vibration.structural.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 and 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} 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:

Longitudinal rod: Z = ρ c_{L} S, Y = \frac{1}{ρ c _{L} S}

Slender beam, bending, centre: Z = 2 m^{'} c_{B} (1 + j), Y = \frac{1 - j}{4 m ^{'} c _{B}}

Slender beam, bending, end: Z = \frac{m ^{'} c _{B}}{2} (1 + j), Y = \frac{1 - j}{m ^{'} c _{B}}

Thin plate, bending, centre: Z = 8 B^{'} m^{''}, Y = \frac{1}{8 B ^{'} m ^{''}}

Thin plate, bending, edge: Z = 3.5 B^{'} m^{''}, Y = \frac{1}{3.5 B ^{'} m ^{''}}

The thin-plate driving-point impedance $Z = 8 B^{'} m^{''}$ 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 $c_{B} = (B ω^{2} / m^{'})^{1/4}$ (the bending wave speed), so its mobility falls as $ω^{- 1/2}$ ; 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) $Y_{M} = ω (1 + j) / (4 B k_{B})$ with $k_{B} = ω / c_{B}$ the bending wavenumber.

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beam_bending_wave_speed

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

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

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

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).

$Z = 2 m^{'} c_{B} (1 + j)$ for a force at the centre of the beam and $Z = (m^{'} c_{B} /2) (1 + j)$ for a force at a free end, with the bending wave speed $c_{B} = (B ω^{2} / m^{'})^{1/4}$ (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

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).

$Y = (1 - j) / (4 m^{'} c_{B})$ for a force at the centre and $Y = (1 - j) / (m^{'} c_{B})$ for a force at a free end, the reciprocal of infinite_beam_impedance. The mobility falls as $ω^{- 1/2}$ .

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

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).

$Y_{M} = ω (1 + j) / (4 B k_{B})$ with the bending wavenumber $k_{B} = ω / c_{B}$ , 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

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.

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 (driving point).

infinite_plate_impedance

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).

$Z = C B^{'} m^{''}$ 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`).
`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

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

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

The reciprocal of 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

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.

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 (driving point).

injected_power

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

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

$W = 0.5 ∣ F ∣^{2} Re {Y}$ : 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

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).

$Z = ρ c_{L} S$ , 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

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

Point mobility of an infinite rod $Y = 1/ (ρ c_{L} S)$ (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

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).

$B^{'} = E h^{3} / (12 (1 - ν^{2}))$ , 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.

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 $∣ ν ∣ \geq 1$ .

plate_bending_wave_speed

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

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

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.

Created and maintained by José Manuel Requena Plens ORCID All projects