Stabilization of turning processes using spindle feedback with state-dependent delay

We develop a stabilization strategy of turning processes by means of delayed spindle control. We show that turning processes which contain intrinsic state-dependent delays can be stabilized by a spindle control with state-dependent delay, and develop analytical description of the stability region in the parameter space. Numerical simulations stability region are also given to illustrate the general results.

Delay differential equations were introduced to model machine-tool vibrations as early as the work of Koenigsberger and Tlusty [12]. It has been observed in the work [2,10] that the time delay between two succeeding cuts depends on the speed of the workpiece rotation and on the workpiece surface generated by the earlier cut. Namely, the time delay is generically state-dependent. This observation has led to modeling the turning processes with state-dependent delay differential equations. We refer to [6] for a review of this type of delay differential equations and refer to [10] for a state-dependent model of turning processes.
Spindle speed control in the turning process is an important method for suppression of machine-tool chatter and has been extensively investigated in the literature [9,15,16]. In the work of [7,8] we proposed a spindle speed control law for the state-dependent model and determined the stability region where stabilization can be achieved in high speed turning processes. However, such a spindle speed is a real-time control law which depends on the instant status of the system and may be difficult to accomplish for practical realization since usually there must be certain response time to activate the state-dependent control. In this paper, we propose a delayed spindle speed control law based on a state-dependent delay differential equation. Such a delayed spindle control will be dependent on historical state of the system. We are interested whether or not the delayed spindle control is feasible to achieve stability, and if feasible, how to analytically describe the stability region in the parameter space.
The significance of analytical descriptions of the stability regions includes that it indicates where in the parameter space the working machine-tool may achieve stability during the process, and that, if stability is not achievable, namely, persistent vibrations are unavoidable, we can predict the vibrations through bifurcation analysis by observing the crossing of the stability boundaries.
We organize the remaining part of the paper as following: In Section 2 we introduce a model for turning processes with state-dependent delay and propose a stabilization of delayed spindle velocity. We show that the resulting nonlinear systems with statedependent delay can be transformed into a system with constant and distributed delays for which we develop linear stability analysis and analytical description of the stability region in the parameter space. In Section 3, we analyze a model of turning processes with instantaneous spindle control strategy, for comparison of the stability region of the corresponding delayed spindle control. Numerical simulations are provided in Section 4 to illustrate the results obtained in Sections 2 and 3. We conclude the paper and discuss open problems in Section 5.

Turning processes with delayed state-dependent spindle control
Our starting point is the turning process as shown in Figure 1. The tool is assumed to be compliant and has bending oscillations in directions x and y. The governing equations read mẍ(t) + c xẋ (t) + k x x(t) = F x , (2.1) mÿ(t) + c yẏ (t) + k y y(t) = −F y . (2. 2) The x and y components of the cutting process force can be written as 3) F y = K y ωh q , (2.4) where m is the mass of the tool; K x and K y are the cutting coefficients in the x and y directions; k x , k y are stiffness; c x , c y are damping coefficients; ω is the depth of cut; h is the chip thickness and q is an exponent with empirical value 0.75. The chip thickness h is determined by the feed motion, the current tool position, and the earlier position of the tool and is given as follows h(t) = ντ (t) + y(t) − y(t − τ (t)), (2.5) where ν is the speed of the feed. Ω(s) ds = 2Rπ + x(t) − x(t − τ (t)), (2.6) where R is the radius of the workpiece. Then the time delay τ is implicitly determined by the oscillation in the x direction. Equations (2.1-2.6) form a model of turning processes with state-dependent delay.
We rewrite the governing equation (2.6) for the delay as Letẋ(t) = u(t),ẏ(t) = v(t). System (2.1-2.6) can be re-written as which is a system of differential equations with threshold type state-dependent delay [4]. We are concerned about stabilization of turning processes with a delayed spindle speed control. We assume that where c ∈ R is a parameter. Now we show that system (2.8) coupled with the spindle speed control (2.9) can be transformed into a system of integral-differential equations. Assuming that RΩ > u(t) for all t ∈ R, we consider the following process of change of variables for system (2.8): (2.14) k(η) = τ (t). (2.15) Then by (2.7) and (2.10) we have The second equation of (2.8) for τ can be rewritten as It follows that Furthermore, taking derivative with respect to t on both sides of (2.11) we have dr dη .
Therefore, system (2.8) can be rewritten as ds, (2.17) which is an integral-differential equation with both discrete and distributed delays.

Characteristic equation of system (2.17)
Let (r * , ρ * , k * , j * , l * ) be a stationary state of system (2.17). The the right hand sides of the differential equations lead to j * = l * = 0 and The unique stationary point of system (2.17) is , we obtain the linearization of system (2.17) near the stationary state: where M, N, P and Q are 4 × 4 matrices given by To have the stationary state of system (2.17) the same as the corresponding system with constant spindle velocity Ω 0 , we require k * equal to the stationary state τ * of τ . Then by (2.6) and (2.15) we need the spindle control parameter c to satisfy , (2.20) which leads to In the following we treat Ω 0 > 0 satisfying (2.20) as a virtual constant spindle speed.
We denote by k r the cutting force ratio Ky Kx , by K 1 the dimensionless depth of cut qKyω(2πR) q−1 kx , and by p the dimensionless feed per revolution ν RΩ 0 , then we have: (2.22) Thus system (2.19) can be rewritten explicitly as follows: For the convenience of computing the characteristic equation, we transform system (2.23) into a set of second order scalar equations of (x 1 , x 2 ). Notice that 0 ). We differentiate the first two equations of (2.23) and substituteẋ 3 ,ẋ 4 to obtain: (2.24) Assume that the tool is symmetric, namely c x = c y , k x = k y . Inspecting the equations in system (2.24) we know that system (2.24) has no nonconstant solution of the form (x 1 , x 2 ) = (c 1 , c 2 )e λη , (c 1 , c 2 ) ∈ R 2 with a zero component. Otherwise, the second order scalar equationÿ(η) + k * cx mẏ (η) + k * 2 kx m y(η) = 0 has nonconstant 1-periodic solution, which is impossible because k * 2 kx m > 0.
By setting (x 1 , x 2 ) = (c 1 , c 2 )e λη with c 1 = 0 and c 2 = 0 and multiplying the corresponding left and right hand sides of the two equations at (2.24), we obtain the characteristic equation of the linearized system Then the characteristic equation (2.25) can be rewritten as: where P (λ) = λ 2 + ξλ + δ. We have Notice that the roots of the quadratic polynomial P(λ) always have negative real parts since ξ = k * cx m > 0 and δ = k * 2 kx m > 0. In the following we consider the roots of We first notice that λ = 0 is not a removable singularity of P(λ) + δ h 1 + q λe λ ' (1 − e −λ ) such that the limit there is zero since we have if δ = 0. Therefore, if λ = iβ, β ∈ R is an eigenvalue then β = 0 and we have, (2.29) We have Lemma 2.1 Suppose that ξ, q, h 1 and δ are positive with ξ = 2q. If λ = iβ, β ∈ R is a zero of P(λ) + δ h 1 + q λe λ (1 − e −λ ), then β = 0 and the following statements are true: (2.30) iii ) β = 2nπ for every n ∈ Z.
By continuity of f − g in the interval ((2n − 3 2 )π, (2n − 1)π) and by the intermediate value theorem, f − g has at least one zero. To show the uniqueness of the zero in the interval ((2n − 3 2 )π, (2n − 1)π), we consider the inverses of f and g restricted to ((2n − 3 2 )π, (2n − 1)π), where f and g are increasing. We have If there are multiple zeros, then by continuity the derivatives of f −1 − g −1 evaluated at the zeros cannot be all negative or all positive.
Next we consider f, g in the interval (2n − 1 2 )π, 2nπ with the assumption q ≥ To find the local maximum of f , we let f ′ (β) = 0 and obtain . Then by the intermediate value theorem, f − g has at least one zero in the interval ((2n − 1 2 )π, β 1 ).
ii) We note from the first part of the proof of i) that the existence and uniqueness of which is equivalent to 2 )π, β 1 ), there must be at least two and the derivatives of f −1 − g −1 there are not all positive or all negative. But this is contradicting (2.40). Therefore, there is no zero in ((2n − 1 2 )π, β 1 ).

System with instantaneous spindle speed
Even though we knew that delayed spindle velocity control is more practically feasible than an instantaneous one, we are interested what the trade-off between these two approaches could be. In this section, we consider model (2.8) of turning processes with the following instantaneous spindle velocity control where c ∈ R is a parameter, which is the control strategy considered in [8]. We reconsider it here for convenience of comparison. Then equation (2.6) governing the state-dependent delay becomes System (2.1-2.5) with the spindle speed control strategy (3.2) can be rewritten as . The second equation of (2.8) for τ can be rewritten as ds.
It follows that k(η) = 0 −1 2πR c 1 rη(s) ds. With the same process of the last section, we rewrite system (2.8) as The unique stationary point of system (3.4) is the same as that of (2.17) if the parameter c ∈ R for the spindle velocities assumes the same value. Namely, we have the stationary state of system (3.4): (r,ρ,k,j,l) = K x ων q k x k * q , − K y ων q k y k * q , k * , 0, 0 , where k * = 2πR c · kx Kxων q 1 q+1 . Setting x = (x 1 , x 2 , x 3 , x 4 ) = (r, ρ, j, l) − (r,ρ,j,l), we obtain the linearized system of system (3.4) near its stationary state and obtain that By the same token leading to (2.20) we require k * = 2π Ω 0 which is equivalent to which is the same as (2.21), then systems (3.4) and (2.17) both have the same stationary states as that of the corresponding system with constant spindle velocity Ω 0 . In the following we always assume c satisfies (3.7) such that the parameter substitution at (2.22) is still valid. Then system (3.6) can be rewritten as: x 1η (s)ds, x 1η (s)ds. (3.8) Writing (3.8) into second order scalar equations of (x 1 , x 2 ), we have 1)), (3.9) Assuming c x = c y and k x = k y and bringing the ansatz (x 1 , x 2 ) = (c 1 , c 2 )e λη with c 1 = 0 and c 2 = 0 into (3.9), we obtain the characteristic equation of system (3.4): Let δ be defined at (2.26) and let The characteristic equation of system (3.4) can be rewritten as: Since the roots of the quadratic polynomial P(λ) always have negative real parts, we consider the roots of We first notice that λ = 0 is not a removable singularity of P(λ)+δ h 2 + q λ (1−e −λ ) such that the limit there is zero since we have if δ = 0. Therefore, if λ = iβ, β ∈ R is an eigenvalue then β = 0 and we have, We have Lemma 3.1 Suppose that ξ, q, h 2 and δ are positive with ξ = 2q. If λ = iβ, β ∈ R is a zero of P(λ) + δ h 2 + q λ (1 − e −λ ), then β = 0 and the following statements are true: and β = 2nπ for every n ∈ Z. ii ) We have (3.14) iii) β ∈ (2(n − 1)π, 2nπ) or β ∈ (−2nπ, −2(n − 1)π) for some n ∈ N.
Notice that in the domains of F, G and H, we have Then their inverses exist. Notice that we have By continuity of F −1 and G −1 , if F −1 − G −1 has a zero, then there must be at least two and the derivative d dβ (F −1 − G −1 ) evaluated at the zeros cannot be all negative or all positive. To obtain a contradiction, we compute the derivative d dβ (F −1 − G −1 ) evaluated at the zeros and have d dβ Note that for every zero By (3.42) and (3.43), we have Namely, the derivatives of F −1 − G −1 at the zeros are all negative. This is impossible and F −1 − G −1 has no zero.
Next we show the uniqueness of the zero of F −1 − G −1 . Suppose F −1 − G −1 has more than two zeros. Then there must be at least three, counting multiplicity since F −1 −G −1 is continuous in (−q, 0) and (3.45) and (3.46) are satisfied. But by (3.44) the derivatives of F −1 − G −1 at the zeros are all negative, which is impossible. Therefore, the zero of F −1 − G −1 is unique.

Numerical simulations
In this section, we numerically demonstrate the results of Theorem 2.6, Theorem 3.7 and Theorem 3.9. Figure 5 shows that case of item (i) of Theorem 2.6 with q > β 1 √ 2 √ 5−2 4 √ 5−8 and Figure 6 the details of the stability region near the origin (0, 0). Note that if (δ, h 1 ) = (0, 0), the characteristic equation P(λ) + δ(h 1 + q λe λ )(1 − e −λ ) = 0 has only two roots with real parts negative. That is, the equilibrium of system (2.17) is stable. It is known that the equilibrium changes stability as the parameter (δ, h 1 ) varies only if it passes the boundary where purely imaginary roots of the characteristic equation occur. Figure 6 shows that with delayed instead of instantaneous spindle speed control, it is still possible to stabilize the equilibrium, as a neighborhood without intersecting either branches of the parameterized curves of (δ, h 1 ) near the origin exists. Figure 7 shows the case of item (iv) of Theorem 2.6 for P(λ) Figure 8 shows the stability region with ξ < 2q described at Theorem 3.7. Since the branch parameterized with β ∈ (γ * n , β * n ) has a vertical asymptote and has a zero on the horizontal line h 2 = 0, the connected stability region for positive (δ, h 2 ) is enclosed by the first branch with n = 1, δ = 0 and h 2 = 0 and the other branches with n ≥ 2 will not contribute an stability region. Figure 9 shows the stability region with ξ > 2q described at Theorem 3.9. With ξ = 1.62 and q = 0.8, we have which leads to n 0 = 2. According to Theorem 3.9, for n ≥ 3, (δ, h 2 ) is positive if and only if β ∈ (β * n , 2nπ). For n = n 0 = 2, we haveβ n = 9.591212 > q ξ ξ−2q = 7.2 > 2π. Then (δ, h 2 ) is positive if and only if β ∈ (β * n , 2nπ) with β * n = 9.75394647. For n = 1 < n 0 , we haveβ n = 3.581158 < q ξ ξ−2q = 7.2. Then (δ, h 2 ) is positive if and only if β ∈ (γ * n ,γ n ) ∪ (β * n , 2nπ), where β * n = 3.92455245, γ * n = 3.19356076, andγ n = 3.86974862. Note that the extra interval (γ * n ,γ n ) in addition to (β * n , 2nπ) corresponds to the small lobe near (0, 0) in Figure 9.

Concluding Remarks
Starting from the model of turning processes with state-dependent delay, we investigated two spindle control strategies one of which contains a state-dependent delay and the other one an instantaneous PD (proportional-derivative) control. Using a time domain transformation, we show that the controlled model can be reduced into models with constant and distributed delays. Since whose characteristic equations are transcendental equations containing both of the terms λ and e λ , the analytical description of the stability region in the space of the parameters δ and h composite of the intrinsic parameters of the turning processes becomes complicated. The analysis shows that feedback spindle control with state-dependent delay still can stabilize the equilibrium state of the turning processes, even though the simulated stability region does not show improvement of the stability region over the PD control. This means that the practically realizable delayed feedback control is feasible and is able to achieve stability. When the parameters are beyond the stability region, we refer to [4] for analysis of global Hopf bifurcation of turning processes with threshold-type state-dependent delay.
We remark that with the delayed feedback control, h 1 = K 1 p q−1 1 − p kr may not be positive as in in Figure 7. However, it remains unclear whether or not a positive h 1 will remove the bound of the stability region in the δ-direction and the lead to a stability region unbounded in both of the δ and h 1 directions. We also remark that we may improve the stability region by investigating nonlinear type spindle speed control with state-dependent delay, namely how to find Ω(t) = r(x(t − τ (t))), where r : R → R is a nonlinear function instead of the linear one investigated in the current work, in order to improve the stability region. We leave these problems for future work.