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

Proceedings of Vibration Problems, 14, 1, pp. 85-9 4, 1973
Institute o f Fundamental Technological Research, Polish Academy o f Sciences
PARAMETRIC VIBRATIONS OF MECHANICAL SYSTEMS W ITH SEVERAL
DEGREES O F FREEDOM UNDER THE ACTION OF ELECTROMAGNETIC FORCE
NGUYEN VAN DAO (HANOI)
1. Systems with /I Degrees of Freedom
Let us consider a vibrating system with n degrees of freedom which consists of a weight
less cantilever beam carrying n concentrated masses mlf m2, m3, (Fig. 1). The
elastic elements of the vibrating system have stiffness kLy k2y
Fig. I.
Supposing that some s -thmass is subjected to electromagnetic force, the differential
equations of motion of the system considered can be written, in accordance with [1] in
the form:
~dt ^ + c q = £sinv/’
mlxl+ki(xl-x2) = -híỉcl-p l(xl-x1):i,
»iix2+l<i(xi - x i) + k1(x2 -xi) = - h ixi - p i(xI \l)3 - /32(x1- x ìy,
(1-1)
.

mix,+ k t_l(x ,-x ,_ 1)+ k ,(x ,-x 1+1) = - h tx,-0,_ d x j -x ^ i ) 3
SI I ÕL
PẢX, x,+ l) + 2 q ^ ,
mn XH + k„_ t (xn xn _ 1) -f kn xn — hnXH
86
N guyen V un Dao
We assume that
L = I(jrJ = 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 = £sin vt — /j[L0q(— 2 , .Vj+ a2.v|) + ^L0( —ajij + 2a2 Vj.vJ],
.V, ^ A , f-V, - V.,) = fjF\ ,

(1 2) »h.Xi+ x,)-ị k 2(.x2- x j) = ///",,
»'íÀ:, + Ẳ ,_1(v., v.,_l) + Ắr,(.rs v.,+ 1) - - ^ q2 L.0 Z{+ uF,,
™»vj \„ - ,v„_ ,) 4 A,,. Y„ - ///■„.
where
/ - / , - — // , .V, - /#, (a i - Vj)\
/ ' r : - - /?, .V, - /?, (V; - V, )•’ - ,? ,( .v 2 - .Y j )\
(1.3)
/ r , = - //, vs vs —/?»_, (.V, - .r,_ ,)J -< f5(.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 rj) + /:2(.v2 -A 3) = 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
regime of vibrations [2]. To that end, we introduce the normal coordinates f i, by
means of the formulae:
(1.5)
n
X, = y ] c 'f ’t,. 5 = 1 , 2 , . . . ,« ,
Ơ -= 1
where cJ is algebraic supplement of the element placcd in the j-th column and the last
line of the characteristic determinant of the system (1.4).
We can easily verify that the normal coordinates f j , satisfy the following equa
tions:
Param etric vibrations o f mechanical system s with several degrees o f freed om 87
Here
io(?> q, *3 , X,) = L0q(- 3 ^ , + 32.x2,) + L0qx,(-al + Ĩ7ĩot,) + Rq,
0 , = _ !_ ỳ ei»Ft- ĩ i h È L eo>

1 Mj Z j k IpMj ' ’
n
Mi = £
/-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 of (1 .6) and one of re
maining n equations. The choice of the appropriate equation depends on the value of natural
frequency CO in the neighbourhood of which the parametric vibrations are examined.
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 Vi - fiFZt
^ = k

——
ẩm 1
ft
> h c2in
ỵ "s'- 3 >
w?* = ^■-[Pic[ii(c[J'-citiì)3 + ậlc[j)(ct2n -c[J))3+ậìc,j ì(ct2n-c y ì)3+
+ p, - 1 (cịJ> - c\L\y + p, c‘1>(c<'■> - <•<ị>,) + + /?„., c'B''(c'J1 - cii>!)3 + ft ’],
ụFS = L0q (- ilcỳ>Ệj + <x2cV:)ỉỉJ) + L0qcíJ)Ệj{-31+2i1c‘»ỉJ).
The remaining n- 1 normal coordinates fi, 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
Wj’ r ~ Wj ’ 0 ~ loj ’ “* ■ 2Mị, ’
(18)
« r = ~ I o , Ql= T~c> fJ = T T 5 - yj = 7T>
A iJ CO j -*-'0 0 ^ j ^ j
Eqs. (1.7) assume the form:
q " + v>q = ejSinyjZ-
(1.9) ' LoCOi
+ = -ụftỉ,j - / i ^ - a ĩ ^ 2 + fỉatq,2ỉj.
88 N guyen Van Dao
Now, we transform the system (1.9) by means of the formulae:
q = efsinyjT + Bsin(pt
q' = yje* cosyjT + Qj Bcos<p,
(1.10) ỈJ = - Ố - cos2y7T + /4ysinỡ; ,
where
2y b
t ’j = Sin2y,r + /lyy;cos0; ,
e? = -^r— 2, b = — —^ 7 - , 9? = i2 ,r + 0 , 6j = yjT + y>j.
J V ỉ-rỉ' 2
The transformed equations have the form:
ịẲ
Q
J - 7 - = - - T - ^ / ' J C O S ? ,
dĩ LqCOj
i/0 fj.
(ỈT Lịj(uJ
Gj B-j t = T ~ T = ^ 0 sinọ?,
= —/íyU —yj)sinớjC0SỠj —//ScosOj-f-
a j Yj ^j L = /4,(1-y * )sin 20,+ /^ sin0j4-
S=hỉ'j + PỈ*-zĩq'2Ệ,

where the non-written terms vanish when 5 = 0 .
We suppose that yj is in the neighbourhood of 1 and that}/; and arc linearly independ
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 +00 0]-^-= -/< 4 * -
(1.12) ^ = G ( B , 0 , A j , Vj),
d.ij h ụ
y‘ dt' = ^ 2 y^ f 2 Cl J'
yjAllh = ị V -yj + + ị + -ị clAJcos2Vi+ ■■■
where
c, = ỊL +
1 2 l-4 yỷ ’
/1 = q + ĩpb* +~2 p (I_4y2)a *
Param etric vibrations ~vf mechanical system s with several degrees o f freedom
89
Since B -+ 0 when / -> 00, then below we shall take into account only the equations:
dA Ị h . li . . -
Yj ~ -t*ỶYjAJ + Ỷ CijS y)j’
(1.13)
y jA j ~ár' = \ V - rJ + + -*2 CL AJcos2V>J>
from which we obtain the amplitude Aj of vibrations:
(1.14)
and the phase
(1.15) sin 2 yj = — yj, COS 2 yjj = + — Ị^cỉ - h 2 yj.
c 1 C1
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
w = Ị 'ị- fxpAj + 1 — y/ -fi2(cị-h2y}),
and
/Ầ2{h2y}-c2L) + (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 V in the system considered, n resonant
peaks corresponding to the values V = (OỵÌYi = l),y = co2(y2 = 1) ••• arc observed
(Fig. 2 ).
Fio. 2.
7 Probleroy drgaA
90
N g uye n Van Dao
2. Parametric Resonance Id a System with Infinite Number of Degrees of Freedom
W e i n v e s t ig a t e i n t h e C a r te s ia n c o o r d i n a t e s x9y,z a p r is m a t ic b e a m w i t h le n g t h / w h o s e
c r o s s - s e c t io n i s sy m m e tr ic a l w i th r e s p e c t t o t w o m u t u a ll y p e r p e n d ic u la r a x e s . W e assumCị
t h a t t h e a x is o f t h e b e a m in t h e u n d e fo r m e d s t a te c o i n c i d e s w ith t h e a x is X a n d t h a t th e .
s y m m e tr ic a l a x e s a r e p a ra lle l t o t h e a x e s y a n d z ( F i g . 3 ).
I
Fig. 3.
The beam UDder certain conditions of strengthening of its ends is subjected to the action
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 yị = yựị, t)j

(2.1)
L = I O',) = L0( 1 - + “2>'ỉ).
and therefore the electromagnetic force depends on the location of the electromagnet
1 . , C'L
and on the vibrations of the beam, and has intensity a —— .
2 oyl
We -assume that the material of the beam follows the law [3]
°x = A O = E ( l-d E 2el)ex,
where ơx is the longitudinal force and €x is the longitudinal elongation. Then, the equation
of motion of the beam is:
(2.2)
g2M
d x 2
w h e r e Q is th e i n t e n s it y o f m a s s o f t h e b e a m , y = y(x, t)— th e d e fl e c t i o n , P(x, t)— t h e
in t e n s it 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 ) > ’JyJỉ - Eỉ ỉ \ ' - dE'y ' [ ĩ? )
Substituting t h is e x p r e s si o n i n t o (2 .2 ) , w e o b t a i n :
e - i f +EJ s - 3dE »/
St2 ÕX*
\ c*y d2y
1 [ õx* ÕX2
w h e r e ,
(&)]
Ji = / / y*dydz, J = / f y 2dydz.
Param etric vibrations ọ/ mechanical system s w ith several degrees of free do 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 of the system considered
can be represented in the form:
(2.3)
where

(2.4)
q + ũịq = eún v t+ nĩx ịy^ - f t - > >
d2y ,J d*y _
õt2 dx* p 2t
02 L p - E L EJ ME>],
ỉ c' Ĩ 9 /-* ’ p I *
' * - - T & +' [ & & +2( £ ) ì & +7 '
The external load P(x, r) has the form:
(2.5)
P(x, 0 =
2A *
f o r /, + y ,
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 ■ " ÕX*
7 +b
r- = 0
q = e* sinvr + Ssin 9>,
oo
y = y .x M c v * ( j 2 -bt+y*]’
Hm. 1
where Bt 0 , f„, arc arbitrary constants, xn are the eigenfunctions which define the
natural modes of vibrations of the beam and depend on the boundary conditions.
Equations (2.3) are different from the corresponding ones of the system (2.6) only by

small terms fiF1, /iF2. Consequently, it is natural to propose the following form of solution
of the system (2.3):
(2.8)
CO
q = e*sinv/ + 5sin9>, y = JT X,(x)s„(t),
*-1
where (p = i?o/ + 0 and By 0 , sn arc functions of time.
92
N g uyen V a n Dao
Now, instead of determining the functions q and y, we determine the functions
B, SH. To find the differential equations for these variables, we represent F2 in the form
of a series:
To seek the functions of time Vn, we multiply both sides of the equality (2.9) by xl9 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 slts2 ,
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 (0lf0)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:
(2.11)
q + Qịq = esin vt+pFlt

snic>2ns„ = /.trnL0*1q2Sll + /xK„(sl, s 2 , j , , i 2l
,uF2 = - ~ Sj AO+p [xyx]'1+ 2XỊ"2 xyisf + Ĩ - .
Therefore, from (2.10), (2.11) we obtain the equations for q, sn:
(2.12)
q + ũịq = e si n w + /ifh
Jijr
S j+ c o jS j = - — S j + PajSj+bjq2Sj-cqzf
where
(2 .1 3 )
Parametric vibrations of mechanical sy ste m s with several decrees o f freedom 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 near to CO I , then the beam co nsidered vibrates strongly w ith frequency V (p a ra 
metric resonance). T his type o f resonance takes place also when the frequency V is near
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 of electric circuit V when V is in the region 13.5- 14.3 H z were very small.
For the same beam, but when / = 58cm and therefore C0L = 26.3, substantial parametric
resonance when V is in the region 27.1-29.4Hz was observed.

R e fere n ce s
1. N ci;yen v an D a o, On the phm onunon of parametric resonance o f a non-lỉnơur vibrator under the ac
tion of electromagnetic force, P ro c . V ibr. P r o b l., 1 3 , 3, 1972.
2 . I I . I I . Eoro/iiOBO B , KD. A . M iiTPono/ibC K HÌi, AcuMWtiotnuuecKue Memoòbt 8 meopuu ỉte.iuneũiibix KO.ICỔCI-
ftuii, M ocK Ba 19 6 3.
3. H . K a u d e r e r , a ĩichtíinear Mechanik, B e rlin 19 5 8.
Streszczenie
PARAMETRYCZNE DRGANIA UKLAD 0W MECHAN1CZNYCH o WIELU STOPN1ACH
SWOBODY, WYWOLANE DZIA LANIEM ELEKTROM AGNETYCZNEJ S1LY
Niniejsza praca jest kontynuacịạ poprzedniej pracy [1], Z badano w nicj drgania mechanicznego ukia-
du o n stopniach sw obody oraz bclki przy zalozeniu, ze sạ one obci^zonc elek.tromagnetycznạ siỉạ para-
metrycznie pobudzajạcạ drgania. Podobnic ja k w [1] rozpatiyw ane drgania param etryczne majạ cz$sto$ố
rósvnạ czcstoắci đrgaií w elektrycznym obwodzie.
W yznaczono am plitudy drgan i zbadano ich statecznosé.
94
N guyen Van D ao
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PO LY TEC H NIC INSTITUTE, HANOI
Received March 8, 1972