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The vibrations of a textile machine rotor with nonlinear characteristics
Mechanism and Machine Theory Vol. 21, No. 1, pp. 29-32, 1986
0094-114X186 $3.00 + .00 © 1986 Pergamon Press Ltd.
Printed in Great Britain.
THE VIBRATIONS OF A TEXTILE M A C H I N E ROTOR WITH NONLINEAR CHARACTERISTICS L. J. CVETICANIN Faculty of Technical Sciences, V. Vlahovica 3, 21000 Novi Sad Yugoslavia, and Fakultet tehni~kih nauka, V. Vlahovica 3, 21000 Novi Sad Abstractmln this paper the free vibrations of a textile machine rotor which rotates with constant angular velocity are considered. The rotor consists of a weightless shaft and a disc with variable mass. The force in the shaft is assumed to be nonlinear. Mass of the disc is varying due to the winding up of the textile band. Severe vibrations occur due to the mass varying. When the ttonlinearity is small and the variation of the mass is a function of "slow time," the vibrations can be denoted not only numerically but also analytically by use of the multiple scales method. The results are compared.
INTRODUCTION
MATHEMATICAL MODEL
In this paper[l] the vibrations of a textile machine rotor on which the textile band is winding up with constant linear velocity are analysed. In that case the winding up is regular and the force in the textile band is constant. But to get constant velocity, it is necessary to connect the rotor with a variator of velocity. This machinery makes the system more complex, complicated and expensive. Because of that, in everyday use, the textile band is winding up with constant angular velocity. In the present paper the vibrations in the case of winding up of textile band with constant angular velocity are analysed. The model of a textile machine rotor which is a very simplified one compared to the real system is shown in Fig. 1. The rotor is idealized by a rigid disc supported on a horizontally mounted massless shaft. The disc is set in the middle of the shaft between the bearings. The bearings may be isotropic and stiff compared with the shaft and the same applies to the supports. Mass re(t), radius R(t) and the mass centre position r(t) of the disc are varying during the time and are functions of time t. In the case of winding up of the first layer of band these functions are, as it is shown by A. P. Bessonov[2],
For such a model using the principle of impulse the mathematical model is
m(t) = mo + R~pLhfit, R(t) = ~/RZo + Rlhfit/~r, r(t)-
2R2pLh fit m(t) sin 2 '
(1) (2)
~d2rsl
re(t) [ dt 2 + [ I x ( ~ × r) + 211 dr d2r] x ~-~ + dtEj = F + Fr,
with the symbols rsl, the position of the centre of the rotating disc relative to So; r, the position of the mass centre relative to the centre of the rotating disc $1; F, the active force; and Fr, the reactive force. The reactive force Fr is as shown in Fig. 2. dm
Fr = "-i--(Vat -- Va), Ot
F~-
dm dR dt dt ro.
The vibrations occur in Soxy plane.
(6)
Equations (4) and (6) are projected on the axes of inertial frame Soxy. Substituting the complex deflection z = x + iy and complex force Ze = Xe + lYe, it is
rd2z + ifi2r eiflt/2
m(t) L--~
h/2.
(5)
where vat = 1~ x Rro + drsl/dt + dR/dt ro is the absolute velocity of band at the moment of winding up, va = f l x Rr0 + drsl/dt is the absolute velocity of the disc at the moment of winding up or
(3)
where Ro is radius of the disc without band, mo is mass of the disc without band, p is density of the band, h is thickness of the band, L is the width of the band, fl is the angular velocity and R~ = Ro +
(4)
d 2 r eif~t/2
- i~
drt + 2fl-~t ]
eiflt/2
dm dR = Ze
dt dt i,
(7)
where i = V ~- 1 is the imaginary unit, x, y are the coordinates of rotating centre of the disc, and Xe, Ye are the projections of the force on the Sox and 29
30
L. J. CVETICAN1N L/2
Now, substituting eqns (3) and (8) into eqn ~7), it is
L/2
m(t)-~
+ b,z =
- iF2
F,
]zl,
tz],-~,O
, ~ - ~)
- ~,
z
\ I
~y
dm dR i _ 2 i R2 pLh dm em t dt dt re(t) dt
Fig. 1. The model of the rotor.
+ ( 2 RzI~ p L h \dt/(dm'~2
s~
9R2 "-LhD'2) ~ eiat
R~ pLh (dm'] 2 R~ p t h a 2 - 2 m--m-'~ \ d t / + 4
(10)
For computational reasons, we introduce the following dimensionless parameters:
•/•00 7
Oa0 =
Z
12"/
,Z= 7,k-R,pL
~, = Ut
R,h
h
, T = toot,
p
R,
--tOo ' I* = rrR--'~ ' y = --pl 'p = -l- '
Fig. 2. The position vectors and velocities.
c,kk cxk E l = t~2 'C2=c2k"r=DT'C3-----'tOo
Soy axes. The complex force Z~ in the shaft is after Bolotin[3]:
x 1
(11)
y l
C 4 = c4ko)0, X = -7 , Y = T •
[( d,z,
Z~ = - b,z + z F, - iF2
(
)
Izi,---~,fl
[zl,-~,a
- + - +
)]
,
The dimensionless differential equation of vibration is (8)
d2Z 9 2 (1 + y ~ ' r ) 7~ + Z = - -~p~,lxye ifhr
where +1 F,
(
t
Izl,--dT-,a-+
[ (dlzl'~ 2 +
+ Cz 3 \ - - - ~ ]
[z
2
~p",ytx
- I x Z l~:,
~
Z
(,
[z(fl
- •)2
- i~
]
t z l, ---g-r- , n ,
dlzl
dtzt ( ~
+ t z I~
+ c, - +):
I,", -
- +
tx2i [2Py2l~2 e in'r
+ 3 c3 l z l ~
,,~T
=~,lzl 2
[ - ~I (dlzl] ~ \-'-d-7-/
],
a~ y
"~
+ t*3 2 py3f~2 (emir _ 1)
(12)
(1 + ~ly'r) 2 (9)
where
F2 I zl, T , ~ - + x
x
dlzl -~2c21z1~
=m-+) C3tZ
I: +
~, (IzI, dZ______~I I ' d~jT
[ [z12(f~-+)2+
+ C2 +
where lz]
-~)=C']Z]2
C4
= (x e + y2)U2 and q~ = a r c t a n y / x , c,,
C2, C3, C4 are the coefficients of nonlinear terms and b~ is the coefficient of linear term.
3 \
dT /
+ Izt(fh
OIZI [ 1 3C3 IzJ---d-~- + 6 4 ~ - ' ~ dlZl
+ I zl ~
(n, - 6):,
]
- a3)
{dlZl~ 3 dZ
]
(13)
31
Vibrations of a textile machine rotor ~z(
Hence eqn (14) becomes
I Z I ' d l Z [ d T , On - a3)
032 02Z
02Z
[-120)
+~L
= (n, - +) {2 C2 I / I Z-----j d dT I
=-~c°2z[~;'( + C 3 [Z [2 + C4 [ I Z 12 (hi (dlZl)~]) -9)5+
\
dT ] J
00) 0Z|"1
o7o0 + ~ j
Ix2 02Z + 0.)2z +
-07 -z
, n,-+)
I Z l' d l Z
tzl'dlZ-----JdT
-i~;2(
, fh-+)].
(17)
Z = Zo(~b, 7) + Z~(cb, 7) + " ' .
(18)
'
Z can be expanded,
where v = arctan Y/X. PARAMETER ANALYSES
In the case when a textile band is winding up on, the rotor the values of dimensionless parameters are I~ = 2 × 10-4-2 × 10 -2 , ~/~ 0.2, p ~ 10. The Ix is a small parameter and the 7 is regarded as slow varying time. For these numerical values of the various parameters, the coefficients of nonlinear terms are ten or more times smaller than the coefficient of linear term. Hence we can say that the nonlinearities are small. The last two terms on the right side of eqn (12) can be neglected as small values of second order. The dimensionless angular velocity is varying in the interval ll~ = 0.1-2. For fh = 0.1-0.5 the value of the term pfl2/4 is the same order as the Ix and the first two terms of the right side of eqn (12) can be neglected, too. Now the differential equation of motion is
dzZ
Substituting (18) into (17) and equating coefficients of the same powers of Ix, it follows: ~o:
02Zo
(19)
&b2 + Z o = 0 ,
IxI: 0)2(~) ~02Za + 0)2(7) Z, 00) OZo a7 06
£Zo 070~b
-- 20) - -
- JZo [',
- i~ (Izl,
(,
(20)
/l'dl/IdT
, l), - + )
d[ZI dT
n,-+)] '
The general solution of (19) can be assumed in the complex form:
dT 2 + 0)2(7) Z = i~to2(r) Z Zo = X o + iYo, x [~;'(
Xo = A(7)e t* + 3 ( 7 ) e -i*,
' Z ' ' ]Z'--''~dT d , ~1-~)-i~2
(21)
Yo = B(7)e i* + B(a-)e -i't' x ( [ZI'dIZ'dT ,,,-a3)]
,
(14)
where 0)2(7) = 1/1 + fh~7. Equation (14) can be calculated not only numerically but also analytically.
A(7) and B(7) will be determined by eliminating secondary terms in (20). Substituting (21) into (13) and (20) the secondary terms can be eliminated. Let us assume that A and B have the polar form:
a
ANALYTICAL SOLUTION
A = ~ e i~,
Equation (14) can be solved by using the method of multiple scales[4]. An expansion for the solution in terms of the two scales ~b and 7 is shown, where
d,l,
--
dT
= 0)(7).
(15)
In terms of these variables, the time derivatives become
a
B = - i ~ e i~,
(22)
where a and 13 are real functions of T and separating real and imaginary parts, we have -do -+ dr
(1
do)
0)2~)
a3
X [C4 (20) 4 - -0)"3 5 - + ~3 o)2n'~ )
d 0 0 -1- ..., = 0)(7) ~ + Ix -07 d-~
(23)
d2 02 o5 o0)(,) o] Ix 2 0 ) ( ~ ) ~ + dTz - 0)2('0----~ 07 0cb J 0+, + 02 + Ixz_..; + ....
or"
d--7 = o)a: (16)
-
- Y + fh0))C2]. (24)
32
L. J. CVETICANIN Using eqns (24) and (25) and the initial conditior~ 13(0) = [3o, the 13 can be denoted. The amplitude as a function of time 7 Ibr dit~ferent values of the angular velocity ~h is given in Fig. 3 for the case of winding up of the first layer of textile in the time interval 0 ~ T ~ 2 v / ~ . I'/, 1 =0.1
NUMERICAL SOLUTION
~i=0.3 •Q t = 0 . 5
Fig. 3. The amplitude as a function of 1~ and T.
The differential equation (14) can be solved also numerically by the use of the subroutine NIODES[5]. The analytically calculated amplitude aA and numerically calculated amplitude a s as functions of dimensionless time T for different values of the par a m e t e r l'lj are plotted in Fig. 4.
CONCLUSION aN
D,t: 0.1
~
.~,i : 0.5 aN
~T Fig. 4. The amplitudes calculated analytically aA and numerically aN as functions of fir and T. The differential equation (23) for the initial condition a(0) = ao has the following solution: a = (1 + "r)(1 - 2C41~jz)/4 {2(1 + "r) - C4fl~ [ -
2 C4(1 + T)-1/2 REFERENCES
X (½ @ C4~~2) - i @ ( 6 3 -~- 3 64~~12)
× (1 + r) 1/2 (1 - 2 C4112) - I - (C31)j + C41q 3) (1 + r ) ( 2 -
F r o m the a b o v e m e n t i o n e d it can be c o n c l u d e d that the difference b e t w e e n analytical and numerically calculated amplitudes is negligible, that the amplitude of vibration has the t e n d e n c y to d e c r e a s e during the time and that the decrease is faster as the angular velocity of rotating is larger, for this kind of rotor balancing is not necessary because the vibrations are d a m p e d during the time. In this p a p e r only the case of winding up of the first layer of the band is analysed. F o r the other layers, the differential equations have the same form. But the values in the equations for mass, radius and mass centre variation are not the same because the mass and the radius of all previously full layers must be taken into account. The starting values for winding up of the next layer of the band are the same as the values at the m o m e n t when the p r e v i o u s layer is full.
2 ~ 2 C4) - 1 ] + a o 2 + 4 C4(½
@ C4~-~2)-I _ ( C 3 .~_ 3 C4~'~12)
x (~ - C 4 f ~ ) -~ + (C3f~ -- C 4 ~ l 3) (l -- C 4 ~ 2 ) - i } - i / 2
(25)
1. L. J. Cveticanin, The oscillations of a textile machine rotor on which the textile is wound up. Mechaniam and Machine Theory, to be published. 2. A. P. Bessonov, Osnovii dinamiki mehanizmov s peremennoj massoj zvenjev. Nauka, Moskva (1974). 3. V. V. Bolotin, Dynamic Stability o f Elastic Systems. Holden-Day, San Francisco (1964). 4. A. H. Nayfeh and D. T. Mook, Nonlinear oscillations. Wiley, New York (1979). 5. IBM Systetrff360, Fortran IV library Subprograms (C28-6596-2).
NICHTLINEARE SCHWINGUNGEN EINES TEXTILMASCH1NEN-ROTORS K u r z f a s s u n g ~ I n dieser Arbeit werden die freien Schwingungen eines Textilmaschinen-Rotors
betrachtet, der sich mit konstanter Winkelgescchwindigkeit bewegt. Dos Modell des Rotors besteht aus einer masselosen Welle und einer Scheibe mit ver~nderlicher Masse; die an der Welle angreifende Kraft ist nichtlinear. Die Masse der Scheibe verfindert sich infolge des Aufwindens des Textilbandes. Durch diese Massen~inderung entstehen Schwingungen. Wenn die Nichtlinearit~it klein und die Massen~inderung eine Funktion der "langsamen Zeit" ist, k6nnen die Schwingungen sowohl numerisch als auch analytisch berechnet werden. Die Ergebnisse weden verglichen.
0094-114X186 $3.00 + .00 © 1986 Pergamon Press Ltd.
Printed in Great Britain.
THE VIBRATIONS OF A TEXTILE M A C H I N E ROTOR WITH NONLINEAR CHARACTERISTICS L. J. CVETICANIN Faculty of Technical Sciences, V. Vlahovica 3, 21000 Novi Sad Yugoslavia, and Fakultet tehni~kih nauka, V. Vlahovica 3, 21000 Novi Sad Abstractmln this paper the free vibrations of a textile machine rotor which rotates with constant angular velocity are considered. The rotor consists of a weightless shaft and a disc with variable mass. The force in the shaft is assumed to be nonlinear. Mass of the disc is varying due to the winding up of the textile band. Severe vibrations occur due to the mass varying. When the ttonlinearity is small and the variation of the mass is a function of "slow time," the vibrations can be denoted not only numerically but also analytically by use of the multiple scales method. The results are compared.
INTRODUCTION
MATHEMATICAL MODEL
In this paper[l] the vibrations of a textile machine rotor on which the textile band is winding up with constant linear velocity are analysed. In that case the winding up is regular and the force in the textile band is constant. But to get constant velocity, it is necessary to connect the rotor with a variator of velocity. This machinery makes the system more complex, complicated and expensive. Because of that, in everyday use, the textile band is winding up with constant angular velocity. In the present paper the vibrations in the case of winding up of textile band with constant angular velocity are analysed. The model of a textile machine rotor which is a very simplified one compared to the real system is shown in Fig. 1. The rotor is idealized by a rigid disc supported on a horizontally mounted massless shaft. The disc is set in the middle of the shaft between the bearings. The bearings may be isotropic and stiff compared with the shaft and the same applies to the supports. Mass re(t), radius R(t) and the mass centre position r(t) of the disc are varying during the time and are functions of time t. In the case of winding up of the first layer of band these functions are, as it is shown by A. P. Bessonov[2],
For such a model using the principle of impulse the mathematical model is
m(t) = mo + R~pLhfit, R(t) = ~/RZo + Rlhfit/~r, r(t)-
2R2pLh fit m(t) sin 2 '
(1) (2)
~d2rsl
re(t) [ dt 2 + [ I x ( ~ × r) + 211 dr d2r] x ~-~ + dtEj = F + Fr,
with the symbols rsl, the position of the centre of the rotating disc relative to So; r, the position of the mass centre relative to the centre of the rotating disc $1; F, the active force; and Fr, the reactive force. The reactive force Fr is as shown in Fig. 2. dm
Fr = "-i--(Vat -- Va), Ot
F~-
dm dR dt dt ro.
The vibrations occur in Soxy plane.
(6)
Equations (4) and (6) are projected on the axes of inertial frame Soxy. Substituting the complex deflection z = x + iy and complex force Ze = Xe + lYe, it is
rd2z + ifi2r eiflt/2
m(t) L--~
h/2.
(5)
where vat = 1~ x Rro + drsl/dt + dR/dt ro is the absolute velocity of band at the moment of winding up, va = f l x Rr0 + drsl/dt is the absolute velocity of the disc at the moment of winding up or
(3)
where Ro is radius of the disc without band, mo is mass of the disc without band, p is density of the band, h is thickness of the band, L is the width of the band, fl is the angular velocity and R~ = Ro +
(4)
d 2 r eif~t/2
- i~
drt + 2fl-~t ]
eiflt/2
dm dR = Ze
dt dt i,
(7)
where i = V ~- 1 is the imaginary unit, x, y are the coordinates of rotating centre of the disc, and Xe, Ye are the projections of the force on the Sox and 29
30
L. J. CVETICAN1N L/2
Now, substituting eqns (3) and (8) into eqn ~7), it is
L/2
m(t)-~
+ b,z =
- iF2
F,
]zl,
tz],-~,O
, ~ - ~)
- ~,
z
\ I
~y
dm dR i _ 2 i R2 pLh dm em t dt dt re(t) dt
Fig. 1. The model of the rotor.
+ ( 2 RzI~ p L h \dt/(dm'~2
s~
9R2 "-LhD'2) ~ eiat
R~ pLh (dm'] 2 R~ p t h a 2 - 2 m--m-'~ \ d t / + 4
(10)
For computational reasons, we introduce the following dimensionless parameters:
•/•00 7
Oa0 =
Z
12"/
,Z= 7,k-R,pL
~, = Ut
R,h
h
, T = toot,
p
R,
--tOo ' I* = rrR--'~ ' y = --pl 'p = -l- '
Fig. 2. The position vectors and velocities.
c,kk cxk E l = t~2 'C2=c2k"r=DT'C3-----'tOo
Soy axes. The complex force Z~ in the shaft is after Bolotin[3]:
x 1
(11)
y l
C 4 = c4ko)0, X = -7 , Y = T •
[( d,z,
Z~ = - b,z + z F, - iF2
(
)
Izi,---~,fl
[zl,-~,a
- + - +
)]
,
The dimensionless differential equation of vibration is (8)
d2Z 9 2 (1 + y ~ ' r ) 7~ + Z = - -~p~,lxye ifhr
where +1 F,
(
t
Izl,--dT-,a-+
[ (dlzl'~ 2 +
+ Cz 3 \ - - - ~ ]
[z
2
~p",ytx
- I x Z l~:,
~
Z
(,
[z(fl
- •)2
- i~
]
t z l, ---g-r- , n ,
dlzl
dtzt ( ~
+ t z I~
+ c, - +):
I,", -
- +
tx2i [2Py2l~2 e in'r
+ 3 c3 l z l ~
,,~T
=~,lzl 2
[ - ~I (dlzl] ~ \-'-d-7-/
],
a~ y
"~
+ t*3 2 py3f~2 (emir _ 1)
(12)
(1 + ~ly'r) 2 (9)
where
F2 I zl, T , ~ - + x
x
dlzl -~2c21z1~
=m-+) C3tZ
I: +
~, (IzI, dZ______~I I ' d~jT
[ [z12(f~-+)2+
+ C2 +
where lz]
-~)=C']Z]2
C4
= (x e + y2)U2 and q~ = a r c t a n y / x , c,,
C2, C3, C4 are the coefficients of nonlinear terms and b~ is the coefficient of linear term.
3 \
dT /
+ Izt(fh
OIZI [ 1 3C3 IzJ---d-~- + 6 4 ~ - ' ~ dlZl
+ I zl ~
(n, - 6):,
]
- a3)
{dlZl~ 3 dZ
]
(13)
31
Vibrations of a textile machine rotor ~z(
Hence eqn (14) becomes
I Z I ' d l Z [ d T , On - a3)
032 02Z
02Z
[-120)
+~L
= (n, - +) {2 C2 I / I Z-----j d dT I
=-~c°2z[~;'( + C 3 [Z [2 + C4 [ I Z 12 (hi (dlZl)~]) -9)5+
\
dT ] J
00) 0Z|"1
o7o0 + ~ j
Ix2 02Z + 0.)2z +
-07 -z
, n,-+)
I Z l' d l Z
tzl'dlZ-----JdT
-i~;2(
, fh-+)].
(17)
Z = Zo(~b, 7) + Z~(cb, 7) + " ' .
(18)
'
Z can be expanded,
where v = arctan Y/X. PARAMETER ANALYSES
In the case when a textile band is winding up on, the rotor the values of dimensionless parameters are I~ = 2 × 10-4-2 × 10 -2 , ~/~ 0.2, p ~ 10. The Ix is a small parameter and the 7 is regarded as slow varying time. For these numerical values of the various parameters, the coefficients of nonlinear terms are ten or more times smaller than the coefficient of linear term. Hence we can say that the nonlinearities are small. The last two terms on the right side of eqn (12) can be neglected as small values of second order. The dimensionless angular velocity is varying in the interval ll~ = 0.1-2. For fh = 0.1-0.5 the value of the term pfl2/4 is the same order as the Ix and the first two terms of the right side of eqn (12) can be neglected, too. Now the differential equation of motion is
dzZ
Substituting (18) into (17) and equating coefficients of the same powers of Ix, it follows: ~o:
02Zo
(19)
&b2 + Z o = 0 ,
IxI: 0)2(~) ~02Za + 0)2(7) Z, 00) OZo a7 06
£Zo 070~b
-- 20) - -
- JZo [',
- i~ (Izl,
(,
(20)
/l'dl/IdT
, l), - + )
d[ZI dT
n,-+)] '
The general solution of (19) can be assumed in the complex form:
dT 2 + 0)2(7) Z = i~to2(r) Z Zo = X o + iYo, x [~;'(
Xo = A(7)e t* + 3 ( 7 ) e -i*,
' Z ' ' ]Z'--''~dT d , ~1-~)-i~2
(21)
Yo = B(7)e i* + B(a-)e -i't' x ( [ZI'dIZ'dT ,,,-a3)]
,
(14)
where 0)2(7) = 1/1 + fh~7. Equation (14) can be calculated not only numerically but also analytically.
A(7) and B(7) will be determined by eliminating secondary terms in (20). Substituting (21) into (13) and (20) the secondary terms can be eliminated. Let us assume that A and B have the polar form:
a
ANALYTICAL SOLUTION
A = ~ e i~,
Equation (14) can be solved by using the method of multiple scales[4]. An expansion for the solution in terms of the two scales ~b and 7 is shown, where
d,l,
--
dT
= 0)(7).
(15)
In terms of these variables, the time derivatives become
a
B = - i ~ e i~,
(22)
where a and 13 are real functions of T and separating real and imaginary parts, we have -do -+ dr
(1
do)
0)2~)
a3
X [C4 (20) 4 - -0)"3 5 - + ~3 o)2n'~ )
d 0 0 -1- ..., = 0)(7) ~ + Ix -07 d-~
(23)
d2 02 o5 o0)(,) o] Ix 2 0 ) ( ~ ) ~ + dTz - 0)2('0----~ 07 0cb J 0+, + 02 + Ixz_..; + ....
or"
d--7 = o)a: (16)
-
- Y + fh0))C2]. (24)
32
L. J. CVETICANIN Using eqns (24) and (25) and the initial conditior~ 13(0) = [3o, the 13 can be denoted. The amplitude as a function of time 7 Ibr dit~ferent values of the angular velocity ~h is given in Fig. 3 for the case of winding up of the first layer of textile in the time interval 0 ~ T ~ 2 v / ~ . I'/, 1 =0.1
NUMERICAL SOLUTION
~i=0.3 •Q t = 0 . 5
Fig. 3. The amplitude as a function of 1~ and T.
The differential equation (14) can be solved also numerically by the use of the subroutine NIODES[5]. The analytically calculated amplitude aA and numerically calculated amplitude a s as functions of dimensionless time T for different values of the par a m e t e r l'lj are plotted in Fig. 4.
CONCLUSION aN
D,t: 0.1
~
.~,i : 0.5 aN
~T Fig. 4. The amplitudes calculated analytically aA and numerically aN as functions of fir and T. The differential equation (23) for the initial condition a(0) = ao has the following solution: a = (1 + "r)(1 - 2C41~jz)/4 {2(1 + "r) - C4fl~ [ -
2 C4(1 + T)-1/2 REFERENCES
X (½ @ C4~~2) - i @ ( 6 3 -~- 3 64~~12)
× (1 + r) 1/2 (1 - 2 C4112) - I - (C31)j + C41q 3) (1 + r ) ( 2 -
F r o m the a b o v e m e n t i o n e d it can be c o n c l u d e d that the difference b e t w e e n analytical and numerically calculated amplitudes is negligible, that the amplitude of vibration has the t e n d e n c y to d e c r e a s e during the time and that the decrease is faster as the angular velocity of rotating is larger, for this kind of rotor balancing is not necessary because the vibrations are d a m p e d during the time. In this p a p e r only the case of winding up of the first layer of the band is analysed. F o r the other layers, the differential equations have the same form. But the values in the equations for mass, radius and mass centre variation are not the same because the mass and the radius of all previously full layers must be taken into account. The starting values for winding up of the next layer of the band are the same as the values at the m o m e n t when the p r e v i o u s layer is full.
2 ~ 2 C4) - 1 ] + a o 2 + 4 C4(½
@ C4~-~2)-I _ ( C 3 .~_ 3 C4~'~12)
x (~ - C 4 f ~ ) -~ + (C3f~ -- C 4 ~ l 3) (l -- C 4 ~ 2 ) - i } - i / 2
(25)
1. L. J. Cveticanin, The oscillations of a textile machine rotor on which the textile is wound up. Mechaniam and Machine Theory, to be published. 2. A. P. Bessonov, Osnovii dinamiki mehanizmov s peremennoj massoj zvenjev. Nauka, Moskva (1974). 3. V. V. Bolotin, Dynamic Stability o f Elastic Systems. Holden-Day, San Francisco (1964). 4. A. H. Nayfeh and D. T. Mook, Nonlinear oscillations. Wiley, New York (1979). 5. IBM Systetrff360, Fortran IV library Subprograms (C28-6596-2).
NICHTLINEARE SCHWINGUNGEN EINES TEXTILMASCH1NEN-ROTORS K u r z f a s s u n g ~ I n dieser Arbeit werden die freien Schwingungen eines Textilmaschinen-Rotors
betrachtet, der sich mit konstanter Winkelgescchwindigkeit bewegt. Dos Modell des Rotors besteht aus einer masselosen Welle und einer Scheibe mit ver~nderlicher Masse; die an der Welle angreifende Kraft ist nichtlinear. Die Masse der Scheibe verfindert sich infolge des Aufwindens des Textilbandes. Durch diese Massen~inderung entstehen Schwingungen. Wenn die Nichtlinearit~it klein und die Massen~inderung eine Funktion der "langsamen Zeit" ist, k6nnen die Schwingungen sowohl numerisch als auch analytisch berechnet werden. Die Ergebnisse weden verglichen.