LATHE VIBRATIONS ANALYSIS ON SURFACE LATHE VIBRATIONS ANALYSIS ON SURFACE ROUHHNESS OF MACHINED DETAILSROUHHNESS OF MACHINED DETAILS

* Gennady Aryassov , ** Tauno Otto and *Valeri Kammonen

(* Department of Mechatronics, Tallinn University of Technology** Department of Machinery, Tallinn University of Technology)

 

1. 1. IntroductionIntroduction This paper describes the influence of lathe vibrations on the roughness of machined details.

For the force vibrations analysis two cutting force models are developed.

In the first case on the basis of model with two degrees of freedom the invariable coordinate of the cutting force is evaluated on each step of calculations.

In the second case on the basis of model with one degree of freedom the cutting force is considered as a function of time.

The cutting force is represented as a sum of components determined by the empirical formula and variables. The range of the variables is related to the roughness value and monitored in a wide range.

Fig. 1. Dynamic models with one (a) and two (b) degrees of freedom.

yx

z

l

z=zbsinpt

y=ybsinptb)

AO

y

y

x

y=ybsinpt

l

a)

A

O

The calculation schemes involve systems with one and two degrees of freedom, representing vibrations of the blank as a rigid body, hinged in the spindle and elastically supported in the tailstock of the lathe.

In order to simplify the dynamic model, we eliminated the factors, which have a minor effect on the results.

Fig. 2. Calculation scheme in cutting.

To develop a dynamic calculation model, first, we formulated the research problem. In machining of the detail on the lathe, the cutting force F (Fig.2) is not constant. It is determined by several factors as the change in the thickness of the cut-off chips, the change in the mechanical properties of the blank material and tool wear. The input of the lathe system is the cutting force F as the function of time and the output is the displacement of the cutter or the blank (Fig.2).

y

xy=ybsinpt

l1F

lF

2. Dynamical model with one degree of freedom without damping2.1 Solution in the case of cutting force according to the first model

The differential equation of forced vibrations caused by the cutting force F (Fig.2) according to the theorem about the kinetic moment is

(1)

where Jo is the moment of inertia of the blank around the headstock (spindle),

φ is the declination angle of the blank,

ky is the horizontal spring constant of elastic support of the blank,

l is the length of the blank,

where m is the mass of the blank,

, f is the frequency of foundation vibrations,

yb is the amplitude of the foundation vibrations.

 

12

..

0 cossin ltFFtpMlkJ aryy

2/2 lympM by

fp 2

The solution of Eq. (1) can be expressed in the form of the displacement of the blank end in relation to initial conditions y0 and v0

The cutting force F is reproduced as a sum of the following items: the constant component Fr determined in practice by the simplified empirical formula [3] and the variable component (l1 is the coordinate of the cutting force). The amplitude of the variable component of the cutting force is related to the roughness value and changes in a rather wide range.

tFacos

)2(cossinsincos 1100

1100 tllElFptlDtplDvtlEllFyly

,)( 22

0

0

pJMD

,0 lk

FF r )3()( 22

0

J

FE a

where is the natural frequency of the lathe system.02 / Jlky

2.2. The differential equation of forced vibrations caused by the moving cutting force F In the previous section (2.1) we have been obtained the solution in the case of the first model cutting force with regard to the blank. Now we have dealing with motion cutting force (Fig.3).

The differential equation of forced vibrations caused by the moving cutting force F

where is the velocity of the cutting tool,

s is the feed of the cutting tool,

nb is the rotational speed of the blank.

The general solution of the differential equation (4) was obtained in the similar way. However, it is not presented here because it is massive.

)4(sinsinsin **2 tutFltFutlFptMlkJ aaryyo

60/10 3 bsnu

Figure 3. Calculation scheme in the case of motion cutting.

l

y u

O A

y=y bs

inpt

F

l1

Fu

3. Dynamical model with one degree of freedom with damping

The differential equations of forced vibrations with damping varying as the velocity in much the same way as Eq.(4)

where is the damping factor.

The general solution of Eq. (5) represents free vibrations

where is the natural frequency of free damping vibrations.

A particular solution of Eq. (5), depending on the foundation vibrations and disturbing force (moving cutting force),

(7)

Adding the general solution (Eq. (6)) and partial solution (Eq. (7)), a general solution of the differential equation (7) for the declination angle of the blank

The constants of integration C1 and C2 are determined from the initial conditions .

)5(sinsinsin **22 tutFltFutlFptMlklJ aaryyo

)6(sincosexp 12111 tCtCnt ,/2 2

oJln 221 n

**2

*22

tCtCnt 1211 sincos)exp(

)8(cossincossincos2sin4

****222222 tKttEttDtCBtAptnpptp

ppnJM

o

y

00 0,0

4. Dynamical model with two degrees of freedom with damping

Fig. 4. Gyroscope system with two degrees of freedom in cutting.

y

C

A

zzC

yC

c

abz y

x

O

N

y

x

z

l

z=zbsinpt

y=ybsinpt

AO

Such a model (Fig.1, b and 4) enables us to take into account the effect of gyroscopic forces resulting from the rotation of the blank.

Our analysis shows that with an increase in the value of the difference between the higher and the lower frequencies, ω1 and ω2, is increased (Fig.5).

NATURAL FREQUENCIES VERSUS FREQUENCY OF ROTATION

Frequency of rotation of the blank, rpm (rpm)

Va

lues

of n

atur

al

Freq

uenc

ies

P1 a

nd P

2, s

-1

1 - for direct precession

2- for reverse precession

PRINCIPAL MODE OF VIBRATION (at the frequency of rotation of the blank 2000 rpm)

Second mode - reverse precession, (2=106,6 s-1)z

y

y=yA /a , (a - amplitude of the principal mode of vibration)z=z A

/a a

mpl

itude

of t

he p

rinci

pal m

ode

of

vibr

atio

n)

First mode - direct precession, (1=117,05 s-1)

Fig. 5. Principal modes of vibration with gyroscopic forces, corresponding to two different natural frequencies.

Where the first mode (so-called direct precession) with the higher frequency ω1 and the second mode (the so-called reverse precession) with the lower frequency ω2. In the first mode of vibration, point of the blank axis moves on the circle in direction of its own rotation, and in the second mode it moves in the opposite direction to rotation (Fig.4).

5. Experimental analysis5.1. Experimental test of the spring constant of the lathe

The accuracy of the accepted models was tested on the lathe 1K62.

During the theoretical analysis, calculation accuracy depends on both the degree of fitness of the accepted models of the real system and on how accurately mechanical characteristics of the lathe are determined. One of these characteristics is the spring constant of the lathe. To decrease of the influence of the resistance force on the test results, the static rigidity was measured with weak vibrations of the lathe, excited by running the electric motor and other mechanisms without a load. Two load positions were involved in the test (Fig.6).

Figure 6. Experimental test of the spring constant of the lathe.

N1N2

Indicators

N3

Applying force

F F

Figure 7. Correlation function between the load and static displacement for horizontal (a) and vertical (b) loads.

Figure 7 shows the results of statistical analysis of the experiment data in the form of correlation functions, where the coefficient of direct regression is the unknown rigidity [9]. The coefficient of correlation was obtained close to a unit that indicates the linear correlative function between the load and the displacement.

y = 577.28xR2 = 0.9367

-300

-200

-100

0

100

200

300

400

500

600

700

800

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Displacement, mm

Load

, N

y(x)=39.5+498.1x

y+ p

y- p

test resultsy = 999.18xR2 = 0.9974

-1500

-1000

-500

0

500

1000

1500

2000

2500

3000

3500

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Displacement, mmLo

ad, N

y(x)=-29.6+1018.5x

y+ p

y- p

test results

The differential equations of forced vibrations with damping, caused by the cutting force F, according to the theorem about the kinetic moment, are presented in the following form

,sin220 ptlMzlkzlyAzJ zzb

)9(coscos *11

220 tllFllFptlMylkylzAyJ aryyb

tDtDtntDtDtnz 2422213111 sincosexpsincosexp ,cossincos *

222 tcptbpta

tDtDtntDtDtny 2422213111 cossinexpcossinexp )10(sincos/sincos *

1*

1111 tdtclklFptbpta yr

The general solution of Eqs.(3)

where D1, D2, D3 and D4 are constants of integrations to be determined from the initial conditions,

1 and 2 are the natural frequencies of vibrations with gyroscopic forces and damping.

where b is the angular velocity of the rotation of the blank,

yb and zb are the amplitudes of the foundation vibrations,

kz and ky are spring constants,

A is the moment of inertia of the blank in relation to the axes of rotation,

5.1.2 Experimental analysis of vibration on idling of the lathe

Frequency, Hz

Velo

city

, m

m/s

0

0,1

0,2

0,3

0 100 200 300 400

theoretical

experimental

Frequency, Hz

Velo

city

,mm

/s

0

0,1

0,2

0,3

0 100 200 300 400

theoretical

experimental

Figure 8 shows the results of vibration measurements without rotation of the blank in the horizontal and vertical planes, theoretical reference results of vibration velocity are also given .

Figure 8. Experimental and theoretical results about horizontal (a) and vertical (b) the vibrations of the blank without rotation.

Figure 9. Experimental and theoretical results about horizontal (a) and vertical (b) vibrations in the case of the blank rotation with gyroscopic forces.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 100 200 300 400 500

Frequency, Hz

Velo

city

, mm

/s

with one degree of freedomwith giroscopic forcesexperiment

n=1600 rpm

Frequency, HzVe

loci

ty, m

m/s

0

0.2

0.4

0.6

0.8

0 100 200 300 400 500

with one degree of freedomwith giroscopic forcesexperiment

n=1600 rpm

Test results in horizontal and vertical planes and the corresponding theoretical results of the vibration velocity are shown in Fig.9. The frequency of the rotation of the spindle was 1600 rpm. As can be seen in Fig.9, the theoretical results, obtained taking into account gyroscopic forces, are in agreement with the experimental results.

5.2 Measuring of vibration by cutting

Fig. 10. Comparative analysis of experimental and theoretical results (dynamical model with one degree of freedom) about horizontal (a) and vertical (b) vibrations by cutting.

Experimental measuring was performed at different cutting speeds, feeds and depths of cut. After every cutting, surface roughness was measured by the profilograph “Surtronic 3+”. The amplitude value of the variable component of the cutting force in Eq. (1) was taken according to the experimental value of roughness. Test results and results of the calculation taking into account the dynamical model with one degree of freedom are presented in Fig.10.

0.3

0.4

0.5

0.6

0.7

test1 test2 test3 test4 test5 test6

Velo

city

, m

m/s

experimentfirst model of force without dampingfirst model of force with dampingmotion force without dampingmotion force with damping

a)

0.15

0.2

0.25

0.3

test1 test2 test3 test4 test5 test6

Velo

city

, m

m/s

b)

experimentfirst model of force without dampingfirst model of force with dampingmotion force without dampingmotion force with damping

Fig. 11. Comparative analysis of experimental and theoretical results (dynamical model with two degrees of freedom) about horizontal (a) and vertical (b) vibrations by cutting.

Test results and results of the calculation taking into account the dynamical model with two degree of freedom are presented in Fig.11. Like in the previous case, the results of calculation with gyroscopic forces are in better agreement with the experimental results.

0.3test1 test2 test3 test4 test5 test6

Velo

city

, mm

/s

0.4

0.5

0.6

experiment

without damping

with damping

a)

test1 test2 test3 test4 test5 test6Ve

loci

ty,

mm

/s

0.15

0.2

0.25

0.3b)

experiment

without damping

with damping

5. Conclusion1. Analysis of roughness measurements data confirmed the accuracy of the dynamical calculation model.

2. Surface roughness parameters of the blank quite satisfactorily agreed with the corresponding data of the theoretical investigation.

3. The calculation model with two degrees of freedom was used to analyse the influence of gyroscopic forces on surface roughness.

4. The results of calculation with taking into account the damping forces correspond with the experimental results better.

5. Analogically the results of calculation for the dynamical model with one degree of freedom and second model cutting force are in better agreement with the experimental results in comparison with the cutting force first model.

6. The results of experimental and theoretical investigations showed that this model is adequate.

Acknowledgement

This work was supported by Estonian Science Foundation (grants No. 5161 and 5636).