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Abstract

Chatter is an undesirable dynamic instability phenomenon, mainly due to the low dynamic stiffness, that, in turning processes, results in poor surface quality and in a reduction in the cutting tool life. To overcome this problem, viscoelastic dynamic absorbers can be employed. Those absorbers are able to introduce reaction forces and dissipate vibration energy. In previous works, the main difficulty in the use of viscoelastic dynamic absorbers was in tool modelling uncertainties and in inaccurate viscoelastic material models. To overcome those difficulties, this article presents a new methodology to identify the machine tool structure dynamic properties by using 1-degree-of-freedom equivalent model combined with a fractional derivative model to describe the behaviour of the viscoelastic material. As a result, it was possible to design an optimal viscoelastic dynamic absorber for chatter mitigation in internal turning using non-linear optimization techniques. The use of generalized equivalent parameters for the absorber allows obtaining a simple equation of motion for the compound system (primary system plus absorber). The numerical and experimental tests were performed, showing the efficacy of the proposed controlling method design.

Keywords

Chatter viscoelastic dynamic absorbers turning process

Introduction

As is well known, the introduction of a high mechanical damping in structures with resonance problems or dynamic instability can reduce, considerably, its vibration levels making also possible to extend the stable operation range of the machine or structure. An example can be found in the internal turning of deep holes. This critical operation uses a tool with a high aspect ratio (length/diameter), resulting in a low stiffness system leading to an excessive vibration between the tool and the workpiece for certain operation conditions. This effect is even more critical since the tool holder have a low damping ratio, which can cause high vibration amplitudes and/or instability problems. As a result, poor surface finishing, geometrical errors and a reduced life of the chipping tool can be observed.^1,2 This stability limit depends on several factors such as the tool holder, clamping conditions, cutting parameters among others.³

In order to avoid a chatter condition, several solutions were developed, such as the use of materials with a high Young’s modulus and a high damping ratio,^4,5 the use of passive^6,7 and active vibration control methods.^8–13 In the passive technique, the objective is to suppress chatter by changing the system behaviour by either improving the design of the machine tool or the use of additional devices that can absorb extra energy or disrupt the regenerative effect.¹⁴ In the active technique, as the name suggests, the chatter vibrations are actively eliminated by continuous monitoring and diagnosis of the turning process and by executing necessary changes in parameters such as speed, feed and depth of cut.¹⁵

When compared with active control devices, passive controls have the advantages of easy implementation, greater reliability, low cost and no need for external energy.⁶

One example of a passive control method is the use of vibration absorbers. These systems are simple devices that, when attached to mechanical systems or structures, called primary systems, reduces their vibration or irradiated noise when these structures are working in a resonance frequency.

Rivin and Kang¹⁶ demonstrated that the application of a combined structure approach, with dynamic analysis, optimization and development of suitable vibration control means, results in significant enhancement of productivity and range of use of cantilever boring bars. Tarng et al.⁶ proposed the use of a piezoelectric inertia actuator mounted in a cutting tool and acting as a tuned vibration absorber for the suppression of chatter in turning operations. Ema and Marui⁷ showed that improvement of the damping capability of boring tools and suppression of chatter vibration may be attempted using impact dampers. Lee et al.¹⁷ developed a passive vibration control system using a dynamic vibration absorber mounted on a cutting tool to suppress vibrations in turning operations. It was shown that the dynamic response of the cutting tool can be greatly improved due to the presence of the dynamic vibration absorber attached to the cutting tool. However, the dynamic vibration absorber must satisfy two conditions: one is the natural frequency that must be equal or close to the natural frequency of the cutting tool and the other is a high damping ratio. Rashida and Nicolescu¹⁸ developed tuned viscoelastic dampers for vibration control through their application on a workpiece in milling operations. This work targeted workpiece held on a palletized workholding system for the control of unwanted vibration. However, the viscoelastic model used was not accurate in a broad frequency band for a given temperature resulting in a not optimal design. Saffury and Altus¹⁹ considered the use of a viscoelastic beam for increasing the resistance against regenerative chatter of turning bars. The optimum structural parameters were found by maximizing the most negative real part of the frequency response function (FRF).

The use of modern viscoelastic materials makes the vibration absorbers easy to produce and suitable for use on almost any structure. Over the last years, the theory of vibration absorbers, originally developed by Ormondroyd and Den Hartog²⁰ for single-degree-of-freedom structures, has been reformulated in order to be applied to complex structures. One contribution was given by Espíndola and Silva,²¹ which presented a general methodology for the optimal design of an absorber system attached to a geometric complex structure, which can be applied for a single frequency or a large frequency band. The introduction of genetic algorithms and non-linear optimization techniques (hybrid algorithm) by Espíndola and Bavastri²² and Bavastri et al.²³ made this technique more general. By using this methodology, one or more vibration modes, in a particular frequency band, can be controlled with one or more viscoelastic dynamic absorbers.

According to Siddhpura and Paurobally¹⁵ for good performance, some passive dampers require very accurate tuning. In previous works, the main difficulty in the use of viscoelastic dynamic absorbers was in tool modelling uncertainties and in inaccurate viscoelastic material models. This article presents a new methodology to identify the machine tool structure dynamic properties by using 1-degree-of-freedom equivalent model and design an optimal viscoelastic dynamic absorber for chatter mitigation in internal turning. The four-parameter fractional derivative was used to characterize the viscoelastic material, and the generalized equivalent parameters (GEPs), as presented by Espíndola and Silva,²¹ were reviewed and implemented to model the dynamic absorber. The primary system (tool holder plus clamping system of the machinery) is modelled, in the frequency of interest, as an equivalent 1-degree-of-freedom linear system for more realistic representation, being the damping, stiffness and mass obtained by an inverse identification problem.

The accuracy in identification of the primary system and the model used for the viscoelastic material, together with the use of the GEP, allows obtaining the physical parameters of the absorber in the optimal way, introducing in the primary system the maximum dynamic impedance.

The passive vibration control method

Viscoelastic dynamic vibration absorbers (VDVAs) are easy to make and apply to a structure of any size and shape. This is, in part, possible, thanks to modern technology of viscoelastic materials, which makes it easy to mould in any shape and to tailor it to meet almost any specifications.

With this concept, it is possible to write down the equations for the movement of the composite system (primary plus absorbers) in terms of the generalized coordinates (degrees of freedom), previously chosen to describe the configuration space of the primary system alone, in spite of the fact that the composite system has additional degrees of freedom introduced by the attached absorbers.

These concepts will be applied in a particular case where the primary system is taken as a 1-degree-of-freedom system.

Mathematical model to viscoelastic material

Since the absorbers to be discussed here are of a viscoelastic nature, it is appropriate to provide a simple introduction to this class of materials using the fractional derivatives model. For these materials, the complex shear modulus is a function of frequency and temperature and is given by^24–27

G_{c} = \frac{G_{0} + {(i b_{0} Ω α_{T})}^{α} G_{\infty}}{1 + {(i b_{0} Ω α_{T})}^{α}}

(1)

where $i$ is the complex unit, $G_{0}$ represents the low-frequency asymptotic value of the shear modulus, $G_{\infty}$ the high-frequency asymptotic value of the shear modulus, $b_{0}$ the relaxation time of the viscoelastic material at the reference temperature $T_{0}$ (see below),²⁸ $α$ the fractional derivative order of the viscoelastic material model, $α_{T}$ the Williams–Landel–Ferry (WLF) shift factor function and $Ω$ the excitation frequency. This is a function of the absolute temperature $T$ and may be defined as

\log_{10} α_{T} = \frac{- θ_{1} (T - T_{0})}{θ_{2} + T - T_{0}}

(2)

where $θ_{1}$ and $θ_{2}$ are constants and $T_{0}$ is a reference absolute temperature.²⁹ Note that $α_{T_{0}} = 1$ .

All constants $G_{0}, G_{\infty}, b_{0}, α, θ_{1}, θ_{2} and T_{0}$ can, comfortably, be identified experimentally.

Thus, at a given temperature, the complex modulus can be written as

G_{c} (Ω) = G (Ω) [1 + i η (Ω)]

(3)

where $η (Ω)$ is known as the loss factor of the material, the relation between the imaginary and real part of the complex shear modulus, and $G (Ω)$ is the dynamic shear modulus, the real part of the complex shear modulus. The loss factor is a measure of the ability of the material to transform dynamic stressing into heat. In this article, the working temperature is considered constant, so the parameter $T$ is omitted for simplicity of notation.

Equivalent generalized quantities for a sample absorber

As introduced by Espíndola and Silva,²¹ a simple absorber can be represented by a single lumped mass ( $m_{a}$ ) connected to a rigid massless base through a resilient viscoelastic device (spring) (Figure 1) with complex stiffness $K_{c} (Ω)$ equal

K_{c} (Ω) = ℓ G_{c} (Ω) = ℓ G (Ω) [1 + i η (Ω)]

(4)

where $ℓ$ is a geometric factor that depends on the geometry of the viscoelastic material in the absorber. In Figure 1(a), $X (Ω)$ , $X_{b} (Ω)$ and $F (Ω)$ are the Fourier transforms of the displacement of the mass, $x (t)$ , of the base, $x_{b} (t)$ , and the applied force $f (t)$ , respectively. This applied force results from the interaction between the absorber and the part of the structure where it is applied.

Figure 1.

(a) A simple viscoelastic absorber and (b) the equivalent model.

As introduced by Espíndola et al.,²⁷ the equivalent viscous damping is defined as the real part of the mechanical impedance at the base of the absorber ( $Z_{b} (Ω) = F (Ω) / i Ω X_{b} (Ω)$ )

c_{eq} (Ω) = m_{a} Ω_{a} \frac{r_{a} (Ω) η (Ω) ε_{a}^{3}}{{[ε_{a}^{2} - r_{a} (Ω)]}^{2} + {[r_{a} (Ω) η (Ω)]}^{2}}

(5)

In the above equation, $Ω_{a}$ is the natural frequency of the system of Figure 1(a), that is the frequency considering the viscoelastic spring with no damping, $ε_{a} = Ω / Ω_{a}$ , $r_{a} = G (Ω) / G (Ω_{a})$ and $η (Ω)$ is the loss factor of the viscoelastic material of the spring.

In the same way, the equivalent generalized mass is the real part of the dynamic mass, $M_{b} (Ω) = F (Ω) / - Ω^{2} X_{b} (Ω)$ , represented by equation (6)

m_{eq} (Ω) = - m_{a} \frac{r_{a} (Ω) {ε_{a}^{2} - r_{a} (Ω) [1 + η^{2} (Ω)]}}{{[ε_{a}^{2} - r_{a} (Ω)]}^{2} + {[r_{a} (Ω) η (Ω)]}^{2}}

(6)

As can be seen in Espíndola and Silva, the two systems shown in Figure 1 are dynamically equivalent.²¹ The primary system ‘feels’ the absorber as being an equivalent, frequency-dependent mass $m_{eq} (Ω)$ attached to it and an equivalent viscous damper with constant $c_{eq} (Ω)$ connected to the ground.

The response of the compound system

As shown in Figure 2, in the particular case that an absorber is connected to the 1-degree-of-freedom primary system, the equation of motion can be written, in the frequency domain, as

[- Ω^{2} (m + m_{eq} (Ω)) + i Ω (c + c_{eq} (Ω)) + k] X (Ω) = F (Ω)

(7)

Figure 2.

(a) Compound system with traditional model and (b) equivalent model.

Note that the effect of attaching an absorber is to modify both the mass and damping elements of the primary system, whose vibration response is $X (Ω)$ . Then, the frequency response (receptance) of the compound system, measured in the primary system, where the excitation is also applied, is given by

H (Ω) = \frac{X (Ω)}{F (Ω)} = \frac{1}{[- Ω^{2} (m + m_{eq} (Ω)) + i Ω (c + c_{eq} (Ω)) + k]}

(8)

where $H (Ω)$ is the receptance matrix of the compound system.

Primary system identification and optimal control

In order to obtain an optimal design of the absorber, first, the primary system must be identified. Since a simple test with the tool holder shows (see Figure 3 below) that in the band of frequencies of interest, there is only an important one, the primary system will be modelled as a 1-degree-of-freedom equivalent system, allowing the use of the model (Figure 2) presented in the previous section.

Figure 3.

Experimental and adjusted FRF curves.

Primary system identification

The primary system, the tool holder, is modelled, as justified above, as an equivalent 1 degree-of-freedom system, with m, c and k identified in an inverse problem. To this end, 1-degree-of-freedom FRF (inertance) is adjusted, in a non-linear optimization scheme, to the correspondent experimental FRF, measured in the primary system. The design variable is a vector consisting of the parameters m, c and k and is defined as

x = {(m, k, c)}^{T}

(9)

The objective function was defined as a quadratic error between the FRF measured and its equivalent mathematical model of 1-degree-of-freedom system and is given by

f_{obj} (x) = erro r^{*} error

(10)

where ‘*’ stands for the transpose conjugate of

error = {\begin{matrix} ⋮ \\ FR F_{\exp} (Ω_{k}) - FR F_{mat} (Ω_{k}) \\ ⋮ \end{matrix}}

(11)

where $Ω_{k}$ is the kth frequency, $FR F_{\exp}$ is the experimental inertance and $FR F_{mat}$ is the mathematical inertance. The mathematical inertance is obtained using equation (8) with $m_{eq} (Ω)$ and $c_{eq} (Ω)$ equal to 0 and the numerator multiply by $- Ω_{k}^{2}$ .

Optimal design of the DVA

When optimizing the physical parameters of the absorber, the natural frequencies are considered as design parameters, while the absorber masses are pre-established in the similar form as Den Hartog³⁰ and the loss factor is known once the viscoelastic material is pre-defined. In this case, the design variable is given by

x = Ω_{a}

(12)

The objective function and inequality constraints used in the optimization methodology are

f_{o b j} (x) = \max_{Ω 1 < Ω < Ω 2} | H (Ω, x) |

(13)

where $H (Ω, x)$ is given by equation (8), || is the modulus of the complex quantities, $x^{L} < x < x^{U}$ , where $x^{L}$ is the lower constrain and $x^{U}$ is the upper constrain of the design variable,³¹ and $Ω 1$ and $Ω 2$ are the lower and upper limits of the frequency band of interest that contains the natural frequencies of the compound system.

The optimization process aims at minimizing the objective function (equation (13)), that is, at minimizing the frequency response level modulus of the primary system in the frequency band of interest.

After the optimization process $Ω_{a}$ is obtained, then the spring stiffness of the absorber can be computed by

k_{a} = m_{a} Ω_{a}^{2}

(14)

where m_a is the mass of the absorbers. This mass $m_{a}$ is fixed outside of the optimization process in the similar form as Den Hartog theory²⁹

μ = \frac{m_{a}}{m}

(15)

where $m$ is the mass of the primary system and $μ$ is the mass relation whose value can be adopted between 0.1 and 0.25.

Once $k_{a}$ and $m_{a}$ are known, by using equation (4), it is possible to manufacture the absorber.

Numerical and experimental results

Primary system identification

The primary system is a tool holder (E2S-SDUPR-11 L-C3-95) with 175 mm overhang. The identification was performed using the methodology presented in primary system identification. Figure 3 shows the quality of the identification by comparing two curves: the measured curve and the regenerated one. The regenerated curve was obtained by using equation (8), with $m_{eq} (Ω)$ and $c_{eq} (Ω)$ equal to 0, and the numerator multiply by $- Ω^{2}$ (excitation frequency). The optimal equivalent parameters of the 1-degree-of-freedom model are presented in Table 1. The measured FRF shows a large peak, much greater than the others (almost 20 dB greater), which justifies the theoretical modelling of the primary system as a 1 degree-of-freedom one. This process of identifying the primary system is far more realistic than the usual practice of taking the tool holder as a free-clamped continuous beam because we are identifying the tool holder, the influence of the clamping system and the damping of the system.

Table 1.

Equivalent parameters of the primary system.

Equivalent mass (kg)	0.2582
Equivalent stiffness (N/m)	2729654.2268
Equivalent damping ratio $ξ = c / 2 m Ω_{n}$	0.0209

The parameter $Ω_{n}$ is the undamped natural frequency $\sqrt{k / m} = 3251.44 rad/s$ (f_n = 517.48 Hz) and c is the viscous damper coefficient, which was obtained in the optimization process = 35.09 N s/m.

Numerical simulation and optimal design

Once the primary system is known, the optimal anti-natural frequency $Ω_{a}$ of the absorber can be founded. The mass $m_{a}$ was fixed using equation (15) with $μ = 0.1$ .

The resilient elements used in the absorber for simulations were the Dyad 601 elastomer³² and a type of the neoprene material whose four fractional derivative parameters are given in Table 2, where the influence of temperature is also considered. The dynamic characteristics of these elastomers can be found in Figure 4, which represents the dynamic shear modulus and the loss factor, both of them as a function of frequency and temperature variable. These curves were obtained through equation (1) and the parameters shown in Table 2.

Table 2.

Four-fractional derivative parameters and its influence on temperature for Dyad 601 and Neoprene.

Parameters	Dyad 601	Neoprene
$G_{0} (N / m^{2})$	$3.4753 \times 10^{5}$	$4.55 \times 10^{6}$
$G_{\infty} ({N/m}^{2})$	$7.886 \times 10^{8}$	$4.18 \times 10^{8}$
$α$	$0.48344$	$0.319$
$b_{0}$	$1.0892 \times 10^{- 5}$	$9.2869 \times 10^{- 9}$
$θ_{1}$	$16.1581$	$5.09$
$θ_{2}$	$84.944$	$46.5$
$T_{0} (K)$	$278.824$	$303$

Figure 4.

Shear modulus and loss factor: (a) Dyad 601 and (b) Neoprene.

The optimal physical parameters of the dynamic viscoelastic absorber are found using the particular methodology presented in section ‘Optimal design of the DVA’ for a frequency band from 0 to 2245 Hz.

After the optimization procedure, a natural frequency of $f_{a} = Ω_{a} / (2 π) = 430.7986 Hz$ and a mass m_a = 25.82 g were obtained for the absorber. Based on the numerical analysis, the FRFs for a system with and without the absorber were generated. Figure 5 shows that there is a remarkable 14 dB reduction in numerical inertance when the optimal absorber is attached to the primary system. When the optimal absorber is composed by Dyad 601 elastomer, the FRF of the compound system has a one pick due to the high damping and when the absorber is composed with neoprene, as viscoelastic material, the FRF has the normal form with 2 degrees of freedom. This behaviour of the Dyad 601 elastomer can be explained due to its high damping, practically 10 times larger than the damping of the neoprene. In this work, Dyad 601 elastomer was used as resilient material.

Figure 5.

Numerical inertance with and without the optimal absorber.

Experimental analysis

After determining the optimal physical parameters (natural frequency and mass), and knowing the dynamic characteristics of the viscoelastic material, the control device can be physically realized according to Espindola and Bavastri.²² The design of the absorber is shown in Figure 6.

Figure 6.

Control device.

The dimensions of the viscoelastic material, as well as those of the absorber, are shown in Table 3.

Table 3.

Absorber dimensions.

Viscoelastic material of each sector
Height, h (mm)	4
Thickness, b (mm)	3.3
Width, t (mm)	10
Tuning mass
External diameter (mm)	24.8
Internal diameter (mm)	14
Thickness (mm)	10
Central core
Diameter (mm)	8

The tool holder was mounted in the machine with a 175 mm overhang. The machine tool used was the Mazak Super Quick Lathe 10M. For practical reasons, external turning operations were accomplished to simulate deep hole internal turning operations. Brass workpieces 32 mm in diameter and 70 mm long were machined using the tool holder with and without the absorber. The cutting speed and the feed rate used were 100 m/min and 0.05 mm/rev, respectively. The depth of cut ranged from 1.0 to 4.0 mm. These parameters were selected to induce chatter condition. The vibratory response was measured during the machining process by using an accelerometer ICP 352C65 attached to the tool holder (50 mm from its end) and a dynamic signal analyser HP 3560A. The sampling frequency was 5120 Hz.

Figure 7 shows a picture of the experimental set-up, including the tool holder and the absorber attached to it.

Figure 7.

Set-up for the experiments with the tool holder and the absorber.

Figure 8 shows the vibratory response in the frequency domain measured during the machining process for a depth of cut 1 mm. In the spectrum of the acceleration for the tool holder without the absorber, the highest peak corresponds to a dynamic instability, which occurs close to the first natural frequency of the system (Figure 8(a)). The process was unstable and high-level vibrations occurred. There are also visible peaks related to the second and third harmonics of the dominating frequency. Usually, there is a single well-defined chatter frequency for an unstable turning process. However, harmonics of the chatter frequency may appear in the vibration energy spectrum for unstable cuts with boring bars, probably due to the non-linearities, as reported by Pratt and Nayfeh,³³ Andrén et al.³⁴ and Daghini et al.³⁵

Figure 8.

System response: (a) without the absorber and (b) with the absorber.

This reduction in vibration level resulted in an improvement in the surface finish of the workpiece, as can be seen in Figure 9. The measured surface roughness (R_a) of the workpiece machined by the tool holder without the absorber was 4.54 µm, while the measured surface roughness (R_a) of the workpiece machined by the tool holder with the absorber was 1.27 µm.

Figure 9.

Workpieces machined by tool holder: (a) without the absorber and (b) with the absorber.

Figure 10 shows the surface roughness in function of the depth of cut for the workpieces machined using the tool holder with and without the absorber. The surface roughness values are higher for the workpieces machined with the tool holder without the absorber due to the high-amplitude vibrations that occurred during the process. The workpieces machined with the tool holder with the absorber have relatively low surface roughness, and their values do not change significantly with the increase in the depth of cut. Thus, the dynamic absorber was effective to avoid chatter and allowed for an increase in the material removal rate. Greater depths of cut were not used because of the geometrical limitations of the tool inserts.

Figure 10.

Surface roughness against depth of cut of the workpieces machined using the tool holder with and without the absorber.

Conclusion

A new methodology for the optimal design of viscoelastic dynamic absorber to mitigate chatter in turning operation is presented.

The four-parameter fractional derivative model for the viscoelastic material was used. The GEPs were reviewed and implemented to model the dynamic absorber.

The primary system was modelled as an equivalent 1-degree-of-freedom linear system for more realistic representation, in the frequency band of interest. The damping, stiffness and mass that represent the tool holder and its clamping system were obtained through inverse identification problem.

The accuracy in identification of the primary system and the accurate model used for the viscoelastic material, together with the use of the GEP, allowed to obtain the physical parameters of the absorber in an optimal way.

The present methodology was used to design a viscoelastic dynamic absorber to mitigate chatter for internal turning process. The experimental results show that the passive control device was effective in reducing the vibration during the process due to the introduction of high reaction forces and capability to dissipate vibration energy. Thus, a higher material removal rate can be achieved by using tool holder with viscoelastic dynamic absorber designed by this methodology.

Footnotes

Appendix 1 Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

C.A.B. acknowledges the financial support of CNPq. D.R.V. acknowledges the financial support of PRH24.

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A methodology to mitigate chatter through optimal viscoelastic absorber

Abstract

Keywords

Introduction

The passive vibration control method

Mathematical model to viscoelastic material

Equivalent generalized quantities for a sample absorber

The response of the compound system

Primary system identification and optimal control

Primary system identification

Optimal design of the DVA

Numerical and experimental results

Primary system identification

Numerical simulation and optimal design

Experimental analysis

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References