Experimental Dynamic Behaviour and Pedestrian Excited Vibrations Mitigation at Ceramique Footbridge (Maastricht, NL) Alain FOURNOL Florian GERARD AVLS, bureau d études en dynamiques des structures - Orsay (France) Vincent DE VILLE Yves DUCHENE BE GREISCH - Liège (Belgium) Michel MAILLARD GERB France - Marly-le-Roi (France)
Experimental Dynamic Behaviour and Pedestrian Excited Vibrations Mitigation at Ceramique Footbridge (Maastricht, NL) Alain FOURNOL Florian GERARD AVLS, bureau d études en dynamiques des structures. Orsay (France) Vincent DE VILLE Yves DUCHENE BE GREISCH. Liège (Belgium) Michel MAILLARD GERB France. Marly-le-Roi (France)
Presentation Context : Design practice of footbridges with intensive use of lightweight materials and long spans Study case : New Ceramic footbridge Link upon the river Meuse (Maas) in the city of Maastricht, NL
Table of contents 1. Conception 2. Experimental Dynamic Behaviour of the Footbridge without TMD 3. Sizing of TMDs 4. Experimental Dynamic Behaviour of the Footbridge with TMD 5. Conclusions Perspectives
1. Conception
Conception Œuvre of René Greisch Total length : 261 m entirely made of steel The 164m main span is a bowstring bridge with a central boxed arch, a box-girder and 14-inclined full locked cables
Conception A new modern ward of high qualitative architecture The bridge has been opened end 2003 and was awarded the 2004 Dutch steel prize. photo- daylight.com In order to anticipate for low structural damping, and thus mitigate the pedestrian induced vibration, it was decided at early stage of design to allow installation of Tuned Mass Dampers.
2. Experimental Dynamic Behaviour of the Footbridge without TMD
Experimental Modal Analysis using residual vibration levels, an Operating Deformation Shape (ODS) measurement was conducted using natural excitation (microseismic excitation, air motion). ODS are performed by measuring Frequency Response Functions (FRF) between each point of a mesh and a fixed reference point. These FRFs contain phase and amplitude relationship between all points of the mesh. A reference vibration transducer was placed at 45 degrees in the vertical transversal plane. Two roving bi-dimensional (Vertical Transversal) velocimeters were then successively moved on the 44 points representing the structure. Thus a total of 44 Frequency Response Functions (FRFs) were computed between the reference point and the roving points. The FRFs were FFT computed in the 0 25Hz frequency range with a resolution of 31.2mHz.
Natural Frequencies Natural frequencies emerge from residual vibration levels (excitation: residual wind or microseismic excitation). The picture below shows the acceleration frequency content calculated from a 10 minutes acquisition of residual vibrations. The first natural frequencies are found at : 0.97 Hz in the horizontal direction, and 1.47 Hz, 1.66 Hz and 2.34 Hz in the vertical direction : 2.5 x 10-3 Vertical acceleration (m/s2) mi-travée quart de travée 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10
Natural frequencies x 10-4 Horizontal acceleration (m/s2) 7 mi-travée quart de travée 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10
Mode Shapes 1 st transversal 0.97 Hz 1 st vertical 1.47 Hz 2 nd vertical 1.66 Hz 3 rd vertical 2.34 Hz 2 nd transversal 2.66 Hz 1 st torsion 3.19 Hz
Mode Shapes
Mode Shapes
Mode Shapes
Damping Ratio H1 Excitation of the 1 st horizontal mode at 0.97 Hz Acceleration spectrum [m/s² RMS] - window: Hanning - Resolution: 0.013 Hz 0.1 max=0.34 m/s² Vert. mid-span 0.05 Vert. mid-span 2 1 0-1 Acceleration [m/s²] max=0.34 m/s² 0 0 2 4 6 8 10-2 0 20 40 60 80 100 Hori. mid-span 0.1 0.05 max=0.43 m/s² Hori. mid-span 2 1 0-1 max=0.43 m/s² 0 0 2 4 6 8 10 Frequency [Hz] -2 0 20 40 60 80 100 Time [s]
Damping Ratio V1 Excitation of the 1 st vertical mode at 1.47 Hz Vert. mid-span Acceleration spectrum [m/s² RMS] - window: Hanning - Resolution: 0.013 Hz 0.2 0.15 0.1 0.05 max=0.96 m/s² 0 0 2 4 6 8 10 Vert. mid-span 2 1 0-1 Acceleration [m/s²] max=0.96 m/s² -2 0 50 100 150 Hori. mid-span 0.2 0.15 0.1 0.05 max=0.02 m/s² 0 0 2 4 6 8 10 Frequency [Hz] Hori. mid-span 2 1 0-1 max=0.02 m/s² -2 0 50 100 150 Time [s]
Damping Ratios Vibration amplitude decreases after harmonic excitation (10 to 15-persons group bending knees simultaneously) is stopped. Several methods are usually used for damping estimation : Logarithm decrement if only one mode is visible Hilbert transform method or Prony-Pisarenko algorithm if several modes co-exist. Acceleration [m/s 2 ] 0.8 0.6 0.4 0.2-0.2-0.4-0.6 Decrement logarithm method - ζ=0.28 % - f 0 =1.45 Hz - error=5.91 % 1 0 original signal detected min-max identified envelop Frequency Damping Mode 1 0.97 Hz 1.2 % ζ Magnification factor Q 42 Mode 2 1.47 Hz 0.3 % 167 Mode 3 1.66 Hz 0.6 % 83-0.8-1 0 50 100 150 200 Time [s] Mode 4 2.34 Hz 0.5 % 100 The couples of frequency / damping parameters of the first modes provide a fairly complete characterization of vibration harshness of the structure.
Frequencies and damping of some footbridges VERTICAL MODES, WITHOUT_TMD 2 1.8 Damping (% critical) 1.6 1.4 1.2 1 0.8 0.6 0.4 Solférino V1 Solférino Tr1 Solférino Tr2 Solférino Tr3 Maastricht V1 Maastricht V2 Maastricht V3 Seoul V2 Melun V1 Melun V2 Suresnes V1 Epinal V1 Epinal V2 0.2 0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Frequency (Hz)
Frequencies and damping of some footbridges HORIZONTAL MODES, WHITOUT TMD 2 1.8 1.6 Damping (% critical) 1.4 1.2 1 0.8 0.6 Maastricht H1 Maastricht H2 Seoul H1 Melun H1 Suresnes H1 Epinal H1 Solférino H1 0.4 0.2 0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Frequency (Hz)
Acceleration levels Some of these values (corresponding to case of vandalism) exceed the acceptance criteria usually accepted for footbridges (both for low frequency motion, typically 0-3Hz) : 0.2 m.s -2 in horizontal directions, 0.7 m.s -2 in the vertical direction. The measurements confirm that these levels may be easily reached on the structure. Mid-span Quarter-span Z [ms -2 ] T [ms -2 ] Z [ms -2 ] T [ms -2 ] Random walk of 10 persons Random walk of 25 persons 0.12 0.18 0.02 0.02 - - - - Synchronised walk of 25 persons with Flexion H1 mode 0.28 0.09 - - Synchronised walk of 25 persons with Flexion V1 mode 0.56 0.02 - - Synchronised walk of 25 persons with Flexion V2 mode 0.62 0.04 - - Specific excitation of Flexion V1 with 10 persons 1.42 0.06 0.92 0.08 Specific excitation of Flexion V2 with 10 persons 0.43 0.04 0.92 0.05 Specific excitation of Flexion V3 with 10 persons 1.69 0.04 1.64 0.10 Specific excitation of Flexion H1 with 10 persons 0.44 0.46 - - Specific excitation of Torsion T1 with 10 persons 2.02 0.15 1.52 0.20
Dynamic Diagnosis of the Structure HORIZONTAL DIRECTION : Damping of the first horizontal mode is satisfactory (1.2%) VERTICAL DIRECTION : the frequency / damping couple is clearly unfavourable : The three first natural frequencies in the vertical direction are inside the frequency range of pedestrian excitation (walking fundamental frequency is mainly between 1.5 Hz and 2.5 Hz). The related damping ratios are very low, from 0.3% to 0.6%: a damping ratio of 0.3% induces a magnification factor of roughly 160 at resonance (compared to static response). DECISION : add damping with TMDs to the first three vertical modes, since : Their frequencies are located inside the walking frequency range : 1.47 Hz, 1.66 Hz, 2.34 Hz Their damping is very low : 0.3%, 0.6 %, 0.5%. Vertical acceptance criteria was easily approached with only 25 persons randomly walking, Their frequencies are mainly represented in the measured response spectrum during random walking.
2. Calculations
Dynamic Model Dynamic model using Finelg, by Greisch Material : Concrete and steel Elements : beams and shells Rayleigh Damping Model Improvement of the accuracy of the model : by making softer links (spring connection) between beams and posts
First results Results of modal analysis : Measurement Calculation Deviation Modal Mass Mode 1 0.97 Hz 1.04 Hz 7 % 157 tons Mode 2 1.47 Hz 1.50 Hz 2 % 255 tons Mode 3 1.66 Hz 1.77 Hz 6 % 232 tons Mode 4 2.34 Hz 2.47 Hz 5 % 212 tons
First results 1 st horizontal 0.97 Hz 1 st vertical 1.47 Hz 2 nd vertical 1.66 Hz Measured modes : 3 rd vertical 2.34 Hz 2 nd transversal 2.66 Hz 1 st torsion 3.19 Hz 1 st horizontal 1.04 Hz 1 st vertical 1.50 Hz 2 nd vertical 1.77 Hz Calculated modes : 3 rd vertical 2.47 Hz 2 nd horizontal 2.94 Hz 1 st torsion 3.68 Hz
3. Design of Tuned Mass Damper
TMD Design 2 DOFs Model TMD k m M c Footbridge Mode Vi Mi Ki Ci
TMD Design 0.6 0.5 F=2.08Hz F=2.10Hz F=2.12Hz F=2.14Hz F=2.16Hz F=2.18Hz 0.6 0.5 d=8% d=10% d=12% d=13% d=14% d=15% d=17% 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 1.8 1.9 2 2.1 2.2 2.3 2. 4 2.5 2.6 0 1.8 1.9 2 2.1 2.2 2.3 2. 4 2.5 2.6 Parameter : TMD frequency Parameter : TMD damping
TMD position ¾ span Midspan ¼ span
Design of Tuned Mass Dampers Optimisation of the TMD parameters (mass, frequency, damping ratio) was carried out using analytical equations of 2-DOF mass-spring systems (see Mechanical Vibrations, Den Hartog, 1956) : Number of TMD Unit Mass Tuned Frequency Spring Coefficient Damping Coefficient % Modal Mass Mode 2: vertical flexion order 1 1 3 300 kg 1.31 Hz 0.28 kn/mm 4.1 kn.s/m 1.3 Mode 3: vertical flexion order 2 1 3 000 kg 1.63 Hz 0.32 kn/mm 4.2 kn.s/m 1.3 Mode 4: vertical flexion order 3 1 4 240 kg 2.29 Hz 0.88 kn/mm 10.3 kn.s/m 2.0 A set of 3 TMD units (with a total mass of 10 540 kg) was installed at mid-span and quarter span, in order to damp the three first vertical modes of the structure, using the available space. The TMDs were designed and built by GERB Company in France.
4. Performance Checking of the TMD
Performance Checking of the TMD Testing was carried out after TMD setup : damping ratios were measured. Frequency ζ without TMD ζ with blocked TMDs ζ with free TMDs Flexion H1 0.97 Hz 1.2 % - 2.4 % Flexion V1 1.47 Hz 0.3 % 0.3 % 1.6 % Flexion V2 1.66 Hz 0.6 % 0.6 % 1.7 % Flexion V3 2.34 Hz 0.5 % 0.6 % 2.3 % Torsion T1 3.19 Hz 0.2 à 0.3 % - 0.4 % The structural damping of the first vertical modes was about 3 to 5 times larger with TMD than without TMD. Notice that the installed TMDs increase damping of the first horizontal mode further on.
TMD ¼ span ½ span TMD V1 TMD V2 TMD V3 Excitation of mode V2
TMD efficiency on footbridges (sample) VERTICAL MODES, WITH_TMD 3 Damping (% critical) 2.5 2 1.5 1 0.5 constant mobility 2e-6 m/s/n Solférino V1 Solférino Tr1 Solférino Tr2 Solférino Tr3 Maastricht V1 Maastricht V2 Maastricht V3 Seoul V2 Melun V1 Melun V2 Suresnes V1 Epinal V1 Epinal V2 Maastricht V1b Maastricht V2b Maastricht V3b Solférino Tr2b Solférino Tr3b V/F criteria, M=250t Seoul V2b 0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Frequency (Hz)
TMD efficiency on footbridges (sample) HORIZONTAL MODES, WITH TMD 4 3.5 3 Maastricht H1 Damping (% critical) 2.5 2 1.5 no Horizontal TMD!!! -> side effect Maastricht H2 Seoul H1 Melun H1 Suresnes H1 Epinal H1 Solférino H1 Seoul H1b Solférino H1b Maastricht H1b 1 0.5 0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Frequency (Hz)
5. Conclusion
Conclusion (case of TMD installation on an existing footbridge) It was shown that the application of TMDs efficiently brings damping to an underdamped structure. Here, the TMD mass was particularly low (1.3 to 2 % of modal mass), however damping was noticeably increased (x3 to x5). Owners usually ask for a maximum acceleration level guarantee. This is uneasy, since the excitation is not under our control (pedestrians) it is difficult to accurately predict (by computing) the future damping, although its should be a contractual goal for this type of project The prediction of TMDs effect on Footbridge s damping needs further investigations.
Experimental Dynamic Behaviour and Pedestrian Excited Vibrations Mitigation at Ceramique Footbridge (Maastricht, NL) Thank you for your attention Alain FOURNOL Florian GERARD AVLS, bureau d études en dynamiques des structures. Orsay (France) Vincent DE VILLE Yves DUCHENE BE GREISCH. Liège (Belgium) Michel MAILLARD GERB France. Marly-le-Roi (France)
Performance Checking of the TMD Forced single-degree-of-freedom oscillator with damping 10 3 X F = 2 ω 1 2 ω0 1 k 2 ω + 2ζ ω0 2 X / X stat 10 2 10 1 ζ=0.25% ζ=1% ζ=2% ζ=5% ζ=10% Dynamic amplification ω = ω X F 0 = 1 2kζ 10 0 ζ=100% Static limit 10-1 0 2 4 6 8 10 Frequency [Hz]
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