# Research on Non-Random Vibration Analysis of Concrete Pump Truck Boom Based on Dynamic Excitation
## 2 Boom Dynamics Analysis  
This article focuses on the boom of a concrete pump truck, seen in Figure 1, which comprises six components. The mathematical model of the pump truck boom system was formulated utilizing the Denavit-Hartenberg (D-H) approach [20]. The velocity vectors of the center of gravity for each rigid body in the boom system were computed in the base coordinate system using the rotation transformation matrix derived from the D-H approach, in accordance with the principles of mechanical system dynamics equations. Additionally, utilizing these velocity vectors, the formulations for the total kinetic energy and total potential energy of the pump truck system were established, and the dynamic equations of the pump truck boom system were then obtained from these formulations [21]. The dynamic model of the pump truck boom system is ultimately formulated in matrix form, establishing a foundation for the comprehensive investigation of the system's dynamic characteristics. The shape of the boom, as presented in Table 1, yields the D-H parameters for the boom, which are detailed in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Connecting Rod Number** & **Mass/kg** & **Length/m** \\ \hline
1 & 1 & 1 \\
2 & 3352.785 & 11.85 \\
3 & 2241.339 & 9.25 \\
4 & 1457.113 & 9 \\
5 & 1144.211 & 11.635 \\
6 & 457.417 & 7.31 \\
7 & 207.387 & 4.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The D-H parameters. Figure 1: Concrete pump truck boom structure diagram. The total kinetic energy of the boom system of a concrete pump truck comprises two components: the translational kinetic energy resulting from the linear motion of the rigid body mass, and the rotational kinetic energy arising from the body's rotation about the joint. The total kinetic energy of the flexible boom system in a certain kind of concrete pump truck can be articulated as  
\[T=T_{1}+T_{2}=\sum_{j=1}^{13}\frac{1}{2}m_{j}V_{j}^{T}V_{j}+\sum_{j=1}^{13} \left(\frac{1}{2}I_{xj}\dot{\theta}_{xj}^{2}+\frac{1}{2}I_{yj}\dot{\theta}_{yj} ^{2}+\frac{1}{2}I_{zj}\dot{\theta}_{zj}^{2}\right) \tag{1}\]  
where \(V_{j}\) is the velocity of each mass relative to the base coordinate, \(V_{j}=[v_{xj}\ v_{yj}\ v_{zj}\ 0]^{T}=\frac{dR_{j}}{dt}\). If the coordinates of the center of mass of the \(i\) rigid body are \(R_{i}\) = [\(x_{i}\), \(y_{i}\), \(z_{i}\), 1]\({}^{T}\), then its coordinates relative to the base coordinate \(R_{0}\) are given by \(R_{0}\) = \({}^{0}T_{i}\ R_{i}\). When \(j=1\), its calculation expression is \(I_{1}=\frac{1}{2}md^{2}\); when \(j\
eq 1\), its calculation formula is \(I_{2}=\frac{1}{3}mx^{2}\). The aggregate potential energy of a concrete pump truck boom system is expressed by the equation:  
\[V=V_{g}+V_{e}=\sum_{i=1}^{21}m_{i}\cdot g\cdot h_{i}+\sum_{i=1}^{3}\frac{1}{2} k_{i}\dot{\theta}_{i}^{2}+\sum_{i=4}^{21}\frac{1}{2}k_{i}(\theta_{i}-\theta_{i-3})^{2} \tag{2}\]  
where \(V_{g}\) represents the entire gravitational potential energy of the system, and \(V_{e}\) denotes the elastic potential energy of the system. The Lagrangian technique \(\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_{i}}\right)-\frac{\partial L }{\partial\dot{q}_{i}}=f_{i}\) is utilized to model the system's dynamics [22], with \(f_{i}\) computed as \(f_{i}=-c_{i}\cdot\theta_{i}\). The following is a mathematical expression for the equation of motion:  
\[[I]\left\{\dot{\theta}(t)\right\}+[C]\left\{\dot{\theta}(t)\right\}+[K]\left\{ \theta(t)\right\}=\left\{\tau(t)\right\} \tag{3}\]  
The Lagrange method is employed to resolve the equations of motion for the mechanical system, thereby establishing the dynamic model of the pump truck boom system. This, in turn, enhances comprehension of the dynamic characteristics of the pump truck boom system and aids in the calculations for subsequent analytical methods. When creating the mechanical model of the boom system, the classical beam bending theory provides a quantitative description of the boom's stiffness properties.