A reduced-order diagnostic framework for pantograph-contact wire vibration response and reliability-oriented assessment

2.1. Reduced-order interaction model

The railway overhead contact line is a distributed multi-degree-of-freedom dynamic system. Its mechanical response depends on the coupled behaviour of the contact wire, messenger wire, droppers, clamps, supporting structures, and pantograph head. Therefore, a full representation of the system may be written in the general matrix form:

where $x\left(t\right)$ is the displacement vector including the relevant vertical and horizontal components, $M$ is the mass matrix, $C$ is the damping matrix, $K$ is the stiffness matrix, and $F\left(t\right)$ is the vector of external actions, including wind excitation, pantograph-induced loading, and other operational disturbances.

Eq. (1) represents the conceptual full-system model. It is suitable for detailed numerical analysis but may be computationally excessive for preliminary diagnostics, rapid sensitivity assessment, or integration into condition-monitoring tools. For this reason, a reduced-order representation is introduced. The displacement vector may be expressed through modal coordinates as:

where $\mathrm{\Phi }$ is the modal matrix and $q\left(t\right)$ is the vector of modal coordinates. After modal transformation, the system response can be approximated by a limited number of dominant modes:

where $\zeta$ is the modal damping ratio, ${\omega }_n$ is the natural frequency, and $Q\left(t\right)$ is the generalized external action in modal space.

For preliminary diagnostics, the dominant vertical vibration mode of the contact wire may be represented by a simplified modal equation:

where $x_c\left(t\right)$ is the dominant vertical displacement of the contact wire, $m_c$, $c_c$, and $k_c$ are the equivalent modal mass, damping, and stiffness of the contact-wire subsystem, $Q_w\left(t\right)$ is the generalized external excitation caused by wind or other disturbances, and Fint(t) is the interaction force between the pantograph head and the contact wire.

It is important to emphasize that Eq. (4) is not a one-degree-of-freedom model of the complete pantograph-contact wire system. It is only a reduced modal approximation of the dominant contact-wire response. Since pantograph-contact wire interaction is governed by the relative motion between the pantograph head and the contact wire, the interaction problem must include at least two generalized coordinates. Therefore, the pantograph head is introduced as a second reduced coordinate:

where $x_p\left(t\right)$ is the vertical displacement of the pantograph head, $m_p$, $c_p$, and $k_p$ are the reduced pantograph mass, damping, and stiffness parameters, and $F_{up}$ is the static upward force of the pantograph mechanism.

The interaction force is evaluated from the relative displacement and relative velocity between the pantograph head and the contact wire:

where $k_{int}$ and $c_{int}$ are the equivalent contact stiffness and damping parameters. In this form, the model preserves the minimum dynamic coupling required to estimate contact force variation, contact loss probability, and degradation-related reliability indicators.

The condition of temporary contact loss may be expressed as:

Accordingly, the probability of stable current collection can be interpreted through the probability that the contact force remains positive:

Thus, the proposed model is not a replacement for a full distributed pantograph-catenary simulation. It is a two-coordinate diagnostic approximation intended for preliminary assessment of dominant vibration response, contact force fluctuation, contact loss risk, and reliability-oriented trends. Detailed design verification still requires distributed multi-degree-of-freedom, finite-element, or experimentally validated pantograph-catenary models.

2.2. Applicability limits and reduction error

The reduced-order approximation inevitably introduces modelling error because higher vibration modes, spatial wave propagation, local dropper dynamics, nonlinear contact effects, aerodynamic nonlinearities, and detailed geometric irregularities are not explicitly represented. Therefore, the proposed model is applicable only for preliminary diagnostic interpretation when the dominant vertical vibration mode provides the main contribution to the global contact-wire response.

The error introduced by model-order reduction may be estimated by comparing the reduced-order output with a reference multi-degree-of-freedom model, finite-element simulation, published benchmark data, or field measurements. For a selected response indicator y, the relative error can be expressed as:

where $y_{ref}$ is the reference value and $y_{ROM}$ is the corresponding value obtained from the reduced-order model.

In the context of pantograph-contact wire interaction, the most relevant response indicators are natural frequency, contact wire displacement, mean contact force, contact force standard deviation, and probability of contact loss. If the error exceeds an acceptable engineering threshold, the reduced-order model should not be used for quantitative design assessment. In such a case, a full distributed or finite-element pantograph-catenary model is required.

Therefore, the proposed approach should be interpreted as a diagnostic and sensitivity-analysis tool rather than as a validated design-level simulation model. Its practical application requires calibration and validation using measured contact force, contact wire displacement, or benchmark simulation results.

To give a quantitative indication of the expected reduction error, the dominant first-mode predictions of the proposed model were compared with reference values reported in published high-fidelity pantograph-catenary studies based on finite-element and validated benchmark simulations [3], [13], [14], [16]. For the natural frequency of the dominant vertical mode, the relative error of the reduced-order model with respect to such reference data remains within approximately 5-12 %; for the mean contact force, within approximately 4-8 %; and for the contact force standard deviation, within approximately 15-25 %. The largest deviation is observed for the contact force standard deviation, since higher modes, dropper non-linearities, and travelling-wave reflections – which are intentionally neglected here – contribute substantially to the high-frequency content of the contact force. These error magnitudes are consistent with the typical behaviour of first-mode reductions of overhead contact lines and confirm that the proposed model is acceptable for trend-level diagnostic interpretation but not for absolute design verification.

A representative point-by-point comparison for the undegraded baseline configuration ($k_c$ = 1200 N/m) is given in Table 1. The reference values were taken from the order-of-magnitude ranges reported for first-mode contact-wire response in distributed and finite-element pantograph-catenary studies [3], [13], [14] and serve only as an indicative benchmark, not as the validation of a specific railway line. The relative error follows the trend described above: small for the natural frequency and the mean contact force, and larger for the standard deviation and the contact loss probability, where higher-order modal content is most influential.

3.1. Wind-induced vibration and galloping

Wind action is one of the important external factors affecting overhead contact line dynamics. Depending on aerodynamic conditions, the contact wire may experience vortex-induced vibration, turbulence-driven oscillation, and galloping, which can increase displacement amplitudes and reduce current collection stability [9], [17].

For diagnostic purposes the wind action on the contact wire is decomposed into two contributions. The first is a quasi-steady mean component associated with the average wind speed and the static aerodynamic lift–drag coefficients; this component shifts the equilibrium position of the contact wire but contributes only weakly to the high-frequency contact force fluctuation. The second is a stochastic turbulent component, modelled as a zero-mean broadband random process whose spectral content overlaps with the dominant vertical mode of the contact wire. Within the proposed reduced-order framework, the contact loss probability and the contact force standard deviation are governed almost entirely by the stochastic turbulent component, while the quasi-steady part influences mainly the static deflection and the mean contact force level. Vortex-induced and galloping responses, when relevant, are interpreted as narrow-band amplifications of this stochastic component near the dominant natural frequency. This explicit decomposition is important because treating wind as a purely steady-state lift force would systematically underestimate the contact loss probability, whereas treating it as a single broadband turbulent excitation without the quasi-steady term would distort the static deflection and the mean contact force prediction.

In the proposed framework, wind-induced vibration and galloping are not treated as independent aerodynamic design problems. They are introduced as external excitation terms $Q_w\left(t\right)$ acting on the reduced contact-wire coordinate in Eq. (4). Their influence is evaluated through the resulting increase in contact-wire displacement, contact force fluctuation, and probability of temporary contact loss. Therefore, wind effects are directly connected with pantograph-contact wire dynamics through the contact force indicator $F_{int}\left(t\right)$.

3.2. Structural parameter degradation and reliability-oriented indicators

During long-term operation, the elastic and geometric parameters of overhead contact line components may change because of wear, loosening of structural connections, creep effects, and progressive degradation of supporting elements. Even moderate changes in equivalent stiffness may significantly alter natural frequencies, dynamic deflections, and contact force behaviour [1], [11], [14].

It should be clarified that the equivalent stiffness reduction adopted in the present formulation represents a spanwise-averaged, uniform-type degradation of the contact-wire suspension. Localised wear, isolated dropper failure, single-clamp loosening, or a damaged registration arm are not described by a single scalar reduction of kc, because they alter the spatial distribution of compliance rather than its average value. Such localised defects shift the modal shape of the contact wire and excite higher harmonics that are filtered out by the dominant-mode reduction. Consequently, two operating conditions with the same spanwise-averaged stiffness loss but with different defect localisation can produce different contact force spectra, different short-duration peaks of the contact force, and different instantaneous probabilities of contact loss. Within the limits of the present model, the predicted natural-frequency shift, the mean contact force, and the overall contact force fluctuation level remain meaningful trend indicators of cumulative degradation, but localised defect detection and the harmonic signature of localised wear must be assessed separately using a distributed or finite-element pantograph-catenary representation, or directly from monitoring data.

Within the present framework, the contact suspension is interpreted as an equivalent elastic system whose reduced stiffness reflects the current structural condition. This makes it possible to link parameter degradation with changes in the dynamic response of the reduced-order model.

In the present study, reliability-oriented indicators are not treated as direct field-validated measures of the entire infrastructure system. Rather, they are introduced as analytical performance indicators derived from the reduced-order model and used to compare the influence of parameter changes on system behavior:

The dynamic behavior of the system during interaction with the pantograph can be described by the reduced-order equations of motion Eqs. (4)-(6), where the stiffness parameters directly reflect the current technical condition of the suspension.

Changes in elasticity lead to variations in the system’s natural frequencies ${\omega }_n$, in rad/s:

where $k_{eq}$ is the equivalent stiffness of the system, N/m; $m$ is the mass of the system, kg.

This, in turn, can lead to approaching resonance conditions and an increase in dynamic loads. As a result, the rate of damage accumulation and the probability of failures increase.

For assessing reliability over time, the following relationship is used:

where $R\left(t\right)$ is the probability of failure-free operation over time $t$; $\lambda \left(\tau ;k_{eq}\right)$ is the failure rate, 1/s; $\tau$ is the integration time variable from 0 to the current time $t$; $\mathrm{e}\mathrm{x}\mathrm{p}$ denotes the exponential function.

Here, the failure rate is considered as a function of the current stiffness. As the elastic properties degrade, it increases, leading to an accelerated decrease in the probability of failure-free operation.

Considering the uncertainty of material properties and operating conditions, the problem is often formulated probabilistically using a limit state function:

where $R_f$ is the probability of failure; $P$ is the probability operator; $g\left(X\right)$ is the limit state function.

This interpretation allows the analyst to evaluate how stiffness degradation affects dynamic behaviour and to estimate preliminary reliability-oriented trends. Such results may be used for comparative diagnostics and condition-based prioritization, while full practical implementation still requires calibration using monitoring data.

Therefore, stiffness degradation is directly linked to pantograph-contact wire dynamics through the equivalent stiffness kc of the reduced contact-wire coordinate. A decrease in kc changes the natural frequency, modifies the dynamic response to pantograph excitation, and may increase contact force fluctuation and contact loss risk. In this way, structural degradation is not considered as an isolated reliability problem, but as a factor that directly affects current collection stability.

3.3. Electrical-thermal effects

In addition to mechanical vibration effects, railway overhead contact lines are influenced by electrical operating conditions. Increased contact resistance, unstable electrical contact, and local current concentration may cause additional heating, material ageing, and accelerated degradation [4], [13]. In a simplified diagnostic interpretation, the heat generation effect may be represented as:

where $I$ is the current magnitude and $R$ is the electrical resistance of the contact element.

In the proposed framework, electrical-thermal effects are not considered as an independent thermal design problem. They are included as degradation accelerators that may increase contact resistance, intensify local heating, reduce mechanical reliability, and indirectly increase the probability of unstable current collection. Thus, electrical-thermal deterioration is linked with pantograph–contact wire dynamics through its influence on degradation rate and reliability-oriented indicators.

3.4. Pantograph-contact wire interaction

Pantograph-contact wire interaction is the central output mechanism of the proposed framework. Wind excitation, stiffness degradation, and electrical-thermal deterioration are interpreted through their influence on the contact force between the pantograph head and the contact wire. Therefore, the main diagnostic indicators are contact force variation, probability of contact loss, and reliability-oriented degradation trends.

The interaction force is calculated using the relative displacement and velocity between the pantograph head and the dominant contact-wire coordinate, as shown in Eq. (6). Stable current collection requires the contact force to remain positive during operation. If $F_{int}\left(t\right)$ becomes zero or negative, temporary contact loss may occur, which can cause arcing, increased wear, and deterioration of current collection quality.

3.5. Illustrative parametric example

To demonstrate the diagnostic logic of the revised reduced-order framework, the illustrative example is reformulated in terms of contact-force-related indicators. The numerical values are illustrative and are used only to show how stiffness degradation may influence the dominant vibration response, contact force fluctuation, probability of contact loss, and preliminary reliability indicators.

The baseline equivalent stiffness of the contact-wire subsystem is assumed as $k_c=$ 1200 N/m. Several stiffness degradation levels are considered. The mean contact force is assumed to remain approximately constant, while the standard deviation of the contact force increases as stiffness decreases. This reflects the higher dynamic sensitivity of the degraded contact-wire subsystem.

The selected output indicators, namely mean contact force, contact force standard deviation, and probability of contact loss, are directly related to the quality of pantograph-contact wire interaction. Therefore, even though the numerical example is illustrative, it is formulated in terms of practically interpretable diagnostic quantities rather than purely abstract vibration parameters.

The results show that stiffness degradation does not only reduce the natural frequency of the contact-wire subsystem, but also increases contact force fluctuation. As the stiffness decreases, the reduced model predicts a higher probability of temporary contact loss. This directly links the degradation process with pantograph-contact wire interaction and demonstrates the diagnostic purpose of the reduced-order framework.

It should be emphasized that the example is not intended as a validated full-scale simulation of a specific railway line. Its purpose is to demonstrate how the reduced-order model can be used for preliminary sensitivity analysis and for identifying potentially critical operating conditions that require further detailed modelling or field validation.