General Investigation We shall find the solution o f Eq... The steady-state response can be found by setting a, xp equal to zero in equations 2.4... The characteristic equation o f the s
Trang 1P A R A M E T R IC R ESO N A N C E O F 4th O R D E R IN A N O N L IN E A R V IBRATING
S Y S T E M U N D ER T H E IN F L U E N C E O F F R IC T IO N S
NGUYEN VAN D A O (HANOI)
I Introduction
A m o n g the parameters having considerable influence on parametric oscillations it
is worth mentioning the nonlinear frictions [I] This paper is concerned with the study o f the influence o f som e kinds o f frictions on the nonlinear oscillations described by the equation with the cubic term at modulation depth:
(1.1) X -f ( 0 2x 4- eịc.x 4- (i.Y3) COS y t -f f a.\3 + e R (.x , x ) = 0 ,
in the resonant zone o f 4th order:
where e9 (O, c, d, a are constants, R(.x, v) is a function characterizing the frictions con
sidered.
It h a s tu r n e d o u t th a t th e p a r a m e tr ic o s c i ll a t io n s a n d th e ir s t a b ility in th e r e s o n a n t
z one (1.2) are essentially dependent on the friction forces But, the resonant curves in the
c a s e s o f lin e a r f r i c t i o n , d ry fr ic tio n a n d th e tu r b u le n t o n e are n o t d iffe r e n t in q u a lity
2 General Investigation
We shall find the solution o f Eq (1.1) in the form
X = ứsinớ,
0 = r t + v ,
where a and y} are new unknown variables which will be determined later By substituting (2.1) into (1.1) and transforming it, we obtain the following equations for a and ip:
y a = — 4 e [ Ax + (c.Y4-i/.Y3)COSy/ -f (XX2 -I- R ( x , i ) ] c o s ỡ ,
yaỳỉ = 4 8 [ Ax + (cx + d x 3) COS y t -f- (XX3 4- R ( x , x)] sin 0 ,
Trang 24 3 6 Nguyen van Dao
which is the standard form for the method o f averaging [2] Following this method
let
where Ỡ* and y* satisfy (he averaged equations:
= — 4e
(2.4)
G(ơ > y) ~ s*n
4y-ỵơỳ; = 4t'
here for short we om it the subscript (*) and
y— a-f ~ art3 -f P( a, y ) + rt3 COS4y»
(2.5)
2n
G ( a , y ) = - ~~ Ị cosớ* /? Ị í7 S Ìiiỡ , ^ tfcosỡjí/ớ,
In
P ( a , y ) = I sin0 • / W ứ s i n ỡ , flcosOjrfO.
0
The functions /1 and (Ị) in (2.3) will be obtained from the right-hand sides o f (2.2) (sec
Sect 3).
The steady-state response can be found by setting a, xp equal to zero in equations
(2.4) T h u s :
d ,
"16 ^oSin4v*o = G ( a 0 , y ) ,
(2.6)
- a0 c o s 4 V o = y f l o + y « « 0 + ^ ( « 0 , v) ■
Eliminating the phase 7*0 yields
(2.7)
where
(2.8) W( a 0 , y ) - G 2(a0 , }') +
W'teo, ỳ) = 0,
2 a° + 8 0 + (a° ’ (16)* •
T o study the stability o f stationary oscillations, the variational equations based on tlie
Stationary solution will be considered By setting in (2.4)
a = ơ0 + ôa, y) ~ ipo + ồyỉ,
where Ỗa and ôy) are small perturbations, and using the steady-state equations (2.6),
we have
d
(2.9)
y () a = 4e
cit
y a 0 - ỗ f = 4e
r", \ , 3í/ứổ ■ ,
G ( «0 y ) + - s i n 4 Vo
y + “ Oftfo + p (a0 , 7 ) + ~ - cos4^ o ô a ~ d a l s i n 4 y v ' V ’| \ , 1
here a prime denotes partial differentiation with respect to o0
Trang 3The characteristic equation o f the system (2.9) is o f the form:
The stability condition is given by the Routh-Hurwitz criterion, that is
In Fig I the branches o f unstable solution are represented by dotted lines in which the conditions (2.10) are Violated It should also be noted that the boundary between the
stable and unstable regions shows the locus o f vertical tangency ( d w / d a 0 — 0) with the
resonant curves.
3 L inear Friction
Let us consider the case
where h is a positive constant Ill this case the equations (2.4) becom e
I h ‘to2 : A \
ya = e a \ - y y + -4- sin4yj I,
(3.2)
L i 3 2 d a 2 _ A \
yaỹ) = ea 12/1 -f — <xa H— - COS 4 y ) j,
and the relation (2.7) is o f the form:
(3.3) (/?2 - Z ) > ắ + 2 / ? ( l - / < 2K + ( l - /u2) 2 + / / V = 0 ,
in which
From (3.3), we obtain
(3.5) flg = — [ P f r * -1 )+ Ị '7 ; > ( 1 - V ) 2 i ( / > j - m v |
-It should be pointed out that the most interesting case is that for which p 2 > D 2 This case is show n in Fig 1 by plotting the Eq (3.5) for ft = ± 1 • 10“ 3, D = 3 • 10~3 with various values o f h : H = 0 (straight lines “0 ”), H 2 — 5* 10~4 (curve 1), H 2 — 10~3
(curve 2).
T o find the refinement o f the first approximation we represent (2.2) in the following expanded form
r
ha d a 3
sin 4^ 4- y c o s 2 ớ + M -1- 2 1
Trang 4438 Nguyen van Dao
( r + ^2 j si 11 (2Ơ — 4 v ) — si l l 4Ỡ + “ !<• + ‘y - j s i n ( 6 0 - 4 y ) - ^ s i n ( 8 f l - 4 y > )
iA/j 16e
(10 = V 2 ' JZ-G+ 0 aơ3 + , cos4ĩ/»— ( I -f atf2) c o s 2 0 + 7 ơysin2ớ
— 4 " ( r + ^ứ2) c o s ( 2 0 - 4 y ) - ị - g ếJ3cos4ỡ + 2 Ịr + 4 í/ứ2| c o s ( 4 0 - 4 y )
— ^ (c + <Vtf2) c o s ( 6 ỡ —.4y) + rá*r-cos(8ớ — 4?/>)
Fi g I
Since Ơ and V’ do not v a r y very rapidly, we shall integrate the right-hand sides o f (3.6)
as if a and w were constant We thus obtain
n - n - ì 6 f
(3.7) a = a 0
-y
16e
v> = Vo +
y s in 2 0 — + y o ^ j c o s 2 0 + - y | c + I COS ( 2 0 - % , ) + ~ cỏ cos 4 0 — | c + d ~ ~ j cos(60-4y>„) + cos(80-4y>o)
a y 2 - - y COS 2 0 - (zl + 0íữ ẳ)sin2ớ - ơ-° (c + í/ữ ẩ )sin (2 0 -4 y io)
+ ~ ứẵ sin4ớ + — ■ Ịc + sin(4ỡ - 4y>o) - - | ị - (c + í/ữẳ) sin (60 - 4 Vo)
+ -^ -s in ( 8 ớ -4 Vo)
Trang 5T h e new refined first a p p r o x i m a t i o n will be given by
4 fte r s o m e sim ple c a lc u la tio n s this leads to
4 Coulomb Friction
Let
Ĩ th e n o n l i n e a r fric tio n c o n s id e re d T h e a v e ra g e d e q u a t i o n s (2.4) n o w a re o f th e f o r m
k2)
y_
4
y - _ 2/i0 d a 3 \
-— a ỳ = £ I -— -— h a a + —j - £ - C O S 4 ^ I , fl > 0
T h e a m p l i t u d e s o f s t a t i o n a r y o s cillatio n s a re d e t e r m in e d by th e e q u a t i o n :
k3) D 1a ị — a ị { \ —fx2 + ậ a ị ) 2 — H ị = 0 , / / 0 4e/i°
T h e c u r v e s 3, 4 p r e s e n te d in F ig 1 a re o b t a i n e d by p l o t tin g e q u a t i o n (4.3) for ệ
5 Turbulent Friction
Let us c o n s id e r n o w t h e case
u b s tit u tin g (4.1) in to (2.4) gives
• _ A i 11 2 y 2 2 _ ciaĩ ■ A , \
y i = ~ 4 e 1 2^ - “ - 16 s in 4 v '
5.2)
he equation (2.7) f o r the am plitude o f stationary oscillations then becomes
Trang 6440 Parametric resonance o f 4th order in nonlinear vibrating system
T h e resonant curves for /3 = ± 7 • 10 - 3 , D = 3 • 10 3 are presented in F ig 1 w ith H 2 = 0 (straight lines “0 ”), H \ = 5 • 10 “ 5 (curve 5), H ị = 10- 4 (curve Ố).
t
Conclusion
T he results o b ta in ed s h o w that
1) Param etric o sc illa tio n s and their sta b ility in th e reson an t zo n e (1 2 ) essen tia lly
d epend on the fr ictio n
2) T h e r e s o n a n t c u rv e s in th e cases o f lin e a r fric tio n , d r y fric tio n a n d th e t u r b u l e n t
o n e are sim ilar in q u ality.
3) W ith the g ro w th o f friction the resonant curves m o v e up.
References
1 G Schmidt, Parametererregte Schwingungen, Berlin 1975.
2 H H BoromoốoB, K> A MHTPonojiLCKHH, AcuMnmomimecKue Memoỏbi e meopĩiu HCAUHCÙHbix KOJie-ôamíũy M o c K B a 1963.
S t r e s z c z e n i e
PARAMETRYCZNY REZONANS CZWARTEGO RZẸDU w NIELINIOWYM
DRGAJACYM UKLADZIE POD WPLYWEM TARC
w artykule rozpatrzono w ptyw tare n a p a ra m e try c z n y re z o n a n s czwartego rzẹdu, kiedy m a miejsce zwiạzek (1.2), w nieliniowym ukfadzie z uwzglẹdnieniem czionu trzeciego stopnia przy gỉẹbokosci modulacji (1.1) Okazuje siẹ, ze drgania parametryczne i ich statecznosc w obszarze rezonansu znacznie zaỉezạ od tare Jednak krzywe rezonansowe w przypadku liniowego, suchego i turbulentnego tarcia ja- kosciowo sạ podobne do siebie i ze zwiẹkszeniem tare rezonansowe krzywe podnoszạ siẹ.
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HANOI POLYTECHNICAL INSTITUTE (DHBK)
FACULTY OF MATHEMATICS AND PHYSICS
Received April 16, 1976.