Dimensionless complementarity modelling for disengagement screening of a spring-loaded railway pawl latch under vibration and shock

1. Introduction

Rolling stock is secured against spontaneous movement on station, siding, marshalling and dead-end tracks by a portable wheel chock placed against the wheel. The chock is positioned by hand or with a setting fork. In 1520 mm gauge practice the device is commonly called a brake shoe, and that term is used here. In North American regulatory usage the same term usually denotes the friction element acting on the wheel tread, so the device considered here corresponds to a portable wheel chock in that terminology. Traceable patent literature documents rail-mounted safety chocks, automatic anti-theft locking arrangements, remotely released track blocks and portable wheel chocks [1-4]. These mechanisms secure a vehicle or retain the chock on the rail, but they do not provide the self-locking fork-to-chock coupling analysed in this study.

The closest known device combines a brake shoe with a catch fitting and a setting tool. The tool is a lever with a handle at one end and a pin-type receiving element at the other. The pin is introduced into a mating hole in the catch fitting, so the shoe can be positioned behind the wheel and later lifted off while the shunter remains to the side, outside the hazardous zone behind the wagon [4]. The pin-hole joint, however, does not form a mechanical self-locking grip and is held only by the alignment of the pin with the hole. Removal is performed by lifting the shoe rather than by tractive extraction under guaranteed retention, and the joint provides neither automatic latching on forward insertion nor controlled release by rotation. Reliable retention of the shoe while it is pulled off the rail under load is therefore not assured, and precise positioning of the pin relative to the hole is required.

A recently developed kit addresses these limitations. The brake shoe carries a catch assembly on its non-working upper part, with a transverse catch element and an undercut. The working head of the setting fork carries a mating self-locking mechanism with a spring-loaded pawl and a cam surface. The catch element and the pawl form a complementary self-locking joint. The tractive force transmitted through the fork shaft increases the engagement instead of releasing it, so the shoe can be extracted from the rail without the shunter’s hand approaching it. Disengagement requires a deliberate rotation of the fork shaft about its longitudinal axis. That rotation drives the catch element against the cam surface of the pawl and withdraws it from engagement. Because the catch assembly sits above the working sole, the braking properties of the sole are unaffected. The kit reduces the biomechanical load on the hands, shoulder girdle and lumbar region of the shunter and shortens the time spent in the hazardous zone. This is in line with the stated aim of the underlying invention and with computational estimates of the mechanical power expended by train assemblers during manual coupling-related operations [5]. In a broader sense the kit follows the engineering-controls approach to occupational risk reduction, in which the hazard is removed at its source rather than managed by administrative measures alone. Such source-level elimination of exposure is consistent with the international trend, reflected in “Vision Zero”-type safety programmes, towards removing personnel from the hazardous zones of shunting operations.

The static self-locking property of the pawl–catch joint, that is, retention which increases under tractive load and cannot be released by that load alone, has been established analytically from the geometry of the engagement and the friction properties of the mating steel surfaces. Reliable operation of a hand-held railway tool also depends on its behaviour under vibration and shock. Hand-transmitted and vehicle-induced vibration is a recognised occupational hazard for railway workers [6]. The measurement and evaluation of such exposure is standardised by ISO 5349-1 and ISO 5349-2 [7, 8], and limit and action values for hand-transmitted vibration are prescribed in the European Union by Directive 2002/44/EC [9]. The question is therefore whether transverse vibration or shock transmitted through the fork shaft during coupling, transport and extraction could cause a transient loss of contact or a parasitic rotation of the pawl, even though the joint is stable under a purely static axial load. This paper answers that question by treating the latch as a unilateral-contact system with a preload threshold. The engagement geometry is formulated explicitly, the contact conditions are separated into three cases, and the seated case with a lift-off threshold is written as a linear complementarity problem. The harmonic problem is reduced to a criterion in the plane of the frequency ratio $r=\omega /{\omega }_n$ and an excitation severity number $\mathrm{\Lambda }$, and the shock problem is treated as the response to a half-sine acceleration pulse of dimensionless duration $\mathrm{\Theta }$ = ${\omega }_n\tau$. The main contribution is the preload number that governs the transition between the seated and the free behaviour, together with the conservative algebraic screening bound and the guard-banded decision rule that follow from it. The brake-shoe kit is the engineering case study to which the framework is applied.

The contribution of this paper is not the oscillator model of a pawl, which is standard. Nor is it the half-sine shock response, which is classical, or the wedge form of the self-locking condition, which dates from the study of self-locking mechanisms. What is new is the treatment of the latch as a unilateral-contact system with a preload threshold, and the consequences that follow from it. A preload number is introduced that governs the transition between the seated and the free behaviour. The severity number and the preload number are shown to combine into a single decision variable whose uncertainty is smaller than that of either, because the spring rate and the engagement angle cancel in their ratio. The complementarity formulation unifies two results that are usually reported separately, a lift-off margin and a dynamic magnification, and shows that they are the two limits of one model. It also reverses two conclusions that the preload-free treatment suggests: damping has only a weak influence on the Case C boundary within the examined preloaded range, and the governing frequency ratio is sub-resonant. Finally, the screening criterion is stated as a guard-banded decision rule rather than as a comparison of point estimates. Acceptance of a design then depends on the uncertainty of the four quantities that can be measured for a specific installation and excitation scenario. The resulting contribution is a mathematical model for an engineering retention mechanism: it combines unilateral contact, dimensionless reduction, numerical integration and measurement-decision uncertainty in one auditable screening procedure.

2. Design of the self-locking coupling

The catch assembly of the brake shoe is designed as a reinforced bracket with two lateral cheeks, guiding chamfers and a transverse catch element placed above the working surface of the sole (Fig. 1, Fig. 2). The working head of the fork is U-shaped and contains two guiding cheeks, a support hook and a spring-loaded pawl (Fig. 3(a)); the guiding chamfers and cheeks reduce the need for precise alignment when the head is introduced into the catch assembly. On forward motion of the head the transverse catch element is received in the throat formed by the support hook and the guiding cheeks and automatically engaged by the spring-loaded pawl (Fig. 3(b)), after which the tractive force required to extract the shoe from the rail can be transmitted through the fork shaft without manual contact with the shoe.

Fig. 1General view of the kit. 1 – brake shoe, fitted with the catch assembly; 2 – fork, with working head and T-handle

Fig. 2Catch assembly of the brake shoe: a) arrangement on the non-working upper part (1 – support body of the shoe; 2 – sole (working friction surface); 3 – catch assembly; 4 – transverse catch element; 5 – mounting (seating) surface of the catch assembly, for a removable-module version); b) enlarged view (1 – lateral cheeks of the catch assembly; 2 – guiding chamfers; 3 – transverse catch element; 4 – lock undercut)

a)

b)

The retaining surface of the pawl bears on the transverse catch element so as to form an undercut. The undercut is characterised by the angle $\alpha$, defined in Section 3.1 for the actual geometry of the device. The inclination is chosen so that the line of action of the resultant contact force, under tractive load, passes on the engagement side of the pawl pivot. The reaction then produces a moment that presses the pawl against the catch element, so an increase in the tractive force increases the engagement rather than opening the pawl. This is the self-locking principle that governs single-degree-of-freedom self-locking mechanisms with rigid links in general [10-12]. As a local contact approximation, the interface may be treated as an equivalent undercut wedge, for which the static self-locking condition is:

where $\mu$ is the coefficient of friction between the pawl and the catch element. For a typical steel-on-steel pair, $\mu$ = 0.15-0.20, the self-locking angle arctan $\mu$ is about 8.5-11°, which corresponds to the lower part of the claimed undercut range of 5-15°. Adverse operating conditions such as lubrication, oil products, frictional dust, moisture or icing all reduce $\mu$. The design value of $\alpha$ within the claimed range is therefore chosen close to its lower bound, preferably 5-8°, so that the margin given by inequality Eq. (1) survives a reduced effective friction coefficient.

Fig. 3Working head of the fork: a) design (1 – shaft; 2 – working head; 3 – guiding cheeks; 4 – support hook; 5 – spring-loaded pawl; 6 – cam surface of the pawl; 7 – pawl spring; 8 – pivot axis of the pawl; 9 – retaining surface of the pawl); b) locked position on the transverse catch element of the shoe (1 – spring-loaded pawl, engaged with the undercut; 2 – position indicator, signalling full engagement)

a)

b)

Fig. 4Latch unit: a) locked position (1 – spring-loaded pawl; 2 – cam surface; 3 – retaining surface, engaged with the undercut of the catch element; 4 – pawl spring; 5 – pivot axis; 6 – lock undercut, providing self-tightening of the grip under tractive force F); b) deliberately released (rotated) position (1 – spring-loaded pawl, rotated about the pivot axis against the spring, the initial locked position shown by the dashed line; 2 – cam surface, now in contact with the catch element; 3 – retaining surface, withdrawn from the undercut; 4 – position indicator, showing the unlocked state)

a)

b)

A single nominal steel-on-steel value cannot represent all service conditions: experimental studies of static friction show significant sensitivity to surface roughness, traces of lubricant and wear debris [13]. Criterion Eq. (1) is accordingly evaluated over a friction-coefficient interval rather than at a single nominal value. Release of the joint requires two deliberate actions. The residual tractive preload is first relieved by a small forward motion of the shaft. The shaft and the working head are then rotated about the longitudinal axis through 30-90°. That rotation displaces the contact zone from the retaining surface onto the cam surface of the pawl, converts the imposed rotation into a retraction against the spring, and withdraws the retaining surface from the undercut. A free-body confirmation that the contact reaction produces a closing moment under tractive load is given in Section 3 (Fig. 7).

Eq. (1) is written here in its equivalent-wedge form. It is not an independent assumption: Section 3.2 derives the general contact condition for the locked configuration and shows that Eq. (1) is its planar, stationary-contact limit, and Table 1 states the circumstances in which the wedge form ceases to be conservative.

3. Dynamic model of the pawl under vibration and shock excitation

The overall structure of the model developed in this section is summarised in Fig. 5. Eq. (1) guarantees that the joint cannot be opened by a purely axial tractive force; Fig. 4(a) and Fig. 4(b) illustrate the locked and the forcibly unlocked states of the pawl. During shunting the fork–shoe assembly may also be subjected to transverse vibration or shock. It is transmitted through the shunter’s hands to the shaft, through contact of the fork or the coupled shoe with the rail and rolling stock, or through the coupling manoeuvre itself. Vehicle-induced acoustic and vibrational loads of this kind have been shown to reach levels relevant to occupational exposure assessment [6], which gives a plausible order of magnitude for the shock component. These disturbances act as a base excitation at the pivot of the pawl. Eq. (1) excludes opening under a static load, so the question is whether the excitation can produce a transient rotation large enough to reduce the engagement margin. Analogous safety problems arise in pawl-based automotive latch systems, where inertial forces can generate an opening moment about the pawl pivot and cause unintended release; combined theoretical and numerical analyses have been used to evaluate the inertia-load resistance of such latches [14]. Virtual-testing studies have likewise shown that latch failure may involve inertial or vibration-induced activation that conventional quasi-static retention assessment does not reveal [15].

Fig. 5Logical roadmap of the modelling approach, from the latch geometry through the mathematical and dimensionless formulation to the stability criterion, shock analysis and resulting design recommendations. Source: authors’ own work

3.1. Engagement geometry, sign convention and the critical opening angle

The pawl rotates about the pivot axis O. The right-handed frame is defined with the $z$ axis along the pivot and the $x$ axis along the direction of the tractive force $F$ transmitted through the fork shaft. The rotation $\phi$ is measured from the seated position and is positive in the opening direction, that is, in the direction that withdraws the retaining surface of the pawl from the undercut of the catch element. All moments about the pivot are positive in the same sense, so the spring preload moment is written $-M_0$ with $M_0$ > 0.

The geometry is defined in Fig. 6, drawn as a generic parameterised model rather than from the released drawing. The catch element of the kit carries a transverse cylindrical pin, and the retaining surface of the pawl bears on that pin rather than on a plane flank. Two consequences follow and are used below. The contact normal passes through the pin axis P, so its direction is fixed by the angular position of the contact point on the pin and not by a manufactured surface angle. The withdrawal direction w is the common tangent at C, perpendicular to that normal. The angle $\psi$ between the tangential direction of the pawl at C and the withdrawal direction is therefore the angle between the two radii OC and PC, a purely geometric quantity. Three further quantities are read from the figure. The engagement depth $h_e$ is the overlap between the retaining surface and the undercut, measured along the direction of withdrawal. The engagement radius $r_e$ runs from the pivot to the contact point C. The contact position $\alpha$ is measured at the pin axis, from the line of action of the tractive force to the radius PC. Disengagement occurs when the rotation consumes the whole overlap. The velocity of C due to a rotation about O is perpendicular to OC and has magnitude $r_e\dot{\phi }$, and only its projection on the withdrawal direction reduces the overlap. The critical opening angle therefore follows as:

2

${\phi }_{cr}=\frac{h_e}{r_e\mathrm{c}\mathrm{o}\mathrm{s}\psi } .$

Only the ratio of the two lengths enters, so with the relative engagement depth $\eta$ = $h_e$/$r_e$ the critical angle is ${\phi }_{cr}$ = $\eta$/$\mathrm{c}\mathrm{o}\mathrm{s}\psi$. No absolute dimension of the engagement is needed anywhere in the analysis, and none is assumed below.

The equality $\psi$ = $\alpha$ holds only for a plane flank whose contact radius is perpendicular to the withdrawal direction. For the cylindrical pin of this device the two angles differ, as Section 3.2 shows. Eq. (2) replaces the previously assumed value of ${\phi }_{cr}$ by a ratio that can be measured on the manufactured pair without disclosing any absolute dimension.

Fig. 6Engagement geometry of the pawl-catch joint, drawn as a generic parameterised model: a) locked configuration, with the engagement detail enlarged; b) incipient-opening configuration, at which the contact reaction vanishes. O – pivot; P – axis of the cylindrical catch pin; C – contact point; G – centre of mass of the pawl; re, rG, rs – lever arms; he – engagement depth measured along the withdrawal direction w; α – angular position of the contact on the pin, measured from the load line; ψ – angle between the radii OC and PC, equal to the angle between the tangential direction t at C and w; a – pin radius; δ – offset of the pin axis from the line OC; c – pawl-slot clearance; the guiding-cheek clearance is included in the tolerance allowance Δt; Nn and Tt – normal and tangential components of the contact reaction; F – tractive force on the catch element; F0, ks – spring preload and rate. The rotation φ and all moments are positive in the opening sense

a)

b)

In service the effective overlap is smaller than the nominal one. Three contributions are subtracted: the depth lost to wear of the retaining surface and of the seat, ${\mathrm{\Delta }}_w$; the worst-case tolerance stack on the overlap, ${\mathrm{\Delta }}_t$; and the elastic approach under load, ${\mathrm{\Delta }}_e$, comprising the local contact deformation and the deflection of the bracket cheeks. The effective critical angle is:

3

${\phi }_{cr,eff}=\frac{h_e-{\mathrm{\Delta }}_w-{\mathrm{\Delta }}_t-{\mathrm{\Delta }}_e}{r_e\mathrm{c}\mathrm{o}\mathrm{s}\psi }=\kappa {\phi }_{cr},$

where $\kappa$ ≤ 1 is the engagement-retention factor. No absolute dimension is needed at any point below. Eq. (2) contains the two lengths only through their ratio, so the reference value ${\phi }_{cr}$ = 0.10 rad used in Section 4 corresponds to an engagement ratio $h_e$/$r_e$ = $0.10\mathrm{c}\mathrm{o}\mathrm{s}\psi$, that is, an engagement depth of about one tenth of the engagement radius. The retention factor is a ratio in the same way: an allowance for wear, tolerance stack and elastic approach amounting to 27 % of the engagement depth gives $\kappa$ = 0.73 and ${\phi }_{cr,eff}$ = 0.073 rad. Both quantities are stated as ratios throughout, so the results transfer to any set of absolute dimensions that satisfies them, and the paper does not depend on the released drawing.

3.3. Lumped-parameter model of the pawl

The pawl is represented as a rigid body of mass m and mass moment of inertia $J$ rotating about its pivot axis O (Fig. 7). This follows the lumped-parameter approach used for single-degree-of-freedom self-locking mechanisms [10], for dynamic modelling of self-locking components under non-stationary excitation [16], and for latch-mediated spring-actuated systems more generally [17]. Dynamic modelling of self-locking parking pawls has similarly shown that pawl reliability should be evaluated at the system-dynamics level, and that wear-sensitive contact geometry may substantially influence the engagement behaviour [18]. The pawl is restrained by its compression spring, which provides a preload moment $M_0=F_0{r}_s$ about the pivot in the closing direction and an incremental restoring moment for rotations $\phi$ from the locked position. An equivalent viscous damping $c\phi$ represents internal friction and the effect of sand, frictional dust or grease in the mechanism. By the engagement geometry, the contact moment about the pivot produced by the reaction at C, $M_C=\left[rC\times \left(Nn+\mu Nt\right)\right]z$, is of closing sign under tractive load (Fig. 7), so the analysis below considers the excitation acting against the spring alone. The latch is a unilateral-contact system: while the opening component of the disturbing moment stays below the combined closing preload and contact moments, the pawl remains seated and no relative rotation occurs. The linear oscillator model that follows neglects this threshold and assumes that relative rotation can begin immediately under the disturbance torque; it is retained as an upper bound and is identified as Case B in Section 3.4. For translational base excitation transmitted to the working head, linearization about the locked equilibrium gives:

10

$J\ddot{\phi }+c\dot{\phi }+k_{\phi }\phi =M_b\left(t\right),$

11

$M_b\left(t\right)=-{\left[{\mathbf{r}}_G\times m{\mathbf{a}}_b\left(t\right)\right]}_z,$

where $k_{\phi }$ is the torsional stiffness of the latch referred to the pivot and $r_G$ is the position vector of the pawl centre of mass $G$ relative to the pivot. The term $a_b\left(t\right)$ is the translational acceleration of the working head, and the subscript $z$ denotes the component about the pivot axis. The peak inertial moment is therefore bounded by $M_{b,\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{ }}$ ≤ $m{r}_G{a}_{max}$, where $a_{max}$ is the peak transverse acceleration. For angular excitation of the working head the disturbing moment is $M_{b\left(t\right)}=-J\ddot{\theta }b\left(t\right)$ instead. Translational and angular excitation must not be mixed in a single scalar estimate. The translational case is analysed here because it is the one directly associated with measured hand-transmitted accelerations. The spring preload and the spring rate play distinct roles. The preload sets the threshold moment $M_0$ that an opening disturbance must overcome before the pawl can leave its seat, whereas the incremental stiffness governs the response after relative rotation begins. Under the small-angle approximation, and for an approximately constant effective spring lever arm $r_s$, the equivalent torsional stiffness referred to the pivot is:

12

$k_{\phi }={\left.\frac{d{M}_s}{d\phi }\right|}_{\phi =0}\approx k_s{r}_s^2,$

13

${\omega }_n = \sqrt{\frac{k_{\phi }}{J}}, f_n = \frac{1}{2\pi }\sqrt{\frac{k_{\phi }}{J}},$

where ${\omega }_n$ and $f_n$ are the undamped natural circular frequency and natural frequency of the pawl about its locked position. The free-body and dynamic model is shown in Fig. 7.

To expose the governing dimensionless parameter groups explicitly, the angular displacement and time are normalised as $q$ = $\phi$/${\phi }_{cr}$ and $s$ = ${\omega }_{n}t$, respectively. For harmonic base acceleration $a_b\left(t\right)=a_0\mathrm{s}\mathrm{i}\mathrm{n}\omega t$, Eq. (10) then takes the dimensionless form:

14

$q^{″}+2\zeta q^{\text{'}}+q=-\mathrm{\Lambda }\mathrm{s}\mathrm{i}\mathrm{n}\left(rs\right),$

where primes denote differentiation with respect to the dimensionless time $s$, $r=\omega$/${\omega }_n$ is the frequency ratio, $\zeta =c/\left[2\right({Jk}_{\phi })1/2$] is the damping ratio, and $\mathrm{\Lambda }={mr}_G{a}_0$/($k_{\phi }{\phi }_{cr}$) is the dimensionless excitation severity number. Thus, the harmonic response of geometrically different pawl mechanisms is governed by only three dimensionless parameters, $r$, $\zeta$ and $\mathrm{\Lambda }$.

Fig. 7Free-body and dynamic model of the spring-loaded pawl under transverse base acceleration ab(t): O – pivot axis; C – contact point at the undercut; G – centre of mass; N, μN – normal and friction components of the contact reaction; Ms(φ) – spring moment (preload M0 plus incremental restoring moment kφφ); F – tractive force on the catch element Source: authors’ own work

The previous formulation contains four dimensionless groups. A fifth is required, because the pawl is pressed onto its seat by the spring and cannot move until the opening moment overcomes the preload. Introducing the preload moment $M_0$ = $F_0{r}_s$, the preload number is:

15

$\mathrm{\Pi }=\frac{M_0}{k_{\phi }{\phi }_{cr}},$

that is, the ratio of the spring preload moment to the moment required to rotate the pawl through the whole engagement against the incremental stiffness. It is the group that distinguishes the contact cases of Section 3.4: $\mathrm{\Pi }$ → 0 recovers a preload-free oscillator, and $\mathrm{\Pi }$ → ∞ a rigidly seated pawl. Two identities follow. The ratio of the severity number to the preload number is:

16

$\frac{\mathrm{\Lambda }}{\mathrm{\Pi }}=\frac{m{r}_G{a}_0}{M_0},$

that is, exactly the ratio of the peak inertial moment to the preload moment. Eq. (16) is independent of $k_{\phi }$ and of ${\phi }_{cr}$, so the seating decision requires only the pawl mass, the mass-centre offset, the installed preload force and the spring lever arm. Replacing ${\phi }_{cr}$ by ${\phi }_{cr,eff}$ = ${\kappa \phi }_{cr}$ multiplies both $\mathrm{\Lambda }$ and $\mathrm{\Pi }$ by 1/$\kappa$, so their ratio is unaffected by wear and tolerances.

The steady-state amplitude $\mathrm{\Phi }$ of the pawl rotation follows from Eqs. (10-12) as:

17

$\mathrm{\Phi }=\frac{m{r}_G{a}_0}{k_{\phi }}D\left(r, \zeta \right), D\left(r, \zeta \right)=\frac{1}{\sqrt{{\left(1 - r^2\right)}^2 + {\left(2\zeta r\right)}^2}}.$

The factor $D$($r$, $\zeta$) is the dynamic magnification factor for harmonic torque excitation of the rotational oscillator; the term “transmissibility” is deliberately avoided, since the classical base-motion transmissibility contains an additional (2$\zeta r$)² term in the numerator and refers to a different response quantity. Accidental unlatching requires the retaining surface of the pawl to leave the undercut, i.e. the rotation amplitude must reach the critical opening angle ${\phi }_{cr}$ determined by the engagement depth, so the condition for stable retention under harmonic excitation is $\mathrm{\Phi }<{\phi }_{cr}$. Recalling the excitation severity number introduced in Eq. (14):

18

$\mathrm{\Lambda }=\frac{m{r}_G{a}_0}{k_{\phi }{\phi }_{cr}},$

the retention condition against vibration-induced disengagement takes the compact dimensionless form:

19

$\mathrm{\Lambda }D(r, \zeta ) < 1,$

and the angular stability factor may be defined as $S_{\phi }={\phi }_{cr}/\mathrm{\Phi }=1/\left[\mathrm{\Lambda }D\left(r,\zeta \right)\right]$. For each damping ratio this defines a stability boundary in the $\left(r,\mathrm{\Lambda }\right)$ plane, that is, a dynamic stability map of the latch. Combinations of design and excitation lying below the boundary retain the shoe, whereas the region above the boundary near resonance corresponds to potential disengagement. The criterion neglects both the preload threshold $M_0$ and the closing contact moment $M_C$, so any positive preload and any positive tractive force shift the actual behaviour further into the stable region.

Eq. (19) holds for a pawl that possesses a relative degree of freedom at all times, that is, for a latch whose installed preload has been lost. It is retained below as Case B and is not a description of the nominal latch.

4. Results and discussion

The values below are not manufacturing data and are not taken from the released drawing of the kit. They are a nominal set, chosen to be mechanically feasible and representative of hand-held railway tooling of this class. Their only purpose is to place the device within the dimensionless domain over which the criteria of Section 3.4 are evaluated. The parametric ranges are the more important entries, since every conclusion is reported over them rather than at the nominal point. Four quantities are taken from published sources rather than assumed: the mass and the length of the setting fork and the mass of the brake shoe, from the technical specifications cited in Section 1, and the frequency band and the shock severity, from ISO 5349-1 and IEC 61373. Every other entry of Table 5, and the engagement ratios of Section 3.1, is assumed for the worked example.

The values of Table 5 are design-stage estimates, and Table 6 assigns to each a relative standard uncertainty together with its sensitivity coefficient in the derived groups. No measurement of any parameter was carried out for the present study. The entries of Table 6 are therefore declared inputs, of the order that drawing tolerances and spring tolerance grades produce for components of this class, and none of them is a measurement result. They are stated explicitly so that the propagation can be reproduced and so that a reader can substitute the values appropriate to a particular design. Because they are declared rather than measured, the analysis is presented in two forms below: the propagation for the assigned values, and the inverse question of how small the uncertainty would have to be to change the decision. Table 6 assigns relative standard uncertainties to the available design parameters and reports their sensitivity coefficients. No experimental uncertainty is assigned to the prescribed acceleration amplitudes or to the assumed grip-to-pivot transfer factors. The resulting combined values are therefore conditional uncertainty estimates rather than a complete measurement uncertainty budget.

The relative standard uncertainties of the prescribed acceleration amplitudes and the grip-to-pivot transfer factors were not assigned because no calibrated measurements were available. Therefore, the combined uncertainties reported in Table 6 are conditional values that exclude the uncertainties of $a_0$, $a_p$, $\beta$, and ${\beta }_p$. They must not be interpreted as a complete measurement uncertainty budget.

The combined value for the natural frequency corresponds to $f_n=$ 66.9 Hz with an expanded uncertainty of 11.1 Hz at $k=$ 2, that is a plausible interval of roughly 56 to 78 Hz. That interval is wide enough that the position of the pawl mode relative to the shaft modes of Table 3 cannot be settled by calculation alone.

For the reference set, Eq. (13) gives ${\omega }_n=$ 420 rad/s, i.e. $f_n$ ≈ 67 Hz. Across the full parametric domain of Table 5 the natural frequency ranges from about 13 Hz (heaviest pawl with the softest spring) to about 165 Hz (lightest pawl with the stiffest spring). This spread confirms that the latch cannot, in general, be assumed to operate far below resonance for all admissible designs, and justifies the dimensionless treatment of Eqs. (17-19) in place of a blanket $r$ ≪ 1 assumption.

Fig. 8 shows the dynamic magnification factor $D$($r$, $\zeta$) for $\zeta$ = 0.05, 0.10 and 0.20. For the reference design, the 16-50 Hz band associated with pronounced hand–arm resonant response [19] corresponds to $r$ = 0.24-0.75, where $D$ ≤ 2.14 for $\zeta$ = 0.10. The excitation severity number of the reference case is $\mathrm{\Lambda }$ = ${mr}_G{a}_0$/($k_{\phi }{\phi }_{cr}$) = 0.072 at $a_0$ = 30 m/s², so that $\mathrm{\Lambda }D$ ≤ 0.154 within this band and the angular stability factor is $S_{\phi }$ ≥ 6.5. Even for sustained excitation exactly at resonance ($r$ = 1), $\mathrm{\Lambda }D$ = 0.72, 0.36 and 0.18 for $\zeta$ = 0.05, 0.10 and 0.20 respectively, i.e. $S_{\phi }$ = 1.4-5.6, and the criterion of Eq. (19) is satisfied throughout. The damping ratios considered here are scenario values rather than experimentally identified properties of the latch. At exact resonance, Eq. (19) reduces to $\mathrm{\Lambda }$/(2$\zeta$) < 1, so that the preload-free reference case is dynamically stable only when $\zeta$ > $\mathrm{\Lambda }$/2. For the reference excitation severity $\mathrm{\Lambda }$ = 0.072, the critical damping ratio is ${\zeta }_{cr}$ = 0.036. Thus, the lowest analysed value $\zeta$ = 0.05 remains above the theoretical stability threshold, whereas a nearly undamped degraded mechanism with $\zeta$ < 0.036 could reach the disengagement boundary under sustained resonant excitation. This result identifies damping as an additional parameter to be determined during prototype modal testing. The corresponding quasi-static rotation ${mr}_G{a}_0$/$k_{\phi }$ = 0.0072 rad (0.41°) is small compared with ${\phi }_{cr}$ = 0.10 rad (5.7°). The computed characteristics are summarised in Table 9.

Fig. 8Dynamic magnification factor D(r, ζ) of the pawl for three damping ratios; the shaded band marks the frequency-ratio interval corresponding to 16-50 Hz for the reference design (fn = 67 Hz)

The complete picture is given by the dynamic stability map of the latch (Fig. 9): the curves $\mathrm{\Lambda }D\left(r,\zeta \right)=$ 1 separate the region of stable retention from the region of potential disengagement in the ($r$, $\mathrm{\Lambda }$) plane. The reference operating segment ($\mathrm{\Lambda }$ = 0.072, $r$ = 0.24-0.75) and the worst-case resonance point lie deep inside the stable region for all three damping ratios. At resonance with $\zeta$ = 0.05, disengagement of the preload-free reference model would require $\mathrm{\Lambda }$ ≥ 0.1, corresponding to a harmonic acceleration amplitude $a_0$ ≥ 42 m/s² sustained exactly at $f_n$. This value should not be interpreted as a direct occupational-exposure threshold, because $a_0$ is the unweighted single-frequency acceleration amplitude applied to the latch housing, whereas Refs. [19, 20] concern hand-arm biodynamic response and transmitted vibration. The comparison therefore serves only to identify the resonance sensitivity of the latch model; prototype measurements are required to establish the actual excitation spectrum at the working head. Away from resonance, the admissible $\mathrm{\Lambda }$ increases rapidly.

Fig. 9Dynamic stability map of the self-locking latch: boundaries ΛD(r, ζ) = 1 for ζ = 0.05, 0.10 and 0.20; the shaded region above the boundaries corresponds to potential disengagement. The thick segment marks the reference design (Λ = 0.072) over the 16-50 Hz band; the open circle marks sustained excitation at resonance

Figs. 8 and 9 describe Case B. The corresponding boundaries for the complete formulation, Eq. (22), were obtained by integrating the complementarity model with an explicit fourth-order Runge-Kutta scheme. The step was fixed at $\mathrm{m}\mathrm{i}\mathrm{n}(2\pi ,2\pi /r)/400$ in dimensionless time, and the seated and free phases were switched on the conditions of Eq. (20) and on the sign of q. For the harmonic case the response was integrated over 90 excitation cycles and the peak taken over the last 35 % of the record. The scheme reproduces the closed-form Case B results to five significant figures. It returns the resonant amplitude $1/2\zeta$ and the peak shock amplification $A\mathrm{ }$= 1.647, 1.542 and 1.374 at $\mathrm{\Theta }$ = 5.2-5.5 for $\zeta$ = 0.05, 0.10 and 0.20, consistent with published response spectra for half-sine impact pulses [27].

Two features of Table 7 change the qualitative picture. The Case C boundary is between four and twenty-three times higher than the Case B boundary, and the gap is widest exactly at resonance. More importantly, the damping ratio has only a weak influence within the examined Case C range: at $r$= 1 a factor of four in $\zeta$ changes the boundary by 3.6 %. The reason is structural – a preloaded pawl is seated over most of the excitation cycle, so no resonant build-up can occur and the excursion is limited by the impulse delivered while $\mathrm{\Lambda }f\left(s\right)$ exceeds $\mathrm{\Pi }$. The critical damping ratio ${\zeta }_{cr}$ = $\mathrm{\Lambda }$/2 is therefore a property of Case B alone. The same effect moves the worst case away from resonance: in Case C the minimum of ${\mathrm{\Lambda }}_{crit}$ lies in the sub-resonant range $r$≈ 0.2-0.5, where the excitation is slow enough for the pawl to follow it quasi-statically after lift-off. Table 8 gives this minimum as a function of the preload number, together with the quasi-static bound $\mathrm{\Pi }+1$ obtained by requiring the static opening moment to exceed the preload plus the incremental restoring moment at full engagement.

Over the whole range examined the computed boundary never falls below 0.64 of the quasi-static bound, which supports a conservative algebraic screening bound requiring no further integration. Restoring the engagement-retention factor of Eq. (3), and recalling that $\kappa$ multiplies the additive term only, the criterion for retention in Case C is:

valid over 0.1 ≤ $\mathrm{\Pi }$ ≤ 40, 0.05 ≤ $\zeta$ ≤ 0.20, 0.1 ≤ $r$≤ 2 and $e$= 0. For $\mathrm{\Pi }$ below about 0.1 the seating threshold ceases to protect the mechanism and Case B must be used. The transition is abrupt. At $\zeta$ = 0.05 and $r$= 1 the boundary rises from $\mathrm{\Lambda }$ = 0.20 at $\mathrm{\Pi }$ = 0 to $\mathrm{\Lambda }$ = 0.81 at $\mathrm{\Pi }$ = 0.1, so even a small installed preload raises the disengagement threshold by roughly a factor of four.

For the nominal set the preload number is $\mathrm{\Pi }$ = 3.33, so $\mathrm{\Lambda }$/$\mathrm{\Pi }$ = 0.022 under the harmonic excitation of Table 5 and 0.072 under the reference pulse. The seating Eq. (20) is satisfied throughout and the pawl does not leave its seat, so the peak rotation predicted by Case C is identically zero for both excitations. The rotations reported below for Case B describe the preload-free idealisation only. The informative indicator for the nominal configuration is therefore the lift-off acceleration of Eq. (21), which here gives $a_{lift}$ = 1389 m/s², about 142 g. Over the parametric domain of Table 5 the indicator is far less comfortable. The ratio $\mathrm{\Lambda }/\mathrm{\Pi }=m{r}_{G}a/\left(F_0{r}_s\right)$ is maximised by the heaviest pawl with the largest mass-centre offset, combined with the smallest preload force and the shortest spring lever arm. That corner, $m$= 0.15 kg, $r_G$ = 0.030 m, $F_0$ = 50 N and $r_s$ = 0.015 m, gives $a_{lift}$ = 167 m/s². Against the bounding pulse $a_p$ = 150 m/s² the ratio ${\mathrm{\Lambda }}_p/\mathrm{\Pi }$ reaches 0.90, so the no-lift-off margin falls from 13.9 at the nominal point to 1.11 at the corner. This identifies the preload moment $M_0=F_0{r}_s$, rather than the damping ratio, as the parameter that governs the design.

The shock response is summarised in Fig. 10. Fig. 10(a) gives the dimensionless amplification $A\left(\mathrm{\Theta },\zeta \right)$ of the peak pawl rotation as a function of the half-sine pulse duration $\mathrm{\Theta }={\omega }_n\tau$. The response is impulse-limited for short pulses. It reaches maxima of 1.65, 1.54 and 1.37 for $\zeta$ = 0.05, 0.10 and 0.20 at $\mathrm{\Theta }$ ≈ 5.2-5.6, that is $\tau$ ≈ 12-13 ms for the nominal $f_n$, and tends to the quasi-static value A → 1 for long pulses. For the worked example and $a_p$ = 100 m/s², the shock severity number is $\mathrm{\Lambda }p=$0.24 and the peak rotation at the most unfavourable pulse duration is ${\phi }_{max}$ = 0.037 rad = 0.37${\phi }_{cr}$, shown in Fig. 10(b). This gives $S_{\phi ,shock}=$ 2.7 for $\zeta$ = 0.10 and 2.5 for $\zeta$ = 0.05, again with the stabilising preload and tractive load neglected. Even at the bounding value of the parametric domain, $a_p$ = 150 m/s² with the same stiffness and geometry, the peak rotation stays below 0.56${\phi }_{cr}$.

Fig. 10Response of the pawl to a half-sine shock pulse: a) dimensionless peak-rotation amplification AΘ,ζ versus dimensionless pulse duration Θ=ωnτ; b) time history of the normalized rotation φ(t)/φcr for the reference parameter set at the most unfavourable pulse duration (Θ= 5.4, ζ = 0.10, ap = 100 m/s2)

a)

b)

The worst pulse duration in Case C is again not the one that maximises the oscillator amplification. In Case B the peak occurs at $\mathrm{\Theta }$ ≈ 5.2, whereas the minimum of ${\mathrm{\Lambda }}_{p,crit}$ in Case C lies near $\mathrm{\Theta }$ ≈ 15. Those are pulses long compared with the natural period, for which the pawl remains off its seat long enough to approach the quasi-static excursion.

Clearance-induced impacts are representable within Eq. (22) through the coefficient of restitution of the $r_{\mathrm{e}}$-seating law. Raising e allows the pawl to rebound and to accumulate rotation over successive cycles, with the effect shown in Table 11.

As the restitution coefficient increases the disengagement threshold falls towards the lift-off threshold $\mathrm{\Lambda }$ = $\mathrm{\Pi }$ = 3.33 itself, and for $e$≥ 0.6 the two coincide over most of the frequency range. Vibro-impact therefore does not lower the threshold at which the pawl first leaves its seat, but it removes the post-lift-off reserve, which for the reference design is about 18 % of the threshold. Under sustained vibration with hard, lightly damped contacts the governing criterion reduces to the seating condition ${\mathrm{\Lambda }}_{eff}$ < $\mathrm{\Pi }$. For shock loading the same computation gives ${\mathrm{\Lambda }}_{p,crit}$ = 4.19 at $\mathrm{\Theta }$ = 5.25 and 3.94 at $\mathrm{\Theta }$ = 15 for every value of e between 0 and 0.9, since a single pulse produces one excursion; the shock results of Table 10 are thus insensitive to the vibro-impact assumption. Two mechanisms remain outside the model: impacts against the walls of the pawl slot, which form a two-sided constraint, and broadband random excitation distributed over the modes of Table 3.

These results are consistent with the effect already established on static grounds. Locating the catch assembly on the non-working upper part of the shoe preserves the braking properties of the sole, while the self-tightening geometry prevents release under tractive load. The dynamic model provides a quantitative design margin for the preload-free limiting case. Within the 16-50 Hz comparison band the worked example retains an angular stability factor of at least 6.5 for $\zeta$ = 0.10. At exact resonance the margin becomes damping-dependent, and the theoretical disengagement boundary is reached only when $\zeta$ falls below ${\zeta }_{cr}=$0.036 for that excitation severity. Under half-sine shock loading the corresponding shock stability factor is 2.7 for $\zeta$ = 0.10 at the most unfavourable pulse duration. These quantities are model-based margins for Case B and not occupational-exposure thresholds, and they require prototype-measured excitation spectra for operational validation.

Because the harmonic response is governed solely by the dimensionless group ($r$, $\zeta$, $\mathrm{\Lambda }$), the sensitivity of the margins to the individual physical parameters enters only through $\mathrm{\Lambda }$ and the natural frequency $f_n$. The severity number $\mathrm{\Lambda }$ is first-order proportional to the pawl mass, the mass-centre offset and the base-acceleration amplitude, and inversely proportional to the torsional stiffness and the engagement-loss angle, whereas $f_n$ scales as the square root of stiffness over inertia. A ±20 % change in any single governing parameter therefore alters $\mathrm{\Lambda }$ by at most ±20 % and $f_n$ by at most about ±10 %. The resonant angular stability factor varies as $S_{\phi }=2\zeta$/$\mathrm{\Lambda }$, so its reference value of 2.8 at $\zeta$ = 0.10 ranges between roughly 2.3 and 3.5 and stays above unity. The natural frequency remains within about 60-75 Hz, so the 16-50 Hz comparison band stays sub-resonant and the in-band stability factor of at least 6.5 is preserved. The margins of Case B are thus robust to parameter uncertainties of this order.

Because every criterion is linear in $\mathrm{\Lambda }$, the consequence of an unknown grip-to-pivot amplification can be stated exactly. Table 12 gives the value of $\beta$ at which each criterion is reached for the reference set.

The table reorders the priorities of the study. The preload-free idealisation loses its margin at an amplification of only 2.8, which is plausible for a hand-held tube near one of its own modes, and that is why Case B is labelled an idealisation. The nominally preloaded latch tolerates about 14 under the reference shock. At the worst corner, the lift-off threshold is reached at $\beta$ = 1.11. Thus, amplification exceeding approximately 11 % relative to the assumed value $\beta$ = 1 would cause loss of seating under the bounding pulse.

Applying the decision rule Eq. (30) with $u_{rel}$ = 10.9 % places the acceptance limit on the point estimate at ${\mathrm{\Lambda }}_{eff}$/$\mathrm{\Pi }$ < 0.822, with the outcome shown in Table 13.

At the worst corner, acceptance requires $u_{rel}$< 5.56 %. With only $F_0$ known exactly, the remaining combined uncertainty is 7.35 %, so the criterion is not satisfied. If both $F_0$ and $r_s$ are known exactly, the remaining combined uncertainty is 5.39 %, which gives an upper bound of 0.997 and therefore only marginally satisfies the criterion. This narrow margin would disappear once the uncertainties of the excitation amplitude and the grip-to-pivot transfer factor are included. The present evidence therefore does not justify acceptance of the corner configuration.

A robust remedy is to increase the preload moment from 0.75 to at least 0.82 N·m. Reducing the uncertainty of $F_0$ alone is insufficient; both $F_0$ and $r_s$, together with the applied acceleration and transfer factor $\beta$, must be characterised before a measurement-based decision can be made.

The framework is analytical, no measurement of any parameter was carried out for this study, and the framework has not been experimentally validated. Every quantity reported above is a model prediction conditional on the parameter values of Table 5, on the uncertainties assigned in Table 6 and on the assumption $\beta$ = 1. None is an operational safety limit. The damping ratio, the torsional stiffness and the critical opening angle rest on design estimates, and the friction coefficient on a literature interval. The influence of each is now bounded: $\zeta$ governs Case B directly and weakly affects the computed Case C boundary, $\mu$ enters only the static Eq. (6), and $\phi$cr cancels in the decision variable of Eq. (16). The validation roadmap follows from the analysis itself and is ranked by the value of each measurement. First, the transfer $\beta$ from grip to pivot over 5-100 Hz, from two accelerometers on an instrumented fork, because Table 12 shows that this quantity can change the margin from large to near its threshold. Second, the installed spring data, $F_0=k_s\left(L_{free}-L_{inst}\right)$, together with the force–displacement curve and the lever arm over the engagement angle, from which the preload number and the correction of Eq. (28) follow. Third, the modal frequencies of Table 3 under the actual grip conditions. Fourth, the damping ratio, which ranks last because Table 7 shows that a fourfold change in it moves the Case C boundary by less than 4 %. Comparison of measured peak rotations with the Case C predictions of Eq. (22) is the closing step. Prototype testing of the assembled latch under the spectra of IEC 61373 remains necessary before any indicator reported here is used as a design value.

5. Conclusions

1) The engagement geometry of a spring-loaded pawl latch has been formulated explicitly. The critical opening angle follows from the engagement depth and the engagement radius, Eq. (2), and a retention factor collects the wear, tolerance and elastic-approach allowances, so that the angle is traceable to drawing dimensions rather than assumed.

2) The contact forces and moments have been derived for the locked and the incipient-opening configurations with a stated sign convention. The equivalent-wedge condition is the plane-flank limit of the general Eq. (6). For the cylindrical catch pin of the device the retention condition, exact within the planar rigid-contact model, is $\mathrm{t}\mathrm{a}\mathrm{n}\psi \le \mu$, Eq. (7), in which $\psi$ is the angle between the two contact radii and is the same angle that fixes the critical opening angle; the wedge form is non-conservative whenever $\psi$ exceeds the contact position $\alpha$.

3) The latch has been treated as a unilateral-contact system. Three cases have been separated and are reported separately: the seated latch, the preload-free oscillator retained as an upper-bound idealisation, and the seated latch with a lift-off threshold, formulated as the linear complementarity problem (22).

4) A preload number $\mathrm{\Pi }$ has been introduced. The decision variable $\mathrm{\Lambda }/\mathrm{\Pi }$ is the ratio of the peak inertial moment to the preload moment; it depends on four measurable quantities and not on the stiffness or the engagement angle, and it carries a smaller relative uncertainty than either severity measure separately.

5) Numerical integration of the complementarity model supports the conservative algebraic screening bound $\mathrm{\Lambda }<0.6(\mathrm{\Pi }+\kappa )$ over the ranges examined. Within these ranges the damping ratio moves the boundary by less than 4 % and the governing frequency ratio is sub-resonant, so the critical damping ratio obtained for the preload-free case is a property of that idealisation alone.

6) For sustained vibration with a restitution coefficient above about 0.6 the disengagement threshold collapses onto the seating criterion $\mathrm{\Lambda }<\mathrm{\Pi }$, so the post-lift-off reserve should not be relied upon in design. Within the examined single half-sine pulse model, the predicted shock response is weakly sensitive to the assumed restitution coefficient.

7) The explicitly defined Euler-Bernoulli/Rayleigh shaft model gives a calculated first bending-frequency range of 3.2-62.0 Hz across the stated boundary and tip-mass cases. This range brackets the 16-50 Hz comparison band and demonstrates that the grip-to-pivot transfer factor cannot be fixed at unity without measurement. The elastic-yield estimate indicates that purely elastic shaft twist cannot independently produce the full release rotation; a smaller torsional contribution remains unresolved until the torsional transfer path is measured. The grip-to-pivot amplification is therefore the measurement of highest value, ahead of the damping ratio.

8) All numerical results are model predictions conditional on the parameter values assumed in Table 5 and on the amplification factor being unity. They are screening indicators for design comparison and not operational safety limits. The framework transfers to other pivoted spring-loaded pawl mechanisms provided that retention is characterised by a critical opening angle, that the seat is a one-sided constraint, and that a preload number and an amplification factor can be established for the mechanism concerned; these conditions, rather than the geometry, delimit its applicability.