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# Mechanical mobility and the FRF family (ISO 7626-1)

Mechanical **mobility** is the complex ratio of a velocity response to the
force that produces it, `Y = v/F`. It is one member of a family of
motion-per-force **frequency-response functions** (FRFs): which one is used
depends only on whether the motion is a displacement, a velocity or an
acceleration, and each has a force-per-motion reciprocal. **ISO 7626-1:2011**
(*Mechanical vibration and shock — Experimental determination of mechanical
mobility — Part 1: Basic terms and definitions, and transducer specifications*)
defines the whole family (Table 1, with the 3.1.2 mobility definition), and the
classic closed-form single-degree-of-freedom (SDOF) resonator serves as the
reference for those definitions. **ISO 7626-2:2015** adds the measurement side:
FRF estimation from measured signals and its acceptance criteria. This FRF
backbone underpins the structure-borne source and transmission standards:
ISO 9611, ISO 10846, EN 15657 and EN 12354-5.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/mechanical_mobility_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/mechanical_mobility.svg" alt="Normalized receptance, mobility and accelerance magnitudes of a single-degree-of-freedom resonator on a log-log frequency axis, all peaking at the resonance where the mobility is stiffness-controlled below and mass-controlled above" width="82%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import vibration

m, k, c = 2.0, 8000.0, 5.0
f = np.logspace(np.log10(0.5), np.log10(200.0), 500)
w = 2.0 * np.pi * f
h = vibration.sdof_receptance(f, m, k, c)
for label, frf in (("receptance |H|", np.abs(h)),
                   ("mobility |Y|", np.abs(1j * w * h)),
                   ("accelerance |A|", np.abs(-(w**2) * h))):
    plt.loglog(f, frf / frf.max(), label=label)
plt.axvline(vibration.resonance_frequency(m, k), ls="--", color="0.6")
plt.xlabel("Frequency [Hz]"); plt.ylabel("Normalized magnitude")
plt.legend(); plt.show()
```

</details>

## 1. The frequency-response-function family (Table 1)

For a harmonic motion `x·e^{jωt}` the velocity is `jω·x` and the acceleration
`−ω²·x`, so all three motion-per-force FRFs follow from the receptance `H` by a
power of `jω`, and each has a force-per-motion reciprocal:

| Motion | FRF (motion / force) | Unit | Reciprocal (force / motion) | Unit |
|---|---|---|---|---|
| displacement | receptance `H = x/F` | m/N | dynamic stiffness `1/H` | N/m |
| velocity | mobility `Y = jω·H` | m/(N·s) | impedance `1/Y` | N·s/m |
| acceleration | accelerance `A = −ω²·H` | 1/kg | apparent mass `1/A` | kg |

`convert_frf` moves between any two of the six FRFs, pivoting through the
receptance. A **driving-point** FRF has the response and force at the same point
(`i = j`); a **transfer** FRF has them at different points. Note that the
force-per-motion kinds are element-wise reciprocals: the *free* quantities of
ISO 7626-1, 3.1.4; the *blocked* matrix quantities of Table 1 do not invert
element-wise for multi-coordinate systems (Table 1 also names `F/a` the
"effective mass", the quantity called apparent mass here).

```python
from phonometry import vibration

# A mobility of 2e-3 m/(N.s) at 80 Hz, expressed as the other FRFs:
Y = 2e-3
print(round(abs(vibration.convert_frf(Y, 80.0, "mobility", "impedance")), 1))     # 500.0  N.s/m
print(f"{abs(vibration.convert_frf(Y, 80.0, 'mobility', 'accelerance')):.3f}")    # 1.005  1/kg
```

The choice between the three motion FRFs is one of convenience, not physics:
they carry the same information and `convert_frf` moves between them exactly.
Accelerance is what an accelerometer-based measurement delivers directly;
mobility is the natural currency of the structure-borne power standards
(power is force times velocity, so `P = ½·Re{Y}·|F|²` at a contact); the
reciprocals appear whenever a source is described by what it imposes rather
than by how it responds. Reading a driving-point mobility plot is a
structural diagnosis in itself: below a resonance the magnitude climbs
proportionally to frequency along a **stiffness line** (`|Y| ≈ ω/k`), above
it the magnitude falls along a **mass line** (`|Y| ≈ 1/(ωm)`), and the height
of the peak between them reflects the damping: it equals `1/c` for the
isolated viscously damped resonator of the next section, while on real
structures with overlapping modes damping is instead estimated by modal
fitting or from the half-power bandwidth.

## 2. The SDOF reference resonator (closed form)

The canonical closed-form reference, expressed in the Table 1 / 3.1.2 FRF
taxonomy, is a mass `m`, viscous damping `c` and stiffness `k`, whose
receptance is

$$
H(\omega) = \frac{1}{k - \omega^2 m + j\,\omega c}, \qquad
\omega_0 = \sqrt{k/m}.
$$

At the resonance `ω0` the driving-point mobility is **purely real** and equal to
`1/c` (the mobility peak measures the damping) while the static receptance
(`ω → 0`) is the compliance `1/k`:

```python
import numpy as np
from phonometry import vibration

m, k, c = 2.0, 8000.0, 5.0
f0 = vibration.resonance_frequency(m, k)             # 10.07 Hz

y0 = complex(vibration.sdof_mobility(f0, m, k, c))
print(round(y0.real, 4), round(y0.imag, 6))   # 0.2 0.0   -> |Y(f0)| = 1/c
print(round(complex(vibration.sdof_receptance(1e-6, m, k, c)).real, 7))  # 0.000125 = 1/k
```

## 3. Measured FRFs and their acceptance criteria (ISO 7626-2)

In the usual ISO 7626-2 arrangement the structure hangs on a suspension soft
enough that its rigid-body modes fall well below the first elastic resonance
(the standard admits freely suspended or grounded structures; clause 5 asks
for a support representative of the intended application), an
exciter drives one point through an **impedance head** (a transducer stack
measuring force and acceleration at the same point, which is what gives the
attached-exciter setup its driving-point FRF), and accelerometers pick up
the response
elsewhere for the transfer FRFs. ISO 7626-5 covers the alternative of impact
excitation with an exciter that is not attached to the structure, in
practice usually an instrumented hammer: it trades the attached exciter's
controlled spectrum for speed, with an excitation spectrum set by the
impactor mass and tip stiffness.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_mobility_rig_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_mobility_rig.svg" alt="ISO 7626 mobility measurement: a free-free beam on soft suspension driven by an exciter through an impedance head at the driving point, an accelerometer at a transfer point, and an impact hammer as the alternative excitation" width="92%"></picture>

Processing measured random-excitation records per ISO 7626-2, 8.1.3 (the H1
estimator `Ĥ = G(response, force)/G(force, force)`) and the ordinary coherence
`γ² = |Gxy|²/(Gxx·Gyy)` used for its data-quality checks are the library's
existing spectral estimators [`transfer_function` and
`coherence`](electroacoustics.md) (H1 is their default). On top of them,
two ISO 7626-2 acceptance criteria are provided:

* **Operational rigid-mass calibration (7.5.2).** The measured FRF of a freely
  suspended rigid block of known mass must agree within ±5 % with `|A| = 1/m`
  (accelerance) or `|Y| = 1/(2πf·m)` (mobility).
* **Random error (Annex A + 8.1.3).** Enough spectra must be averaged that the
  normalized random error `ε = √((1−γ²)/(2nγ²))` at each resonance of a
  driving-point mobility is below 5 %.

```python
import numpy as np
from phonometry import vibration

# A 10 kg calibration block: |A| must be 1/m = 0.100 1/kg at every frequency.
f = np.array([20.0, 100.0, 500.0])
res = vibration.rigid_mass_calibration_check([0.100, 0.102, 0.097], f, mass=10.0)
print(res.passed, res.within_tolerance.tolist())   # True [True, True, True]

# The Annex A example: coherence 0.8 needs about 75 averages for < 5 %.
print(round(float(vibration.random_error_percent(0.8, 75)), 2))   # 4.08  %
```

The calibration check returns a `RigidMassCalibrationResult` carrying the
per-frequency deviation and pass flags, and a `.plot()`: the measured FRF
magnitude against the rigid-mass line with its ±5 % tolerance band (upper
panel) and the relative deviation against the same band (lower panel, where a
few-percent tolerance is actually readable). A calibration that drifts out of
the band towards a few kHz points at a transducer or attachment-compliance
error, exactly what the check is meant to catch:

```python
import numpy as np
from phonometry import vibration

m = 10.0                                                  # calibration block mass
f = np.logspace(np.log10(20.0), np.log10(5000.0), 400)
drift = 0.05 * (f / 2500.0) ** 2                          # high-frequency drift
measured = (1.0 / m) * (1.0 + 0.015 * np.sin(2 * np.pi * np.log10(f)) + drift)
res = vibration.rigid_mass_calibration_check(measured, f, mass=m)
print(res.passed)                                         # False (drift exceeds 5 %)
res.plot()
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/rigid_mass_calibration_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/rigid_mass_calibration.svg" alt="Rigid-mass calibration check of a 10 kg block: the measured accelerance magnitude follows the flat rigid-mass line inside the plus-or-minus five percent tolerance band across most of the range, then drifts above the band towards a few kilohertz where the out-of-tolerance points are marked, and the lower panel shows the same deviation in percent crossing the plus five percent limit" width="82%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import vibration

m = 10.0
f = np.logspace(np.log10(20.0), np.log10(5000.0), 400)
drift = 0.05 * (f / 2500.0) ** 2
measured = (1.0 / m) * (1.0 + 0.015 * np.sin(2 * np.pi * np.log10(f)) + drift)
res = vibration.rigid_mass_calibration_check(measured, f, mass=m)
bad = ~res.within_tolerance

fig, (top, bot) = plt.subplots(2, 1, sharex=True, figsize=(10, 7),
                               gridspec_kw={"height_ratios": [1.5, 1.0]})
top.fill_between(f, res.expected * 0.95, res.expected * 1.05, color="C1",
                 alpha=0.15, label="±5 % tolerance band")
top.semilogx(f, res.expected, "--", color="C1", label="expected |A| = 1/m")
top.semilogx(f, res.measured, color="C0", label="within tolerance")
top.semilogx(f[bad], res.measured[bad], "o", color="C1", label="out of tolerance")
top.set_ylabel("Accelerance |A| [1/kg]"); top.legend()

bot.axhspan(-5.0, 5.0, color="C1", alpha=0.15)
bot.semilogx(f, 100.0 * res.deviation, color="C0")
bot.semilogx(f[bad], 100.0 * res.deviation[bad], "o", color="C1")
bot.set_xlabel("Frequency [Hz]"); bot.set_ylabel("Deviation [%]")
plt.show()
```

</details>

## 4. The `MobilityResult` bundle

`sdof_mobility_result` bundles the FRF over frequency into a `MobilityResult`,
which exposes `.magnitude`, `.phase`, `.to(target)` (any Table-1 kind) and a
`.plot()` of `|Y(f)|` with the resonance marked:

```python
import numpy as np
from phonometry import vibration

f = np.logspace(np.log10(0.5), np.log10(200.0), 400)
res = vibration.sdof_mobility_result(f, mass=2.0, stiffness=8000.0, damping=5.0)
z = res.to("impedance")                        # impedance = 1/Y per frequency
print(res.frequencies[int(np.argmax(res.magnitude))].round(1))   # ~10.1 Hz
```

**Test-report fiche.** `MobilityResult.report(path)` renders a one-page
mechanical-mobility measurement report (ISO 7626-1:2011 FRF definitions,
measurement per ISO 7626-2:2015). Mobility is a continuous frequency-response
function, not an octave-band quantity, so the sheet presents it honestly as the
`|Y(f)|` magnitude spectrum plus a compact table of characteristic points (the
FRF type, driving-point or transfer, the frequency range, the peak frequency,
the peak mobility magnitude and the phase there), and a boxed peak mobility
`|Y|` at the frequency it occurs at (for a driving-point FRF a resonance, where
`|Y| = 1/c` measures the damping). It is a characterisation, so there is no
pass/fail verdict; `language="es"` renders the Spanish fiche. The fiche always
embeds the `|Y(f)|` spectrum, so it needs both the report and plot extras
(`pip install "phonometry[report,plot]"`).

[![ISO 7626 mechanical-mobility example report: a metadata header, a table of the FRF characteristic points (type, frequency range, peak frequency, peak mobility and phase) beside the mobility magnitude spectrum, and the boxed peak mobility](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso7626_mobility_example.webp)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso7626_mobility_example.pdf)

## References

- Cremer, L., Heckl, M., & Petersson, B. A. T. (2005). *Structure-borne
  sound: Structural vibrations and sound radiation at audio frequencies*
  (3rd ed.). Springer. ISBN 978-3-540-22696-3.
  [doi:10.1007/b137728](https://doi.org/10.1007/b137728).
  The standard monograph on structural vibration: point and transfer
  mobilities of beams and plates, and the power flow `P = ½·Re{Y}·|F|²` that
  makes mobility the working quantity of this page.
- International Organization for Standardization. (2011). *Mechanical
  vibration and shock — Experimental determination of mechanical mobility —
  Part 1: Basic terms and definitions, and transducer specifications*
  (ISO 7626-1:2011).
  [iso.org catalogue](https://www.iso.org/standard/50426.html).
  The Table 1 FRF family and the free/blocked distinctions implemented here.
- International Organization for Standardization. (2015). *Mechanical
  vibration and shock — Experimental determination of mechanical mobility —
  Part 2: Measurements using single-point translation excitation with an
  attached vibration exciter* (ISO 7626-2:2015).
  [iso.org catalogue](https://www.iso.org/standard/62483.html).
  The measurement side: H1 processing, rigid-mass calibration and the
  random-error criterion.

## Standards

ISO 7626-1:2011, *Mechanical vibration and shock — Experimental
determination of mechanical mobility — Part 1: Basic terms and definitions, and
transducer specifications*: the FRF family and its reciprocals (Table 1, the
3.1.2 mobility and 3.1.4 free-quantity definitions) and the driving-point /
transfer distinction. ISO 7626-2:2015, *Part 2: Measurements using single-point
translation excitation with an attached vibration exciter*: the 8.1.3 H1
processing of random excitation, the 7.5.2 rigid-mass operational calibration
(±5 %) and the Annex A random-error criterion (< 5 % at resonances).
Conformance is anchored on the closed-form SDOF identities (consistent with the
Table 1 / 3.1.2 definitions): the driving-point mobility peak `|Y(ω0)| = 1/c`,
the static receptance `H(0) = 1/k`, the exact Table-1 reciprocity
`impedance·mobility = 1`, the rigid-mass calibration values (`|A| = 0.100` 1/kg
for 10 kg; `|Y| = 1.59155e-4` m/(N·s) at 100 Hz) and the Annex A example
(γ² = 0.8, n = 75 → ε = 4.08 %).
