Tuning vibration absorbers to mitigate simultaneous regenerative and mode-coupling chatter

This paper examines a set of parametric tunings for vibration absorbers to enhance stability in the event of simultaneous chatter during machining and proposes a novel tuning criterion of minimizing the definite integral of frequency response function, while accounting for damping of the absorber’s base component. Chatter vibrations are an inherent characteristic of machining processes, and a common tool for mitigating chatter is a tuned mass damper (TMD), which functions effectively when its dynamic characteristics are appropriately tuned to the natural frequency of the vibrating base component. Robotic machining is a recent cost-effective method of material removal, but simultaneous regenerative and mode-coupling chatter can arise on robot tools. TMD tuning criteria for mode-coupling and regenerative chatter are different, and tuning to suppress one may prompt the other. Moreover, conventional analytical tunings do not consider damping of the base component, which may be a valid assumption for low-damped machine tools but not for highly-damped robot tools. The possibility of simultaneous chatter and base component damping effect necessitates the exploration of alternative tuning criteria.


Introduction
Machining is one of the most widely used manufacturing procedures. In addition to conventional machine tools, robot tools have recently been employed for machining as well. However, both robot and machine tools face restrictions in productivity due to the onset of undesired vibrations. 1 The combination of machine/robot tool, cutting tool, and workpiece creates a complex dynamic system that can vibrate under certain conditions. Undesired machining vibrations are categorized into three main types: transient, forced, and self-excited vibrations. Among these, self-excited vibration is considered the most damaging. 2,3 The primary causes of self-excited vibrations in machining are regenerative and mode-coupling chatter. 4 Chatter vibrations are dynamic instabilities that occur during the cutting process, and they can have the most severe impact around the fundamental frequency and other harmonics of the system. Chatter has the potential to harm the cutting tool, workpiece surface integrity, and various components of the machine tool. [5][6][7] The application of vibration absorbers can effectively suppress chatter by dissipating energy from the cutting process and vibrating components. 8 Tuned mass dampers (TMDs) are a popular type of vibration absorber used in machining. 9 However, the lack of design criteria makes it challenging to accurately implement TMDs. 10 The effective performance of a TMD is achieved by adjusting or tuning its stiffness and damping properties. For this purpose, two well-known analytical tuning methods are available. The first method, introduced by Den Hartog 11 (as cited in), 12,13 minimizes the magnitude of the system's receptance frequency response function (FRF), which is a measure of the output displacement amplitude to input force ratio. This method is a suitable approach for general vibrations and mode-coupling chatter suppression. 3 The other tuning method, which maximizes the negative realpart of the receptance FRF, increases machining stability against regenerative chatter, facilitating a higher stable depth of cut. Sims 14 suggested an analytical solution for this tuning method. In addition, Rivin and Kang 15 proposed another analytical tuning method that requires the input value of chatter frequency information. However, this method is undesirable as it requires a series of expensive machining experiments to determine the chatter frequency. Nevertheless, this method has an advantage over Den Hartog's and Sims' methods, as it considers the base component damping.
The analytical tunings proposed by Den Hartog and Sims are based on a two-degree-of-freedom system with no damping on its primary body. This assumption could be acceptable for low-damped machine tool structures, but robot tools have significant damping factors that causes deviations in the results of the conventional analytical tunings from the optimal values.
In an experimental investigation for mode-coupling chatter suppression on a turning process using vibration absorbers, Nakano et al. 16 implemented several TMDs on a rotating workpiece and demonstrated the effectiveness of the absorbers on reducing vibrations. Similarly, Zhang et al. 17 implemented an active vibration absorber on a robot tool to mitigate undesired process vibrations. Both papers used minimizing FRF magnitude (Den Hartog's criterion) as the common principle for tuning and comparing the level of vibration suppression. However, these papers did not consider the effect of regenerative chatter.
Wang et al. 7 investigated a nonlinear TMD that included an elastic support to introduce dry friction to the tool post of a lathe machine. This improved machining stability by reducing regenerative chatter. Tarng et al. 6 also achieved regenerative chatter suppression in the turning process using a fixed-passive TMD. In both studies, the TMD was tuned to maximize the negative real part of the FRF. However, neither study took mode-coupling chatter into account. Sorby 3 has discussed the design and tuning methods of TMDs for boring bars. Using theoretical analysis, he demonstrated that a TMD can suppress either modecoupling chatter by minimizing FRF magnitude or regenerative chatter by maximizing FRF negative real-part. Wu et al. 18 discussed chatter phenomena and geometric errors during machining on thin-walled workpieces in their literature review paper. Both regenerative and modecoupling chatter were analyzed, and different strategies to tackle chatter vibrations were discussed. Both pieces of literature discussed tuning methods to suppress regenerative and mode-coupling chatter separately, without investigating any tuning technique to mitigate different types of chatter simultaneously.
Yuan et al. 19 investigated reduction of mode-coupling chatter during robotic machining using a semi-active TMD with variable stiffness. However, the study employed transmissibility of base vibration as the tuning criterion and neglected regenerative chatter effect.
The majority of studies conducted so far have focused on analyzing the two types of chatter mechanisms separately, with regenerative chatter often being considered the primary source of instability. 20 However, Zhang et al. 20 demonstrated that both regenerative and mode-coupling chatter can occur simultaneously during the machining process on machine tools.
Simultaneous chatter can occur on robot tools, resulting in significant vibrations due to the low structural stiffness of robots. Robot tools are more prone to mode-coupling chatter, which makes them less stable compared to the robust structures of conventional machine tools. 21 Due to the possibility of simultaneous chatter occurring on both machine and robot tools, it is desirable to tune TMDs to mitigate both chatter scenarios. However, this versatile tuning is not feasible because the analytical tunings have different criteria. To the authors' knowledge, no published work is available that proposes any tuning to suppress simultaneous chatter. Furthermore, the available analytical tunings are suitable for systems with negligible damping. To bridge those gaps, this study introduces a new parametric tuning method that minimizes the definite integral of FRF to mitigate simultaneous chatter. Additionally, this method considers damping of the base component.

Theory of machining chatter
As discussed, chatter vibrations are the main sources of instability during the machining process. The following are brief explanations of the major chatter mechanisms: Mode-coupling chatter refers to the relative vibration between the tool and workpiece, which occurs as a consequence of close-coupled modes of vibration. This type of chatter can cause exponential growth in vibration magnitude until it is balanced by damping or results in damages. Mode-coupling chatter frequency is the same as the system mode-shape that prompts it and cannot be changed by variation of cutting parameters like feed rate, width of cut, or spindle speed. 2 Regenerative chatter occurs when a tool cuts a wavy surface machined during the previous cutting pass. Besides stiffness and damping characteristics of a machine structure, machining stability has a dependence on the amount of phase shift (delay) between the two successive surface waves that generate dynamic chip thickness. When the chip thickness shifts dynamically, the cutting force oscillates. The oscillatory cutting force might grow exponentially and result in regenerative chatter. 2 The influential elements on chatter phenomena are depth of cut (chip width) and spindle speed as cutting variables plus dynamic stiffness of cutting system (machine/robot and cutting tool structures). Decreasing cutting variables for chatter mitigation is not desirable because of reduction in material removal rate and productivity. 8

Tuned mass dampers for chatter suppression
One practical technique for suppressing machining chatter is to enhance damping of the connected mode to the chatter frequency. This approach can be achieved by increasing the dynamic stiffness of the oscillating part through the addition of a vibration absorber. 22 A vibration absorber is a spring-mass component that is fitted to the primary body (the unstable part). The chatter mitigation is accomplished by substantial oscillation of the absorber's mass. 23 The chatter suppression ability of a TMD is determined by its mass, natural frequency, and damping ratios. The mass and natural frequency ratios refer to the ratio of the TMD's mass and natural frequency to those of the primary body it is attached to. Increasing the mass ratio will cause the system's first and second natural frequencies to split. The precise adjustment of a TMD's natural frequency and damping ratios is critical to successfully reduce machining chatter. 14,23

Conventional tuning methods for vibration absorbers
To ensure the effective performance of a TMD in suppressing chatter, proper tuning of its natural frequency and damping ratio is necessary after defining its mass. Two conventional analytical tunings have been developed using a two-degree-of-freedom system with an undamped primary body. A schematic of the system is shown in Figure 1.
Using Figure 1, the equations of motion of the depicted spring-mass-damper can be written as follows :: Here, m 1 , k 1 , c 1 , and x 1 represent the mass, stiffness, damping, and displacement of the primary body, respectively, while subscript 2 represents the TMD's similar dynamic characteristics. The system is subjected to an external harmonic load Fe jωt , where j is the imaginary unit, ω is the excitation frequency, and t is the time.
The following are the two conventional analytical tunings for TMDs in accordance with the discussed model.

Minimizing FRF magnitude
To mitigate mode-coupling chatter on a cutting system's base component, it is recommended to adjust the dynamic characteristics of the fitted TMD to minimize the frequency response function (FRF) magnitude of the primary body. This is achieved through classical tuning against free and forced vibrations. Den Hartog 11 proposed an analytical solution to satisfy this criterion, considering a system with no damping in the base component (c 1 = 0). According to Den Hartog, the optimal frequency and approximate damping ratio of the TMD can be calculated as follows where ω 2,opt,A , ω 1 , μ, and ζ 2,opt are TMD optimized natural frequency, stand-alone primary body natural frequency, mass ratio, and TMD optimized damping ratio, respectively. Using Equations (2) and (3) stiffness and damping coefficient of a TMD can be computed. 3 Figure 2 illustrates the effect of adding a TMD, tuned based on Den Hartog's criterion, on the primary body's FRF.

Maximizing FRF negative real-part
To improve the stability of the cutting process against regenerative chatter using TMDs, it is essential to maximize the negative real part of the primary body's FRF. 8 Sims 14 proposed an analytical tuning method for this criterion, considering a system with no damping in the base component (c 1 = 0), which is as follows Here, ω 2,opt,N is the TMD's optimized natural frequency. The damping ratio formula is the same as Den Hartog's, equation (3); however, the resulting damping values are different due to their dissimilar natural frequencies. Figure 3 illustrates the effect of adding a TMD adjusted to maximize the negative real-part of the primary body's FRF.
The Den Hartog and Sims tunings assume that the primary body has only one undamped mode of vibration and generate approximate values for the optimal damping ratio.

Stability lobe diagram
When regenerative chatter develops, vibrations between the tool and workpiece can grow as large as the chip thickness, which may cause the tool to lose contact with the part and jump out of the cut. 24 The stability lobe diagram (SLD) is a tool used to adjust cutting variables for maximizing material removal rate without experiencing regenerative chatter. It helps visualize the stability behavior of the cutting process by predicting the approximate regenerative chatter-free depth of cut as a function of spindle speed. 8 The vertical axis of the SLD represents the maximum chatter-free depth of cut or the limiting chip width that can be estimated by Here, b lim represents the limiting chip width, K s is the cutting coefficient, u is the directional orientation factor that maps the cutting force to the direction of the tool tip's FRF, ω is the chatter frequency, and Re(G( jω)) denotes the tool's receptance FRF's real-part in scalar form. 24 Horizontal axis of SLD is a representation of spindle speed that is evaluated using where N is spindle speed in rotations per minute, q is the integer number of imprinted waves on the workpiece surface due to regenerative chatter during one full rotation of spindle, and G( jω) is the tool's receptance FRF in scalar form. SLD illustrates a series of lobes of depth of cuts versus spindle speed, the theoretical boundary between stable and unstable areas on the graph. Pairs of (N, b lim ) located above the boundary are unstable and below are stable. 24

Conventional tunings facing damped structures and simultaneous chatter
The two conventional analytical tunings are suitable for base components with insignificant damping ratios. Machine tools typically have negligible damping values since they move on low friction linear and ball-screw mechanisms. 25 However, robot tools have highly damped structures, 1 and as a result, traditional analytical tunings may not lead to accurate TMD tunings for robot tools.
In addition to the damping issue there is a risk of simultaneous chatter, in robotic machining, the proximity of the robot tool's structural and cutting process stiffnesses can lead to mode-coupling chatter even during light cutting conditions, which can precede or occur in conjunction with regenerative chatter. 21 Furthermore, as previously mentioned, simultaneous chatter can also occur on machine tools, particularly when flexible tooling systems are employed. 20

Contradictory effects of conventional analytical tunings
As previously discussed and introduced, in order to address mode-coupling chatter, it is necessary to minimize the magnitude of FRF (as shown by the solid line in Figure 4(a)), which will result in a decrease of its real-part (as shown by the solid line in Figure 4(b)). However, this decrease has a negative impact on the suppression of regenerative chatter. On the other hand, to tackle regenerative chatter, it is necessary to maximize the negative real-part of    Figure 4(b)), which can cause an increase in the magnitude of FRF (as shown by the dashed line in Figure 4(a)). This increase in magnitude may then trigger mode-coupling chatter.
Therefore, in the situations that both regenerative and mode-coupling chatter are possible to arise, it is sensible to employ a tuning technique to control FRF and its negative real-part curves at the intervals of the two conventional analytical tunings.

System modelling method
The receptance FRF is a commonly used tool for studying the vibration behavior of multi-degree of freedom systems. 8 In this study, receptance FRF is employed to investigate several parametric tunings for mitigating simultaneous chatter.
To obtain FRFs of a two-degree-of-freedom system (Figure 1), the equation of motion (equation (1)) needs to be transformed from time to frequency domain as follows Zð jωÞ ¼ " here, the variable Z( jω) refers to the complex dynamic stiffness matrix. This matrix can be inverted to produce a complex receptance response functions matrix the matrix H( jω) represents complex receptance FRFs and is obtained by inverting the complex dynamic stiffness matrix Z( jω). The mass, damping, and stiffness matrices M, C, and K are presented in equation (7). The complex receptance matrix provides four FRFs of the assembly, and the corresponding displacements of each mass can be computed as In equation (9), the four FRFs in the matrix share the same denominator, which is the characteristic equation or the determinant of the dynamic stiffness matrix. The roots of the characteristic equation represent the resonance frequencies. In the FRF matrix, each H ab denotes a scalar receptance FRF at the subscript a per unit force F at the subscript b . 23 This study focuses on the receptance behavior at H 11 with a specific annotation of G( jω), which represents the receptance response of the primary body under external harmonic excitation.

Parametric tuning criteria
The required criteria for the conventional and new parametric tunings are presented as follows. Figure 5 shows two points and three areas on the FRF and its real-part graphs. The maximum and minimum points serve as objective functions for conventional tunings, whereas the manipulating areas are objective functions for new parametric tunings.
In Figure 5(a), E F represents the maximum magnitude of the FRF and Δ F is the integrated area below the FRF curve in the closed interval of ½ ω 1a , ω 1b .
In Figure 5(b), on the graph of the FRF real-part, E Rn is the minimum value of the negative real-part of the FRF, Δ Rp is the integrated area below the positive real-part of the FRF curve in the closed interval of ½ ω 1a , ω Int and Δ Rn is the integrated area between the frequency axis and the negative real-part of the FRF curve in the closed interval of ½ ω Int , ω 1b . On the graphs, ω Int represents the frequency at which the real-part curve intersects the frequency axis, and ω 1 represents the natural frequency of the stand-alone primary body.
The closed interval of ½ ω 1a , ω 1b approximates a frequency range where FRF and its real-part graphs exhibit different behaviors due to various tunings. The extent of this divergence region is dependent on the mass ratio, with an increase in mass ratio resulting in an expansion of the region, as shown in Figure 6.
Equations (10)-(12) represent three planned parametric tunings that aim to suppress simultaneous chatter by controlling the FRF and its negative real-part curves between Den Hartog's and Sims' tunings. The integrated area below the FRF curve at the closed interval of ½ ω 1a , ω 1b , denoted by Δ F in Figure 5(a), can be calculated as follows Gð jωÞdω (10) The integrated area between FRF real-part curve and frequency axis at the closed interval of ½ ω 1a , ω 1b can be expressed as follows ReðGð jωÞÞdω (11) here, Δ R represents the integrated area between the FRF real-part curve and the frequency axis, which is equal to the sum of Δ Rp and the absolute value of Δ Rn as shown in Figure 5(b)).
The sum of the integrated area below the FRF curve at the closed interval of ½ ω 1a , ω 1b and the area between the frequency axis and the FRF negative real-part curve at the closed interval of ½ ω Int , ω 1b can be expressed as follows ReðGð jωÞÞdω (12) here, Δ FRn represents the sum of Δ F and absolute value of Δ Rn , which can be visualized in Figure 5.

Implementation of parametric tunings
The input data required for the planned parametric tunings are the mass, stiffness, and damping of the machine or robot tool's unstable component, as well as the mass of the TMD. It is assumed in this study that the unstable component has only one dominant mode, namely the fundamental mode, and that it is not coupled to other modes.
To mitigate simultaneous chatter, three parametric studies are considered with objective functions of minimizing the resultants of Equations (10) to (12). In addition, two other parametric studies with the same objective functions as the analytical tunings are defined to show their deviations due to the damping of the primary body. Figure 7 illustrates the steps of the numerical parametric studies.
The first step is to model the selected unstable component of the machine/robot tool as a one-degree-offreedom spring-mass-damper, i.e., the primary body. Modal analysis is a reliable technique to estimate the equivalent dynamic characteristics. For this study, the fundamental modal parameters of the primary body, namely modal stiffness, mass, and damping at the first resonance frequency, are required. The excitation load must be applied to the tool tip, and the response should be measured from the attachment point to the TMD. The TMD mass ratio should be decided based on physical limitations, as shown in block (1) of Figure 7.
Next, preliminary stiffness and damping values are calculated using Den Hartog's analytical method, as described in block (2). Then, the half and double values of the Den Hartog results are used as the starting and ending points for two vectors of trial stiffness, k ! 2s , and damping, c ! 2s . These vectors define the parameter space. While wider parameter spaces are possible, these limits are sufficient. A higher number of iterations n for the parameter spaces produces more accurate tuning results, as described in block (3). All combinations of stiffness and damping (parameter spaces) are implemented into equations (7) and (8) to calculate possible FRFs, as shown in block (4). When the desired objective function for any of the designated parametric studies is satisfied, the corresponding stiffness and damping values will be the tuning results for that parametric study, as indicated in block (5).
This study compares a total of seven different tuning methods, numbered from I to VII, as presented in Table 1.

Effects of base component damping on tuning results
The conventional analytical tunings were developed using a two-degree-of-freedom system with an undamped primary body. Figure 8 compares the resulting FRFs and their real-parts for a bare and equipped primary body with a TMD that is tuned according to the I Den Hartog analytical tuning, II Sims analytical tuning, III minimizing FRF top magnitude E F (the parametric version of Den Hartog's method), and IV maximizing FRF negative realpart least magnitude E Rn (the parametric version of Sims' method). Here, x st is the displacement under the static load.
Regarding the damping ratio of the base component, as reported by, 26 the damping ratio of an industrial manipulator axis could be around 16%. Therefore, this value is selected for the primary body damping ratio in this study. The TMD mass ratio is set to 25% in this study; in practice, the TMD mass ratio is chosen based on the physical limitations of the system. As illustrated in Figure 8, there are significant deviations between the conventional analytical tunings I and II with their parametric versions III and IV. Figure 8(b) depicts the fourth quadrant of FRF real-part because it is the area of interest for regenerative chatter suppression.
In Figure 8(a), it can be observed that I Den Hartog analytical tuning fails to minimize the FRF magnitude like the undamped primary body case presented in Figure 2. However, III minimizing FRF top magnitude produces two equal crests and maintains the lowest FRF magnitude.
Similarly, II Sims analytical tuning also faces a similar issue as illustrated in Figure 8(b), whereas IV maximizing FRF negative real-part least magnitude creates the desired behavior by forming two equal troughs with the highest magnitude.  In the case of an undamped primary body, the analytical tunings produce their expected results. However, in actual circumstances, where the primary body possesses damping, a TMD can be tuned more accurately by employing parametric methods.

Parametric tunings to mitigate simultaneous chatter
To account for the possibility of simultaneous chatter, employing a tuning method that controls FRF and its negative real-part at the intervals of the two standardized analytical tunings could be more effective for stability enhancement. Figure 9 illustrates the resultants of the five appointed parametric studies; each curve is generated by one of the discussed objective functions.
In Figure 9, the results of parametric studies V, VI, and VII are shown, generated using Equations (10)-(12)  as the objective functions, respectively. These three tuning methods appear to be more effective in situations where there is a risk of simultaneous regenerative and mode-coupling chatter. This is because the generated FRFs and negative real-parts curves are between those of the two single-aimed tuning methods (III and IV), i.e., the parametric versions of the conventional analytical tunings.
Machining processes typically generate shifting frequencies. Thus, to enhance machining stability, it is essential to consider a domain around the fundamental frequency of the unstable component. Figure 10 evaluates the discussed parametric studies on modecoupling chatter suppression based on FRF integrated area Δ F . A smaller Δ F indicates stronger mode-coupling chatter mitigation effects in the defined frequency domain. Figure 10 shows that the parametric version of Den Hartog has the largest FRF integrated area Δ F , indicating the weakest mode-coupling chatter suppression capabilities for the defined domain of frequency compared to the other tunings. Although this method is well-known for its effectiveness in mitigating mode-coupling chatter at the resonance frequency by generating the lowest FRF top magnitude, when considering a domain of shifting frequencies, minimizing the integrated area below FRF produces the lowest FRF magnitude. Therefore, method V, which minimizes the integrated area below FRF, demonstrates more generality compared to the other tunings.
In the following, a more tangible assessment of the regenerative chatter mitigation capabilities of the conventional analytical tunings, along with the IV and V parametric tunings, is presented using SLD. The other three  parametric tunings are not shown because they make the diagram chaotic.
Shifting the FRF negative real-part section towards the frequency axis lifts the SLD lobes up to a higher stable depth of cut, making the cutting process more stable against regenerative chatter. Figure 11 presents the SLD of the four selected tuning methods.
In Figure 11, it can be observed that IV maximizing FRF negative real-part least magnitude generates the most stable region, followed by II Sims analytical tuning. The proposed method of V minimizing integrated area below FRF curve, which produces excellent results for mode-coupling chatter mitigation at a domain of frequencies, creates a stable region that is much higher than I Den Hartog's analytical tuning and stands after II Sims analytical criterion. The tuning method V performs better than VI and VII against both mode-coupling and regenerative chatter.
To fill the left sides of the II Sims analytical tuning and V minimizing integrated area below FRF curve, more lobes can be added to the graphs. The second troughs of these two methods are at much higher depth of cut, which are not depicted in this figure. However, this does not imply that they create a more stable depth of cut at low spindle speeds because the first trough of the next q (the integer number of imprinted waves on the workpiece surface introduced by equation (6)) will bring down the stable region.

Conclusions
This study introduced new parametric tuning criteria for tuned mass dampers to mitigate simultaneous regenerative and mode-coupling chatter during machining processes on robot tools. The objective functions of the discussed tunings were to minimize integrated areas defined by FRF and its real-part curves at a certain domain of frequency around the fundamental resonance frequency of the unstable component of a cutting system. The investigated tunings were compared with the conventional analytical tuning methods of Den Hartog (suitable for mitigating mode-coupling chatter) and Sims (recommended for regenerative chatter suppression).
Among the three newly investigated parametric tuning methods, minimizing integrated area below FRF curve was found to be the most versatile tuning to mitigate the phenomenon of simultaneous chatter. Theoretical analysis demonstrated that this proposed parametric tuning can be more effective in mitigating simultaneous chatter compared to the single-aimed conventional analytical tuning methods. At a defined domain of frequency, it generated the lowest FRF magnitudes that mitigated mode-coupling chatter more effectively and caused shorter settling times compared to the analytical tunings. Furthermore, its tuning result was reasonably close to the effective tunings to mitigate regenerative chatter. Additionally, this tuning considered the damping of the base component, which is an important advantage, especially in the case of highly damped structures like robot tools.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.