Source: https://docs.jmrp.io/phonometry/vibration/structural/transfer-stiffness/ # Dynamic transfer stiffness of resilient elements (ISO 10846) The vibro-acoustic transfer property of a resilient element (a vibration isolator, mount, bellows or hose) is its **dynamic transfer stiffness** $k_{2,1}$, the frequency-dependent ratio of the *blocking force* on the output (receiver) side to the displacement on the input (source) side (ISO 10846-1, 3.7): $$ k_{2,1} = \frac{F_{2,\mathrm{b}}}{u_1} \quad [\mathrm{N/m}]. $$ Because a vibration isolator is only effective between structures of large driving-point stiffness, the force it delivers to the receiver approximates this blocking force (ISO 10846-1, Eq. 7), so $k_{2,1}$ is the quantity that characterises the isolator's transmission. $k_{2,1}$ feeds the structure-borne source and building prediction standards: ISO 9611, EN 15657 and EN 12354-5.

Show the code for this figure

```python import matplotlib.pyplot as plt import numpy as np from phonometry import vibration # Kelvin-Voigt isolator k + jwc loaded by an 8 kg blocking mass. k, c, m2 = 1.0e6, 120.0, 8.0 f0 = np.sqrt(k / m2) / (2.0 * np.pi) f = np.logspace(np.log10(f0 / 5.0), np.log10(f0 * 40.0), 600) k_true = k + 1j * 2.0 * np.pi * f * c t = vibration.base_transmissibility(f, m2, k, c) k_indirect = vibration.transfer_stiffness_indirect(f, t, m2) # warns where T is not small # One line: the indirect determination bundled as a result draws its own # Lk(f) level spectrum: res = vibration.indirect_transfer_stiffness_result(f, t, blocking_mass=m2) res.plot() plt.show() # By hand, the true element stiffness against the indirect-method estimate: fig, ax = plt.subplots() ax.semilogx(f, vibration.transfer_stiffness_level(k_true), label="true $L_k$ of $k+j\\omega c$") ax.semilogx(f, vibration.transfer_stiffness_level(k_indirect), "--", label="indirect method $-(2\\pi f)^2 m_2 T$") ax.axvline(f0, color="0.6", linestyle=":", label="resonance $f_0$") ax.set(xlabel="Frequency [Hz]", ylabel="Transfer stiffness level $L_k$ [dB re 1 N/m]") ax.grid(True, which="both", alpha=0.3) ax.legend() plt.show() ```

## 1. The transfer-stiffness level and loss factor Results are reported as a **level** re the reference stiffness $k_0 = 1\ \text{N/m}$ (ISO 10846-2 and -3, 3.17), 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): $$ L_k = 10\log_{10}\frac{|k_{2,1}|^2}{k_0^2} = 20\log_{10}\frac{|k_{2,1}|}{k_0}, \qquad \eta = \frac{\mathrm{Im}(k_{2,1})}{\mathrm{Re}(k_{2,1})}. $$ ```python from phonometry import vibration # A resilient mount with |k2,1| = 1 MN/m and a 5 % loss factor: k = 1e6 * (1.0 + 0.05j) print(round(float(vibration.transfer_stiffness_level(k)), 2)) # 120.01 dB re 1 N/m print(round(float(vibration.loss_factor(k)), 3)) # 0.05 ``` ## 2. Direct and indirect determination Why a *blocked* force rather than, say, the isolator's transmissibility? Because a transmissibility is a property of a whole assembly: it changes with whatever masses and stiffnesses the isolator happens to connect, so data measured on one rig would not transfer to another installation. The blocked force per unit input displacement is a property of the element alone, and it predicts the force the element delivers to any receiver that is much stiffer than the element itself, which is exactly the situation a vibration isolator is designed for. ISO 10846 therefore sandwiches the isolator between a driven input mass and an output that is either rigidly blocked or loaded with a known mass: The **direct method** (ISO 10846-2) measures the blocked output force and the input displacement, $k_{2,1} = F_{2,\mathrm{b}}/u_1$. The **indirect method** (ISO 10846-3) loads the output with a compact blocking mass $m_2$ and measures the vibration transmissibility $T = u_2/u_1$; the blocking force is then the inertia force of the mass (ISO 10846-3, Eq. 1): $$ k_{2,1} = -(2\pi f)^2\,(m_2 + m_\mathrm{f})\,T \qquad (T \ll 1), $$ with $m_\mathrm{f}$ the mass of the output flange. The approximation is valid well above the mass/spring resonance, where $T$ is small. ```python import numpy as np from phonometry import vibration # Indirect method: a 10 kg blocking mass, transmissibility 0.01 at 500 Hz. k = vibration.transfer_stiffness_indirect(500.0, 0.01, blocking_mass=10.0) print(f"{abs(complex(k)):.3e}") # 9.870e+05 N/m # Bundle a swept measurement into a result carrying its level and loss factor: f = np.logspace(1.5, 3.3, 200) t = vibration.base_transmissibility(f, mass=8.0, stiffness=1e6, damping=120.0) res = vibration.indirect_transfer_stiffness_result(f, t, blocking_mass=8.0) print(round(float(res.levels[-1]), 1)) # 125.1 dB re 1 N/m (high-f) res.plot() # the Lk(f) level spectrum, as in the figure above (needs matplotlib) ``` The `TransferStiffnessResult` carries the complex $k_{2,1}$ and exposes `.levels`, `.loss_factor`, `.magnitude`, `.to("impedance"/"apparent_mass")`, `.band_average()` (section 4) and `.plot()`. **Test-report fiche.** `TransferStiffnessResult.report(path)` renders a one-page dynamic-transfer-stiffness characterisation report for a resilient element (ISO 10846-1:2008 definition; determined by the direct method, ISO 10846-2:2008, or the indirect blocking-mass method, ISO 10846-3:2002). The sheet shows the $L_k(f)$ level spectrum beside a compact table of characteristic points (the determination 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 there), then the one-third-octave band levels $L_{k,\mathrm{av}}$ of section 4 that the test report of both parts presents (ISO 10846-2 9 m), ISO 10846-3 10 j)), and a boxed low-frequency $L_k$. The characteristic points are read at the lowest *valid* line and the spectrum draws the excluded lines apart; a band of fewer than five valid lines prints its line count instead of a level. It is a characterisation, so there is no pass/fail verdict; `language="es"` renders the Spanish fiche. The fiche always embeds the $L_k(f)$ spectrum, so it needs both the report and plot extras (`pip install "phonometry[report,plot]"`). [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso10846_transfer_stiffness_example.pdf) The two methods split the frequency axis between them. The direct method works from 1 Hz, the lower bound of the ISO 10846-2 scope (in practice the floor is set by the rig and its instrumentation), up to where the test rig's own resonances intrude (typically a few hundred hertz for large elements); the indirect method only becomes valid well above the blocking-mass/spring resonance, where the transmissibility is small, and extends the characterisation into the kilohertz range. A full isolator dataset is usually the two spliced together. Hydraulic mounts are outside the single-spectrum description altogether: their stiffness is amplitude-dependent by design, so one $L_k(f)$ does not characterise them. ## 3. Validity of the indirect method ISO 10846-3 (clause 6) requires the $T \ll 1$ approximation to be accurate within **1 dB** (12 % of the stiffness magnitude), which bounds the usable frequency range on both sides: * **Impedance mismatch (Inequality 2).** Valid only where $\Delta L_{1,2} = L_{a1} - L_{a2} \ge 20\ \text{dB}$, i.e. $|T| \le 0.1$, the constant `TRANSMISSIBILITY_LIMIT`. `transfer_stiffness_indirect` computes $|T|$ at every frequency line and emits a `TransferStiffnessWarning` when any line exceeds it (routine near or below the mass/spring resonance, as in the figure above); the result it builds marks those lines as not valid, and its band average leaves them out. * **Rigid blocking mass (Inequality 3).** Above an upper frequency $f_3$ the blocking mass no longer moves as a rigid body; results are valid only while its measured effective mass $m_{2,\text{eff}} = 2F_2/(a'_1 + a''_1)$ (Eq. 4) stays within 1 dB of the rigid mass: $|10\log_{10}(m_{2,\text{eff}}^2/m_2^2)| \le 1\ \text{dB}$. `effective_blocking_mass` finds $f_3$ from that measurement (section 5). * **Linearity (clause 7.6).** Two input spectra 10 dB apart must give transfer-stiffness levels within 1.5 dB. Inequality 2 is computed with the indirect method itself and Inequality 3 through the effective mass of section 5; the linearity criterion (ISO 10846-3 clause 7.6, and clause 7.7 of Parts 2, 4 and 5) is described here and left to the operator, as are the preload, creep and temperature conditioning. The blocking-force idealisation itself is quantified by ISO 10846-1, Eq. (6): for an isolator of output driving-point stiffness $k_{2,2}$ on a termination of stiffness $k_\mathrm{t}$, the delivered force is $F_2/F_{2,\mathrm{b}} = 1/(1 + k_{2,2}/k_\mathrm{t})$, within 10 % of the blocking force for $|k_{2,2}| < 0.1\,|k_\mathrm{t}|$ (Eq. 7): ```python import warnings from phonometry import vibration # |T| = 0.5 violates Inequality (2): the indirect result is flagged. with warnings.catch_warnings(record=True) as caught: warnings.simplefilter("always") vibration.transfer_stiffness_indirect(50.0, 0.5, blocking_mass=10.0) print(caught[0].category.__name__) # TransferStiffnessWarning # Blocking-force approximation at the 10 % limit (ISO 10846-1, Eq. 6): print(round(abs(complex(vibration.blocking_force_ratio(1e5, 1e6))), 4)) # 0.9091 ``` ## 4. One value per one-third-octave band A swept or stepped measurement gives $k_{2,1}$ at hundreds of lines, and every part of the series reports it as one value per one-third-octave band: the squared magnitude averaged over the $n$ lines the band holds (ISO 10846-2 Formula (6), -3 Formula (7), -4 Formula (11), -5 Formula (6)), $$ k_\mathrm{av} = \left\{ \frac{1}{n} \sum_{i=1}^{n} \lvert k_{2,1}(f_i) \rvert^2 \right\}^{1/2}, \qquad L_{k,\mathrm{av}} = 10 \lg \frac{k_\mathrm{av}^2}{k_0^2}, $$ "where the summation is performed over a minimum of $n = 5$ frequencies". Averaging the square rather than the level weights the peaks, and the phase is lost. The five lines are also a requirement on the measurement (the analyser shall resolve at least five distinct frequencies per band), so `band_averaged_stiffness` gives a band of one to four lines no value (NaN) and a `TransferStiffnessWarning`. It assigns each line to the base-ten band that encloses it, names the bands by their ISO 266 centres, and leaves out first the lines that failed an adequacy condition: the indirect result marks the lines with $|T| > 0.1$, so its `.band_average()` averages only the valid part of the sweep. ```python import warnings import numpy as np from phonometry import vibration # Direct method, a Kelvin-Voigt mount k + jwc, lines every 2 Hz: f = np.arange(90.0, 1120.0, 2.0) k21 = 1.0e6 + 1j * 2.0 * np.pi * f * 80.0 direct = vibration.TransferStiffnessResult( frequencies=f, transfer_stiffness=vibration.transfer_stiffness_direct(k21 * 1e-6, 1e-6) ) bands = direct.band_average() print(bands.nominal_frequencies[[0, -1]], bands.line_counts[[0, -1]]) # [ 100. 1000.] [ 12 114] print(round(float(bands.levels[-1]), 2)) # 120.99 dB re 1 N/m # Indirect method on an 8 kg blocking mass: |T| falls to 0.1 at 187.5 Hz, so the # lines up to 186 Hz fail Inequality (2) and never reach the average; the 200 Hz # band averages its 18 lines from 188 Hz up. t = vibration.base_transmissibility(f, mass=8.0, stiffness=1.0e6, damping=120.0) with warnings.catch_warnings(): warnings.simplefilter("ignore", vibration.TransferStiffnessWarning) indirect = vibration.indirect_transfer_stiffness_result(f, t, blocking_mass=8.0) bands = indirect.band_average() print(bands.nominal_frequencies[0], bands.line_counts[0]) # 200.0 18 bands.plot() # the band levels, the undetermined bands marked (needs matplotlib) ``` ## 5. Checking the rig from the measured spectra Three of the adequacy conditions of the series are level differences or a mass, and can be judged from what the rig measured. Each check returns the condition frequency by frequency with an overall `.passes`, draws itself with `.plot()`, and warns (`TransferStiffnessWarning`) where the condition fails. * **The output is blocked**, $\Delta L_{1,2} = L_{a1} - L_{a2} \ge 20$ dB (ISO 10846-5 Inequality (1); Inequality (1) of Part 2, (2) of Parts 3 and 4): `check_blocked_output`. * **The input moves in one direction**, the excitation direction at least 15 dB above each perpendicular one (Part 5 Inequality (2); (3) of Part 2, (5) of Part 3, (7) of Part 4): `check_unwanted_input`, the loudest unwanted direction deciding. ISO 10846-2 7.6.1 sends the exclusion of the lines that fail it to "6.1, Inequality (1)", the blocked-output condition, where 6.4, Inequality (3), is meant (see the [errata](https://docs.jmrp.io/phonometry/reference/errata/)). * **The mass in front of the output force transducers is light enough** for the direct method (ISO 10846-4 Inequality (3), ISO 10846-2 Inequality (2)), $m_0 \le 0{,}06 \times 10^{L_{F2}/20} / 10^{L_{a2}/20}$ kg with $L_{F2}$ re 1 µN and $L_{a2}$ re 1 µm/s², that is $0{,}06\,|F_2|/|a_2|$. On the bound the force levels differ by 0,51 dB (inertia force in phase) to 0,54 dB (against it): the 0,5 dB of NOTE 1, which ISO 10846-4 prints as "05 dB" (see the [errata](https://docs.jmrp.io/phonometry/reference/errata/)). `check_output_mass` returns the limit and that worst-case bias, `bias_bound_db`. For the indirect method, ISO 10846-4 Formula (6) (ISO 10846-3 Formula (4)) measures the effective mass of the blocking mass, $m_{2,\mathrm{eff}} = |2F_2/(a'_1 + a''_1)|$, and its upper limit $f_3$ is the lowest frequency at which it departs from $m_2$ by more than 1 dB, Inequality (5), a departure below 40 Hz being ignored. `effective_blocking_mass` interpolates that crossing. ```python import numpy as np from phonometry import vibration f = [63.0, 125.0, 250.0] check = vibration.check_blocked_output(f, [110.0, 110.0, 110.0], [85.0, 88.0, 86.0]) print(check.holds, check.passes) # [ True True True] True # m0 = 0.4 kg (output flange, force distribution plate, half the transducers): mass = vibration.check_output_mass([125.0], 0.4, [120.0], [100.0]) print(mass.mass_limit_kg, mass.passes, mass.bias_bound_db.round(3)) # [0.6] True [0.355] fe = np.geomspace(20.0, 5000.0, 400) m_eff = 20.0 * (1.0 + (fe / 3000.0) ** 2) ones = np.ones(fe.size, dtype=complex) block = vibration.effective_blocking_mass(fe, m_eff * ones, ones, ones, blocking_mass_kg=20.0) print(round(block.upper_frequency_limit_hz, 1)) # 1047.9 Hz ``` ## 6. The driving-point method (ISO 10846-5) With the output of the element blocked, the input force and the input acceleration give the driving-point stiffness (Formula (3)), $k_{1,1}(f) = F_1/u_1 = -(2\pi f)^2 F_1/a_1$, which equals the transfer stiffness only at low frequencies. The upper limiting frequency $f_\mathrm{UL}$ of clause 6.2 is the lowest frequency at which the driving-point stiffness level is 2 dB below its average from 1 Hz to 20 Hz; up to it the band averages of $k_{1,1}$ stand for those of $k_{2,1}$ within 2 dB (Formula (7)). `driving_point_stiffness` computes $k_{1,1}$, finds $f_\mathrm{UL}$, checks Inequalities (1) and (2) when given the output or the unwanted accelerations, and its `.band_average()` averages only the valid lines at or below $f_\mathrm{UL}$ (8.3: "if $f \le f_\mathrm{UL}$"). The 2 dB for those bands is the standard's statement, which Annex B assumes again (B.3.5); the 6.2 criterion only watches $k_{1,1}$ fall below its own low-frequency value. ```python import numpy as np from phonometry import vibration f = np.arange(1.0, 200.0, 0.2) # the 0.2 Hz lines of 7.5 w = 2.0 * np.pi * f k21 = 1.0e6 * (1.0 + 0.05j) res = vibration.driving_point_stiffness(f, (k21 - w**2 * 2.0) * 1e-6, -(w**2) * 1e-6) print(round(res.upper_limiting_frequency_hz, 1)) # 52.2 Hz # Annex B: u from signal 0.3, instrumentation 0.5, repeatability p/sqrt(3), # test rig 1/(2 sqrt 3), driving-point discrepancy 2/sqrt 3 and linearity # 1.5/(2 sqrt 3) dB, and U = 2u (B.3). budget = vibration.driving_point_uncertainty(119.3, repeatability_range_db=0.6) print(round(budget.combined_uncertainty_db, 3)) # 1.405 dB print(round(budget.expanded_uncertainty_db, 2)) # 2.81 dB ``` The three rectangular terms of the budget are the expressions Annex B prints, 0,289, 1,155 and 0,433 dB. Table B.1 carries them rounded up to one decimal, 0,3, 1,2 and 0,5 dB, the conservative rounding an uncertainty may take (ISO/IEC Guide 98-3, 7.2.6); the library keeps the expressions, and every term can be passed in to reproduce the table. ## 7. Relation to the FRF family The dynamic stiffness is a member of the frequency-response-function family (ISO 10846-1, Annex A / Table A.2): it is the reciprocal of the receptance and relates to the mechanical impedance $Z$ and effective mass $m_\text{eff}$ by $k = j\omega Z = -\omega^2 m_\text{eff}$. These conversions are the same as the [mechanical-mobility](https://docs.jmrp.io/phonometry/vibration/structural/mechanical-mobility/) `convert_frf` pivot: ```python from phonometry import vibration k = 1e6 + 5e4j # N/m, at 250 Hz Z = vibration.convert_frf(k, 250.0, "dynamic_stiffness", "impedance") print(abs(complex(vibration.convert_frf(Z, 250.0, "impedance", "dynamic_stiffness")))) # 1.0012e6 ``` ## 8. Test-report fiche `TransferStiffnessResult.report(path)` renders a one-page dynamic-transfer-stiffness characterisation report for a resilient element (ISO 10846-1:2008 definition; determined by the direct method, ISO 10846-2:2008, or the indirect blocking-mass method, ISO 10846-3:2002). The sheet shows the $L_k(f)$ level spectrum beside a compact table of characteristic points (the determination 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 there), then the one-third-octave band levels $L_{k,\mathrm{av}}$ of section 4 that the test report of both parts presents (ISO 10846-2 9 m), ISO 10846-3 10 j)), and a boxed low-frequency $L_k$. The characteristic points are read at the lowest *valid* line and the spectrum draws the excluded lines apart; a band of fewer than five valid lines prints its line count instead of a level. It is a characterisation, so there is no pass/fail verdict; `language="es"` renders the Spanish fiche. The fiche always embeds the $L_k(f)$ spectrum, so it needs both the report and plot extras (`pip install "phonometry[report,plot]"`). ```python import numpy as np from phonometry import ReportMetadata, vibration # Ten lines in every one-third-octave band from 20 Hz to 2 kHz: freqs = 1000.0 * 10.0 ** ((np.arange(-175, 35) + 0.5) / 100.0) k21 = 1e6 + 1j * (2 * np.pi * freqs) * 80.0 # Kelvin-Voigt element k + jwc u1 = 1e-6 + 0j k = vibration.transfer_stiffness_direct(k21 * u1, u1) # direct method: k2,1 = F2,b/u1 res = vibration.TransferStiffnessResult(frequencies=freqs, transfer_stiffness=k) res.report( "transfer_stiffness.pdf", metadata=ReportMetadata( specimen="Rubber vibration isolator", measurement_standard="ISO 10846-2", ), ) # one-page fiche (needs phonometry[report,plot]) ``` The example fiche is regenerated with `make reports` and kept in the repository. Click the preview to open the PDF: [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso10846_transfer_stiffness_example.pdf) *Dynamic-transfer-stiffness fiche (`TransferStiffnessResult.report`): the FRF characteristic points, the transfer-stiffness level spectrum and its one-third-octave band levels.* ## See also - [Mechanical mobility and the FRF family (ISO 7626-1)](https://docs.jmrp.io/phonometry/vibration/structural/mechanical-mobility/): the source/receiver mobility rule that says when a blocked force is the right description at all, and the FRF vocabulary this page's conversions belong to. - [Structure-borne sound power of equipment (EN 15657)](https://docs.jmrp.io/phonometry/buildings/design/structure-borne-power/): the source characterisation that sits on the other side of the isolator. - [Installed structure-borne sound (EN 12354-5)](https://docs.jmrp.io/phonometry/buildings/design/installed-structure-borne/): the prediction that consumes $k_{2,1}$ together with both mobilities and turns it into a level in a receiving room. - API reference: [`vibration.structural.transfer_stiffness`](https://docs.jmrp.io/phonometry/reference/api/vibration/transfer-stiffness/) and [`vibration.structural.mechanical_mobility`](https://docs.jmrp.io/phonometry/reference/api/vibration/mechanical-mobility/). - Theory: [Point mobilities and radiation efficiency](https://docs.jmrp.io/phonometry/reference/theory/vibration/#point-mobilities-and-radiation-efficiency-cremer-5-hopkins-29): the mobility and impedance definitions the transfer stiffness is the reciprocal of. ## References - Cremer, L., Heckl, M., & Petersson, B. A. T. (2005). *Structure-borne sound: Structural vibrations and sound radiation at audio frequencies* (3rd ed.). Springer. ISBN 978-3-540-22696-3. [doi:10.1007/b137728](https://doi.org/10.1007/b137728). Vibration isolation theory: why an isolator's performance depends on the source and receiver mobilities, the physics behind the blocked-force characterisation. - International Organization for Standardization. (2008). *Acoustics and vibration — Laboratory measurement of vibro-acoustic transfer properties of resilient elements — Part 1: Principles and guidelines* (ISO 10846-1:2008). [iso.org catalogue](https://www.iso.org/standard/38936.html). The principles part of the series: the blocking-force idealisation and the FRF relations this page implements. ## Standards ISO 10846 (parts 1-5), *Acoustics and vibration — Laboratory measurement of vibro-acoustic transfer properties of resilient elements*: the dynamic transfer stiffness $k_{2,1} = F_{2,\mathrm{b}}/u_1$ and its FRF relations (Part 1, clause 5 and Annex A / Table A.2), the level $L_k$ re 1 N/m and the loss factor (Parts 2 and 3, clauses 3.8/3.17), the direct method (Part 2) and the indirect method $k_{2,1} = -(2\pi f)^2 (m_2 + m_\mathrm{f}) T$ (Part 3, Formula 1) with its validity conditions (Part 3, clause 6: Inequalities 2 and 3; clause 7.6 linearity) and the blocking-force approximation (Part 1, Eqs. 6/7). Part 4 (2003) extends the same quantities to elements other than supports, with the output-mass limit of its Inequality (3), the effective blocking mass of its Formula (6) and the band average of its Formula (11); Part 5 (2008) is the driving-point low-frequency method, with Formula (3), the upper limiting frequency of 6.2, Formula (7) and the Annex B budget. Conformance is anchored on the standard's closed-form definitions: the level of a decade of stiffness, the indirect inertia relation, the Table-A.2 identity $k = j\omega Z$, the $|T| = 0.1 \leftrightarrow \Delta L_{1,2} = 20\ \text{dB}$ validity limit and its 1 dB (12 %) accuracy bound, the Eq. (6) force ratio $1/1.1$, and the linearity criterion (7.7 of Part 2, 7.6 of Part 3); Parts 4 and 5 print no worked example, and their rows are closed forms too (a band of identical lines averages to itself, a massless spring gives a flat stiffness, the output mass on its bound biases the force by 0,51 to 0,54 dB, the 0,5 dB of NOTE 1).