<!-- canonical: https://docs.jmrp.io/phonometry/reference/api/vibration/point-mobility/ -->
Source: https://docs.jmrp.io/phonometry/reference/api/vibration/point-mobility/

Point mobilities and impedances of infinite structures (Cremer, Heckl &
Petersson 2005, Chapter 5, Table 5.1).

The **point mobility** `Y` of a structure is the complex ratio of the
velocity response at a driving point to the point force that produces it, and
its reciprocal is the **point impedance** $Z = 1/Y$ (the same
motion-per-force / force-per-motion pair as ISO 7626-1 mechanical mobility, so
these theoretical values slot straight into
[`MobilityResult`](/phonometry/reference/api/vibration/mechanical-mobility/#mobilityresult) and
[`convert_frf`](/phonometry/reference/api/vibration/mechanical-mobility/#convert_frf)). For an *infinite* structure the
driving point never sees a reflected wave, so the mobility is the free-field
value that sets the vibrational power a source injects (Cremer 5.5): with a
point force of amplitude `F` the time-averaged injected power, in watts, is
(Cremer Eq. 5.23):

$$
W = 0.5\, |F|^2 \operatorname{Re}\{Y\}
$$

These are the theoretical companions of the *measured* driving-point mobilities
of ISO 7626 and the isolator transfer stiffnesses of ISO 10846, and they supply
the receiver mobility that the installed structure-borne prediction of
EN 12354-5 needs when no measurement is available.

**Compilation (Cremer Table 5.1).** With `m'` the mass per unit length
(kg/m), `m''` the mass per unit area (kg/m^2), `B` the bending stiffness of
a beam (N.m^2) and `B'` the bending stiffness of a plate *per unit width*
(N.m), the impedance `Z` and mobility `Y` for a point force are:

$$
\text{Longitudinal rod:} \quad Z = \rho c_\mathrm{L} S, \qquad Y = \frac{1}{\rho c_\mathrm{L} S}
$$

$$
\text{Slender beam, bending, centre:} \quad Z = 2 m' c_\mathrm{B} (1 + j), \qquad Y = \frac{1 - j}{4 m' c_\mathrm{B}}
$$

$$
\text{Slender beam, bending, end:} \quad Z = \frac{m' c_\mathrm{B}}{2} (1 + j), \qquad Y = \frac{1 - j}{m' c_\mathrm{B}}
$$

$$
\text{Thin plate, bending, centre:} \quad Z = 8 \sqrt{B' m''}, \qquad Y = \frac{1}{8 \sqrt{B' m''}}
$$

$$
\text{Thin plate, bending, edge:} \quad Z = 3.5 \sqrt{B' m''}, \qquad Y = \frac{1}{3.5 \sqrt{B' m''}}
$$

The thin-plate driving-point impedance $Z = 8 \sqrt{B' m''}$ is real
and frequency independent (the plate behaves as a pure resistance to a point
force), so a plate absorbs power like a matched resistance. The beam
impedance grows as $c_\mathrm{B} = (B \omega^2 / m')^{1/4}$ (the bending wave
speed), so its mobility falls as $\omega^{-1/2}$; the
$(1 - j)$ factor means half the input goes into a reactive near field.
A moment excitation of the beam has the mobility (Cremer Eq. 5.75)
$Y_\mathrm{M} = \omega (1 + j) / (4 B k_\mathrm{B})$ with $k_\mathrm{B} = \omega / c_\mathrm{B}$ the
bending wavenumber.

> Auto-generated from the source docstrings by `scripts/generate_api_docs.py` (`make api-docs`). Do not edit by hand.

## beam_bending_wave_speed

```python
beam_bending_wave_speed(
    frequency: ArrayLike,
    bending_stiffness: float,
    mass_per_length: float,
) -> NDArray[np.float64]
```

Free bending wave speed of a beam
$c_\mathrm{B} = (B \omega^2 / m')^{1/4}$.

**Parameters**

| Name | Description |
| :--- | :--- |
| `frequency` | Frequency `f`, in hertz (scalar or array, > 0). |
| `bending_stiffness` | Beam bending stiffness $B = E I$, in N.m^2. |
| `mass_per_length` | Mass per unit length `m'`, in kg/m. |

**Returns:** The bending phase speed `cB`, in m/s.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive stiffness, mass or frequency. |

## infinite_beam_impedance

```python
infinite_beam_impedance(
    frequency: ArrayLike,
    bending_stiffness: float,
    mass_per_length: float,
    *,
    location: str = 'centre',
) -> NDArray[np.complex128]
```

Point impedance of an infinite beam in bending (Cremer Table 5.1).

$Z = 2 m' c_\mathrm{B} (1 + j)$ for a force at the centre of the beam and
$Z = (m' c_\mathrm{B} / 2)(1 + j)$ for a force at a free end, with the
bending wave speed $c_\mathrm{B} = (B \omega^2 / m')^{1/4}$
([`beam_bending_wave_speed`](/phonometry/reference/api/vibration/point-mobility/#beam_bending_wave_speed)).

**Parameters**

| Name | Description |
| :--- | :--- |
| `frequency` | Frequency `f`, in hertz (scalar or array, > 0). |
| `bending_stiffness` | Beam bending stiffness $B = E I$, in N.m^2. |
| `mass_per_length` | Mass per unit length `m'`, in kg/m. |
| `location` | `"centre"` or `"end"`. |

**Returns:** The complex point impedance `Z`, in N.s/m.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive input or unknown location. |

## infinite_beam_mobility

```python
infinite_beam_mobility(
    frequency: ArrayLike,
    bending_stiffness: float,
    mass_per_length: float,
    *,
    location: str = 'centre',
) -> NDArray[np.complex128]
```

Point mobility of an infinite beam in bending (Cremer Table 5.1).

$Y = (1 - j) / (4 m' c_\mathrm{B})$ for a force at the centre and
$Y = (1 - j) / (m' c_\mathrm{B})$ for a force at a free end, the reciprocal
of [`infinite_beam_impedance`](/phonometry/reference/api/vibration/point-mobility/#infinite_beam_impedance). The mobility falls as
$\omega^{-1/2}$.

**Parameters**

| Name | Description |
| :--- | :--- |
| `frequency` | Frequency `f`, in hertz (scalar or array, > 0). |
| `bending_stiffness` | Beam bending stiffness $B = E I$, in N.m^2. |
| `mass_per_length` | Mass per unit length `m'`, in kg/m. |
| `location` | `"centre"` or `"end"`. |

**Returns:** The complex point mobility `Y`, in m/(N.s).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive input or unknown location. |

## infinite_beam_moment_mobility

```python
infinite_beam_moment_mobility(
    frequency: ArrayLike,
    bending_stiffness: float,
    mass_per_length: float,
) -> NDArray[np.complex128]
```

Moment (rotational) mobility of an infinite beam (Cremer Eq. 5.75).

$Y_\mathrm{M} = \omega (1 + j) / (4 B k_\mathrm{B})$ with the bending wavenumber
$k_\mathrm{B} = \omega / c_\mathrm{B}$, the angular velocity per unit applied moment
at the driving point.

**Parameters**

| Name | Description |
| :--- | :--- |
| `frequency` | Frequency `f`, in hertz (scalar or array, > 0). |
| `bending_stiffness` | Beam bending stiffness $B = E I$, in N.m^2. |
| `mass_per_length` | Mass per unit length `m'`, in kg/m. |

**Returns:** The complex moment mobility `Y_M`, in rad/(N.m.s).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive input. |

## infinite_beam_point_mobility

```python
infinite_beam_point_mobility(
    frequency: ArrayLike,
    bending_stiffness: float,
    mass_per_length: float,
    *,
    location: str = 'centre',
) -> MobilityResult
```

Infinite-beam point mobility bundled as a [`MobilityResult`](/phonometry/reference/api/vibration/mechanical-mobility/#mobilityresult).

**Parameters**

| Name | Description |
| :--- | :--- |
| `frequency` | Frequencies `f`, in hertz (array, > 0). |
| `bending_stiffness` | Beam bending stiffness $B = E I$, in N.m^2. |
| `mass_per_length` | Mass per unit length `m'`, in kg/m. |
| `location` | `"centre"` or `"end"`. |

**Returns:** The [`MobilityResult`](/phonometry/reference/api/vibration/mechanical-mobility/#mobilityresult) (driving point).

## infinite_plate_impedance

```python
infinite_plate_impedance(
    bending_stiffness: float,
    mass_per_area: float,
    *,
    location: str = 'centre',
) -> float
```

Point impedance of an infinite thin plate (Cremer Table 5.1).

$Z = C \sqrt{B' m''}$ with $C = 8$ for a force at the plate
centre and $C = 3.5$ for a force at a free edge. The impedance is
purely real and
frequency independent: an infinite plate presents a matched resistance to a
point force.

**Parameters**

| Name | Description |
| :--- | :--- |
| `bending_stiffness` | Plate bending stiffness per unit width `B'`, in N.m (see [`plate_bending_stiffness`](/phonometry/reference/api/vibration/point-mobility/#plate_bending_stiffness)). |
| `mass_per_area` | Mass per unit area `m''`, in kg/m^2. |
| `location` | `"centre"` ($C = 8$) or `"edge"` ($C = 3.5$). |

**Returns:** The point impedance `Z`, in N.s/m.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive stiffness/mass or unknown location. |

## infinite_plate_mobility

```python
infinite_plate_mobility(
    bending_stiffness: float,
    mass_per_area: float,
    *,
    location: str = 'centre',
) -> float
```

Point mobility of an infinite thin plate
$Y = 1 / (C \sqrt{B' m''})$.

The reciprocal of [`infinite_plate_impedance`](/phonometry/reference/api/vibration/point-mobility/#infinite_plate_impedance) (real, frequency
independent).

**Parameters**

| Name | Description |
| :--- | :--- |
| `bending_stiffness` | Plate bending stiffness per unit width `B'`, in N.m. |
| `mass_per_area` | Mass per unit area `m''`, in kg/m^2. |
| `location` | `"centre"` ($C = 8$) or `"edge"` ($C = 3.5$). |

**Returns:** The point mobility `Y`, in m/(N.s).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive stiffness/mass or unknown location. |

## infinite_plate_point_mobility

```python
infinite_plate_point_mobility(
    frequency: ArrayLike,
    bending_stiffness: float,
    mass_per_area: float,
    *,
    location: str = 'centre',
) -> MobilityResult
```

Infinite-plate point mobility bundled as a [`MobilityResult`](/phonometry/reference/api/vibration/mechanical-mobility/#mobilityresult).

The plate mobility is frequency independent, so the returned spectrum is
constant across *frequency*; bundling it lets it be plotted and converted
with the ISO 7626 mobility machinery.

**Parameters**

| Name | Description |
| :--- | :--- |
| `frequency` | Frequencies `f`, in hertz (array, > 0). |
| `bending_stiffness` | Plate bending stiffness per unit width `B'`, in N.m. |
| `mass_per_area` | Mass per unit area `m''`, in kg/m^2. |
| `location` | `"centre"` or `"edge"`. |

**Returns:** The [`MobilityResult`](/phonometry/reference/api/vibration/mechanical-mobility/#mobilityresult) (driving point).

## injected_power

```python
injected_power(force: ArrayLike, mobility: ArrayLike) -> NDArray[np.float64]
```

Time-averaged vibrational power injected by a point force (Cremer 5.23).

$W = 0.5\, |F|^2 \operatorname{Re}\{Y\}$: only the real part
(conductance) of the mobility carries power; the reactive part stores
near-field energy.

**Parameters**

| Name | Description |
| :--- | :--- |
| `force` | Point-force amplitude `F` (peak, scalar or array), in N. |
| `mobility` | Complex point mobility `Y` (broadcast with *force*), in m/(N.s). |

**Returns:** The injected power `W`, in W.

## longitudinal_rod_impedance

```python
longitudinal_rod_impedance(
    density: float,
    longitudinal_wave_speed: float,
    cross_section_area: float,
) -> float
```

Point impedance of an infinite rod in longitudinal motion (Table 5.1).

$Z = \rho c_\mathrm{L} S$, real and frequency independent.

**Parameters**

| Name | Description |
| :--- | :--- |
| `density` | Material density `rho`, in kg/m^3. |
| `longitudinal_wave_speed` | Longitudinal wave speed `cL`, in m/s. |
| `cross_section_area` | Cross-section area `S`, in m^2. |

**Returns:** The point impedance `Z`, in N.s/m.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive input. |

## longitudinal_rod_mobility

```python
longitudinal_rod_mobility(
    density: float,
    longitudinal_wave_speed: float,
    cross_section_area: float,
) -> float
```

Point mobility of an infinite rod $Y = 1 / (\rho c_\mathrm{L} S)$
(Table 5.1).

**Parameters**

| Name | Description |
| :--- | :--- |
| `density` | Material density `rho`, in kg/m^3. |
| `longitudinal_wave_speed` | Longitudinal wave speed `cL`, in m/s. |
| `cross_section_area` | Cross-section area `S`, in m^2. |

**Returns:** The point mobility `Y`, in m/(N.s).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive input. |

## plate_bending_stiffness

```python
plate_bending_stiffness(
    youngs_modulus: float,
    thickness: float,
    poisson_ratio: float = 0.0,
) -> float
```

Bending stiffness of a thin plate per unit width (Cremer Eq. 4.22).

$B' = E h^3 / (12 (1 - \nu^2))$, the plate bending stiffness `B'`
in N.m used throughout this module and by the coincidence frequency of
[`phonometry.vibration.structural.radiation_efficiency.coincidence_frequency`](/phonometry/reference/api/vibration/radiation-efficiency/#coincidence_frequency).

**Parameters**

| Name | Description |
| :--- | :--- |
| `youngs_modulus` | Young's modulus `E` of the plate material, in Pa. |
| `thickness` | Plate thickness `h`, in m. |
| `poisson_ratio` | Poisson's ratio `nu` (Default: 0.0). |

**Returns:** The bending stiffness per unit width `B'`, in N.m.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive modulus/thickness or $\lvert\nu\rvert \ge 1$. |

## plate_bending_wave_speed

```python
plate_bending_wave_speed(
    frequency: ArrayLike,
    bending_stiffness: float,
    mass_per_area: float,
) -> NDArray[np.float64]
```

Free bending wave speed of a plate
$c_\mathrm{B} = (B' \omega^2 / m'')^{1/4}$.

**Parameters**

| Name | Description |
| :--- | :--- |
| `frequency` | Frequency `f`, in hertz (scalar or array, > 0). |
| `bending_stiffness` | Plate bending stiffness per unit width `B'`, in N.m. |
| `mass_per_area` | Mass per unit area `m''`, in kg/m^2. |

**Returns:** The bending phase speed `cB`, in m/s.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive stiffness, mass or frequency. |