## 3 Dynamic Response Analysis by Utilizing the Interval Quantitative Results
### Uncertainty Estimation with the Series Expansion Approach  
Consider the following differential equation of motion about a general dynamical system with \(r\) degree as  
\[\left\{ \begin{array}{l}\mathbf{M}\ddot{\mathbf{u}}(t)+\mathbf{P}\dot{\mathbf{u}}(t)+\mathbf{K} \mathbf{u}(t)=\mathbf{F}(t)\\ \mathbf{u}(t_{0})=\mathbf{u}_{0},\dot{\mathbf{u}}(t_{0})=\dot{\mathbf{u}}_{0}\end{array} \right. \tag{19}\]  
where \(\mathbf{M}=(m_{ij})\in\mathbb{R}^{r\times r}\), \(\mathbf{P}=(p_{ij})\in\mathbb{R}^{r\times r}\), \(\mathbf{K}=(k_{ij})\in\mathbb{R}^{r\times r}\) are the mass, damping, and stiffness matrices, respectively, \(\mathbf{F}(t)=(f_{i}(t))\in\mathbb{R}^{r\times 1}\) means the external load vector, \(\mathbf{u}(t),\ \dot{\mathbf{u}}(t),\ \ddot{\mathbf{u}}(t)\in\mathbb{R}^{r\times 1}\) are the response vectors of displacement, velocity and acceleration, and \(\mathbf{u}(t_{0})\) and \(\dot{\mathbf{u}}(t_{0})\) are the corresponding initial conditions. When the uncertainty influences are taken into consideration, the elements either in constitutive matrixes (\(\mathbf{M}\), \(\mathbf{P}\), and \(\mathbf{K}\)) or in external excitation vector \(\mathbf{F}(t)\) may be valued by the quantitative results of uncertainties \(\mathbf{x}\) and \(\mathbf{y}(t)\). In this study, the fluctuation of inherent structural parameters lies on the static uncertainty items, whereas the dispersion of loading conditions is derived from the integration of both static and dynamic issues. Thus, Equation (19) can be rewritten as  
\[\mathbf{M}(\mathbf{x})\cdot\ddot{\mathbf{u}}(\mathbf{x},\mathbf{y}(t),t)+\mathbf{P}(\mathbf{x})\cdot\dot{ \mathbf{u}}(\mathbf{x},\mathbf{y}(t),t)+\mathbf{K}(\mathbf{x})\cdot\mathbf{u}(\mathbf{x},\mathbf{y}(t),t)=\mathbf{ F}(\mathbf{x},\mathbf{y}(t),t) \tag{20}\]  
In point of fact, the uncertainty propagation problem can be stated as: Input the quantitative models of structural and loading uncertainties as cited in Section 2 (from original samples), and seek a set \(\Gamma\in\mathbb{R}^{r+1}\) of response being consistent with Equation (20), where \(\Gamma\) is the feasible domain of the known vector \(\mathbf{x}\) and \(\mathbf{y}(t)\), i.e.,  
\[\Gamma=\left\{ \begin{array}{l}\mathbf{u}(\mathbf{x},\mathbf{y}(t),t):\mathbf{M}(\mathbf{x}) \cdot\ddot{\mathbf{u}}(\mathbf{x},\mathbf{y}(t),t)+\mathbf{P}(\mathbf{x})\cdot\dot{\mathbf{u}}(\mathbf{x },\mathbf{y}(t),t)\\ \quad\quad+\mathbf{K}(\mathbf{x})\cdot\mathbf{u}(\mathbf{x},\mathbf{y}(t),t)=\mathbf{F}(\mathbf{x},\mathbf{y}( t),t),\\ x_{i}\in x_{i}^{l}=\left[x_{i}^{l},\ x_{i}^{l}\right],\ i=1,2,\cdots,m\\ y_{k}(t)\in y_{k}^{l}(t)=\left[y_{k}^{l}(t),\ y_{k}^{l}(t)\right],\ k=1,2, \cdots,n\end{array} \right\} \tag{21}\]  
Then, the following expressions can be obtained by ignoring the high order items, namely,  
\[\dot{\mathbf{q}}_{\mathbf{state}}\big{(}\mathbf{x}^{\mathbf{c}},\mathbf{y}^{\mathbf{c}}(t_{j+1}),t_{j+1} \big{)} =\mathbf{A}(\mathbf{x}^{\mathbf{c}})\cdot\mathbf{q}_{\mathbf{state}}\big{(}\mathbf{x}^\mathbf{{c}},\mathbf{y}^{\mathbf{c}}(t_{j}),t_{j}\big{)}+\mathbf{B}(\mathbf{x}^{\mathbf{c}})\cdot\mathbf{F} \big{(}\mathbf{x}^{\mathbf{c}},\mathbf{y}^{\mathbf{c}}(t_{j}),t_{j}\big{)} \tag{31}\]  
and  
\[\delta\dot{\mathbf{q}}_{\mathbf{state}}(t_{j+1}) =\mathbf{A}(\mathbf{x}^{\mathbf{c}})\cdot\delta\mathbf{q}_{\mathbf{state}}(t_{j})+ \delta\mathbf{A}\cdot\mathbf{q}_{\mathbf{state}}\big{(}\mathbf{x}^{\mathbf{c}},\mathbf{y}^{\mathbf{c}}(t_{ j}),t_{j}\big{)} \tag{32}\] \[+\mathbf{B}(\mathbf{x}^{\mathbf{c}})\cdot\delta\mathbf{F}(t_{j})+\delta\mathbf{B} \cdot\mathbf{F}\big{(}\mathbf{x}^{\mathbf{c}},\mathbf{y}^{\mathbf{c}}(t_{j}),t_{j}\big{)}\]  
With the precise time integration method, \(\mathbf{q}_{\mathbf{state}}(\mathbf{x}^{\mathbf{c}},\mathbf{y}^{\mathbf{c}}(t),t)\) and \(\delta\mathbf{q}_{\mathbf{state}}(t)\) can be explicitly reconstructed.