Effect of Modal Parameters on Both Delay-Independent and Global Stability of Turning Process

The model fo r regenerative vibration of linear orthogonal turning process is a second order time-invariant delay differential equation. Stability analysis resulted in lobes that combine to give transition curve that separates the paramete r space of spindle speed and depth of cut into stable and unstable subspaces. It is found that there is a subspace of the stable subspace in which the turning process is delay-independent stable. The size of this subspace is found to be a function of modal parameters and increases with damping ratio of the tool. Non-linear analysis of turning by some investigators suggests that subcritical bifurcations always occur thus the need to design a portion of the subspace of delay -independent stability for global stability. The subspace of global stability is also theoretically and quantitatively demonstrated to increase faster than the driving increase in damping ratio.


Introduction
The general state space time-delay system with constant real matrix coefficients is = + − + + d 0 − (1) where > 0, , , ∈ ℝ × ℝ , ∈ ℝ + and the history is on -, 0 . The coefficient matrices and respectively captures the effects of discrete delay and distributed delay. Stability analysis has been carried out for the scalar version of this equation on the parameter space of coefficients of the current and distributed delay terms [1]. The interest in this wo rk is on systems without distributed delay such that equation (1) becomes = + − .
(2) Equation (2) is said to be delay-independent stable if stability persists for all delay values belonging to ℝ + wh ile it is said to be delay-dependent stable if stability is only retained for all delay values belonging to a subspace of ℝ + [2]. Regenerative vibration of linear turning will be demonstrated in the next subsection to be governed by equation (2). It is known that one of the parameters affecting stability of turning operation is the spindle speed which is inversely proportional to discrete delay. This means that turning operation is delay-dependent stable.
Trigonometric ideas are emp loyed in a way unique to this work in detailed stability analysis of linear turning space regenerative vibration leading to demarcation of parameter of spindle speed and dept of cut into stable and unstable domains. It results that a portion of the stable domain of damped turning that lies below certain dept of cut is delay-independent stable being that asymptotic stability is retained within it at all spindle speeds; a notion first s een in this work.
Subcritical nature of turning bifurcat ion has been experimentally and analytically established. This means that in practical setting, the space of delay-independent stability is not globally stable thus the need to establish a design procedure that enables the delineation of a portion of g lobal stability in which regenerative chatter is deemed eliminated. The method of estimat ion of size and delineation of boundary of domain of g lobal stability as outlined in this work is another major contribution of this paper.
Among the contributions of this paper is the d iscovery that sizes of sub-space of delay-independent and global stability increase considerably with increase in damping ratio since it is theoretically established that fractional increase in these sizes are always greater than fractional increase in damp ing rat io. In other words, the pain involved in achieving greater damping will be more than compensated in gains in global stability.

Equation of Regenerative Vibration of
Turning Tool Figure 1 represents the turning of an external cylindrical surface. The workpiece at the rotational speed Ω in revolutions per minute of the spindle is clamped in a chuck while the tool is made to transverse it. The mechanical model in figure1 represents an orthogonal turning process. In orthogonal cutting process, the cutting edge of the tool is perpendicular to the feed motion [3] as can be observed fro m figure1. The modal parameters of the turning process are; mass of tool, is the equivalent viscous damping coefficient of the tool system and the stiffness of the tool system. Chatter is an unstable vibration in machin ing due regenerative effects that are originally triggered by internal and external perturbations. Regenerative effect as seen enlarged in figure1 is the effect of waviness created on a mach ined surface due to perturbed dynamic interaction between the tool and the workpiece. The present tool pass that is indicated as full curve has waviness that is not in phase with the last tool p rofile indicated as dashed curve. A variation in chip thickness causes cutting force variation that results in vibration which subsequently builds up to chatter if cutting parameter co mbination is unfavourable. In this model a single degree of freedom vib ration is assumed in − direction (feed direction). The tool is fed into the workp iece at a speed . Response ( ) of the tool system is measured relative to the unloaded equilibriu m position of tool (or tool holder axis).The response of the tool ( ) satisfies the equation of motion that is as derived in what follows; is the -component of cutting force. could have the empirical form found in the works of Tlusty [4]; = a (4) where is the cutting coefficient (a property of the workp iece material), is the depth of cut, a is the actual feed rate and is an exponent that has typical values 0.8 and 3 4 . The latter exponent spells the three-quarter rule [4,5]. Unifo rm feed rate is the prescribed movement of the tool's cutting edge in meters per revolution of the workp iece thus the actual feed rate a could be defined as difference of present and one period delayed position of tool if discrete delay equal to period of revolution is adopted. Thus from figure1 it could be seen that (6) To make the derivation mo re co mpact the following notations are used; = and − = . The notation also applies to any subsequent variable that involves delay. Applying the notation and re-arranging, equation (6) beco mes The motion of the tool is a linear superposition of prescribed feed motion , static transverse deflection of the tool system t ( ) and perturbation ( ) [4] then = + t + . (11) The motion of the tool as described by equation (11) is seen to be a delay differential equation .The absolute value of the coefficient of − in equation (11) is called the specific cutting force variation and is given as (12) Equation (12) could be put in modal form + 2 + 2 = − − (13) where the natural frequency and damping ratio of the tool system are given in terms of modal parameters , and respectively as = and = 2 . Equation (13) is seen to represent a delayed oscillator when re-written thus + 2 + 2 + = . (14) Equation (14) is the general equation governing linear regenerative vibration of the tool in turning process. The nature of the solution of equation (14) is a reflection of stability condition of an operation. If the perturbation motion or its derivatives rises with time, there is chatter (unstable operation) while bounded perturbation response with time imp lies non-chatter operation. With the substitutions 1 = and 2 = made, equation (14) could be put in state differential equation form where , = ( − ) for = 1 and 2 . The time domain equation (15) is the substance of stability characterizat ion of turning process.

Chatter Stability Analysis of Turning
Equation (15) Expansion of the exponential term of equation (19) in Maclaurin's series shows that the characteristic equation has infinitely many solutions or eigen-values also called characteristic roots, with each having the form = + i . Eigen-values of the system migrate on the complex plane as the cutting parameters are varied. All the roots must have negative real parts for the turning process to be stable thus operation is critical whenever there exist roots on the imaginary axis. Bifu rcation in turning operation could occur when a pair of co mplex conjugate characteristic roots crosses fro m the left -half plane to right-half plane of the complex p lane .This occurrence is called the Hopf bifurcation of a corresponding non-linear system. The trivial solution to equation (14) is = e where , ∈ [7]. Fo r any pair o f co mplex conjugate roots 1,2 there exists a solution = 1 e 1 + 2 e 2 being that equation (14)  . Thus is seen to be the frequency of the arising chatter vibrations. This bifurcation of Hopf type has been proven experimentally by Shi and Tobias [8] and analytically by Stepan and Kalmar-Nagy [9] to have subcritical nature. This subcritical nature of turning bifurcation has imp lication in design engineering of mach ine tools as will be seen later.
The stability boundary curves also called the D-curves or Stability lobes are drawn to separate the stable cutting domain (at which all < 0) fro m the unstable one ( at which some > 0 ). On the D-curves exist parameter combinations that produce a pair of roots of characteristic equation that are pure imag inary without any root existing in the right-hand plane. The D-curves could be tracked based on equation (19) by making the substitution = ± i . The two equations resulting are (21) is vio lated when = 0 suggesting that bifurcation of saddle node type is not expected in turning operation. Equations (21) and (22)   .
This becomes re-arranged to give the expression for critical depth of cut as .
Equations (35) and (36) are co mbined to generate a stability lobe for a particu lar . Stability lobes of various ′ on the same parameter space are combined to give the stability transition or boundary curve for the linear turning process. It is seen from equation (36) that negative depth of cut results only when < mean ing that chatter frequencies are above the fundamental natural frequency of the tool since negative depth of cut has no practical meaning in turning. Also any value of results in the same critical depth of cut since does not appear in equation (36). For illustration, stability lobes for = 1 to 10 are generated to form the stability transition curve for a system with the parameters = 5700 rads −1 and = 0.02. Th is is shown in figure3. Any point on the region below a combination of all the lobes is stable while those above are for chatter as is illustrated by six trajectories of figure4 generated at selected points of parameter space of figure3. Trajectories of figure4 are produced for a system with parameters; = 5700 rads −1 and = 0.02 , = 0.0431kg , = 5 × 10 7 Nm −7 4 , = 0.75 , feed speed = 0.0025ms −1 assuming a history 1 ( ) = 0.000001m and 2 ( ) = 0.0001m/s in the interval − ≤ ≤ 0. Each trajectory is determined by the corresponding cutting parameters of caption of figure4. It is obvious from figure3 that there is a sub-region of the stable subspace in which the operation is delay-independent stable since stability is retained no matter how high spindle speed gets.
The nature of the solution of (15) is a reflection of stability condition of an operation. If the perturbation motion or its derivatives rises with time, there is chatter (unstable operation) while bounded perturbation response with time implies non-chatter operation. It is seen from figure4 a and b that a point 0.000001 , 0.0001 on the phase plane at t=0 traces the trajectory towards the origin as indicated by arrow as time changes. This means that the points 2,0.04 and 0.45,0.1 are stable in conformity with their location as marked star on the stability chart. The same point spirals away fro m initial position as time progresses for the operating points 1.5,0.06 and 0.7,0.4 as seen in figure4 c and d. This implies instability which is expected since they are placed above the transition curve as marked with circle. The trajectories seem reluctant to get to origin in figure4e and d thus suggesting that the points almost lie on the stability t ransition curve as marked with diamond on the stability chart.

Effect of Choice of Modal Parameters on Delay-independent Stability
Fro m the equation  results in a quantity , that is a function of modal parameters at the turning point boundary frequency of interest 2 = 2 + 1. , is a positive quantity for positive damping. The mean ing is that the countable infinite number of turning points 2 + 1 − tan −1 1 2 + 1 , 2 + 1 are minima on the Ω -parameter plane. It is seen that the minimu m crit ical depth of cut is the same for all the stability lobes. It is also seen that increase in damping rat io results in increase in the min imu m critical depth of cut. If the designations Ω t = 2 + 1 − tan −1 1 2 + 1 and t = 2 + 1 are made, a p lot of local minimu m points as damping ratio varies fro m 0 to 0.1 is shown for = 1 in figure5. This means that size of sub-region of delay-independent stability increases with damp ing ratio. Increase in damping ratio is effected by favourable variation of the tool modal parameters. It can be read from figure5 that increase in damping ratio will increase the sub-area of delay-independent stability by shifting the minimu m critical points somewhat upwards and rightwards. If damping ratio is changed by the amount ∆ , the fractional increase in sub-area of delay-independent stability is (40) Since damp ing rat io of mach ine tools are small, it could be writing that ∆ = causing equation (40) to be It is seen fro m equation (41) that fractional increase in the area of delay-independent stability is greater than that of damping ratio. In relative terms this means that there is considerable increase of delay-independent stable subspace of turning by increase of damp ing ratio.

Design Implications
It is already seen that linear turning is delay-independent stable in the sub-region below the line = 2 + 1 . In practical situation highest spindle speed of a turning machine normally lies either within low o r high spindle speed domains. High spindle speed domain encompasses spindle speeds that are co mparable with tool fundamental natural frequency. It lies within the first three lobes. For this reason the area of delay-independent stability enclosed by the lines Ω = 2 + 1 − tan −1 1 2 + 1 and = 2 + 1 together with the Ω and axes of size where takes value depending on attainable spindle speed, is a function of positive slope of as is easily seen fro m figure6. Th is means that this area of delay-independent stability of turning increases as damping ratio increases.   Closed form bifu rcation analysis has been conducted at the min imu m points [9] in which it is found that though linear stability analysis suggests global stability at non-dimensionless depth of cut below 2 + 1 that bifurcation could still occur at a subcritical point due to non-linearity. This means that if min imu m unstable depth of cut is specified for a turn ing tool and t wo modal parameters are known, the third modal parameter can be designed using the equation This is re-written to give the design equation where is the design factor of safety that ensures global stability and is specified by the first min imu m critical speed to be greater than the maximu m spindle speed of the mach ine. Alternatively equation (44) could be used to establish the depth of cut below which g lobal stability of a g iven turning is ensured.
is chosen to preclude the possibility of non-linearit ies and perturbations triggering unstable vibrations at subcritical points. It has been measured experimentally that at the minimu m points, chatter does not occur below 87% of critical depth of cut t = 1+ t 2 Ω t 1− [8]. Based on this result it will amount to reliable design to specify that ≤ 0.87. It is seen that as increases, the maximu m depth of cut that ensures delay independent stability increases. This means that the depth of cut designed by equation (44) increases as maximu m spindle speed specification of the machine decrease. This result is seen clearly fro m figure7 in which is plotted against for different values of = 1, 2, 3 … … . .10 with = 0.7 . Figure7 highlights the positive effect of damping rat io on global stability of turning. It also shows the importance of use of the biggest that specifies crit ical speed higher than the maximu m spindle speed in equation (44) since slope rises with . It should be noted fro m the way the two terms "delay-independent stability" and "global stability" are used that the latter is a sub-domain of the former.
To simu ltaneously quantify the notion that global stability is a sub-domain of the delay-independent stability and the finding that variation of modal parameters in a way that reflect as increase in damping rat io increases delay-independent stability and its subset of global stability, stability charts are generated for systems with parameters; = 5700 rads −1 and damping ratios = 0, 0.01, 0.02 and 0.025 respectively as seen in figures 8a, b, c, and d. It is seen from figure8.a that there is no delay-independent stable subspace for a turning tool with zero damp ing ratio. Delay-independent stable subspace is seen to exist when the damping ratio is non-zero as seen in figures 8b, c and d. Every stability chart of nonzero damp ing ratio on the right hand side is magnified portion of the adjacent chart to further reveal the subspace of delay-independent stability that lies below the line = t = 2 + 1 . The increase in this area with damp ing ratio is very noticeable as is expected fro m equation (41). The portion of this area for g lobal stability depend on the maximu m spindle speed and choice of the design factor ≤ 0.87. For examp le if = 0.7, = 5700 rads −1 , = 0.0431kg , = 5 × 10 7 Nm −7 4 , = 0.75 , = 0.0025ms −1 and maximu m non-dimensionless spindle speed of the mach ines Ω m = 1 3 , use made of equation (44) results in what is presented in the table1 below. Colu mn 4 of table1 is for area of global stability gs . Th is area for each chart of figure8 is enclosed within a b lack rectangle OABC and is seen to increase considerably with damping ratio according to equation (41). In conclusion, choice or variation of modal parameters of turning tool in a way that results in increase in damping rat io considerably improves both delayindependent stability and global stability. Through the method outlined in this work, the domain of global stability could be delineated for a real turn ing machine with known modal parameters.

Conclusions
Modelling and stability analysis of linear turning is carried out. It is found that variation of modal parameters in a way that results in increase in damping rat io improves stability. Ideas borrowed fro m results of non-linear analysis of turning are utilized in design to determine the domain and size of global stability wh ich is a portion of the subspace of delay-independent stability in wh ich subcritical chatter is not expected. The size of sub-space of global stability at any value of damping ratio depends on how conservative the design factor of safety is where the basic condition satisfied is ≤ 0.87 . It is established that sizes of sub-space of delay-independent and global stability increase considerably with increase in damping ratio since it is theoretically d iscovered in this work that their fract ional increase is always greater than that of damping rat io causing it as equation (41)