Theoretical calculation and experiment of the impact characteristics in the plane ultrasonic rolling

To achieve high machining efficiency in optimal surface integrity manufacturing, a relationship model between rolling depth and rolling force was established based on Hertz contact theory, and a theoretical model of impact characteristics was established according to the indentation geometry to evaluate the machining efficiency. Subsequently, the plane ultrasonic rolling experiment of 7075 aluminum alloy was carried out to verify the relationship between rolling depth and rolling force; meanwhile, the mapping between process parameters and surface characteristics, impact characteristics, and surface morphology were studied, respectively. On this basis, the surface integrity prediction model was established by using nonlinear curve fitting method, and the optimal parameter solution was obtained by using quantum genetic algorithm (NSGA-II). The results show that the rolling force increases linearly with the increase in rolling depth. The impact characteristics increase with the increase in static force and amplitude, and decreases with the increase in step and feed speed, and the impact characteristics is negatively correlated with the processing efficiency. The optimization results provide a reference for engineering applications.


Introduction
The fatigue life of metallic components is determined largely by surface quality, such as surface roughness, micro-hardness, and residual stress [1][2][3]. To date, compared with shot peening [4], laser intensification [5], and supersonic particle bombardment [6], the ultrasonic rolling processing (USRP) has been regarded as an outstanding technique to enhance material surface quality and maintain invariant the interior chemical composition of the material [7,8]. It is mainly reflected in two aspects: one is to improve the surface characteristics; for example, Zhu et al. [9], Bozdana et al. [10], Wang et al. [11], and Liu et al. [12] research the effect of process parameters on surface integrity, and illustrated that better surface quality could be obtained at the static force, and bad influence would occur beyond a certain range. The other is to improve the fatigue life; for instance, Zhao et al. [13], Zhao et al. [14], and Ye et al. [15] demonstrated that ultrasonic rolling leads to improved fatigue life, and can be of great help to improve the stability of materials.
From the above overview, many researchers [16,17] have investigated the influence of processing parameters on surface integrity. However, when optimal surface characteristic is obtained, the process parameters have been shown to be low spindle speed and slow feed rate, which affect to some extent workpiece processing efficiency. In order to alleviate the above problem, optimization and prediction of high processing efficiency and optimal surface characteristic are researched.
Firstly, the relationship between rolling depth and rolling force in planar ultrasonic rolling was established based on Hertz contact theory. Subsequently, according to the indentation geometry, the impact characteristic model is established to evaluate the machining efficiency. Then, the prediction model of surface integrity was established, and the optimal parameters of machining efficiency and surface performance were obtained.

Rolling depth model for in-plane ultrasonic rolling
During in-plane ultrasonic rolling, mechanical vibration of the roller is generated by an ultrasonic generator impulses from the static force and from the ultrasonic vibration are transferred to the workpiece surface by the roll, as is shown in Fig. 1.
Based on the mechanism of ultrasonic rolling, the workpiece surfaces will undergo elastic-plastic deformation as the roller comes into contact with the workpiece. According to Hertz contact theory, an indentation is formed whose depth is as same as the sum of the reduction and the amplitude of the applied vibration. When the roller is removed, the depth of the indentation decreases because of the elastic recovery, as is shown in Fig. 2.
According to relevant geometric relations, Eq. (1) can be obtained: where h a , h 0 , A 0 , and h e are actual depth, nominal depth, amplitude, and magnitude of elastic recovery, respectively.
The unloading process can be assumed to be an elastic process because the reduction is not too large. Based on the elastic half-space volume, the magnitude of elastic recovery is expressed in Eq. (2).
where E * = E 1 /(1 − μ 1 ) + E 2 /(1 − μ 2 ) and E 1 and E 2 are the elastic modulus of the roller and the plane, respectively. μ 1 and μ 2 are the Poisson ratios of the roller and the plane, respectively; F is the rolling force; and Hv is the workpiece surface micro-hardness. Therefore, the relationship between the rolling force and the nominal rolling depth should be established to obtain the actual rolling depth.

Mechanical analysis of plane ultrasonic rolling
When the roller comes into contact with the workpiece, the workpiece surfaces will undergo the elastoplastic deformation. However, compared with the plastic deformation, the elastic deformation is smaller and will rebound after application of USRP. Therefore, the displacement of elastic recovery is ignored and only the plastic deformation is analyzed. The specific force analysis is shown in Fig. 3. where F is the static force; N is the pressure exerted by the workpiece on the roller; f is the friction force exerted by the workpiece on the roller; R T is the radius of the roller; θ and ø are the contact angle in the x-z plane and in the x-y plane, respectively. According to the Hertz contact theory, the contact shape is an approximately spherical cap [18]. In order to obtain the N acting on the roller in the contact zone, the contact area is divided into j discrete equally spaced units. The area of each interval is very small, so the N i is assumed to be approximately constant: dN in the ith interval. The N acting on the roller can be calculated by integrating over the contact zone: where dN = P m dA = P m (R T dθ)(R T sinθdφ) = P m R T 2 sinθdθdφ and P m is the average surface stress in the contact zone. Then, the N is distributed along the axes: where the minus sign means that �� ⃗ N x and �� ⃗ N y are opposite to the x-axis and y-axis, respectively.
According to Coulomb's law, the forces of friction along the axis are calculated as follows.
N are perpendicular, and μ is the friction coefficient between the roller and the workpiece.
According to the relationship of the Newtonian law, the forces acting on the roller are balanced in USRP, so the net force along the y-axis is zero: Substituting Eq. (6) and Eq. (11) into Eq. (12): Because h a is much less than 2R T , h 0 2R T is approximated to 0. µ is very small, and P m is approximately set as the workpiece surface micro-hardness Hv [19]; then, Eq. (13) is simplified as follows: It can be seen from Eq. (14) that there is a linear relationship between the rolling depth and the rolling force.

The geometric contact model of ultrasonic rolling
The geometric contact model between the roller and the workpiece is shown in Fig. 4. According to the relevant mathematical relations, Eq. (15) can be obtained: Considering that h a 2 is a high-order infinitesimal, Eq. (15) can be simplified to Eq. (16):

The model of the impact characteristics
As is shown in Fig. 5a, the actual contact trajectory K between the roller and the workpiece is a sine curve in the USRP. As is shown in Fig. 5b, L and b are the step and the processing length, respectively. The processing area S C = L × b, the actual impact area S U = 2a × K. The impact characteristic W, identified as the number of shocks per unit area, is expressed in Eq. (17): When the rolling force decreases, the step increases, and the feed speed increases. It is possible to cause W to be less than 1, which leads to intermittent processing. In contrast, when W is larger than 1, the unit area acting on the workpiece surface was subject to W (> 1) shock. It is not hard to see that the more significant the impact characteristics, the lower the machining efficiency. In other Lb words, it is necessary to pursue the lowest impact number and achieve the optimal surface quality.

The platform of ultrasonic rolling
The ultrasonic rolling experiments were carried out on vertical machining center, type VMC-850E. The experimental system was composed of a Kistler dynamometer (9257b) and a bespoke wireless transmission ultrasonic vibration-assisted rolling system, as is shown in Fig. 6. The material of the roller was YG8 cemented carbide, and the reinforced material was 7075 aluminum alloy with a Vickers hardness of 325 Hv. Their mechanical properties are shown in Table 1.
After the USRP, the residual stress σ, surface roughness Ra, and surface hardness Hv are measured by the PROTO X-ray unit, Taylor Hobson rough meter (Subtonic 3 +), and micro-hardness instrument (MH-5), respectively. In particular, a Cu target and the force of 30 N last for 10 s are accepted in the measurements of residual stress and hardness, respectively.

Results and analysis
The corresponding test parameters and test plan are shown in Tables 2 and 3, respectively. As the residual compressive stress on the workpiece surface increases by increasing the force and amplitude or decreasing the feed speed and step, the impact characteristics increase, which is caused by the increase of the impact times in the unit area of the workpiece surface. This conclusion is consistent with reference [19].

Relationship between rolling force and rolling depth
It can be seen from Fig. 7 that the rolling force increases linearly with the increase of the rolling depth, which is consistent with the conclusions in the literature [14,20]. The values obtained from the experimental were smaller than the theoretical values, and the reason is that ultrasonic vibration usually can reduce the applied contact force. However, the discrepancy between the values was within 10%.

Influence of the parameters on the surface properties and the impact characteristics
The influence of the parameters on the surface properties and the impact characteristics is presented in Fig. 8. It can be observed that, on a whole, the residual compressive stress, surface hardness, and the impact times rise with the increase in the static force and amplitude. On the one hand, according to the literature [13,21], the reason is that the plastic strain energy transmitted to the workpiece increases with the increase of static force and amplitude. On the other hand, according to Eqs. (14), (16), and (17), with the increase in the static force, the overlapping area of two adjacent indentations increases due to the contact radius increases, which gives rise to enhance the impact characteristics. That is to say, as the impact characteristics increases, the number of times of shock (from the ultrasonic pulses) increase per unit area. The times of shock are positively correlated with the    [22]. Therefore, the residual compressive stress can be increased effectively by increasing the static force.
With the increase of step and feed speed, the surface hardness, residual compressive stress, and impact characteristics decrease significantly. The surface roughness is not sensitive to feed speed, but increases sharply with the increase of step. This is because the indentation area is greatly affected by the step. It is the decrease of the indentation area and the increase of the residual height that lead to the deterioration of the surface roughness. However, it is obvious that the processing efficiency has been improved.

Influence of impact characteristics on surface morphology
The influence of impact characteristics on surface morphology is shown in Fig. 9. It is seen from Fig. 9a-d that with the increase of impact times, the surface topography gradually tends to be flat, which is because the workpiece surface is repeatedly impacted and the superposition effect of peak-valley of materials is obviously improved. At the same time, the surface residual compressive stress and hardness are also greatly increased according to the experiment. However, better surface integrity is achieved at the expense of machining efficiency due to the reduction of step. It can be seen from Fig. 9e to h that although the difference of surface morphology is not obvious when the number of impact is constant, the surface integrity and machining efficiency are quite different. For example, the machining efficiency in Fig. 9e is about twice than that in Fig. 9h, but the residual compressive stress and surface hardness in Fig. 9h are about 1.5 times than that in Fig. 9e. This is due to the balance between rolling force and step. Therefore, the surface roughness deterioration caused by step can be improved by appropriately increasing rolling force. The surface morphologies under the two extremes of impact number are shown in Fig. 9I and J, respectively. When the number of impacts is too large, although excellent residual compressive stress and hardness can be obtained, pits are generated due to repeated impacts on the workpiece per unit area, as shown in Fig. 9I. On the contrary, when the impact number is too small, the phenomenon of intermittent processing is occurred because the adjacent indentation area cannot be interfered. In short, it can be seen from Fig. 9 that a larger residual compressive stress, hardness, and smaller surface roughness can be obtained by increasing appropriately the times of impact. However, it is bound to reduce processing efficiency. In order to obtain the process parameters that will give high efficiency and large residual compressive stress, hardness, and small surface roughness, the impact time is taken as a method to evaluate the processing efficiency and to identify the optimal parameters. In other words, it is the ultimate goal that is to achieve both the minimum number of shocks and the optimal surface quality.

The prediction of surface integrity and the impact characteristics
Based on the experimental results, the predictive model for surface integrity and the impact characteristics were established using the nonlinear regression method. By virtue of variance analysis and F tests, the regression in Eq. (18) was demonstrated to be accurate at a confidence level of 96.3%, which shows the strong correlation between predicted values and measured values.
where x 1 , x 2 , x 3 , and x 4 represent the values of F, v, L, and A 0 , respectively.To verify the reliability of the prediction model, the predicted and experimental values were compared, as shown in Fig. 10. The maximum error are 6.6%. These results illustrate that the predicted value was close to test results. Therefore, the model was demonstrated to have a high prediction precision within the scope of the applied tests.

Multi-objective optimization results and discussion
NSGA-II has been used in machining process because of fast non-dominated sorting approach, fast crowded distance estimation procedure, and simple crowded comparison operator among a lot of multi-objective optimization method. The mathematical equations and constraint conditions of optimization procedure were established. First of all, construct optimization variables: x = (x 1 , x 2 , x 3 , x 4 ). Then, according to NSGA-II and experiment design, constraint conditions were constructed, as shown in Eqs. (19) to (22). In order to obtain better surface quality and higher processing efficiency, the multi-objective function considers optimization of residual stress, surface hardness, surface roughness, and processing efficiency simultaneously. The optimization model is given in Eq. (23).
The optimal solution of surface quality is that the machining efficiency reaches the maximum when the The partial optimal solution is given in Table 4, when the force is in the range of 228 to 276 N, the feed speed is in the range of 94 to 112 mm min −1 , the step is in the range of 0.095 to 0.12 mm, and the amplitude is in the range of 3.3 to 3.4 μm; the outstanding surface integrity and machining efficiency can be achieved simultaneously.

Conclusions
(1) The rolling force increases linearly with the increase of rolling depth, and the experimental value is about 10% less than the theoretical value due to ultrasonic vibration. (2) The residual compress stress and surface hardness increase with the increase in static force and amplitude, and decrease with the increase in step and feed speed. The surface roughness increases with the increase of step and decreases with the increase of force and amplitude, and it is not sensitive to the feed speed. (3) The impact number decreases as the rolling force decreases, the step increases, and the feed speed increases. When the number of impact is greater than 1 and less than 1, it is the criterion of intermittent and continuous machining, respectively. The impact characteristics is negatively correlated with the processing efficiency. (4) The optimal parameters of the outstanding surface integrity and machining efficiency are in the range of 228 to 276 N, the feed speed is in the range of 94 to 112 mm min −1 , the step is in the range of 0.095 to 0.12 mm, and the amplitude is in the range of 3.3 to 3.4 μm.

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