Bifurcation and chaotic vibration of frictional chatter in turning process

Abstract

A comprehensive 3-degree-of-freedom dynamic model of turning process is proposed to overcome the shortage of consideration on the feed velocity in Rusinek’s work; nonlinear dynamics behaviors of the cutting tool are investigated by means of displacement bifurcation diagram, time history, and power spectrum with cutting velocity as the bifurcation parameter. These results show that there are many kinds of nonlinear dynamic phenomena under the effect of cutting velocity, such as period, chaos, and crisis; furthermore, periodic chaos and crisis, and the high cutting velocity seem to be the effective means to suppress the vibration amplitude of the cutting tool.

Keywords

Bifurcation chaotic turning process frictional chatter cutting velocity

Introduction

Chatter as one of an important topic because it frequently results in poor surface finish, reduce efficiency, increase the tool wear, and limited productivity. It is widely accepted that there are four different types of cutting chatters in cutting process,¹ they are frictional chatter, regenerative chatter, mode coupling chatter, and thermo-mechanical chatter. After years of research found that the most important and common causes of chatter were the friction or regenerative factors. Frictional chatter is mainly attributed to the nonlinear dry friction force and velocity-dependent cutting force.^2–4 Regenerative chatter occurs because of the unstable cutting force that caused by the variation in the effective chip thickness or the rate of penetration of the tool.^5–8

Chatter is considered to be an important factor that can affect cutting stability and machining productivity by Taylor.⁹ Then, a fundamental theory of cutting chatter was first proposed by Tobias¹⁰ and Koenigsberger and Tlusty¹¹ who provide the initial basic framework within which the cutting chatter problem is usually formulated. Moreover, Arnold¹² studied cutting chatter take into account the effect of the negative damping that mainly due to the dry friction between the workpiece and the tool. Furthermore, the nonlinear dynamical model of the cutting process was developed by Grabec¹³ and he investigated the frictional chatter take into account the velocity-dependent cutting forces and the dry friction between the workpiece and the chip. All in all, a variety of nonlinear dynamic cutting models have been established and studied in the past, the main nonlinear factors considered in the cutting systems are the nonlinear friction coefficient,¹⁴ the separation of the tool and workpiece,¹⁵ and the tool and workpiece geometry.^15–17 Especially, a general dynamical model of the cutting process account for the vibrating tool leaving the workpiece was developed by Rusinek et al.¹⁸ The bifurcation analysis has been applied to the milling and grinding process.^19,20 Moreover, a novel dynamical model of the cutting process was developed by Rusinek et al.,²¹ in which the tool vibrations are generated by a combined action of the velocity-dependent cutting forces and the dry frictional force acting on the tool face as well as on the tool flank. However, a 2-degree-of-freedom model does not fully show the nonlinear dynamic characteristics of the cutting system. On the basis of previous work, a comprehensive 3-degree-of-freedom model of turning process was established and the feed velocity factor was considered in this work.

Dynamic model of the turning system

In this work, a comprehensive 3-degree-of-freedom model of turning process is proposed, as shown in Figure 1. The cutting tool is modeled as a beam that can vibrate in the X, Y, and Z directions while the workpiece is assumed to be a rigid one. Hence, the governing equations of the cutting tool vibration in the three directions are given as

\begin{matrix} m \overset{\cdot\cdot}{x} + c_{x} \overset{\cdot}{x} + k_{x} x = F_{x} + N_{x} \\ m \overset{\cdot\cdot}{y} + c_{y} \overset{\cdot}{y} + k_{y} y = F_{y} + N_{y} \\ m \overset{\cdot\cdot}{z} + c_{z} \overset{\cdot}{z} + k_{z} z = - F_{z} - N_{z} \end{matrix}

(1)

where $m$ is an equivalent mass of the cutting tool, $c_{i}$ and $k_{i}$ are equivalent damper coefficients and equivalent spring coefficients, the tool vibration displacements x, y, and z are measured from their equilibrium positions; particularly, $N_{i}$ represents the normal force, $F_{i}$ represents the dry friction force;^2,21 here, i = x, y, z means X, Y, and Z directions; they are defined in equations (2) and (3), respectively, as follows

\begin{matrix} N_{x} = Q_{0} h (c (v_{x} - 1)^{2} + 1) H (h) H (v_{x}) \\ N_{y} = K_{cony} hH (h) \\ N_{z} = K_{conz} zH (z) \end{matrix}

(2)

where $N_{x}$ is the normal force between the tool and the chip in X direction, $N_{y}$ is the normal force between the tool and the workpiece in Y direction, $N_{z}$ is the normal force between the tool and the workpiece in Z direction, $Q_{0}$ is the specific cutting force modulus, $h$ is the instantaneous thickness of cut, $c$ is a constant controlling the dependence of the cutting force on the relative velocity between the tool and the workpiece, $K_{cony}$ and $K_{conz}$ are the contact stiffness between the tool and the workpiece in the Y and Z directions, respectively. H(.) is the Heaviside function, sgn(.) is the sign function, and $v_{x}$ is the relative velocity between the tool and the workpiece in the X direction.

Figure 1.

A 3-degree-of-freedom model of turning process, where $m$ is the equivalent mass of the cutting tool. $k_{x}$ , $k_{y}$ , and $k_{z}$ , $c_{x}$ , $c_{y}$ , and $c_{z}$ are equivalent spring coefficients and equivalent damper coefficients in the X, Y, and Z directions, respectively. $h$ is the instantaneous chip thickness, $ϕ$ is the shear angle of the workpiece material, $v_{c}$ is the nominal cutting velocity, $v_{f}$ is the feed velocity of the tool, $v_{x}$ is the relative velocity between the tool and the workpiece in the X direction, $v_{y}$ is the relative velocity between the tool and the chip in the Y direction, and $v_{z}$ is the relative velocity between the tool and the workpiece in the Z direction. $N_{x}$ , $N_{y}$ , and $N_{z}$ , and $F_{x}$ , $F_{y}$ , and $F_{z}$ are the normal forces and the dry friction forces on the tool surfaces of the tool in the three directions, respectively.

In addition, $F_{x}$ , $F_{y}$ , and $F_{z}$ are the dry friction forces between the tool, the workpiece, and the chip in the three directions have the form

\begin{matrix} F_{x} = & N_{y} μ_{x} (sgn (v_{x}) - a_{x} v_{x} + β_{x} {v_{x}}^{3}) \\ + N_{z} μ_{x} (sgn (v_{x}) - a_{x} v_{x} + β_{x} {v_{x}}^{3}) \\ F_{y} = & N_{x} μ_{y} (sgn (v_{y}) - a_{y} v_{y} + β_{y} {v_{y}}^{3}) \\ + N_{z} μ_{y} (sgn (v_{f} + \overset{\cdot}{y}) - a_{y} (v_{f} + \overset{\cdot}{y}) + β_{y} (v_{f} + \overset{\cdot}{y})^{3}) \\ F_{z} = & N_{x} μ_{z} (sgn (v_{z}) - a_{z} v_{z} + β_{z} {v_{z}}^{3}) \\ + N_{y} μ_{z} (sgn (v_{z}) - a_{z} v_{z} + β_{z} {v_{z}}^{3}) \end{matrix}

(3)

where $F_{x}$ is the dry friction force between the tool and the workpiece in X direction, $F_{y}$ is the dry friction force between the tool and the workpiece, between the tool and the chip in Y direction, $F_{z}$ is the dry friction force between the tool and the workpiece, between the tool and the chip in Z direction, $v_{x}$ is the relative velocity between the tool and the workpiece in the X direction, $v_{y}$ is the relative velocity between the tool and the chip in the Y direction, $v_{z}$ is the relative velocity between the tool and the workpiece in the Z direction, $v_{f}$ is feed velocity in the Y direction, $μ_{x}$ , $μ_{y}$ , and $μ_{z}$ are the static coefficient of friction in the three directions, respectively, and $a_{x}$ , $β_{x}$ , $a_{y}$ , $β_{y}$ , $a_{z}$ , and $β_{z}$ are constants which regulate the nonlinear characteristics of the friction forces between the respective surfaces in contact.

In addition, the instantaneous thickness of cut $h$ is determined by the equation

h = v_{f} τ - y, τ = 2 π / Ω

(4)

where $v_{f}$ is feed velocity in the Y direction, $τ$ is a workpiece revolution period, and $Ω$ is the angular velocity of the workpiece as well as the function of the cutting velocity.

As follows, $v_{x}$ is the relative velocity between the tool and the workpiece in the X direction, $v_{y}$ is the relative velocity between the tool and the chip in the Y direction, and $v_{z}$ is the relative velocity between the cutting tool and the workpiece in the Z direction, they can be expressed as

v_{x} = v_{c} - \overset{\cdot}{x}, v_{y} = v_{x} \tan ϕ - \overset{\cdot}{y} + v_{f}, v_{z} = \overset{\cdot}{z}

(5)

where $v_{c}$ is the nominal cutting velocity, $v_{c} = Ω R$ , $R$ is the radius of the workpiece, and $ϕ$ is the shear angle of the workpiece material. From above equations (4) and (5), it is obvious that the feed velocity will directly affect the instantaneous cutting thickness and the relative velocity between the tool and the chip in the Y direction in our model. These effects can lead to changes in the instantaneous cutting force and affect the dynamic characteristics of the cutting system.

Substituting equations (2) and (3) into equation (1) and rearranging them, so the governing equations of the cutting tool vibration in the three directions are given by

\begin{matrix} m \overset{\cdot\cdot}{x} + c_{x} \overset{\cdot}{x} + k_{x} x = K_{cony} hH (h) μ_{x} (sgn (v_{x}) - a_{x} v_{x} + β_{x} {v_{x}}^{3}) \\ + K_{conz} zH (z) μ_{x} (sgn (v_{x}) - a_{x} v_{x} + β_{x} {v_{x}}^{3}) + Q_{0} h (c (v_{x} - 1)^{2} + 1) H (h) H (v_{x}) \\ m \overset{\cdot\cdot}{y} + c_{y} \overset{\cdot}{y} + k_{y} y = Q_{0} h (c (v_{x} - 1)^{2} + 1) H (h) H (v_{x}) μ_{y} (sgn (v_{y}) - a_{y} v_{y} + β_{y} {v_{y}}^{3}) \\ + K_{conz} zH (z) μ_{y} (sgn (v_{f} + \overset{\cdot}{y}) - a_{y} (v_{f} + \overset{\cdot}{y}) + β_{y} (v_{f} + \overset{\cdot}{y})^{3}) + K_{cony} hH (h) \\ m \overset{\cdot\cdot}{z} + c_{z} \overset{\cdot}{z} + k_{z} z = - Q_{0} h (c (v_{x} - 1)^{2} + 1) H (h) H (v_{x}) μ_{z} (sgn (v_{z}) - a_{z} v_{z} + β_{z} {v_{z}}^{3}) \\ - K_{cony} hH (h) μ_{z} (sgn (v_{z}) - a_{z} v_{z} + β_{z} {v_{z}}^{3}) - K_{conz} zH (z) \end{matrix}

(6)

The cutting force in three directions is composed of the normal force and the dry friction force. In order to simplify the calculation, we nondimensionalize the above equations. By defining a characteristic time scale and make a characteristic length scale L = 1, the dimensionless equations of vibration could be modified as follows

\begin{matrix} \overset{\cdot\cdot}{x} + 2 ξ_{x} \overset{\cdot}{x} + x = K_{cony} hH (h) μ_{x} (sgn (v_{x}) - a_{x} v_{x} + β_{x} {v_{x}}^{3}) \\ + K_{conz} zH (z) μ_{x} (sgn (v_{x}) - a_{x} v_{x} + β_{x} {v_{x}}^{3}) + q_{0} h (c (v_{x} - 1)^{2} + 1) H (h) H (v_{x}) \\ \overset{\cdot\cdot}{y} + 2 ξ_{y} \sqrt{α} \overset{\cdot}{y} + α y = q_{0} h (c (v_{x} - 1)^{2} + 1) H (h) H (v_{x}) μ_{y} (sgn (v_{y}) - a_{y} v_{y} + β_{y} {v_{y}}^{3}) \\ + K_{conz} zH (z) μ_{y} (sgn (v_{f} + \overset{\cdot}{y}) - a_{y} (v_{f} + \overset{\cdot}{y}) + β_{y} (v_{f} + \overset{\cdot}{y})^{3}) + K_{cony} hH (h) \\ \overset{\cdot\cdot}{z} + 2 ξ_{z} \sqrt{β} \overset{\cdot}{z} + β z = - q_{0} h (c (v_{x} - 1)^{2} + 1) H (h) H (v_{x}) μ_{z} (sgn (v_{z}) - a_{z} v_{z} + β_{z} {v_{z}}^{3}) \\ - K_{cony} hH (h) μ_{z} (sgn (v_{z}) - a_{z} v_{z} + β_{z} {v_{z}}^{3}) - K_{conz} zH (z) \end{matrix}

(7)

where $α = k_{y} / k_{x}$ , $β = k_{z} / k_{x}$ are the ratio of the stiffness, ${ω_{x}}^{2} = k_{x} / m$ , ${ω_{y}}^{2} = k_{y} / m = α {ω_{x}}^{2}$ , and ${ω_{z}}^{2} = k_{z} / m = β {ω_{x}}^{2}$ are the natural frequencies, $ξ_{x} = c_{x} / 2 m ω_{x}$ , $ξ_{y} = c_{y} / 2 m ω_{y}$ , and $ξ_{z} = c_{z} / 2 m ω_{z}$ are the dimensionless damping ratios in the three directions, respectively. $q_{o}$ is the dimensionless cutting force coefficient, $a_{c}$ is the instantaneous dimensionless thickness of cut; $c_{1}$ , $a_{x}$ , $β_{x}$ , $a_{y}$ , $β_{y}$ , $a_{z}$ , and $β_{z}$ are the dimensionless constants related to the nonlinear characteristics of the forces. $v_{x}$ is the dimensionless relative velocity between the tool and the workpiece in the X direction, $v_{y}$ is the dimensionless relative velocity between the tool and the chip in the X direction, $v_{z}$ is the dimensionless relative velocity between the tool and the workpiece in the Z direction, and $v_{f}$ is the dimensionless feed velocity in the Y direction; $K_{cony}$ , $K_{conz}$ are the dimensionless contact stiffness between the tool and the workpiece in the Y and Z directions, respectively. There are two functions discontinuous sign function sgn(.) and Heaviside function H(.).

Effect of the cutting velocity on cutting tool dynamics

The nonlinear dynamic behaviors of the cutting tool were investigated using the numerical simulation method in this section. The second-order Euler integration scheme was used to solve the governing equations of the cutting tool vibration, the integration step is 0.001, and the specified tolerance of numerical integration is 1e–8. In the turning process, the cutting velocity is an important cutting parameter that can affect the dynamic characteristics of the cutting system and limiting the production efficiency. Therefore, the dimensionless angular velocity of the workpiece $Ω$ serves as a bifurcation parameter that the values between 1 and 6. Bifurcation diagrams were plotted use the extreme displacements of the cutting tool that means the velocity of the cutting tool being zero in the X, Y, and Z directions, respectively. In this section, the research mainly focuses on the influence of the various dimensionless angular velocity of the workpiece on the cutting tool dynamics and the vibration amplitude of the cutting tool. The initial conditions and other dimensionless parameters are chosen as follows

\begin{matrix} x = \overset{\cdot}{x} = 0, y = 0, \overset{\cdot}{y} = 0.1, z = 0.0001, \\ \overset{\cdot}{z} = 0, α = β = 1, ξ_{x} = ξ_{y} = ξ_{z} = 0.01, \\ a_{x} = a_{y} = a_{z} = 0.3, \\ β_{x} = β_{y} = β_{z} = 0.1, μ_{x} = μ_{y} = μ_{z} = 0.5, \\ K_{cony} = K_{conz} = 0.5, q_{o} = 0.9, \\ c = 0.3, \tan ϕ = 0.45, R = 0.1, v_{f} = 0.1 \end{matrix}

At the first, the nonlinear dynamic behavior of the cutting tool in the three directions is examined by bifurcation diagrams with the dimensionless angular velocity of the workpiece as the bifurcation parameter. The corresponding bifurcation diagrams are presented in Figure 2. The vertical axis represents the vibration amplitude of the cutting tool in the three directions, while the horizontal axis is the dimensionless angular velocity of the workpiece as the velocity increases from 1 to 6. It is clear that there exist period, quasi-periodic, and chaotic vibrations in the X and Y directions but there is only chaotic vibration in the Z direction, as shown in Figure 2. Especially when the dimensionless bifurcation parameters $Ω$ are between 3.8 and 6, the vibration response is still chaotic. It can be found from Figure 3 that it is a very small-amplitude vibration. In addition, as the increase of the dimensionless angular velocity of the workpiece, the range of vibration amplitude of the cutting tool in the X and Y direction gradually increases; at the same time, the range of vibration amplitude in the Z direction gradually decreases; finally, there is hardly no vibration in the three directions because the range of vibration amplitude is small enough.

Figure 2.

Bifurcation diagrams of the extreme displacements of the cutting tool in the three directions as a function of dimensionless angular velocity of the workpiece $Ω$ between 1 and 6.

Figure 3.

Time history and power spectrum of the cutting tool vibration displacements in the three directions while the dimensionless angular velocity of the workpiece $Ω = 4$ .

In the following simulations, time history and power spectrum are used to analyze the dynamic vibration of the cutting tool in detail based on the bifurcation diagrams. We will choose some values of the dimensionless angular velocity of the workpiece to reflect the various nonlinear dynamic responses of the cutting tool. For the case of the dimensionless angular velocity of the workpiece $Ω = 1$ . It can be seen in Figure 4 that the vibration response of the cutting tool in the X and Y directions is periodic response; meanwhile, chaotic response appears in the Z direction. A spike in power spectrum means a periodic response; simultaneously, according to the bifurcation diagram (Figure 2), it can be found that the vibration response of the cutting tool in the X and Y directions is the periodic-1 response.

Figure 4.

Time history and power spectrum of the cutting tool vibration displacements in the three directions while the dimensionless angular velocity of the workpiece $Ω = 1$ .

When the dimensionless angular velocity of the workpiece $Ω = 2$ , it can be observed that the range of vibration amplitude of the cutting tool in the X and Z directions changes obviously. The vibration response of the cutting tool in the X and Y directions is quasi-periodic response because power spectrum consists of fundamental spike and some very small spikes. The vibration response in the Z direction is still chaotic response that can be found by messy time series and continuous and broad-band power spectrum in Figure 5.

Figure 5.

Time history and power spectrum of the cutting tool vibration displacements in the three directions while the dimensionless angular velocity of the workpiece $Ω = 2$ .

For the case of the dimensionless angular velocity of the workpiece $Ω = 3$ . Seen from Figure 6, it can be found that the range of vibration amplitude of the cutting tool in the X direction continues to increase but the vibration amplitude in the Z direction becomes decrease. At the same time, the vibration response of the cutting tool in the X direction is periodic response; in fact from the bifurcation diagram Figure 2, it can be found that it is a periodic-5 response. It is clear that the vibration response in the Z direction is still chaotic response.

Figure 6.

Time history and power spectrum of the cutting tool vibration displacements in the three directions while the dimensionless angular velocity of the workpiece $Ω = 3$ .

For the last case of the dimensionless angular velocity of the workpiece $Ω = 4$ , there is chaotic response of the cutting tool in the three directions. The very narrow continuous background around fundamental spike in the X and Y directions is presented in power spectrum. In addition, the vibration amplitude of the cutting tool in the X directions begin to decrease while the vibration amplitude in the Y directions gradually increased, but at this time the vibration amplitude of the cutting tool in the Z direction decreases significantly which corresponds to almost no vibration. When the dimensionless angular velocity of the workpiece $Ω = 6$ , the vibration amplitude of the cutting tool in all three directions hardly tends to zero, which means that there is almost no vibration.

From the above analysis results, it is obvious that the nonlinear dynamic response and vibration amplitude of the cutting tool will influence by the cutting velocity. The nonlinear dynamic response of the cutting tool in the X and Y directions undergoes a process from periodic to quasi-periodic to chaotic, the whole process is accompanied by several crisis, even periodic chaos and periodic crisis, but vibration response in the Z direction is always chaotic response with the increase of the cutting velocity. In addition, as the increase of the cutting velocity, the vibration amplitude of the cutting tool in the X and Y direction increases first and then decreases corresponds to the vibration amplitude continue to decrease in the Z direction. According to the presented model, the vibration amplitude of the cutting tool in the Z direction will directly affect the machining quality of the machined surface. This implies that high cutting velocity may be effectively suppressed cutting tool vibration to provide machining accuracy.

Conclusion

A comprehensive 3-degree-of-freedom model of turning process was established on the basis of 2-degree-of-freedom turning model proposed by Rusinek et al.²¹ Moreover, the effect of feed velocity on the cutting force is considered in the 3-degree-of-freedom turning model, which is more closely to reality. Here, we mainly focus on the influence of cutting velocity on the nonlinear dynamic characteristics of the cutting system; the results suggest that the cutting velocity is a key element for cutting system that can influence cutting tool nonlinear dynamic response, such as the cutting tool show a periodic response, quasi-periodic response, and chaotic response under different cutting velocities. Furthermore, it also has an effect on the cutting tool vibration amplitude, especially the cutting tool vibration amplitude in the Z direction. For our current model, it can be concluded that high cutting velocity may be an effective way to suppress chatter vibration, although the vibration amplitude in the X and Y directions is increased first but then it decreases. In order to improve efficiency, high velocity cutting is often used in cutting process, so it is necessary to study the effect of high velocity cutting on the stability of the cutting system.

Footnotes

Handling Editor: Anand Thite

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was funded by the National Natural Science Foundation of China (no. 11372122 and no. 51465029) and Science and Technology Program of Gansu Province of China (no. 1610RJYA020).

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