,Modern Technique and Technologies 2002 81 . ,.. ..

MATHEMATICAL MODEL OF THE FORCED FLUCTUATIONS OF THE PREFABRICATED CLTTlNG

TOOL ELEMENTS IN MILLING THE SPATIAL COMPLEX SURFACES ON THE MACHINES WTTH

NUMERICAL PROGRAM CONTROL

D. A Malyshkin. A M. Markov, E. MJ. Tatarkin Altai State Technical University

Modemlevelofthe technique development isrhar- acterized by designing agreat variety of the details with spatial complex shape with the simultaneous require- ments increase to the quality oftheir processing. These details include dies, molds, metallic: models for cast-

Main tool of the mechanical processing automa- tion of the details including spatial complex surfaces are themillerswithnumericalprogramcontrol(NPC). The operating efficiency ofthese tools depends on the quality of finishing operations performance. I t allows one to reduce the time for aflertrealment, the part of which amounts to 4040% of all labor intensiveness of the details manufacturing.

Process of milling having irregular, discrete char- acter is accompanied by the fluctuations of high in- tensity causing a decrease of processing quality. During irregular and discrete cutting an increase of the processing capability is not always accompanied by the required quality. It is due to the growth of fluc- tuations amplitude.

More “weak” link of technological system is cut- ting tool because of a comparatively low.rigidity with respect to the rest elements. Therefore its fluctuations determine the quality of the finished detail surface. Forced fluctuations of the cutting ttol resulting from the action of disturbing force changeable batchwise is accompanied by the resonance phenom.ena (the fre- quency of milling cutter self fluctuations is close or, equals to the disturbing force frequency) that causes unrestricted growth of the fluctuations amplitude and through it the growth of microasperity height:

One ofthe methods for a decrease ofthecutting tool fluctuations intensity is the usage of vibration damper. Spectral analysis oftechnological systcm hasshown that the cutting tool fluctuations are more intensive in’com- parison with those of the rest machine units. , .

In this connection, currently, for metal process- ing a wide spectrum of tools (cutters, face mill, etc) IS used, in.which construction the damping elements are introduced as insertions of elastic material. I n processing spatial complex surfaces the usage ofthe given tbol did not find a wide application. The main reason is the lack of mathematical models describing the operation of prefabricated end-milling cutters. Therefore, the task of mathematical model obtain- ing, describing the fluctuation of the prefabiicated cutting tool for spatial complex surfaces processing

, ing, etc.

. .

is actual and opportune. As the model imitating the process of the forced

fluctuations of the prefabricated end-milling cutter elementsksuiting in milling spatid complex surfaces from the action of disturbing forces changeable batch- wise (cutting force) the scheme of two-mass dynamic system is chosen.

The system has two levels offreedom: the first one corresponds to the fluctuations of prefabricated and- milling cutter case with the respect to spindle (gener- alized coordinate XJ; the second system corresponds to the fluctuations ofmiller.cutting plate with respect to its case (generalized coordinate xJ. To obtain the equations of the given dominating technological sys- tem moving we use the theorem on the change of ki- netic energy in generalized coordinates (Lagranzh’ equations of the I I class) based on generalized coordi- nates xI and xl. The origin of coordinate axes X, and X, coincides with the position ofthe bodies’ static bal- ance included into the system

hereTisthe kineticenergyofthegiven dissipative sys- tem; i,and.i, are generalizedvelocitiesalong the cor- responding axels; xi andx, aregeneralized coordinates; Q,, and Q., are generalized forces calculated by the corresponding genemlized coordinates.

Following differentiating T and determination of the generalized forces by every of generalized cwrdi- nates we obtain the following system

(m,+m,)x, + m , x 2 +c,xI +C,x, = O m,(x, t x , )+c ,x , + G , x , =Hsin(p t tp) (2) i

here m,, mo; C,, C,,, G,, G, are adduced mass, rigidity and damping ofthe case and miiier cutting plate cor- respondently; X I , Y2 are the projections of the miller case and cutting plate acceleration correspondently on the axel of the coordinates x, and xi. To obtain ana- lytical expression of the axel.force component ofcut- ting P=f(Sz, n, V, t).being in the right part ofthe low equation ofthe system (2) the decomposition in Foil- rier series was used 121.

Mathematicai model lhas the following inputs and outputs:

Inputs: cutting conditions (millern rotation frequen- CY, feeding to miller tooth S , velocity of cutting V,). dynamic coefficients (mass m, rigidity C , damping G).

a t p u t s : the values of vibration displacements x, and x? of the technological system elements miller case-cutting plate.

Dynamic model ofthe given technological system is the model, which is constructed with the following assumptions: . :

. . ~ ~.

Section 3: Technology, Equipment and Machine-Building Production Automation 0-7803-7373-1/02/$1.17 t92002 IEEE

82 Modern Technique and Technologies 2002

- the forced fluctuations of the given system are considered as linear fluctuations of small values; - masses of the miller case and cutting plate are

concentrated in the mass centers; - fluctuations are accounted only in axel direc-

tion, in which surface asperity forms; - we neglect the fluctuations of the rest elements

because of their insignificant action on the surface quality formation; - based on the linear task we shall regard that in

the given system the damping is in direct proportion to the fluctuations velocity; - cutting force changes batchwise with the fre-

quency equal to the miller rotation' frequency; - elastic deformations ofthe technological system

elements are considered within the Hooke's law ful- fdment.

REFERENCES I . V. A Batuev. Increase of the Spatial Complex

Surfaces Milling Capacity and Precision on the Ma- chines with Numerical Program Control by MCdilS of Cutting Forces Stabilization, Dissertation, Cheiyab- insk, 1986

2. A V. Kolyadin. Stability of the Machine Tech- nological System.Motion in Step-by-step Milling Hard-to -Machine.Materials, Dissertation, L. , LPI, 266 P, 1982

3. A A Yablonsky, S. S. Noreika Cource of Fluc- tuations Theory, Textbook, M., Vysshaya Shkol4 258 P, 1966

WAYS OF JAW BREAKERS CONSTRUCTIONS IMPROVEMEXT

AV. Makarov I Siberian State Industrial University

Search for new the most effective kinematic schemes ofjaw breakers is one of the main problems of ore mining and processing industry.'To solve the

0

6 2

X

Fig. I. Calculated scheme of the dominating technologi- cai system fluctuations (miller case-cutting plate) 1 - spindle unit, 2 - miller case, 3 - cutting plate

Solution of the differential equations (2) sys- tem is based on the Laplace transformation. As a result of transformations the transfer function of dominating function was obtained, considering x, and x,and the expression for amplitude frequency characteristic of technological systems and ' i ts graphical interpretation as ou!put signals by'turns (Fig. 1).

To automate calculations and their graphical in- terpretation obtaining the software Mathcad PLUS (Math.SoR.lnc) was used.

Developed mathematical model allows one at the stage of milling operations designing to calcu- late the values of vibration displacement forming the processed surface asperity and to use it in CAP.

problem we use the method developed by the Profes- sor L. T, Dvoniil\ov [ I j .

kcording to the structural theory any kinematic schemes can be described by the structural system:

'[CPIrh(LTm)=r+(r-l).o.., +...+ i *

1 h(k-m)={ 1. m < k , 0. m L k .

here P,is the number of kinematic pairs of k'class; h(k-m) is the unit function; t i s the number ofpairs , by means of which the most complex link is connect- ed with other links; ni is the number of links adding by i kinematic pairs; n is the numher of links: W is mo- bility; m is the number of total connections imposed onto the system. . .

W consider the case when the breaker scheme coli- sists of Assur group (W,=O) and the dog attached to it. Total mobility of W is equal to 1, I% construct simple machines with closed change-

able contour. For this purpose we specify the follow- ing conditions:

1. t=3 2. scheme consists of kinematic pairs of 5 class

3. the number of general imposed connections is (p,,?,3,4=0)

equal toihree (m=3)

Section 3: Technology, Equipment and Machine-Building Product ion Automation 0-7803-7373-1/02/$1.17 Q2002 IEEE