Mathematical modelling and parametric evaluation of a detachable vibration-damping ergonomic handrail cover for railway shunters

3.1. Hand-transmitted vibration metrics

According to ISO 5349-1 [9], the root-mean-square acceleration measured at the grip is frequency-weighted in third-octave bands from 6.3 to 1250 Hz with the weighting $W_h$, which is close to unity in the 8-16 Hz bands (0.87-0.96) and decreases approximately inversely with frequency above 16 Hz. The weighted accelerations of the three basicentric axes are combined into the vibration total value:

and the daily exposure is normalized to 8 h:

where $T$ is the daily exposure duration and $T_0$ = 8 h. $W_h$ strongly de-emphasizes the range above 100 Hz, where resilient interfaces often have their largest dynamic effect. The supplementary weighting $W_p$ of ISO/TR 18570 [22] gives more weight to higher frequencies. Weighted and unweighted quantities are therefore reported side by side. The standard contact-surface metric is formed from $T_1$, while $T_2$/$T_{base}$ is reported separately as a reduced-order biodynamic response index. The numerical study represents the dominant radial axis. The same equations may be applied independently to other axes only after axis-specific inputs and parameters are identified; the scalar results alone do not determine the three-axis values $a_{hv}$ or A(8).

3.2. Two-degree-of-freedom transmission model

The transmission tract from the handrail to the hand is represented by the lumped model in Fig. 2. Models of this class are routinely applied to transport-machinery units containing compliant elastomer elements, such as rotor systems on polyurethane support layers [23]. Comparable reduced-order vibration models are also used for freight-vehicle suspensions under vibration-induced fatigue [24]. The handrail is an ideal prescribed-motion source because its mechanical impedance is much higher than that of the hand. The first moving body is the rigid shell plus the dynamically participating elastomer mass $m_c$, with absolute displacement $x_1\left(t\right)$. The second is the effective hand-forearm mass $m_h$, with displacement $x_2\left(t\right)$. The cover layer forms stage 1 ($k_c$, $c_c$), and the glove plus palm tissue forms stage 2 ($k_s$, $c_s$). The reference mass ratio $m_c$/$m_h$ is about 0.14, rather than negligible, and produces the shell mode examined in Section 4.

The coordinates in Fig. 2 are measured from static equilibrium along one selected basicentric direction. The numerical results below use the dominant radial direction of the grip. Applying Newton’s second law to each moving body gives the two coupled equations:

Fig. 2Two-degree-of-freedom model of the handrail-cover-glove-hand tract. Each coordinate arrow starts directly on its body. x0t is the prescribed handrail displacement and the sole kinematic excitation; x1t and x2t are the absolute displacements of the shell and effective hand mass along the selected radial axis

In matrix form, with $x=(x_1,x_2)T$, the system reads $M\ddot{x}+C\dot{x}+Kx=f_0\left(t\right)$, where the excitation vector $f_0\left(t\right)=(c_c{\dot{x}}_0+k_c{x}_0,0)T$ collects the terms transferred from the moving base. For harmonic excitation $x_0\left(t\right)=X_0{e}^{i\omega t}$ the response is $X\left(\omega \right)=Z^{-1}\left(\omega \right)F_0\left(\omega \right)$ with the dynamic stiffness matrix $Z\left(\omega \right)=-{\omega }^{2}M+i\omega C+K$. For an elastomer the viscous element is replaced by the hysteretic (complex-stiffness) description, which reproduces the frequency-independent loss factor of rubber-like materials [25, 26]:

where ${\eta }_c$ and ${\eta }_s$ are the loss factors of the cover layer and of the glove–tissue stage. The equivalent viscous damping ratio of an isolated stage is $\zeta =\eta /2$. Solving the two-by-two system by Cramer’s rule gives the two transfer functions in closed form. The transmissibility to the hand and the transmissibility to the shell are, respectively:

where the common denominator is:

The two outputs have different physical meanings. $T_2$ is the modelled motion of the effective hand mass and is used below as a reduced-order biodynamic response. $T_1$ is the motion of the rigid cover shell and can be measured relative to the rail by accelerometers bonded to those bodies. A palm adapter inside a glove under ISO 10819 includes the glove transmission and is not automatically equal to either coordinate in this lumped model. Section 4.4 maps the outputs to the proposed measurements. The undamped natural frequencies follow from $det(K-{\omega }^{2}M)=$ 0:

Whose two roots give the tract mode $f_1$, dominated by the hand on the compliant glove-tissue stage, and the shell mode $f_2$, dominated by the cover mass on the elastomer layer. Setting $m_c$$\to$ 0 in Eq. (8) collapses the denominator and recovers the single-degree-of-freedom result used in earlier treatments of this problem:

The two-degree-of-freedom model therefore contains the earlier description as a special case, and the difference between them is controlled by the mass ratio $\mu =m_c/m_h$. The lumped description itself remains admissible only while the elastomer layer is acoustically thin, that is, while the excitation frequency stays well below the first internal wave resonance of the layer:

where $f_w$ is the first internal shear-wave resonance and $f_{lim}$ = $f_w$/4 is adopted as a conservative screening limit for the lumped layer. For V1-V4, $f_{lim}$ is approximately 202, 253, 417 and 814 Hz. Each predicted shell mode lies below its own limit, but only 8-200 Hz is common to all four variants. Band-synthesized comparisons therefore use 8-200 Hz. Curves and tabulated points above a variant’s $f_{lim}$ are shown only as trend extrapolations and require a distributed-parameter or finite-element model for confirmation.

The compression stiffness of the cover layer over the loaded contact arc is estimated as:

where $A_0$ is the full area of the loaded contact arc (for a palm grip of about 90 mm length on a rail of 30 mm diameter, $A_0$$\approx$ 30 cm²), $\kappa$$\le$ 1 is the relief area ratio, that is the fraction of $A_0$ actually carried by the rib crests of the friction relief, $h$ is the layer thickness and $E_0$ is the Young’s modulus of the elastomer. Because the relief subdivides the layer into low, narrow ribs, the shape factor of the loaded elements remains small and the apparent modulus stays close to $E_0$ [25, 26]. The modulus is estimated from the Shore A hardness $H_A$ by the Gent relation [27]:

which gives $E_0$ = 3.61, 5.52 and 9.35 MPa for 60, 70 and 80 Shore A respectively. The effective mass of the hand coupled at a cylindrical grip is taken as $m_h$ = 2.0 kg. This value is consistent both with the standard body-segment parameters of Winter [28], according to which the forearm-and-hand segment amounts to 0.022 of body mass, about 1.5-1.7 kg for a 70-75 kg worker, and with the apparent mass of the hand–arm system at the palm in the 20-80 Hz region given by the biodynamic data of ISO 10068 [29], namely 1.5-2.5 kg. The adopted value also accounts for the part of the forearm dynamically coupled at the grip. For the glove-and-tissue stage $k_s\approx$5×10⁴ N/m is adopted as an order-of-magnitude estimate [19]; with $m_h$ = 2.0 kg this places the driving-point resonance of the bare hand-rail contact at 25.2 Hz, in agreement with the hand-arm resonance region reported for the palm. Because this single parameter governs the whole conclusion of the study, its influence is examined explicitly in Section 4.6. For damping, high-damping polyurethanes and thermoplastic elastomers have loss factors ${\eta }_c$$\approx$ 0.3-0.6; the hand tissue is more heavily damped and ${\eta }_s$ = 0.6 is adopted. High-loss elastomers and polymer composites for vibration-damping elements of transport vehicles are an area of active development [30, 31], so the upper part of this range is practically attainable.

3.3. Effective mass of the cover

The effective mass $m_c$ of the first body is obtained from a mass budget of the parts that move with the shell. For a cover of 200 mm working length installed on a rail of 15 mm radius, with an elastomer layer of 8 mm and half-shells of 4 mm wall thickness, the polyurethane layer ($\rho$ = 1150 kg/m³) weighs 0.220 kg, the polyamide half-shells ($\rho$ = 1140 kg/m³) 0.143 kg, and the flanges with the fixators about 0.050 kg. Following the Rayleigh procedure for a distributed spring, one third of the layer mass is added to the moving body:

A mass ratio of one seventh is not negligible, and Section 4 shows that it determines the position of the second resonance of the tract. The sensitivity of the results to $m_c$ over the range 0.10-0.45 kg is reported in Fig. 3(c) and Section 4.6.

3.4. Anti-rotation retention model

The anti-rotation function of the friction relief is described by a torque balance. When the fixators are tightened to a total radial clamping force $N$ distributed over the contact arc, the cover is retained on the rail by the friction torque:

where ${\mu }_{eff}$ is the effective friction coefficient of the relieved elastomer-steel pair and $r_r$ is the handrail radius; for a relief forming several distinct contact zones, the products of the zone forces and friction coefficients are summed. Reliable retention requires:

$M_t$ = 10 N·m, ${\mu }_{eff}$ = 0.6, $r_r$ = 15 mm and $n_s$ = 1.5 are used as screening values. They give $N$$\ge$ 1.67 kN. Two M6 flange fixators tightened to 3-5 N·m may generate a total preload of roughly 5-8 kN, but the torque-preload relation has wide scatter and must be measured for the actual joint. At $N$ = 1.67 kN and $\kappa$ = 0.20, the nominal crest pressure $N$/($\kappa A_0$) is about 2.8 MPa. Because long-term admissible pressure has not yet been established for the selected elastomer, temperature and contamination, $p_{adm}$ = 2-3 MPa is treated only as a sensitivity range in Section 4.5. Reducing $\kappa$ lowers stiffness but raises crest pressure, so the two functions are coupled by Eqs. (12), (15) and (16).

3.5. Excitation spectra, reference configuration and evaluation ratios

Three idealized third-octave rail-input spectra are defined over 8-1000 Hz for measurement planning. $S_1$ is flat. $S_2$ is constant to 16 Hz and then proportional to 16/$f$, representing ride- and jerk-dominated motion. S3 is proportional to ($f$/16)^1/2 and capped above 250 Hz, representing impacts and structural ringing. The lumped-model band synthesis uses only the common conservative range 8-200 Hz established by Eq. (11). Full-band workplace exposure must be obtained experimentally. The alternative to the device is a bare handrail gripped through the same glove and palm tissue, represented by:

The bare configuration has a resonance at 25.2 Hz for the adopted parameters. Two output-specific references are required. The modelled hand response with the cover, $T_2$, is compared with the bare-hand response $T_{base}$. The cover-surface motion, $T_1$, is compared with the bare rail surface, whose transmissibility relative to the prescribed rail input is unity. Using $T_{base}$ as the denominator of a $T_1$ ratio would compare different physical outputs and create a spurious result. The two band-synthesized ratios are therefore:

where $aᵢ$ is the rail-input band acceleration. $W_i=W_{h,i}$ gives the weighted ratios and $W_i=$ 1 the unweighted ratios. $R_{surf}$ is the contact-surface quantity relevant to a same-duration, one-axis ISO 5349 comparison. $R_{bio}$ is an internal model-response index and is not itself $a_{hv}$ or A(8). For three-axis exposure, the measured axis-specific accelerations must first be combined as in Eq. (1). Frequency-specific changes are reported as:

4.1. System modes and the stiffness of the cover layer

Table 1 presents four design variants of the cover layer spanning the claimed material and geometry range, from a soft, thick layer with a deep relief (V1) to a hard, thin layer with full contact (V4). The layer stiffness varies by a factor of 32, from 216 to 7014 kN/m. The decisive quantity, however, is not $k_c$ itself but its ratio to the stiffness of the glove-tissue stage: $k_c$/$k_s$ ranges from 4.3 to 140. Because the two stages act in series, the compliance of the tract is set by the softer of them, and the series stiffness therefore stays within 81-99 % of $k_s$ for every variant. The first natural frequency consequently moves only from 25.2 Hz for the bare handrail to 22.6-25.1 Hz with the cover fitted. This explains, and corrects, the illustrative value $f_0$$\approx$ 20 Hz adopted in the patent application [21]: it characterizes the hand on its own tissue rather than any property of the cover. The second natural frequency is new. It is the shell mode, in which the mass $m_c$ oscillates on the elastomer layer, and it falls between 158 and 814 Hz. For the softer variants it lies inside the frequency range that carries the impact content of the excitation. A single-degree-of-freedom model cannot reproduce this mode at all, which is the principal reason for retaining $m_c$.

Fig. 3 maps the main levers acting on $f_2$. Increasing $h$, reducing hardness or $\kappa$, and increasing $m_c$ lower the shell-mode frequency. Reducing mass raises $f_2$ without increasing crest pressure. Each V1-V4 shell mode remains below its own conservative $f_{lim}$, although V4 lies close to that boundary. Section 4.5 therefore treats stiffness and retention as a conditional feasibility problem, while Section 4.6 avoids a full-band ranking until the high-frequency model and material data are validated.

4.2. Transmissibility and insertion loss relative to the bare handrail

Table 2 and Fig. 4 compare the modelled hand response of V1 with the bare-hand response of Eq. (17). Below about 30 Hz both configurations amplify the input, and the cover lowers the tract resonance from 25.2 to 22.6 Hz. The biodynamic insertion loss ${IL}_{bio}$ is therefore –0.1 to –1.4 dB in this range. Between 30 and 100 Hz the curves are close and ${IL}_{bio}$ remains within ±2.2 dB. The 158 Hz shell mode raises $T_2$ above $T_{base}$, giving –5.9 dB at 160 Hz. Values at 250 and 500 Hz illustrate the formal lumped-model roll-off, but exceed the conservative V1 limit $f_{lim}$$\approx$ 202 Hz and are not used in the integrated metrics. These figures describe the internal reduced-order response, not the ISO 5349 contact-surface acceleration.

Table 3 extends the $T_2$ comparison to all four variants at octave-band centre frequencies. At 16, 31.5 and 63 Hz every variant remains close to the bare-hand response. At 125 and 250 Hz the softer variants are influenced by their shell modes, while V4 is close to the bare biodynamic reference. A systematic reduction in $T_2$ appears only at 500 Hz and is largest for V1. Higher damping flattens the modal peaks but slows the high-frequency roll-off (Fig. 4(b)). Entries above the variant-specific $f_{lim}$ values in Section 3.2 are screening extrapolations.

Fig. 4Modelled transmissibility to the effective hand mass: a) V1-V4 and the bare configuration; b) loss-factor sensitivity for V1. Curve segments above each variant’s flim are trend extrapolations

a)

b)

4.3. Frequency-weighted evaluation

Table 4 separates the two comparisons defined by Eq. (18) over the common 8-200 Hz validity range. For the modelled biodynamic response, $R_{bio,Wh}$ is 1.00-1.07 and $R_{bio,unw}$ is 1.00-1.06 across the four variants and three spectra. The contact-surface result is different. $R_{surf,Wh}$ is 1.00-1.06, while $R_{surf,unw}$ ranges from 1.00 to 1.48 because the shell modes alter unweighted motion within the common band. These ranges apply only to the four discrete variants, adopted hand parameters and three idealized radial spectra. Fig. 5 shows the modelled S3 band levels and how both metric families vary with $k_c$/$k_s$. The curves demonstrate why a design optimized only for $T_2$ can amplify $T_1$.

The small change in $R_{bio,Wh}$ is consistent with anti-vibration-glove studies [19, 20], but it must not be presented as a direct prediction of $a_{hv}$ or A(8). The standard quantity is based on measured contact-surface acceleration, represented here by $R_{surf}$ only for the studied radial axis. Three consequences follow. First, the model alone does not justify a general exposure-reduction claim; field measurements over all three axes are required. Second, high loss remains desirable because it limits the shell resonance. Third, $W_h$-weighted, unweighted, peak and $W_p$-weighted results should be reported together so that a small weighted change does not hide high-frequency amplification.

Fig. 5Common-validity evaluation (8-200 Hz): a) Wh-weighted modelled hand-response band levels for S3, comparing T2 of V1 with Tbase; b) correctly referenced biodynamic and contact-surface S3 ratios versus kc/ks. Vertical lines mark V1-V4

a)

b)

4.4. Model outputs and measurement correspondence

Fig. 6 makes the output distinction explicit. A sensor bonded to the cover shell and referenced to a sensor on the rail measures $T_1$. The corresponding bare-surface reference is unity. The modelled hand coordinate $T_2$ must instead be compared with $T_{base}$. For V1, $\left|T_1\right|/\left|T_2\right|$ corresponds to about 26 dB at 125 Hz and 31 dB at 160 Hz. The formal 38 dB result at 250 Hz lies beyond the conservative V1 limit and is shown only as an extrapolated trend. This divergence is caused by inertial decoupling; it is not an ISO 10819 palm-adapter result. In ISO 5349 workplace measurements, the transducer estimates acceleration at the hand-contact surface and is used to calculate $a_{hv}$ and A(8). In ISO 10819 glove testing, the palm adapter lies inside the glove and measures the vibration transmitted through the glove relative to the test handle. The present model combines the glove and palm tissue in stage 2, so its two coordinates cannot reproduce that adapter signal. An additional interface coordinate and separately identified glove and tissue parameters would be required.

Fig. 6Output-specific transmissibilities: cover-shell motion T1 is compared with the bare rail surface (= 1), whereas modelled effective-hand motion T2 is compared with the bare biodynamic reference Tbase: a) V1; b) V4. Shading begins at the conservative variant-specific limit flim

a)

b)

4.5. Necessary stiffness condition and retention constraint

The model gives a useful order-of-magnitude screening condition. Because the cover layer and glove–tissue stage act in series, the cover cannot materially reduce the combined quasistatic stiffness unless its stiffness is comparable with or smaller than $k_s$. Thus the following is a necessary, but not sufficient, condition for appreciable resonance shift:

Anti-rotation retention acts in the opposite direction. Eqs. (15) and (16) require $N$$\ge$$n_s{M}_t/\left({\mu }_{eff}{r}_r\right)$. With $n_s$ = 1.5, $M_t$ = 10 $N$·$m$, ${\mu }_{eff}$ = 0.6 and $r_r$ = 15 mm, the illustrative minimum is 1.67 kN. If the rib crests carry this force over $\kappa A_0$, the admissible long-term contact pressure $p_{adm}$ gives the lower bound:

Substituting Eq. (21) into Eq. (20) eliminates $\kappa$ and yields an explicit upper bound on the modulus of the elastomer, and hence, through the Gent relation Eq. (13), on its Shore hardness:

For $p_{adm}$ = 3 MPa and $h$ = 12 mm, Eq. (22) gives $E_0$$\le$ 1.08 MPa, or about 29 Shore A by Eq. (13). For 2 MPa and 10 mm it gives 0.60 MPa, or about 16 Shore A. These values are sensitivity examples because $p_{adm}$, $M_t$ and ${\mu }_{eff}$ have not yet been measured for the proposed material, temperature and contamination states. An equivalent way to expose that dependence is to solve Eq. (22) for pressure. V1-V4 would require about 12, 15, 31 and 78 MPa, respectively, before Eq. (20) could be met; each exceeds the assumed 2-3 MPa range. Accordingly, the necessary stiffness and retention conditions do not overlap under the adopted assumptions, but the analysis is not a universal impossibility proof. At $k_c$ = $k_s$ the S3 biodynamic calculation gives $R_{bio,unw}$$\approx$ 0.92 and $R_{bio,Wh}$$\approx$ 1.09, which also shows that satisfying the necessary condition would not by itself guarantee a favourable standard exposure result.

Fig. 7Illustrative feasibility map for the adopted retention parameters. Contours show log₁₀(kc/ks) after κ is set by Eq. (21): a) padm = 2 MPa; b) padm = 3 MPa. Hatching marks the necessary condition kc≤ks. The map is a pressure-sensitivity calculation, not a universal material limit

a)

b)

4.6. Parametric sensitivity and design recommendations

The influence of the design parameters on the two natural frequencies is quantified by logarithmic sensitivities:

For $k_c\gg k_s$, the shell mode is approximately $f_2=(1/2\pi )(k_c/m_c)$. Eqs. (9), (12) and (13) then give $S\kappa =S{E}_0=$ +0.5 and $S_h=S_{mc}$ = −0.5. The Shore-hardness sensitivity $S_{HA}$ is about 1.2, 1.6 and 2.5 at 60, 70 and 80 Shore A. Thus one Shore unit changes $f_2$ by roughly 2-3 %. Across the considered ranges, $\kappa$, $h$, hardness and $m_c$ shift $f_2$ by factors of approximately 2.2, 1.6, 1.6 and 2.1. Mass is the only one of these levers that does not directly tighten the contact-pressure constraint. For the tract mode, sensitivity to $k_c$ is small when $k_c$/$k_s$ is large, while sensitivity to $k_s$ approaches 0.5. Raising ${\eta }_c$ from 0.2 to 0.8 reduces the V1 $T_2$ peak and improves its S3 biodynamic ratios, but the corresponding $R_{surf}$ metrics must be checked separately. Varying $k_s$ from 25 to 400 kN/m changes the predicted benefit substantially, confirming that $k_s$ must be identified experimentally rather than treated as a universal hand parameter.

A defensible design objective must control both physical outputs. Let $\theta$ = ($H_A$, $h$, $\kappa$, $m_c$, ${\eta }_c$) belong to the feasible set $\Theta$ defined by retention, pressure and geometry constraints. For the three spectra used here, the combined vibration-neutrality problem is:

where $D_{out}$ is the outer grip diameter, $D_{max}$ its ergonomic and clearance limit, and $f_{2,min}$ the prescribed lower shell-mode bound. The constraint $f_{lim}$($\theta$) $\ge$ 200 Hz keeps every candidate within the common admissible band. Eq. (24) is a discrete minimax screen, not a claimed continuous optimization. On the four-variant 8-200 Hz grid, V4 has the smallest maximum rounded departure (about 2 %) across the twelve ratios in Table 4. This is not a full-band optimum: its shell mode is near 814 Hz, and rankings change when extrapolated bands are included. Final material selection therefore requires measured spectra and a distributed high-frequency model. Table 5 summarizes the provisional guidance.

4.7. Discussion of the expected technical effect

The analysis supports three distinct functions. First, the friction relief provides a calculable anti-rotation mechanism that is absent from the closest prototype [17]. Its predicted margin still depends on the torque–preload relation and on friction under temperature, moisture and contamination. Second, anatomical zoning, drainage, anti-slip texture and low thermal conductivity can improve grip usability without modifying the standard rail. Third, the vibration target is neutrality rather than assumed isolation. Within the common 8-200 Hz band, V4 keeps both output families closest to unity; this does not establish a full-band optimum because its shell mode lies near 814 Hz. A very stiff shell can appear neutral in $T_2$ while amplifying high-frequency $T_1$, and a soft shell can place its mode near 158-207 Hz. The retrofit character remains an operational advantage: the two half-shells are installed without welding, drilling or replacement of the rail. Limitations are the linear lumped model, one-axis radial numerical study, idealized spectra, combined glove-tissue stage, estimated $m_c$, $k_s$, $p_{adm}$, $M_t$ and ${\mu }_{eff}$, and the absence of prototype data. These limitations define the verification tasks below.

4.8. Proposed experimental verification procedure

Workplace measurements should follow ISO 5349-2 [32]. A tri-axial transducer or validated adapter should measure acceleration at the hand-contact surface in the basicentric axes, with instrumentation conforming to ISO 8041-1 [33]. The same operators, wagons and speed ranges should be tested with and without the cover. At least three operators and at least twenty coupling-uncoupling cycles per condition are proposed for the pilot stage. The confirmatory sample size should then be set by an a priori power analysis using the observed between-operator and cycle-to-cycle variance. Report $a_{hwx}$, $a_{hwy}$, $a_{hwz}$, $a_{hv}$, A(8), peaks and third-octave spectra, together with uncertainty. Compact MEMS sensors may be used only after appropriate calibration and uncertainty evaluation [34, 35]. Risk-oriented processing of the recorded vibration signals, of the kind developed for predictive machine diagnostics [36], can be adapted to classify individual coupling events by severity rather than to average them.

A laboratory test should excite a standard handrail section from 8 Hz to at least the shell mode. Comparison with the lumped model is restricted to each variant’s $f_{lim}$ from Eq. (11); a distributed or finite-element model is required above it. One accelerometer is bonded to the rail and another to the rigid cover shell in the palm zone. Their ratio is:

where $a_{cov}$ and $a_{rail}$ are the complex cover-shell and rail acceleration spectra, respectively. Eq. (25) validates $T_1$, $k_c$ and ${\eta }_c$; it does not directly validate $T_2$. $T_2$ requires a calibrated inertial hand surrogate or an identified hand-arm impedance model. The predicted $f_1$ and $f_2$ are the primary modal quantities to confirm. A separate ISO 10819 test should be used when glove transmissibility is required. Its palm-adapter result must not be labelled $T_1$ or $T_2$ unless the model is extended with a glove-palm interface coordinate and independently identified parameters. Finally, a slip-torque test should record the onset of angular motion for smooth and relieved surfaces under dry, wet, cold and contaminated conditions. These tests provide the axis-specific data needed before any three-axis A(8) claim is made.