Stability for Bridge Cable and Cable-Deck Interaction. Dr. Maria Rosaria Marsico D.J. Wagg
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1 Stability for Bridge Cable and Cable-Deck Interaction Dr. Maria Rosaria Marsico D.J. Wagg S.A. Neild
2 Introduction The interaction between cables and structure can lead to complex vibration response Cables are highly non linear due to the interaction of geometric non linear effects and extremely lightly damping
3 Experimental part One possible solution is RTDS: Real-time dynamic substructuring The nonlinear components of the structure are physically tested. The dynamic response of the remaining part of the structure is computed numerically. The interface between physical and numerical components is provided by a set of transfer systems, typically actuators.
4 Modelling: the cable-deck interaction u, axial displacements v, out-of-plane displacements w, in-plane displacements angle of inclination measured from the horizontal line in the gravity plane The cable is rigidly supported at the upper end and an oscillator is included at the lower end to simulate the deck mode.
5 Equation to capture the modal behaviour The sag is small compared to the length of the cable The dynamics along the cable (u) are insignificant The amplitude of vibration is small compared with the sag such that the compatibility equation can be linearized The mode shapes for the linearized system in both the in plane and out-of-plane directions were calculated
6 Deck The deck is modelled as a single degree-of-freedom system, so the resulting equation of motion is
7 Equation for 3 modes and deck mode Other modes are assumed to be insignificant. The oscillator is excited vertically by an external force of amplitude F and frequency Ω the first out-of-plane mode the second out-of-plane mode the first in-plane mode the equation of motion for the deck The forcing frequency Ω and the oscillator (i.e. global mode) frequency ω g are close to twice the first out-of-plane natural frequency
8 Analysis 1.98 m long, steel cable inclined mm diameter mass of 0.67 kg/m (attaching lead weights at 60 mm intervals) T static 205 N CABLE NATURAL FREQUANNCIES
9 Substructuring Experiments Deck displacement num Forward prediction INSTRUMENTATION a. load cell b. LVDT displacement transducer c. high speed vision system support motion to the bottom
10 Localized stability In order to calculate the first out-of-plane stability boundary, we have to solve simultaneously both the equations for the second in-plane mode and for the first out-of-plane mode.
11 Stability Boundaries for three Different Global Modes FORCING AMPLITUDE vs FREQUENCY PLANE Each line represents the stability boundary of the semi-trivial solution, which physically corresponds to the point at which the cable starts to have out-of-plane response when both the input and previous response were in-plane
12 Experimental analysis EXPERIMENTAL AND THEORETICAL STABILITY BOUNDARIES. FIRST OUT-OF-PLANE MODE. (a) q=0.96; (b) q=1; (c) q=1.04 Analytical model Unstable experimental points * Stable experimental points
13 Earthquake Engineering Lab
14 Input and Output Systems
15 Analytical model for a new cable 2 nd in plane l=5.40 m α= Φ= 0.8 mm T static 286 N
16 Theoretical stability boundaries FORCING AMPLITUDE vs FREQUENCY PLANE
17 Test #10: Δ=2.16m, f=6.735rad/sec, delay 17.1
18 Experimental stability boundaries
19 Conclusions The in-plane and out-of-plane modes for a cable are considered to know its stability boundaries A series of real-time dynamic substructuring experiments were performed in order to experimentally validate this observation Analytical and experimental results show good agreement
20 THANK YOU FOR YOUR KIND ATTENTION Stability for Bridge Cable and Cable-Deck Interaction Dr. Maria Rosaria Marsico D.J. Wagg S.A. Neild
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