Transcript arjassov_OST_05

LATHE VIBRATIONS ANALYSIS ON SURFACE
ROUHHNESS 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. Introduction
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.
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.
a)
y
y=ybsinpt
y
j
O
x
A
l
b)
z=zbsinpt
y=ybsinpt
z
y
x
O
l
A
Fig. 1. Dynamic models with one (a) and two (b) degrees of freedom.
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=ybsinpt
y
F
l1
F
Fig. 2. Calculation scheme in cutting.
x
l
2. Dynamical model with one degree of freedom without damping
2.1 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
J 0 j  k yj l 2  M y sin p t  Fr  Fa cos t  l1
..
(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,
M y  mp2 ybl / 2
p  2f ,
yb
where m is the mass of the blank,
f is the frequency of foundation vibrations,
is the amplitude of the foundation vibrations.
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 Fa cos t (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.
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
 v  Dl p 

y  j l   y0  F0l1  El1l cos t   0
 sin  t  Dl sin pt  F0l1  El l1 cos t



Fa
Fr
M0
E

D
,
F0  ,
2
J 0 ( 2  p 2 )
kl
J 0 ( 2    )
where
  k yl 2 / J 0
is the natural frequency of the lathe system.
(2)
(3)
2.2. The differential equation of forced vibrations caused by the moving cutting force F
j
y=ybsinpt
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).
y
F
F
u
u
A
O
l1
l
Figure 3. Calculation scheme in the case of motion cutting.
The differential equation of forced vibrations caused by the moving cutting force F
J oj  k yl 2j  M y sin pt  Fr l  ut   Fa sin *t  l  Fa  ut sin *t
(4)
where u  snb10 3 / 60 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.
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)
J oj  j l 2  k yl 2j  M y sin pt  Fr l  ut   Fa sin *t  l  Fa  ut sin *t
where  is the damping factor.
(5)
The general solution of Eq. (5) represents free vibrations
j1  exp  nt C1 cos 1t  C2 sin 1t 
1   2  n2
2n   l 2 / J o ,
where
(6)
is the natural frequency of free damping vibrations.
A particular solution of Eq. (5), depending on the foundation vibrations
(moving cutting force),
*
**
and disturbing force
j2  j2  j2
(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
j  exp( nt )C1 cos 1t  C2 sin 1t  

J o 4n p
2

   p 
My
2
2
2
2

 p 2 sin pt  2np cos pt  A  Bt  C sin  *t  D cos *t  Et sin  *t  Kt cos *t
(8)
The constants of integration C1 and C2 are determined from the initial conditions j 0  j0 , j 0  j0.
4. Dynamical model with two degrees of freedom with damping
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.
z
z=zbsinpt
y=ybsinpt
y
O
x
A
l
z
a
g
O
b
c
y



A
g
C
zC
yC
z
y
N
x
Fig. 4. Gyroscope system with two degrees of freedom in cutting.
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).
PRINCIPAL MODE OF VIBRATION
NATURAL FREQUENCIES VERSUS FREQUENCY OF ROTATION
1 - for direct precession
2- for reverse precession
Frequency of rotation of the blank, rpm
(rpm)
z=zA /a amplitude of the principal mode of
vibration)
Values of natural
Frequencies P1 and P2, s-1
(at the frequency of rotation of the blank 2000 rpm)
Second mode - reverse precession, (2=106,6 s-1)
z
y
First mode - direct precession, (1=117,05 s-1)
y=yA /a , (a - amplitude of the principal mode of vibration)
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 analysis
5.1. 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).
Indicators
N2
N3
F
N1
F
Applying force
Figure 6. Experimental test of the spring constant of the lathe.
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.
800
3500
700
3000
y+ p
600
y = 577.28x
R2 = 0.9367
400
test results
300
200
-200
0.1
0.2
0.3
0.4
0.5
y(x)=39.5+498.1x
-300
0.6
0.7
R2 = 0.9974
1500
1000
y- p
0
0
-100 0
test results
500
y- p
100
y = 999.18x
2000
Load, N
Load, N
500
y+ p
2500
0.8
-500
-1000
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y(x)=-29.6+1018.5x
-1500
Displacement, mm
Displacement, mm
Figure 7. Correlation function between the load and static displacement for
horizontal (a) and vertical (b) loads.
1.6
1.8
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
J 0 z  Ab y   l 2 z  k z zl 2  M zl sin pt,
J 0 y  Ab z   l 2 y  k y yl 2  M yl cos pt  Fr l1l  Fal1l cos*t
(9)
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,
The general solution of Eqs.(3)
z  exp  n1t D1 cos 1t  D3 sin 1t   exp  n2t D2 cos 2t  D4 sin 2t  
 a2 cos pt  b2 sin pt  c2 cos *t ,
y  exp  n1t D1 sin 1t  D3 cos 1t   exp  n2t  D2 sin 2t  D4 cos 2t  
 a1 cos pt  b1 sin pt  Fr l1 / k yl  c1 cos*t  d1 sin *t
(10)
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.
5.1.2
Experimental analysis of vibration on idling of the lathe
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 .
0,3
0,3
theoretical
theoretical
experimental
experimental
Velocity,mm/s
Velocity, mm/s
0,2
0,1
0
0,2
0,1
0
0
100
200
Frequency, Hz
300
400
0
100
200
300
400
Frequency, Hz
Figure 8. Experimental and theoretical results about horizontal (a) and vertical (b) the
vibrations of the blank without rotation.
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.
n=1600 rpm
1.4
with one degree of freedom
with giroscopic forces
experiment
1
0.8
0.6
0.4
with one degree of freedom
with giroscopic forces
experiment
0.6
Velocity, mm/s
Velocity, mm/s
1.2
n=1600 rpm
0.8
0.4
0.2
0.2
0
0
0
100
200
300
Frequency, Hz
400
500
0
100
200
300
400
Frequency, Hz
Figure 9. Experimental and theoretical results about horizontal (a) and vertical
(b) vibrations in the case of the blank rotation with gyroscopic forces.
500
5.2 Measuring of vibration 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.
experiment
first model of force without damping
first model of force with damping
motion force without damping
motion force with damping
0.7
0.6
0.3
b)
Velocity, mm/s
Velocity , mm/s
a)
0.25
0.5
0.2
experiment
first model of force without damping
first model of force with damping
motion force without damping
motion force with damping
0.4
0.3
0.15
test1
test2
test3
test4
test5
test6
test1
test2
test3
test4
test5
test6
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.
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.
0.5
0.25
Velocity, mm/s
b) 0.3
Velocity, mm/s
a) 0.6
0.4
experiment
0.2
experiment
without damping
without damping
with damping
with damping
0.15
0.3
test1
test2
test3
test4
test5
test6
test1
test2
test3
test4
test5
test6
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.
5. Conclusion
1. 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).