# Distributed Dynamic Load Identification of Beam Structures Using a Bayesian Method
## 2 The Bayesian Formulation of the Distributed Dynamic Load Identification  
### Identification Model for Simply Supported Beams in the Frequency Domain  
For the simply supported beam shown in Figure 1, the connection between the distributed dynamic load and the response in the frequency domain is given by:  
\[\int_{0}^{l}H(x_{k},x,\omega)F(\omega,x)\mathrm{d}x=X(x_{k},\omega) \tag{1}\]  
Here, \(H(x_{k},x,\omega)\) denotes the frequency response function (FRF), \(x_{k}\) is the response position, and \(X(x_{k},\omega)\) is the frequency response. \(F(\omega,x)\), which is rewritten as Equation (2) in the complex field, is the fast Fourier transform (FFT) of \(f(x,t)\). Substituting Equation (2) into Equation (1), Equation (1) is rewritten as Equation (3). \[ \begin{array}{c}F^{\prime}(x)=\left[\overline{P}_{1}(x)\quad\overline{P}_{2}( x)\quad\dots\quad\overline{P}_{j}(x)\right]\left\{\begin{array}{l}a_{1}\\ a_{2}\\ \vdots\\ a_{j}\end{array}\right\}\\ F^{i}(x)=\left[\overline{P}_{1}(x)\quad\overline{P}_{2}(x)\quad\dots\quad \overline{P}_{j}(x)\right]\left\{\begin{array}{l}b_{1}\\ b_{2}\\ \vdots\\ b_{j}\end{array}\right\}\\ \end{array}  \tag{2}\]  
Figure 1: Simply supported beam model under distributed forces. \[\int_{0}^{l}\left[ \begin{array}{cccccc}H^{\prime}\overline{P}_{1}(x)&\cdots&H^{ \prime}\overline{P}_{j}(x)&-H^{i}\overline{P}_{1}(x)&\cdots&-H^{i}\overline{P}_ {j}(x)\\ H^{i}\overline{P}_{1}(x)&\cdots&H^{i}\overline{P}_{j}(x)&H^{\prime}\overline{P}_ {1}(x)&\cdots&H^{\prime}\overline{P}_{j}(x)\\ \end{array} \right]dx \begin{Bmatrix}a_{1}\\ \vdots\\ a_{j}\\ b_{1}\\ \vdots\\ b_{j}\end{Bmatrix} =\left\{ \begin{array}{c}X^{\prime}\\ X^{i}\\ X^{i}\end{array} \right\} \tag{3}\]  
where \(\overline{P}_{j}(x)\) is the normalized one-dimensional Legendre polynomials, and \(a_{j},b_{j}\) are the polynomial coefficients. \(H^{\prime},H^{i}\) are the real and imaginary parts of FRF, while \(X^{\prime},X^{i}\) denote the system responses. Therefore, Equation (3) is converted as Equation (4) in the case of a multipoint response and then abbreviated as Equation (5). \[\int_{0}^{l}\left[ \begin{array}{cccccc}H_{1}^{\prime}\overline{P}_{1}&\cdots &H_{1}^{\prime}\overline{P}_{j}&-H_{1}^{i}\overline{P}_{1}&\cdots&-H_{1}^{i} \overline{P}_{j}\\ H_{1}^{i}\overline{P}_{1}&\cdots&H_{1}^{i}\overline{P}_{j}&H_{1}^{\prime} \overline{P}_{1}&\cdots&H_{1}^{i}\overline{P}_{j}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ H_{n}^{\prime}\overline{P}_{1}&\cdots&H_{n}^{\prime}\overline{P}_{j}&-H_{n}^{i} \overline{P}_{1}&\cdots&-H_{n}^{i}\overline{P}_{j}\\ H_{n}^{i}\overline{P}_{1}&\cdots&H_{n}^{i}\overline{P}_{j}&H_{n}^{\prime} \overline{P}_{1}&\cdots&H_{n}^{\prime}\overline{P}_{j}\end{array} \right]dx \begin{Bmatrix}a_{1}\\ \vdots\\ a_{j}\\ b_{1}\\ \vdots\\ b_{j}\end{Bmatrix} =\left\{ \begin{array}{c}X_{1}^{\prime}\\ X_{1}^{i}\\ \vdots\\ X_{n}^{\prime}\\ X_{n}^{i}\end{array} \right\} \tag{4}\]  
\[Q\left\{ \begin{array}{c}A_{1}\\ B_{1}\end{array} \right\}=X \tag{5}\]  
In Equation (4), \(H_{n}^{r},H_{n}^{i}\) are the representations of the symbols \(H^{r}(x_{n},x,\omega)\), \(H^{i}(x_{n},x,\omega)\), while \(Q,A_{1},B_{1}\) in Equation (5) are given as:  
\[Q=\int_{0}^{l}\left\{ \begin{array}{c}\underline{Q}_{1}\\ \underline{Q}_{2}\\ \vdots\\ \underline{Q}_{n}\end{array} \right\}dx,Q_{n}=\left[ \begin{array}{cccccc}H_{n }^{\prime}\overline{P}_{1}&\cdots&H_{n}^{\prime}\overline{P}_{j}&-H_{n}^{i} \overline{P}_{1}&\cdots&-H_{n}^{i}\overline{P}_{j}\\ H_{n}^{i}\overline{P}_{1}&\cdots&H_{n}^{i}\overline{P}_{j}&H_{n}^{\prime} \overline{P}_{1}&\cdots&H_{n}^{\prime}\overline{P}_{j}\end{array} \right] \tag{6}\]  
\[A=\left\{ \begin{array}{c}a_{1}\\ \vdots\\ a_{j}\\ b_{1}\\ \vdots\\ b_{j}\end{array} \right\},A_{1}=\left\{ \begin{array}{c}a_{1}\\ \vdots\\ a_{j}\end{array} \right\},B_{1}=\left\{ \begin{array}{c}b_{1}\\ \vdots\\ b_{j}\end{array} \right\},X=\left\{ \begin{array}{c}X_{1}^{\prime}\\ X_{1}^{\prime}\\ \vdots\\ X_{n}^{\prime}\\ X_{n}^{i}\end{array} \right\} \tag{7}\]  
where \(Q\) is the dynamic calibration matrix [3].