[A] System coefficient matrix [A]+ Pseudo inverse of the matrix [A]
[A(t)] State matrix of coupled vehicle-bridge system [A(ω)] System matrix in frequency domain
As Amplitude of cosine wave
bi Distance of ith wheel from centre line of vehicle body {b} Time series vector of measured response of bridge deck
{b(ω)} Frequency domain function of the measured bridge response at selected station
cv Suspension damping of Model-1 cv1 Front suspension damping of Model-2 cv2 Rear suspension damping of Model-2 cv11 Front-right suspension damping of Model-3 cv21 Front-left suspension damping of Model-3 cv12 Rear-right suspension damping of Model-3 cv22 Rear-left suspension damping of Model-3 ct Tyre damping of Model-1
ct1 Tyre damping of Model-2
ct2 Rear tyre damping of Model-2 ct11 Front-right tyre damping of Model-3 ct21 Front-left tyre damping of Model-3 ct12 Rear-right tyre damping of Model-3 ct22 Rear-left tyre damping of Model-3
Cb Viscous dampingper unit length of bridge Cbr Rotational damping per unit length of bridge
Cv Viscous damping per unit length of the vehicle body
CvT Rotational viscous damping per unit length of the vehicle body [C (t)] Time dependent damping matrix of coupled vehicle-bridge system D1 Vehicle centroid distance from the leading edge of the vehicle body D2 Vehicle centroid distance from the trailing edge of the vehicle body
ex Load eccentricity
EbIb Flexural rigidity of the bridge Ek(xk,tk) Standard error of the mean (SEM) EvIv Flexural rigidity of flexible vehicle body f1(t) Moving load function for front axle f2(t) Moving load function for rear axle
fl(.) Non linear function that relates the measurements to the system state {f(ω)} Frequency variation of moving force
fb(x,t) Impressed vertical force on the bridge due to vehicle interaction fT(x,t) Imposed torque on the bridge due to vehicle interaction
F(t) Input excitation
Fobj Objective function for optimization based indirect inverse scheme {F(t)} Generalized stochastic force vector for coupled vehicle-bridge system {F(ω)} Fourier transform of the input excitation
gl(.) System transition function g0(t) Gaussian stochastic process GbJb Torsional rigidity of bridge
GvJv Torsional rigidity of the vehicle cross section h(x) Bridge deck profile
h0 Central rise of mean parabolic surface h1 Magnitude of approach road settlement hm(x) Deterministic mean surface profile hr(x) Random road surface unevenness h(t) Impulse response function [H(ω)] Frequency Response Function
Hj( ,t) Transient frequency response function
Imv Polar moment of inertia per unit length of vehicle body cross section Imb Polar moment of inertia per unit length of bridge cross section
[I] Identity matrix
kv Suspension stiffness of Model-1 kv1 Front suspension stiffness of Model-2 kv2 Rear suspension stiffness of Model-2 kv11 Front-right suspension stiffness of Model-3 kv21 Front-left suspension stiffness of Model-3
kv12 Rear-right suspension stiffness of Model-3 kv22 Rear-left suspension stiffness of Model-3 kt Tyre stiffness of Model-1
kt1 Tyre stiffness of Model-2 kt2 Rear tyre stiffness of Model-2 kt11 Front-right tyre stiffness of Model-3 kt21 Front-left tyre stiffness of Model-3 kt12 Rear-right tyre stiffness of Model-3 kt22 Rear-left tyre stiffness of Model-3
[K(t)] Time dependent stiffness matrix of coupled vehicle-bridge system lv Length of vehicle body
L Bridge span
Lr Ramp length
L(.) Likelihood function
m Roughness shape coefficient mb Mass per unit length of bridge mqv Sprung mass of Model-1
mv Mass per unit length of flexible vehicle body mw Unsprung mass of Model-1
mw1 Front wheel mass of Model-2 mw2 Rear wheel mass of Model-2 mw11 Front-right wheel mass of Model-3 mw21 Front-left wheel mass of Model-3 mw12 Rear-right wheel mass of Model-3 mw22 Rear-left wheel mass of Model-3
[M] Mass matrix of coupled vehicle-bridge system Mvk Generalized mass of vehicle body in bending
Mvk Generalized mass of vehicle body in bending n Sizes of the coupled vehicle-bridge system matrix
N Number of terms used to build up the road surface roughness Nl Total number of time step
Np Random samples of vehicle parameters Nt Number of time instants considered
Ns Number of time history samples inside the ensemble output set p(.) Probability density function
{p(t)} State vector of coupled vehicle-bridge system Piden Identified vehicle parameters
Ptrue True vehicle parameters
{P(t)} Augmented load vector in state-space equation Qbk(t) Generalized force in the kth mode of bridge in flexure
Qvk(t) Generalized force in the kth mode of transverse vibration of vehicle body
rm Measured response time histories rs Simulated response time histories
{r (t)} Response vector of coupled vehicle-bridge system
rPm(t) Particular integral of the coupled vehicle-bridge system differential equation
Rg0g0( ) Autocorrelation function
Sg0g0( ) Temporal power spectral density function
Sg0g0( ) Two-sided power spectral density function in space domain SGG ( ) One sided spectral density
uj(t) Time dependent Eigenvector of coupled vehicle-bridge system state matrix
[U(t)] Modal matrix of coupled vehicle-bridge system
V Vehicle speed
ws Normalized weight of particle W(.) Importance weights
{x(t)} Time series vector
{X(ω)} Fourier transform of the vector of response co-ordinates
X0j Constants of integration to be determined from the initial conditions y(t) Measured output at different stations
y(x,t) Bridge deflection measured at location x from the reference point at time instant t
Ydynamic Maximum response due to fluctuating load imposed on the bridge due to vibratory motion of the vehicle excited by road unevenness
Ystatic Response of the bridge at the mid span location for adverse position of static wheel loads
z1 Heave motion of sprung mass z2 Vertical motion of un-sprung mass
z(u,t) Vertical deflection of the vehicle body measured at location u from the reference point at time instant t
Zl Bridge response measurement
j(t) Time dependent eigenvalues of coupled vehicle-bridge system state matrix
(.) Dirac-delta function
Spatial frequency bands width
ηbk(t) Time dependent generalized coordinate of bridge in bending ηTk(t) Time dependent generalized coordinate of bridge in torsion
ηvk(t) Time dependent generalized coordinate of vehicle body in bending
identified value Identified vehicle parameters value
reference value True vehicle parameters value
L Lower bound of the vehicle parameters for construction of prior PDF
U Upper bound of the vehicle parameters for construction of prior PDF Wavelength of road unevenness
(.) Mean value of identified vehicle parameters
F Mean of bridge vehicle interaction force
Y(tk) Mean of the output samples
k(xk,tk) Mean response of bridge )
l (t
γ Time dependent generalized coordinate of vehicle body in torsion
Y(tk) Standard deviation of the output samples φb(x) Bridge mode shapes in bending
φvk(u) Vehicle mode shapes in bending )
vk(u
φ Vehicle mode shapes in torsion
{ } System parameters
(.) Probability density function
ξbk Modal damping ratio of bridge in bending ξvk Modal damping ratio of vehicle body in bending
ξvTk Modal damping ratio of vehicle body in torsion ξTl Modal damping ratio of bridge in torsion
)
k(t
ψv Time dependent generalized coordinate used to describe torsional motion
(.) Standard deviation of identified vehicle parameters
F Standard deviation of bridge vehicle interaction force θs Random phase angle
Tb(u) Bridge mode shapes in torsion Temporal frequency
ωb Bridge natural frequency in bending ωT Bridge natural frequency in torsion
ωvk Vehicle body natural frequency in bending ωvk Vehicle body natural frequency in torsion
Wave number / spatial frequency
0 Discontinuity frequency of road
s Road spatial frequency
L Lower cut-off frequencies of spatial unevenness
U Upper cut-off frequencies of spatial unevenness