## 5 Dynamic Analysis
### Simulation Results  
In this section the results of the dynamic simulations of the proposed system in different working conditions are presented. The main aim of the simulations is to verify that the mechanism works properly and, in particular, it is important to check the following conditions:  
1. If the projection of the cutting force along the sliding direction is less than the elastic force threshold of the mechanism, then the grinding wheel support should not move. This happens for small burr heights. With the considered setup (specified in Table 3), the elastic force threshold of the mechanism, which is equal to the elastic reaction force for \(x\to 0\), is 149.01 N). This force threshold corresponds to a burr height of 2.5 mm with the considered setup.
2. If the projection of the cutting force along the sliding direction exceeds the elastic force threshold, the grinding wheel support should move backwards according to the compliance of the mechanism and the dynamics of (27); moreover, for a constant burr height, an equilibrium condition should be achieved (after a transient) between the cutting force projection and the elastic reaction force and, therefore, a cut profile with constant height should be obtained.
3. If the burr height decreases, the compliant mechanism should move forward, and return towards the TDC. This condition is very important since it assures the stability of the system. Of course, even if the burr becomes very small, the mechanism can never reach the TDC due to the mechanical stop.  
Condition 1 assures the cutting precision of the system, if the cutting force (and the burr height) is below a given threshold. Condition 2 assures that the mechanism is compliant if the cutting forces are too high, so that they are reduced to an acceptable value; nevertheless, the cutting precision is jeopardized and an additional cutting cycle is necessary to completely remove the burr. Condition 3 assures that the mechanism returns back (up to the initial configuration) if the burr height decreases.  
In the simulations the parameters listed in Table 3 are used. A steel burr is considered with \(\mu=0.75\)[31]. The mechanical stop has been modeled as discussed in Section 5.2.  
Different burr profiles are used as an input to simulations. The outputs of simulations are the position \(x\) (directly related to the cut profile) and velocity \(\dot{x}\) of the grinding wheel support as a function of time. In order to verify the conditions 1-3, simulations are carried out with step variations of burr height. Between different steps the burr height is constant in order to verify that an equilibrium condition (cut profile with constant height) is obtained after an initial transient. Different test cases with different profiles of step burr height  
Figure 9: Normalized shape of function \(f\) (29).  
are analyzed as detailed below (Sections 5.3.1-5.3.5). Finally, this study will include a simulation in which the burr height is defined by a random function (Section 5.3.6).  
The dynamic simulations start (for \(t=0\) s) with the grinding wheel support in contact with the mechanical stop. For this reason, the position \(x\) is 0.3 mm for \(t=0\) s in all the simulations performed, which corresponds to the position of the mechanical stop. This means that the mechanism initial configuration is very near to the TDC. Due to the initial length of the springs, a small elastic force is present that presses the grinding wheel support against the mechanical stop.