PSI - Issue 84

840 Enrique García-Macías et al. / Procedia Structural Integrity 84 (2026) 837–844 Geometrically, signal ( ) and its instantaneous envelope ( ) have common tangents at contact points. Therefore, the HT can be used to remove the rapid oscillations from the amplitude modulated signal. Following this principle, in García-Macías and Martínez-Castro (2020), we developed a meta-model for the fast estimation of maximum response envelope based on a time sub-sampled approximation of the instantaneous envelope. The HT of the semi-analytic solution previously outlined in Section 2.1 is obtained by the integral operator in Eq. (5) at every time step throughout the complete time domain as: ̃( ) = 1 ∫ −( ) d + ∞ − ∞ = 1 {∑ PV ∫ ℎ ( ) + ( ) τ− d / 0 =1 } + 1 {PV ∫ ℎ τ −( ) d + ∞ 0 }, (8) where subscripts have been omitted for notational convenience. Note that the first term in the right-hand side relates the Hilbert integral throughout the load lane elements ( ∈ [1, ] ), while the second term accounts for the free vibration solution once the load has left the structure. Given that the HT is a linear operator, the HT in Eq. (8) can be constructed from the transform of the homogeneous and particular solutions separately. It was show in García-Macías and Martínez-Castro (2020) that the HT of the homogeneous solution reads: ℎ̃ (τ) = 1 −ζ ω |τ| PV∫ ℎ τ −( ) d / 0 , (9) with ℎ̃ (τ) = 1 ( lim → / −ε ℎ (τ, ) − lim →ε ℎ (τ, )), (10) ℎ ( , ) = Si( ( − ))[ sin( ) − cos( )] −Ci( ( − ))[ cos( ) + sin( )], (11) where Si and Ci denote the sine and cosine integral functions, and is a small positive number tending to zero. On the other hand, the HT of the particular solution reads: ̃ (τ) = 1 ( lim → / −ε (τ, ) − lim →ε (τ, )), (12) with ( , ) = − {( − )[6 ( 1 ) + (3 ( 2) ( + 3 ) + ( 3) (2 2 + 5 + 11 2 ) )]/6 + [ ( 1 ) + ( ( 2) + ( 3) )] ln| − |}. (13) Finally, the Cauchy principal value of the second term in the right-hand side of Eq. (8), i.e. the free vibration term, can be readily obtained as: 1 PV ∫ ℎ τ− ( s ) ∞ 0 = − | | [ +1 sin(ω τ)− +1 cos(ω τ)] − 1 lim → − | | ℎ ( , ). (14) Once the semi-analytic solution, ( , ), and its HT, ̃ ( , ), have been obtained, the analytical signal is constructed and its module is computed to obtain the instantaneous envelope ( , ) as in Eq. (7). This formulation is computationally demanding because evaluating the Hilbert transform in Eq. (8) requires knowledge of the signal over the entire time domain. As a result, each time step involves accounting for all train axles across the full load lane, including free-vibration effects. To reduce this cost, the authors proposed a reduced Hilbert transform approach based on time-window filtering around the evaluation instant and an approximation of the homogeneous solution using infinitely extended sinusoidal terms above a user-defined cut-off frequency. Further mathematical details are omitted here for brevity and can be found in García-Macías and Martínez-Castro (2020). 3. Proposed metamodel For a given train configuration, estimating maximum response envelopes generally requires multiple dynamic analyses over a range of train speeds, with peak responses extracted from time-sampled results. Unlike step-by-step