PSI - Issue 84

Allan Larsen et al. / Procedia Structural Integrity 84 (2026) 1103–1110 1107 The black line indicates the wind speed for maximum response ( ≈ 0.18) while the red line indicates the wind speed for onset of vortex induced vibrations ( ≈ 0.20) and blue line indicates exit of the lock-in range ( ≈ 0.15). The natural frequency of the 5 th mode of the cable is then obtained as 5 = 3.8 Hz is expected to enter into lock-in at a wind speed = 5 ⁄0.2≈ 1.6 m/s, receive maximum excitation at = 5 ⁄0.18 ≈ 1.8 m/s and exit lock-in at = 5 ⁄0.15 ≈ 2.1 m/s, marked by the horizontal red and blue lines in Fig. 3. The 6 th mode of the same cable an eigenfrequency 6 = 4.56 Hz clearly falls in the lock-in range of the 5 th mode. Thus, adjacent modes may fall in the lock-in range for one given mode to form beating vibrations. From Fig. 3 it is noted that the higher the mode of the cable the winder the lock-in wind speed range becomes making it theoretically possible for an increasing number of modes to combine in beating vibrations.

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Fig. 5. Typical time trace and frequency spectrum of hanger vibrations measured at the Hålogaland bridge, Norway. An example of beating hanger vibrations measured at the Hålogaland bridge 1.3 m above the deck anchorage, is shown in Fig. 5, Larsen (2021). From the time trace (left) of the normalized displacements ⁄ measured at a wind speed of 2.4 m/s it is noted that the vibrations are irregular sinusoidal displaying a clear beating. The frequency spectrum of the time trace (right) clearly identifies two major peaks at 3.8 Hz and 4.56 Hz with smaller contributions at 3.08 Hz and 5.32 Hz as discussed in the example above. Beating vibrations of cables are not only observed for bridge hanger cables in the field. Wind tunnel tests of transmission line models report similar phenomena, Muggiasca (2018). Besides the occurrences of beating of hanger cables, the Hålogaland bridge measurements demonstrated that vortex induced vibrations endure with similar vibration amplitudes at all wind speeds up to 18 m/s which was the highest wind speed encountered during the measurement campaign. 3. Galloping vibrations (rain-wind) Galloping vibrations of cable structures, in contrast to vortex induced vibrations, are understood to be excited by a variation of the crosswind static aerodynamic force as the relative inflow angle changes over a cycle of the vibration. Galloping excitation requires that the cross-section geometry of the structure is asymmetric with respect to the direction of the wind. Circular cable cross sections are, by definition, symmetric with respect to the wind no matter the inflow angle and are thus not prone to galloping vibrations. However, the presence of ice, snow or a rivulet of rainwater adhering to the cable surface may temporarily change the apparent geometry of the circular cross section into a shape which fulfills the criteria for onset of galloping vibrations. Galloping vibrations in contrast to vortex induced vibrations may develop large amplitudes several times the cable diameter. Fig. 6 (left) shows galloping vibrations of hanger cables of the 1915 Çanakkale bridge in the 5 th mode at a frequency of 2.7 Hz which were observed to be due to the combined action of wind and rain. Rain-wind oscillations are commonly found in cable-stayed bridges having stay-cables inclined relative to vertical and has attracted considerable attention in research and codes of practice. In contrast, rain-wind vibrations of vertical hanger cables have not, to the knowledge of the author, been reported for vertical hanger cables. The non-dimensional amplitude of the 1915 Çanakkale bridge vibrations is estimated to ⁄ ≈ 2 – 3 from Fig. 6 (left).