Response of a joint passive crowd-SDOF system subjected to crowd jumping load Jackie Sim, Dr. Anthony Blakeborough, Dr. Martin Williams Department of Engineering Science University of Oxford University of Oxford
Jan 04, 2016
Response of a joint passive crowd-SDOF system subjected to crowd
jumping load
Jackie Sim, Dr. Anthony Blakeborough, Dr. Martin WilliamsDepartment of Engineering Science
University of Oxford
University of Oxford
Vibration problem on cantilever grandstand
Flexible structure with large span and lightweight
Synchronised crowd loadings 1.5 ~ 2.8 Hz
+
Dynamic analysis of cantilever grandstand
Human-structure interaction
Passive crowdCrowd model
Active crowd
Load model
Contents
Outline
1. Passive crowd model
How to model the seated and standing crowds?
2. Active crowd model
How to model the jumping crowd?
3. Analysis of active + passive crowds on SDOF structure
What is the structural response?
4. Results
5. Case study
Passive crowd model (1)
• Griffin et al. – experimental tests and model development
• Measure the apparent mass of 24 seated and 12 standing men:
ix
iFim
gapp
DOF 1
m2
k2
c1
k1
m1
c2
F
y2
y1
DOF 2
Passive crowd model (2)
• Curve-fitting the crowd apparent mass response
• Crowd model represented as transfer function
• Seated:
• Standing
0 5 10 15 200.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Frequency (Hz)
No
rmal
ized
ap
par
ent
mas
s SeatedStanding
0 5 10 15 20-100
-80
-60
-40
-20
0
20
Frequency (Hz)P
ha
se
(d
eg
ree
)
01.38109673.13293948.602533.7400.1
28.379423974.13467327.318126.3213.0234
234
ssss
ssss
20.889932707.26219875.967565.9600.1
62.893300667.25837809.504010.4200050.0234
234
ssss
ssss
Fourth order polynomial i.e. 2DOF system
Active crowd model (1)
Experimental tests
- University of Surrey
- 100 test subjects
- Each individual jumping on rigid force plates
- Metronome at 4 beat frequencies (1.5, 2, 2.67 and 3.5 Hz)
- Synchronised test results were analysed
Active crowd model (2)
0 1 2 3 4 5-500
0
500
1000
1500
2000
Load
(N
)
Time (s)0 0.1 0.2 0.3 0.4 0.5
-500
0
500
1000
1500
2000
2500
Lo
ad
(N
)
Time (s)
Load-time history at 2 HzAverage impulse of
each individual
Average impulse of all individuals => Crowd jumping load
Active crowd model (3)
Crowd jumping load
0 1 20
1
2
3 1.5 Hz
Time (s)0 1 2
0
1
2
3 2 Hz
Time (s)
0 1 20
1
2
3 2.67 Hz
Time (s)0 1 2
0
1
2
3 3.5 Hz
Time (s)
F'
F'
F'
F'
Beat Frequency
(Hz)1st 2nd 3rd
1.5 0.911 0.150 0.034
2 1.193 0.337 0.040
2.67 1.228 0.311 0.032
3.5 1.020 0.157 0.008
Fourier coefficients
High FC => Better synchronisation
ms
F
x
Analysis (1)
Passive crowd-SDOF system subjected to crowd jumping
load
SDOF structure
Seated / Standing
crowd
Crowd jumping load
Interaction force
_
+
Feedback system representation
Displacement
Acceleration
Analysis (2)
Joint passive crowd-SDOF system
H()
Structuralresponse
R()
Crowdjumping load
F()
Frequency domain analysis
Parameters
Natural frequency of empty structure: 1 ~ 8Hz
Structural damping ratio: 2%
Passive crowd mass ratio, : 0 ~ 0.3 (increment of 0.05)
Subjected to crowd jumping load at 1.5, 2, 2.67 and 3.5Hz
Results - Maximum displacement
Results – RMS Acceleration
Case study – Cardiff Millennium Stadium
• First mode at 2.9 Hz• Crowd mass = 16800 kg per
bay• Assume = 0.3• Structure mass = 56000 kg for
one bay• Structure stiffness;
MN/m6.185600029.2 22 sns mk
Results
• Rugby match between Australia and France in Nov 1999
• Displacement of approximately 50mm reported after the match
• Half full capacity
• Mass ratio, = 0 ~ 0.15
Maximum displacement (mm)
RMS Acceleration (times g = 9.81m/s2)
Crowd jumping frequency (Hz)
1.5 2 2.67 3.5
0 14.2 12.8 38.2 14.0
0.05 7.3 9.0 34.9 8.2
0.1 3.3 4.8 27.1 3.7
Crowd jumping frequency (Hz)
1.5 2 2.67 3.5
0 0.20 0.14 0.70 0.34
0.05 0.08 0.09 0.65 0.2
0.1 0.03 0.05 0.52 0.08
Concluding remarks
• Passive crowd adds significant damping to the system and alters the resonance frequency
• Preliminary analysis on the Cardiff Millennium Stadium gave good results
• Current work – statistical model of the crowd jumping load – taking into account the timing of each individual