Explore the half-power bandwidth (3 dB) method, which measures the damping ratio ζ from the width of a resonance peak in the frequency response function of a single-degree-of-freedom system. Change the natural frequency, damping ratio, mass and excitation force to watch the peak shape, the half-power points f1 and f2, the quality factor Q and the bandwidth Δf update in real time.
Parameters
Natural frequency fn
Hz
Frequency at which the resonance peak appears
Damping ratio ζ
The value the half-power method should recover
Mass m
kg
Equivalent system mass. Sets stiffness k and static deflection
Excitation force amplitude F
N
Amplitude of the sinusoidal force applied during the sweep
The FRF magnitude curve with its resonance peak and the dashed half-power level (peak ÷ √2). The two crossings f₁ and f₂ are marked and the bandwidth Δf between them is shaded. A marker sweeps the frequency axis.
Dimensionless magnitude H (displacement normalised by the static deflection) and quality factor Q. r = f/fn is the frequency ratio, ζ the damping ratio. For light damping the peak height is close to Q.
Damping ratio from the half-power bandwidth. f₁ and f₂ are the half-power points where the FRF magnitude falls to peak ÷ √2 (−3 dB) on each side of resonance, and Δf = f₂−f₁ is the bandwidth.
Half-power frequency ratios r₁ and r₂ (f₁=fn·r₁, f₂=fn·r₂) and the system stiffness k. m is the mass and the static deflection is x_st = F/k.
What is the Half-Power Bandwidth Method?
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I keep hearing "measure damping with the half-power method". The damping ratio is the number that says how fast vibration dies out, right? How do you measure that from a "width"?
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Good question. The damping ratio ζ is a dimensionless number for how quickly vibration settles — the smaller it is, the longer the vibration rings on. The key idea of the half-power method is that the amount of damping is directly tied to "how sharp the resonance peak is". Shake the structure while slowly changing the frequency and plot the response amplitude: you get a hill at the natural frequency fn. With light damping the hill is tall and narrow; with heavy damping it is low and gentle. So measure the "width" of the hill and you read off the damping in reverse.
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The width of the hill, I see. But where exactly do you measure it from and to? The skirt of the hill stretches all the way down to zero response.
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That is the clever part. You do not measure down to the skirt — you measure the width at "the height that is 1/√2 of the peak". Draw a horizontal line at the peak value divided by √2, and the hill crosses that line twice, once on each side. Those two frequencies are called f₁ and f₂, and the gap Δf = f₂−f₁ is the half-power bandwidth. On the canvas above, the dashed line is the half-power level, the red and green dots on it are f₁ and f₂, and the shading is Δf. Raise the damping ratio and you see the hill fatten and Δf widen.
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Why 1/√2? Half should be at the 1/2 height, surely. That number looks oddly specific.
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"Half" here means "half the power", not half the amplitude. Vibration energy and power are proportional to the square of amplitude. So when the amplitude is 1/√2 of the peak, its square — the power — is exactly (1/√2)² = 1/2 of the peak. In decibels that is 10·log10(0.5) ≈ −3 dB, which is why the half-power method is also called the "3 dB method" or "half-power bandwidth method". It is exactly the same idea as the bandwidth of an electrical filter.
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Once I have f₁ and f₂, how do I get the damping ratio out of them?
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This is the beautiful part: ζ = (f₂−f₁)/(f₂+f₁), just a subtraction and an addition. For light damping f₂+f₁ ≈ 2fn, so you can remember it as ζ ≈ Δf/(2fn). With the default values in this tool (fn=50 Hz, ζ=0.030) you get f₁≈48.48 Hz, f₂≈51.48 Hz, Δf≈3.00 Hz. Then Δf/(2fn) = 3.00/100 = 0.030 — exactly the ζ you put in. Pulling an invisible material property, damping, out of visible data, the FRF — that is the value of the half-power method.
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It sounds handy. Can I always use it? If it is that good, I wonder why other methods exist.
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Sadly it is not universal. The half-power method is at its best when damping is light and the modes are cleanly separated. With heavy damping (ζ above about 0.1) the peak becomes flat and the half-power points get fuzzy, and the approximation error grows. Worse is the case where a neighbouring mode's peak and tail overlap: the skirt of one hill rides up onto the other's half-power line, and f₁ and f₂ can no longer be read correctly. Then you switch to more advanced curve-fit methods such as circle fit or rational-polynomial fitting. In practice it is reassuring to cross-check with the time-domain logarithmic-decrement method too.
Frequently Asked Questions
The half-power bandwidth method is a frequency-domain technique that extracts the damping ratio from the width of the resonance peak in the frequency response function (FRF) of a single-degree-of-freedom system. You drive the structure across a frequency sweep, measure the resonance peak amplitude, then read off the two frequencies f1 and f2 on each side where the response has dropped to the peak value divided by the square root of two (−3 dB). The damping ratio follows directly as ζ = (f2−f1)/(f2+f1) ≈ Δf/(2·fn). A sharper peak means a narrower Δf and smaller damping.
Because 1/√2 (≈0.707) is the point where the power is exactly halved. Vibration energy and power are proportional to the square of amplitude, so when the amplitude is 1/√2 of the peak, the power is (1/√2)² = 1/2 of the peak. In decibels this corresponds to 10·log10(0.5) ≈ −3 dB, which is why the names half-power bandwidth, 3 dB bandwidth and half-power point all refer to exactly the same location on the curve.
The half-power bandwidth method works in the frequency domain and the logarithmic-decrement method in the time domain; the two are complementary. The logarithmic-decrement method finds ζ from how fast the amplitude of free vibration decays cycle by cycle, so it only needs a single impact and a recorded response — quick and simple. The half-power method needs an FRF, obtained from a frequency sweep or from an FFT after impulse excitation, but it handles several modes at once and is the standard step in experimental modal analysis. For a sharp, isolated mode both agree; for closely spaced modes the half-power method is preferred, while in noisy field measurements the logarithmic-decrement method is often chosen.
The half-power method is accurate when damping is light and the modes are well separated. When damping is high (ζ above roughly 0.1), the approximations r1 = √(1−2ζ) and r2 = √(1+2ζ) break down and the peak itself becomes flat, making the half-power points hard to locate. When neighbouring modal peaks overlap, the tail of one mode interferes with the half-power level of the other, so f1 and f2 cannot be read correctly. A coarse frequency resolution captures a narrow Δf with only a few spectral lines, which adds large error. In these cases more advanced modal-parameter estimators such as circle-fit or rational-polynomial curve fitting are used.
Real-World Applications
Experimental modal analysis: The half-power method is the most basic way to estimate damping in experimental modal analysis. After exciting a structure with an impact hammer or shaker and measuring the response with accelerometers to obtain the FRF, the method is applied to each resonance peak to extract the trio of natural frequency, damping ratio and mode shape. For bridges, building structures, aircraft wings, vehicle bodies — almost every structure — it is the first tool reached for when characterising dynamic behaviour.
Machine fault diagnosis and condition monitoring: In rotating machinery and plant equipment, loose bolts, cracks and changes in support stiffness alter the shape of resonance peaks. By acquiring the FRF periodically and tracking the damping ratio from the half-power bandwidth, a sudden change in damping or a widening of Δf can be detected as an early sign of structural degradation. Changes in damping are often more sensitive than shifts in natural frequency, making them a valuable parameter for predictive maintenance.
Verifying vibration-damping designs: When damping material, a dynamic absorber or anti-vibration mounts are added, the half-power method confirms whether damping increased as designed. Take the FRF before and after the countermeasure; a lower resonance peak and a wider Δf prove the treatment is working. If the peak barely changes, you know the damper mounting location or tuning frequency needs to be revised.
A common language with acoustics and electrical systems: The half-power (3 dB) bandwidth concept is not limited to mechanical vibration. Speaker and microphone resonances, RLC filter circuits, the quality factor Q of a laser cavity — every field that deals with resonance uses the same definition. The half-power method learned for mechanical systems carries over directly to understanding Q in acoustics and electronics.
Common Misconceptions and Pitfalls
The most common mistake is measuring the width at the 1/2 height of the peak. The "half" in half-power means half the power, which in amplitude terms is the 1/√2 ≈ 0.707 height. If you carelessly read f₁ and f₂ at the 1/2 (= 0.5×) height, the bandwidth comes out too wide and you overestimate the damping ratio. Always check whether the FRF is displayed as amplitude or as power (amplitude squared) or in decibels: for an amplitude display, read at 0.707×peak; for a dB display, read at the points 3 dB below the peak. This is where people first stumble with the half-power method.
Next, measuring a narrow peak without considering the frequency resolution. The resonance peak of a lightly damped structure is extremely sharp, and a half-power bandwidth Δf of only a few Hz is not unusual. If the FFT frequency resolution Δf_FFT is of the same order, the peak is captured by only two or three spectral lines and the error in locating the half-power points becomes enormous. As a rule of thumb, lengthen the measurement time so that at least five to ten spectral lines fall within the half-power bandwidth. A damping ratio obtained at coarse resolution often comes out several times the true value.
Finally, assuming the half-power method can be applied to any structure as is. The half-power method is an approximation built on a sharp, single resonance of a single-degree-of-freedom system. When damping is high (ζ above 0.1) the peak becomes flat and the half-power points blur, and the approximation error of r₁ = √(1−2ζ) and r₂ = √(1+2ζ) can no longer be ignored. For structures whose neighbouring modes sit close together and whose peak tails overlap, the response of one mode interferes with the half-power level of the other, so f₁ and f₂ cannot be read correctly. In such cases use circle fitting (Nyquist-circle fit) or rational-polynomial curve fitting, and treat the half-power method only as a way to get an initial estimate.
How to Use
Enter natural frequency (fnNum) in Hz—typically 5–500 Hz for mechanical systems like vehicle suspensions or building structures.
Set damping ratio ζ (zetaNum) between 0.01 and 1.0; values near 0.05–0.10 represent lightly damped steel frame oscillations.
Input mass (kg) and excitation force (N) to establish the system dynamic response.
Observe the resonance peak and identify the half-power points (−3 dB from peak) where bandwidth Δf = f₂ − f₁ is calculated.
Read Quality factor Q and measured damping ζ directly from output labels to verify energy dissipation characteristics.
Worked Example
A 50 kg aluminum cantilever beam with natural frequency fn = 12 Hz and damping ratio ζ = 0.08 is excited with a 200 N sinusoidal force. The simulator computes: resonance amplification ≈ 6.25, Quality factor Q ≈ 6.25, bandwidth Δf ≈ 1.92 Hz. The half-power frequencies are f₁ ≈ 11.04 Hz and f₂ ≈ 12.96 Hz. Using ζ = 1/(2Q) confirms the measured damping ratio of 0.08, validating the 3 dB bandwidth method for structural health monitoring.
Practical Notes
Underdamped systems (ζ < 0.2) show sharp peaks and narrow Δf; use accelerometers to capture fine resonance details in wind-turbine blade testing.
The half-power bandwidth method is immune to baseline drift—critical for field vibration surveys on aging concrete bridges where dc offset contaminates direct amplitude readings.
In viscous damper design for seismic isolation, ensure measured ζ matches analytical predictions; deviations exceeding 15% indicate nonlinear friction or seal degradation.
For lightly damped systems (Q > 10), small frequency resolution errors compound; use fixed ±2% frequency sweep rates to minimize bandwidth measurement uncertainty below 5%.