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Random excitation of a vibratory system with autoparametric interaction
Journal of Sound and Vibration (1980) 69(l), 10 l- 116
RANDOM EXCITATION OF A VIBRATORY SYSTEM WITH AUTOPARAMETRIC INTERACTION J. W.
ROBERTS
Department of’Mechanical Engineeriq, University of Edinburgh, Edinburgh EH9 3JL. Scotland (Received 1 March 1979, and in revised,form 77 September 1979)
The paper is concerned with the broad band random excitation of a two degree of freedom vibratory system with non-linear coupling of autoparametric type. A general equation for the evolution of the moments of any order of the response co-ordinates is derived by using stochastic calculus and found to represent an infinite hierarchy set. Consideration is given to the determination of the mean square stability boundary for unimodal response with no transverse motion of the coupled system. Two approximate solutions are obtained. These are first of all a solution based on a Gaussian closure technique applied to the system moment equations which allows the stability condition to be determined from the eigenvalues of a four by four matrix, and secondly a perturbation solution which leads to a simple analytical expression for the stability boundary. The two methods give results in close agreement for low values of system damping, but which differ appreciably at high damping levels. Finally, results are obtained from an investigation of the response regions of a laboratory model excited from a random noise generator. The experimental results are found to give excellent correlation with the predicted instability boundaries in the close neighbourhood of internal resonance but show a distinct indication of a wider instability region than predicted by both analytical methods.
1. INTRODUCTION This paper describes a particular form of to as autoparametric, discrete form of two
a continuation of previous work on random excitation of systems with non-linear coupling between modes of vibration of the type referred being closely related to systems with parametric excitation. The basic
degree of freedom system is shown in Figures l(a) and l(b) and consists essentially of a cantilever beam attached to a primary vibratory system in such a way that response of the primary system imposes an axial motion on the cantilever, thus generating a parametric excitation term in the equation of transverse motion of the coupled cantilever. The transverse motion in turn induces a reaction force on the primary system. The system is a common structural arrangement, representing for example fuselage-tailplane coupling in aircraft structures. The discrete parameters of the primary system of Figure 1 may be interpreted as generalized parameters of a transverse elastic mode of a wide range of linear elastic structures, thus extending the area of application of the present study. This will be made explicit in a further paper. Here broad band random excitation of the primary system is considered as a forced vibration as in Figure l(a), representing for example turbulence or buffet excitation, or a random imposed ground motion as in Figure l(b) representing for example stochastic excitation of a rolling system by a surface profile. In earlier papers [l, 21 approximate techniques and some numerical results for the response parameters of this system were presented and also previous work in this area was discussed. In addition, a recent series of review articles [3] contains an extensive bibliography of work on parametric and autoparametric systems including stochastic excitation. The equations of motion of the system of Figure 1 were first derived (for deterministic excitation) 101 OO22-460)3/80/050101+ 16 $02.00/O
0 1980 Academic Press Inc. (London) Limited
102
J. W. ROBERTS
by Haxton and Barr [4] who showed that large transverse motions of the coupled system could be induced by harmonic forcing of the main system under conditions of internal resonance: that is, when the ratio of the natural frequency of transverse motion of the coupled cantilever to the natural frequency of the primary system was in the neighbourhood of 0.5. Under this condition, the response of the primary system is modified by the reaction of the coupled system in the manner of a vibration absorber.
(a) Figure
1. System model. (a) Random
(b)
force excitation;
(b) ground
motion
excitation.
The significance of the internal resonance condition was found to carry over to the random excitation case by Ibrahim and Roberts [l], who evaluated the system response moments by an approximate numerical integration technique based upon a Markov vector approach combined with a Gaussian approximation of the response moment equations. It was established that large random motions of the coupled system could be generated by random excitation of the primary system for a range of system parameters in the neighbourhood of internal resonance. Otherwise, unimodal response of the primary system was obtained. In a further study [2] the stability boundary for the unimodal response was obtained by linearizing the approximated moment equations and solving numerically the resulting eigenvalue problem. The results showed the significance of the internal tuning ratio, system damping factors and excitation parameter. In the present paper, the results of measurements of the system stability boundary for a laboratory model are compared with predicted levels. The analysis of the system stability is considered by two separate methods. First of all the Gaussian approximation used previously [2] is extended and shown to give results which are independent of the mass ratio parameter. Secondly, an analytical solution for the unimodal stability boundary is obtained from a perturbation solution of the system equations. Comparison is then made with the experimental results.
SYSTEM WITH AUTOPARAMETRIC
103
INTERACTION
2. SYSTEM EQUATIONS OF MOTION For the system of Figure l(a), the equations methods as
of motion may be derived by standard
(M -t m) 2 -t c,1 -k K,x -. (6/51) m(j2 -t yj) = f(t)
(1)
for the primary system and mj -\ C,j -t [K, -. (6/51) mjt] y -1 (36/2512) my(yj -I- 3’) = 0
(2)
for the coupled cantilever. In the case of the ground motion excitation. Figure l(b), the form of equation (1) remains valid if x denotes relative motion of the mass M, (M -)_m)T -t- C,st -b K,x
-. (6/51) m(j’ -1 yj) = -(M
-k m)ii(t).
(3)
while the cantilever equation is modified by the ground acceleration, mj -k C,j
-k [K,
- (6/51) m(Z + ii)] y -F-(36/251*) my(yjj -t j2) = 0.
(4)
The form of autoparametric coupling action is clear from equations (1) and (2). In equation (2) the x motion appears as a time dependent modification of the cantilever stiffness, i.e.. a parametric excitation, while in equation (1) the cantilever motion appears as a quadratic non-linearity representing the reaction force on the primary mass. In equation (4) the parametric excitation term also contains the ground acceleration and thus the ground motion excitation case produces a slightly different equation of motion. It is considered that for a lightly damped primary system the additional term will have a negligibly small effect on the interaction between the two systems under internal resonance conditions. Accordingly, equations (1) and (2) are taken as representative of both forms of system excitation. It is desirable to write equations (1) and (2) in a dimensionless form. Upon making the transformations x = x/x,, Y = y/x,, r = or t,
(5)
where x0 is some convenient reference displacement, the equations become X” -t 1(,X’ -1 X - ER(Y’~ -1 YY”) = W(z), Y” -t- 2[,rY’ -I- (r2 - .5X”) Y -1 E*( Y2 -1 YY”) Y = 0,
(‘7)
where 0): = K,I(M
-b m),
wz = K,/m,
cl = C,/2(M
-F m)o,,
;, = C2/2mw2.
In equations (6) and (7) the following system parameters have been introduced: mass ratio R = m/(M -k m); tuning ratio r = w,/co 1; coupling parameter E = 6x,/51 (assumed small). The prime denotes differentiation o.r.t. z. The excitation function in equation (6) is w(r) = {(M -k m) co: x,} -I
f‘(s/o,).
(8)
It is assumed that the real time excitation f(t) is a zero mean, stationary Gaussian process with a smooth spectral density S, extending to some frequency well above o,, where S, = 2x L Then, from equation (8),
W(T)
00 E{f(t) f’(t -t t,)} e-‘“‘I dt,. (W s 1 may be regarded as ideal Gaussian white noise having auto-
104
J. W. ROBERTS
correlation
function R(u) = E[ W(z) W(z -1 u)] = 20 6(u),
(l(J)
D = ~cS~/(M -1 m)2 cofxi
(III
where D is given by and 6( ) denotes the Dirac delta function. The reference displacement x0 may now be chosen with advantage to be the root mean square response of the primary system for the case of zero coupled system motion. With y(t) = 0 in equation (1) one obtains: [E{~~(t)}]i’~ = [l/(M -f m)] [rcS,/2[,~~]‘~~ = x0.
(12)
Then D = 25,,
F = 6x,/51 = [6/51(M + m)] [zS~/~<,O~]‘~~.
(13)
In this way the standardized excitation W(r) is made independent of the practical system excitation level, whereas the latter is embodied in the coupling parameter E. 3. SYSTEM
EQUATIONS
IN TERMS
OF STOCHASTIC
INTEGRALS
The assumption of ideal white noise for W(r) is convenient analytically, but implies that the system equations are not strictly valid, since W(r) is not integrable in the ordinary sense. Formally, one may write W(r) = dB/dz where B(z) is a Brownian motion process. The system equations (6) and (7) may then be written in terms of stochastic integrals. To do this it is necessary to eliminate the non-linear acceleration terms and this may be done by a process of repeated substitution up to any order of parameter E, Carrying this out to order s2 and letting X, = X, X2 = I: X, = X’ and X, = Y’, one obtains dX, = X, dr,
dX, = X, dz,
dX, = (-2,“,X,
-k s2R[-2:rX;X3
dX, = { -25,rX, -t s2p[2<,r X:X,
- X, -k ER[X;
- 2r[,X,X,
- ?X;]
- X,X;]} dz -t- (1 + s2RX;} dB(z), - r2X2 + E[ -x,x, -t- r2Xi - X,X:]}
- 2&x,X,] dr -1 &X2dB(r),
where p = 1 - R. In equations (14), dB(r) is an incremental with properties
(14)
Brownian motion process
E(dB’} = 20 dz.
E{dB} = 0,
(15)
Since W(z) represents the limiting case of a smooth wide band process the system equations (14) should be interpreted as a set of stochastic differential equations of Stratonovich type corresponding to the general form [5] dX, = f;.(X, z) dz + $
Gij(X, z) dB,,
i = 1,2 )...)
II.
(16)
j=l
Equation (16) may in turn be transformed into a stochastic equation of It8 type: 6(X, z) -t $ f k=l
1
dr -1 2 GJX, r) dB,,
$ Gkjg j=l
k
i = 1,2 ,...,
n.
(17)
j=l
It is seen that for the case of the system equations (14) the additional terms in equations (17) are zero, so that equations (14) may be interpreted directly as a set of equations of It6 type. It follows that the system response vector X(r) constitutes a Markov process.
105
SYSTEM WITH AUTOPARAMETRIC INTERACTION
A general solution for the transition probability of X is not possible. However. a set of differential equations for the joint moments of any order of the components of X may be obtained directly from equations (14) by using the methods of stochastic calculus [5]. With introduction of the scalar function (18) 4 = (x;xjxk,x;) and the joint moment of the system response co-ordinates
of order N = ti + ,,i + k -t IL
* m..tJkl =
X;XjX;X;
SSR
p(X. z)
dX = E{c$}.
(1’))
d--l
the fundamental theorem of It8 may be used in the form
’{dx,} -k :Trace[G]
d4 =
[Q] [GIT[&]
(20)
dr, I
I
where [Q] dz = E{dB,} {dB,}‘. Substituting in equation (20) from equation (14) and taking expectations of both sides leads to rn&, =: im.r-l.j.k+l.l
-‘jmi,j-l,k.l+l -
-’ -’
2,,rlm,
‘b2~lrni
.j+l
2;lkmi,j.k.I
-
kmi+l.j.k~l.l
. j . k, i - r21mi . j+, .k. ,_, -k Dktk -
E{Rk[mi,j,k-l,l+2
_I- E2.(Rk[-2~lmi.j+2.k,i -
-
-
,k+l.l-I -
2~2rmi.j+l,k-l.l+l -
nLi+
L.j+l,k.I&ll
"'i+l,j+2.I-I,11
m.l.J+l.k,1+11 -’ D’(’ -
-
1)mi,j+2,k,1-2)
1) m,, j . k ‘2mi,j+2.k-l.ll
-' 2Dk'mi,j+l.k~l.~&l~
-' d[2~2rmi.j+2.k,I -’
2.1
2RDk(k
-
-' r2mi.j+3,k.I 1)mi.j+2.k-2.1’
-1 (71)
The system of linear first order differential equations (21) determines the evolution in time of the system response moments of any order. Inspection of equations (21) shows that for non-zero coupling parameter E the equations for moments of any specific order are coupled with higher order moments, so that the system (21) represents an infinite hierarchy of moment equations and is incapable of explicit solution. This situation was discussed in the previous papers [l, 21 and an approximate truncation scheme was introduced based on a Gaussian approximation of moments higher than second order which permitted evaluation of the response first and second moments by numerical integration. Further inspection of system (21) shows that it is satisfied by the following stationary solution: m 2,joo =
E{X2j = D/25, = 1, m,,o2o = E(X”)
= D,i25, = 1, all other moments zero. (22)
This solution represents unimodal response of the primary system with zero transverse motion of the cantilever. The mean square stability threshold of this solution in terms of the system characteristics and excitation parameter is of obvious practical interest and significance, and is investigated in the following sections. An alternative set of system equations in terms of stochastic integrals may be obtained by application of the method of stochastic averaging. The method is an extension of the well known averaging method for non-linear systems [6] developed for problems with randomly varying parameters by Stratonovich and Romanovskii [7]. The method is fully described in reference [8], and has been applied extensively to obtain useful results in a number of random parametric problems [9-121. In the present problem the resulting system equations depend upon joint moments of the averaged response co-ordinates and do not lead to a useful solution.
106
J. W. ROBERTS
4. MEAN
SQUARE
STABILITY
OF THE UNIMODAL APPROXIMATION
SOLUTION-GAUSSIAN
To investigate the behaviour of the system (21) in the neighbourhood solution one may introduce mijkl
=
of the unimodal (23)
m$, -1Sijk,
is the stationary solution (22) and 6 represents a set of small in equations (21) where rn$), perturbations. For small variations of the system moments in the neighbourhood of the stationary solution it is assumed that the response processes are well represented as Gaussian. Using the standard relations for third and fourth moments into (21) and neglecting products of the 6’s leads to the following linearized equation for the moment perturbations of first and second order [2] :
‘ijkl
=
(24)
“ijkl>
where C is a 14 x 14 matrix. Introducing the form of solution 6..IJkl = 6!‘ ) e!.’ rjkl rJki’
(25)
C&O’ = 1/6’ (J) ijkl’ LJkl
(26)
leads to
It is clear that the stability of the stationary unimodal solution may be determined from the nature of the eigenvalues of the square matrix C. By ordering the components of the vector Bijklin a specific way the eigenvalue problem (26) may be formulated as ~
$0)
0
1
g(O) 2
0
= A SO,
$0’ 3
0
(27)
go’ 4
C (6 x446)_ where
sy = [S 1000~00101=~e
= C~2000 6,010~00201=~8:o) = PO200~0,01~00021=~
Sk”’= [6 0100 60001 6 1100 60110 6 1001 60011 y, 0
1
Cl1 =
I
02
c,,
=
-1
0
0 -21, -2
1 -41,
i -1
> c,,=
1 -2;,
(28)
1’
0
-r2 r:24[1
2
0
-2ri,
1
-2r2
-4rr,
I ,
SYSTEM WITH AUTOPARAMETRIC
0
-V2
c44=
1 -2r[,
0 -E
0
INTERACTION
0
0
-2;,E
0
0
0
0
0
1
1
0
0
0
-1
-2:,
0
1
--6
0
2c,t
0
--r2 0
0
-2r[,
-Y2
-1
107
1 -2(:“,
-t r(,)
r
The block diagonal form of equation (27) with zero or one-way coupling between the element matrices allows a considerable simplification of the initial (14 x 14) eigenvalue problem. First of all C,, is seen to be uncoupled from the remaining elements and its eigenvalues contribute independently to the set of eigenvalues of C. For the remaining eighth order matrix, it is seen that C,,, C,, and C,, are coupled in the forward direction only. A property of this matrix structure is that the eigenvalues of the complete matrix are simply the set of eigenvalues of the diagonal blocks. The diagonal block matrices of C may therefore be examined independently to investigate stability of the moment perturbations. It may be shown without difficulty that the eigenvalues of C,, and C,, are stable for non-zero [I and need not be considered further. In the case of C,, application of the RouthHurwitz criterion leads to a simple condition for stable eigenvalues in terms of the coupling parameter: E2 < r3 :,/:,.
(30)
This condition does not reflect the sensitivity to internal resonance found in the experimental work and corresponds in fact to parametric instability of the coupled system due to the non-resonant spectral density of the axial acceleration term %.The result is analogous to that of Ariaratnam [12] for the classical linear stochastic parametric problem. Finally, the eigenvalues of C,, need to be considered. Numerical evaluation is required but these indeed give rise to an instability boundary, sensitive to the internal resonance condition r CY0.5. Identical results for stability boundaries were obtained from C,, to those illustrated in the previous paper [2], obtained by direct computation from the 14 x 14 characteristic matrix. Additionally, it is clear from equations (29) that the system mass ratio R plays no part in the elements of C,, and cannot influence the stability boundary. Consequently, the reaction force on the primary system due to the coupled system motion has no effect on the stability threshold so that for stability analysis the system is analogous to the problem of the cantilever excited parametrically by filtered noise, the primary system acting essentially as a linear filter. 5. PERTURBATION SOLUTION FOR THE INSTABILITY THRESHOLD OF THE UNIMODAL RESPONSE The stochastic stability of the unimodal response may be investigated by an alternative method to the Gaussian technique of the previous section. Wedig has used a perturbation method to study the stability of a linear system parametrically excited by band pass filtered white noise [13], and his method may be extended to the present system by considering
108
J. W. ROBERTS
small motions of the coupled system in the vicinity of the unimodal response Y s 0 for small values of coupling parameter E. The system equations (6) and (7) may then be approximated by X” + 25,X’ -t- x -. &R(Y’Z-t YY”) = W(r),
(31)
Y” -I- 2<,rY’ -k (r’ -. EX”) Y = 0,
(32)
where cubic terms in e2 have been neglected. It is assumed that Y(r) is small in equation (32). Take X(z) - X,(r) + X,(r) where X,(r) is a variational term, assumed small and X,(7) is the unimodal response, satisfying xb’ + 2(,X;
-t x, = W(z).
(33)
Then equation (32) becomes, with neglect of the small quadratic term, Y,’-1 2r[,Y’ -k (7’ - &Xi) Y = 0.
(34)
Approximated in this way for small motions of the coupled system the system equations (33) and (34) show that the primary system acts essentially as a linear filter whose acceleration response appears as a parametric excitation of the coupled system. The involvement of acceleration in the parametric term means that the system differs in detail from that studied by Wedig. Upon introduction of the transformation Y(7) = Z(7) exp( - rc2z), equation (34) becomes Z” -t r:[l
- e,Xi] Z = 0,
s1 = s/r2(1 - $,
(35)
rf = r(1 - 5:).
(36)
Equation (35) may be further written as two first order equations by transforming Z into amplitude and phase components, rewritten in complex exponential form as Z = C cos @ = $ ei@ -t $C emi@s Z, -k Z,, E ir,(Z, - Z,),
z’ = r,Csin @
(37)
where Z*1 = Z 2’ Then Zl, = i rl[Z,
- E~+Xi(Z,
-t Z,)],
Z2 = -i r1[Z2 - Ed$Xg(Z,
-b Z,)].
(38)
It is useful to write the primary system equation in the form: xb’ + 2:,x;
r2 + $)X0 -t (,i
=
W(T),
(39)
where q ’ = (1 - CT)2 1. Equation (39) may then be factored into complex components in the following way. First one writes x, = z, -k z,,
zx = z,.
(40)
Then Z; -k ([, -t iv) Z, = iW/2r], Zk -k ([, - iv) Z, = -i W/2r7.
(41)
Using expression (41) in equation (39) one obtains, for X$ xb’ = w + cc; - q2 -1 i2@r) Z, -I- (I: - q2 - i2qcl) Z, = W - (1 - i2@,) Z, - (1 -I- i21!,) Z,,
(42)
it being assumed that 21: @ 1. If it is now assumed that W(7) is an ideal white process then equations (41) may be replaced
109
SYSTEM WITH AUTOPARAMETRIC INTERACTION
by stochastic differential equations of It6 type. With use made of the formal relationship W(z) = c1dB/dr, 143) where c( is a constant and /I(z) is a Brownian motion process with unit variance parameter. then. from equations (10) and (13) c( = [20] i/z = 2[;,2. 144)
By using equation (42) in equations (38) a set of equations of It6 type in the transformed system co-ordinates Zj may be written as follows: dZ, == ir,Z, i
-F r:,i%(l
+ Eli $(l
- i2n<,)Z,Z,
+ i2q[l)Z,Z,
-1 sii?(l
-1 s,i?
- i2q
(l+i2q<,)Z,Z,
dz - is,+(Z,
-F Z,)a dfi.
1 d%, == -ir,Z,
- ic, + (1 - i2r][,) Z,[Z,
-k Z,] - it;, 2 (1 -!- i2q;i) Z,[Z,
-1 Z,]
1
d7 I
-b
is1 ? (Z, -F Z,) a dp, dZ, = -- (; 1 - iv) Z, dz - i 2; d/I.
-t iv) Z, dr -t i $ dfl.
dZ, = -(:,
(45)
Now one can define a vector of joint moments of the complex response co-ordinates Zi : M,‘, = E{Z:Z;Z;,
Z,Z,Z;Z;,
Z;Z”,Z;}.
(46)
Applying the It6 theorem (20) to the elements of the vector (46) one obtains ultimately a linear first order differential equation for M,, in terms of lower and higher order moments :
;M;,=A,,&,
?G{(l-
+
- &
1
{k(k - l)M,-2.1
-t %G{kM,,,,
p2a2r - -lHM,.l ‘I 8
-F iv) - (1/2r,)(i, - i - (k/2r,)(<,
G=
I_:
-t- 1(1- l)M,,&
- 2klM,-,,,-,
- M,,,_,}
where A,, = Diag{i - (k/2r,)(!,
-k (1 -F i26,rl)M,,,+,)
Z,vl)M,+,,,
(47)
- ig), - (k/2r,)([,
-t iv) - (I/2r,)(<,
_p
_;],H=
[-!
-1 iv) - (1/2r,)(;,
- irr).
- iv)),
-;
-3.
(48)
Taking a solution to equation (47) in the form M,, = C,, exp(2r,A7) one obtains AC,, = A&
-k i?G
(1 - i2i1rl)C,+,,,
-F (1 -t- i2!,q)Ck.1+1
i a2 - 16$r1 -t*G 4rl
k(k - l)Ck_2,, - 2kK,_,,,_,
-F I(/ - l)C,,,_,
i E2u2r - ‘HC,
8
I.
q
(49)
110
J. W. ROBERTS
The coupling of C,, with higher order components of C in the term of order .si in equation (49) means that direct solution for the eigenvalues 1 is not possible. For the case [
Then substituting in equation (49) and equating terms in like powers of s1 one obtains, for terms in E?, [HOE - Akl] Cl, -F f:
A%;;'=
kG{(l
-
i2!,rl)C;!:,l
-t- (1 + i2i,rl)C;;:,)
j=l
In principle, equation (51) allows the determination of the eigenvalues to any order of the perturbation series (50) by an iterative procedure. The corresponding eigenvectors have, of course, infinite dimension and may be likewise determined recursively from equation (51). Since E{ Y’} = E((Z, -t 2,)‘) exp(--2r[,r)
= { 1,2, l} C,, exp(2r,A - 2r<,) z,
(52)
the condition for mean square stability of Y(r) is seen to be Re(l) < (r/ri) i,
or
Re(;l) < [,( 1 - c:)-“2,
(53)
where A is any characteristic root of equation (49) having a non-zero component vector C 00’ Upon investigating the approximations for the eigenvalues given by equation (51) up to order s* it is found that the eigenvalue corresponding to A.’= 0 has the largest real part, with the series expansion given by A0 = 0, 12
=
r151
2(1 - i:,
(1
-t
4rf)
A’ = 0,
(1 9!:) - (1 - 4r:) (3 -t 5@ _~ 2(1 _ [(l - 4rf)2 -I- 16rfif]
(54)
The condition for mean square stability may then be expressed in terms of the original parameters of equation (34) as E2 16r*(l - <:)(l - ci) - 2(1 + 7# + 2(1 -. it) (1 - 4r2)* -t 16r2(it + ii/2 - 2r2!z)
< 2r3$(l - $(l -
SYSTEM WITH AUTOPARAMETRIC
ill
INTERACTION I
0.07
Figure Gaussian
2. Computed solution.
i
\\ \ :\
0.08
Oc6
-
0.05
-
0.04
-
0.03
-
xx?
-
00
-
mean square
stability
boundaries;
[, = <, = OQO2. ---.
Perturbation
solution;
.
5. All the curves have a characteristic “V” shape with a minimum point at a value of tuning ratio parameter I of 0.5, indicating the strong effect of internal resonance on the stability boundary. In addition, both sets of curves show an instability threshold at r = 0.5 whose value in terms of the coupling parameter E is proportional to the system damping coefficients. However, there are seen to be significant differences between the predicted stability boundaries of the two methods. At I = 05, the stability threshold given by the Gaussian
0.08 -
0.07
0.06
:2
0.05
-
0.04
-
0.03
-
cm2
-
; :: F $ s
0.01
Figure
3. Computed
mean square
stability
boundaries;
i, = i, = 0.01. Key as Figure
2
112
J. W. ROBERTS
1 0.45
1
I
mean square
I
I
1,
0~50 Infernal
Figure 4. Computed
I
stability
imnq
,
,
O-55 rot10,r
boundaries;
(‘, = [, = 0.05. Key as Figure
2
method is consistently 70 y0 higher than that of the perturbation solution. It is also noticeable that the perturbation solution predicts a narrower bandwidth of the instability region than the Gaussian approach. These two effects in combination give rise to good agreement between the two sets of curves over a portion of the range of frequency ratio; but not at the precise condition of internal resonance where there is a sizeable discrepancy.
6. EXPERIMENTAL INVESTIGATION Experimental work was carried out on the model shown schematically in Figure 5. A cantilever beam AB of rectangular cross section was employed as an alternative to the discrete form of primary system shown in Figure 1. This allowed the use of a forcing point where the beam motion was compatible with the range of travel of available vibration
Vlbratlon
Accelerometer
generotor
Current
Figure
5. Experimental
model and excitation
drive
system.
SYSTEM WITH AUTOPARAMETRIC
INTERACTION
113
generators. The coupled system was in the form of a spring steel cantilever BC. A sliding mass D allowed adjustment of the natural frequency ratio in the range 0+06 A requirement for the tests was an essentially flat spectral density of excitation, and the following scheme was adopted to avoid the effect of the complex variation in load impedance inherent in the normal arrangement of a vibration generator coupled to a highly resonant load. First of all, the vibrator armature mass, suspension stiffness and dissipation were treated as integral parts of the dynamical system and were included in the evaluation of the system characteristics. Secondly, a specially designed power amplifier with current feedback control was used to drive the vibrator from a Hewlett-Packard random noise generator. This ensured that the electrodynamic force at the vibrator armature (i.e., the system boundary). being proportional to current, had the smooth spectral density characteristic of the signal source independent of the system motion and load impedance. Finally, an oversize vibrator was used to guarantee linearity of the force characteristic over the range of drivmg current. The current drive system was tested and calibrated by operating it into a blocked load through a force gauge. The low pass filter shown in the excitation system had a cut-off frequency of 35 Hz which was a factor of four times the fundamental primary system resonance frequency. Its inclusion was found necessary to avoid exciting complex coupled motions of the system involving modes of vibration higher than the fundamental. The cut-off frequency was considered high enough to allow an adequately broad band of excitation. The level control was an indexed, lockable ten-turn potentiometer which allowed a precise and repeatable setting of the excitation level. 14n accelerometer located at the tip of the primary beam was coupled to a charge amplifier and double integration circuit to detect primary system response. For the coupled system a pair of strain gauges located at the root of the spring steel cantilever gave a dynamic strain output which was calibrated in terms of the transverse displacement of the coupled system mass. The existence of a transition boundary for the onset of random motions of the coupled system under random force excitation of the primary system was readily established in preliminary tests, and a series of tests was carried out to determine this boundary over a range of values of internal tuning ratio for two values of system mass ratio parameter. There was considerable difficulty in determining a precise boundary. Three characteristic forms of coupled system behaviour were observed at different excitation levels. These were (a) zero motion over the whole observation period, (b) continuous random motion. and (c) partially developed random motion with periods of zero motion, the latter tending to dominate with reducing excitation level. This characteristic made difficult the precise determination of changeover to region (a) since observations of the coupled system mean square response over time intervals of up to twenty minutes showed a large variability. Ultimately, the procedure evolved to determine a boundary for coupled system motion was as follows. The excitation level was increased to a point where definite random motions of the coupled system were observed. The excitation level setting was reduced in stages. measurements of the coupled system mean square response being taken at each stage with observation times of one hour or more used to obtain a consistently decreasing set of estimates. Finally a graph of these readings was extrapolated to locate an excitation level representing the transition to unimodal random response of the system with zero motion of the coupled cantilever. The results gave good agreement with the purely visual observation of the transitional behaviour of the system, but the procedure was very time consuming. The excitation levels determined for the stability boundary were converted to equivalent values of coupling parameter E by making use of an excitation calibration graph of the system with the coupled system constrained to have zero motion. i: was determined directly
114
J. W. ROBERTS
from E = 6x0,/51 (see equations (13)) where x0 is the mean square displacement of the primary system at any excitation level for the case of zero motion of the coupled system. The results are presented in Figures 6 and 7 for two values of system mass ratio. The graphs show the recorded values of coupling parameter F at the instability threshold for a range of values of the internal tuning parameter r. The system natural frequencies and damping coefficients were measured from decay records and the mass ratios were determined by a mass perturbation technique.
7. DISCUSSION
OF RESULTS
Figures 6 and 7 show comparisons between experimental measurements of the system stability boundaries for unimodal response, and the predicted mean square stability boundaries derived from the Gaussian solution of section 5, and from the perturbation solution of section 6. In general, it is seen that the experimental results show the characteristic V-shaped distribution centred in the vicinity of the internal resonance condition r = 0.5, and correlate well with the stability boundaries predicted by both methods. In fact, in the immediate neighbourhood of r = 05, excellent agreement is obtained between the observed and predicted behaviour for both mass ratios. This is a significant feature of the results and gives confidence in using the analytical results to predict the minimum threshold level of coupling parameter for development of random motions of the coupled system. For values of tuning ratio r above and below the internal resonance case and outside its immediate vicinity agreement between experimental results and predicted boundaries is less good and indeed the experimental points give a distinct indication of a wider instability region than predicted by the analytical methods in both graphs. The effect is seen to be more pronounced for the smaller of the two mass ratios used (Figure 6). This is
L 0.44
-,
0.46
I 0.48
/ oeo
Internal
+un,ng
1 ocl2
1 0’54
1 0.5
rotl0.l
Figure 6. Comparison of measured and predicted stability boundaries. (‘, = 09085, Perturbation solution; ----, Gaussian solution; 0, experimental results.
[, = 00030,
R = 0.16.
SYSTEM WITH AUTOPARAMETRIC INTERACTION
115
Figure 7. Comparison of measured and predicted stability boundaries. i, = 0.0070. i, = 09032. R = 0.34. Key as Figure 6.
an interesting result which suggests a degree of inadequacy in both analytical methods. No explanation can be offered at the present time. After changing the system mass ratio by adopting a larger sliding mass on the spring steel cantilever, it was found that a small change in damping constants had taken place. This explains why the predicted curves vary slightly between Figures 6 and 7. The only visible effect of the mass ratio change on the experimental results is the narrowing of the width of the unstable region as discussed in the previous paragraph. There is no distinct change in the instability threshold level in the vicinity of internal resonance. The assumption that the mass ratio term has negligible effect on the stability boundary is supported by results for tuning ratio close to 0.5, but not elsewhere. Finally, it is noted that the values of damping coefficients evaluated from the model lead to predicted stability boundaries from the Gaussian and perturbation solutions which are in very close agreement, the latter giving only a slightly lower instability threshold than the former. The calculations illustrated in Figures 24 of section 5 show that this is not a generai property of the two solutions. Clearly the reported results are insufficient to explore fully the comparative accuracy of the analytical solutions. A further programme of tests carried out with a range of system damping ratios would be required.
8. CONCLUSIONS
Consideration has been given to the determination of the mean square stability boundary for unimodal response of a certain type vibratory system with autoparametric interaction between the modes of vibration. Two analytical solutions based upon approximate treatments of the stochastic equations of motion have been obtained and compared. The two methods give results which agree closely for small values of system damping constants, but which differ appreciably at high damping levels. In both cases the solutions are independent
116
J. W. ROBERTS
of the system mass ratio. Results obtained from measurements tory model under random noise excitation gave extremely predicted stability boundaries in the close neighbourhood indicated a distinctly wider instability region than predicted.
of the responses of a laboragood correlation with both of internal resonance, but
ACKNOWLEDGMENT The investigation reported in this paper was funded by a grant from the Science Research Council. The author gratefully acknowledges this support.
REFERENCES 1. R. A. IBRAHIMand J. W. ROBERTS1976 Journal of Sound and Vibration 44,335-348. Broad band random excitation of a two-degree-of-freedom system with autoparametric coupling. 2. R. A. IBRAHIMand J. W. ROBERTS1977 Zeitschrtft fur Angewandte Mathematik and Mechanik 57, 643-649. Stochastic stability of the stationary response of a system with autoparametric coupling. 3. R. A. IBRAHIMand J. W. ROBERTS1978 The Shock and Vibration Digest 10, 17-38. Parametric vibration, Part V: Stochastic problems. 4. R. S. HAXTON and A. D. S. BARR 1972 Journal of Engineering for Industry, Transactions of the American Society of Mechanical Engineers, Series B 94, 119-125. Autoparametric vibration absorber. 5. L. ARNOLD 1974 Stochastic Differential Equations. New York: John Wiley. 6. I. A. MITROPOLSKII1967 International Journal of Nonlinear Mechanics 2, 69-96. Averaging methods in nonlinear mechanics. 7. R. L. STRATONOVICHand I. M. ROMANOVSKII1965 in Nonlinear Transformations of Stochastic Processes, Paper No. 26 (P. I. Kuznetsov, R. L. Stratonovich and V. I. Tikhonov, Editors). Oxford: Pergamon Press, pp. 322-326. Parametric effect of a random force on linear and nonlinear vibrating systems. 8. R. L. STRATONOVICH1967 Topics in the Theory ofRandom Noise, Volume 2. New York: Gordon and Breach. 9. S. T. ARIARATNAM 1971 in Proceedings of IUTAM Symposium, Herrenalb 1969 Instability of Continuous Systems (H. Leipholz, Editor). Berlin: Springer-Verlag. pp. 78-84. Stability of structures under stochastic disturbances. 10. G. K. BAXTER 1971 Ph.D. Thesis, Syracuse University, New York. The nonlinear response of mechanical systems to parametric random excitation. 11. G. SCHMIDT 1977 in Proceedings of IUTAM Symposium, Stochastic Problems in Dynamics, 1976 (B. L. Clarkson, Editor). London: Pitman. Probability densities of parametrically excited random vibrations. 12. S. T. ARIARATNAM 1967 in Proceedings International Conference on Dynamic Stability of Structures (G. Herrmann, Editor). Oxford: Pergamon, pp. 255-265. Dynamic stability of a column under random loading. 13. W. WEDIG 1972 in Proceedings oj’IUTAM Symposium on Stability of Stochastic Dynamical Systems (R. D. Curtain, Editor). Berlin: Springer-Verlag, pp. 160-172. Regions of instability for a linear system with random parametric excitation.
RANDOM EXCITATION OF A VIBRATORY SYSTEM WITH AUTOPARAMETRIC INTERACTION J. W.
ROBERTS
Department of’Mechanical Engineeriq, University of Edinburgh, Edinburgh EH9 3JL. Scotland (Received 1 March 1979, and in revised,form 77 September 1979)
The paper is concerned with the broad band random excitation of a two degree of freedom vibratory system with non-linear coupling of autoparametric type. A general equation for the evolution of the moments of any order of the response co-ordinates is derived by using stochastic calculus and found to represent an infinite hierarchy set. Consideration is given to the determination of the mean square stability boundary for unimodal response with no transverse motion of the coupled system. Two approximate solutions are obtained. These are first of all a solution based on a Gaussian closure technique applied to the system moment equations which allows the stability condition to be determined from the eigenvalues of a four by four matrix, and secondly a perturbation solution which leads to a simple analytical expression for the stability boundary. The two methods give results in close agreement for low values of system damping, but which differ appreciably at high damping levels. Finally, results are obtained from an investigation of the response regions of a laboratory model excited from a random noise generator. The experimental results are found to give excellent correlation with the predicted instability boundaries in the close neighbourhood of internal resonance but show a distinct indication of a wider instability region than predicted by both analytical methods.
1. INTRODUCTION This paper describes a particular form of to as autoparametric, discrete form of two
a continuation of previous work on random excitation of systems with non-linear coupling between modes of vibration of the type referred being closely related to systems with parametric excitation. The basic
degree of freedom system is shown in Figures l(a) and l(b) and consists essentially of a cantilever beam attached to a primary vibratory system in such a way that response of the primary system imposes an axial motion on the cantilever, thus generating a parametric excitation term in the equation of transverse motion of the coupled cantilever. The transverse motion in turn induces a reaction force on the primary system. The system is a common structural arrangement, representing for example fuselage-tailplane coupling in aircraft structures. The discrete parameters of the primary system of Figure 1 may be interpreted as generalized parameters of a transverse elastic mode of a wide range of linear elastic structures, thus extending the area of application of the present study. This will be made explicit in a further paper. Here broad band random excitation of the primary system is considered as a forced vibration as in Figure l(a), representing for example turbulence or buffet excitation, or a random imposed ground motion as in Figure l(b) representing for example stochastic excitation of a rolling system by a surface profile. In earlier papers [l, 21 approximate techniques and some numerical results for the response parameters of this system were presented and also previous work in this area was discussed. In addition, a recent series of review articles [3] contains an extensive bibliography of work on parametric and autoparametric systems including stochastic excitation. The equations of motion of the system of Figure 1 were first derived (for deterministic excitation) 101 OO22-460)3/80/050101+ 16 $02.00/O
0 1980 Academic Press Inc. (London) Limited
102
J. W. ROBERTS
by Haxton and Barr [4] who showed that large transverse motions of the coupled system could be induced by harmonic forcing of the main system under conditions of internal resonance: that is, when the ratio of the natural frequency of transverse motion of the coupled cantilever to the natural frequency of the primary system was in the neighbourhood of 0.5. Under this condition, the response of the primary system is modified by the reaction of the coupled system in the manner of a vibration absorber.
(a) Figure
1. System model. (a) Random
(b)
force excitation;
(b) ground
motion
excitation.
The significance of the internal resonance condition was found to carry over to the random excitation case by Ibrahim and Roberts [l], who evaluated the system response moments by an approximate numerical integration technique based upon a Markov vector approach combined with a Gaussian approximation of the response moment equations. It was established that large random motions of the coupled system could be generated by random excitation of the primary system for a range of system parameters in the neighbourhood of internal resonance. Otherwise, unimodal response of the primary system was obtained. In a further study [2] the stability boundary for the unimodal response was obtained by linearizing the approximated moment equations and solving numerically the resulting eigenvalue problem. The results showed the significance of the internal tuning ratio, system damping factors and excitation parameter. In the present paper, the results of measurements of the system stability boundary for a laboratory model are compared with predicted levels. The analysis of the system stability is considered by two separate methods. First of all the Gaussian approximation used previously [2] is extended and shown to give results which are independent of the mass ratio parameter. Secondly, an analytical solution for the unimodal stability boundary is obtained from a perturbation solution of the system equations. Comparison is then made with the experimental results.
SYSTEM WITH AUTOPARAMETRIC
103
INTERACTION
2. SYSTEM EQUATIONS OF MOTION For the system of Figure l(a), the equations methods as
of motion may be derived by standard
(M -t m) 2 -t c,1 -k K,x -. (6/51) m(j2 -t yj) = f(t)
(1)
for the primary system and mj -\ C,j -t [K, -. (6/51) mjt] y -1 (36/2512) my(yj -I- 3’) = 0
(2)
for the coupled cantilever. In the case of the ground motion excitation. Figure l(b), the form of equation (1) remains valid if x denotes relative motion of the mass M, (M -)_m)T -t- C,st -b K,x
-. (6/51) m(j’ -1 yj) = -(M
-k m)ii(t).
(3)
while the cantilever equation is modified by the ground acceleration, mj -k C,j
-k [K,
- (6/51) m(Z + ii)] y -F-(36/251*) my(yjj -t j2) = 0.
(4)
The form of autoparametric coupling action is clear from equations (1) and (2). In equation (2) the x motion appears as a time dependent modification of the cantilever stiffness, i.e.. a parametric excitation, while in equation (1) the cantilever motion appears as a quadratic non-linearity representing the reaction force on the primary mass. In equation (4) the parametric excitation term also contains the ground acceleration and thus the ground motion excitation case produces a slightly different equation of motion. It is considered that for a lightly damped primary system the additional term will have a negligibly small effect on the interaction between the two systems under internal resonance conditions. Accordingly, equations (1) and (2) are taken as representative of both forms of system excitation. It is desirable to write equations (1) and (2) in a dimensionless form. Upon making the transformations x = x/x,, Y = y/x,, r = or t,
(5)
where x0 is some convenient reference displacement, the equations become X” -t 1(,X’ -1 X - ER(Y’~ -1 YY”) = W(z), Y” -t- 2[,rY’ -I- (r2 - .5X”) Y -1 E*( Y2 -1 YY”) Y = 0,
(‘7)
where 0): = K,I(M
-b m),
wz = K,/m,
cl = C,/2(M
-F m)o,,
;, = C2/2mw2.
In equations (6) and (7) the following system parameters have been introduced: mass ratio R = m/(M -k m); tuning ratio r = w,/co 1; coupling parameter E = 6x,/51 (assumed small). The prime denotes differentiation o.r.t. z. The excitation function in equation (6) is w(r) = {(M -k m) co: x,} -I
f‘(s/o,).
(8)
It is assumed that the real time excitation f(t) is a zero mean, stationary Gaussian process with a smooth spectral density S, extending to some frequency well above o,, where S, = 2x L Then, from equation (8),
W(T)
00 E{f(t) f’(t -t t,)} e-‘“‘I dt,. (W s 1 may be regarded as ideal Gaussian white noise having auto-
104
J. W. ROBERTS
correlation
function R(u) = E[ W(z) W(z -1 u)] = 20 6(u),
(l(J)
D = ~cS~/(M -1 m)2 cofxi
(III
where D is given by and 6( ) denotes the Dirac delta function. The reference displacement x0 may now be chosen with advantage to be the root mean square response of the primary system for the case of zero coupled system motion. With y(t) = 0 in equation (1) one obtains: [E{~~(t)}]i’~ = [l/(M -f m)] [rcS,/2[,~~]‘~~ = x0.
(12)
Then D = 25,,
F = 6x,/51 = [6/51(M + m)] [zS~/~<,O~]‘~~.
(13)
In this way the standardized excitation W(r) is made independent of the practical system excitation level, whereas the latter is embodied in the coupling parameter E. 3. SYSTEM
EQUATIONS
IN TERMS
OF STOCHASTIC
INTEGRALS
The assumption of ideal white noise for W(r) is convenient analytically, but implies that the system equations are not strictly valid, since W(r) is not integrable in the ordinary sense. Formally, one may write W(r) = dB/dz where B(z) is a Brownian motion process. The system equations (6) and (7) may then be written in terms of stochastic integrals. To do this it is necessary to eliminate the non-linear acceleration terms and this may be done by a process of repeated substitution up to any order of parameter E, Carrying this out to order s2 and letting X, = X, X2 = I: X, = X’ and X, = Y’, one obtains dX, = X, dr,
dX, = X, dz,
dX, = (-2,“,X,
-k s2R[-2:rX;X3
dX, = { -25,rX, -t s2p[2<,r X:X,
- X, -k ER[X;
- 2r[,X,X,
- ?X;]
- X,X;]} dz -t- (1 + s2RX;} dB(z), - r2X2 + E[ -x,x, -t- r2Xi - X,X:]}
- 2&x,X,] dr -1 &X2dB(r),
where p = 1 - R. In equations (14), dB(r) is an incremental with properties
(14)
Brownian motion process
E(dB’} = 20 dz.
E{dB} = 0,
(15)
Since W(z) represents the limiting case of a smooth wide band process the system equations (14) should be interpreted as a set of stochastic differential equations of Stratonovich type corresponding to the general form [5] dX, = f;.(X, z) dz + $
Gij(X, z) dB,,
i = 1,2 )...)
II.
(16)
j=l
Equation (16) may in turn be transformed into a stochastic equation of It8 type: 6(X, z) -t $ f k=l
1
dr -1 2 GJX, r) dB,,
$ Gkjg j=l
k
i = 1,2 ,...,
n.
(17)
j=l
It is seen that for the case of the system equations (14) the additional terms in equations (17) are zero, so that equations (14) may be interpreted directly as a set of equations of It6 type. It follows that the system response vector X(r) constitutes a Markov process.
105
SYSTEM WITH AUTOPARAMETRIC INTERACTION
A general solution for the transition probability of X is not possible. However. a set of differential equations for the joint moments of any order of the components of X may be obtained directly from equations (14) by using the methods of stochastic calculus [5]. With introduction of the scalar function (18) 4 = (x;xjxk,x;) and the joint moment of the system response co-ordinates
of order N = ti + ,,i + k -t IL
* m..tJkl =
X;XjX;X;
SSR
p(X. z)
dX = E{c$}.
(1’))
d--l
the fundamental theorem of It8 may be used in the form
’{dx,} -k :Trace[G]
d4 =
[Q] [GIT[&]
(20)
dr, I
I
where [Q] dz = E{dB,} {dB,}‘. Substituting in equation (20) from equation (14) and taking expectations of both sides leads to rn&, =: im.r-l.j.k+l.l
-‘jmi,j-l,k.l+l -
-’ -’
2,,rlm,
‘b2~lrni
.j+l
2;lkmi,j.k.I
-
kmi+l.j.k~l.l
. j . k, i - r21mi . j+, .k. ,_, -k Dktk -
E{Rk[mi,j,k-l,l+2
_I- E2.(Rk[-2~lmi.j+2.k,i -
-
-
,k+l.l-I -
2~2rmi.j+l,k-l.l+l -
nLi+
L.j+l,k.I&ll
"'i+l,j+2.I-I,11
m.l.J+l.k,1+11 -’ D’(’ -
-
1)mi,j+2,k,1-2)
1) m,, j . k ‘2mi,j+2.k-l.ll
-' 2Dk'mi,j+l.k~l.~&l~
-' d[2~2rmi.j+2.k,I -’
2.1
2RDk(k
-
-' r2mi.j+3,k.I 1)mi.j+2.k-2.1’
-1 (71)
The system of linear first order differential equations (21) determines the evolution in time of the system response moments of any order. Inspection of equations (21) shows that for non-zero coupling parameter E the equations for moments of any specific order are coupled with higher order moments, so that the system (21) represents an infinite hierarchy of moment equations and is incapable of explicit solution. This situation was discussed in the previous papers [l, 21 and an approximate truncation scheme was introduced based on a Gaussian approximation of moments higher than second order which permitted evaluation of the response first and second moments by numerical integration. Further inspection of system (21) shows that it is satisfied by the following stationary solution: m 2,joo =
E{X2j = D/25, = 1, m,,o2o = E(X”)
= D,i25, = 1, all other moments zero. (22)
This solution represents unimodal response of the primary system with zero transverse motion of the cantilever. The mean square stability threshold of this solution in terms of the system characteristics and excitation parameter is of obvious practical interest and significance, and is investigated in the following sections. An alternative set of system equations in terms of stochastic integrals may be obtained by application of the method of stochastic averaging. The method is an extension of the well known averaging method for non-linear systems [6] developed for problems with randomly varying parameters by Stratonovich and Romanovskii [7]. The method is fully described in reference [8], and has been applied extensively to obtain useful results in a number of random parametric problems [9-121. In the present problem the resulting system equations depend upon joint moments of the averaged response co-ordinates and do not lead to a useful solution.
106
J. W. ROBERTS
4. MEAN
SQUARE
STABILITY
OF THE UNIMODAL APPROXIMATION
SOLUTION-GAUSSIAN
To investigate the behaviour of the system (21) in the neighbourhood solution one may introduce mijkl
=
of the unimodal (23)
m$, -1Sijk,
is the stationary solution (22) and 6 represents a set of small in equations (21) where rn$), perturbations. For small variations of the system moments in the neighbourhood of the stationary solution it is assumed that the response processes are well represented as Gaussian. Using the standard relations for third and fourth moments into (21) and neglecting products of the 6’s leads to the following linearized equation for the moment perturbations of first and second order [2] :
‘ijkl
=
(24)
“ijkl>
where C is a 14 x 14 matrix. Introducing the form of solution 6..IJkl = 6!‘ ) e!.’ rjkl rJki’
(25)
C&O’ = 1/6’ (J) ijkl’ LJkl
(26)
leads to
It is clear that the stability of the stationary unimodal solution may be determined from the nature of the eigenvalues of the square matrix C. By ordering the components of the vector Bijklin a specific way the eigenvalue problem (26) may be formulated as ~
$0)
0
1
g(O) 2
0
= A SO,
$0’ 3
0
(27)
go’ 4
C (6 x446)_ where
sy = [S 1000~00101=~e
= C~2000 6,010~00201=~8:o) = PO200~0,01~00021=~
Sk”’= [6 0100 60001 6 1100 60110 6 1001 60011 y, 0
1
Cl1 =
I
02
c,,
=
-1
0
0 -21, -2
1 -41,
i -1
> c,,=
1 -2;,
(28)
1’
0
-r2 r:24[1
2
0
-2ri,
1
-2r2
-4rr,
I ,
SYSTEM WITH AUTOPARAMETRIC
0
-V2
c44=
1 -2r[,
0 -E
0
INTERACTION
0
0
-2;,E
0
0
0
0
0
1
1
0
0
0
-1
-2:,
0
1
--6
0
2c,t
0
--r2 0
0
-2r[,
-Y2
-1
107
1 -2(:“,
-t r(,)
r
The block diagonal form of equation (27) with zero or one-way coupling between the element matrices allows a considerable simplification of the initial (14 x 14) eigenvalue problem. First of all C,, is seen to be uncoupled from the remaining elements and its eigenvalues contribute independently to the set of eigenvalues of C. For the remaining eighth order matrix, it is seen that C,,, C,, and C,, are coupled in the forward direction only. A property of this matrix structure is that the eigenvalues of the complete matrix are simply the set of eigenvalues of the diagonal blocks. The diagonal block matrices of C may therefore be examined independently to investigate stability of the moment perturbations. It may be shown without difficulty that the eigenvalues of C,, and C,, are stable for non-zero [I and need not be considered further. In the case of C,, application of the RouthHurwitz criterion leads to a simple condition for stable eigenvalues in terms of the coupling parameter: E2 < r3 :,/:,.
(30)
This condition does not reflect the sensitivity to internal resonance found in the experimental work and corresponds in fact to parametric instability of the coupled system due to the non-resonant spectral density of the axial acceleration term %.The result is analogous to that of Ariaratnam [12] for the classical linear stochastic parametric problem. Finally, the eigenvalues of C,, need to be considered. Numerical evaluation is required but these indeed give rise to an instability boundary, sensitive to the internal resonance condition r CY0.5. Identical results for stability boundaries were obtained from C,, to those illustrated in the previous paper [2], obtained by direct computation from the 14 x 14 characteristic matrix. Additionally, it is clear from equations (29) that the system mass ratio R plays no part in the elements of C,, and cannot influence the stability boundary. Consequently, the reaction force on the primary system due to the coupled system motion has no effect on the stability threshold so that for stability analysis the system is analogous to the problem of the cantilever excited parametrically by filtered noise, the primary system acting essentially as a linear filter. 5. PERTURBATION SOLUTION FOR THE INSTABILITY THRESHOLD OF THE UNIMODAL RESPONSE The stochastic stability of the unimodal response may be investigated by an alternative method to the Gaussian technique of the previous section. Wedig has used a perturbation method to study the stability of a linear system parametrically excited by band pass filtered white noise [13], and his method may be extended to the present system by considering
108
J. W. ROBERTS
small motions of the coupled system in the vicinity of the unimodal response Y s 0 for small values of coupling parameter E. The system equations (6) and (7) may then be approximated by X” + 25,X’ -t- x -. &R(Y’Z-t YY”) = W(r),
(31)
Y” -I- 2<,rY’ -k (r’ -. EX”) Y = 0,
(32)
where cubic terms in e2 have been neglected. It is assumed that Y(r) is small in equation (32). Take X(z) - X,(r) + X,(r) where X,(r) is a variational term, assumed small and X,(7) is the unimodal response, satisfying xb’ + 2(,X;
-t x, = W(z).
(33)
Then equation (32) becomes, with neglect of the small quadratic term, Y,’-1 2r[,Y’ -k (7’ - &Xi) Y = 0.
(34)
Approximated in this way for small motions of the coupled system the system equations (33) and (34) show that the primary system acts essentially as a linear filter whose acceleration response appears as a parametric excitation of the coupled system. The involvement of acceleration in the parametric term means that the system differs in detail from that studied by Wedig. Upon introduction of the transformation Y(7) = Z(7) exp( - rc2z), equation (34) becomes Z” -t r:[l
- e,Xi] Z = 0,
s1 = s/r2(1 - $,
(35)
rf = r(1 - 5:).
(36)
Equation (35) may be further written as two first order equations by transforming Z into amplitude and phase components, rewritten in complex exponential form as Z = C cos @ = $ ei@ -t $C emi@s Z, -k Z,, E ir,(Z, - Z,),
z’ = r,Csin @
(37)
where Z*1 = Z 2’ Then Zl, = i rl[Z,
- E~+Xi(Z,
-t Z,)],
Z2 = -i r1[Z2 - Ed$Xg(Z,
-b Z,)].
(38)
It is useful to write the primary system equation in the form: xb’ + 2:,x;
r2 + $)X0 -t (,i
=
W(T),
(39)
where q ’ = (1 - CT)2 1. Equation (39) may then be factored into complex components in the following way. First one writes x, = z, -k z,,
zx = z,.
(40)
Then Z; -k ([, -t iv) Z, = iW/2r], Zk -k ([, - iv) Z, = -i W/2r7.
(41)
Using expression (41) in equation (39) one obtains, for X$ xb’ = w + cc; - q2 -1 i2@r) Z, -I- (I: - q2 - i2qcl) Z, = W - (1 - i2@,) Z, - (1 -I- i21!,) Z,,
(42)
it being assumed that 21: @ 1. If it is now assumed that W(7) is an ideal white process then equations (41) may be replaced
109
SYSTEM WITH AUTOPARAMETRIC INTERACTION
by stochastic differential equations of It6 type. With use made of the formal relationship W(z) = c1dB/dr, 143) where c( is a constant and /I(z) is a Brownian motion process with unit variance parameter. then. from equations (10) and (13) c( = [20] i/z = 2[;,2. 144)
By using equation (42) in equations (38) a set of equations of It6 type in the transformed system co-ordinates Zj may be written as follows: dZ, == ir,Z, i
-F r:,i%(l
+ Eli $(l
- i2n<,)Z,Z,
+ i2q[l)Z,Z,
-1 sii?(l
-1 s,i?
- i2q
(l+i2q<,)Z,Z,
dz - is,+(Z,
-F Z,)a dfi.
1 d%, == -ir,Z,
- ic, + (1 - i2r][,) Z,[Z,
-k Z,] - it;, 2 (1 -!- i2q;i) Z,[Z,
-1 Z,]
1
d7 I
-b
is1 ? (Z, -F Z,) a dp, dZ, = -- (; 1 - iv) Z, dz - i 2; d/I.
-t iv) Z, dr -t i $ dfl.
dZ, = -(:,
(45)
Now one can define a vector of joint moments of the complex response co-ordinates Zi : M,‘, = E{Z:Z;Z;,
Z,Z,Z;Z;,
Z;Z”,Z;}.
(46)
Applying the It6 theorem (20) to the elements of the vector (46) one obtains ultimately a linear first order differential equation for M,, in terms of lower and higher order moments :
;M;,=A,,&,
?G{(l-
+
- &
1
{k(k - l)M,-2.1
-t %G{kM,,,,
p2a2r - -lHM,.l ‘I 8
-F iv) - (1/2r,)(i, - i - (k/2r,)(<,
G=
I_:
-t- 1(1- l)M,,&
- 2klM,-,,,-,
- M,,,_,}
where A,, = Diag{i - (k/2r,)(!,
-k (1 -F i26,rl)M,,,+,)
Z,vl)M,+,,,
(47)
- ig), - (k/2r,)([,
-t iv) - (I/2r,)(<,
_p
_;],H=
[-!
-1 iv) - (1/2r,)(;,
- irr).
- iv)),
-;
-3.
(48)
Taking a solution to equation (47) in the form M,, = C,, exp(2r,A7) one obtains AC,, = A&
-k i?G
(1 - i2i1rl)C,+,,,
-F (1 -t- i2!,q)Ck.1+1
i a2 - 16$r1 -t*G 4rl
k(k - l)Ck_2,, - 2kK,_,,,_,
-F I(/ - l)C,,,_,
i E2u2r - ‘HC,
8
I.
q
(49)
110
J. W. ROBERTS
The coupling of C,, with higher order components of C in the term of order .si in equation (49) means that direct solution for the eigenvalues 1 is not possible. For the case [
Then substituting in equation (49) and equating terms in like powers of s1 one obtains, for terms in E?, [HOE - Akl] Cl, -F f:
A%;;'=
kG{(l
-
i2!,rl)C;!:,l
-t- (1 + i2i,rl)C;;:,)
j=l
In principle, equation (51) allows the determination of the eigenvalues to any order of the perturbation series (50) by an iterative procedure. The corresponding eigenvectors have, of course, infinite dimension and may be likewise determined recursively from equation (51). Since E{ Y’} = E((Z, -t 2,)‘) exp(--2r[,r)
= { 1,2, l} C,, exp(2r,A - 2r<,) z,
(52)
the condition for mean square stability of Y(r) is seen to be Re(l) < (r/ri) i,
or
Re(;l) < [,( 1 - c:)-“2,
(53)
where A is any characteristic root of equation (49) having a non-zero component vector C 00’ Upon investigating the approximations for the eigenvalues given by equation (51) up to order s* it is found that the eigenvalue corresponding to A.’= 0 has the largest real part, with the series expansion given by A0 = 0, 12
=
r151
2(1 - i:,
(1
-t
4rf)
A’ = 0,
(1 9!:) - (1 - 4r:) (3 -t 5@ _~ 2(1 _ [(l - 4rf)2 -I- 16rfif]
(54)
The condition for mean square stability may then be expressed in terms of the original parameters of equation (34) as E2 16r*(l - <:)(l - ci) - 2(1 + 7# + 2(1 -. it) (1 - 4r2)* -t 16r2(it + ii/2 - 2r2!z)
< 2r3$(l - $(l -
SYSTEM WITH AUTOPARAMETRIC
ill
INTERACTION I
0.07
Figure Gaussian
2. Computed solution.
i
\\ \ :\
0.08
Oc6
-
0.05
-
0.04
-
0.03
-
xx?
-
00
-
mean square
stability
boundaries;
[, = <, = OQO2. ---.
Perturbation
solution;
.
5. All the curves have a characteristic “V” shape with a minimum point at a value of tuning ratio parameter I of 0.5, indicating the strong effect of internal resonance on the stability boundary. In addition, both sets of curves show an instability threshold at r = 0.5 whose value in terms of the coupling parameter E is proportional to the system damping coefficients. However, there are seen to be significant differences between the predicted stability boundaries of the two methods. At I = 05, the stability threshold given by the Gaussian
0.08 -
0.07
0.06
:2
0.05
-
0.04
-
0.03
-
cm2
-
; :: F $ s
0.01
Figure
3. Computed
mean square
stability
boundaries;
i, = i, = 0.01. Key as Figure
2
112
J. W. ROBERTS
1 0.45
1
I
mean square
I
I
1,
0~50 Infernal
Figure 4. Computed
I
stability
imnq
,
,
O-55 rot10,r
boundaries;
(‘, = [, = 0.05. Key as Figure
2
method is consistently 70 y0 higher than that of the perturbation solution. It is also noticeable that the perturbation solution predicts a narrower bandwidth of the instability region than the Gaussian approach. These two effects in combination give rise to good agreement between the two sets of curves over a portion of the range of frequency ratio; but not at the precise condition of internal resonance where there is a sizeable discrepancy.
6. EXPERIMENTAL INVESTIGATION Experimental work was carried out on the model shown schematically in Figure 5. A cantilever beam AB of rectangular cross section was employed as an alternative to the discrete form of primary system shown in Figure 1. This allowed the use of a forcing point where the beam motion was compatible with the range of travel of available vibration
Vlbratlon
Accelerometer
generotor
Current
Figure
5. Experimental
model and excitation
drive
system.
SYSTEM WITH AUTOPARAMETRIC
INTERACTION
113
generators. The coupled system was in the form of a spring steel cantilever BC. A sliding mass D allowed adjustment of the natural frequency ratio in the range 0+06 A requirement for the tests was an essentially flat spectral density of excitation, and the following scheme was adopted to avoid the effect of the complex variation in load impedance inherent in the normal arrangement of a vibration generator coupled to a highly resonant load. First of all, the vibrator armature mass, suspension stiffness and dissipation were treated as integral parts of the dynamical system and were included in the evaluation of the system characteristics. Secondly, a specially designed power amplifier with current feedback control was used to drive the vibrator from a Hewlett-Packard random noise generator. This ensured that the electrodynamic force at the vibrator armature (i.e., the system boundary). being proportional to current, had the smooth spectral density characteristic of the signal source independent of the system motion and load impedance. Finally, an oversize vibrator was used to guarantee linearity of the force characteristic over the range of drivmg current. The current drive system was tested and calibrated by operating it into a blocked load through a force gauge. The low pass filter shown in the excitation system had a cut-off frequency of 35 Hz which was a factor of four times the fundamental primary system resonance frequency. Its inclusion was found necessary to avoid exciting complex coupled motions of the system involving modes of vibration higher than the fundamental. The cut-off frequency was considered high enough to allow an adequately broad band of excitation. The level control was an indexed, lockable ten-turn potentiometer which allowed a precise and repeatable setting of the excitation level. 14n accelerometer located at the tip of the primary beam was coupled to a charge amplifier and double integration circuit to detect primary system response. For the coupled system a pair of strain gauges located at the root of the spring steel cantilever gave a dynamic strain output which was calibrated in terms of the transverse displacement of the coupled system mass. The existence of a transition boundary for the onset of random motions of the coupled system under random force excitation of the primary system was readily established in preliminary tests, and a series of tests was carried out to determine this boundary over a range of values of internal tuning ratio for two values of system mass ratio parameter. There was considerable difficulty in determining a precise boundary. Three characteristic forms of coupled system behaviour were observed at different excitation levels. These were (a) zero motion over the whole observation period, (b) continuous random motion. and (c) partially developed random motion with periods of zero motion, the latter tending to dominate with reducing excitation level. This characteristic made difficult the precise determination of changeover to region (a) since observations of the coupled system mean square response over time intervals of up to twenty minutes showed a large variability. Ultimately, the procedure evolved to determine a boundary for coupled system motion was as follows. The excitation level was increased to a point where definite random motions of the coupled system were observed. The excitation level setting was reduced in stages. measurements of the coupled system mean square response being taken at each stage with observation times of one hour or more used to obtain a consistently decreasing set of estimates. Finally a graph of these readings was extrapolated to locate an excitation level representing the transition to unimodal random response of the system with zero motion of the coupled cantilever. The results gave good agreement with the purely visual observation of the transitional behaviour of the system, but the procedure was very time consuming. The excitation levels determined for the stability boundary were converted to equivalent values of coupling parameter E by making use of an excitation calibration graph of the system with the coupled system constrained to have zero motion. i: was determined directly
114
J. W. ROBERTS
from E = 6x0,/51 (see equations (13)) where x0 is the mean square displacement of the primary system at any excitation level for the case of zero motion of the coupled system. The results are presented in Figures 6 and 7 for two values of system mass ratio. The graphs show the recorded values of coupling parameter F at the instability threshold for a range of values of the internal tuning parameter r. The system natural frequencies and damping coefficients were measured from decay records and the mass ratios were determined by a mass perturbation technique.
7. DISCUSSION
OF RESULTS
Figures 6 and 7 show comparisons between experimental measurements of the system stability boundaries for unimodal response, and the predicted mean square stability boundaries derived from the Gaussian solution of section 5, and from the perturbation solution of section 6. In general, it is seen that the experimental results show the characteristic V-shaped distribution centred in the vicinity of the internal resonance condition r = 0.5, and correlate well with the stability boundaries predicted by both methods. In fact, in the immediate neighbourhood of r = 05, excellent agreement is obtained between the observed and predicted behaviour for both mass ratios. This is a significant feature of the results and gives confidence in using the analytical results to predict the minimum threshold level of coupling parameter for development of random motions of the coupled system. For values of tuning ratio r above and below the internal resonance case and outside its immediate vicinity agreement between experimental results and predicted boundaries is less good and indeed the experimental points give a distinct indication of a wider instability region than predicted by the analytical methods in both graphs. The effect is seen to be more pronounced for the smaller of the two mass ratios used (Figure 6). This is
L 0.44
-,
0.46
I 0.48
/ oeo
Internal
+un,ng
1 ocl2
1 0’54
1 0.5
rotl0.l
Figure 6. Comparison of measured and predicted stability boundaries. (‘, = 09085, Perturbation solution; ----, Gaussian solution; 0, experimental results.
[, = 00030,
R = 0.16.
SYSTEM WITH AUTOPARAMETRIC INTERACTION
115
Figure 7. Comparison of measured and predicted stability boundaries. i, = 0.0070. i, = 09032. R = 0.34. Key as Figure 6.
an interesting result which suggests a degree of inadequacy in both analytical methods. No explanation can be offered at the present time. After changing the system mass ratio by adopting a larger sliding mass on the spring steel cantilever, it was found that a small change in damping constants had taken place. This explains why the predicted curves vary slightly between Figures 6 and 7. The only visible effect of the mass ratio change on the experimental results is the narrowing of the width of the unstable region as discussed in the previous paragraph. There is no distinct change in the instability threshold level in the vicinity of internal resonance. The assumption that the mass ratio term has negligible effect on the stability boundary is supported by results for tuning ratio close to 0.5, but not elsewhere. Finally, it is noted that the values of damping coefficients evaluated from the model lead to predicted stability boundaries from the Gaussian and perturbation solutions which are in very close agreement, the latter giving only a slightly lower instability threshold than the former. The calculations illustrated in Figures 24 of section 5 show that this is not a generai property of the two solutions. Clearly the reported results are insufficient to explore fully the comparative accuracy of the analytical solutions. A further programme of tests carried out with a range of system damping ratios would be required.
8. CONCLUSIONS
Consideration has been given to the determination of the mean square stability boundary for unimodal response of a certain type vibratory system with autoparametric interaction between the modes of vibration. Two analytical solutions based upon approximate treatments of the stochastic equations of motion have been obtained and compared. The two methods give results which agree closely for small values of system damping constants, but which differ appreciably at high damping levels. In both cases the solutions are independent
116
J. W. ROBERTS
of the system mass ratio. Results obtained from measurements tory model under random noise excitation gave extremely predicted stability boundaries in the close neighbourhood indicated a distinctly wider instability region than predicted.
of the responses of a laboragood correlation with both of internal resonance, but
ACKNOWLEDGMENT The investigation reported in this paper was funded by a grant from the Science Research Council. The author gratefully acknowledges this support.
REFERENCES 1. R. A. IBRAHIMand J. W. ROBERTS1976 Journal of Sound and Vibration 44,335-348. Broad band random excitation of a two-degree-of-freedom system with autoparametric coupling. 2. R. A. IBRAHIMand J. W. ROBERTS1977 Zeitschrtft fur Angewandte Mathematik and Mechanik 57, 643-649. Stochastic stability of the stationary response of a system with autoparametric coupling. 3. R. A. IBRAHIMand J. W. ROBERTS1978 The Shock and Vibration Digest 10, 17-38. Parametric vibration, Part V: Stochastic problems. 4. R. S. HAXTON and A. D. S. BARR 1972 Journal of Engineering for Industry, Transactions of the American Society of Mechanical Engineers, Series B 94, 119-125. Autoparametric vibration absorber. 5. L. ARNOLD 1974 Stochastic Differential Equations. New York: John Wiley. 6. I. A. MITROPOLSKII1967 International Journal of Nonlinear Mechanics 2, 69-96. Averaging methods in nonlinear mechanics. 7. R. L. STRATONOVICHand I. M. ROMANOVSKII1965 in Nonlinear Transformations of Stochastic Processes, Paper No. 26 (P. I. Kuznetsov, R. L. Stratonovich and V. I. Tikhonov, Editors). Oxford: Pergamon Press, pp. 322-326. Parametric effect of a random force on linear and nonlinear vibrating systems. 8. R. L. STRATONOVICH1967 Topics in the Theory ofRandom Noise, Volume 2. New York: Gordon and Breach. 9. S. T. ARIARATNAM 1971 in Proceedings of IUTAM Symposium, Herrenalb 1969 Instability of Continuous Systems (H. Leipholz, Editor). Berlin: Springer-Verlag. pp. 78-84. Stability of structures under stochastic disturbances. 10. G. K. BAXTER 1971 Ph.D. Thesis, Syracuse University, New York. The nonlinear response of mechanical systems to parametric random excitation. 11. G. SCHMIDT 1977 in Proceedings of IUTAM Symposium, Stochastic Problems in Dynamics, 1976 (B. L. Clarkson, Editor). London: Pitman. Probability densities of parametrically excited random vibrations. 12. S. T. ARIARATNAM 1967 in Proceedings International Conference on Dynamic Stability of Structures (G. Herrmann, Editor). Oxford: Pergamon, pp. 255-265. Dynamic stability of a column under random loading. 13. W. WEDIG 1972 in Proceedings oj’IUTAM Symposium on Stability of Stochastic Dynamical Systems (R. D. Curtain, Editor). Berlin: Springer-Verlag, pp. 160-172. Regions of instability for a linear system with random parametric excitation.