vibration.structural.experimental_sea

Experimental statistical energy analysis: coupling loss factors from measured energies (Norton & Karczub Ch. 6).

Statistical energy analysis (SEA) has two routes to the coupling loss factor eta_ij. The predictive route derives it from a wave transmission coefficient at the junction, which is what coupling_loss_factor does with the closed-form coefficients of junction_transmission. The experimental route, implemented here, inverts the steady-state power balance from measured subsystem energies: it needs no model of the junction at all, and it is the only route open for real joints (welds, bolt rows, spot welds, adhesives) whose wave behaviour is not tractable.

For two subsystems the steady-state power balance is (Norton Eqs. 6.10 and 6.11, generalised to a drive on either subsystem):

$${\Pi }_1=\omega \left[({\eta }_1+{\eta }_{12})E_1-{\eta }_{21}{E}_2\right]\text{\hspace{1em}(6.10)}$$ $${\Pi }_2=\omega \left[({\eta }_2+{\eta }_{21})E_2-{\eta }_{12}{E}_1\right]\text{\hspace{1em}(6.11)}$$

with $E_i=M_i⟨v_i^2⟩$ the band energy of subsystem i (mass times the space- and time-averaged mean-square velocity), eta_i its internal loss factor and omega the band centre frequency in rad/s. Two inversions follow.

Single drive plus reciprocity (power_injection_clf). Drive subsystem 1 only. The second equation with ${\Pi }_2=0$ gives one relation between eta_12 and eta_21; the SEA consistency (reciprocity) relationship ${\eta }_{12}{n}_1={\eta }_{21}{n}_2$ (Eq. 6.8) supplies the second, so with the modal densities n_1, n_2 known (Eq. 6.15):

$${\eta }_{12}=\frac{{\eta }_2{E}_2}{E_1-E_2{n}_1/n_2}$$ $${\eta }_{21}={\eta }_{12}\frac{n_1}{n_2}$$ $${\Pi }_{\text{in}}=\omega ({\eta }_1{E}_1+{\eta }_2{E}_2)$$

The input power reduces to the total dissipated power, as it must in the steady state: substituting the balance of subsystem 2 into Eq. (6.10) cancels the two coupling terms exactly. The bracket $E_1-E_2{n}_1/n_2$ is positive exactly when the modal energy of the driven subsystem exceeds that of the receiver, $E_1/n_1>E_2/n_2$; a measurement that violates it is not a two-subsystem SEA system and is rejected.

Two drives, no reciprocity assumed (power_injection_matrix). The classical power-injection method drives each subsystem in turn and measures both energies each time, giving four equations for the four unknowns eta_1, eta_2, eta_12, eta_21 with no prior assumption at all. Reciprocity then becomes a check on the measurement rather than an input: PowerInjectionResult.modal_density_ratio compares the measured ${\eta }_{12}/{\eta }_{21}$ with the expected $n_2/n_1$.

Modal densities for the usual subsystems are provided as well (flat_plate_modal_density, cylindrical_shell_modal_density, bar_modal_density, beam_modal_density), following Norton Eqs. 6.23-6.29. The flat-plate expression $n(f)=S\sqrt{12}/(2c_{L}t)$ is the same quantity as EN 12354-4’s $n=\pi S{f}_c/c_0^2$ (modal_density), only parametrised by the plate itself rather than by its critical frequency; the two agree identically and a regression test pins that.

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

bar_modal_density

bar_modal_density(length: float, longitudinal_wave_speed: float) -> float

Modal density of a uniform bar in longitudinal vibration (Eq. 6.23).

$n(f)=2L/c_L$, independent of frequency.

Parameters

Name	Description
length	Bar length L, in m (> 0).
longitudinal_wave_speed	Bar wave speed $c_L=\sqrt{E/\rho }$, in m/s (> 0).

Returns: The modal density n(f), in modes per hertz.

Raises

Exception	When
ValueError	for a non-positive input.

beam_modal_density

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

Modal density of a uniform beam in flexure (Norton Eq. 6.24).

$n(f)=L(\rho A/EI{)}^{1/4}/\sqrt{2\pi f}$: unlike every other subsystem here, it decreases with frequency.

Parameters

Name	Description
frequency	Band centre frequency f, in hertz (scalar or array, > 0).
length	Beam length L, in m (> 0).
mass_per_length	Mass per unit length rho A, in kg/m (> 0).
bending_stiffness	Flexural stiffness E I, in N.m^2 (> 0).

Returns: The modal density n(f), in modes per hertz.

Raises

Exception	When
ValueError	for a non-positive input.

cylindrical_shell_modal_density

cylindrical_shell_modal_density(
    frequency: ArrayLike,
    area: float,
    thickness: float,
    mean_radius: float,
    longitudinal_wave_speed: float,
    *,
    band: str = 'octave',
) -> NDArray[np.float64]

Average modal density of a thin-walled cylinder (Norton 6.27-6.29).

The semi-empirical approximations of Szechenyi, as collected by Clarkson & Pope, in three regimes of $x=f/f_r$ around the ring frequency ring_frequency:

$$n=\frac{5S}{\pi c_{L}t}\sqrt{x},x\le 0.48\text{\hspace{1em}(6.27)}$$ $$n=\frac{7.2S}{\pi c_{L}t}x,0.48<x\le 0.83\text{\hspace{1em}(6.28)}$$ $$n=\frac{2S}{\pi c_{L}t}\left[2+\frac{0.596}{F-1/F}\left(F\arccos \frac{1.745}{F^2{x}^2}-\frac{1}{F}\arccos \frac{1.745F^2}{x^2}\right)\right],x>0.83\text{\hspace{1em}(6.29)}$$

with the bandwidth factor $F=\sqrt{f_{\text{upper}}/f_{\text{lower}}}$. These are average values: they do not resolve the large fluctuations that the cut-on of successive circumferential orders produces below the ring frequency, which for long thin shells can be substantial.

Parameters

Name	Description
frequency	Band centre frequency f, in hertz (scalar or array, > 0).
area	Shell surface area S, in m^2 (> 0).
thickness	Wall thickness t, in m (> 0).
mean_radius	Mean shell radius a_m, in m (> 0).
longitudinal_wave_speed	Plate wave speed cL, in m/s (> 0).
band	Analysis bandwidth, "octave" (Default, $F=1.414$) or "third" ($F=1.122$).

Returns: The modal density n(f), in modes per hertz.

Raises

Exception	When
ValueError	for a non-positive input or an unknown band.

flat_plate_modal_density

flat_plate_modal_density(
    area: float,
    thickness: float,
    longitudinal_wave_speed: float,
) -> float

Modal density of a flat plate in flexure (Norton Eq. 6.25).

$n(f)=S\sqrt{12}/(2c_{L}t)$, independent of frequency, with the plate (quasi-longitudinal) wave speed $c_L=\sqrt{E/(\rho (1-{\nu }^2))}$.

Parameters

Name	Description
area	Plate surface area S, in m^2 (> 0).
thickness	Plate thickness t, in m (> 0).
longitudinal_wave_speed	Plate wave speed cL, in m/s (> 0).

Returns: The modal density n(f), in modes per hertz.

Raises

Exception	When
ValueError	for a non-positive input.

power_injection_clf

power_injection_clf(
    frequency: ArrayLike,
    energy1: ArrayLike,
    energy2: ArrayLike,
    internal_loss_factor1: ArrayLike,
    internal_loss_factor2: ArrayLike,
    modal_density1: ArrayLike,
    modal_density2: ArrayLike,
) -> PowerInjectionResult

Coupling loss factors from a single-drive energy measurement.

Subsystem 1 is driven and subsystem 2 receives power only through the junction. Inverting the steady-state balance of subsystem 2 together with the reciprocity relationship ${\eta }_{12}{n}_1={\eta }_{21}{n}_2$ gives

$${\eta }_{12}=\frac{{\eta }_2{E}_2}{E_1-E_2{n}_1/n_2}$$ $${\eta }_{21}={\eta }_{12}\frac{n_1}{n_2}$$ $${\Pi }_{\text{in}}=\omega ({\eta }_1{E}_1+{\eta }_2{E}_2)$$

Energies are $E_i=M_i⟨v_i^2⟩$ with M_i the subsystem mass and $⟨v_i^2⟩$ the space- and time-averaged mean-square velocity in the band.

Parameters

Name	Description
frequency	Band centre frequencies f, in hertz (> 0).
energy1	Band energy E_1 of the driven subsystem, in J (> 0).
energy2	Band energy E_2 of the receiving subsystem, in J (> 0).
internal_loss_factor1	eta_1 (> 0), scalar or per band.
internal_loss_factor2	eta_2 (> 0), scalar or per band.
modal_density1	n_1, in modes per hertz (> 0).
modal_density2	n_2, in modes per hertz (> 0).

Returns: A PowerInjectionResult (method "single-drive").

Raises

Exception	When
ValueError	for a non-positive input, mismatched band lengths, or a measurement with $E_1/n_1\le E_2/n_2$ (the receiving subsystem holds at least as much modal energy as the driven one, so no two-subsystem SEA model fits it).

power_injection_matrix

power_injection_matrix(
    frequency: ArrayLike,
    energies: ArrayLike,
    input_powers: ArrayLike,
) -> PowerInjectionResult

Full two-drive power-injection inversion (no reciprocity assumed).

Each subsystem is driven in turn with a known injected power while both band energies are measured, giving four equations for the four unknowns eta_1, eta_2, eta_12, eta_21. Because reciprocity is not used, PowerInjectionResult.modal_density_ratio becomes an independent check on the measurement.

Parameters

Name	Description
frequency	Band centre frequencies f, in hertz (> 0), shape (nb,).
energies	Measured band energies, in joules, shape (2, 2, nb): energies[i][j] is the energy of subsystem i while subsystem j is driven (all > 0).
input_powers	Injected powers, in watts, shape (2, nb): input_powers[j] is the power injected into subsystem j during test j (all > 0).

Returns: A PowerInjectionResult (method "two-drive"); its energy1/energy2 and input_power1/input_power2 report the first test (subsystem 1 driven).

Raises

Exception	When
ValueError	for a bad shape, a non-positive value, or a singular energy matrix in some band.

PowerInjectionResult

PowerInjectionResult(
    frequencies: NDArray[np.float64],
    coupling_loss_factor12: NDArray[np.float64],
    coupling_loss_factor21: NDArray[np.float64],
    internal_loss_factor1: NDArray[np.float64],
    internal_loss_factor2: NDArray[np.float64],
    energy1: NDArray[np.float64],
    energy2: NDArray[np.float64],
    input_power1: NDArray[np.float64],
    input_power2: NDArray[np.float64],
    modal_density1: NDArray[np.float64] | None,
    modal_density2: NDArray[np.float64] | None,
    method: str,
)

Loss-factor budget of a two-subsystem SEA model, per band.

All arrays share the band axis frequencies.

Attributes

Name	Description
frequencies	Band centre frequencies f, in hertz.
coupling_loss_factor12	eta_12, subsystem 1 to 2.
coupling_loss_factor21	eta_21, subsystem 2 to 1.
internal_loss_factor1	eta_1 of subsystem 1.
internal_loss_factor2	eta_2 of subsystem 2.
energy1	Band energy E_1, in joules.
energy2	Band energy E_2, in joules.
input_power1	Power injected into subsystem 1, in watts.
input_power2	Power injected into subsystem 2, in watts.
modal_density1	Modal density n_1, in modes per hertz, or None when the inversion did not need it.
modal_density2	Modal density n_2, or None.
method	"single-drive" (reciprocity assumed) or "two-drive" (full power-injection matrix).

PowerInjectionResult.coupling_strength

property

Coupling ratio ${\eta }_{12}/{\eta }_1$.

SEA subsystems should be weakly coupled: values well below 1 mean the junction leaks far less power than the subsystem dissipates, the regime where the two-subsystem model is trustworthy.

PowerInjectionResult.dissipated_power

property

Total power dissipated internally, in watts.

$\omega ({\eta }_1{E}_1+{\eta }_2{E}_2)$, which equals input_power in the steady state and is the round-trip check of the inversion.

PowerInjectionResult.input_power

property

Total power injected into the system, in watts.

PowerInjectionResult.modal_density_ratio

property

Ratio $n_1/n_2$ implied by the measured coupling loss factors.

From the reciprocity relationship ${\eta }_{12}{n}_1={\eta }_{21}{n}_2$, so it equals ${\eta }_{21}/{\eta }_{12}$. For a "two-drive" inversion, where reciprocity was never assumed, comparing this with the modal densities computed from the geometry is the consistency check on the measurement.

PowerInjectionResult.plot()

PowerInjectionResult.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Plot the loss-factor budget against frequency.

The two coupling loss factors and the two internal loss factors on one log axis: their ordering is the whole diagnosis, because SEA is only valid where the coupling stays below the internal damping.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Parameters

Name	Description
ax	Existing axes, or None to create a figure.
language	Label language, "en" (default) or "es".
kwargs	Forwarded to the eta_12 curve.

PowerInjectionResult.transmitted_power

property

Net power flowing from subsystem 1 to 2, in watts.

${\Pi }_{12}=\omega ({\eta }_{12}{E}_1-{\eta }_{21}{E}_2)$; negative when the net flow runs the other way.

ring_frequency

ring_frequency(mean_radius: float, longitudinal_wave_speed: float) -> float

Ring frequency of a cylindrical shell (Norton Eq. 6.26).

$f_r=c_L/(2\pi a_m)$: the frequency at which the shell vibrates uniformly in the breathing mode. Above it a cylinder behaves like a flat plate; below it the modes group by circumferential order and the modal density is no longer a simple function of frequency.

Parameters

Name	Description
mean_radius	Mean shell radius a_m, in m (> 0).
longitudinal_wave_speed	Plate wave speed cL, in m/s (> 0).

Returns: The ring frequency fr, in hertz.

Raises

Exception	When
ValueError	for a non-positive input.