## 5 Assembly Contact Forces and Resulting Part Deviations
### Part Deviations Due to Assembly Contact Forces  
In the theoretical case, assuming that there is a point \(A_{1}\) on part 1 for the riveted or welded connection position, there must exist a point \(A_{2}\) on part 2 which is connected to part 1. Points \(A_{1}\) and \(A_{2}\) coincide in space, and points \(A_{1}\) and \(A_{2}\) are recorded as a pair of assembly connection points in the assembly process, whether there is interference or a gap between the two assembly planes. According to the law of linear elasticity, assuming that a unit force \(FA\) is applied to any point \(A\) on the part before riveting or welding, the displacement change at point \(A\) can be obtained as \(\delta_{A}\)=\(C_{A}\cdot F_{A}\). \(C_{A}\) is the flexibility coefficient of point \(A\), which indicates the displacement generated when point \(A\) is subjected to a unit force. From this, by dividing the part into a finite number of nodes, the relationship between the amount of displacement at all nodes and the force applied can be obtained:  
\[[\delta]=[C]\cdot[F] \tag{30}\]  
\[[F]=[K]\cdot[\delta] \tag{31}\]  
Figure 9: Part 1 is fully penetrated in part 2.  
In Equations (30) and (31), \([\delta]=[\delta_{A},\delta_{B},\delta_{C},\cdots,\delta_{N}]^{T}\) denotes the amount of displacement of all nodes, \([F]=[F_{A},F_{B},F_{C},\cdots,F_{N}]^{T}\) denotes the forces acting on all nodes; \([C]\) is the multidimensional flexibility matrix of the part, representing the displacement produced by each node when all nodes on the part are subjected to a unit force; \([K]\) is the multidimensional stiffness matrix of the part, \([K]=[C]^{-1}\).  
Equations (30) and (31) can accurately explain the relationship between nodal forces and nodal displacements, where \([\delta]\) and \([F]\) contain information on the displacements and bearing forces of all nodes. At workstation \(k\), part 1 and part 2, the two parts to be connected, are set to be assembled; in the global coordinate system, the flexibility matrix of the two are \([C_{1}\ ]\) and \([C_{2}\ ]\), the two parts to be connected by the assembly forces are \([F_{1}(k)]\) and \([F_{2}(k)]\), then the two parts' displacement \([\delta_{1}(k)]\) and \([\delta_{2}(k)]\) is:  
\[\big{[}\delta(k)\big{]}\!=\!\begin{bmatrix}\delta_{1}(k)\\ \delta_{2}(k)\end{bmatrix}\!=\!\begin{bmatrix}C(k)\ \end{bmatrix}\!\cdot\!\! \begin{bmatrix}F_{1}(k)\\ F_{2}(k)\end{bmatrix}\!=\!\begin{bmatrix}C_{1}\\ C_{2}\end{bmatrix}\!\cdot\!\!\begin{bmatrix}F_{1}(k)\\ F_{2}(k)\end{bmatrix} \tag{32}\]  
In Equation (32), \(F_{1}(k)=\big{[}F_{A_{1}}(k)\ F_{B_{1}}(k)\ \cdots\ F_{N_{1}}(k)\big{]}^{T}\), \(F_{A_{1}}=\big{[}f_{A_{1}x}(k)\ \ f_{A_{1}y}(k)\ \ f_{A_{1}z}(k)\big{]}^{T},\{A,B, \cdots N\}\) is the key point number on part1, \(\big{\{}f_{A_{1}x}f_{A_{1}y}f_{A_{1}z}\big{\}}\) is the component of the assembly force at point \(A\) on part 1 on the xyz axis. \(\delta_{1}\!=\!\big{[}\delta_{A_{1}}(k)\ \ \ \delta_{B_{1}}(k)\ \ \cdots\ \ \ \delta_{N_{1}}(k)\big{]}^{T},\delta_{A_{1}}\!=\!\Big{[}\Delta x_{A_{1}}(k)\ \ \ \Delta y_{A_{1}}(k)\ \ \ \Delta z_{A_{1}}(k)\Big{]}^{T}\) indicates part deviation due to assembly forces at point \(A\) on part 1.