Chung Cheng Institute of Technology, National Defense University, Ta-Hsi, Tao-Yuan, Taiwan, R.O.C.
NomenclatureĀ
Cross section of the beam
A,BConstant matrices
BsSensitivity matrix
CDamping matrix
EElastic modulus
FModel force
FKMoving tank force input
FmMissile impact force input
HMeasurement matrix
IUnit matrix
IaMoment of inertia
KTotal number of tank input forces
KaKalman gain
KbCorrection gain
KnModel stiffness matrix
kTime (discretised)
lLength of the beam element
MTotal number of missile impact forces
MnModel mass matrix
MsSensitivity matrix
NTotal number of estimation time steps
PState's error covariance matrix
PbFilter's error covariance matrix
QProcess noise covariance matrix
QwScalar of process noise covariance
RMeasurement noise covariance matrix
RvScalar of measurement noise covariance
rWeighting factor
SInnovation covariance
tTime (continuous)
uDisplacement of beam
XState vector
YDisplacement vector
Y˙Velocity vector
ŸAcceleration vector
ZObservation vector
ΓInput matrix
δKronecker delta
ΔtSampling time (interval)
ρsMass per unit length of the beam
σStandard deviation
ϑMeasurement noise vector
wProcess noise vector
ΦState transition matrix
Superscripts∧Estimated
–Estimated by filter
TTranspose of matrix
Subscriptsi, jIndices
A method to determine the moving tank and missile impact forces on a bridge is developed. The present method is an online adaptive recursive inverse algorithm, which is composed of the Kalman filter and the recursive least square estimator (RLSE), to estimate the force inputs on the bridge structure. The state equations of the bridge structure were constructed by using the model superposition and orthogonal technique. By adopting this inverse method, the moving tank and missile impact force inputs acting on the bridge structure system can be estimated from the measured dynamic responses. Besides, this work presents an efficient weighting factor applied in the RLSE, which is capable of providing a reasonable estimation results. The results obtained from the simulations show that the method is effective in determining the moving tank and missile impact forces, so that the acceptable results can be obtained.
Missile impact, Kalman filter, RLSE, recursive least square estimator, bridge structure, impact studies, impact forces
