ISSN 1068-798X, Russian Engineering Research, 2008, Vol. 28, No. 10, pp. 1007–1009. © Allerton Press, Inc., 2008.Original Russian Text © V.V. Kuts, I.V. Kucheryaev, 2008, published in STIN, 2008, No. 10, pp. 14–17.

1007

Modern CAD/CAM systems for composite millsare based on sets of interrelated models, such as themodel of the working components of the mill, themodel of the mill structure, and the model of the millhousing. A key problem in the development of suchCAD/CAM systems is the transition from such modelsto simulation of the manufacture of the mill housing,taking account of the characteristics of the equipmentemployed.

Simulation of composite-mill manufacture entailscalculating the adjustment of the machine tool andattachments in the machining of housing element

q

ofthe mill—for example, the socket beneath the cuttingplate [1]. By appropriate adjustment, the housing mustbe set in a position where the control program for thenumerically controlled machine tool may be imple-mented.

To this end, one of three conditions must be satis-fied. Specifically, one of the three unit vectors of thecoordinate system

X

q

Y

q

Z

q

(

Σ

q

) of housing element

q

ofthe mill must be parallel to the axes

Z

U

of the machinetool’s coordinate system. The three conditions are as

follows: 1) unit vector is parallel to the axis

Z

q

;

2) unit vector is parallel to the axis

X

q

; 3) unit vec-

tor is parallel to the axis

Y

q

.

When using five-coordinate machine tools formanufacture of the mill housing, six parameters maybe employed in machine-tool adjustment:

α

1

q

, …,

α

nq

(

n

= 6). Two correspond to the table of the machine toolfor orientation of the mill housing, which is attached tothe table’s base plate; four correspond to second-ordermotion of the tool.

There are many models of five-coordinate multi-purpose and universal machine tools. These machinetools may have various types of tables, such as thosein Fig. 1.

Determining the parameters of machine-tool adjust-ment reduces to calculating the orientation matrix

M

Uq

of coordinate system

Σ

q

relative to the machine tool’scoordinate system

X

U

Y

U

Z

U

(

Σ

U

). It is assumed here thatthe following parameters are known:

M

Pq

, the orienta-

kq

iq

jq

tion matrix of housing element

q

relative to the coordi-nate system

X

P

Y

P

Z

P

(

Σ

P

) of the mill housing;

M

VP

, theorientation matrix of system

Σ

P

relative to the coordi-nate system

Σ

V

of the attachment;

M

2

V

, the orientationmatrix of the attachment relative to the machine tool’scoordinate system; and

M

U

2

, the orientation matrix ofthe coordinate system of the machine tool’s table rela-tive to the machine tool’s coordinate system

Σ

U

.

Machine-Tool Adjustment in Machining the Housingsof Composite Mills

V. V. Kuts and I. V. Kucheryaev

DOI:

10.3103/S1068798X08100195

(a)

(b)

Z

2

,

Z

U

,

Z

P

,

Z

V

X

q

H

1

Y

P

Y

1

Z

q

α

2

Y

2

,

Y

U

,

Y

V

L

1

Z

1

α

1

Y

2

,

Y

U

,

Y

V

Z

2

,

Z

U

,

Z

P

,

Z

V

X

q

Z

q

α

2

L

1

45°

Z

1

Y

1

H

1

α

1

Y

P

Fig. 1.

Mobile circular table (a) and rotating two-coordinatetable (b).

1008

RUSSIAN ENGINEERING RESEARCH

Vol. 28

No. 10

2008

KUTS, KUCHERYAEV

MOBILE CIRCULAR TABLE

Consider the calculation scheme for the adjustmentparameters of the mobile circular table in Fig. 1a. Thetable is adjusted by inclining its base plate at an angle

α

1

and rotating it around its axis by an angle

α

2

. Thisensures the necessary orientation of the mill housing.To determine

α

1

and

α

2

, we specify the coordinate sys-tems

X

1

Y

1

Z

1

(

Σ

1

) and

X

2

Y

2

Z

2

(

Σ

2

). For the sake of sim-plicity, we assume that the position of system

Σ

2

corre-sponds to the coordinate system of the table as a wholewhen

α

1

= 0 and

α

2

= 0. Then, the transition matrix

M

12

between these systems is [1]

where

x

,

y

, and

z

are linear displacements of the coordi-nate system

Σ

2

relative to system

Σ

1

;

β

x

,

β

y

, and

β

z

arethe angles of subsequent rotations of coordinate systemΣ2 around their axes (these angles are positive if therotation from the positive section of the axis is clock-wise); L1 and H1 are the distances from the axis of plateinclination to the axis of rotation of the table.

On rotation by an angle α1, the coordinate system Σ1

takes a new position (X1Y1Z1 ),

determined by the matrix = M(βx = α1). Coordi-nate system Σ2 also takes a new position relative tocoordinate system ΣU, determined by the matrix

When the table rotates by an angle α2, coordinatesystem Σ2 takes the new position , determined by the

matrix = M(βz = α2), and a new position relative to

coordinate system ΣU, determined by the matrix =

MU2 (βz = α2).

M12 M x = 0; y = –L1; z = –H1;(=

βx = 0; βy = 0; βz = 0),

Σ1'M11' X1' Y1' Z1' Σ1'

M11'

MU2' MU2M121– M11' βx = α1( )M12.=

Σ2'

M22'

MU2'

M22'

Then, with simultaneous rotation by α1 and α2, theposition of the table relative to coordinate system ΣU isdescribed by the matrix

To satisfy the first condition noted earlier, we mustsolve the following system for α1 and α2

(1)

To satisfy the second condition noted earlier, wemust solve the following system for α1 and α2

(2)

To satisfy the third condition noted earlier, we mustsolve the following system for α1 and α2

(3)

ROTATING TWO-COORDINATE TABLE

The calculation scheme for a rotating two-coordi-nate table (Fig. 1b) differs from the preceding case onlyin that the transition matrix in the initial position (α1 = 0,α2 = 0) is

In the manufacture of the mill housing using a three-coordinate machine tool and attachment (for example,a division head), the adjustment of the machine toolwhen machining housing element q may be describedby four parameters: α1q, …, αnq (n = 4). These parame-ters are implemented by the control program.

The division head permits the implementation oftwo adjustment parameters: β1q, the angle of rotation ofthe mill housing around the axis of the head; and β2q,the angle of inclination of the division-head axis to theattached housing.

In implementing these two parameters, the millhousing must occupy a position satisfying one of thethree conditions noted earlier.

In the adjustment of the division head (Fig. 2), theposition of an element of the mill housing relative to the

MU2' α1 α2,( )

= MU2M121– M11' βx = α1( )M12M22' βz = α2( ).

kqMPq1– MVP

1– M2V1– MU2'( ) 1– α1 α2,( ) iU = 0;⋅

kqMPk1– MVP

1– M2V1– MU2'( ) 1– α1 α2,( ) jU = 0.⋅ ⎭

⎪⎬⎪⎫

iqMPk1– MVP

1– M2V1– MU2'( ) 1– α1 α2,( ) iU = 0;⋅

iqMPk1– MVP

1– M2V1– MU2'( ) 1– α1 α2,( ) jU = 0.⋅ ⎭

⎪⎬⎪⎫

jqMPk1– MVP

1– M2V1– MU2'( ) 1– α1 α2,( ) iU = 0;⋅

jqMPk1– MVP

1– M2V1– MU2'( ) 1– α1 α2,( ) jU = 0.⋅ ⎭

⎪⎬⎪⎫

M12 M x = 0; y = L1; z = H1;(=

βx π/4; βy = 0; βz = 0).=

Rd

Z 'PZ ''P

X ''P

X 'P

β2q

Fig. 2. Adjustment of division head.

RUSSIAN ENGINEERING RESEARCH Vol. 28 No. 10 2008

MACHINE-TOOL ADJUSTMENT IN MACHINING THE HOUSINGS OF COMPOSITE MILLS 1009

machine tool’s coordinate system is determined by thematrix

where Rd is the radius of rotation of the mill housing oninclination of the division head’s axis (Fig. 2).

Then, to satisfy the three conditions, we must writethree systems of equations analogous to Eqs. (1)–(3)and solve them for β1q and β2q. In these systems (notshown here), we introduce the matrices

and the unit vectors and .

The corresponding calculation schemes permit esti-mation of the positioning error of element q of the millhousing and the orientation error of the coordinate sys-tem Σq after manufacture.

Estimating the positioning error of housing ele-ments entails calculation of probabilistic estimates ofthe mean displacements (xPq, yPq, zPq) of the coordinatesystem Σq after manufacture of element q relative to itsinitial position (expressed in the coordinate system ΣP)and estimates of the mean quadratic deviations of thesedisplacements (∆xPq, ∆yPq, ∆zPq).

Estimating the orientation error entails calculationof probabilistic estimates of the mean angular displace-ments (βxPq, βyPq, βzPq) of the coordinate system Σq

after manufacture of element q relative to its initialposition (expressed in the coordinate system ΣP) andestimates of the mean quadratic deviations of these dis-placements (∆βxPq, ∆βyPq, ∆βzPq).

The positioning and orientation errors of the hous-ing elements after machining may be estimated by sim-ulation, which relies on a series of S experiments withadjustment of the machine tool and attachment inaccordance with the relevant distributions, followed bycalculation of xPqs, yPqs, zPqs and βxPqs, βyPqs, βzPqs foreach experiment s and statistical analysis of the results.

The sequence of calculations in step (experiment) sis as follows.

1. Calculate the random errors in implementing theadjustment parameters of the machine tool , …,

and the attachment , …, , taking account of the

distribution of each parameter and of the values ∆α1, …,∆αn and ∆β1, …, ∆βm.

2. With known , …, and , …, , calcu-late the orientation matrix of element q with respect tothe system ΣU

3. Determine the new position of element q relativeto the mill housing

4. From the known matrices MPq and , calculatethe orientation parameters of coordinate system Σq rel-ative to the mill housing in the initial (xq, yq, zq, βxq, βyq,βzq) and specified ( , , , β , β , β ) posi-tions.

5. Calculate the linear displacements in experiment s:xPqs = – xq, yPqs = – yq, zPqs = – zq and the angu-

lar displacements βxPqs = β – βxq, βyPqs = β – βyq,

βzPqs = β – βzq.

6. Repeat the calculation (steps 1–4) S times.

On the basis of the results, we calculate the meansand mean square deviations, such as

These calculation schemes have been implementedin the CAD/CAM system for F-CAD composite mills[2].

REFERENCES

1. Lapshev, S. I., Borisov, A.N., and Emel’yanov, S.G.,Geometricheskaya teoriya formirovaniya poverkhnosteirekushchimi instrumentami (Geometric Theory of Sur-face Shaping by Cutting Tools), Kursk: Kursk. GTU,1997.

2. Russian Registration Certificate 2006613368 for aComputer Program, 2006.

MP 'P '' β1q β2q,( ) M x = Rd β2q;sin(=

z = Rd 1 β2qcos–( ); βy = β2q; βz = β1q ),

MPq1– MP 'P ''

1– MVP1– MUV

1–, , ,

iU jU

δα1δαn

δβ1δβm

δα1δαn

δβ1δβm

MUq' MUq α1q δα1… αnq, , δαn

,+ +(=

β1q δβ1… βmq, , δβm

).+ +

MPq' MPqMUq1– α1q … αnq β1q … βmq, , , , ,( )=

× MUq α1q δα1… αnq, , δαn

,+ +(

β1q δβ1… βmq, , δβm

).+ +

MPq'

xq' yq' zq' xq' yq' zq'

xq' yq' zq'

xq' yq'

zq'

xPq1S--- xPqs; ∆xPq

s 1=

S

∑ 1S 1–----------- xPqs xPq–( )2

s 1=

S

∑ ;= =

βxPq1S--- βxPqs;

s 1=

S

∑=

∆βxPq1

S 1–----------- βxPqs βxPq–( )2

s 1=

S

∑ .=