Damping Estimation from Free Decay Responses of Cables with MR Dampers

Abstract

This paper discusses the damping measurements on cables with real-time controlled MR dampers that were performed on a laboratory scale single strand cable and on cables of the Sutong Bridge, China. The control approach aims at producing amplitude and frequency independent cable damping which is confirmed by the tests. The experimentally obtained cable damping in comparison to the theoretical value due to optimal linear viscous damping reveals that support conditions of the cable anchors, force tracking errors in the actual MR damper force, energy spillover to higher modes, and excitation and sensor cables hanging on the stay cable must be taken into consideration for the interpretation of the identified cable damping values.

Semi-Active Cable Damping to Compensate for Damping Losses Due to Reduced Cable Motion Close to Cable Anchor

“…This means that the cable damping ratio due to a transverse damper is at least 20% to 40% smaller than expected from the closed-form solution 1 /2 a/L (a: damper position, L: cable length, valid for a/L ≈ 1 %) being valid for a taut string behaviour. This estimation agrees well with the experiments reported in [25][26][27][28][29][30][31] which show that the measured additional cable damping ratio due to the transverse damper, i.e., the total cable damping ratio minus the inherent cable damping ratio, is often around 50% to 70% of the theoretical value 1 /2 a/L being valid for a taut string behaviour. Besides the effect of the reduced damper motion, other effects may also lead to reduced cable damping ratios such as insufficient damper activation due to manpower excitation of real stay cables with great modal mass [32], friction damping [33][34][35], excitation of higher order modes necessitating low pass filtering during post-processing of the measurement data [29,35], force tracking errors in case of controllable dampers [24][25][26][27][36][37][38][39][40] and the fact that the damping estimation 1 /2 a/L is valid for a/L ≈ 1 % but transverse dampers are commonly positioned at 2% to 3% of the cable length.…”

“…For the simulation of the cable model with transverse damper, the partial differential Equations ( 1) and (3) are discretized, adopting a finite truss element modelling approach with the spatial sampling interval Δ𝑥 ≪ 𝐿 that is selected small enough to ensure precise approximation of the partial differential Equations ( 1) and (3) [24,34,35,45] M v̈+ C v̇+ K v = φ 𝑓(𝑡) + F ex (8) where M, C and K denote the mass, damping and stiffness matrices, v is the vector of the transverse displacements, φ is the connectivity vector of the transverse damper force and the excitation force vector Fex is introduced to excite the cable model harmonically at the eigenfrequencies of the considered modes. The inherent cable damping ratio 𝜉 𝑐𝑎𝑏𝑙𝑒 is assumed to be 0.4% which is a typical value of the first few modes of stay cables [2,29,30].…”

“…where M, C and K denote the mass, damping and stiffness matrices, v is the vector of the transverse displacements, ϕ is the connectivity vector of the transverse damper force and the excitation force vector F ex is introduced to excite the cable model harmonically at the eigenfrequencies of the considered modes. The inherent cable damping ratio ξ cable is assumed to be 0.4% which is a typical value of the first few modes of stay cables [2,29,30].…”

“…Because of these nonlinearities the peak-to-peak damping ratio is not fully constant as for linear transverse cable dampers. Therefore, the damping of the free decay response is assessed from the exponential fit of the free decay response between decay start and 10% of the steady state amplitude [29,30,34]. Figure 7 depicts examples of the exponential fits for a passive damper being optimized to the excited mode (10), a passive damper being optimized to the first four targeted modes (12) and for the semi-active damper with the consideration of the control force constraints.…”

“…Frequency detection from the peaks of the displacement signal resulting in a maximum time delay of half a period. This time delay is more than acceptable considering that resonant vibrations are long standing vibrations, see cable vibration measurements presented in [30].…”