# Research on Non-Random Vibration Analysis of Concrete Pump Truck Boom Based on Dynamic Excitation
## 4 Non-Random Vibration Analysis Method
### Dynamic Response Boundary Solving  
This work uses the Monte Carlo simulation method to forecast the dynamic response boundaries of the concrete pump truck boom system under unpredictable excitation. This method effectively estimates the response range of the system under unclear parameters, hence establishing the upper and lower bounds of the system [27]. When the system experiences a time-varying unknown external force, this force is regarded as an interval process \(f(t)\). In this instance, explicitly solving the equations is challenging because of the time-dependent uncertainty stemming from both the external force and the system parameters. To address this issue, we initially discretize the continuous time-varying variables, and the dependent interval process can be converted into mutually independent interval processes, referred to as \(M_{I}(t)\), \(K_{I}(t)\), and \(C_{I}(t)\), using the transformed system parameters. The following is a mathematical expression for the equation of motion [28]:  
\[m^{I}(t)\ddot{x}(t)+C^{I}(t)\dot{x}(t)+K^{I}(t)x(t)=f^{I}(t) \tag{15}\]  
\(x(t)\), \(\dot{x}(t)\), \(\ddot{x}(t)\), and \(f^{I}(t)\) represent displacement, velocity, acceleration, and external excitation, respectively. \(m^{I}(t)=[m^{m}(t)-m^{I}(t)\), \(m^{m}(t)+m^{I}(t)]\), \(k^{I}(t)=[k^{m}(t)-k^{I}(t)\), \(k^{m}(t)+k^{I}(t)]\), \(c^{I}(t)=[c^{m}(t)-c^{I}(t)\), \(c^{m}(t)+c^{I}(t)]\), \(f(t)=[f^{m}(t)-f(t),f^{m}(t)+f(t)]\). The Newmark-\(\beta\) algorithm can be employed to solve Formula (15). The solution statement is reformulated into the following matrix equation:  
\[ \begin{bmatrix}x_{i+1}\\ \dot{x}_{i+1}\\ \ddot{x}_{i+1}\end{bmatrix} =B \begin{bmatrix}x\\ f\end{bmatrix} \!,B= \begin{bmatrix}\begin{pmatrix}(B_{x})_{1}&(B_{f})_{1}\\ (B_{x})_{2}&(B_{f})_{2}\\ (B_{x})_{3}&(B_{f})_{3}\end{bmatrix}  \tag{16}\]  
To facilitate description and analysis, the uncertainty domain of the external force can be represented by an ellipsoidal model, expressed as  
\[(\mathbf{f}-\mathbf{f}^{m})^{T}\mathbf{P}_{f}^{-1}(\mathbf{f}-\mathbf{f}^{m})\leq 1 \tag{17}\]  
The aforementioned equations indicate that determining the upper and lower bounds of the system response can be reframed as two optimization tasks. These two optimization problems are employed to identify the extreme values of the system response under uncertainty, hence establishing the boundary range of the dynamic response. The particular optimization challenges are delineated as follows:  
\[ \begin{array}{l}\max\ x_{i+1}=(\mathbf{B}_{x})_{l}\mathbf{x}+(\mathbf{B}_{f})_{ l}\mathbf{f}\\ \mathrm{s.t.}\ \ \left(\mathbf{f}-\mathbf{f}^{m}\right)^{T}\mathbf{P}_{f}^{-1}( \mathbf{f}-\mathbf{f}^{m})\leq 1\\ \min\ x_{i+1}=(\mathbf{B}_{x})_{l}\mathbf{x}+(\mathbf{B}_{f})_{l}\mathbf{f}\\ \mathrm{s.t.}\ \ \left(\mathbf{f}-\mathbf{f}^{m}\right)^{T}\mathbf{P}_{f}^{-1}( \mathbf{f}-\mathbf{f}^{m})\leq 1\end{array}  \tag{18}\]  
According to optimization theory, the optimal solution to the problem can be attained when the following equations are formulated:  
\[(\mathbf{f}-\mathbf{f}^{m})^{T}\mathbf{P}_{f}^{-1}(\mathbf{f}-\mathbf{f}^{m})=1 \tag{19}\]  
Consequently, the optimization problem can be addressed via the Lagrange multiplier approach [29]. Formulate the expression for the Lagrange function as follows:  
\[L=(\mathbf{B}_{x})_{l}\mathbf{x}+(\mathbf{B}_{f})_{l}\mathbf{f}+\lambda( \mathbf{f}-\mathbf{f}^{m})^{T}\mathbf{P}_{f}^{-1}(\mathbf{f}-\mathbf{f}^{m}) \tag{20}\]  
Obtain the ideal solution:  
\[f=f^{m}\pm\frac{\mathbf{P}_{f}(\mathbf{B}_{f})_{l}^{T}}{\sqrt{\left(\mathbf{ B}_{f}\right)_{l}\mathbf{P}_{f}(\mathbf{B}_{f})_{l}^{T}}} \tag{21}\]  
Consequently, the upper and lower bounds of the displacement at each time interval are expressed as  
\[ \begin{array}{l}x_{i+1}^{U}=(\mathbf{B}_{x})_{l}x+(\mathbf{B}_{f})_{l}f^{m} +\sqrt{(\mathbf{B}_{f})_{l}\mathbf{P}_{f}(\mathbf{B}_{f})_{l}^{T}}\\ x_{i+1}^{L}=(\mathbf{B}_{x})_{l}x+(\mathbf{B}_{f})_{l}f^{m}-\sqrt{(\mathbf{B}_{ f})_{l}\mathbf{P}_{f}(\mathbf{B}_{f})_{l}^{T}}\end{array}