Parametric vibrations of mechanical system with several degrees of freedom under the action of electromagnetic force

Systems with /I Degrees of Freedom Let us consider a vibrating system with n degrees of freedom which consists of a weight­ elastic elements of the vibrating system have stiffness k Ly k

Institute o f Fundamental Technological Research, Polish Academy o f S cien ces

PARAMETRIC VIBRATIONS O F M ECHANICAL SYSTEM S W IT H SEVERAL

DEGREES O F FREEDOM UNDER T H E ACTION O F ELEC TR O M A G N ETIC F O R C E

N G U Y E N V A N D A O (HA N O I)

1 Systems with /I Degrees of Freedom

Let us consider a vibrating system with n degrees of freedom which consists of a weight­

elastic elements of the vibrating system have stiffness k Ly k 2y

F ig I.

Supposing that some s -thmass is subjected to electromagnetic force, the differential

the form:

ml x l + k i (xl - x 2) = - h í ỉcl - p l (xl - x 1):i,

»iix2 + l<i(xi - x i) + k 1(x2 - x i ) = - h ix i - p i (xI - \ l)3 - /32(x1- x ì y ,

(1-1) .

mi x , + k t _ l ( x , - x , _ 1) + k , ( x , - x 1+1) = - h t x , - 0 , _ d x j - x ^ i ) 3

86 N g u y e n V u n Dao

We assume that

L = I ( j r J = z.0( l - » , * , + OTjX*), and that the friction forces and the non-linear terms in (1.1) are small with respect to the remaining terms Then, Eqs (1.1) can be rewritten as:

L-a'q + c = £ s i n v t — / j [ L 0q ( —2, Vj+ a 2.v|) + ^ L 0( —a j i j + 2 a 2 Vj.vJ],

.V, ^ A , f-V, - V.,) = f j F\ ,

(1 2) » h X i + x , ) - ị k 2(.x2 - x j ) = / / / " , ,

»'íÀ:, + Ẳ , _ 1( v , - v , _ l ) + Ắr,(.rs - v , + 1) - - ^ q 2 L.0 Z { + uF,,

™»vj \„ - ,v„_ ,) 4 A,, Y„ - ///■„.

where

/ - / , - — // , V, - /#, ( a i - V j ) \

(1.3)

/ r , = - / / , v s v s — / ? »_, (.V, - r , _ , ) J - < f 5(.v., - V, + , ) 3 ,

3

n •

/'f» = ^/1 X ~ I ( ^ fi —\ It - I )^ —fttt •'

We suppose that the characteristic equation of the homogeneous system

m, V, + / r , ( r , - -V2) = 0 ,

J v2+ * i ( * 2 - - r j ) + /:2(.v2 - A3) = 0,

^ ^ n - 1 ( n J ) -+■ k„ A „ —0 >

has no multiple roots and that its rootsw ,, , 0)n are linearly independent Then, to study

the system (1.2), wc shall analyze its particular solution corresponding to the one-frequency

means of the formulae:

(1.5)

n

X, = y ] c ' f ’t , 5 = 1 , 2 , , « ,

Ơ -= 1

where cJ is algebraic supplement o f the element placcd in the j-th column and the last

line of the characteristic determinant of the system (1.4).

tions:

P a ra m etric vibrations o f m ech a n ica l syste m s w ith several d egrees o f fr e e d o m 87

Here

io (? > q, * 3 , X,) = L 0 q ( - 3 ^ , + 32.x2 , ) + L 0q x , ( - a l + Ĩ 7 ĩ ot,) + Rq,

0 , = _ ! _ ỳ ei » F t - ĩ i h È L e o>

n

M i = £

/-1

In the first approximation, the investigation of one-frequency regime in the system considered can be reduced to a study of two equations: the first o f (1.6) and one of re­

maining n equations The choice of the appropriate equation depends on the value o f natural

Supposing that the frequency V of external force is near the natural frequency 0)j Then

we shall investigate the equations:

( 1 7 )

where

L0q + ~ q = £ sin V i - fiFZt

^ = k —— ẩm 1

ft

> h c2in

ỵ "s'- 3 >

w?* = ^■-[Pic[ii(c[J' - c itiì)3 + ậ l c[j)(ct 2n - c [ J))3+ậì c,j ì(ct2n - c y ì)3+ .

+ p, - 1 (cịJ> - c\L\y + p, c‘1>(c<'■> - <•<ị>,) + + /?„., c'B''(c'J1 - cii> ! ) 3 + f t ’],

ụFS = L 0q ( - i l cỳ>Ệj + <x2c V :)ỉ ỉJ) + L 0qcíJ)Ệ j { - 3 1+ 2 i 1c‘» ỉ J).

The remaining n - 1 normal coordinates f i, fy_j, £/+1, .» fn arc far from the reso­ nance, their vibrations will be small in comparison with the resonant vibrations considered

of the coordinate ỈJ, and in the first approximation they may be disregarded.

Equations (1.7) describing the one-frequency regime of vibrations have the same structure as the equations of motion of the system with single degree of freedom [1]

This gives reason to expect that in each resonant region the same peculiarities of motion

will be displayed as were found in the system with a single degree of freedom.

Introducing the notations

(18)

« r = ~ I o A i J , Ql = T ~ c > CO j fJ = T T 5 - -*-'0 0 ^ j yj = 7T>^ j

Eqs (1.7) assume the form:

q " + v>q = e j S i n y j Z

+ = - ụ f t ỉ ,j - / i ^ - a ĩ ^ 2 + fỉatq,2ỉj.

N g u y e n V a n D ao

Now, we transform the system (1.9) by means of the formulae:

q' = yje* cosyjT + Qj Bcos<p,

(1 1 0 ) ỈJ = - Ố - c o s 2 y 7T + /4ys in ỡ ; ,

where

2y b

t ’ j = S in 2 y ,r + /lyy; cos0; ,

e? = - ^ r — 2, b = — —^ 7 - , 9? = i2,r + 0 , 6j = yjT + y>j.

The transformed equations have the form:

ịẲ

Q J - 7 - = - - T - ^ / ' J C O S ? ,

G j B - j t = T ~ T = ^ 0 sinọ?,

= — /íy U —yj)sinớjC0SỠj

—//ScosOj-f-a j Y j ^ j L = / 4 , ( 1 y * ) s i n 20 , + / ^ s i n 0 j 4

-S = h ỉ ' j + P Ỉ * - z ĩ q ' 2Ệ,

where the non-written terms vanish when 5 = 0.

ent Then, in the first approximation the solution of the system (1.9) satisfies the equations obtained from (1.9) by averaging in time its right-hand part:

[1 + 0 0 0 ]^ = / < 4 *

where

c, = ỊL +

1 2 l-4 y ỷ ’

/1 = q + ĩpb* +~2 p (I_4y2)a *

P a ra m etric vib ra tio n s ~vf m echanical sy s te m s with several d eg rees o f freedom 89

Since B -+ 0 when / -> 0 0, then below we shall take into account only the equations:

(1.13)

y j A j ~ á r ' = \ V - rJ + + -*2 CL A J c o s 2 V>J>

from which we obtain the amplitude Aj of vibrations:

(1.14)

and the phase

(1.15) sin 2 y j = — y j , COS 2 yjj = + — Ị ^ c ỉ - h 2 y j

Equations (1.12)— (1.15) are different from the corresponding ones in the system with

a single degree of freedom [1] only by the values of the constant coefficients The method

used enabled us to reduce the more complicated problem to the whole complex oĩn prob­

lems of the type considered earlier In spite of this, in the first approximation each of such problems can be investigated independently of the others, because according to the con­ ditions of the problem, the resonant processes cannot be developed at the same time in more than on resonant region.

The stability of stationary regimes of vibrations may be found by analysing Eqs (1.12) The criteria of stability formed in [1] are:

c w

> 0 for Aj ^ 0,

cAj

and

/Ầ2{h2y } - c 2L) + ( y } - l -ỊẲỔ)2 > 0 for Aj = 0.

The study made in [1] concerning the stability of stationary regimes of motion will be

suitable for the character of resonant processes described by Eqs (1.12) in qualitative

relation This removes the necessity of analysis in detail the criteria of stability Here we

note only that for very slow change of frequency Vin the system considered, n resonant

peaks corresponding to the values V = (OỵÌYi = l),y = co2(y2 = 1) ••• arc observed

(Fig 2).

Fio 2.

7 P ro b lero y drgaA

90 N g u y e n V a n D ao

2 Parametric Resonance I d a System with Infinite Number of Degrees of Freedom

W e i n v e s t i g a t e i n t h e C a r t e s i a n c o o r d i n a t e s x 9y , z a p r i s m a t i c b e a m w i t h l e n g t h / w h o s e

c r o s s - s e c t i o n i s s y m m e t r i c a l w i t h r e s p e c t t o t w o m u t u a l l y p e r p e n d i c u l a r a x e s W e assumCị

t h a t t h e a x i s o f t h e b e a m i n t h e u n d e f o r m e d s t a t e c o i n c i d e s w i t h t h e a x i s X a n d t h a t t h e

s y m m e t r i c a l a x e s a r e p a r a l l e l t o t h e a x e s y a n d z ( F i g 3 )

I

F ig 3.

of electromagnetic force which is ìị distant from the origin of the coordinates and directed

to the axis y We assume that the inductance L is a function of distance y ị = y ự ị , t)j

and therefore the electromagnetic force depends on the location o f the electromagnet

1 , C'L

We -assume that the material of the beam follows the law [3]

° x = A O = E ( l - d E 2e l ) e x ,

where ơx is the longitudinal force and €x is the longitudinal elongation Then, the equation

of motion of the beam is:

d x 2

w h e r e Q i s t h e i n t e n s i t y o f m a s s o f t h e b e a m , y = y ( x , t ) — t h e d e f l e c t i o n , P ( x , t ) — t h e

i n t e n s i t y of e x t e r n a l load, — t h e b e n d i n g m o m e n t :

M - f í f ( y w ) > ’Jy Jỉ - Eỉ ỉ \ ' - dE' y ' [ ĩ ? )

Substituting t h i s e x p r e s s i o n i n t o ( 2 2 ) , w e o b t a i n :

e - i f +EJ s - 3dE » /

\ c*y d2y

1 [ õx* Õ X 2

P a ra m etric v ib ra tio n s ọ / m ech a n ica l syste m s w ith several d e g re e s o f fr e e d o m 91

We assume that the non-linear terms and the terms characterizing friction are small in comparison with the Unear terms Then the equations of motion o f the system considered can be represented in the form:

(2.3)

where

(2.4)

q + ũ ị q = e ú n v t + n ĩ x ị y ^ - f t - > >

' * - - T & +' [ & & +2( £ ) ì & + 7 '

The external load P(x, r) has the form:

where A is the length of that element of the beam which is directly subjected to the action

of electromagnetic force.

To solve the system (2.3), we note first that the generative equations ( / 1 = 0)

(2.6)

have the solution:

(2.7)

<7 +-0ẳ<7 = e sin V/, C/2y

<?r2 ■7 + b " ÕX*r- = 0

q = e*sinvr + Ssin 9>, oo

y = y x M c v * ( j2- bt+y*]’

Hm. 1

Equations (2.3) are different from the corresponding ones o f the system (2.6) only by

of the system (2.3):

(2.8)

CO

q = e * sin v / + 5sin9>, y = J T X,(x)s„(t),

* - 1

where (p = i?o/ + 0 and By 0 , sn arc functions o f time.

92 N g u y e n V a n D a o

of a series:

To seek the functions of time Vn, we multiply both sides of the equality (2.9) by x l9 and

integrate the result over the total length of the beam; due to the orthogonality of the eigen­ functions there remains only one term on the right-hand side which corresponds to the

number n, so that

where Kn are polynomials of third degree, relatively of s l t s2, .

We consider now parametric resonance when the frequency of the electric circuit V

is in the neighbourhood of 0 ) j assuming that the natural frequencies (0l f 0)2 are inde­ pendent Then we retain in (2.11) only the coordinate Sj The remaining coordinates

Sỵ Sj_l9 Sj+ị are far from the resonance and their values will be small in comparison

with S j and in the first approximation we can disregard them Thus, following the expres­ sions (2.4), (2.8), we have:

00

(2.9)

I I

(2.10)

Substituting (2.8), (2.9) into (2.3) and equating the coefficients ỵ„, we arrive at:

snic>2 ns„ = /.trnL 0 *1q2S ll + /xK„(sl , s2 , j , , i 2l

,uF2 = - ~ Sj AO+ p [ x y x ] ' 1 + 2 XỊ "2 x y i s f + Ĩ - Therefore, from (2.10), (2.11) we obtain the equations for q, sn:

(2.12)

q + ũ ị q = e si n w + / i f h

Jijr

S j + c o j S j = - — S j + Pa jSj +bj q2S j - c q zf

where

( 2 1 3 )

P aram etric vib ra tio n s o f m echanical s y s te m s with several d ecrees o f fre e d o m 93’

It is easily seen that the system of Eqs (2.12) is the complete analogy of the differen­ tial equations of vibrations of a system with single degree of freedom To avoid repeti­ tion, we shall refer below to the paper [1], where the problem of construction of a solu­ tion of the system of equations of the form (2.12) is considered in detail.

Thus, following the results of [1], we conclude that when the frequency V of an electric

circuit is n e a r to CO I , th en th e b eam c o n s id e re d v ib rates stro n g ly w ith freq u en cy V ( p a r a ­

m etric re so n an ce) T h is type o f reso n an ce ta k e s place also w hen th e fre q u e n c y V is n e a r

to w2,a>3 However, it must be emphasized that in the system with distributed para­ meters the vibrations with ‘the lowest frequency (i'jj) play the main role.

Some experiments were performed with beams and systems of several degrees of free­ dom The experimental results in the cases considered were in good agreement with the theoretical results This fact testifies to the acceptability of the limitations used in the prob­ lem and shows that the approximate solutions found by using the assumption concerning the one-frequency regime of vibrations in the regions of resonance can be adopted for practical purpose.

For the cantilever beam with parameters E = 2 • 107 N/cm2, J = 16' 10“3cm4, rp =

= 10 4N • s2/cm2, I = 46cm; therefore, 0)i = 14.8, 0)2 = 93.7, strong vibrations with frequency o f electric circuit V when V is in t h e regio n 1 3 5 - 14.3 H z were very sm a ll

For the same beam, but when / = 58cm and therefore C 0L = 26.3, substantial parametric

resonance when V is in the region 27.1-29.4Hz was observed.

R e f e r e n c e s

1 N c i;y e n v a n D a o , On the ph m o n u n o n o f param etric resonance o f a non-lỉnơur vibrator under the ac­

tion of electromagnetic force, P ro c V ib r P r o b l , 1 3 , 3, 19 7 2

2 I I I I E oro/iiO B O B , KD A M iiT Pono/ibC K H Ìi, AcuMWtiotnuuecKue Memoòbt 8 meopuu ỉte.iuneũiibix KO.ICỔCI-

ftuii, M o c K B a 1 9 6 3

3 H K a u d e r e r , aĩichtíinear Mechanik, B e rlin 1 9 5 8

S t r e s z c z e n i e

PA R A M ETR Y C ZN E D RGANIA U K L A D 0 W M ECHAN1CZNYCH o W IELU STOPN1ACH SWOBODY, WYW OLANE D Z IA L A N IE M ELEK TR O M A G N ETY C ZN EJ S1LY

N iniejsza p raca je s t kontynuacịạ poprzedniej p ra c y [1], Z b ad an o w nicj d rg a n ia m echanicznego ukia-

d u o nsto p n iach sw obody oraz bclki przy zalo z en iu , ze sạ one obci^zonc elek.trom agnetycznạ siỉạ para-

m etrycznie pobudzajạcạ drgania P odobnic ja k w [1] ro zp atiy w an e d rgania p aram etryczne m ajạ cz$sto$ố rósvnạ czcstoắci đrgaií w elektrycznym obw odzie.

W y znaczono am plitudy drgan i z b ad an o ich statecznosé.

94 N g u y e n V an D ao

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rp y > K e H b i 3JICKTpoMarHHTH0A c m io ii napaM eTpHMccKH B 0 3 ố y > K ju iK )in eìi KOJieỐaHHH A H aaorM M H o KOK

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d e p a r t m e n t o f m a t h e m a t i c s a n d p h y s i c s

P O L Y T E C H N IC INSTITUTE, HANOI

Received March 8, 1972