# Vibration Damage Analysis of Bottom Hole Assembly Under Axial Impact Based on Dynamic Analysis
## 2 Impact Simulation Model
### Drill String Vibration Model  
To investigate the vibration-induced damage of the drill string system under impact loading, a simplified drill string model was established, as shown in Figure 4.  
In the simplified drill string model, the drill pipe, BHA, and drill bit are connected by linear springs with torsional stiffness and damping. During the drilling process, the WOB drives the bit into the formation while the rotary table rotates the BHA by actuating the Kelly, thereby enabling the drill bit to rotate and cut through the rock. The lower BHA and bit are modeled as a simplified flywheel. The drilling response of the bit is primarily characterized by the interaction between the weight on bit (W), torque (\(T_{\text{m}}\)), moment of inertia (\(J\)), penetration rate (V), and angular velocity (\(\omega\)).  
The drill string is divided into multiple components, with each element within a single component having constant geometric and material properties. Each node is assigned three translational degrees of freedom and three rotational degrees of freedom. Set \(w\), \(u\), \(v\) represent the axial displacements of the drill string in three directions, \(\theta_{x},\theta_{y},\theta_{z}\), denote the rotational angles (twist) in each direction, and \(q\), \(m\) indicate the shear forces and bending moments experienced by the drill string. The finite element force model for a single drill string element is illustrated in Figure 5.  
Figure 4: BHA simplified model.  
Figure 5: Stresses and deformations of beam elements.  
In the drill string system, the displacements at each node constitute the generalized coordinates of the system. Based on the assumed displacement field, the translational deformation of the element is expressed using shape functions as follows:  
\[\begin{array}{l}\omega(z,t)=[N_{w}]\{u_{1}\}\\ u(z,t)=[N_{w}]\{u_{2}\}\\ v(z,t)=[N_{v}]\{u_{3}\}\\ \theta(z,t)=[N_{\theta}]\{u_{4}\}\end{array} \tag{2}\]  
where:  
\[\begin{array}{l}\{u_{1}\}=\{u_{i},\theta_{yi},u_{j},\theta_{yi}\}^{T}\\ \{u_{2}\}=\{v_{i},-\theta_{xi},v_{j},-\theta_{xj}\}^{T}\\ \{u_{3}\}=\{w_{i},w_{j}\}\\ \{u_{4}\}=\{\theta_{zi},\theta_{zj}\}\end{array} \tag{3}\]  
\([N_{w}],[N_{u}],[N_{v}],[N_{\theta}]\) denotes the displacement interpolation (shape) function, and \(u_{1},u_{2},u_{3},u_{4}\) represents the displacement vector. The total energy of the element consists of kinetic energy and potential energy. Using time derivatives, the expression for kinetic energy is given by:  
\[\begin{array}{l}T_{e}=\frac{1}{2}\{\dot{u}_{1}\}^{T}[m_{SA}]\{u_{1}\}\frac{1} {2}\{\dot{u}_{2}\}^{T}([m_{SD}]+[m_{SR}])\{\dot{u}_{2}\}+2\Omega\{\dot{u}_{1} \}[J_{s}]\\ \{u_{2}\}-\frac{1}{2}\{\dot{u}_{3}\}^{T}([m_{SD}]+[m_{SR}]\{\dot{u}_{3}\})+ \frac{1}{2}\{\dot{u}_{4}\}^{T}[m_{SA}]\{\dot{u}_{4}\}+\frac{1}{2}/Q^{2}\end{array} \tag{4}\]  
where \(m_{SA}\) denotes the axial translational mass matrix of the element; \(m_{SD}\) is the transverse (lateral) mass matrix; \(m_{SR}\) represents the rotational mass matrix; and \(J_{s}\) is the inverse mass matrix of the element.  
By integrating the drill string vibration model with the wellbore vibration equations, a coupled dynamic model of the wellbore-drill string-bit-rock system is established:  
\[[M].\Big{\{}\ddot{U}\Big{\}}+[C].\Big{\{}\dot{U}\Big{\}}+[K].\{U\}=\{F(t)\} \tag{5}\]  
\([M],~{}[K],~{}[C]\) are the mass, stiffness, and damping matrices, respectively; \(\{U\},~{}\Big{\{}\dot{U}\Big{\}},\)\(\Big{\{}\ddot{U}\Big{\}}\) are the displacement, velocity, and acceleration vectors, respectively; and \(\{F(t)\}\) is the external force vector acting on the system.