VEHICLE PARAMETER IDENTIFICATION
2.2 Methods of Moving Load Identification
Direct measurement of vehicle load time history over the bridges is not feasible at all.
However, measurements of structural response under moving load may be utilized to estimate the vehicle load in indirect way that has motivated many researchers to develop analytical, numerical and computational techniques to find load time history registered by a passing vehicle. This area of research falls under the class commonly called as “inverse engineering” which are commonly used for state or parameters estimation of the dynamic
system. Concept of solving such inverse problem is to determine the system inputs if the system outputs are known. In this regard, the inverse solution method can be grouped into two major heads such as direct inverse method and indirect inverse method.
2.2.1 Direct Inverse Method
Time history of moving load is found directly from measured bridge response. Strain gauge and accelerometers are common transducers used in acquiring response time history of bridge at several locations. In this category, there are two distinct approaches, one is time domain and another is frequency domain method. Frequency domain, however, is restricted to linear problems.
2.2.1.1 Time domain method
In linear dynamics, the system output to an arbitrary excitation can be written adopting Duhamel integral (Inman, 2001) as
−
= th t F d t
y
0
) ( ) ( )
( τ τ τ (2.1)
where h(t) is the impulse response function and F(t) is the input excitation. The aim is to find F(t) from the measured output y(t) at different stations. Careful reading of existing literatures in time domain method of moving load identification (Law et al, 1997; Chan et al, 1998; Zhu and Law, 2003) reveals that identification problem finally could be transformed to a linear algebraic equation of the form
} { )}
( ]{
[A xt = b (2.2)
where {x(t)}∈Rn is a time series vector of unknown force F(t), {b}∈Rk is the time series vector of measured response of bridge deck. The system coefficient matrix A∈Rn×k is associated with bridge-vehicle system. In general number of measured stations (k) is not equal to the number of unknown time dependent force. Such type of situation occurring in moving load identification is falling into class of ill-posed problem (Uhl, 2007). For this, unique solution does not exist and there is no globally defined solution on the given data.
But when n>k, the equation have a solution given in the least square sense as }
{ ] [ )}
(
{xt = A+ b (2.3)
where [A]+ denotes pseudo inverse of the matrix [A], defined by
T
T A A
A
A] {[ ] [ ]} [ ]
[ + = −1 (2.4) Some authors (Yu and Chan 2003, Jankawshi 2013) recommended using singular value decomposition (SVD) instead of pseudo-inverse solution (PI). Stevens (1987) suggested regularization method to overcome the difficulties of ill-conditioned matrix. Regularization process is the technique to transform ill-posed problem to well posed problem. Popular regularization method is Tikhonov method (Tiknonov and Arsenin, 1977) which has been successfully applied in moving load identification problem.
2.2.1.2 Frequency domain method
In linear dynamics, output and input is related in frequency domain as )}
( )]{
( [ )}
(
{X ω = H ω F ω (2.5) In which {X(ω)} is the Fourier transform of the vector of response co-ordinates, {F(ω)} is the Fourier transform of the input excitation and [H(ω)] is a complex matrix called
“Frequency Response Function”, which depends on inertia, stiffness and damping properties of the system. The expression for [H(ω)] is given below
1
2[ ] [ ] [ ]}
{ )]
(
[H ω = −ω M +iωC + K − (2.6) With the knowledge of bridge-vehicle system model, frequency domain representation of moving force estimation can be expressed as before in time domain method,
)}
( { )}
( )]{
(
[Aω f ω = bω (2.7)
Where [A(ω)] is the system matrix in frequency domain, {f(ω)} is frequency variation of moving force and {b(ω)} is the frequency domain function of the measured bridge response at selected station. The number of unknown forces is usually less than the number of stations at which response is measured. Under such circumstances, the system is over determined and usually pseudo inverse of the matrix [A(ω)]+ post-multiplied by {b(ω)}
yields the Fourier transform of unknown moving forces. If matrix [A(ω)] is square, then [A(ω)]+=[A(ω)]-1, in which case unknown force vector can be directly found from Eq.(2.7).
If matrix [A(ω)] is singular, then problem is ill-posed. A regularization technique as in time
domain method was tried by various authors (Zhu and Law, 2002; Yu and Chan, 2003).
Such problems in inverse techniques demands tremendous efforts in obtaining correct estimates. Finally the moving load time history is to be obtained by inverse Fourier Transformation process.
2.2.2. Indirect Inverse Method
In this inverse method, first vehicle parameters are identified from measured bridge response signal. The response may be strain, displacement or acceleration picked up by the sensors while the vehicle is in motion. On obtaining an estimate of system parameters, mathematical model of the coupled system is solved to reconstruct axle load time history Following indirect inverse schemes have been used for estimating vehicle parameters from bridge dynamic response
2.2.2.1 Optimization based indirect inverse scheme
In an optimization based indirect inverse scheme, an objective function was constructed based on the residual between the measured and simulated bridge response using Least square method given by Deng and Cai (2009).
=
−
= n
i m s
obj r i r i
F
1
)}2
( ) (
{ (2.8) where i and n are the time-point number and total number of time points in the response time history respectively; rm and rs are the measured and simulated response time histories, respectively.
Vehicle axle load could be reconstructed from the identified vehicle parameters using stochastic global search technique, called Genetic Algorithm (GA) based on the mechanics of natural genetics.
Setting proper upper bound and lower bound as well as a proper set of initial values for the parameters to be identified in the GA program, the objective function (Fobj) can then be optimized. The identified error can be calculated as
%
×100
= −
true true iden
P P Error P
Identified (2.9) where Piden and Ptrue are the identified parameter and the true parameter, respectively.
2.2.2.2 Probabilistic based indirect inverse scheme
Another technique adopted in direct inverse scheme was based on Bayesian inference, called particle filter method (Nasrellah and Manohar, 2010). Noisy measured bridge response had been utilized for vehicle parameters estimation so that axle load time history can be re-constructed. Probabilistic based indirect inverse scheme is elaborated in the next section.
In the light of past research works, more realistic vehicle model has been found to be needed for better estimation of vehicle parameters and axle load identification. When bridge idealization is concerned, a beam model instead of a plate mode had been found to be accurate enough in the identification when the lower modes of the bridge deck are dominated by vibrational modes along the longitudinal axis (Zhu and Law, 2000). Also, it has been found that Finite element beam model requires a lot more of the measured information to have the same accuracy as the modeling of bridge as a continuous system (Zhu and Law, 2003). Identification using Direct inverse scheme is limited to simple forward scheme due to ill condition comes into picture. Even though regularization method had been employed to overcome this deficiency, significant identification error have been reported when bridge random surface profile was taken into consideration for forward scheme. In view of these, probabilistic based indirect inverse method has been adopted for identification of vehicle parameters as well as axle load estimation in the present study which has capability to incorporate uncertainties present in the forward scheme.
Uncertainties may be related to modelling error of the physical system, measurement noise, vehicle path eccentricity and variation of vehicle speed during bridge responses measurement. The Particle filter method involves large number of iteration. In order to overcome this drawback, a closed form solution has been proposed for the sample generations in forward scheme.
For the purpose of vehicle load identification, the modeling of vehicle - bridge coupled system and determination of dynamic response has been referred to as a “Forward scheme”. In the present work, three different vehicle models such as Quarter Car model, Flexible Half Car model and Flexible Full Car model have been considered in bridge- vehicle interaction dynamics. Bridge has been idealized as simply supported Euler Bernoulli’s beam incorporating flexural and torsional vibration mode. In addition to this, bridge deck surface has been considered as non homogeneous random process in space
domain. Detail formulation and method of solution have been given in Chapter-3 and Chapter-4.