vibration.structural.transfer_stiffness

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Dynamic transfer stiffness of resilient elements (ISO 10846-1/-2/-3).

The vibro-acoustic transfer property of a resilient element (a vibration isolator, mount, bellows or hose) is its dynamic transfer stiffness: the frequency-dependent ratio of the blocking force phasor F2,b on the output (receiver) side to the displacement phasor u1 on the input (source) side, with the output blocked (ISO 10846-1, 3.7), in N/m:

k_{2, 1} = \frac{F _{2, b}}{u _{1}}

For an isolator between two structures of large driving-point stiffness, the force delivered to the receiver approximates this blocking force (ISO 10846-1, Equation 7), so $k_{2, 1}$ characterises the isolator’s transmission. Results are reported as a level, in dB, re the reference stiffness $k_{0} = 1$ N/m (ISO 10846-2 and -3, 3.17):

L_{k} = 10 lo g_{10} (\frac{∣ k _{2, 1} ∣ ^{2}}{k _{0}^{2}}) = 20 lo g_{10} (\frac{∣ k _{2, 1} ∣}{k _{0}})

and, in the low-frequency range where inertial forces in the element are negligible, the loss factor is the tangent of the phase angle of $k_{2, 1}$ (ISO 10846-1, 3.8): $η = Im (k_{2, 1}) / Re (k_{2, 1})$ .

Two laboratory methods determine $k_{2, 1}$ :

Direct method (ISO 10846-2): measure the blocked output force F2,b and the input displacement u1 directly: $k_{2, 1} = F_{2, b} / u_{1}$ .
Indirect method (ISO 10846-3): load the output with a compact blocking mass m2 and measure the vibration transmissibility $T = u_{2} / u_{1}$ ; the blocking force is the mass’s inertia force (ISO 10846-3, Equation 1): $k_{2, 1} = - (2 π f)^{2} (m_{2} + m_{f}) T$ for $T ≪ 1$ , where mf is the mass of the output flange of the test element. The approximation is valid only where $∣ T ∣ \leq 0.1$ (Inequality (2): $Δ L_{1, 2} \geq 20$ dB) and while the blocking mass still behaves rigidly, $10 lo g_{10} (m_{2, eff}^{2} / m_{2}^{2}) \leq 1$ dB (Inequality (3)); see transfer_stiffness_indirect.

The dynamic transfer stiffness is a member of the frequency-response-function family (ISO 10846-1, Annex A / Table A.2): $k = j ω Z = - ω^{2} m_{eff}$ , so it converts to mechanical impedance and effective mass through phonometry.convert_frf ("dynamic_stiffness" <-> "impedance" <-> "apparent_mass"). This module feeds the structure-borne source and building prediction standards (ISO 9611, EN 15657, EN 12354-5).

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base_transmissibility

base_transmissibility(
    frequency: ArrayLike,
    mass: float,
    stiffness: float,
    damping: float = 0.0,
) -> np.ndarray

Transmissibility of a mass on an ideal resilient element (model).

The output mass m on a massless Kelvin-Voigt element (spring k in parallel with a viscous damper c) driven at the input has the base-excitation transmissibility

T = \frac{u _{2}}{u _{1}} = \frac{k + j ω c}{k - ω ^{2} m + j ω c}

This ideal-element model is the counterpart of the indirect-method test arrangement (ISO 10846-3): feeding T into transfer_stiffness_indirect with the same mass recovers the element’s transfer stiffness $k + j ω c$ in the high-frequency limit $T ≪ 1$ .

Parameters

Name	Description
`frequency`	Frequency `f`, in hertz (scalar or array).
`mass`	Output mass `m`, in kg.
`stiffness`	Element stiffness `k`, in N/m.
`damping`	Viscous damping `c`, in N.s/m (Default: 0.0).

Returns: The complex transmissibility T.

blocking_force_ratio

blocking_force_ratio(
    driving_point_stiffness: ArrayLike,
    termination_stiffness: ArrayLike,
) -> np.ndarray

Ratio of the delivered force to the blocking force (ISO 10846-1, Eq. 6).

For an isolator driving a receiving structure, the output force for a given source displacement u1 is $F_{2} = k_{2, 1} u_{1} / (1 + k_{2, 2} / k_{t})$ (Equation (6)), where k2,2 is the isolator’s output driving-point stiffness (output blocked at the input) and kt the dynamic driving-point stiffness of the termination. This function returns

\frac{F _{2}}{F _{2, b}} = \frac{1}{1 + k _{2, 2} / k _{t}}

the factor by which the delivered force deviates from the blocking force $F_{2, b} = k_{2, 1} u_{1}$ of Equation (7). For $∣ k_{2, 2} ∣ < 0.1∣ k_{t} ∣$ the ratio is within 10 % of unity ( $1/1.1 = 0.909$ at the limit), which is the stiffness mismatch that justifies characterising an isolator by its blocked transfer stiffness alone.

Parameters

Name	Description
`driving_point_stiffness`	Output driving-point stiffness `k2,2` of the isolator (complex, scalar or array), in N/m.
`termination_stiffness`	Driving-point stiffness `kt` of the receiving structure (complex, scalar or array, non-zero), in N/m.

Returns: The complex ratio F2/F2,b.

Raises

Exception	When
ValueError	for a zero termination stiffness.

indirect_transfer_stiffness_result

indirect_transfer_stiffness_result(
    frequency: ArrayLike,
    transmissibility: ArrayLike,
    blocking_mass: float,
    *,
    flange_mass: float = 0.0,
) -> TransferStiffnessResult

Indirect-method transfer stiffness bundled as a TransferStiffnessResult.

See transfer_stiffness_indirect for the ISO 10846-3 validity conditions (Inequalities (2) and (3)); bands with $∣ T ∣ > 0.1$ trigger a PhonometryWarning.

Parameters

Name	Description
`frequency`	Frequencies `f`, in hertz (array).
`transmissibility`	Vibration transmissibility $T = u_{2} / u_{1}$ (complex).
`blocking_mass`	Blocking mass `m2`, in kg (> 0).
`flange_mass`	Output-flange mass `mf`, in kg (Default: 0.0).

Returns: The TransferStiffnessResult (indirect method).

Warns

Warning	When
PhonometryWarning	where any $∣ T ∣ > 0.1$ (Inequality (2) violated).

loss_factor

loss_factor(stiffness: ArrayLike) -> np.ndarray

Loss factor $η = Im (k_{2, 1}) / Re (k_{2, 1})$ (ISO 10846-1, 3.8).

Valid in the low-frequency range where inertial forces in the element are negligible; it is the tangent of the phase angle of the transfer stiffness.

Parameters

Name	Description
`stiffness`	Dynamic transfer stiffness $k_{2, 1}$ (complex, scalar or array, with a non-zero real part), in N/m.

Returns: The loss factor eta (dimensionless).

Raises

Exception	When
ValueError	for a purely imaginary stiffness ( $Re (k_{2, 1}) = 0$ ), for which the loss factor is undefined.

REFERENCE_STIFFNESS

Constant (float).

REFERENCE_STIFFNESS = 1.0

transfer_stiffness_direct

transfer_stiffness_direct(
    blocking_force: ArrayLike,
    input_displacement: ArrayLike,
) -> np.ndarray

Dynamic transfer stiffness by the direct method (ISO 10846-2).

$k_{2, 1} = F_{2, b} / u_{1}$ , the blocked output force phasor over the input displacement phasor.

Parameters

Name	Description
`blocking_force`	Blocked output force phasor `F2,b` (complex), in N.
`input_displacement`	Input displacement phasor `u1` (complex, non-zero), in m.

Returns: The dynamic transfer stiffness $k_{2, 1}$ , in N/m.

Raises

Exception	When
ValueError	for a zero input displacement (dead input channel).

transfer_stiffness_indirect

transfer_stiffness_indirect(
    frequency: ArrayLike,
    transmissibility: ArrayLike,
    blocking_mass: float,
    *,
    flange_mass: float = 0.0,
) -> np.ndarray

Dynamic transfer stiffness by the indirect method (ISO 10846-3, Eq. 1).

$k_{2, 1} = - (2 π f)^{2} (m_{2} + m_{f}) T$ : the blocking force is the inertia force of a compact blocking mass m2 (plus the output flange mass mf), derived from the measured vibration transmissibility $T = u_{2} / u_{1}$ . Valid for $T ≪ 1$ (i.e. well above the mass/spring resonance).

Validity (ISO 10846-3, clause 6). The $T ≪ 1$ approximation of Formula (1) is required accurate within 1 dB, i.e. within 12 % of the calculated stiffness magnitude. This holds only where Inequality (2) is met: $Δ L_{1, 2} = L_{a 1} - L_{a 2} \geq 20$ dB, i.e. $∣ T ∣ \leq 0.1$ (TRANSMISSIBILITY_LIMIT). Bands with |T| above that limit (routine near or below the mass/spring resonance) trigger a PhonometryWarning; treat those bands as outside the valid frequency range of the test arrangement. The upper frequency limit f3 additionally requires the blocking mass to vibrate as a rigid body: results are valid only while its effective mass m2,eff, measured per Formula (4) as $m_{2, eff} = 2 F_{2} / (a_{1}^{'} + a_{1}^{''})$ (two accelerometers spaced $D = S$ across the contact area), stays within 1 dB of the rigid mass, $10 lo g_{10} (m_{2, eff}^{2} / m_{2}^{2}) \leq 1$ dB (Inequality (3), 6.2.3).

Parameters

Name	Description
`frequency`	Frequency `f`, in hertz (scalar or array).
`transmissibility`	Vibration transmissibility $T = u_{2} / u_{1}$ (complex, scalar or array; velocity and acceleration ratios have the same value).
`blocking_mass`	Blocking mass `m2`, in kg (> 0).
`flange_mass`	Output-flange mass `mf`, in kg (Default: 0.0).

Returns: The dynamic transfer stiffness $k_{2, 1}$ , in N/m.

Raises

Exception	When
ValueError	for a non-positive frequency or blocking mass.

Warns

Warning	When
PhonometryWarning	where any $∣ T ∣ > 0.1$ (Inequality (2) violated).

transfer_stiffness_level

transfer_stiffness_level(
    stiffness: ArrayLike,
    *,
    reference: float = 1.0,
) -> np.ndarray

Level of the dynamic transfer stiffness (ISO 10846-2/-3, 3.17).

$L_{k} = 20 lo g_{10} (∣ k_{2, 1} ∣/ k_{0})$ dB, with k0 the reference stiffness.

Parameters

Name	Description
`stiffness`	Dynamic transfer stiffness $k_{2, 1}$ (complex or real, scalar or array, non-zero), in N/m.
`reference`	Reference stiffness `k0` (Default: 1 N/m), in N/m.

Returns: The level L_k, in dB re k0.

Raises

Exception	When
ValueError	for a non-positive reference or a zero stiffness magnitude (a dead channel has no level).

TransferStiffnessResult

TransferStiffnessResult(
    frequencies: np.ndarray,
    transfer_stiffness: np.ndarray,
    blocking_mass: float | None = None,
)

A dynamic transfer stiffness over frequency (ISO 10846).

Attributes

Name	Description
`frequencies`	Frequencies, in hertz.
`transfer_stiffness`	Complex $k_{2, 1}$ per frequency, in N/m.
`blocking_mass`	Blocking mass `m2` used (indirect method), in kg, or `None` for the direct method.

TransferStiffnessResult.level

property

Transfer-stiffness level L_k re 1 N/m, in dB (3.17).

TransferStiffnessResult.loss_factor

property

Loss factor $η = Im / Re$ per frequency (3.8).

TransferStiffnessResult.magnitude

property

Transfer-stiffness magnitude $∣ k_{2, 1} ∣$ , in N/m.

TransferStiffnessResult.plot()

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

Plot the transfer-stiffness level L_k(f).

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

TransferStiffnessResult.report()

TransferStiffnessResult.report(
    path: str,
    *,
    metadata: ReportMetadata | None = None,
    engine: str = 'reportlab',
    verbose: bool = False,
    language: str = 'en',
) -> str

Render a dynamic-transfer-stiffness fiche to a PDF (ISO 10846).

Writes a one-page transfer-stiffness characterisation report for a resilient element: the standard-basis line naming the determination method (direct, ISO 10846-2:2008, or indirect blocking-mass, ISO 10846-3:2002; definition per ISO 10846-1:2008), an optional metadata header, a two-panel body with a compact table of the FRF’s characteristic points (the method, the blocking mass for the indirect method, the frequency range, and the low-frequency stiffness plateau $∣ k_{2, 1} ∣$ , its level L_k and the loss factor eta there) beside the transfer-stiffness level spectrum L_k(f), a boxed low-frequency L_k with the stiffness magnitude and method alongside, and a footer identity/disclaimer block.

Dynamic transfer stiffness is a continuous frequency-response function, so the fiche presents it as a spectrum plus a table of characteristic points; a transfer-stiffness determination is a characterisation, so there is no pass/fail verdict.

Parameters

Name	Description
`path`	Destination path of the PDF file.
`metadata`	Optional `ReportMetadata` supplying the header identity (`specimen` is the tested resilient element) and the footer identity; the `requirement` field is ignored.
`engine`	Rendering back end; only `"reportlab"` is supported.
`verbose`	Accepted for a uniform `.report()` signature; the transfer-stiffness fiche has a single body layout, so it has no effect.
`language`	Fiche language: `"en"` (default, English) or `"es"` (Spanish, with a comma decimal separator).

Returns: The written path as a str.

Raises

Exception	When
ValueError	If `engine` is not `"reportlab"` or `language` is unknown.
ImportError	If reportlab or matplotlib is not installed. The fiche always embeds the `L_k(f)` spectrum, so both are required (`pip install "phonometry[report,plot]"`).

TransferStiffnessResult.to()

TransferStiffnessResult.to(target: str) -> np.ndarray

Convert $k_{2, 1}$ to an FRF (ISO 10846-1 Annex A / Table A.2).

target is "impedance" ( $Z = k / (j ω)$ ) or "apparent_mass" ( $m_{eff} = - k / ω^{2}$ ); see phonometry.convert_frf.

TRANSMISSIBILITY_LIMIT

Constant (float).

TRANSMISSIBILITY_LIMIT = 0.1

Creado y mantenido por José Manuel Requena Plens ORCID Todos los proyectos