**4. Design**

In normal cutting conditions, the mechanism should have high stiffness to ensure high cutting precision. On the other hand, when the burr is too high the mechanism should exploit its compliance to reduce the cutting forces and, as a consequence, a second cutting cycle is necessary to completely remove the burr.

The desired regressive behavior of the mechanism is related to its geometrical parameters and to the springs stiffness. If the link lengths (*l*1 and *l*2) are defined by the scale of the application (related to the workpiece dimensions and maximum burr height), there are 5 other parameters to be chosen (see Figure 5): *k*1, *k*2, *k*3, *a*, and *b*. It is not straightforward to find a good set of values by trial-and-error, so an optimization algorithm has been used. The algorithm aims at finding the parameters whose resulting mechanism forces *F* fit a desired (regressive) curve.

In this work the desired force profile is set to *F* = 150 − 300*x* [*N*] (blue dotted line in Figure 7) with *l*1 = 0.15 m and *l*2 = 0.3 m. The *fmincon* MATLAB function has been used to find the optimal parameters that minimize the error between the actual and desired force profiles. The optimization algorithm yields the values listed in Table 2. It is worth to notice from Figure 7 that the system is stable: the normal component of the deburring force (*Fb*) decreases with *x* faster than the elastic force on the slider (*F*):

$$\frac{\partial F\_b}{\partial x} < \frac{\partial F}{\partial x} \tag{20}$$

**Figure 7.** The optimization algorithm finds the mechanism parameters whose *<sup>F</sup>*(*x*) (red line) fits at best the objective curve (blue dotted line). Black line shows how *Fb*(*x*) decreases with the movement of the mechanism (with *h* = 5 mm).

**Table 2.** Parameters computed by the optimization algorithm.


The black line of Figure 7 is the deburring force (normal component, as discussed in Section 2). This curve moves to the right for higher burr heights (with higher cutting forces for *x* = 0), and vice versa. The intersection between this line and the optimized curve (red line in figure) yields the equilibrium point, i.e., the *x* value at which the mechanism moves to balance the deburring force and the elastic force of the mechanism. For very low values of *h* the black line does not intersect the optimized curve, so the mechanism does not move. In this case, a precision cut is ensured, with cutting forces below the force threshold that activates the mechanism compliance. For the parameters of Table 2 and Figure 7 (see Table 3 in Section 5.3 for the complete list of parameters) the burr height (steel) that activates the mechanism compliance is 2.5 mm.


**Table 3.** Simulation parameters.

It is important to notice that it is possible to perform the optimization starting from a different desired force curve and, in particular, different force thresholds can be set depending on the application. This will result in a different set of springs and anchor points.

From a mechanical point of view, it is important to include a mechanical end stop for the slider to avoid possible changes in configuration of the slider-crank (i.e., to keep the rocker arm link always above the slider axis) and to avoid that the mechanism works in the TDC. Indeed, the mechanism is designed to work near the TDC, where it can generate a finite elastic force, but not exactly in the TDC, which could be dangerous for the structural integrity of the mechanism if the burr height is very large. The detailed model of the end stop will be presented in Section 5.2.

#### **5. Dynamic Analysis**

#### *5.1. Dynamic Model*

The proposed mechanism has one degree of freedom and can thus be modeled using a single equivalent mass *m*<sup>∗</sup> that is excited via a single equivalent force *F*<sup>∗</sup>. The equivalent mass and force clearly depend on the mechanism configuration (*x*) and on its main parameters.

The equations of motion are obtained by recalling that the power equals the derivative of the kinetic energy:

$$P = \frac{dE\_k}{dt} \tag{21}$$

Since friction between the members of the mechanism and their potential energy can be neglected. The kinetic energy of the equivalent mass equals by definition the kinetic energy of all the members of the mechanism:

$$E\_k = \frac{1}{2}m^\*\dot{\mathbf{x}}^2 = \frac{1}{2}\left(m\dot{\mathbf{x}}^2 + m\_r\left(\dot{\mathbf{x}}\_{Gr}^2 + \dot{y}\_{Gr}^2\right) + I\_r\dot{\boldsymbol{\alpha}}^2 + m\_\mathbf{c}\left(\dot{\mathbf{x}}\_{\mathbf{Gc}}^2 + \dot{y}\_{\mathbf{Gc}}^2\right) + I\_\mathbf{c}\dot{\boldsymbol{\theta}}^2\right) \tag{22}$$

where .*x* is the velocity of the equivalent mass (equal to the one of the slider); *m* is the mass of the slider (included the spindle and the grinding wheel); *mr*, *Ir*, .*xGr*, .*yGr* are the mass, barycentric moment of inertia, barycenter velocity along the *x* axis, and barycenter velocity along the *y* axis of the rocker arm link, respectively; *mc*, *Ic*, .*xGr*, .*yGr* are the mass, barycentric moment of inertia, barycenter velocity along the *x* axis, and barycenter velocity along the *y* axis of the crank, respectively. The coordinates of the centers of mass of the rocker arm link and of the crank are (respectively):

$$\left\{ \begin{array}{c} \mathbf{x}\_{Gr} \\ \mathbf{y}\_{Gr} \end{array} \right\} = \left\{ \begin{array}{c} \mathbf{x} + \frac{l\_{2}}{2} \cos(a) \\ \frac{l\_{2}}{2} \sin(a) \end{array} \right\}, \left\{ \begin{array}{c} \mathbf{x}\_{\mathbb{G}c} \\ \mathbf{y}\_{\mathbb{G}c} \end{array} \right\} = \left\{ \begin{array}{c} l\_{1} + l\_{2} - \frac{l\_{1}}{2} \cos(\theta) \\ \frac{l\_{1}}{2} \sin(\theta) \end{array} \right\} \tag{23}$$

The corresponding velocities are:

$$\begin{Bmatrix} \dot{\mathbf{x}}\_{Gr} \\ \dot{y}\_{Gr} \end{Bmatrix} = \left\{ \begin{array}{c} \dot{\mathbf{x}} - \frac{l\_{2}}{2} \sin(a)\dot{a} \\ \frac{l\_{2}}{2} \cos(a)\dot{a} \end{array} \right\} \; \left\{ \begin{array}{c} \dot{\mathbf{x}}\_{Gc} \\ \dot{y}\_{Gc} \end{array} \right\} = \begin{Bmatrix} \frac{l\_{1}}{2} \sin(\theta)\dot{\theta} \\ \frac{l\_{1}}{2} \cos(\theta)\dot{\theta} \end{Bmatrix} \tag{24}$$

By introducing (24) and the speed ratios of (11), the final form of (22) is:

$$\begin{split} \frac{1}{2}m^\*\dot{\mathbf{x}}^2 = \frac{1}{2} \left( m + m\_r \left( 1 + \frac{l\_1^2}{4} \mathbf{r}\_{ax}^2 - l\_2 \sin(a) \mathbf{r}\_{ax} \right) + I\_r \mathbf{r}\_{ax}^2 + m\_c \left( \frac{l\_1^2}{4} \mathbf{r}\_{\theta x}^2 \right) \\ + I\_c \mathbf{r}\_{\theta x}^2 \right) \dot{\mathbf{x}}^2 \end{split} \tag{25}$$

The power *P* of *F*∗ can be calculated from the deburring force and the springs forces:

$$P = F^\* \dot{\mathbf{x}} = -(k\_1 \mathbf{s}\_1)\dot{\mathbf{s}}\_1 - (k\_2 \mathbf{s}\_2)\dot{\mathbf{s}}\_2 - (k\_3 \mathbf{x})\dot{\mathbf{x}} - (\mathbf{c}\_{aq}\dot{\mathbf{x}})\dot{\mathbf{x}} + F\_b \dot{\mathbf{x}} \tag{26}$$

in which the force of a damper to be connected in parallel to spring *k*3 (with damping *ceq*) is introduced in order to avoid undesired vibrations.

The equation of motion is found by differentiating (25) and putting it equal to (26):

$$\begin{aligned} m^\* \ddot{\mathbf{x}} + \frac{1}{2} \left( \left( 2I\_r + \frac{l\_2^2}{2} m\_r \right) \mathbf{r}\_{ax} \dot{\mathbf{r}}\_{ax} + \left( 2I\_c + \frac{l\_1^2}{2} m\_c \right) \mathbf{r}\_{\theta x} \dot{\mathbf{r}}\_{\theta x} \\ -m\_r l\_2 \left( \mathbf{r}\_{ax} \cdot \mathbf{a} \cdot \cos(a) + \dot{\mathbf{r}}\_{ax} \sin(a) \right) \right) \dot{\mathbf{x}}^2 = F^\* \dot{\mathbf{x}} \end{aligned} \tag{27}$$

The differential Equation (27) is implemented and solved numerically in MATLAB for different working scenarios. The simulation results and the simulation parameters used are presented in Section 5.

#### *5.2. Mechanical Stop Model*

To avoid the singular configuration at the TDC and changes in the mechanism configuration, a mechanical stop has been introduced. The mechanical stop should be made of soft material to reduce unnecessary shocks to the mechanism structure. The mechanical stop is modeled as a one degree of freedom system that is activated as soon as *xsup* < *xstop*, in which *xsup* is the coordinate of the surface of the slider and *xstop* is the coordinate of the mechanical stop when the slider is not in contact with it (see Figure 8).

**Figure 8.** Model of the mechanical stop of the mechanism.

The force transferred by the mechanical stop to the mechanism is:

$$F\_{stop} = k\_{stop} \left(\mathbf{x}\_{sup} - \mathbf{x}\_0\right)^\varepsilon + f\left(\mathbf{x}\_{sup}, \mathbf{x}\_0, h\_0, \mathbf{x}\_1, h\_1\right) \dot{\mathbf{x}} \tag{28}$$

where:


.Function *f* is described by a polynomial function, whose normalized shape is shown in Figure 9:

$$f\left(\mathbf{x}\_{\text{sup}}, \mathbf{x}\_{0}, h\_{0}, \mathbf{x}\_{1}, h\_{1}\right) = \begin{cases} h\_{0} & \mathbf{x}\_{\text{sup}} \ge \mathbf{x}\_{0} \\ h\_{0} + a\Delta^{2}(3 - 2\Delta) & \mathbf{x}\_{1} \le \mathbf{x}\_{\text{sup}} \le \mathbf{x}\_{0} \\ h\_{1} & \mathbf{x}\_{\text{sup}} \le \mathbf{x}\_{1} \end{cases} \tag{29}$$

where:

$$a = h\_1 - h\_0 \quad , \quad \Delta = \frac{\mathbf{x}\_{sup} - \mathbf{x}\_0}{\mathbf{x}\_1 - \mathbf{x}\_0} \tag{30}$$

**Figure 9.** Normalized shape of function *f* (29).

#### *5.3. 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:


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 *μ* = 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 .*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 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.

#### 5.3.1. Test Case 1—1-Step Profile

A 1-step burr profile with a constant height (*h* = 5 mm, see solid line in Figure 10 (top)) is used as an input for the dynamic simulation. The burr height is sufficiently high to make the projection of the cutting force along the sliding direction exceed the elastic force threshold of the mechanism (dashed line in Figure 10). The simulation results, i.e., the position and velocity of the grinding wheel support in function of time, are presented in Figure 10 (middle and bottom). From the analysis of Figure 10, it can be noticed that the grinding wheel support moves backwards (positive values of *x*) thanks to the compliance of the mechanism; moreover, an equilibrium condition is achieved (after a transient of about 0.25 s with a small overshoot) between the cutting force projection and the elastic reaction force and, therefore, a cut profile with constant height is obtained. Therefore, the simulation results for this test case are in line with what was expected in this case (Condition 2 of Section 5.3). The burr height, initially equal to 5 mm, after processing is about 2.5 mm. This means that a second cutting cycle is necessary to completely remove the burr. The maximum (normal) deburring force reduces to 148 N from the nominal value of 300 N (corresponding to the nominal height of 5 mm), i.e., roughly a 50% (automatic) force reduction is obtained, which is related to the 50% reduction of the cutting height (2.5 instead of 5 mm).

**Figure 10.** Step burr profile (*h* = 5 mm, **top**); position (**middle**) and velocity (**bottom**) of the grinding wheel support.

5.3.2. Test Case 2—3-Steps Burr Profile "Low–Low–High"

A 3-steps burr profile "low–low–high" (see Figure 11 (top)) is used as an input for the dynamic simulation. "Low"/"high" means that the burr height is not/is sufficiently high to make the projection of the cutting force along the sliding direction exceed the elastic force threshold of the mechanism. In this test case, steps with increasing height are investigated. In the first two steps the cutting force is below the elastic force threshold, whereas in the third step the cutting force is sufficiently high (above the threshold) to make the mechanism work in the compliant mode of operation. The simulation results are presented in Figure 11 (middle and bottom). It can be noticed that the grinding wheel support does not move while the first two steps are processed (precision cut), and it moves backwards thanks to the compliance of the mechanism while the third step is processed. Similarly to the previous test case, an equilibrium condition is achieved (after a transient of about 0.3 s after the beginning of step 3) and a cut profile with constant height is obtained for step 3, as it can be noticed in Figure 11. Additionally, in this test case the simulation results confirm the expected dynamic behavior of the system: the grinding wheel support moves backward only when the cutting force projection exceeds the elastic force threshold, and when this happens a constant cut profile is obtained, after an initial transient (Condition 1 and Condition 2 of Section 5.3). From another point of view, for small burr heights the mechanism is (ideally) infinitely stiff, whereas it becomes compliant for high burr heights. Some small oscillations can be noticed when the burr height changes value but is not sufficiently high to allow the mechanism to work in the compliant mode (i.e., the grinding wheel support does not move and the cutting precision is ensured). This is due to the presence of the mechanical stop, which is not infinitely stiff. Nevertheless, this effect causes a negligible effect on the grinding wheel position, as it can be noticed in Figure 11 (middle). The maximum (normal) deburring force reduces again to 148 N from the nominal value of 240 N (corresponding to the nominal height of 4 mm), i.e., roughly a 38% (automatic) force reduction is obtained, which is related to the 38% reduction of the cutting height (2.5 instead of 4 mm).

**Figure 11.** Three-steps burr profile ("low–low–high" (**top**)); position (**middle**) and velocity (**bottom**) of the grinding wheel support.

5.3.3. Test Case 3—3-Steps Burr Profile "High–High–High"

A 3-steps burr profile "high–high–high" (see Figure 12 (top)) is used as an input for the dynamic simulation. Similarly to the previous test case, 3 steps with increasing height are investigated. Differently with respect to the previous test case, in this case, for all the 3 steps the cutting force is sufficiently high to make the mechanism work in the compliant mode of operation. The aim of this test case is to verify that, when it passes from one step to another, the system moves from an equilibrium condition to a new equilibrium condition. The simulation results are presented in Figure 12 (middle and bottom). It can be noticed that, at each step, the grinding wheel moves backwards thanks to the compliance of the mechanism. Three different equilibrium conditions are achieved (after a transient of about 0.25 s each time) with increasing height of the residual burr and, inside each step, a cut profile with constant height is obtained (after the transient), as it can be noticed in Figure 12. The simulation results of this test case confirm the expected dynamic behavior of the system: each time that the system encounters a higher step, it moves to a new equilibrium condition with a higher height of residual burr (in particular, Condition 2 of Section 5.3 is satisfied for all the 3 steps). The maximum (normal) deburring force reduces to 147.3 N from the nominal value of 360 N (corresponding to the nominal height of 6 mm), i.e., roughly a 60% (automatic) force reduction is obtained, which is related to the 60% reduction of the cutting height (2.5 instead of 6 mm).

**Figure 12.** Three-steps burr profile ("high–high–high" (**top**)); position (**middle**) and velocity (**bottom**) of the grinding wheel support.

5.3.4. Test Case 4—3-Steps Burr Profile "High–Highest–High"

A 3-steps burr profile "high–highest–high" (see Figure 13 (top)) is used as an input for the dynamic simulation. The burr height is *h* = 4 mm in the first step, then it becomes *h* = 6 mm in the second step, and finally it returns to *h* = 4 mm in the third step. In all the three steps the burr height is sufficiently high to make the mechanism work in the compliant mode of operation. The main aim of this test case is to verify that, if the burr height decreases, the compliant mechanism moves forward and returns towards the TDC, as specified in Condition 3 of Section 5.3. The simulation results are presented in Figure 13 (middle and bottom). It can be noticed that, during the first two steps, the grinding wheel moves backwards thanks to the compliance of the mechanism, and two different equilibrium conditions are achieved with increasing height of the residual burr. Moreover, in the third step, the compliant mechanism moves forward and rapidly returns towards the TDC. In this step, a new equilibrium condition is obtained, with the same height of residual

burr as in the step 1 (as expected, since the burr height is equal in steps 1 and 3). Therefore, the mechanism works properly and, in particular, Condition 3 of Section 5.3 is satisfied.

**Figure 13.** Three-steps burr profile ("high–highest–high" (**top**)); position (**middle**) and velocity (**bottom**) of the grinding wheel support.

5.3.5. Test Case 5—3-Steps Burr Profile "High–Low–High"

A 3-steps burr profile "high–low–high" (see Figure 14 (top)) is used as an input for the dynamic simulation. In the first and third step the burr height is sufficiently high to make the mechanism work in the compliant mode of operation, whereas in the second step the burr height is sufficiently low to generate a cutting force projection below the elastic force threshold. The main aim of this test case is to verify that, when the burr height decreases and generates a cutting force projection below the elastic force threshold, the mechanism returns in the initial configuration, i.e., with the grinding wheel support in contact with the mechanical stop. The simulation results are presented in Figure 14 (middle and bottom). It can be noticed that, during the first step, the grinding wheel moves backwards thanks to the compliance of the mechanism, and an equilibrium condition is achieved with a certain height of the residual burr. In the second step, the compliant mechanism moves forward and rapidly returns towards the TDC (Condition 3 of Section 5.3 is satisfied). Since the cutting force projection is below the elastic force threshold, the mechanism would tend to go to the TDC, but due the mechanical stop this is not possible. Indeed, an impact between the grinding wheel support and the mechanical stop takes place, as it can be noticed in Figure 14, in which small bounces are reported after *t* = 1.5 s. Of course, the amplitude and damping of these bounces depend on the dynamic parameters of the mechanism (masses, inertias, springs stiffness, and damping) and on the characteristics (stiffness and damping) of the mechanical stop. After the bounces, the mechanism remains in the initial configuration (*x* is about 0.3 mm, as it can be noticed in Figure 14), up to the beginning of the third step. Finally, in the third step, a new equilibrium condition is obtained, with a height of residual burr higher than in the first step (as expected, since the burr height is higher).

**Figure 14.** Three-steps burr profile ("high–low–high" (**top**)); position (**middle**) and velocity (**bottom**) of the grinding wheel support.

#### 5.3.6. Test Case 6—Generic Burr Profile

A set of generic burr profiles have been generated using a random variation of burr height in order to test the mechanism in more realistic scenarios. The random burr profiles are then used as an input for the dynamic simulations. Random profiles have been generated starting from the power spectral density (PSD) of the random process [33]. The three most common shapes for the noise have been assumed, namely white noise (flat PSD), flicker-noise (PSD with slope -1 in a log-log diagram), and random-walk noise (PSD with slope-2). The latter is usually adopted when it come to the roughness of road surfaces (ISO 8608:2016). The shape of the PSD is related to the frequency distribution of the profile, while the area under the PSD is related to the RMS (squared) of the random profile. It has been assumed that the profiles, which are generated with a frequency of 1000 Hz, have RMS between 1 and 4 mm.

A random burr profile using a PSD slope of −2 and a RMS of 2.5 mm and the related simulation results are presented in Figure 15. It can be noticed that the proposed mechanism performs well also for a generic burr profile. In particular, the compliance of the mechanism is exploited to reduce the cutting forces and, as a consequence, a second cutting cycle is necessary to completely remove the burr (see Figure 16). Similarly to the Test case 5 (Section 5.3.5) small bounces on the mechanical stop are visible in the middle plot of both Figure 15; Figure 16 (for example after *t* = 0.55 s, after *t* = 1.9 s, etc.). From the analysis of Figure 16, it can be noticed that after the second cutting cycle a very high precision is ensured: most of the burr profile is at 2.85 \* 10−<sup>4</sup> mm that is the equilibrium position of the mechanical stop, and very small variations (<1 \* 10−<sup>5</sup> m) with respect to this value are present. Indeed, in the second cutting cycle the grinding wheel support is always in contact with the mechanical stop, and the small variations in the burr profile are due to the (low) compliance of the mechanical stop. In this case the compliance of the mechanism is not exploited, since the maximum burr height is below 2.5 mm, which is the minimum burr height that is necessary to make the mechanism work in the compliant mode, as explained in Section 5.3. Very similar conclusions can be drawn when other values of PSD slope and RMS are used (as above specified).

**Figure 15.** Generic burr profile, first cutting cycle (**top**); position (**middle**) and velocity (**bottom**) of the grinding wheel support.

**Figure 16.** Generic burr profile, second cutting cycle (**top**); position (**middle**) and velocity (**bottom**) of the grinding wheel support. Please notice that the scale of these graphs are the same of Figure 15 for comparison purposes.
