PSI - Issue 84

Enrique García-Macías et al. / Procedia Structural Integrity 84 (2026) 837–844

839

Fig. 1. (a) General bridge structure subjected to a single moving load, including cubic spline interpolation of displacements along the load lane. (b) Geometric representation of an analytic signal ( ) and its envelope ( ) . The FE mesh is assumed to allow the representation of equivalent one-dimensional beam-type DOFs along the load lane. On this basis, the displacement field can be characterised by a Hermite shape functions ℎ ( ) , so that Eq. (1) can be rewritten without explicit dependence on the and coordinates as: ( , ) = ∑ ( ) ∑ ( )ℎ ( ) 4 =1 , =1 (2) where the matrix coefficients represent the evaluation of the mode shapes along the load lane. Functions ( ) in Eq. (2) were obtained in Martínez-Castro, A. E. et al. (2006) in closed-form as the sum of a homogeneous and a particular solution, ( ) = ℎ ( ) + ( ) , given by: ℎ (τ)=e -ζ n ω n τ [ cos(ω τ)+ sin(ω τ)]; (τ)=α ( 0 ) +α ( 1 ) ( τ)+α ( 2 ) ( τ) 2 +α ( 3 ) ( τ) 3 , (3) with = − / being the local time at segment , and in Eq. (3) the damped natural angular frequency and the damping ratio of the n -th mode. The coefficients ( ) in (τ) , provided in Martínez-Castro et al. (2006), only depend on the modal properties (i.e. mode shapes, natural frequencies and damping ratios), the speed , and the length of the load lane segment. Moreover, the coefficients and of the homogeneous solution in Eq. (3) are obtained from the initial conditions 0 = (0) and 0̇ = ̇ (0) as: = 0 −α 0 , = 0̇ + − ( 1) . (4) The complete solution is defined piecewise, with an analytical expression per element. At-rest conditions are commonly imposed for the initial time =0 , i.e. 0 =0 , 0̇ =0 . For the following elements, the initial conditions for element +1 are given by the end values of element , i.e. ( )| =+01 = ( )| = / , ̇ ( )| =+01 = ̇ ( )| = / . Finally, velocity and acceleration are obtained by time differentiation of ( , ) in Eq. (1). 2.2. Hilbert transform of the semi-analytic solution The HT of a function ( ) is a real-valued function ̃( ) is defined as (Feldman (1994), Huang and Nii O (2005)): ℋ[ ( )] = ̃( ) = 1 PV ∫ −( ) d + ∞ − ∞ , (5) where PV denotes the Cauchy principal value, required due to the singularity at = . The main interest of the HT lies in its ability to extend real-valued signals into analytic signals, defined as (Schreier and Louis (2010)): ( ) = ( ) + ̃( ) = ( ) ( ) , (6) defining a phasor rotating in the complex plane (Fig. 1(b)). The instantaneous amplitude ( ) (envelope) reads: ( ) = ±| ( )| = ±√ 2 ( ) + 2̃ ( ). (7)