.

Ref: MEC-29
Rev: v1
Field: Mechanical
Result: ζ

δ = \frac{\ln (\frac{x_{i}}{x_{i+n}})}{n}, ζ = \frac{δ}{\sqrt{4 π^{2} + δ^{2}}}

Measure positive peak displacements from equilibrium, n full cycles apart. Applies to a free underdamped response dominated by one viscously damped mode.

The faster the peaks shrink in a ring-down, the more damping the system has.

Calculator

Solve for

ζDamping ratioComputed

x_iStarting peak

0.1100

x_i+nLater peak

0.0150

nCycles between peaks

120

Log decrement δ0.3054
Peak ratio per cycle1.357

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.

Damping ratio () current value 0.04855

x_i+nLater peakKey driver

−0.08510.344

x_iStarting peak

−0.1920.168

nCycles between peaks

0.007290.144

For light damping ζ ≈ δ/(2π), so a 5% damping ratio shrinks each peak by about 27%.
Averaging over several cycles reduces the error from reading single peaks.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
x_i+n Later peak	0.01 to 50 mm	−0.08507	0.3441	0.4292	+10%	−10.4 %
x_i Starting peak	0.1 to 100 mm	−0.1921	0.1683	0.3604	+10%	+10.4 %
n Cycles between peaks	1 to 20	0.007291	0.1443	0.137	+10%	−9.07 %

Sweep of Later peak

Later peak (mm)	Damping ratio
0.01	0.3441
0.029	0.2961
0.08409	0.2457
0.2438	0.1933
0.7071	0.1392
2.05	0.08377
5.946	0.02757
17.24	−0.02889
50	−0.08507

Sweep one variable

Vary

Damping ratio ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Damping ratio

Result ·

What the equation gives you. Everything else feeds it.

x_i

Starting peak

mm · typical 0.1 to 100

Displacement of the first peak. Larger starting peaks mean more decay.

x_i+n

Later peak

mm · typical 0.01 to 50

Displacement n cycles later. A smaller later peak means more damping.

Cycles between peaks

· typical 1 to 20

Number of full cycles. More cycles for the same ratio mean less damping per cycle.

Worked example with the values above

δ = ln(10/4) / 3 = 0.3054.
√(4π² + δ²) = 6.291.
ζ = 0.3054 / 6.291 = 0.04855.

Assumptions and limits

Free underdamped response with viscous damping.
One dominant mode.

Common mistakes

Using the logarithm to base 10.
Measuring peak-to-peak instead of from equilibrium.

Statistics and uncertainty

A 2% error in reading each peak leads to about ±0.03 in δ over one cycle; measure over several cycles to reduce it.

Where it is used

Measuring damping from tap or snap tests.
Checking damper degradation.

Maths you need

Natural logarithm: ln
Free vibration: decaying oscillation

Related formulas

Practise this formula