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):

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

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:

$$\text{Longitudinal rod:}Z=\rho c_{\mathrm{L}}S,Y=\frac{1}{\rho c_{\mathrm{L}}S}$$ $$\text{Slender beam, bending, centre:}Z=2m^{\prime }{c}_{\mathrm{B}}(1+j),Y=\frac{1-j}{4m^{\prime }{c}_{\mathrm{B}}}$$ $$\text{Slender beam, bending, end:}Z=\frac{m^{\prime }{c}_{\mathrm{B}}}{2}(1+j),Y=\frac{1-j}{m^{\prime }{c}_{\mathrm{B}}}$$ $$\text{Thin plate, bending, centre:}Z=8\sqrt{B^{\prime }{m}^{\prime \prime }},Y=\frac{1}{8\sqrt{B^{\prime }{m}^{\prime \prime }}}$$ $$\text{Thin plate, bending, edge:}Z=3.5\sqrt{B^{\prime }{m}^{\prime \prime }},Y=\frac{1}{3.5\sqrt{B^{\prime }{m}^{\prime \prime }}}$$

The thin-plate driving-point impedance $Z=8\sqrt{B^{\prime }{m}^{\prime \prime }}$ 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 $c_{\mathrm{B}}=(B{\omega }^2/m^{\prime }{)}^{1/4}$ (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) $Y_{\mathrm{M}}=\omega (1+j)/(4B{k}_{\mathrm{B}})$ with $k_{\mathrm{B}}=\omega /c_{\mathrm{B}}$ 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 $c_{\mathrm{B}}=(B{\omega }^2/m^{\prime }{)}^{1/4}$.

Parameters

Name	Description
frequency	Frequency f, in hertz (scalar or array, > 0).
bending_stiffness	Beam bending stiffness $B=EI$, 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).

$Z=2m^{\prime }{c}_{\mathrm{B}}(1+j)$ for a force at the centre of the beam and $Z=(m^{\prime }{c}_{\mathrm{B}}/2)(1+j)$ for a force at a free end, with the bending wave speed $c_{\mathrm{B}}=(B{\omega }^2/m^{\prime }{)}^{1/4}$ (beam_bending_wave_speed).

Parameters

Name	Description
frequency	Frequency f, in hertz (scalar or array, > 0).
bending_stiffness	Beam bending stiffness $B=EI$, 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).

$Y=(1-j)/(4m^{\prime }{c}_{\mathrm{B}})$ for a force at the centre and $Y=(1-j)/(m^{\prime }{c}_{\mathrm{B}})$ for a force at a free end, the reciprocal of 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 $B=EI$, 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).

$Y_{\mathrm{M}}=\omega (1+j)/(4B{k}_{\mathrm{B}})$ 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 $B=EI$, 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 $B=EI$, 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).

$Z=C\sqrt{B^{\prime }{m}^{\prime \prime }}$ 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 $Y=1/(C\sqrt{B^{\prime }{m}^{\prime \prime }})$.

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).

$W=0.5|F{|}^2\mathrm{Re}\{Y\}$: 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).

$Z=\rho c_{\mathrm{L}}S$, 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

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

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).

$B^{\prime }=E{h}^3/(12(1-{\nu }^2))$, 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.

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 $|\nu |\ge 1$.

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 $c_{\mathrm{B}}=(B^{\prime }{\omega }^2/m^{\prime \prime }{)}^{1/4}$.

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.