1 Introduction
2 Bridge–vehicle dynamic interaction
2.1 Bridge–vehicle system equations
2.2 Bridge deck roughness
2.3 Discretization of bridge equations of motion
2.4 Solution of forward problem
3 Identification of vehicle parameters
4 Results and discussions
4.1 Response statistics of bridge
4.2 Comparison of vehicle load estimation with published results
4.2.1 Comparison with the results of numerical study
Vehicle speed (m/s) | Percentage error in estimation of gross vehicle weight (%) | |||||
---|---|---|---|---|---|---|
Class B (good condition) | Class C (average condition) | Class D (poor condition) | ||||
Law et al. [41] | Present study | Law et al. [41] | Present study | Law et al. [41] | Present study | |
10 | 11.57 | 3.18 | 13.71 | 5.82 | 23.49 | 8.95 |
15 | 11.55 | 4.13 | 15.35 | 7.38 | 25.46 | 11.51 |
20 | 12.46 | 4.96 | 16.73 | 8.94 | 27.79 | 13.03 |
25 | 13.45 | 8.55 | 19.47 | 13.33 | 30.01 | 16.22 |
4.2.2 Comparison with the results of experimental study
4.3 Influence of various factors on vehicle parameter identification
4.3.1 Effect of bridge response measurement location
Vehicle parameters | % Error | Vehicle parameters | % Error | ||
---|---|---|---|---|---|
Mid span | Quarter span | Mid span | Quarter span | ||
m
v
| 5.07 | 8.53 |
k
t
| 4.33 | 9.35 |
m
w
| 2.20 | 9.29 |
c
v
| 5.30 | 7.69 |
k
v
| 4.72 | 7.17 |
c
t
| 6.24 | 8.78 |
Vehicle parameters | 5 % noise | 10 % noise | ||
---|---|---|---|---|
No. of iteration | Error (%) | No. of iteration | Error (%) | |
m
w
| 31 | 5.07 | 54 | 8.75 |
m
v
| 42 | 2.20 | 38 | 6.04 |
k
v
| 28 | 4.72 | 53 | 6.92 |
k
t
| 36 | 4.33 | 56 | 9.35 |
c
t
| 25 | 5.31 | 31 | 7.74 |
c
v
| 63 | 6.24 | 39 | 9.63 |
4.3.2 Effect of assumption of the prior PDF p(Φ0)
Range | Vehicle mass (m
v) kg | Wheel mass (m
w) kg | ||
---|---|---|---|---|
Case 1 | Case 2 | Case 1 | Case 2 | |
\(\varPhi_L\)
| 0.4 × 104
| 1.7 × 104
| 0.3 × 103
| 1.4 × 103
|
\(\varPhi_U\)
| 3.6 × 104
| 1.9 × 104
| 3.5 × 103
| 1.6 × 103
|
Range | Vehicle suspension stiffness (k
v) N/m | Wheel stiffness (k
t) N/m | ||
---|---|---|---|---|
Case 1 | Case 2 | Case 1 | Case 2 | |
\(\varPhi_L\)
| 0.3 × 107
| 2.8 × 107
| 5.2 × 106
| 0.6 × 107
|
\(\varPhi_U\)
| 8.5 × 107
| 4.0 × 107
| 8.5 × 107
| 1.1 × 108
|
Range | Vehicle suspension damping (c
v) N-s/m | Wheel damping (c
t) N-s/m | ||
---|---|---|---|---|
Case 1 | Case 2 | Case 1 | Case 2 | |
\(\varPhi_{\text {L}}\)
| 1.5 × 104
| 6.5 × 104
| 5.2 × 103
| 0.5 × 104
|
\(\varPhi_{\text{U}}\)
| 12.3 × 104
| 8.3 × 104
| 6.4 × 104
| 0.9 × 104
|
4.3.3 Effect of different vehicle velocities
Vehicle parameters |
V = 60 km/h |
V = 80 km/h | ||
---|---|---|---|---|
No. of iteration | Error (%) | No. of iteration | Error (%) | |
m
w
| 26 | 1.21 | 12 | 8.15 |
m
v
| 43 | 2.20 | 18 | 7.04 |
k
v
| 35 | 1.72 | 22 | 6.63 |
k
t
| 22 | 3.12 | 42 | 3.45 |
c
t
| 21 | 3.72 | 19 | 3.93 |
c
v
| 65 | 2.31 | 18 | 8.63 |
4.3.4 Effect of different roughness conditions
Road condition | Mass of vehicle (kg) | Mass of wheel (kg) | ||
---|---|---|---|---|
No. of iteration | Error (%) | No. of iteration | Error (%) | |
Good | 16 | 3.21 | 19 | 2.02 |
Average | 38 | 3.59 | 25 | 3.97 |
Poor | 49 | 6.06 | 43 | 5.95 |
Road condition | Suspension stiffness (N/m) | Suspension damping (Ns/m) | ||
---|---|---|---|---|
No. of iteration | Error (%) | No. of iteration | Error (%) | |
Good | 20 | 2.16 | 23 | 3.95 |
Average | 25 | 2.72 | 44 | 4.31 |
Poor | 43 | 4.02 | 72 | 6.14 |
Road condition | Tyre stiffness (N/m) | Tyre damping (Ns/m) | ||
---|---|---|---|---|
No. of iteration | Error (%) | No. of iteration | Error (%) | |
Good | 36 | 1.98 | 19 | 2.24 |
Average | 41 | 2.33 | 44 | 4.59 |
Poor | 85 | 8.02 | 68 | 10.27 |
4.4 Comparison of CPU processing time for estimation of vehicle parameters with numerically generated samples
Parameters | Sampling frequency (Hz) | Analytically generated samples | Numerically generated samples | ||
---|---|---|---|---|---|
Processing time (s) | % error | Processing time (s) | % error | ||
m
v
| 300 | 20.21 | 5.98 | 54.37 | 8.81 |
500 | 21.47 | 5.07 | 73.66 | 6.22 | |
700 | 26.52 | 3.82 | 102.88 | 5.47 | |
m
w
| 300 | 13.47 | 4.09 | 45.65 | 6.50 |
500 | 19.79 | 2.20 | 77.16 | 4.13 | |
700 | 22.73 | 1.66 | 117.67 | 3.06 | |
k
v
| 300 | 16.00 | 6.21 | 39.22 | 8.39 |
500 | 20.63 | 4.72 | 63.66 | 5.64 | |
700 | 23.58 | 2.99 | 167.18 | 3.36 | |
c
v
| 300 | 22.31 | 6.54 | 42.44 | 5.99 |
500 | 25.26 | 5.30 | 115.74 | 5.85 | |
700 | 32.42 | 5.29 | 158.82 | 5.01 | |
k
t
| 300 | 20.63 | 7.51 | 46.30 | 9.30 |
500 | 23.16 | 4.33 | 57.23 | 6.07 | |
700 | 25.68 | 4.02 | 83.59 | 5.84 | |
c
t
| 300 | 11.79 | 7.25 | 32.15 | 10.11 |
500 | 12.63 | 6.24 | 52.73 | 7.79 | |
700 | 13.89 | 2.89 | 91.31 | 5.83 |