## Online Dynamic Load Identification Method
### Infinite Number of Degrees of Freedom and Modal Truncation  
The computations involved in the above algorithm increases exponentially with the increase in the number of degrees of freedom. Hence, depending on the computational power the above algorithm may not achieve real-time. Here it should be noted that by real time we mean that the CPU time needed to evaluate the dynamic load is an order of magnitude shorter than the time interval \(\delta t\). Although the algorithm can easily achieve that for relatively small number of degrees of freedom, but this becomes more challenging when studying a complex structure. In this subsection we aim to introduce some changes so that the algorithm will remain efficient even at a large degree of freedom. To this end the modal analysis theory is used to achieve the modal transformation. Hence, the n-order dominant modal characteristic of a structure is used to replace the full modal characteristic. Thus, the computations can be significantly reduced and higher efficiency can be achieved so that the algorithm can still identify the dynamic load in real time even for complex structures. Figure 3, shows a simply supported beam subjected to dynamic load at a given position \(l_{a}\). The beam length is \(l\) and the dynamic load functions is \(u(t)\). Using the modal transform method, the response of the beam can be given as  
\[\mathbf{p}(t)=\mathbf{w}\mathbf{q}(t) \tag{31}\]  
where \(\mathbf{q}(t)\) is the modal displacement vector and \(\mathbf{w}=[\varphi_{1},\varphi_{2}\cdots\varphi_{n}]\) the modal matrix. Substituting Equation (31) into (1), we get  
\[\mathbf{w}^{\top}\mathbf{M}\mathbf{w}\bar{\mathbf{q}}(t)+\mathbf{w}^{\top} \mathbf{C}\mathbf{w}\bar{\mathbf{q}}(t)+\mathbf{w}^{\top}\mathbf{K}\mathbf{ w}\mathbf{q}(t)=\mathbf{w}^{\top}\mathbf{B}_{u}\mathbf{u}(t) \tag{32}\]  
Since the mode shapes are orthogonal we can write  
\[\mathbf{w}^{\top}\mathbf{M}\mathbf{w}=\mathbf{I} \tag{33}\]  
\[\mathbf{w}^{\top}\mathbf{K}\mathbf{w}=\mathbf{I} \tag{34}\]  
where \(\mathbf{I}=\mathrm{diag}(\lambda_{1},\lambda_{2}\cdots\lambda_{n})\) and \(\lambda_{i}=\omega_{i}^{2}\) is the \(i\)th-order eigenvalue. Then Equation (32) can be simplified as  
\[\bar{\mathbf{q}}(t)+\mathbf{g}\bar{\mathbf{q}}(t)+\mathbf{l}\mathbf{q}(t)= \mathbf{w}^{\top}\mathbf{B}_{u}\mathbf{u}(t) \tag{35}\]  
where \(\mathbf{g}=\mathbf{w}^{\top}\mathbf{C}\mathbf{w}\) is modal damping matrix where we assume proportional damping in the system i.e., \(\mathbf{g}=\mathrm{diag}(2\zeta_{1}\omega_{1},2\zeta_{2}\omega_{2},\cdots 2 \zeta_{n}\omega_{n})\) with \(\zeta_{i}\) being the damping ratio of the \(i\)th-mode shape. After transforming the continuous system to a discrete one the approach proposed in the previous section can now be applied where in the modal space the state vector can be defined as  
\[\mathbf{x}(t)=\left[ \begin{array}{c}\mathbf{p}(t)\\ \dot{\mathbf{p}}(t)\end{array} \right] \tag{36}\]  
Figure 3: Simply supported beam model under single-point concentrated dynamic load. while the system \(\mathbf{A}_{c}\) and the control \(\mathbf{B}_{c}\) matrices are rewritten as  
\[\mathbf{A}_{c}=\left[ \begin{array}{cc}\mathbf{0}&\mathbf{I}\\ -\mathbf{1}&-\mathbf{g}\end{array} \right],\qquad\mathbf{B}_{c}=\left[ \begin{array} []{cc}\mathbf{0}\\ \mathbf{w}^{\top}\mathbf{B}_{u}\end{array} \right] \tag{37}\]  
Again, if the measured response is acceleration we can obtain the matrices \(\mathbf{H}\) and \(\mathbf{D}\) by  
\[\mathbf{H}=\left[ \begin{array}{cc}-\mathbf{H}_{0}\mathbf{w}\mathbf{l}&- \mathbf{H}_{0}\mathbf{w}\mathbf{g}\end{array} \right],\qquad\mathbf{D}=\mathbf{ H}_{0}\mathbf{w}\mathbf{w}^{\top}\mathbf{B}_{u} \tag{38}\]  
Once the matrices \(\mathbf{A}_{c},\mathbf{B}_{c},\mathbf{H}\) and \(\mathbf{D}\) are obtained for the continuous system the online-identification algorithm of dynamic loads can be used in the same way as before. But again it should be noted that the number of critical mode shapes to be considered will affect the speed of computations in achieving real time.