## 3 Definition of the Problem  
During the machining process of an aero-engine casing, the part is clamped by a set of fixtures, which in general can be divided into two kinds, i.e., one that provides the position and clamping function, and the other that is used to enforce rigidity. As shown in Figure 1a, the positioning and clamping elements are arranged on the lower end face. Multiple supports are arranged evenly on the inside of the part to increase the stiffness of the part. For radial deformation force monitoring, the challenging problems are the complexity of part deformation as well as the coupling effect of the change of stiffness. When the part is separated from the fixture due to deformation, it becomes impossible to monitor the change of clamping force. Therefore, it is necessary to exert an extra supporting force to maintain the contact between the part and the fixture. Obviously, the supporting force cannot be too small; less it would be unable to monitor the change in clamping force due to the machining deformation. Conversely, it cannot be too large, as that would cause a large clamping deformation and hence affect the machining accuracy. Figure 1: Clamping of an aero-engine casing and mechanism of residual stress release. (**a**) Clamping of an aero-engine casing; (**b**) Blank; (**c**) Part; (**d**) Clamping state before processing; (**e**) Clamping state after processing. The part deformation caused by the pre-support force \(F_{i}\) is calculated by (1):  
\[ \begin{split} d_{1}=F_{1}\cdot C_{1,1}+F_{2}\cdot C_{2,1}\cdots F_{n }\cdot C_{n,1}\\ \vdots\\ d_{n}=F_{1}\cdot C_{1,n}+F_{2}\cdot C_{2,n}\cdots F_{n}\cdot C_{n,n} \end{split}  \tag{1}\]  
where \(d_{i}\) is the deformation generated by the action of all the auxiliary supports at the clamping point \(i\), \(F_{i}\) is the pre-support force applied by the auxiliary support \(i\), and \(C_{i,j}\) denotes the equivalent flexibility of the force acting normal to the clamping point \(i\) at the clamping point \(j\). The auxiliary supports are arranged evenly along the circumference, and to reduce the effect of uneven clamping deformation on machining, a same amount of pre-support force is applied at each auxiliary support. Therefore, (1) can be simplified to (2):  
\[d=F\cdot C \tag{2}\]  
where \(F\) is the support force applied at each clamping point, \(C=\sum_{1}^{n}C_{,i}\) is the equivalent flexibility, and \(d\) is the clamping deformation. After one step of machining, the clamping force will change because of the rebalancing of the internal residual stress and the change of stiffness of the part. Therefore, the support force of each auxiliary support becomes \(F^{\prime}\), and the auxiliary support still works under the conditions of (3):  
\[F^{\prime}\geq 0 \tag{3}\]  
The part tends to deform after machining, but the fixture constrains its deformation. The deformation relationship between the part and the fixture after the machining is given by (4):  
\[d^{\prime}=F^{\prime}\cdot C^{\prime} \tag{4}\]  
where \(F^{\prime}\) is the support force at each clamping point, \(C^{\prime}=\sum_{i=0}^{n}C^{\prime}_{,i}\) is the equivalent flexibility, and \(d^{\prime}\). is the clamping deformation, all after the machining. As fixtures constrain the part deformation, the amount of part deformation \(\Delta\) can be expressed as in (5):  
\[\Delta=d-d^{\prime} \tag{5}\]  
According to the elastic mechanics, the deformation force can be expressed as in (6):  
\[F_{b}=\frac{\Delta}{C^{\prime}} \tag{6}\]  
The delta of the support force is given as in (7):  
\[\Delta F=F-F^{\prime} \tag{7}\]  
As a result, the change of clamping force can be expressed as in (8):  
\[\Delta F=\frac{d\times(C^{\prime}-C)}{C\times C^{\prime}}+F_{b} \tag{8}\]  
where \(C\) is the equivalent flexibility of the part before machining, \(C^{\prime}\) is the equivalent flexibility of the machined part, \(d\) is the clamping deformation, and \(F_{b}\) is the deformation force.