Optimal Semiactive Damping Control for a Nonlinear Energy Sink Used to Stabilize Milling

Improving product quality of machining components has always met with problems due to the vibration of the milling machine’s spindle, which can be reduced by adding a vibration absorber. The tuned vibration absorber (TVA) has been studied extensively and found to have a narrow bandwidth, but the cutting force possesses wide bandwidth in the process of machining parts. Introducing nonlinearity into the dynamic vibration absorber can effectively increase the bandwidth of vibration suppression and can significantly improve the robustness of the vibration absorber. In addition, a semiactive TVA has proved to be more effective than a passive TVA for many applications, so the main purpose of this study is to find some appropriate semiactive control methods for a nonlinear energy sink (NES), a nonlinear vibration absorber, in structural vibration applications. Two semiactive control methods are considered in this study: continuous groundhook damping control based on velocity and on-off groundhook damping control based on velocity. To fairly compare these vibration absorbers, the optimal parameters of a passive TVA, a passive NES, and two semiactive NESs are designed using numerical optimization techniques to minimize the root-mean-square acceleration. Two cutting forces are introduced in this study, a periodic force and an aperiodic force, and the four vibration absorbers are compared. When the primary structure is excited with aperiodic cutting force, the amplitude of the primary structure decreased by 17.73% with the passive TVA, by 72.29% with the passive NES, by 73.54% with the on-off NES, and by 87.54% with the continuous NES. When the primary structure is excited with periodic cutting force, the amplitude of the primary structure decreased by 49.01% with a passive TVA, by 86.93% with a passive NES, by 96.38% with an on-off NES, and by 99.23% with a continuous NES. The results show that the passive NES is better than the passive TVA; the semiactive NES provides more effective vibration attenuation than the passive NES, and the continuous control is more effective than the on-off control.


Introduction
A tuned vibration absorber (TVA) is a vibration absorbing device that is attached to a primary system to control vibration. It is usually composed of a mass, a linear stiffness, and a viscous damper. In 1909, Frahm [1] invented the first simple shock absorber. en, Ormondroyd and Hartog [2] evolved it into a Voigt-type dynamic vibration absorber with damping. e passive TVA described above has simple structure but poor control effect, so the active TVA and semiactive TVA were successively studied by scholars. An active system can provide good performance, but it becomes more complex at the same time. Many problems arise such as the selection of sensors, power supply, weight constraints, stability, and robustness [3]. So, passive control and active control has been replaced by semiactive control for some applications [4,5]. A number of studies have introduced the advantages of semiactive control over passive control [6][7][8][9][10][11][12][13][14].
In the 1970s, Karnopp et al. [4] applied semiactive dampers to vehicle suspension systems, and the results showed that semiactive systems can provide many of the performance gains of active systems. en, the variable damping system was proposed by Hrovat et al. [15], who studied a variable damping semiactive control applied to the suppression of wind-induced vibration of structures, and it also confirmed that semiactive TVA performed better than passive and active TVA. Four semiactive damping control laws [16] based on skyhook control and balance control are introduced, and the results show that they have better vibration suppression effects than the traditional passive control in the high frequency range. Koo et al. [17] introduced four semiactive control policies and optimized the parameters of a TVA according to them and then compared the transmissibility of the four systems and concluded that semiactive TVAs better reduce vibration. Ihsan et al. [18] used the root-mean-square value (RMS) of the acceleration as an indicator for evaluating the comfort of a car seat. Both skyhook and groundhook controls were developed. is paper mainly discusses groundhook control, which is divided into on-off groundhook control and continuous groundhook control. For on-off groundhook control, the damper is switched between two values, which are called onstate damping and off-state damping, as shown in Figure 1. Which state the damping is switched to is determined by the product of the relative velocity across the semiactive damper and the absolute velocity of the primary structure. For continuous groundhook control, the damping is not limited to just two values. It can change continuously between the two damping states, and the value is again determined by the velocity product. e figure shows the relationship between damping force and relative velocity of the primary structure and the absorber. It can be seen that c on and c off represent on-state damping and off-state damping, and c represents passive damping.
In its simplest form, a nonlinear energy sink refers to a mass that is connected to the main structure by strongly nonlinear stiffness and viscous damping. Because linear dynamic vibration absorbers can effectively suppress vibration only when the excitation has narrow bandwidth, Roberson [19] points out that introducing nonlinearity into the dynamic vibration absorber can effectively increase the bandwidth of vibration suppression and can significantly improve the robustness of the vibration absorber. e passive NES has been studied by many scholars. Vakakis et al. [20] studied the response of a system with a linear oscillator coupled to a grounded nonlinear oscillator and found the phenomenon of targeted energy transfer. McFarland et al. [21] and Kerschen et al. [22] verified experimentally the phenomenon of targeted energy transfer and the existence of a threshold input energy to initiate effective TET. Zhang et al. [23] presented a new design for an NES which can realize vibration control and energy harvesting and realized efficient vibration control. Yao et al. [24] studied the vibration attenuation performance of the NES for the unbalanced rotor system and verified the NES has better performance over a wider frequency range than the linear absorber.
Milling is a widely employed material removal process, and many researchers have studied the effects of cutting parameters to improve the machining accuracy. Bolar et al. [25] studied feed per tooth, tool diameter, axial and radial depth of cut, and their influence on surface finish. Wang et al. [26] introduced a hybrid optimization method which can help users to find optimal cutting parameters, which provide better efficiency and lower power consumption for a milling process. Ding et al. [27] also studied the effects of cutting parameters on surface roughness in hard milling of AISI H13 steel.
As mentioned above, a semiactive TVA performs better than a passive TVA and a passive NES acts better than a passive TVA when the cutting force of the spindle possesses wide bandwidth. So, this paper investigates the vibration suppression effect of a semiactive nonlinear energy sink. By equipping the spindle of a milling machine with a passive TVA, a passive NES, and two semiactive NESs, the performance of these absorbers is evaluated. e performance index is the root-mean-square value of the acceleration of the primary structure (i.e., the spindle). Finally, the simulation results are analyzed and compared. is work provides new ideas about an NES application and the theoretical basis for the study of a semiactive nonlinear energy sink.

Description of the Semiactive Control Methods
In this section, two semiactive control methods are introduced. e ideal damping structural model is shown in Figure 2(a), where a passive damper is connected between the primary structure and the ground. e damper always applies to the main structure a force that is opposite to the direction of its velocity. e semiactive control methods proposed imitate this model and generate a similar force. e equivalent model is shown in Figure 2(b). e state of the main structure and the NES are input to the controller, and then the damping coefficient can be adjusted. e difference between ideal damping model and equivalent damping model is that the damping force generated by c ground is replaced by that generated by c control . ese two models are derived from automobile suspension systems where semiactive control methods are used to reduce vibration of unsprung and sprung masses. m PS , c PS , and k PS are the mass, damping coefficient, and stiffness of the primary structure, respectively; m NES and k NES are the mass and stiffness of the NES; c ground is the damping coefficient of the groundhook damper; and c control is the controllable damping coefficient. x PS and x NES are the displacements of the two masses, and _ x PS and _ x NES are their velocities. F is the excitation, a force which acts directly on the primary mass.

Continuous Groundhook Damping Control Based on
Velocity. Now considering the ideal structural model, the damping force of c ground can be written as where F ground is the groundhook damping force. For the equivalent model, the controllable damping force can be written as where F c is the equivalent damping force. In order for the damper to dissipate vibration energy, the product of the 2 Shock and Vibration equivalent damping force F c and the relative velocity _ x PS − _ x NES must satisfy the inequality us, the inequality can be derived e desired force is c ground _ x PS , but the semiactive damper can generate this force only when _ x PS and _ x PS − _ x NES have the same sign. Otherwise, the semiactive damper can provide only a force opposite to the desired force, in which case it is better to let the damping force be zero. e continuous semiactive groundhook control algorithm is thus given by Setting equation (2) equal to equation (6) when possible, c control can be expressed as It is obvious that c control will tend to infinity when the relative velocity _ x PS − _ x NES is very small, so there must be both an upper bound and a lower bound. e damping coefficient in equation (7) is therefore rewritten as Equation (8) is the continuous groundhook damping control based on velocity.

On-Off Groundhook Damping Control Based on Velocity.
To simplify equation (8), an on-off groundhook control law is proposed, given by

Shock and Vibration
where c max is the on-state damping constant of the on-off damper. Setting equation (2) equal to the first case in equation (9), c control can be expressed as Equation (10) is the on-off groundhook control based on velocity. Equations (1)-(10) are derived from reference [12].

Numerical Models
In this section, a passive TVA, a passive NES, and a semiactive groundhook equivalent NES are introduced. Figure 3(a) shows a 2-DOF system with a passive TVA, Figure 3(b) shows a 2-DOF system with a passive NES, and Figure 2(b) shows the groundhook equivalent NES configuration.
e equations of motion of the passive TVA model can be written as In addition, those of the passive NES model can be written as e difference between the passive TVA and the passive NES is that the passive NES has a cubic stiffness. e equations of motion of the groundhook equivalent NES configuration are where c control is replaced by either of the two semiactive control methods during numerical simulation. e difference between the passive NES and the semiactive NES is that the damping coefficient of the semiactive NES changes according to the state of the system. ere are two kinds of cutting force to input. e aperiodic force [19] is shown in Figure 4(a) and its wavelet transform in Figure 5. From 0s to 0.5 s, the frequency of the excitation changes linearly from 300 Hz to 200 Hz during the initial engagement of the cutting process; from 0.5 s to 1.5 s, the frequency remains at 200 Hz, which is the steady cutting state; and from 1.5 s to 2 s, the frequency of the excitation changes linearly from 200 Hz to 300 Hz during disengagement from the workpiece. e periodic cutting force [28] is shown in Figure 6. ere are two stages within each cycle, which are cutting and noncutting. e period is 0.005 s, and the main frequency is 200 Hz. e purpose of the wavelet transform is to visually depict the variation of frequency components of the cutting force with time. e parameters of the system are shown in Table 1. e primary structure mass, stiffness, and damping and the absorber mass are fixed; the stiffness and damping coefficient of the absorbers are to be optimized.

Optimization
is section describes the performance of the two semiactive NESs and compares both of them with the passive NES. Also, the passive NES is compared with the passive TVA. In order to evaluate every vibration absorber fairly, relevant parameters of each are optimized: the damping of the TVA (c TVA ), the stiffness of the TVA (k TVA ); the damping of the passive NES (c NES ), the stiffness of the passive NES (k NES ); the on-state damping of the on-off NES (c max ), the stiffness of the on-off NES (k NES ); the damping coefficient of the groundhook damper of the continuous NES (c ground ), the stiffness of the continuous NES (k NES ), and the upper damping bound of the continuous NES (c max ). During the optimization process, a multidimensional nonlinear constrained minimization function called "globalsearch.m (fmincon.m)" in the Matlab optimization toolbox is used. e "fmincon.m" is more likely to find a local minimum, but "globalsearch.m" can locate the global minimum.

Optimization Routine.
e optimization process composed of three steps. e first step is to define the system parameters, and the second step is to set initial values and ranges of parameters to be optimized in the simulation. e initial values of the parameters and their ranges for the optimization are summarized in Table 2. e most important part of the optimization progress is calculating the objective function, defined here as e minimum value of the objective function is returned along with the corresponding optimal parameters.    Shock and Vibration

Optimal Results with the Aperiodic Cutting
Force. e displacement of the primary structure (i.e., the spindle) of every optimal system is shown in Figure 7, from which we can see the amplitude of displacement with a semiactive NES attenuates faster than that with a passive NES, and the amplitude of displacement with a passive NES also attenuates faster than that with a passive TVA during the steady cutting stage. During steady cutting, the amplitude of the primary structure decreased by 17.73% with the passive TVA, by 72.29% with the passive NES, by 73.54% with the on-off NES, and by 87.54% with the continuous NES. During the initial engagement of the cutting process and the final disengagement, the amplitude of the spindle with the passive TVA is larger than that with the passive NES and that with the passive NES is larger than that with the semiactive NES. e optimal parameters are summarized in Table 3.
e semiactive damping of the on-off NES is shown in Figure 8(a) while Figure 8(b) shows the semiactive damping of the continuous NES.
To verify the optimization results, the accelerations of the main structure versus the stiffness of the TVA and the damping coefficient of the TVA were plotted. For example, Figure 9(a) shows the relation of the three variables for the system with a passive TVA. In order to see the minimum value more clearly, Figure 9(b) magnifies the region near the minimum. e corresponding damping c TVA−opt , stiffness k TVA−opt , and acceleration are 160 Ns/m, 8.4 × 10 6 N/m, and 2.07 m/s 2 , respectively.

Optimal Results with the Periodic Cutting Force.
e displacement of the primary structure of every optimal system is shown in Figure 10, from which we can see the displacements of main structures with semiactive NESs        attenuate faster than that with a passive NES. Furthermore, the displacements of primary structures with NESs attenuate faster than that with a TVA. Within three seconds, the amplitude of the primary structure decreased by 49.01% with a passive TVA, by 86.93% with a passive NES, by 96.38% with an on-off NES, and by 99.23% with a continuous NES. e optimal parameters are summarized in Table 4. e semiactive damping of the NESs is shown in Figure 11. e control methods in this study come from the existing research, and the optimization routine comes from Koo et al.'s [17] research. e results in this study are reliable because the results from Koo et al.'s research have been reproduced with the same mechanisms. e analytic solutions are not easy to get, so there will be further study about semiactive NES.

Conclusions
In this study, systems based on a passive TVA, a passive NES, and two semiactive NESs are compared. e two semiactive control policies are on-off groundhook control based on velocity and continuous groundhook control based on velocity. We compare the performance of the passive TVA, the passive NES, and the semiactive NESs using parameters optimized to reduce the root-mean-square acceleration of the primary structure (spindle). When the primary structure is excited with an aperiodic cutting force, the amplitude of the primary structure decreased by 17.73% with the passive TVA, by 72.29% with the passive NES, by 73.54% with the on-off NES, and by 87.54% with the continuous NES. When the primary structure is excited with periodic cutting force, the amplitude of the primary structure decreased by 49.01% with a passive TVA, by 86.93% with a passive NES, by 96.38% with an on-off NES, and by 99.23% with a continuous NES.
e results indicate that the semiactive NES performs better than the passive NES, and the passive NES performs better than the passive TVA. Further analysis shows that the continuous NES performs better than the onoff NES. As shown in this study, the conclusions are valid when the system is excited by two completely different cutting forces, which means the methods have good robustness.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.