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

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 t

I N S T I T U T E

F U N D A M E N T A L

T E C H N I C A L

R E S E A R C H

P O L I S H

A C A D E M Y

C F S C I E N C E S

Proceedings

of Vibration Problems

rwOVE WYDAWNICTWO NA UKO WE — POLISH SCIENTIFIC PUBLISHERS

I n s tit u te o f F u n d a m e n ta l T ech n ica l R e se a rc h , P o lis h A c a d e m y o f S c ie n c e s

ON T H E PHENO M ENO N OF P A R A M ETRIC RESONANCE OF A N O N LIN EA R VIBRATO R

U N D ER TH E ACTIO N OF E LEC T R O M A G N ET IC FO R C 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 D A O (H A N O I)

1 Introduction • Fquntions of Motion Ị

/ = -A\v-/?,.v3,

0

F i g 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 0 ( l - a , * + a 2 x 2 + a 3 A;3 + )

282 sg u yen van Dơo

In the present paper, we take

L(x) = L0(\ - a1.r + a2A':).

The equations of oscillation of the system under consideration can be written as:

- ị - (Lq) + Rq+ ~ q = £sinr/,

m x - t ỉ ĩ ỵ x - t k x + p i A*-5 = -sỹ-q - - ,

where £sini7 is* the external periodic excitation applied to the circuit, I q2 ~ is the force

2 tf.Y

a c t i n g o n t h e mass /7/, a n d ĨỊ i s t h e e l e c t r i c a l c h a r g e

Substituting the expression for L ự ) into (1.1) and introducing the notations:

CJ 5 = , CJ°2 = — T = r j f ậ = , h

(1.2)

we obtain the following equations of oscillation of the electromechanical system under consideration:

_ -\-Q2q + & ~_ - + ( — + ( - * ! + 2y.2x ) ~ ~ j - = esiny r ,

(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:

~ r + Q-q + % = esinyr,

r/r

(1.4)

dr* * dr = ~ 2/fj The forced oscillation of the electrical circuit, clearly, is

Substituting this value of q into the second Eq (1.4), we obtain:

-J~T + x + h — — y - e*2[ \ — cos( 2yr — 2Ố)].

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 / l x + {or — 2//(£> 0 + £i s i n T — è 2 c o s 2 r ) ] v = ịi(bo + bi s i n T — £ 2 c o s 2 r )

Assuming L(.v) = z.0e 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 11.3) 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 i n y T + /?sin<f,

x = —bị — — —r T cos2ỵr + A sinỡ,

1 —4 y

vhere

284 Sguyen van Dao

Substituting (2.1) into (1.5), we obtain:

Q - j - = -/'ổ C v, .V , <?', <7")cos<r,

(2.3) "

y - j = 4 ( y 2 — l ) s i n ớ c o s ớ - ^ A ' c o s ớ — B Q ( Q B COS qr + 2 y ổ * c o s y r ) c o s ọ : c o s ớ ,

w h ere

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 * - Ỉ Ị Ĩ L xq'2

Considering the parametric resonance, we assume that the frequency y takes values

in t h e n e i g h b o u r h o o d o f u n it y a n d t h a t y a n d Q a r e i n d e p e n d e n t — i c , b e t w e e n t h e m

here is no relation of the form

nl y + n1Q = 0,

where n1 and n2 are integers.

In the first approximation, the quantities b,(Ị>yAy\ịỉ satisfy averaging equations,

w h i c h are o b tain ed f r o m (2.3) by averagin g their right-h and sid es on t :

L +/i 2 -J dr - ~ 2~ ’

~ ~ = G{B,ệ,A,ỳ)>

(2 4 )

(ỈA ụ hy 1 1 À ■ ~ ,

y - ~ = - -^-CÁ sin2y>+ ,

yA = 4* (1 - y2 + //.d)/! + -ị-ụậA3 + cA COS 2V + ,

w h e re

c = j S r + 2 ’ = ố2 + 3^ ỉ+ 2 ^ 0 - 4 ^ 7 ’

2-T e r m s n o t w ritte n in (2.4) will be equal to zero w hen 5 = 0.

T h e first eq u a tio n o f (2 4 ) is in d ep en d en t w ith resp ect to th e rem ain in g e q u a tio n s,

from which it follows that the quantity B tends asymptotically to zero Consequently,

On the phenomenon o f param etric 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

- f ỉ - i r - A + ~ c A sin2^,

This system has two stationary solutions A0 = const, If'0 = const The first solution

of it is

The second solution different from zero is determined as the roots of the equations:

ịỉhy = nc sin 27*0,

Hence, we obtain:

(2.10) sin2y.’o = — y, cos 2 V o = ± — \/ cĩ - h fỹ 2,

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

286 sg u yen van Dao

we can write Eqs (2.9), (2.10) in the form:

Q~ = — () + — 1 ± I ^ 2y 2,

(2.12)

s i n 2 v o = ^ r 7 , c o s 2 v o = + ) ỵ I v 2- ' j r 2y 2.

In Fie 2 the resonant curve is represented for values Ổ = 0.15, # 2 = 0.15, J/i1 = 0.1

3 S ta b ility of S tationary 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 Ệand ÌỊarc small p e r t u r b a t i o n s 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)

-y = 4 «pVl2 ■ 5-/<cv10sin2y<0 • Y).

The characteristic equation of this system is:

y 2A0}.2 + [ẨcyAQsw\2ìị'0 ) - ^ /Ẩ2cfiA3cos2y0 = 0,

or, according to (2.8):

y2.40 /.2+fihy2A 0 /.+ ~ M ^oỊl - ỵ2 + — upAị^ = 0 Hence follows the condition o f stability of the stationary solution A 0,y.'0 in the fori

(^ > 0, /; > 0, /J0 > 0):

For geometrical interpretation of this condition, we note that the equation of tl resonant curve (2.9) can be rewritten in the form:

Consequently, the inequality (3.2) is equivalent to

d w

-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 M ow '2) ‘ n t 0 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 c Wj c a 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 S T 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, y 0 is an indefinite constant We have

y - r - = y ( - / / y - R ' S Ì n 2 y ’o) • Ẹ ,

0 — ( 1 —y ZTị.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 / i 2c2 - (y2 - 1 - aJT2] í •

The quantity Ệ will be asymptotically tending to zero, and therefore the solution A0 = 0

is stable, if

ỊÁ2h 2y 2 > / r c 2 - ( ỵ 2 — 1 — ỊẦ I)2,

/u2(h2y 2 — c2) + (y2 - 1 - ịầ A)2 > 0 The left-hand side of the last inequality coincides with the free term of the quadratic

Eq (3.3) Consequently, the solution A q = 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 A 0 = 0 is stable if the value y 2 does not lie in the interval y] ^ y 2 ^ y ị of the axis y 2, 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 > y H) 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 A 0 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 S t the mass m accomplishes the forced oscillation with frequency 2y At the point B(yn = 1/2), the principal resonance takes place From y = y s to y = y D 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 y s < y < y Di the interference of frequencies occurs The parametric oscillation with frequency y becomes overwhelming.

For y > y D, 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 < y p the mass m will vibrate with two frequencies y and 2y, and the parametric

oscillation will become overwhelming For the remaining values of y, the mass m accom­

plishes forced oscillations with frequency 2y.

4 Experim ental 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, 6b) are obtained.

F ig 4.

a

W v / V W V W W V W

F i g 5.

/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 s t r o n g l y 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).

F ig 7.

For values V from 36 Hz to 40 Hz, 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

in sid e t h e r e s o n a n c e (v = 3 0 H z ) ; cu rve 2 for th e valu e V = 3 7 H z , a n d curve 3 f o r 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 i o n , t h e a u t h o r w i s h e s t o e x p r e s s a p p r e c i a t i o n f o r t h e a s s i s t a n c e o f c o l ­

l e 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 i n e x e c u t i n g t h e e x p e r i m e n t a l

work.

References

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2 A E H echokob, K tneopuu u pa cu em y 3AeK»ipoMcuHwiiHOĩo eu6pa m opa 3 3/ieKTpH'iecTBO, 1 2 , 1961.

3 D D K a n a , Parametric coupling in a nonlinear eìectro-mechanicơl system, Transactions o f the ASME\

Proc of the V ibrations Conference, 2, 1967.

4 N N B o g o liu b o v an d Yu A M i tr o p o l s k i , A sym ptotic methods in th e theory o f non-linear oscillations, Moscow 1963.

5 N g u y e n v a n D a o , Fundamental m ethods o f the theory o f non-linear oscil/ationsy H a n o i 1971.

S t r e s z c z c n i c ZJA W ISK O P A R A M ET R Y C Z N E G O REZONANSU N IEL IN IO W E G O O SC Y LA TO R A

P O D DZIALAN1EM ELEK T R O M A G N E TY C ZN E J SILY

w pracy rozpatrzono zjawisko parametrycznego rezonansu nieliniowego sprẹzystego oscylatora po- budzanego elektromagnetycznie (rys 1) Ustalono, ze w rozpatrywanym ukladzie oprócz znanego pod- staw ow ego rezo n an su , gdy czẹstoấé drgart w elektrycznym obw odzic je s t dw a razy m niejsza ođ czẹstoấci