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6 I E H * Y P P 0 M G I R W I

In uence of tool speed movement during cutting honeycomb block on its vibration resistance and cutting accuracy

Xxxxxxx Xxxxxxx 1*, Xxxxxx Xxxxxxxxx 1, Xxxxxxxx Xxxxxx 1, Xxxxxxx Xxxxx 1, Xxxxxxxx Xxxxxxxxx 1

1* Research laboratory of robotic systems, Reshetnev Siberian State University of Science and Technology, Krasnoyarsky Xxxxxxx Xx. 00, Xxxxxxxxxxx, 000000, Xxxxxxx Xxxxxxxxxx.

*Corresponding author(s). E-mail(s): xxxxxxxx@xxxxx.xxx ; Contributing authors: xxxxxxxx@xxxxx.xxx ; xxxxxxx@xxxxxx.xx ; xx00000@xxxx.xx; xxxxxxxx00000000@xxxxx.xxx ;

Abstract

The state of cutting blanks from honeycomb fabric methods is asso ciated with di culties of rectilinear cutting of cell walls, which leads to a non-straight edge after cutting with a cutting tool. Such a form of curvilinear trajec tory of the cutting tool movement is a consequence of the cutting-edge v ibration in the pro- cess of moving towards cutting. In order to eliminate this e ec t, it is necessary to determine optimal cutting speed at which the blade and the h oneycomb wall will not be in the resonance.

Keywords: cutting tool, honeycomb block, cutting edge shape, spectru m frequencies, waveforms, critical speed

The method of cutting blanks from a honeycomb sheet, when a cutting tool is made in the form of a thin blade, performed in the direction normal to the s urface to be treated, the plunge, which is accompanied by the impact of the cuttin g edge on the cell wall of the honeycomb, is associated with di culties in rectil inear movement due to vibrations [ 1] and as a result of elastic deformations. Manifestation of the coordinate relationship is that under the action of the cutting force, the top of the cutting tool blade is generally displaced not only in the direction of c utting, but

also in the perpendicular direction to it [ 2]. When cutting the wall of a honeycomb block, the top of the cutting edge of the cutting tool is in conditions of all-round compression, where any movements other than tangential are constraine d. The forces that prevent the movement of the cutting tool in orthogonal (perpendi cular to the walls of the cutting tool blade) directions are determined by the re action forces from the walls of the honeycomb block, acting in the vicinity of the top of th e cutting edge of the cutting tool, which retains elastic properties [ 3]. At the moment of shifting the element, the wall of the honeycomb block is destroyed, the reac tion from the side of the cutting tool and the wall of the honeycomb block disappears, whic h leads to a violation of the balance of forces. The top of the cutting edge of the cutti ng tool rushes to a new equilibrium position, but not along the tangential dir ection, but along a complex trajectory, which has an o set along the normal to the cutting surface. The amount of potential energy accumulated by the moment of shear depends on the cutting conditions, the wear of the cutting edge of the cutting tool, the rigidity of the elastic system, etc. The more accumulated energy is spent during s hearing, the greater is the amplitude of oscillations. The part of the potential energy that has accumulated due to deformations of the elastic system along the normal to the cuttin g surface will tend to create a deviation of the vertex trajectory from the tangential direction. As a result, a signi cant error in the geometric shape and waviness of the m achined surface is formed. In order to reduce the energy intensity of the cutting p rocess, to improve the quality of the cut and the productivity of the equipment, to obt ain the required parameters of the shape of the material processing surface, it is nece ssary to solve the problem of optimizing the shape of the cutting edge of the cutting tool and determine the rigidity of the cross section, in order to exclude deviations whe n the cutting tool blade enters the material being cut. The speci c strain energy rel eased per unit on deformable volume of the cut honeycomb block must be constant. Taking i nto account that dynamic shock loads occur for high speeds of the cutting tool, as we ll as from the cutting force momentum conservation theorem, it can be assumed that th e amount of deformation is inversely proportional to the cutting speed of the cut ting edge of the cutting tool. For arbitrary deformation of the cut wall of the honeycomb xx xxx in the direction of the cutting speed vector Vcut , one can write as

= Vcut T; (1)

then the deformation time can be determined by the relation:

T = Vcut (2)

Thus, the time of action of the cutting edge of the cutting tool on the c ut wall of the honeycomb block is inversely proportional to the cutting speed , and the force of action is determined by the value of the cutting force Fcut required to cut the wall. It can be concluded that the time of action on the wall of the honeycomb block decreases with the removal of the cutting edge of the cutting tool f rom the beginning of the cut for the same type of linear deformation. At the same time, the am ount of motion transmitted to the material at a constant cutting force also decr eases, and the

amount of deformation of the honeycomb block wall is indeed inversely pr oportional to the linear velocity of the cutting tool, and it is also known that t he value of the velocity vector prevails over its direction. The speed of cuttin g a cutting tool into the wall of a honeycomb block can be described by two critical cutting xx xxxx. The rst critical speed determines such a frequency of the cutting tool, which coincides with the frequency of natural oscillations of the wall of the honeycomb block def ormed during the cutting process. At speeds less than the rst critical spee d, the process of quasi- static deformation of the honeycomb wall takes place. At speeds greater th an the rst critical one, there is a shock cut of the cutting edge of the cutting tool on the wall of the honeycomb block. The description of the cutting process in this c ase is associated with the use of impact theory. In this case, there are a number of signi cant features, taking into account which allows us to simplify the mathematical modeling of the process.

So, in work [ 4] it is noted that under the impact action on the material, the value of

all nite forces acting on the object during the same period of time can be neglected. In addition, the application of the Xxxxxx theorem [ 5] becomes extremely e ective for determining the work of cutting forces, which is equal to the scalar product of the force impulse by half a sum of the initial and nal speeds of the point of t he cutting edge of the cutting tool. The second critical speed is associated wit h the occurrence of shock waves and is characterized by the speed of sound propagation in a gi ven material, therefore, this speed is not relevant for cutting the wall s of a honeycomb block with a cutting tool. When determining the rst critical s peed, it is necessary to take into account that the frequency of natural vibrations of the mater ial mass deformed during cutting can be determined from known dependenc ies [6], while the frequency of natural vibrations of the mass of the honeycomb block wall mat erial m

is determined by the relation:

! 0 = C

(3)

To estimate the elasticity xxx cient C, let’s write Xxxxx’x law in scalar form for the elastic force that deforms the material [ 3, 4]:

F = C x; (4)

where x is the amount of linear displacement. In stress-strain coord inates, equation

(4) has the form:

= E "; (5)

where is the internal stress; " relative deformation;

E is the modulus of elasticity of the rst kind for the deformable mater ial. Considering that = F=S , and " = x=L, one gests the expression for C:

C = E S ; (6)

where S is the area of the deformable cross section of the material;

L is the linear size of the deformable object or the height of the honeycomb block.

Fig. 1 Geometric parameters of the cutting tool (1) and thickness o f the cut wall (2)

Figure 1 shows the model of the initial state of the process of cuttin g cutting tool 1 into the wall of honeycomb block 2.

For the cutting edge of the cutting tool blade (Figure 1), the area S of th e deformable cross section of the material can be written like:

S = h t; (7)

where h is the thickness of the cutting tool blade; t - is the wall thickness of the honeycomb block. Then expression (6) is transformed:

C = E h t

(8)

The process of cutting the honeycomb wall is accompanied by deformation , the mass of the deformed material is determined as follows:

md = V; (9)

where is the density of the honeycomb material; V - is the volume of the deformed material of the honeycomb block. The volume V of the deformable wall material of the honeycomb block is determined by the value of its contact area:

V = L S (10)

Fig. 2 Calculation scheme

The expression for the frequency of natural vibrations of the mass of the honeycomb block wall being cut with a cutting tool blade will take the followi ng form [7]:

! 01 = 1

E (11)

For the material used for the manufacture of honeycomb wall E = 70 109 P a; = 2700kg=m3 and wall length L = 0 :02m:

! 01 = 254587 s 1 = 40519 Hz

For the second critical speed, let’s determine the value of the vel ocity of propagation of shock waves (sound) in the wall of the honeycomb block, which can be calculated

[8] by the relation:

a = E ; (12)

but for a particular case, the second critical speed is irrelevant du e to the fact that it is much greater than the cutting speed. In the blade of the cutting t ool, when cutting the wall of the honeycomb block, there are two types of vibrations - longi tudinal and transverse.

For longitudinal vibrations, when determining the natural frequenc ies of the cutting tool blade, the wave equation is used:

E @2 u(x; t ) = @2 u(x; t )

(13)

@x2 @2t

Natural frequencies in this case of pinning are determined by the de pendence [9]

! 0cutting tool = (2n 1)

EA ; (14)

m 0

Fig. 3 Cutting tool blade waveforms

where A is the area of the cross section of the cutting tool; m 0 { is mass of cutting tool; l - is the length of the cutting tool blade. Provided that the mass of the cutting tool m 0 = h b l 1 cutting tool A = h b expression (13) is converted to the form

! 0cutting tool = (2n 1)

E ; (15)

where 1 is the density of the cutting tool blade material. For the modes of vib rations, there is a dependency

n x

u (x) = sin (2n 1) n

(16)

the rst three forms of which are shown in Figure 3. First (lowest) f requency at n = 1:

! 0cutting tool 1 = 2l

E (17)

Second frequency at n = 2:

! 0cutting tool 2 = 3

E (18)

etc.

For a cutting tool blade made of steel with characteristics E = 2 :1 1011 P a; = 7800kg=m3 , a graph (Figure 4) of the dependence of natural oscillation frequencies ! 1 and ! 2 of its length l = 0 :02 0:07m For transverse vibrations [ 9] in the absence of

Fig. 4 Graph of the rst ! 1 and the second ! 2 natural frequencies of the cutting tool from its length

perturbing forces, the di erential equation is used:

m @2 uy + EJ

0 @2t

@4 uy z @x4

= 0 ; (19)

Taking into account the boundary conditions, the natural frequencies of transverse oscillations are:

! = i

EJ z = i

E h2 ; (20)

0transi

2 l2

h b 2 l2 12

where the vibration modes for the rst, second and third modes are eq ual to, respectively 1 = 3 :52; 2 = 22 :04; 3 = 61 :17.

Figure 5 shows a graph of natural frequencies of transverse oscillation s for three values of the rst transverse natural frequencies, Hz : ! 0trans 1 = 85 :7; ! 0trans 2 = 536:4; ! 0trans 3 = 1488 :8. For eld testing, a stand was prepared, the schematic diagram

of which is shown in gure 6. The stand consists of a base 1 xed on support s 2, in which the guide 3 of the cutting tool blade 4 is located, the heel 5 is xed to its end, the load 6 with guides 7, the pointer 8 is connected to the sensor 9 and th e sample being cut 10. The principle of operation is in creating the calculated i mpact force due to the moving load 6 along the guides 7 under the action of gravity and hit ting the striker 5 transmits the movement through the striker 5 to the cut ting tool 4 moving in its own guide 3 rigidly connected by supports 2 to the base 1. The cu tting edge of the cutting tool cuts into the wall honeycomb transmits a shock wav e, the impact of which on the wall of the honeycomb block is recorded by the sensor 9 and r ecorded in the vibrometer 8. Figure 7 shows a stand with the result of cuttin g the wall of a honeycomb block when a cutting tool blade is inserted at an estimated speed of 5 m=s.

Fig. 5 Graph of natural frequencies of transverse vibrations ! 0transi with a cutting tool blade length from 0.02 to 0.07m

Based on the results of three tests, using the obtained displacemen t spectra using the fast Fourier transform method, amplitude-frequency characterist ics were built (Figure 8) Processing of experimental data revealed the following spectru m of frequencies that occur when a cutting tool blade is cut when it hits the edge of the hon eycomb block wall: 4.96; 9.93; 10; 10.8; 14.35; 24.7; 31.66; 33.75; 34.6; 46.2; 47.5; 52.5; 64.32; 70;

71.65; 75; 86.37; 96.83; 107.89; 117.36; 130.14; 1488.68; 165; 166.25; 167.75; 190; 333.75

Hz .

The powers in the frequency bands for each of the three tests are sho wn in Figure

The resonance phenomenon occurs when any of the forced frequencies ! fors falls

into the resonant band [ 10]

0; 7 ! 0i ! fors 1; 3 ! 0i (21)

Therefore, when cutting a honeycomb block with a cutting tool blad e, it is necessary to strive to ensure that the forced frequency of the cutting tool i s either in the pre- resonance or after the resonant region of this range and thereby ensure a st raight cut of the honeycomb block wall.

Figure 10 shows the results of a study of natural frequencies and mode shapes of a cantilevered cutting tool moving at a speed of 5 m=s in ANSYS Explicit Dynamics [11, 12]. As a result, the spectrum of natural frequencies is obtained: 164.55; 1030. 1;

1623.3; 2883; 3098.6; 5648.7; 9332.7; 9741.4; 13945 Hz . From the range (20), one can

select the critical natural frequency and thereby determine the frequency of the vertical 8

Fig. 6 Schematic diagram of the experimental stand 1 - base; 2 - supp ort; 3 - cutting tool guide; 4- cutting tool blade; 5 - heel; 6 - drummer; 7 - guide drummer; 8 - pointer; 9 - vibration sensor; 10 - cell structure

Fig. 7 Experimental stand and result

reciprocating movement of the cutting tool blade. i.e., the requ ired cutting speed, at which the straightness of the cut edges of the honeycomb block wall wil l be ensured. To determine the forced frequency of the impact of the cutting tool blade on the wall of the honeycomb block, the following dependence was used:

! fors = Vcut ; (22)

2 Sl

where Sl is the movement of the cutting tool blade.

The degree of removal of natural frequencies from disturbing ones can b e estimated using resonant frequency bands of in uences. The resonant band of an osc illatory link is considered to be the range of frequencies lying within the limi ts determined by dependence (21). As a result, it is possible to determine the possi ble cutting speed, which excludes the resonance phenomenon in the \cutting tool blade { honeycomb wall" system. Since the longitudinal vibrations of the cutting tool b lade ! 0cuti and the

Fig. 8 Frequency response 1, 2 and 3 tests

Fig. 9 Results of power measurements in frequency bands

walls of the honeycomb block ! 01 are much higher than the forced frequency ! forced , they can be excluded from consideration, leaving only the transvers e vibrations of the cutting tool blade. If the calculated natural frequency ! 0i of the dynamic sys- tem "cutting tool blade-wall of the honeycomb block" falls into the re sonant band of the disturbing frequency ! forced , out, then this fact indicates that the cut becomes

non-rectilinear relative to the edges of the honeycomb block wall. An an alysis of the dynamic quality when cutting the honeycomb wall shows the need to e xclude ! 0i from falling into the resonant band. In this case, the dynamic quality can b e estimated by the value of the frequency ratio xxx cient:

K fi = ! 0i ; (23)

! fors

a) b)

c) d)

e) f)

Fig. 10 Numerical simulation results to determine the natural freq uencies and modes of oscillation of the cutting tool

Table 1 Values of the xxx cient of proximity to resonance

Meaning K Di Quality control K Di < 0:75 satisfactory

0:75 K Di < 0:8 bad

K Dli 0:8 unacceptable

Proximity to resonance within 70{100% can be estimated using the xxx xx ent [ 13]:

K Di = (1 j1 K fi j) 100%; (24)

The dynamic cut quality according to this indicator can be estimated b y comparing the K Xx xxx cient with the standard values selected according to Table 1.

According to this criterion, the forced frequency of the action of the cutting tool blade on the cut wall of the honeycomb block is determined, excludin g falling into the resonant band and from formula (22), the corresponding cutting spee d. Thus, the straightness of the cut faces of the honeycomb block wall was obtained at a c onstant cutting force.

1 CONCLUSION

When cutting the wall of the honeycomb block with a sharpened cutti ng tool blade, shock loads occur, leading to oscillations of the cutting tool-wall of t he honeycomb block, leading to resonance, which signi cantly a ects the straight ness of the cut. At the same time, for the cutting tool, it is necessary to take into accou nt transverse and longitudinal vibrations, and for the wall of the honeycomb block, the rs t critical speed is due to the frequency of the cutting tool, coinciding with the f requency of natural vibrations of the wall of the honeycomb block deformed during cutting. It is proposed to evaluate the quality of the cut using the xxx cient of proximity to resonance and compare it with standard values.

Declarations

Funding. This work was supported by the Ministry of Science and Higher Education of the Russian Federation [State Contract No. FEFE-2020-0017]

Competing interests. The authors declare no competing interests.

References

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