On the phenomenon of parametric resonance of a nonlinear vibrator under the action of electromagnetic force

INSTITUTE
FUNDAMENTAL
TECHNICAL
RESEARCH
POLISH
ACADEMY
CFSCIENCES
Proceedings
of Vibration
Problems
Quarterly Vol. 13 No. 3 Warszawa 1972
rwOVE WYDAWNICTWO NA UKO WE — POLISH SCIENTIFIC PUBLISHERS
Proceedings o f Vibrations Problem s, 13. 3. pp. 281 -2 91 , 1972.
In stitu te o f F und a m enta l Technical Rese arch, Polish A ca de m y o f S c ie n ces
ON TH E PHENOMENO N OF PA RA MET RIC RESONANCE OF A N ON LIN EA R VIBRATO R
UND ER TH E ACTION OF ELE CTR OMAG N ET IC FORC E
This paper deals with the parametric resonance of a nonlinear vibrator in an electro
mechanical system (Fig. 1). It was found that in the system under consideration, in addi
tion to ihe well-known principal resonance which takes place when the frequency of
the electrical circuit is equal to half the natural frequency of the vibrator, there exists
an interval of frequencies, in which the mass m vibrates strongly with the frequency of
the electrical circuit (parametric resonance). Tile results of an experiment conducted
arc in conformity with the theoretical analysis.
Let us consider an electromechanical system, the scheme of which is represented in
Fie. 1. The vibrator consists of a cantilever beam and a block of mass /?/, which is made
of magnetic material. The elastic force of the beam is assumed to have a nonlinear char
acteristic
where A is the coordinate of the mass m. The inductance u .x ) of the coil in the electrical
circuit R-L-C is a variable quantity depending on the position A of the mass m swinging
N G l'V E N V A N DAO (HAN OI)

1. Introduction • Fquntions of Motion Ị
/ = -A\v-/?,.v3,
0
F ig . 1.
under the coil L and thus modifying the reluctance of its magnetic circuit. We assume
that the function L(x) may be expanded in a power of x:
L (x ) = L 0( l - a , * + a 2x 2 + a 3A;3 + ) .
282 sguyen van Dơo
In the present paper, we take
L(x) = L0(\ - a1.r + a2 A':).
The equations of oscillation of the system under consideration can be written as:
-ị- (Lq) + Rq+ ~ q = £sinr/,
dt c
( u ) , V
■■ I I Q 3
_
Jy2 ^
m x-tỉĩỵ.x-tkx + pi A*-5 = -sỹ-q - - ,
where £sini7 is* the external periodic excitation applied to the circuit, I q2 ~ is the force
2 tf.Y
acting on the mass /7/, and ĨỊ is the electrical charge.
Substituting the expression for Lự) into (1.1) and introducing the notations:
k rìQ 1 _ J 1.
CJ5 = , CJ°2 = — T = rjf. ậ = , h
1)1 L0 c k mo
(1.2)
Jf= *- = e = -■SI — , / » -ie » tr — 2 ’
Lq(0 0) >') L00J
we obtain the following equations of oscillation of the electromechanical system under
consideration:

d2q da _ ế/:f/ _ - rfv da
_ -\-Q2q + &~_-+ ( — + ( - * ! + 2y.2x ) ~ ~ j- = esiny r ,
f/r i/r dĩ2 dr (ir
(1.3)
c/2.t . <7.v - . L0 ,,
.^ * + x + * “ + ^ - g V (-» ,+ 2 * v > ,
where the stroke (') designates the derivative with respect to the variable T.
If we confine ourselves to investigation of the linear problem, then Eqs. (1.3) become:
d2q dq
~ r + Q-q + % = esinyr,
r/r
(1.4)
d2,Y i/.v o
dr* * dr = ~ 2 /fj
The forced oscillation of the electrical circuit, clearly, is
<7 = e*sin(/T— Ố).
Substituting this value of q into the second Eq. (1.4), we obtain:
-J~T + x + h — — y e*2[\— cos( 2yr — 2 Ố)].
r/r2 dr 4m
From this equation we see that in the system (1.4) the mass m always vibrates with fre
quency 2y, which is double the frequency y of the electrical circuit.
As far as is known, under certain conditions in a nonlinear system (1.3) there exists,
in addition to the oscillation refered to, also oscillation of the mass m with the frequency y
of the electrical circuit. This oscillation is called the parametric oscillatiòn.
On the phenomenon o f parametric resonance o f a nonlinear vibrator
283
It is to be noted that the linear oscillation of the mass m in a similar electromechanical
system has been investigated by certain researchers. Thus, using the small parameter
method, L. G. Etkin considered the linear equation of the motion of the mass m in the
form [1 ]:

A + 2 /lx + {or — 2//(£>0 + £i s in T — è 2 c o s 2 r ) ] . v = ịi(bo + bi s in T — £ 2 c o s 2 r ) .
Assuming L(.v) = z.0 e x p (-ó.x-r o2 .v:). Ộ = 0 (linear spring), A. E. Chesnokov [2]
investigated the principal resonance of the mass m in the case where y = \'2.
Recently, D. D. Kana [3] has investigated the parametric oscillation of the clectrical
circuit, taking L(ỵ) in the form:
where A is a constant.
Now, we return to tlie system of Fiqs. 1 1.3) considered in the present study. Assuming
that the nonlinear terms and the terms characterizing the friction forces are small in com
parison with the remaining terms, and introducing a small parameter u% we can write
Eqs. (1.3) in the form:
Bearing in mind the application of the asvmptotic method [4, 5] of nonlinear mechanics
or construction of the approximate solution of Eqs. (1.5), we ưansĩorm them into the
standard form by means of formulae which reduce q, q'y A*, x' to new variables A% ifs tì, (ị)\
L(: Y) = I 0(l + *.v),
where a is a constant. Equations of motion arc written by him in the form:
(1.5)
2. Solution
(2.1)
q = e *s in y T + /?sin<f,
q' = ye*cosyT QBcosẹ ,
x= —bị — ——rT cos2ỵr + A sinỡ,
1 —4 y
vhere
284 Sguyen van Dao
Substituting (2.1) into (1.5), we obtain:
Q -j-= -/'ổCv, .V , <?', <7 ")cos<r,
Q B -f- = ? #/)sinẹ?,
(2.3) "
y - j = 4 (y 2— l)sinớcosớ-^A'cosớ — BQ(QB COS qr + 2 y ổ * c o s y r )c o s ọ : c o s ớ ,
where

A y ~ —■ /4(1 — y2 )sin20 -f-/t/A'sinỡ BQ(QBCOScp - f 2 y e * c o s y r ) COS(f sinỚ,
0 = m ^ + ( - ai X+*ÌX* )^ L + (-* ,+ 2« ,*)-^ - a ,
A' = /i-Ệ. +ậx* - ỈỊ Ĩ L xq'2.
a X m
Considering the parametric resonance, we assume that the frequency y takes values
in t h e n e ig h b o u r h o o d o f u n ity a n d th at y a n d Q a re in d e p e n d e n t — i. c ., b etw e en th em
here is no relation of the form
nly + n1Q = 0 ,
where n1 and n2 are integers.
In the first approximation, the quantities b,(Ị>yAy\ịỉ satisfy averaging equations,
which are obtained from (2.3) by averaging their right-hand sides on t:
r, , àtí
___
H0t
L +/i 2

J dr - ~ 2~ ’
~~ = G{B,ệ,A,ỳ)>
(2 .4)
(ỈA ụ hy . 1 1 À ■ ~ ,
y -~ = - -^-CÁ sin2 y>+ ,
a T 1 2 .
yA = 4* (1 - y2 + //.d)/! + -ị-ụậA3 + cA COS 2V + ,
ClT 2. o z
w here
3/56Ỉ è2 . . , o l2 3 _ ỐỈ
c = j S r + 2 ’ = ố2 + 3^ ỉ+ 2 ^ 0 - 4 ^ 7 ’
(2'5) I I
b> = ế ^ 2e'2-
Terms not written in (2.4) will be equal to zero when 5 = 0.

T he first eq uation o f (2.4) is independent w ith respect to the rem aining equations,
from which it follows that the quantity B tends asymptotically to zero. Consequently,
On the phenomenon o f parametric resonance o f a nonlinear vibrator
285
below we shall be interested in the last two equations of the system (2.4), rejecting the
terms not written:
(2.6)
dA
(i Ĩ
hy u . -
-fỉ-ir-A + ~ cA sin2^,
yA = -ị-(l - y 2 + [Ấắ)A + + -yC/lcos2^.
This system has two stationary solutions A0 = const, If'0 = const. The first solution
of it is
(2.7) /4 0 = 0, V’o — arbitrary.
The second solution different from zero is determined as the roots of the equations:
ịỉhy = nc sin 27*0 ,
(2.8) . . 3
Hence, we obtain:
(2.9)
= -rTT
4 / y2 — 1
- J ± )/~c2 — h2y2 ,
(2.10)
sin2y.’o = — y, cos 2 Vo = ± — \/cĩ-h fỹ2,
where in both expressions (2.9) and (2.10) in every case we must take upper or lower
signs.
M
F i g . 2.
Introducing the notations

(2.11) a2 = ỗ = <4 = yuc, = ịxh,
286 sguyen van Dao
we can write Eqs. (2.9), (2.10) in the form:
Q~ = — () + — 1 ± I ^ 2y2,
(2.12)
sin2vo = ^ r 7 , cos2v o = + )ỵ I v 2- 'j r 2y2.
In Fie. 2. the resonant curve is represented for values Ổ = 0.15, # 2 = 0.15, J/i1 = 0.1
3. Stability of Stationary Solutions
The Stability of the stationary solution A0. Vo (2.8) is determined by means of the
Routh-Hurwitz criterion. Thus, we return to Eqs. (2.6) and substitute
A — A 0 ~ : , ụ' = V’o + rh
ill which Ệ and ÌỊ arc small perturbations. Substituting these values into the system (2.6)
eliminating the stationary ports using Eqs. (2.8), and retaining only first order term:
we obtain the following equation in variation:
. , . . .
y y - /^^Ocos-Vo • *)♦
(3.1)
r/?7 3 , ym , . -
y = 4 «pVl2 ■ 5-/<cv10sin2y<0 • Y).
The characteristic equation of this system is:
y2A0}.2 + [ẨcyAQsw\2ìị'0) ^ /Ẩ2cfiA3cos2y0 = 0,
or, according to (2 .8):
y2.40 /.2+fihy2A0 /.+ ~ M^oỊl - ỵ2 + — upAị^ = 0 .
Hence follows the condition of stability of the stationary solution A0,y.'0 in the fori
(^ > 0, /; > 0, /J0 > 0):
(3.2) /sỊl - y 2-r mJ + > 0.
For geometrical interpretation of this condition, we note that the equation of tl
resonant curve (2.9) can be rewritten in the form:
(3.3) W(A0t y) = \ - y 2 + - fi2(c2 - h2y2) = 0.
Consequently, the inequality (3.2) is equivalent to

dw
<3-4> t ! > 0 -
On Í he phenomenon o f par a metric resonance o f a nonlinear vibrator 287
The resonant curve li'(A0.y) = 0 divides the plane Mow'2) ‘nt0 regions, in each
of which the function IV(A0, y) has a definite sign ( -f or -). When one proceeds aloiiíỊ
a straight line, which is parallel to the axis a and cuts the resonant curve, and crosses
from the region w < 0 to the region w > 0, then, at the point of intersection of the straight
line with the resonant curve (see point XÍ in Fig. 2), the derivative cWjca is positive, and
therefore, according to (3.4), this point corresponds to the stable state of oscillation. In
the opposite case (see point A in Fig. 2), the unstable state occurs. Following this rule,
it can be seen that the branch ST is stable and. by contrast, the branch PT is unstable.
In order to investigate the stability of the solution A0 = 0 (2.7), we return to Eqs.
(2 .6 ). put tin e in them
A = £. V’ = Vo + >/-
where, for the present, y0 is an indefinite constant. We have
y - r - = y ( - / / y - R ' S Ì n 2 y ’o) • Ẹ ,
(3.5) d r 2
0 — (1 —yZTị.i - \ H-//Ccos2 ^0) •
Hence, it follows that
ccosllfo = ( y 2~ 1 -/^J)
fj.
and consequently,
csin2^0 = — Ị /72c2- ( ỵ 2- 1
Then, the first equation of the system (3.5) takes the form:
y~~- = ị[-/i* y + J /i2c2 - (y2 - 1 - aJT2]í •
The quantity Ệ will be asymptotically tending to zero, and therefore the solution A0 = 0
is stable, if
ỊÁ2h2y 2 > /r c 2- (ỵ2— 1 — ỊẦ. I)2,
(3.6) or
/u2(h2y2 — c2) + (y2 - 1 - ịầA)2 > 0 .

The left-hand side of the last inequality coincides with the free term of the quadratic
Eq. (3.3). Consequently, the solution Aq = 0 is stable in that interval, where the quadratic
Eq. (3.3) has either two positive roots, or two complex-conjugate roots, or two negative
roots. Hence, it follows that the zero solution A0 = 0 is stable if the value y2 does not
lie in the interval y] ^ y2 ^ yị of the axis y2, from which the resonant curve emerges.
In Fig. 2, the stable branches are shown by heavy lines. This representation is con
venient for the analysis of amplitudes in the system under consideration. Thus, for instance,
if we start from 7 = 0 and increase the frequency, the amplitude [of the parametric oscilla
tion (the last component in the expression (2.1) for x] follows the stable branch OST.
At point r, the amplitude drops suddenly on the lower branch PQ, and follows it from
288 Xguyen van Dao
D to Q if y continues to increase. If, however, y decreases (say from y > yH) the lower
branch will be followed up to point p, at this point there will be an upward jump PE
on the stable branch ESO.
The resultant oscillation of the mass m [see formula (2.1)] is periodic with the period
T = 2n/y and consists of two oscillating components: forced oscillation with amplitude
V | l - 4 y 2| and frequency 2y> and parametric oscillation with amplitude A0 and fre
quency y. The dependence of these amplitudes on the frequency y is represented by
diagrams in Fig. 3. From this figure it can be seen that, if we start from y — 0 and in-
Fro. 3.
crease the frequency, then, for values y smaller than ys corresponding to the point St the
mass m accomplishes the forced oscillation with frequency 2y. At the point B(yn = 1/2),
the principal resonance takes place. From y = ys to y = yD two oscillations co-exist:
forced oscillation with decreasing amplitude on branch / / / and parametric oscillation
with increasing amplitude (parametric resonance) on the stable branch ST. In the inter
val ys < y < yDi the interference of frequencies occurs. The parametric oscillation with
frequency y becomes overwhelming.
For y > yD, there exists again only forced oscillation with frequency 2y.
If we decrease the frequency y — for instance, from yQ — then, in the interval ys <
< y < yp the mass m will vibrate with two frequencies y and 2 y, and the parametric

oscillation will become overwhelming. For the remaining values of y, the mass m accom
plishes forced oscillations with frequency 2y.
4. Experimental Results
An experiment was undertaken to obtain gross verification of the analytical predic
tions. The experimental rigidity consists of a cantilever beam of stiffness k = 6.2 TV/cm
and a block of mass m = 0.00418 Ns2/cra, and, therefore, the natura Ifrequency of the
vibrator is 0) = ị k/m = 38.5 Hz.
A scheme of the instrumentation used to measure oscillations of the mass m is shown
in Fig. 4. For observation of oscillations of the mass m, we use a piezoelectric sensor
with an oscillograph. For comparison of the frequency of oscillations of the mass m and
of the electrical circuit, an oscilloscope is used. Stationary patterns-Lissajous patterns
(Figs. 5a, 6 b) are obtained.
Fig. 4.
a
W v / V W V W W V W
F ig . 5.
a b
/W ÍW W V W W W V W W W \A
Fio. 6.
[289]
290 Nguyen van Dao
Increasing from zero the frequency of external force V it was observed that, at a value
V - 19Hz the mass m vibrates strongly with doubled frequency V (Fig. 5b). This is the
principal resonance. Figure 5a shows photographs of that oscillation (curve I). For con
venience of comparison, in Fie. 5a is represented the electrical sicnal with frequency
2v = 38 Hz (curve 0).
Fig. 7.
For values V from 36 Hz to 40Hz, there exist parametric oscillations of the mass m
with frequency V (F is. 6b). The curve 7 in Fig. 6 acorresponds to oscillations of the mass m
inside the resonance (v = 30Hz); curve 2 for the value V = 3 7 Hz, and curve 3 for the

value V = 39Hz (paramctric resonance). For comparison, in Fig. 6a is represented the
electrical signal with frequency 2v = 72 Hz.
Thus the experimental results are in conformity with theoretical analysis.
I n c o n c l u s io n , t h e a u t h o r w i s h e s to e x p r e s s a p p r e c ia t io n f o r th e a s s is t a n c e o f c o l 
le a g u e s D r N g u y l n x u a n H u n g a n d P h a m h u u H u n g in e x e c u t in g th e e x p e r i m e n t a l
work.
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ZJAW ISKO PARAM ETRYCZNEGO REZONANSU NIELINIOWEGO OSCYLATORA
POD DZIALAN1EM ELEKTROMAG NETYCZNEJ SILY
w pracy rozpatrzono zjawisko parametrycznego rezonansu nieliniowego sprẹzystego oscylatora po-
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w ta s n e j o s c y l a t o r a , w y s t ẹ p u j c r o w n ie z p r z e d z i a l c z ẹ s t o s c i , w k tó r y m m a s a m in t e n s y w n i e d r g a z c z ẹ s to s c i q
e l e k tr v c z n e g o o b u o d u i r e z o n a n s p a r a m e tr y c z n y ) .
R e z u l ta ty p r z e p r o w a d z o n e g o e k s p e r y m c n t d p o t \v i e r d z a jạ a n a l i z ẹ te o r e t y c z n ạ .
p e 3 Kì V. e
flB.'IF.HIIE riA PAM ETPIIM FCK O rO PE3 0H A H C A H E .1IIH EH H O rO O CI.U I.T IH TO PA
ĨI O :i JIE i'IC T B IIE M S.lH K TPO M A fH l IT H O fl c n . l b l
B pacore paccMOTpeno HB.iemie mpjMeTpimecKoro p e 3 0 H a H c a He.nmeiiiioro yn pyroro ocmi.uiHTopa
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'3 3 a M CH LLUe MCM CO O C T D CHH an M aC TO T a O C U I I.l .lH T O pa . B b lC T \n aC T TO /KC H H T C p B a .I HaC TO T, B KOTOpO.M
i a c c a m H H TC ỈỈC ỈJBHO K o:ie Õ J ie T C fl c -‘lac T O T O Íĩ 3 / i e K T p i m e c K o r o K O H T yp a ( n a p a .M e T p ii 'i e c K i i f i p C 3 0 H a n c ).
Pe3\7ibraTbi npOBc^eniicro 3i\cnepH.\ieiiTa noATBep/K.iaiCT TcoperimecKiiii aHa.1113.
EPARTMENT OF MATHEMATICS AND PHYSICS.
POLYTECHNIC INSTITUTE, Hanoi
On the phenomenon of parametric resonance o f a nonlinear vibrator 291
Received Xovembcr 6, 1971
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