This novel module is a smart element that deals with the vibration mitigation in micro-cutting processes. The main principle is to screen the vibratory frequency in order to control the displacement at the tool tip point of the machine. The proposed approach integrated mechatronic and control theories in the design phase. The device aims at increasing the quality of workpiece finishing through an AVC architecture based on high performance PZT-actuators. It is based on the Stewart platform [
22,
23,
24,
25,
26], but it has only three degrees of freedom (DOFs) (two rotations on X, Y axes and one displacement on the Z axis).
Figure 2 shows an overview of the AVC system that includes two platforms made of an Al alloy. The fixed platform is bound to the machine tool’s vertical axis, and the mobile is linked to the frame of the spindle. Three PZT multi-stack actuators permit the relative movement of these two platforms. The functional concept focuses on the recognition of undesirable displacements at the tool tip point. When a deviation is measured at the tool tip point (TTP), 3 actuators are switched on in order to smooth pulsations and limit their consequences on the surface, as shown in
Figure 3. The piezo-electric element is able to manage compressive axial loads, but precautions are needed with tensile and/or shear forces; for this reason, special flexures have been designed and integrated to prevent any breakage or failure. These flexural joints are able to avoid torsional and shear stresses to the piezo elements. A mechanical system provides the suitable stiffness to the actuators, avoiding undesired stresses and breaks. In additions, two flexural springs are located close to every actuator. They have high torsional and radial stiffness, nevertheless, they are free to move in the axial direction. The module is also equipped with three steel C-shaped plates to increase the stiffness, located between the fixed platform and the actuators. Furthermore, every flexural joint is connected to a cooler joint that provides compressed air against the actuator surface when the temperature exceeds 40 °C.
Figure 4 illustrates the smart actuator designed and developed to satisfy the machining requirements.
In this way, the AVC module may be pointed out as a black block between two interfacings of the machine tool. The movement of the smart block is actuated by three piezo-electric stack actuators, which are located in a strategical position. The piezo electric actuation is regulated by the strains with sub-micron range sensitivity. An accelerometer converts the strokes at the TTP to be transformed into voltage of PZT-displacement. The module is also equipped by a set of sensors (force, temperature, and displacement) useful to permit and monitor the system functionalities. In particular, a temperature sensor is located on every actuator surface and a set of strain gauges is mounted on the steel disk. The main actuators’ technical features are listed in
Table 2.
The proposed module needs to satisfy the Machine Tool life cycle time that is usually close to 30,000 h. In particular, the operative condition is the milling environment with temperature close to 40 °C, machining lubrication, and a frequency between 100 Hz and 300 Hz. The reliability analysis showed MTTF (Mean Time To Failure) of actuators close to 2.80E10 cycles (52,000 h at 150 Hz) satisfying the machine requirements.
This configuration may create some criticalities in terms of loss of stiffness (due to “in-series” arrangement of the piezo actuators). To avoid this problem, a set of parts (e.g., shaped plates, flexural springs) was added to systems to guarantee the required stiffness in XYZ directions. The FE analysis showed a reduction of spindle stiffness close to 6.9% in Z direction and 9.1% in X-Y directions. Nevertheless, this reduction does not impact on the required machining performance.
In order to identify the best location of the triaxial accelerometer to measure the tool tip point deviations, the dynamic stiffness of the mechatronic module was evaluated. In particular, the module architecture guarantees that the dynamic stiffness between the accelerometer point at the bottom ram flange and the tool tip is very high in frequency domain to assume that the reduction of vibration is the same for the two positions.
2.1. The Mechatronic Model
One of the most critical points in the control the AVC device is the identification of the correct mechatronic model; this includes a combination of multidisciplinary fields such as mechanical, electrical, control, and energy engineering. The first step is to evaluate the machine tool behavior using a set of simulations. In this way, a mechanical model may be represented by a Finite Element (FE) analysis. FE aims at describing the main characteristics of the machine tool, focusing on structural parts, links, and piezoelectric actuators (
Figure 5). This study was developed on a commercial milling machine and the simulation results are presented in
Table 3 and
Figure 5. The FE model consists of approximately 150,000 elements.
To validate the simulation results, an experimental modal analysis was performed using the “hummer test”.
Table 3 shows a comparison between the FE model frequencies and experimental data at different modes. The FE frequency modes are congruent with the experimental data, confirming the robustness of the numerical model.
Figure 5 highlights a set of FE analysis examples.
In particular, FE analysis was replicated to evaluate the impact of the AVC device. It was noted that the mechatronic module introduced a frequency mode at 280 Hz, close to the critical frequency range for a standard milling operation. The aim of the control strategies was to suppress this mode and that of all others in order to improve the dynamic behavior of the machine tool.
The model was reduced in accordance to the Craig and Bampton technique [
27,
28,
29] in order to facilitate dynamic simulations and testing. In this way, the degrees of freedom (DOFs) of any substructure were classified as boundary or internal. The basic assumption was that the sub-structure was characterized by defining the constrained modes and a few normal modes. In fact, it was useless to consider a large number of modes, as only a few of them had a physical meaning. Starting from the “FE model”, a “reduced FE” with a small number of DOFs have been identified.
In addition, the energy dispersion quantification of structural objects through vibration depended by several damping criteria. Damping parameters could not be quantified from similar structures or predicted by the Finite Element analysis.
In this study, the damping is assumed to be viscous and influenced by frequency. The assumption is to consider the damping condition as a linear formulation of mass and stiffness properties:
where
α and
β are variables to be identified and [
M] and [
K] are the mass and stiffness matrices, respectively. The benefit of this construction is that the damping formulation is a diagonal matrix. The damping proportion
ξ is associated to
α and
β through the following relation:
where
ω is the Eigen-frequency of the i-th mode number. This method is applied to find damping characteristics within the normalized ranges, this is known as the half-power bandwidth technique. The results show that
α varies from 0.05 to 0.10 while
β is close to 9 × 10
−6. In particular, it is noted that the mass is proportional to damping when α is greater than
β.
Another critical point in defining the mechatronic model is the representation of the state space. This procedure aims at illustrating a system of n-first order differential equations. Therefore, the use of matrices simplifies the mathematical structure of the system equations. The growth of state variables, inputs, and outputs does not influence the complexity of these calculations. An AVC system and regulator need to work in real-time and for this reason the device behavior has to be explored in the time variable.
The state space (SS) model is derived from the corresponding FE model and it describes the dynamic behavior of the architecture using mathematical formulations. The dynamic scheme defines the coherence between the input and output parameters.
Table 4 summarizes the SS model that has been created to control the device starting from the FE results.
In order to perform the simulation, the reduced FE model has been analyzed using Matlab-Simulink software. The reduced form is:
where
z represents the path of movements of real DOFs,
F is the force, {
M} and {
K} are the mass and stiffness matrices, respectively, and {
V} is the damping matrix. Defined vector x(t), the matrices of state
A, input
B, and output
C and
D are deduced from the modal investigation. The dynamics of structure are defined by the SS form as follows:
with
{C} and {D} depend on selected variables, x is the state vector, and u represents the energy vector. The SS model of the AVC device may be created using FORTRAN code. This configuration is assimilated into SIMULINK to provide the dynamic reaction of the exhibited assembly (machine tool and AVC module) under defined perturbations.
2.2. Control Modeling and Strategies
The definition of the control strategy plays a key role in guaranteeing the AVC module functionality. State of the art technology presents a broad range of techniques to control AVC systems that may be classified in two main categories: active feedforward controllers and linear feedback regulators [
30,
31,
32,
33,
34,
35,
36]. The first group of controllers provides stability and a straightforward physical implementation. Nevertheless, feedforward controllers may create a set of issues with a nonlinear feedback. The linear controllers are based on a regulator design and are able to manage any potential differentiation between the plant and the model [
37].
A number of researches prefer to use linear representations for control strategy and simulation when micro-movements are examined [
31,
32,
33,
34,
37]. The speed of the feedback architecture has a critical impact on the vibration dominance of smart assemblies.
In this study, a linear controller has been selected, in so far it is especially useful to control vibrations in flexible structures. As defined in the SS model, linear control methods have been considered using simulations to optimize the controller implemented in the real-time hardware. In order to decrease the vibration effect, a supervisor has been analyzed to manage the required signals (e.g., voltage). In this configuration, the perturbation signal of the TTP is an input of the regulator to be sent to the PZT-actuator in recovering the displacement. Vibration suppression has been evaluated using the multiple input - multiple output (MIMO) layout and the H2-Linear Quadratic Gaussian (LQG) regulator that together provide the chance to drive many actuators and manage huge sensor data [
31,
32] to reduce the white noise troubles.
Linear Quadratic Gaussian (LQG) optimal control is briefly introduced. An adjusted regulator factor may be achieved by reducing the following function:
where
Q represents the power matrix of the device output and R is the device input matrix.
The Linear Quadratic Gaussian (LQG) method may be useful and effective in real manufacturing environments. In addition, the H2 formulation replaces the stochastic reading of the LQG technique using the down size of the two-norm of the closed-loop structure. This choice avoids the need to read factors such as the strength of the white noise that passes through the LQG control [
36,
37,
38,
39]. A closed loop control system needs to guarantee stability, performance, and robustness. These proprieties may be satisfied by evaluating the sensitivity function S and complementary sensitivity function T. In particular, the sensitivity S regulates the disturbance on the output of the control scheme while the complementary sensitivity T is significant for the closed-loop reaction and noise measurement. In particular, a robust control system design has to reduce S and T at low and high frequencies, respectively, and avoid vibration peaks.
In the light of these considerations, the selected regulator is based on LQG-H2 theory [
32,
33,
34,
35,
36,
37,
38,
39,
40]. The main functional concept of the AVC device is to recognize any undesirable displacement on the tool tip point of the machine tool through a three-axial sensor. A regulator panel drives information from the accelerometer managing the dynamic signals to the three PZT actuators. The active structure is summarized as follows:
Equation (8) show the SS calculation, where
x is the state vector,
u is the control vector,
w is the input,
z is the controlled variable (movement of TTP), and
y is the recorded output. The H2 control needs a regulator based on the following transfer function:
such that
is negligible.
In this way, the H2 dynamic regulator assumes that:
where
Ke and
Kc are solutions of the Filter Algebraic Riccati Equation-Control Algebraic Riccati Equation [
39]. The proposed solution is simple to manage, nevertheless, the assumptions restrict H2 optimization to the LQG framework.
2.3. Hardware In the Loop (HIL) Validation
In order to validate the mechatronic model, a set of simulations has been performed using Simulink software. The simulations involved the machine tool, PZT actuator, and controller models. In particular, the PZT actuator linear model has been included into the reduced machine tool representation.
The main objective was to validate any different integrated model-testing control-schemes. A preliminary verification focused on a perturbation created by a digital input (sin-signal) to recreate the TTP displacement plotted in a time-domain. The simulation wanted to represent the undesirable, unbalanced tool rotation at 11,500 rpm. The control model was tested in both conditions (off/on) highlighting the effectiveness of the control system by 30–50% in terms of displacement amplitude reduction, as shown in
Figure 6.
To complete the preliminary validation of the regulator, a test bench was equipped with an electronic board consisting of a Matlab-based-FPGA (Field Programmable Gate Array) and a CPU (Central Processing Unit). The main advantage of applying a FPGA strategy is that the regulator is included as hardware, and it is very fast in comparison to microcontrollers.
Figure 7 presents the block-wise flowchart of the regulator electronics with high-voltage amplifier. The regulator panel includes 12 bit A/D converters for analogue sensors (movement or acceleration) in addition to 16 bit D/A converters to create analogue signals for high-voltage power. The I/O (Input/Output) parts of the integrated circuit technology are conveyed on an I/O board and located in the CPU-FPGA board. The acceleration signal generates a differential control signal which is multiplied by different gains for the three different piezo-actuators in order to impose a displacement along the X direction. This scheme is simple in the parameter setting and easy to implement, nevertheless, it is strongly dependent on the dynamics of the active device (e.g., the sensor and the amplifier dynamics).