35.4 Time-Domain Simulation

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Time-domain simulation (Tlusty and Ismail, 1981, 1983; Tlusty, 1986; Tsai et al., 1990; Lee and Liu,

1991a, 1991b; Smith and Tlusty, 1993; Elbestawi et al., 1994; Tarng and Li, 1994; Weck et al., 1994) is an

alternative method for determining regenerative chatter. In a time-domain simulation, the machining

forces and structural vibrations are simulated in the time domain for a specific set of process parameters

and the resulting signals (i.e., forces and displacements) are examined to determine if chatter is present.

The analyses presented above for the turning and face-milling operations assume that the tool always

maintains contact with the part and that the cutting and thrust pressures are independent of the process

parameters. Further, the face milling analysis approximated the time-varying force process matrix, AðtÞ;

by the zeroth term of its Fourier expansion. With time domain simulations, nonlinear effects may be

directly incorporated into the simulation; thus, more accurate stability prediction is possible. The

disadvantage of time-domain simulations is the extreme computational cost that is required. For a

specific spindle speed, several simulations must be conducted at different depths-of-cut; thus, the

stability boundary for that spindle speed is determined iteratively. This procedure is repeated for a range

of spindle speeds to construct a complete stability lobe diagram.

For turning operations, the machining force is calculated using Equation 35.1, the feed is calculated

using Equation 35.2, and the tool displacement is calculated using Equation 35.4. For face-milling

operations, the feed is calculated using Equation 35.30, the machining forces in the x and y directions are

calculated using Equation 35.33 to Equation 35.35, and the tool and part displacements, respectively, are

calculated using Equation 35.47 and Equation 35.48. To calculate the machining forces in the face-milling

operation, the angular displacement of each tooth is required. The angular displacement of the ith tooth is

uiðtÞ ¼

2p

60

Nst þ

2p

Nt ði 2 1Þ ð35:59Þ

The feed and force equations are static, while the structural displacement equations are dynamic and must

be solved via a numerical integration technique. A sufficiently small time step must be utilized in the

numerical integrations to account for the small system time constants associated with the large structural

frequencies.

35.4.1 Example 3

The feed force for a turning operation is given by FðtÞ ¼ 0:6df 0:7ðtÞ: The structural dynamics are

given by Equation 35.26 with the following parameters: vn ¼ 600 Hz, z ¼ 0:2; and k ¼ 12 kN/mm.

This section presented an analytical method to generate stability lobe diagrams for face-milling

operations. The limiting depth-of-cut in a face-milling operation is given by

dlim ¼ 2

pLR

Nt ð1 þ k2Þ

where Nt is the number of teeth, LR is the inverse eigenvalue of ð2p=NtÞA0½GtðjvcÞ þ GpðjvcÞ􀀉;

A0 is the zeroth term of the Fourier expansion of the force process matrix, GtðjvcÞ and GpðjvcÞ are

the transfer functions relating the tool structural and part structural vibrations, respectively, to

the machining forces evaluated at the chatter frequency vc; k ¼ LI=LR; and LI is the imaginary

part of ð2p=NtÞA0½GtðjvcÞ þ GpðjvcÞ􀀉: The corresponding spindle speed is

Ns ¼

60vc

Nt½p 2 2 tan21ðkÞ þ 2lp􀀉

where l ¼ 0; 1; 2; … is the stability lobe number.

35-14 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

A time-domain simulation of the system including the effect of the tool disengaging from the part is

constructed, and simulations for Ns ¼ 10,000 rpm, d ¼ 8 mm, and fnom ¼ 0.1 mm are conducted.

The simulation is repeated for fnom ¼ 0.2 mm. For both simulations, the time history of the feed

force and the tool displacement are plotted.

To account for the phenomenon of the tool disengaging from the part, the feed in Equation 35.2 must

be modified as follows:

f ðtÞ ¼

fnom þ zðtÞ 2 fpðt 2 TÞ if fnom þ zðtÞ 2 fpðt 2 TÞ $ 0

0 if fnom þ zðtÞ 2 fpðt 2 TÞ , 0

(

ð35:60Þ

where

fpðtÞ ¼

zðtÞ if fnom þ zðtÞ 2 fpðt 2 TÞ $ 0

2fnom þ fpðt 2 TÞ if fnom þ zðtÞ 2 fpðt 2 TÞ , 0

(

ð35:61Þ

If the feed at the current time is calculated to be negative, then the cutting tool has disengaged from the

part and the feed is zero. The term fpðtÞ accounts for feed due to structural vibrations at the previous

spindle rotation, even when the cutting tool disengages from the part. The results for fnom ¼ 0:1 mm and

fnom ¼ 0:2 mm are shown in Figure 35.12 and Figure 35.13, respectively. As the nominal feed is increased,

chatter is suppressed.

35.4.2 Example 4

The cutting and thrust forces in a face-milling operation are given by FCðtÞ ¼ 1:4df 0:6ðtÞ and FTðtÞ ¼

0:4df 0:8ðtÞ; respectively. The lead angle is 458, the entry angle is 2 608; the exit angle is 608, the number of

teeth is four, and the feed per tooth is ft ¼ 0:15 mm. The part is assumed to be perfectly rigid, and tool

structural dynamics for the x and y directions are given by Equation 35.58 and Equation 35.59,

respectively. A time-domain simulation is developed to determine the limiting stable depth-of-cut for

spindle speeds of 1000 and 32,000 rpm. For both spindle speeds, the system is simulated for a depth-ofcut

10% below the limiting stable depth-of-cut and for a depth-of-cut 10% above the limiting stable

depth-of-cut. The cutting force, thrust force, x tool displacement, and y tool displacement are plotted.

0 0.02 0.04 0.06 0.08 0.1

0

0.5

1

1.5

Time (s)

Toll displacement (mm) Feed force (kN)

0 0.02 0.04 0.06 0.08 0.1

−0.15

−0.1

−0.05

0

Time (s)

FIGURE 35.12 Time-domain simulations for Example 3 with fnom ¼ 0.1 mm.

Regenerative Chatter in Machine Tools 35-15

© 2005 by Taylor & Francis Group, LLC

The nonlinear effect of tooth disengagement is included:

x€tðtÞ þ 2ð0:15Þð3000Þx_tðtÞ þ 30002xtðtÞ ¼

30002

15

Fx ðtÞ ð35:62Þ

y€tðtÞ þ 2ð0:1Þð4000Þy_tðtÞ þ 40002ytðtÞ ¼

40002

17

Fy ðtÞ ð35:63Þ

To account for the phenomenon of the tool disengaging from the part, the feed in Equation 35.30 must

be modified as follows:

fiðtÞ ¼

fciðtÞ 2 fpiðt 2 TtÞ if fciðtÞ $ 0

0 if fciðtÞ , 0

(

ð35:64Þ

0 0.02 0.04 0.06 0.08 0.1

0.8

1

1.2

1.4

1.6

Time (s)

Tool displacement (mm) Feed force (kN)

0 0.02 0.04 0.06 0.08 0.1

−0.15

−0.1

−0.05

0

Time (s)

FIGURE 35.13 Time-domain simulations for Example 3 with fnom ¼ 0.2 mm.

0 0.5 1 −1

−0.5

0

0.5

Time (s)

x (mm) Fx (kN)

y (mm) Fy (kN)

0 0.5 1 −1.5

−1

−0.5

0

Time (s)

0 0.5 1 −0.1

−0.05

0

0.05

0.1

Time (s)

0 0.5 1 −0.1

−0.05

0

Time (s)

FIGURE 35.14 Time-domain simulation for Example 4 with Ns ¼ 1000 rpm and d ¼ 2.025 mm.

35-16 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

where

fciðtÞ ¼ ft cos½uiðtÞ􀀉 þ {xtðtÞ 2 xpðtÞ} cos½uiðtÞ􀀉 þ {ytðtÞ 2 ypðtÞ} sin½uiðtÞ􀀉 ð35:65Þ

fpiðtÞ ¼

{xtðt 2 TtÞ2 xpðt 2 TtÞ} cos½uiðtÞ􀀉þ{ytðt 2 TtÞ2 ypðt 2 TtÞ} sin½uiðtÞ􀀉 if fciðtÞ $ 0

2ft cos½uiðtÞ􀀉þfpiðt 2 TtÞ if fciðtÞ , 0

(

ð35:66Þ

Note that uiðtÞ ¼uiþ1ðt 2 TtÞ and i 2 1 !Nt if i ¼ 1: The term fpiðtÞ represents the contribution to the

instantaneous feed when the previous tooth was at the same angular location as the ith tooth. If the tooth

and part are in contact, this contribution is due to the tool and part vibrations. If the tooth and part are

not in contact, this contribution is the previous contribution added to the static portion and the

instantaneous feed is set to zero. Through time-domain simulations, the limiting stable depth-of-cut for

0 0.5 1

−4

−2

0

2

Time (s)

Fx x (mm) (kN)

y (mm) Fy (kN)

0 0.5 1

−4

−3

−2

−1

0

Time (s)

0 0.5 1

−0.4

−0.2

0

0.2

0.4

Time (s)

0 0.5 1

−0.4

−0.2

0

0.2

0.4

Time (s)

FIGURE 35.15 Time-domain simulation for Example 4 with Ns ¼ 1000 rpm and d ¼ 2.475 mm.

0 0.01 0.02 0.03 −2

−1

0

1

Time (s)

Fx (kN)

x (mm)

y (mm) Fy (kN)

0 0.01 0.02 0.03 −2

−1.5

−1

−0.5

0

Time (s)

0 0.01 0.02 0.03

−0.06

−0.04

−0.02

0

0.02

Time (s)

0 0.01 0.02 0.03

−0.2

−0.15

−0.1

−0.05

0

Time (s)

FIGURE 35.16 Time-domain simulation for Example 4 with Ns ¼ 32,000 rpm and d ¼ 3.105 mm.

Regenerative Chatter in Machine Tools 35-17

© 2005 by Taylor & Francis Group, LLC

Ns ¼ 1000 rpm is found to be 2.25 mm and the limiting depth-of-cut for Ns ¼ 32,000 rpm is found to be

3.45 mm. The results are shown in Figure 35.14 to Figure 35.17. The system is stable in Figure 35.14 and

Figure 35.16, while instability is evidenced in Figure 35.15 and Figure 35.17 by the force in the y direction

saturating at 0 kN.

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