Design of a tunable mass damper for mitigating vibrations in milling of cylindrical parts

Heng YUAN,Min WAN,*,Yun YANG

aSchool of Mechanical Engineering,Northwestern Polytechnical University,Xi'an 710072,China

bState IJR Center of Aerospace Design and Additive Manufacturing,Northwestern Polytechnical University,Xi'an,Shaanxi 710072,China

KEYWORDS Chatter;Milling;Mitigation of vibrations;Stability;Tunable mass damper(TMD)

Abstract Milling the free-end of cylindrical parts,which are vertically fixed on the machine table,often suffers from large chatter vibrations.This kind of phenomenon is harmful to the cutting process.Therefore,it is of great importance to develop means to suppress these undesirable chatters.This paper proposes a new idea for designing a tunable mass damper(TMD)to reduce vibrations in milling of cylindrical parts.Frequency response function(FRF)of the milling system is derived to comprehensively reveal the influence of both the dynamic response of the machine tools and the TMD.Critical axial depth of cut,which is usually used to characterize the process stability,is formulated by considering the FRFs of both the milling system itself and the TMD.Maximization of critical axial depth of cut is taken as objective function,while kernel dynamic parameters of TMD,which are involved in the derived expression of critical axial depth of cut,are extracted as designable variables.Optimization procedure is carried out to adjust the parameters of TMD by using sequential quadratic programming algorithm.A series of experiments with a designed passive TMD validate that the design has a good performance in reducing vibrations and improving stability of milling process.

1.Introduction

The free ends of some cylinder-like parts,such as gear shaft and end cam,often have special geometrical configurations,which usually need to be machined by milling process.Among these kinds of parts,some have to be fixed on the machine table in a cantilevered status,and thus,chatter,which results from a kind of self-excited vibration in the generation of chip thickness,is very easy to occur.This phenomenon is harmful to the stability of cutting process,and will decrease the accuracy and efficiency of parts.1-3Thus,it is of great importance to avoid chatters in machining processes of these parts.This paper presents a study on designing a tunable mass damper(TMD)to suppress the vibrations in milling of cylindrical parts with free-ends.

Actually,many research efforts have been made to develop chatter avoidance techniques.4-6Literature review shows that there exist two kinds of methods to improve the stability of cutting processes.

(1)One is to construct stability lobe diagram(SLD),which shows the relationship between cutting stability and process parameters,and then to select the chatter-free parameters from SLD to ensure stable cutting.

(2)The other is to design hardwares,e.g.active or passive TMDs,to suppress chatters.7

With respect to the first means,many excellent works have been done to predict SLDs.8-10Tobias11firstly expressed SLD as a function of depth of cut and spindle speed for the purpose of guiding cutting process.Altintas and Budak12realized predicting SLD by developing a zero-order frequency method,while Insperger and Stepan13did it by establishing a semidiscretization method.Based on direct integration scheme,Ding et al.14proposed a full-discretization method to predict SLD.Wan et al.15presented an efficient scheme,i.e.the lowest envelope method,to predict SLD with relatively low time cost.It should be pointed out that predicting SLD cannot increase the stability of the process,and it is just to obtain the natural dynamic response of the process itself,which is then used as guideline for planning the chatter-free process parameters.

As an important approach for improving the process stability,designing TMDs has been investigated by many researchers.TMDs can be grouped in two classi fi cations,i.e.the active and passive TMDs.The former is mainly concentrated on actively damping the machine's dynamics.Van Dijk et al.16developed a robust control methodology to increase the stability and productivity of milling process.Long et al.17designed an active strategy to control the relative vibration between the tool and workpiece during milling process.Wang et al.18employed a stiffness variation method to modulate the stiffness around a nominal value to suppress milling chatters.Passive TMDs are widely used in machining because of the easy implementation and low cost.Kolluru et al.19used the distributed discrete masses consisting of viscoelastic layer to achieve the required workpiece surface quality of thin walled casings.Bolsunovsky et al.20proposed a TMD to tune the damping with respect to a selected spindle rotation frequency,and realized stably machining a flexible part with good quality.Wan et al.21improved the chatter stability by attaching appropriate additional masses to the workpiece,and thorough studies were also carried out to reveal the effect of additional masses on stability lobes.Wang22proposed a new type of nonlinear TMD,which was equipped with an additional friction-spring element,to suppress chatters for improving machining performance.Yang et al.23,24designed a two-degree-of-freedom of TMD to mitigate milling vibrations,and achieved the aim of improving machining stability.Besides,Zhang et al.25conducted experimental investigations,which assessed the feasibility by submerging the milling system in viscous fluid,to damp milling chatters.Elias and Matsagar26described the dynamic behaviors and distinguishing features of single TMD and multiple TMDs.Shui and Wang27designed a mechanical vibration absorber with tunable piecewise-linear stiffness,which is realized by using a slider with two stop-blocks to constrain the bilateral deflections of the elastic support.The tunable absorber possesses a typical nonlinear characteristic at each given position of the slider,and its stiffness can be tuned over a wide range by adjusting the slider position.Shi et al.28presented a low-cost method of a constrained layer damper for effectively suppressing vibrations in thin-wall milling.Kumbhar et al.29utilized magnetorheological elastomer-shape memory alloy composite to design an adaptive tuned vibration absorber,which is able to shift the stiffness and the natural frequency and then reduce the vibration level of the primary system.Hayati et al.30presented a strategy to increase the machining stability by using a tunable holder with variable mass and stiffness to suppress chatters.Zhang et al.31proposed a tuned mass damper having the capacity of extending the band of working frequency through tuning the stiffness.

There are also some other methods aiming at suppressing the machining chatters.Stepan et al.32studied chatter avoidance in cutting process of highly flexible workpieces by means of finite element method.Mohring and Wiederkehr33presented intelligent fixtures to reduce workpiece vibrations and distortions.Zeng et al.34designed appropriate fixture layout scheme to suppress the machining vibrations of the flexible workpiece.Wan et al.35presented a stability improvement method in thin-wall milling by applying tensile prestress to the workpiece.

To well damp the vibrations,optimal design is usually necessary in the design procedure of TMD.Den Hartog36proposed an optimization criterion to minimize the maximum amplitude of the system's frequency response,and then obtained the expressions for calculating the TMD's optimal parameters.Nishihara and Asami37established a closed-form solution to the exact optimizations of dynamic vibration absorbers through minimizing the maximum amplitude magnification factors.Bandivadekar and Jangid38optimized the multiple tunable mass dampers to control the system's vibrations under external excitation.Asami et al.39presented an analytical solution to the classical optimization problem of TMD.Numerical methods were also employed to get the optimal parameters of TMD for suppressing vibrations in milling.40,41Li and Ni42used gradient theory to optimize TMD parameters without extra restrictive assumptions.It should be pointed out that the above researches on optimization were mainly conducted by taking the magnitude of the frequency response function as the optimal objective.However,studies on optimally tuning strategies to suppress chatters are limited.43Sims44considered the negative real part of frequency response function to propose an analytical tuning methodology.Saffury and Altus45optimized chatter resistance of viscoelastic turning bars by maximizing the negative real part of frequency response function.

It should be pointed out that in the design procedure of the TMD,there exist many design criteria.One typical feature of the proposed method lies in that it is established based on increasing the critical axial depth of cut,while the most existing methods are focused on reducing the magnitude of frequency response function.That is,the proposed method takes axial depth of cut as objective function,and can clearly and directly shows the tuned results.Whereas,the outputs of most existing methods are just the intermediate results,which need to be further processed to assess the desired objectives.

Different from the existing works,which took the frequency response function as optimization objective,this paper proposes a new objective,i.e.axial depth of cut,to realize opti-mally designing a passive TMD for suppressing the chatter vibrations in milling process of cylindrical parts.Extracting the objective function for obtaining better stability is described in detail.Parameters of TMD are optimized by using sequential quadratic programming algorithm.A series of experiments are conducted to validate the effectiveness of the designed TMD.Expressions for calculating the critical axial depth of cut and the optimal parameters are described in Section 2,in which designing the TMD is also detailed.A series of experiments are carried out to conduct verification in Section 3,followed by conclusion in Section 4.

2.Design principle of TMD

2.1.Frequency response function(FRF)of milling system with TMD

Without the loss of generality,FRF of the primary milling system together with an additional damper in X-direction,i.e.φxxwill be taken as an example for illustration.Fig.1 illustratively shows the structure that can reveal the dynamic characteristic of the milling system.It can be seen that it is a system consisting of a primary structure of the machine tools and a single degree-of-freedom passive damper.

The dynamic equation of the primary structure and damper can be described as follows.

where m0,c0and k0represent mass,damping ratio and stiffness of the primary structure's dominant mode.md,cdand kdare the mass,damping ratio and stiffness of the damper.x0and xdare the displacements of the primary structure and the damper,respectively.ω is the angular frequency.The force acting on the primary structure(m0)is considered to have a harmonic form of Feiωt.

Eq.(1)can be transformed into Laplace domain.

Fig.1 Primary milling system damped by TMD.

where X0and Xddenote the displacements of the primary structure and the damper in Laplace domain,respectively.From Eq.(2),the transfer function between the displacement of m0and external force acting on the m0can be expressed as follows.

For the convenience of study,the transfer function can be converted as frequency response function by replacing s with iω as follows.

with

where some involved parameters are de fined as follows.

2.2.Calculation of critical axial depth of cut

For the milling system shown in Fig.2,dynamic cutting forces in X-and Y-directions can be calculated as Eq.(7).1

with where Ktis the cutting coefficient in tangential direction.a is axial depth of cut.N is the teeth number.Δx and Δy denote variation of the dynamic displacements between present and previous tooth period in X-and Y-directions,respectively.is directional coefficient matrix,whose components can be calculated by the following expressions.1

where Kris the ratio of the radial cutting force coefficient to the tangential one.φstand φexare the start and exist angles.

Fig.2 Typical milling system with two degree of freedom.

The dynamic displacement vector｛Δt｝can be converted from time domain to frequency domain as follows.

where T is tooth passing period.Xnow（ω）and Xlast（ω）denote the dynamic displacement vectors in frequency domain at present time and previous tooth period,respectively.They can be calculated as follows.

where φ（iω）denotes the frequency response function of the structure,which is de fined as follows.

Substituting Eq.(9)into Eq.(7)in frequency domain yields

The milling system will be critically stable at the chatter frequency ωc,and thus,the characteristic equation can be determined as follows.

where［ I］is the unit matrix.Λ is eigenvalue with the Definition being as follows.

For systems used for milling cylindrical parts,the dynamic response of the spindle-holder-tool combination is approximately symmetrical.That is,the frequency response functions in X-and Y-directions are very close to each other.

Mathematically,the following equation holds.

Under this condition,the characteristic equation can be rearranged as a quadratic equation as follows.

with

where λ0=αxxαyy-αxyαyxand λ1=αxx+αyy.The root of Eq.(16)can be calculated as follows.

wi th

It should be noted that Γ is a complex number.For the simplicity of derivation,it can be further expressed by a real part ΓRand an imaginary part ΓIaccording to the calculation result shown in Eq.(19).Substituting Eqs.(15)and(19)into Eq.(18),Λ can be rewritten as follows.

with

Substituting Eq.(20)and e-iωcT=cosωcT-isinωcT into Eq.(14),critical axial depth of cut alimcan be expressed as follows.

The imaginary part of alimshould be zero because axial depth of cut is a real number.Under this understanding,alimcan be finally determined as follows.

Fig.3 Result of ΓRG+ΓIH.

where the coefficients(N,Kt,ΓRand ΓI)can be de fined as constants when the cutting conditions are determined.N and Ktand Γ2R+Γ2Iare all positive numbers.Because alimis a real number,it means that ΓRG+ΓIH should be less than zero around chatter frequency ωc.Fig.3 schematically shows one geometrical form of ΓRG+ΓIH.It should be pointed out that Fig.3 does not mean that ΓRG+ΓIH always has the form shown in Fig.3.Actually,the proposed design method aims at obtaining larger value of critical axial depth of cut.It should be pointed out that under a certain condition,ΓRand ΓIare known constants.From the perspective of this fact,Eq.(23)shows that critical axial depth of cut is only related to the minimum of ΓRG+ΓIH.In summary,obtaining larger value of alimdoes not need a unique form of Fig.3,and one only needs to maximize the minimum of ΓRG+ΓIH when it is negative.Thus,ΓRG+ΓIH is selected as optimization objective to design TMD in the following sections.

Substituting real(G)and imaginary(H)parts obtained from Eq.(5)into the concerned objective ΓRG+ΓIH and reducing constant k0leads to

As stated in Section 2.2,to realize maximizing alim,it is needed to maximize the minimum of p.Mathematically,this problem has the following form.

Actually,in Eq.(25),m0,c0and k0are the basic dynamic property of the machine tools,and they are usually some certain values.Thus,only the left parameters,f,ξdand μ,which are directly related to the dynamic parameters md,cdand kdof the damper(i.e.TMD),could be changed to realize the objective of Eq.(25).It should be pointed out that the structure of TMD is usually designed in advance according to the size of machine table and the workpiece to be machined.It means that the mass mdof TMD is not changeable for a selected process.That is,μ is usually considered as constant.Based on this understanding,to realize Eq.(25),it is mainly needed to calculate the optimal value of f and ξd.Then the problem can be described as follows.

This problem is considered as a multi-objective optimization and the method of sequential quadratic programming(SQP)is used to find the optimal values.

2.3.Design of TMD

The above principle can be used to design TMD for milling the end surface of a cylindrical bar.The main steps are as follows.

(1)Determine the size of the workpiece,and then measure the modal parameters ω0,m0and ξ0of the workpiece by conducting the modal test.

(2)Design and manufacture TMD around the natural frequency of the workpiece.It is planned to have a structure shown in Fig.4.

(3)Calculate the mass ratio μ and establish the relationship between the natural frequency ωdof TMD and the adjustable structure parameter ldof the designed TMD.

(4)Determine the damping ratio ξdof TMD under different contact conditions.

(5)Select the cutting parameters and determine the values of ΓRand ΓI,which are required in Eq.(19).

(6)For the concerned cutting conditions,calculate frequency ratio f and damping ratio ξdby using the optimization method described in Section 2.

(7)According to the results of steps(3)and(4),tune the TMD to the optimal position.

In the remaining contents of this section,an illustrative description will be made to explain the main components of the TMD together with how to conduct steps(3)and(4).

Fig.4 Illustration of TMD.

Table 1 Frequency of TMD under different length(ld).

Fig.5 Simulation results of TMD.

In Fig.4,c6 is the workpiece,which is a cylindrical bar.The workpiece(c6)is fixed on the base(c8).The sleeve(c1),which is fixed on the workpiece,is used to provide gap between the workpiece(c6)and the thin walled tube(c4).Four screws(c2)and the sleeve(c3)are used to fix the thin-walled tube(c4).Two long screws(c5)are used to adjust the height of the friction plate(c7).

The thin walled tube(c4)can be moved up and down in vertical direction so that the length(ld)between the fixed point and the top of the thin walled tube(c4)is adjustable.That is,through adjusting the position of c4,different frequency ωdof TMD can be achieved.In this paper,the parts have relatively low rigidity because low rigid part is easy to cause cutting chatters,which need to be suppressed.The size of TMD is determined according to the size of the concerned part.It should be noted that in industrial environment,different parts have quite different sizes and parameters.However,as long as it has low rigidity with the symmetrical column-like form,the basic principle described in this paper can be used to design TMD for the suppression of chatter vibrations.To experimentally validate the proposed idea,the workpiece used in this paper is selected as a cylindrical part with the diameter and the height being 30 mm and 250 mm.When modal test is completed,the size of the TMD is designed around the natural frequency of the workpiece.Through modal analysis,the size of the thin-walled tube(c4)is adjusted and determined.The inner diameter of c4 is 36 mm.The thickness and height of thin wall is 1 mm and 100 mm.The outer diameter of the mass block is 78 mm and its height is 25 mm.To further clarify the relationship between the frequency ωdof the TMD and the length ld,finite element model analysis of the TMD are conducted.Results are summarized in Table.1 and plotted in Fig.5.From Fig.5,it can be seen that with the increase of ld,the frequency ωdof the TMD increases.This phenomenon is due to that once the distance between the fixed point and the bottom of the thin walled tube decreases,the stiffness of the thin walled tube will increase,and as a result,the frequency ωdof the TMD increases correspondingly.Damping ratio of TMD is influenced by the contact condition between the thin walled tube(c4)and the friction plate(c7).The damping ratio can be tuned by using different abrasive papers to change the friction coefficient of contact surface.In this TMD,four series of abrasive papers,i.e.P60,P120,P180 and P320,are selected.The results of the FRFs corresponding to the four abrasive papers are measured by CutProTMand shown in Fig.6,from which the damping ratios related to P60,P120,P180 and P320 are determined as 2.28%,1.73%,1.26%and 0.62%,respectively.

Fig.6 Determination of damping ratios of TMD under different abrasive papers.

Fig.7 Experimental setup.

Table 2 ΓRand ΓIunder different radial depths of cut.

Fig.8 FRFs of workpiece without and with TMD for Case 1.

Table 3 Modal parameters of workpiece.

Table 4 Optimal parameters of TMD.

Fig.9 Experimental results(n=3000 r/min,ae=2 mm,ap=1.2 mm).

Fig.10 Experimental results(n=2000 r/min,ae=3 mm,ap=0.5 mm).

3.Experimental verification

Fig.7 is the actual setup of the TMD manufactured according to Fig.4.A series of down milling tests are conducted to validate the performance of the TMD on a three-axis CNC machining center.A four- fluted carbide flat end mill with tool diameter of 12 mm is used in the experiments.The material of workpiece is AISI1045 steel.A microphone is used to measure the signals of sounds during the process of the experiments.CutProTMis also used to collect and analyze the sound signals of the cutting processes.Three radial depths of cut,i.e.2 mm,3 mm and 6 mm,are adopted.Corresponding to the three used radial depths of cut,ΓRand ΓIrequired in Eq.(19)are calculated,as listed in Table.2.

The measured frequency response function(FRF)of the workpiece without the TMD is shown in Fig.8.The corresponding modal parameters(ω0,m0and ξ0)are extracted using the software of CutProTM,as listed in Table.3.The mass ratio μ of the TMD is calculated as 15%.According to the procedure in Section 3,the optimal frequency ratio f and damping ratio ξdof the TMD are listed in Table.4.Finally,the TMD is tuned to the corresponding optimal position.With respect to Case 1,the theoretical and experimental FRFs of the workpiece with the TMD are illustratively shown in Fig.8.It can be seen that once TMD works,the stiffness of the workpiece is greatly increased.

The measured sound signals and their Fourier transformations with and without TMD are shown in Figs.9-11.TF denotes integer times of tooth passing frequency,while CF denotes chatter frequency.Summary of the experimental results are listed in Table 5.

In Fig.9,spindle speed n is 3000 r/min.Radial depth of cut is 2 mm and axial depth of cut is 1.2 mm.The tooth passing frequency is 50 Hz.It can be seen that the dominant frequency is integer multiples of the tooth passing frequency whether TMD is used or not.That means that for Case 1,the milling process is stable whether TMD is on or off.At the same time,it is clearly seen that the amplitude of the sounds without TMD is obviously larger than that with TMD.What's more,the maximum of the FFT's spectra without TMD is more than twice the value of that with TMD.This phenomenon implies that the cutting process is more stable and smoother with the help of the TMD.

Fig.11 Experimental results(n=3000 r/min,ae=6 mm,ap=0.5 mm).

Another milling test is conducted under the cutting condition of n=2000 r/min,ae=3 mm andap=0.5 mm.The tooth passing frequency is 33.3 Hz.From Fig.10 and Table.5,it can be seen that the dominant frequencies,i.e.515 Hz and 648 Hz,are not integer multiples of the tooth passing frequency when the system is without TMD.

The frequencies,515 Hz and 648 Hz,are around 513.9 Hz(414 Hz plus 3 times of 33.3 Hz)and 647.1 Hz(414 Hz plus 7 times of 33.3 Hz),respectively.That means that chatter occurs in the cutting process.1Whereas,the cutting process is stable with lower sounds and smaller FFT spectra when the system is with TMD.

In the third experiment shown in Fig.11,a cutting condition of n=3000 r/min,ae=6 mm and ap=0.5 mm is chosen.The tooth passing frequency is50 Hz.From the collected signals,it can be seen that there exist many chatter frequencies such as 340 Hz,540 Hz,740 Hz and 940 Hz when the system is without TMD.The frequencies,340 Hz,540 Hz,740 Hz and 940 Hz,are around 364 Hz(414 Hz minus 50 Hz),564 Hz(414 Hz plus 3 times of 50 Hz),764 Hz(414 Hz plus 7 times of 50 Hz)and 964 Hz(414 Hz plus 11 times of 50 Hz),respectively.The maximum amplitude of the sounds is about 0.6 and the maximum of the FFT spectra is up to 0.13.When the system is with TMD,there also exist chatter frequencies such as 340 Hz and 500 Hz.However,the amplitude of the sounds is only about 0.2 and the maximum of the FFT spectra is less than 0.04.This means the TMD has good performance in reducing vibrations.

Table 5 Summary of experimental results.

To show the effectiveness of the designed TMD,stability lobe diagrams,which correspond to the three cases with and without TMD,have been added to Figs.9-11.It can be clearly seen that the critical axial depth of cut has been increased after using TMD.At the same time,it can be seen that the three experimental observations reasonably agree with the predicted SLDs.This means that the designed TMD helps to increase stability domain.From the results and analysis mentioned above,it can be seen clearly that the amplitude of sounds and maximum of the FFT spectra decrease obviously.The designed TMD shows a good performance in reducing vibrations and improving stability of cutting process.

4.Conclusions

(1)A tunable mass damper is designed and manufactured to suppress vibrations in milling process of cylindrical parts.

(2)The FRFs comprehensively considering the dynamic response of the machine tools and the TMD are derived to formulate the critical axial depth of cut,which is then selected as the objective to maximize the stable feasible region.Subsequently,parameters of TMD are designed and optimized by adopting sequential quadratic programming algorithm.

(3)A series of experiments show that the amplitude of sounds and maximum of the FFT spectra decrease obviously when the system is with the TMD.That is,the designed TMD shows a good performance in reducing vibrations and improving stability of cutting process.

Acknowledgements

This research has been supported by the National Natural Science Foundation of China(No.51675440 and 51705427),National Key Research and Development Program of China(No.2017YFB1102800),and the Fundamental Research Funds for the Central Universities of China(No.3102018gxc025).