Nonlinear Dynamics and Stability of a Machine Tool ... - Springer Link

Nonlinear Dynamics 13: 373–394, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Nonlinear Dynamics and Stability of a Machine Tool Traveling Joint I. RAVVE, O. GOTTLIEB, and Y. YARNITZKY Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa, 32000, Israel (Received and accepted: 14 February 1997)

Abstract. In this work, we investigate the nonlinear dynamics and stability of a machine tool traveling joint. The dynamical system considered includes contacting elements of a lathe joint and the cutting process where the onset of instability is governed by mode coupling. The equilibrium equations of the dynamical system yield a unique fixed point that can change its stability via a Hopf bifurcation. The unstable domain is primarily governed by the cutting tool location, the contact stiffness of the joint and the depth of material to be removed. Self excited vibrations due to a mode coupling instability evolve around the unstable fixed point and one or more limit cycles may coexist in the statically unstable domain. Stability and accuracy of the approximate analytical solutions are analyzed by applying Floquet analysis. Perturbation of the dynamical system with weak periodic excitation results with periodic and aperiodic solutions. Key words: Machine tool traveling joint, contact nonlinearity, self excited vibration, mode-coupling instability, quasiperiodic response.

1. Introduction Growth in machine manufacturing and improvement of their quality present high demands to the precision and productivity of machine tools. This, in turn, requires reduction of static and dynamic deformations in the traveling joints of the frame units. Static and dynamic behavior of modern machine tools depend to a large extent on the design of the system guides. An experimental and numerical study performed by Weck et al. [1] reveals via mode shape analysis that the contribution of contact deformations at the joints to the total compliance of the machine may be crucial while elastic deformations of the mating bodies themselves are relatively small. Consequently, noncontact deformations of sliding and stationary elements (support and bed) may be assumed negligible with respect to the contact deformations in their joint. Thus, in this paper the contacting elements are considered rigid bodies with a thin elastic contact layer between their surfaces. The process of machining is usually accompanied by self excited and forced vibration. External forcing is typically due to imbalance of the rotating parts or due to eccentricity caused by misalignment of the machine and the workpiece axes. Self excited vibration, or machine tool chatter, can be generated by a variety of sources: Mode-coupling instability due to cross-coupling of both the elastic system [2] and the cutting force components [3]. The cutting force resultant direction does not coincide with the normal to the machining surface. Possible stick-slip due to dry friction forces [2, 4]. Chip-thickness variation or regenerative effect that occurs when the cutting edge traverses a workpiece surface from which it has taken a previous cut [3, 5]. In this case the chip thickness (and the resulting cutting force) depend on the present and delayed displacements of the cutting tool relative to the workpiece.

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There are several experimental and theoretical studies on contact stiffness [2, 6, 7]. The underlying assumptions governing contact dynamics include: the real area of contact is proportional to the applied normal load and to the number of the contacting points; the average area of each contact remains constant; the contact of two flat rough surfaces is equivalent to that of a flat surface and a rough one with a modified distribution of asperity height. In recent years there has been progressive replacement of metal-metal guides by metal-plastic ones, especially in heavy machines. Some plastics are often used as covering materials for guideways. The reasons are more convenient technology of the guide surface manufacturing, better stick-slip conditions and lower friction coefficients. The static dependence of pressure (p) on contact deformation ( ) for both metal-metal and metal-plastic guides is essentially nonlinear. A variety of empirical relationships have been proposed relating this dependence to the distribution of asperity heights. Greenwood and Williamson in the classical paper [8] assumed an exponential distribution and obtained the corresponding load-compliance relation

p = k exp(=); where k and are exponential distribution parameters.

(1)

Dolbey and Bell [9] assumed a power distribution of asperity height and obtained a power law

p = cm ; (2) where c depends on material and surface finish and m is a power law parameter. Levina

and Reshetov performed additional tests for repeated central loading for various contacting surfaces with different finishing. Their results demonstrate that metal-metal contact (steel, cast iron, bronze) and metal-plastic guides can be described by quadratic (m 2) [10, 11] and cubic (m 3) [11] powers respectively. In this work, we consider free and forced vibrations of a lathe support in the horizontal plane for a metal-plastic guide and study the nonlinear dynamics and stability of system response. We concentrate on investigation of the mode coupling instability related to the contact dynamics of a machine tool guide and incorporate the cross-coupling of the lathe elastic system and the cutting process. We construct a damping matrix based on a linear viscous dissipation model and formulate the system generalized forces via the principle of virtual work. The external loads include both static and dynamic components of the cutting force. We note here that when successive cuts do not overlap, the chip-thickness variation effect does not arise [3]. An example of machining without overlap is thread cutting. Thus, in order to investigate the fundamental threshold of the system instability due to mode coupling, we do not consider stick-slip and regenerative effects. In Section 2 we derive the nonlinear equations of the motion for the traveling unit via Lagrangian approach. In Section 3 we determine the uniqueness of the system fixed point and analyze its stability. An approximate analytical criterion determining the unstable domain is formulated. In Section 4 we approximate the self excited solutions of the nonlinear system via the harmonic balance method. The accuracy and stability of the coexisting limit cycles are verified via Floquet analysis. A fifth order approximation is shown to suffice with an eigenvalue close to unity whereas the remaining eigenvalues govern stability of the solutions. A bifurcation analysis reveals six regions with different system behavior. The solution amplitudes and frequencies are also verified numerically. In Section 5 we perturb

Nonlinear Dynamics and Stability of a Machine Tool Traveling Joint

375

Figure 1. Lathe definition sketch.

the dynamical system with weak periodic excitation defining the machining dynamic load. Periodic and aperiodic responses are obtained in the various bifurcation regions. 2. Equations of Motion A conceptual sketch showing the principle of machining is presented in Figure 1: the lead screw (4) provides the feed motion of the traveling unit (2) along the contact surfaces of the stationary unit (1). The longitudinal turning of the workpiece (5) is being executed by the lathe tool (3). Since the contacting bodies are assumed absolutely rigid, the location and the orientation of the traveling unit in the horizontal plane can be described by three coordinates: Ai (i 1; 2; 3) where A1 is the lateral translatory displacement, A2 is rotation, and A3 is longitudinal translation. We note that a typical guideway design with metal-metal contacts includes possible gaps between the mating surfaces whereas metal-plastic contact surfaces assembled under pressure may result an internal preload prior to the external static load. However, in this work we assume that there are no initial clearances and no contact deformations before application of the static cutting load. The plane frame of reference xy is introduced. Its origin coincides with the traveling unit geometric center and the x-axis is directed opposite to the feed motion. The location of the cutting tool is defined by the coordinates xc and yc of the machining point and the screw location is defined by the parameter ys. The length and width of the traveling joint are L and b, and r is the depth of material to be removed by turning. Size b is not shown in Figure 1 as b is the size of both contact surfaces in the direction normal to the plane of the figure. To obtain the general nonlinear forced and damped equations of motion, we follow a Lagrangian formulation

=

@T @T + @ + @W = Q ; (3) i @ z_i @zi @ z_i @zi where zi are generalized co-ordinates, T is the kinetic energy, W is the potential energy, is a dissipation function and Qi are generalized forces. d dt

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Figure 2. Contact deformation sketch.

2.1. ENERGY OF ELASTIC DEFORMATIONS According to the load-compliance relation (2), for a metal-plastic guide with specific contact energy Wo (per unit length of the contact surface) is 8 > > > > < > > > > :

Wo = b p() d = cb4

for

Wo = 0

for

Z

4

> 0;

0

m=

3, the

(4)

0;

where is the contact deformation in the direction of normal to the contacting surfaces, and is a dummy variable. The scheme of the contact deformations is presented in Figure 2. B1 and B2 denote positive distances while l1 and l2 are co-ordinates and may be of any sign. On the first (upper) contact surface, the contact takes place within the portion l1 x B1 , and on the second (lower) surface – within the portion B2 x l2 . Elastic displacements in the traveling joint of machine are fairly small relative to the joint sizes, therefore B1 B2 L=2, l1 l2 l. Assume both contact surfaces have equal specific stiffness c. Then the energy of the contact elastic deformation is 3 2 L=2 Z Zl

=

Wc = cb4 64

L=2

42(x) dx +

l

41(x) dx75 ;

(5)

where 1 and 2 are the contact deformations on the first and second guideway surfaces respectively. Since the contacting bodies are rigid, there is a linear distribution of the contact deformations along the joint surfaces. For a small angle A2

1 = A1 + A2x; 2 = A1 A2x = 1

so the value of displacements

l

(6)

in Equation (5) does not matter. Introducing the following normalized

A1 ; z = A ; z = A3 ; z1 = L= 2 2 3 2 L=2

(7)


377

we obtain

Wc = cbL (5z14 + 10z12z22 + z24): 320 5

(8)

The lead screw is formulated as a linear spring and its energy is

Ws = cs2s = c2s (A3 ysA2 )2 = cs8L (z3 ysz2 )2 ; 2

2

(9)

= 2ys=L

where cs is the screw stiffness, s is extension or compression of the screw and ys is the screw axis dimensionless co-ordinate. The total potential energy consists of the contact energy and that of the screw:

W = Wc + Ws:

(10)

2.2. KINETIC ENERGY In assumption that the support gravity center coincides with the system geometric center, the kinetic energy is

_2

2 T = m2 (A_ 21 + A_ 23 ) + J A ; 2

(11)

m2 = (L=J2)2

(12)

where m is the support mass and J is its central moment of inertia. Introducing

m1 = m;

_

and using the normalized velocities zi , we obtain

T = L8 (m1z_12 + m2z_22 + m1z_32 ): 2

(13)

2.3. GENERALIZED FORCES The generalized forces are produced by the cutting force which consists of three components whose absolute values Px , Py , and Pz depend on [12]: the depth of material to be removed r the feed rate s the cutting velocity v

Px = Cxsx v x r x ; Py = Cy sy v y r y ; Pz = Cz sz v z r z ;

(14)

where the coefficients depend primarily on the tool geometry and on the workpiece and tool materials. The feed rate (s) depends on the generalized velocities A2 ; A3

(_ _ )

s = so A_ 3 + yc A_ 2; (15) where so is a technological (static) value of the feed rate and yc is a cutting tool co-ordinate. The velocity of machining v depends on the generalized co-ordinates via the load applied to

the electric drive. We assume that variations of the feed rate and cutting velocity are negligible

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when calculating the cutting force (i.e. s and v are constant). Furthermore, the power indices

x , y and z are close to unity [12]. Thus, the cutting force components in (14) simplify to

Px = Qfxr; Py = Qfy r; Pz = Qr:

(16)

Q defines a vertical cutting load per unit depth of material to be removed. For specific tool design and machining conditions Q is considered a constant value. fx and fy characterize the ratios between the Cartesian components of the cutting force. These ratios are also considered constant for design purposes. Example values for lathe turning are fx 0:32 and fy 0:4 [12]. We assumed that during machining, the support moves from the tailstock to the gearbox of the lathe (in the direction opposite to the x-axis). Then the x-component of the cutting force applied to the support is positive. The virtual work done by the cutting force is V = Pxx Py y; (17) where x and y are virtual increments to the machining point displacement components.

Thus, according to Figure 1

x = (A3 ycA2 ); y = (A1 + xcA2 ):

(18)

Normalizing the machining point co-ordinates and the cutting depth results in

xc ; y = yc ; r = r : xc = L= c L=2 2 L=2

(19)

Substitution of Equations (16, 18, 19) into Equation (17) yields 2 V = L 4Qr (fy z1 + fa z2 fxz3 );

(20)

where

fa = fy xc + fxyc :

(21)

The generalized forces are calculated as Qi

= V=zi

2 2 2 Q1 = L f4y Qr ; Q2 = L f4a Qr ; Q3 = L f4x Qr :

In absence of the regenerative effect, the depth of material to be removed three components

(22)

r consists of

r = ro + rs + rd ; (23) where ro is the initial depth which is pre-set with the machining conditions and rs is the elastic component due to the cutting tool elastic displacement.

rs = z1 + xcz2 :

(24)

The dynamic component rd is caused by the workpiece axis eccentricity. Suppose that is the distance between the workpiece geometric axis and rotational axis. For the small values of relative to the machining radius

rd = cos t;

(25)


379

= 2=L is the normalized eccentricity and is the lathe spindle rotational frequency.

where

2.4. ENERGY DISSIPATION

The dissipation function presents a quadratic form of a positive definite symmetric matrix for viscous damping assumed for the metal-plastic contact:

= 12 L4

3 X 3 2 X

i=1 j =1

cij z_i z_j ;

(26)

where L2 =4 is the normalizing factor and cij are the damping matrix coefficients. The dynamic characteristics of machine tool depend largely on the damping in joints. According to [13], the mass and the stiffness matrices of machine tool structures are generally constructed with a reasonable degree of accuracy which is not possible for the damping matrix. We follow Levina and Reshetov [11] who considered the machine tool joint as a simple single-degreeof-freedom oscillatory system and found that a damping ratio of 5% approximates a large range of metal-plastic joints. We note that this equivalent damping ratio is sensitive to surface finish, magnitude of the contact pressure and lubrication. The damping matrix components will be evaluated in the Appendix using the eigenmodes and the damping ratio.

=

2.5. EQUATIONS OF MOTION Substitution of Equations (10, 13, 22, 26) into the Lagrangian formulation (3) and rearranging the terms yields the following equations of motion.

m1 z1 +

3 X

m2 z2 +

3 X

i=1 i=1

c1i z_i + S (5z13 + 5z1 z22) + fy Q(z1 + xcz2 ) = fy Q(ro + cos t) c2i z_i + S (5z12z2 + z23 ) + cs ys (ysz2 z3 ) + fa Q(z1 + xcz2 )

= faQ(ro + cos t) m3 z3 +

3 X

i=1

c3i z_i cs(ysz2 z3 ) fx Q(z1 + xcz2 ) = fxQ(ro + cos t);

(27)

where

S = cbL 20

3

(28)

is the contact stiffness of the joint. We consider the particular case where the lead screw is located symmetrically between the 0). Selection of zero values for the off diagonal first and the second contact surfaces (ys damping terms c13 and c23 results in decoupling of the equations. The equations of motion are reduced to two coupled, damped, nonlinear ordinary differential equations describing the lateral translatory (A1 ) and angular (A2 ) displacements of the support.

=

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m1 z1 + c11 z_1 + c12 z_2 + S (5z13 + 5z1 z22) + fy Q(z1 + xcz2 ) = fy Q(ro + cos t) m2 z2 + c21 z_1 + c22 z_2 + S (5z12z2 + z23 ) + fa Q(z1 + xcz2 ) = faQ(ro + cos t): (29) The longitudinal translatory motion (A3 ), which is decoupled from the other equations,

does not influence compliance of the joint in the direction normal to the machining surface and only affects the wearing rate of the contacting materials. Thus, the longitudinal equation of motion becomes:

m3z3 + c33 z_3 + cs z3 fxQ(z1 + xcz2 ) = fxQ(ro + cos t): (30) Alternatively we define a set of relative motion co-ordinates ui ui = zi z~i ; i = 1; 2; (31) z1 ; z~2 ) is the unique fixed point to be shown in Section 3. Consequently the equations where (~ of motion (29) become

m1u1 + c11 u_ 1 + c12 u_ 2 + (15S z~12 + 5S z~22 + fy Q)u1 + (10S z~1z~2 + fy Qxc)u2 + 15S z~1u21 + 10S z~2u1u2 + 5S z~1u22 + 5Su31 + 5Su1 u22 = fy Q cos t m2u2 + c21 u_ 1 + c22 u_ 2 + (10S z~1z~2 + fa Q)u1 + (5S z~12 + 3S z~22 + fa Qxc)u2 + 5S z~2u21 + 10S z~1u1u2 + 3S z~2u22 + 5Su21u2 + Su32 = faQ cos t:

(32)

3. Stability of the Fixed Point 3.1. UNIQUENESS

= 0), the static (~ ~ ) ( fy (ro + z~1 + xc z~2) + s(5z~13 + 5z~1 z~22) = 0; (33) fa (ro + z~1 + xc z~2) + s(5z~12 z~2 + z~23) = 0; where s = S=Q is the dimensionless contact stiffness (or stiffness related to a specific cutting force). These two cubic equations can be rewritten as a ninth order polynomial in z~1 or in z~2 . In order to show conditions for uniqueness we reduce Equation (33) to a single According to equation (29), where the forcing terms have been set to zero ( equilibrium point z1 ; z2 is defined by the following set

cubic equation employing the following considerations. The static cutting force causes the elastic displacement of the cutting tool. This, in turn, modifies the depth of the material to be removed and leads to the change of the cutting force with respect to its initial value. However, the variance of the material depth due to the cutting load should be small in order to ensure the strict tolerances on the machining conditions. We assume that the change of the material depth is fairly small with respect to its pre-set value

jz1 + xcz2j ro:

(34)


381

Thus, the equilibrium equation (33) reduces to (

~ + 5z~1z~22 + S1 = 0; 5z~12 z~2 + z~23 + S2 = 0;

5z13

(35)

where the following notations are used

S1 = rosfy ;

S2 = rosfa :

(36)

Equation (35) is equivalent to 2000Y 3

+ 200S1Y 2 + (125S22

35S12

)Y + S13 = 0

(37)

and the fixed point is now defined as

p

z~1 = Y ; 3

S2 : z~2 = S 5z~120 z~3 1

(38)

1

The discriminant of Equation (37) is always negative:

D=

[(

6:25 108 S22 2:2S12

5S22

)2 + 0:16S14]:

(39)

Consequently, a unique fixed point exists for any dynamics for which assumption (34) is valid. We also verify this fact numerically for the original Equation (33) for various parameter sets. 3.2. STABILITY Stability of the fixed point is determined by the eigenvalues of the characteristic polynomial obtained from the equations of motion (32).

4 + a3 3 + a2 2 + a1 + a0 = 0;

(40)

where

a0 = k11 k22m mk12 k21 ; 1 2 a1 = k11 c22 + k22 c11m mk12 c21 k21 c12 ; 1 2 a2 = km11 + km22 + c11 c22m mc12 c21 ; 1 2 1 2 c c 11 22 (41) a3 = m + m : 1 2 The damping matrix components cij are obtained in the Appendix and those of the stiffness matrix are obtained from Equation (32).

k11 k12 k21 k22

= S (15z~12 + 5z~22) + Qfy ; = 10S z~1z~2 + Qfy xc; = 10S z~1z~2 + Qfa; = S (5z~12 + 3z~22 ) + Qfaxc:

(42)

Analysis of the eigenvalues obtained from Equation (40) defines the fixed point stability. An unstable domain is defined by:

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Figure 3. Fixed point stability diagram. (a) Cutting tool location ( xc ; yc ). (b) Stiffness s vs. machining depth ro .

a Hopf bifurcation where two of the four eigenvalues have zero real parts; a case when one of the eigenvalues is identically zero.

The latter case can be obtained when the coefficient a0 in Equation (40) vanishes, corresponding to a singular stiffness matrix of the linearized system. We note that an alternative to calculation of eigenvalues as a test of stability, is the classical Routh–Hurwitz criterion which for Equation (40) requires positive coefficients ai > 0 (i 0 : : : 3) and a1 a2 a3 a0 a23 a21 > 0 for asymptotic stability. In thisstudy we calculated the eigenvalues (j , j 0 : : : 3) directly from Equation (40) where 0; m1 m2 k11 k22 k12 k21 > 0; k11 k22 2 + 4k12k21 0: m1 m2 m1 m2

(44)

Criteria (44) indicate that the fixed point stability depends not only on the stiffness terms, but also on the system masses m1 and m2 . However, modified criteria independent of the system masses can be deduced from Equation (44)

k11 k22 k12 k21 > 0;

k12k21 0:

(45)

Note that two conditions of (45) yield also k22 > 0. Furthermore, the criteria (45) are stronger than those of (44): if (45) is satisfied, then (44) is also satisfied for any masses. The limit of (45) is depicted in Figures 3a and 3b by solid lines as a conservative bound of stability for the parameter space considered. We also consider the cases when the first of two conditions (45) is satisfied. Various parameter combinations were tested to verify this assumption numerically. Consequently, the bounds of the unstable region reduce to k12 0 and k21 0. Condition k21 0 yields fa 0 and z2 0. Then it follows from Equation (21) that in the parameter space xc ; yc one of the boundaries for the unstable region is a straight line.

=

~=

fy xc + fxyc = 0:

=

= ( )

=

(46)

Similarly, a second bound may be found from k12

=0

~ ~ + Qfy xc = 0; (47) z1 ; z~2 ) depends on the cutting tool location (xc; yc). Considering the where the fixed point (~ fact that z~2 = 0 on one boundary and that the unstable region is fairly narrow (Figure 3a), we assume z~2 z~1 . Consequently, we repose Equation (35) in the following form 5z~13 1 + z~22 =z~12 + S1 = 0; 5z~12 z~2 5 + z~22 =z~12 + S2 = 0: (48) For z~22 =z~12 1, solution of Equations (47, 48) results in a straight line q q q xc fy 40sro2 + fy + ycfx 40sro2 = 0: (49) 10S z1 z2

3

3

3

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Note that the angle in Figure 3a between the straight lines of (46) and (49) confining the narrow unstable region within the parameter space xc ; yc is

f 2fx+fyf 2 x

s 3

y

( )

fy Q :

40S ro2

(50)

The instability is governed by the inelastic terms representing the static cutting force. Equation (49) also defines the boundary of the unstable region in the technological parameter space s; ro . Assume that the cutting point location xc ; yc is now fixed, but the normalized contact stiffness s of the traveling joint and the cutting depth ro can be varied. Thus, Equation (49) yields

( )

( )

3 f4 sro2 = fxc3 40y = const: a

(51)

( )

This is the boundary of the unstable region in the parameter space s; ro , provided xc and fa are of the opposite sign (otherwise there is no unstable region). The condition for the existence of an unstable region is

xc fa = xc (fy xc + fxyc) < 0:

(52)

If the cutting tool location is such that xc fa and for any depth of material to be removed.

> 0, then the system is stable for any stiffness

4. Self Excited Solutions: Accuracy and Stability The unstable domain is governed by the following system parameters: the cutting tool coordinates xc ; yc , the contact stiffness of the joint s and the depth of material to be removed ro . The Hopf bifurcation theory implies that self excited vibration develops around the singular fixed point. A harmonic balance approach [15] is used to estimate the frequencies and the amplitudes of the various limit cycles. We consider the unforced equation of motion (32) where 0, and assume that there is an approximate periodic solution ui .

( )

=

u~i di +

~

n X

(aij cos j!t + bij sin j!t); i = 1; 2:

(53)

j =1

Since we are looking for the steady state solution, the phase angle for one of the harmonics may be chosen arbitrarily. For example, without any loss of generality, one may set b11 0. Introducing Equation (53) into Equation (32), expanding the powers and products of the trigonometric functions into those of multiple arguments and neglecting those of j!t for j > n, we obtain a set of 4n 2 nonlinear equations with 4n 2 unknowns, including the limit cycle frequency. After solving this set, the system limit cycle radii and biases are obtained. The radius Ri is defined as half the difference between the extreme values, and the bias Bi as half the sum.

=

+

Ri = Max [u~i (t)] 2

=

~

Min [ui (t)]

+

; Bi = Max [u~i (t)] +2 Min [u~i (t)] ;

(54)

where i 1; 2. The limit cycle parameters were studied for different cutting tool locations: xc varied, while yc was kept constant (yc 2).

=


385

Figure 4. Limit cycle radii. (a) Translatory motion R1 ( xc ). (b) Rotatory motion R2 (xc ).

Stability and accuracy of approximate analytical solutions is obtained by Floquet analysis [16]. For an autonomous system, one of the characteristic multipliers (eigenvalues of the monodromy matrix) is identical to unity and can be used as an accuracy norm (u ) while three others indicate to stability of the response. The accuracy norm of the limit cycle (u ) proved to be very sensitive to the order of approximation (n in Equation (53)), particularly near the bifurcation points. Consequently, n 5 was found to suffice. In order to obtain the harmonic balance equations, a symbolic program was developed on Mathematica [17]. The nonlinear equation set was solved numerically also on Mathematica using an iterative Newton–Raphson procedure. Numerical simulation of the dynamical system and evaluation of the monodromy matrix components were carried out by a Runge–Kutta 4 integration scheme. The limit cycle radii for the translatory and rotatory motion are presented in Figure 4 and the biases are presented in Figure 5. The limit cycle frequencies do not coincide with those of the linearized dynamical system, except for the small regions near the bifurcation points. The self excited solution consists of four branches: two are stable and two unstable. Three or one limit cycles may coexist in the statically unstable domain denoted by regions III and IV respectively in Figure 4 where the fixed point changes its stability via a Hopf bifurcation. The subdomain of multiple limit cycles (region III) consists of two stable solutions separated by an unstable one whereas there is a single stable limit cycle in region IV. Furthermore, beyond the statically unstable domain, two limit cycles (stable and unstable) coexist with the

=

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Figure 5. Limit cycle biases. (a) Translatory motion B1 ( xc ). (b) Rotatory motion B2 (xc ).

stable fixed point in regions II and V. There are no limit cycles in regions I and VI. The stable limit cycles and their frequencies are verified by numerical simulation and its Fourier transform. Asterisks in Figure 4 correspond to numerical integration results, solid lines to the stable harmonic balance solutions and dash lines to the unstable solutions. Dependency of limit cycle frequency vs. the lathe tool location (xc ) is plotted in Figure 6. Figure 7 depicts the Floquet multipliers for all the branches. For any branch, one of the multipliers is very close to unity indicating to the accuracy of the approximate solution.

5. Periodic and Aperiodic Forced Response Perturbation of the dynamical system with weak periodic excitation results in further bifurcations including periodic and aperiodic solutions obtained in various regions of parameter space. Figure 8 depicts a period doubled solution in region I xc 1:7; yc 2 via the time series, Poincaré map and power spectra. The period doubled solution in region I results when the small amplitude periodic response loses its stability [18]. Figure 9 depicts a quasiperiodic solution in region III xc 1:5; yc 2 . Figures 10 and 11 depict synchronous and aperiodic response in region IV xc 1:45; yc 2 . The periodic (mode-locking) Poincaré map (Figure 10b) reveals a high period (u1 t u1 t mT , m 52) content whereas loss of periodicity is revealed by the structure of the map (Figure 11b) and by the

( =

( =

= ) ( = = ) ()= ( +

= )

)

=


387

Figure 6. Limit cycle frequency ! ( xc ).

Figure 7. Floquet multipliers. (a) Stable (small) limit cycle: region III. (b) Stable (large) limit cycle: regions II, III, IV, IV. (c) Unstable (large) limit cycle: regions II, III. (d) Unstable (small) limit cycle: region V.

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Figure 8. Period doubled forced response, x c = 1:7 (region I), = 12 10 (b) Phase plane + Poincaré points. (c) Power spectrum.

6

m, = 500 Hz. (a) Time series.


Figure 9. Quasiperiodic forced response, x c (b) Poincaré map. (c) Power spectrum.

=

1:5 (region III),

= 2 10

6

m,

389

= 350 Hz. (a) Time series.

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Figure 10. Synchronous periodic response, x c = 1:45 (region IV), = 5 10 (a) Time series. (b) Poincaré map. (c) Power spectrum.

6

m, = 318:31 Hz = 2000 rad/s.


Figure 11. Aperiodic forced response, x c (b) Poincaré map. (c) Power spectrum.

=

1:45 (region IV),

= 12 10

6

m,

391

= 320 Hz. (a) Time series.

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wide band spectrum (Figure 11c). We note that additional simulations not presented here, reveal coexisting periodic and quasiperiodic solutions and further bifurcations of tori. 6. Closing Remarks Nonlinear dynamics and stability analyses of a machine tool traveling joint was performed. Static stability was found to be governed by a Hopf bifurcation which led to coexisting self excited limit cycles. The self excited solutions were obtained by an approximate harmonic balance scheme where the number of terms and limit cycle stability were analyzed via Floquet theory. Perturbation of the dynamic system with weak periodic excitation revealed further bifurcations including periodic and aperiodic response. The relative motion between the cutting tool and the workpiece governs the precision of machining and surface roughness. Self excited and forced vibrations related to machine tool contact dynamics directly influence wearing and fatigue strength of the system and may have an unfavorable effect on tool life. This analysis reveals the importance of mode-coupling instability as a source of large amplitude self excited and aperiodic forced vibration in lathe dynamics where successive cuts do not overlap. Note that single degree of freedom models will not reproduce the mode coupling instability which may be the source of self excited chatter in the case where stick-slip is negligible and where partial overlap is small. We emphasize that in the broad scope of machine tool dynamics, investigation of instabilities leading to self excited oscillations require incorporation of stick-slip and regeneration effects. However, in the case where partial overlap is small, the threshold of stability will be governed by mode coupling. Additional effects are expected to modify the domain of stability as in the case of combined external and parametric excitation. Acknowlegements The authors are grateful to the reviewers for their comments and suggestions which enabled to clarify the role of mode coupling instability in the analysis of periodic and aperiodic machine tool vibrations. Appendix. The Damping Matrix According to Equation (55), the stiffness matrix coefficients include both elastic and inelastic components: 2 ~ + 5z~22 10z~1z~2 3 fy fy xc 3 5+ 5 K =S4 Q4 : fa fa xc 10z~1 z~2 5z~12 + 3z~22 | {z } | {z } el Inelastic component K in Elastic component K 2

15z12

(55)

The experimental determination of the damping data in [11] was carried out without machining the workpiece. The measured damping ratio corresponds to the free running of the machine tool without cutting. Therefore, in order to establish the damping matrix, consider the linearized system with the elastic part of the stiffness matrix only. We assume here that when the change of the cutting load due to the tool displacement takes place, the damping ratio remains the same. We apply the following calculation sequence: determine the undamped eigenvalues and the normalized eigenmodes;


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calculate the components of the diagonal damping matrix in the principal co-ordinates of the decoupled equations (

= 5%);

transform the above components from the principal co-ordinates to the original ones. The natural frequencies !1 and !2 are the roots of the equation !

k11el + k22el !2 + DetK el = 0: m1 m2 m1m2

!4

(56)

The system eigenmodes are

n i i ni = i ; i = 1; 2;

(57)

where 2 el el ni = m2!ki el k22 = m !k212 kel ; i = 1; 2 (58) 1 i 12 11 and i are the normalizing scaling factors, such that the principal mass matrix is the identity matrix and the principal stiffness matrix includes only !i2 on the diagonal:

i = q

1

m1 n2i + m2

:

(59)

The components of the diagonal principal damping matrix are

cpi = 2!i

(60)

The original damping matrix is

C = N T C pN 1; where N is the transformation matrix 1 n1 2 n2 N = 1 2 :

(61)

(62)

Expanding Equation (61), we obtain 2 cp + 2 cp c11 = 2 22 (1n 1n2)2 ; 2 1 2 1 2 n cp + 2 n cp c12 = c21 = 22 22 (1n 1n 1)22 ; 2 1 2 1 p p 2 2 2 2 c22 = 22n22c(1n+ 1nn1)c22 : 2 1 2 1

(63)

References 1.

Weck, M., Meißen, W., Finke, R., and Müller, W., ‘Anwendung der Methode finiter Elemente bei der Analyse des dynamischen Verhaltens gedämpfter Werkzeugmaschinenstrukturen’, Annals of CIRP 24, 1975, 303–307.

394 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

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