Parametric instability in structures under support motion

J. Sound Rib. (1971) 14 (4), 491-509

PARAMETRIC

INSTABILITY SUPPORT

IN STRUCTURES

UNDER

MOTION-f

A. D. S. BARRAND D. C. MCWHANNELL Department of Mechanical Engineering, University of Edinburgh, Edinburgh, Scotland (Received 5 June 1970)

Vertical ground accelerations are shown to act in a parametric manner on the lateral oscillation of building structures and hence to influence substantially their response under certain conditions. As an initial approach to this type of problem, the parametric action of a periodic vertical support motion on the lateral oscillation of a simple “shear-type” building is examined theoretically. The experimental demonstration of the various instability regions for simple two- and five-degree-of-freedom structures of this type is described and the results of analytical and computational procedures for the estimation of the regions are given. Another type of instability region is also shown to exist when the input is quasiperiodic in the form of two inputs of different frequency; the analysis is restricted to that of a single-degree-of-freedom structure with equal acceleration amplitude inputs. The problem of autoparametric interaction within a vibrating structure is introduced by an analysis of the motion of a simplified portal frame, and some of the possible consequences of internal resonance situations are discussed. 1. INTRODUCTION The dynamic stabiiity of structures under parametric loading is an area of considerable current interest in mechanics. This paper deals with certain simple types of structure in which the parametric loads arise from inertia forces in the structure due to motion of its supports. A practical instance of support excitation arises in the case of buildings subjected to earthquake motion of the ground. The horizontal component of ground motion acts as an ordinary

forcing effect on the lateral displacement of the structure. The vertical component of ground motion, which is commonly disregarded, acts, however, in a parametric manner on the lateral displacement, that is it appears within the coefficients of the structural displacement in the equations of motion. Further, if the vector of ground displacement is periodic in time, then the response problem is mathematically one of the solution of a non-homogeneous system of equations having periodic coefficients. For a linear structure this problem can be solved in principle, but it may in practice present considerable computational difficulties, particularly for a large structure of many degrees of freedom. In many cases the periodic fluctuations in the coefficients will be relatively small and, provided the parameters of the homogeneous equation (i.e. vertical ground motion only) are well removed from any instability region of the parameter space, the coefficients may be regarded as constants, and response under the horizontal ground motion alone considered. If, on the other hand, the parameters of the vertical ground motion and the structure are such as to lie in an unstable zone of the solution of the homogeneous equation, then the structure’s lateral displacement will grow exponentially with time, with what seems in experimental work, t Paper presented at the symposiumon “Structural dynamics”held at The University of Technology, Loughborough,

Leicestershire,

England, 23 to 25 March 1970. 491

492

A. D. S. BARR AND D. C. MCWHANNELL

in extreme cases, to be an almost explosive violence. The final amplitude is then limited only by inertial and elastic non-linearities within the structure. In view of this situation it is necessary to be able to estimate the position and extent of the regions of the parameter space that are associated with unstable solutions of the homogeneous equation, i.e. the equation under vertical excitation only. In the analysis, the damping of the structure has been neglected; there are two reasons for this. First, the mathematics are generally simpler and shorter, and, second, in most cases the resulting predictions are “safe” in the sense that if a point lies outside an instability region when the damping is zero it will also be outside it if the damping is finite though small. This statement is not necessarily true of a combination instability, where the width of the unstable region depends on the ratio of the damping in the two modes involved; that is, the introduction of damping to a different extent in each mode may produce a larger region of instability (see, for instance, reference [l]). This has not yet been demonstrated experimentally. The approximate result quoted for combination regions in section 3.2.2 below is actually applicable only for an equal, though infinitesimal, damping ratio in each mode. Such equality is often approximated in real structures. In the damped situation the instability regions tend to recede so that a finite excitation is required before the region is entered. Points in the area between the undamped instability region and its included damped region are points of reduced effective damping, so that for such points, although the vertical motion does not cause lateral instability, it can reduce the overall lateral damping of the structure. In many actual situations the ground motion is by no means steady sinusoidal, and more difficult transient analysis might be required. As a step towards other realistic input conditions the case of a two-frequency input is considered in section 3.4 for a single-degree-offreedom system. Some earthquake seismograph records show a beating form of motion for which this is a first approximation. Autoparametric effects are discussed briefly in sections 2.2 and 5. The non-linear interaction terms which arise because of the motion of a structure can be important in determining its response if “internal resonance” conditions apply.

2. EQUATIONS

OF MOTION

The equations of motion under vertical ground motion of two types of simple structure are given below. The first case is that of a simple multi-storey shear structure; non-linear inertial and stiffness effects, which become increasingly significant with growth in response amplitude, are not included on the grounds that such terms do not influence the onset of instability-which is the principal concern of this paper. The second case, a simple portal frame, is considered in finer detail and certain more important non-linear terms are retained in the equations because these terms are the basis of the autoparametric action. 2.1.

MULTI-STOREY

SHEAR-TYPE

STRUCTURE

Figure 1 indicates the general form of structure under consideration. For simplicity, the floors are assumed rigid of mass m and the interconnecting columns are all taken to be identical having length I, rigidity EI and negligible mass. For small displacements the floor motion is parallel to the ground, which itself has a vertical motion u(t) which will generally be assumed to be periodic. Due to the acceleration ii(t), the various columns are subjected to axial forces which modify their lateral stiffnesses. This interaction between axial force and lateral stiffness is evaluated by statical considerations on a beam of length 1 restrained to zero slope at each end and subjected to a combined axial load and lateral shear force.

PARAMETRIC

For thejth

floor of a structure

INSTABILITY

493

IN STRUCTURES

having IZfloors the equation

of motion 1
?Wfj=-2kj(Xj-Xj_~)+2kj+~(Xj+~-Xj) where kj is the lateral stiffness of one of the columns found that

can be written as

immediately

below the jth floor. It is

The first term in k, represents the usual lateral stiffness of the column, while the second term brings in the effect of gravity and the axial acceleration ii, both of which add to provide an axial force which varies down the structure, having greatest magnitude at the base.

Figure

1. n-Storey

building.

If, further, the base motion is sinusoidal of the form u = ccos!&, of motion for an n-storey building can be written in the form

then the matrix equation

A%+[C-(cr+pcosQt)S]x=O,

(1)

where x = co1 {x,, x2,. . ., x,>, the mass matrix A = mE (E being the unit matrix), stiffness matrix in the absence of gravity and is in the symmetric tridiagonal form

c _

-I

24E1 13

r -12 The scalars a and /3 are given by

-12 -I

3mg

(2) -12

-1 *.

@$?

“=-J7 and S is the symmetric

r

s=

-

tridiagonal

C is the

-1

-1 1

(3)

matrix

-(4n - 2) (2n - 2) .. . (2n-2j+2)

(2n - 2) -(4n - 6) (4)

494 2.2.

A. D. S. BARR AND D. C. MCWHANNELL SIMPLE PORTAL FRAME

An elementary lumped mass equivalent of a portal frame is considered in order to demonstrate the occurrence of the autoparametric action with as little complication as possible. The frame under consideration is shown in a deflected position in Figure 2. The equal uprights are built in at the base and carry a lumped mass m at the upper end. The cross beam carries a central mass M and is assumed to be simply supported at its junctions with the uprights. All the beams are regarded as massless and inextensible.

Figure 2. Simplifiedportal frame. With fixed x and y coordinates as indicated and with x-wise displacements referred to as w, and y-wise displacements as u, with the subscripts shown, the absolute coordinates of the three masses can be written x,=wi,

x*=l+w*,

y, = h - 2;, + u,

yz = h - v2 + u,

x3 = 31+ 3(% + w,), y3 = h + u - +(uz + c,) - L’,.

(5)

Differentiation of these allows the absolute velocities and hence the kinetic energy to be written. The various displacements are, however, interdependent to some extent and the system can be reduced to two independent coordinates, i.e. two degrees of freedom. Thus, because of the postulated inextensibility, the following relations can be written : v1 = +Aw;;

v2 = +Aw;;

(WI-

~2)

=

tBv:,

(6)

where A and B are constants for the vertical and cross beams, respectively. Using these relations, the equations can be reduced to the two unknowns w, and v3. If the various displacements and their iirst derivatives are regarded as small quantities of the first order, the kinetic energy is expanded to the third order, so that in the final equations of motion only quadratic non-linearities will appear. When w is written as a first-order representation of the “sway” of the portal having a natural frequency pi, and v as the vertical motion of the centre mass in the symmetric mode with natural frequency p2, the Lagrangian equations of motion take the form is + (pi - Aii + (A/R) ii}w + $B(vi; + ti2) = 0, (7) ti + {p: + +BRQ} u + A(wii -i- kit’) = ii, where linear elasticity of the structure has been assumed and R = (2m + M)/M.

3. STABILITY REGIONS The matrix equation (1) is a set of simultaneous differential equations with periodic coefficients. If the natural frequencies of the structure are, in the absence of the parametric

495

PARAMETRIC INSTABILITYINSTRUCTURES

excitation, represented byp,, p2, . . ., pn, that is, the quantitiesp, are the roots of the frequency equation I(C - CLS)- Ap21= 0, (8) then it is found that for very small parametric excitation, instability of the solution x occurs for values of the excitation frequency Q in the neighbourhood of 2pi (where i = 1,2,. . ., n) and also in the neighbourhood of (pi +p,), i #j. These instability regions will be referred to as principal (at Q2w2pJ and combination (at S zpi +pj) regions, respectively. Other unstable regions exist but are less important in the sense that, if the parametric excitation is characterized by a quantity E < 1, they are of order c2 and below, while the principal and combination regions are of order E. Also, the higher order regions are much more influenced by damping within the system which makes them more difficult to obtain experimentally. The main concern of this section is to indicate ways in which the stability boundaries of the equation (1) can be generated numerically for a given system. 3.1. NOTES 0~

GENERAL

SOLUTI~N(PERIODIC

INPUT)

The main features of the solution of periodic systems of the form of equation (1) are well established. Equation (1) falls into the class of Hamiltonian systems which are those whose dynamical equations can be expressed in the form i = M(t) z,

(9)

where z is a column vector of dimension 2n (for an n-degree-of-freedom

system),

E being the unit matrix of order n, and H(t) is a symmetric real (and in this case periodic) matrix. Equation (1) can easily be set into the form of equation (9) by making the substitution Zj=Xj

(j=1,2

)...) n),

Zj =Xj-n

(j=n+l,n+2

(10) ,..., 2n),

which gives i = P(t) z, where P(t) takes the form P(t) =

0 -A-‘[C

- (a + /3cosL’t)S]

(11)

1’

0”

02)

and equation (11) can easily be written in the form of equation (9). Corresponding to any set of 2n linearly independent initial conditions there are 2~ solutions of equation (11). If these solutions are arranged as the various columns of a 2n x 2n matrix, we can refer to this as the solution matrix Z. It then follows that i = P(t) z.

(13) Taking the initial conditions Z(0) such that Z(0) = E, then, since the system matrix P(t) is periodic with period T, it can be seen that Z(t + T) is also a solution of (10) and hence that Z(t+T)=Z(t)M;

(14)

:. Z(T) = Z(0) M = M,

where the constant matrix M, the monodromy matrix of the system, relates the solution at t = 0 and t = T. Continuing the argument over many periods, the boundedness of the solution clearly depends on the powers of M and hence on its eigenvalues. These are referred to as the

496

A. D. S. BARR AND D. C. MCWHANNELL

multipliers of the system. Hamiltonian systems are also known to be reciprocal, which means that for any eigenvalue pi of M the eigenvalue (pi)-’ must also occur and also, from the reality of H(t), the eigenvalue pi, the complex conjugate of pi, must also occur. It is found that if any multiplier has a modulus greater than unity the system is unstable, and since any multiplier inside the unit circle has, due to the reciprocal nature of the system, an equivalent multiplier outside, the system is stable only when all the multipliers lie on the unit circle. This is a necessary condition for stability. Floquet’s theorem gives information on the form of the solution of equation (13). When M = exp B, where B is a constant matrix, then the form of solution is Z(t) = L(t) exp (Bt),

(15)

where L(t) = L(t + T) is a matrix having period T, the same as that of P. Thus, reverting again to equation (l), a solution can be taken in the form x(t) = eht e(r),

(16)

where $(t) is now a periodic vector (period T) and h is the characteristic exponent of the solution. Clearly, if any h has a positive real part for a given system at any particular forcing frequency Sz, an unstable solution results, the growth rate depending on the magnitude of the real part. 3.2.

ESTIMATION

OF STABILITY

BOUNDARIES

Because of the structure of the multipliers mentioned above, they can leave the unit circle only after one or more pairs have coincided. Thus it is found for a principal region, involving as it does only one frequency of the system, that as D approaches 2p, (for small excitation) the boundary of instability is reached when two of the multipliers move into coincidence at the point (-l,O), the corresponding solution then being periodic with a period twice that of the excitation, i.e. 2T. This periodicity on the boundaries of the region can be made use of when the boundaries are being established. As Sz approaches a combination resonance, however, the situation is more complicated: two pairs of appropriate multipliers now move into conjunction symmetrically above and below the real axis, and the solution on the boundary is not generally periodic or is of unknown period. The most obvious way to investigate stability numerically is to compute from the system matrix P(t) [equation (13)] the solution over one period T for 2n different starting conditions col(l,O,O.. .), col(0, 1 ,OO.. .), etc. ; that is, to generate the monodromy matrix M. The eigenvalues of M can then be found by standard methods and examined to see if any are off the unit circle. This procedure is, however, very demanding on computing time for systems having more than a few degrees of freedom. For situations where only the position and nature of the instability zones is sought, it is uneconomical to have to carry out the numerical integration of the equations of motion. For this purpose, more direct approximate methods can be used; suitable procedures are outlined below. 3.2.1. Regions bounded by periodic solutions It was stated above that entry to a principal (2pJ region is through a solution of period 2T ; it is similarly found that entry to the less importantp, region is via a solution of period T. These conditions permit the finding of the stability boundaries by appropriate periodic solution substitution. After Bolotin [2], equation (1) is rewritten as Dii+[E-(cr+~cos~t)G]x=O,

(17)

PARAMETRIC

INSTABILITY

497

IN STRUCTURES

where G= C-‘S;

D=C-‘A;

(18)

hence, a boundary with period 2T= (47r/Q) can be found by taking x(t) =

(ak sin $kQt + b, cos QkQt).

5

(19)

k=1,3,5...

When equation (19) is substituted in equation (17) and cosine and sine terms are equated to zero in the various harmonics, an infinite determinant is generated, and when this is restricted to terms in k = 5, the following determinant is obtained : ;E-aG+@G-+Q*D I

-+G E-aG-%2D

3W 0

-tk+

-$G E-crG--=D2D 4

=O.

(20)

It is now possible to look for zeros of this determinant in the @,/I) parameter plane by direct evaluation. It is possible to recast the problem as an eigenvalue problem which is generally more convenient for computation (cf. procedure outlined in section 3.3). Similar determinantal boundaries can be derived for solutions having period T, even values of k including k = 0, being taken in equation (19). It is not usually necessary to consider terms as high as k = 5 as in equation (20). It is often sufficient to consider only k = 1 when equation (20) will reduce to its first leading diagonal term. 3.2.2. Approximate procedure for combination regions

The boundaries of stability in the case of combination resonance can be found approximately by application of perturbation [ 1] or averaging procedures [3,4]. For present purposes it is sufficient to express the stability boundaries to the first order in the small parameter (c/l) representing the excitation. When equation (1) is transformed to the normal coordinates q1 of the problem with p = 0 x=Rq,

(21)

where R is the modal matrix, the following equation is obtained: ij+ [F-(/3cosf&)L]q=O,

(22)

where F = diag [p:,pi.. .,p,‘] is the frequency matrix. The matrix L = (R-' A-‘SR)

(23)

is a similarity transformation on matrix A-IS, where A and S are symmetric and is thus itself symmetric. If the elements of L are lij ,then the combination instability near Q =pi +pj has upper and lower boundaries given by the relation .Q=(pi

+pj)*Qq3. 2JP,Pj

(24)

The width of the region for a given excitation j3 thus depends, to this order of accuracy, on the appropriate coupling term in L between the two modes concerned. 3.3.

CALCULATION

OF COMBINATION

REGIONS FROM EQUATION

(16)

Since the combination instabilities are known to lie on the vicinity of Q =pi +pj, it is possible to search numerically for their boundaries using the solution form (16) and recognizing a positive real part of h as an indication of instability.

498

A. D. S. BARR AND D. C. MCWHANNELL

Thus, from equation (16), a substitution into system equation (17) can be made in the form (see Bolotin [2]) x(t) = eht +&, + 2 (uk sin ki2t + b,cos kf2t) k=l

c

. I

When coefficients of the various sinkSZt and coskL?t terms are equated to zero, a system of homogeneous equations is given. The related determinant curtailed at k = 1 takes the form (h2-Q2)D+(E-olG) -PG -2hQD

2hQD

--HG h2D+(E-crG) 0

Z 0. (h2 - s2) Do+ (E - CXG)

(25)

For a given point on the (Q, /I) plane, equation (25) defines the appropriate h. The problem can with advantage be recast as an eigenvalue problem. Multiplying the last row of equation (25) by h-’ and the last column by h, the equation can be restated as det(U-h2V)=detVedet(V-‘U-h2E)=0,

(26)

where v+

-;

-71,

(27)

and the form of U can be inferred from equations (25), (26) and (27). The (complex) eigenvalue problem IV-’ U - h2 El = 0 can now be tackled by standard routines. For a chosen 52, the value of /I at which any eigenvalue h2 changes over from real negative to positive or complex, indicates a boundary of the region. Growth contours within the unstable region can also be obtained. 3.4.

QUASI-PERIODIC

INPUT;

STABILITY REGIONS

As was mentioned in the introduction, the ground motion in many practical cases is not steady sinusoidal for any great length of time but has the appearance of a modulated sine wave. The simplest mathematical approximation which can be considered to see if any new phenomena are introduced by this is one in which the vertical ground motion is the sum of two inputs of equal acceleration amplitude but different frequency. Research on this problem is still proceeding, and here the single-degree-of-freedom case only will be considered. The relevant equation of motion can be written from equation (1) as jt+(p2-ycosi&-ycoskfit)x=O,

(28)

where y = +(c/l) D2, and k > 1 is a real number. If a small parameter Eand a non-dimensional time are chosen such that 6c 7=1;2t, E=-5i9 equation (28) can be written x”+(Q2-ecow-•coskT)x=O,

(30)

where dashes indicate differentiation with respect to 7, and (31)

Q=P/~

relates the natural frequency to the lower of the excitation frequencies. Equation (30) can readily be approached by the asymptotic method developed by Struble [5]. The solution is taken in the form X(T) = a(~) ~0s

(QT - +(T)) + E#~(T)+ ?? 2 242(T)

i-

* * *.

(32)

PARAMETRIC

INSTABILITY

499

IN STRUCTURES

Substitution of this in equation (30) and selection out of the various powers of Eresults in two so-called variational equations, (- 2Qa’ + 2~’4 + a#“) = 0,

(2&z& + a” - ad”) = 0,

(33)

an equation of first order in E, U; + Q2 u1 = cos ( QT - +) cos T + cos ( QT - 4) cos k7,

(34)

and of second order, ~;+Q~u~=u,cos~+u,cosk~.

(35)

If any of the terms on the right-hand side of equation (34) or (35) are such as to lead to resonance, they are removed and considered along with the variational equations. Thus in equation (34) with the products of cosines written as sums, it can be seen that resonant terms arise if Q is near to 3 or &k. These are the usual primary or principal resonance situations. If neither of these cases arises then the solutions of (34) can be written as u,=zdcos[(Q+l)~-~]+Bcos[(Q-l)~-~]+Ccos[(Q+k)~-~]+ + Dcos[(Q-k)T-$1,

(36)

where A =-u/2(2Q+

11,

B = a/2(2Q - l),

C = -a/2(2Qk

-t k2),

D = a/2(2Qk - k2).

(37) Solution (36) can now be substituted in equation (35). It is found that resonance in u2 can occur not only for Q = 1 or Q = k, which are the usual regions, but also for Q = 3(k + 1) and Q=$(k1). 3.4.1. Resonance at 24 = (k + 1) In this case the appropriate resonance terms in equation (35) having frequency Q or [Q - (k + l)] are removed and inserted in equation (33) which, after manipulation, become (2Q4 + a"- a$'">= ?? '{+(A+ B + C + D) + +(B + D) cos [(k + 1 - 2Q)

7 +

(-2Qa’ + 2~’4’ + a#“) = e2(-$(B + D) sin [(k + 1 - 2Q) 7 + 241).

24]},

(38)

If the first terms only on the left of each of (38) are retained and [(k + 1 - ~Q)T + 291 = #, the following equations are obtained: d#/dr = (k + 1 - 2Q) + 24’ =(k+l-2Q)+(~2/Qu){3(A+B+C+D)++(B+D)~~s~},

(39)

and da/d-r = (e2/2Q){$(B + D) sin $1. The boundaries of an ungrowing solution are then given by

After manipulation, the boundaries are found to be-for 2Q=(k+l)+x

k(k+

62 2Q=(k+1)-k(k+1)

2Q near (k + l)-

k2 + 3k + 1 1)I ’ 1) [ (k+2)(2k+ k2+k+1 (k+2)(2k+l)

I ’

Values of k near unity are excluded due to interference with secondary resonance.

(40)

500

A. D. S. BARR AND D. C. MCWHANNELL

3.4.2. Resonance at 20 = (k - 1) Proceeding in a similar manner but this time with resonant [Q - (k - l)] leads to boundary equations

which give finally the stability

boundaries

as-for

terms in u2 of frequency

Q or

2Q near (k - 1)~

(41)

Values of k near 2 and 3 are excluded resonances. 3.4.3.

due to interference

with principal

and secondary

Discussion on quasi-periodic input

The two parametric inputs of equation (30) can together be considered as a beating input, and the resonance situations above correspond to stimulation by the carrier frequency and the modulation frequency, respectively. Considering the instability associated with 2Q = k + 1, it is found from equation (40) that for all the values of k the region is relatively narrow and is thus perhaps not of great interest.

0

I O-01

I 002

003

O-04

005

306

0.07

0.08

0 c,9

0.10

F

Figure 3. Instability region near Q = 3(/c - 1) of the equation x” + (Q2 - ECOST- ecosk~)x = 0, for k = 1.1. +, Unstable points; 0, stable points; -, theoretical boundary (order 9). The region associated with the resonance 2Q = k - 1 can, however, be of much more impressive extent. This is particularly true if k is not very far from unity; that is, the system is under parametric excitation by two high and relatively close frequencies. For the purposes of illustration the case k = 1.1 will be considered. The full lines on Figure 3 show the stability boundaries on the plane of 1/2Q s Q/2p against E, the area between the full lines corresponding to unstable solutions. The region begins at @/2p) = 10 and is initially very narrow; however, the upper boundary beyond E of

PARAMETRIC INSTABILITY IN STRUCTURES

501

approximately O-03increases very rapidly having a vertical asymptote at E = 0.06. The lower bound likewise increases to a vertical asymptote for E = O-11. While boundaries calculated from equations (41) may not be very reliable at values of 1/2Q so far from the initial value, the existence of a very substantial unstable region is nevertheless indicated. If the approximation is carried to order E4, further terms arise to modify the shape of the above region and also further regions arise corresponding to basic resonances at Q = k I 1. The parametric excitation is periodic if k is rational, and then the solution within an unstable region is an exponentially growing one with a “period” which is either that of the excitation or double it. To make some evaluation of the precise extent of the unstable region, some computed solutions of equation (30) were carried out using the I.B.M. continuous system modeling program (CSMP). Computations were carried out for k = 1.1 for times 7 from 0 to 600, which was generally long enough to show if any build-up in amplitude was occurring, although of

‘I2-

” ‘n

Figure 4. Computed

-4

-

-6

-

instabilities,

Input

quasi-periodic

1/2Q = 12. (b) Q = (k - 1) instability, E = O.l0,1/2Q

envelope

input, k = 1.1. (a) Q = +(k - 1) instability, = 16.67.

E = 0.03,

course it was not possible to say if the amplitude was tending to an “unbounded” magnitude. Stable and unstable points from the CSMP are indicated in Figure 3 and show the very large extent of the region and the limitation in the accuracy of the predictions of equations (41). The broken lines in Figure 3 indicate the probable position of the actual instability boundaries. In the course of the computation, unstable points of the e4 region from Q = (k - 1) were also obtained and these are shown on the figure; this region has also a considerable width. The computed solution in the Q = (k - 1) region had a “period” equal to that of the input, while in the 2Q = (k - 1) region it was twice that. Typical solutions are drawn in Figure 4. Both solutions start from initial conditions x = 10V3,x’ = 0. The growth rate on the (k - 1) trace is very much higher than that in the +(k - 1) example, but, as can be seen from Figure 3, the former case is very much more “in the heart” of the corresponding instability zone. Equally high growth rates can be found within the *(k - 1) region at, for instance, the point E = 0.059, 1/2Q = 35.7 where the amplitude rises to over 760 x 10e3 in the same time. The CSMP output also indicated a superimposed high frequency ripple at the carrier frequency; this can be seen on the lower trace of Figure 4. 33

502

A. D. S. BARR AND D. C. MCWHANNELL

The situation in which the two inputs are of equal magnitude E has been examined for simplicity but the existence of instability regions of this type is not dependent on this equality. This has been checked by computed examples.

4. EXPERIMENTAL WORK AND COMPARISONS WITH THEORY 4.1.

EXPERIMENTAL

The work described below is limited to the periodic parametric excitation of model twostorey and five-storey buildings of the form shown in Figure 1. The columns of the models were continuous strips of $ in. x O-019 in. spring steel. The floors in the two-storey model were 4 in. long blocks of 2 in. square duralumin attached to each of the columns by two 4BA Allen screws through cover plates, the pitch of the floors was 3a in. In the five-storey case the floors were made from duralumin plate and had dimensions + in. deep x 1%in. wide x 4 in. long, the inter-floor pitch being 3s in. For the experiments, the model was firmly bolted to the head of a large @‘ye-Ling V1006) vibration generator driven by an accurate decade oscillator through a power amplifier. The vibrator head motion was fixed to the vertical as nearly as possible and was also arranged to be along the centre-line of the model. An accelerometer attached to the vibrator head measured its r.m.s. acceleration for any given frequency setting. As only the qualitative nature of the response (i.e. stable or unstable) was required, pick-up instrumentation was not essential, although for some of the tests a strain gauge attached to one of the columns was used. Generally the instabilities were clearly visible, although with some of the weaker cases considerable time was required (several minutes in some instances) before the oscillation reached its final amplitude. The nature of the instability was also determined visually; for instance, in the combination mode (pi +pJ both first and second modes could be clearly seen vibrating in superposition, each at about its own frequency. The stiffness matrix for each of the systems was obtained by inversion of the flexibility matrix which was obtained experimentally by applying loads to each of the lloors in turn and measuring the resulting deflections. The mass matrix was obtained from the measured weight of each floor. 4.2.

COMPUTED RESULTS

From these matrices the normal modes and natural frequencies of the system were calculated on the computer by standard methods. From this information the modal matrix was assembled and inverted and the excitation matrix L [equation (23)] computed. For each of the structures the instability regions in the neighbourhood of p1 and 2p, were calculated for each natural frequency, using the periodic solution determinant of the form of equation (20) taken in eigenvalue form (see section 3.3). This is much simpler than the general problem of equation (26) because the eigenvalues in this case are real. Calculations were carried out using first only one oscillatory component in the Fourier series for x, and again using two components (k = 1,3 for 2p, region). When two components are used, the roots of the next region having periodic solutions of the same kind on its boundaries are also obtained. Thus the determinant (20) would generate the boundaries of unstable regions centred on 2~~~2~,13,2~,15. The two component solutions for the 2p, regions of the five-storey structure are shown in Figure 5. The abscissa of this figure is the small quantity c/Z. The physical excitation is the ground acceleration rather than the amplitude, however, and this is the reason for the apparent dominance of the higher modes. Equal ground acceleration contours at 2g, 4g and 6g are also included; the relative widths of the regions along a particular contour then give a more

503

PARAMETRIC INSTABILITYIN STRUCTURES

realistic picture of the importance of each region. The regions originating at frequencies pi were also calculated and obtained experimentally, but these are omitted from the figure for clarity; they are also extremely narrow. It can be seen that partial overlapping of many of the regions occurs; this is even more evident if all thep,, 2pi and combination regions are drawn on the one figure. The question of coexistence of more than one instability region has not been discussed above; the analysis becomes more complicated because several terms are simultaneously near-resonant. A general treatment of the situation by the averaging method

I

0

1

I

0 Cl

002

,

I

I

003

004

0.05

c/L

Figure 5. Computed 2pi regions for five-storey model. Solution unstable within wedge-shaped region. is given by Hsu [6]. In the structure under discussion the only overlap to occur up to the relatively low (4g) base accelerations used in the experiment, was that of the 2pZ and p4

regions. In general, however, particularly for structures with many relatively close natural frequencies, the complication of overlapping regions will be unavoidable. Non-dimensional plots of the 2p, regions against base acceleration amplitude are shown along with experimental points in Figure 6. Experimental and computed frequencies have been non-dimensionalized to the form (Q/2pi), the experimental frequencies being divided by experimental values of 2pl and computed values of Q by computed values of 2pj. This is felt to be the best form of presentation as it emphasizes that both experimentally and computationally these regions begin right on the appropriate 2pi value. The computed frequencies were within 5 % of the experimental ; the values (rad/sec) are shown in Table 1. TABLET j=

Experimental pr Computed pi

1

2

3

4

5

28.9

83.5 92.7

134.1 130.4

172.5 167.5

200.0 191.2

28.1

With regard to the combination resonances of the five-storey structure, there are apparently 10 possible regions. These are not of equal importance, however, as can be seen by examining the L matrix of the sytsem. The off-diagonal terms of L are much larger than the others, which

504

A. D. S. BARR AND D. C. MCWHANNELL

means

that

ml,J/l/p,pj

the combination for each possible

modes

(pi + p,,,)

are generally

the widest.

The values

of

pair is shown in Table 2. TABLE 2

Mode no.

2

1 2 3 4

1.129 -

3 0.156 -2.388 -

4 -0.147 0.092 3.324 -

5 0.029 -0,205 0022 3.706

1

Values of (ml*jlz/piPj)

x lo-’

I

It is clear from this that instabilities other than (pi + pj+,), i = 1, . . ., 4 are relatively insignificant. This was reflected in the experimental work where only four main combination regions were easily obtained; most of the others were undetectable.

Figure 6. 2pi instability regions for five-storey model. -, Computed boundaries; boundary points. (a) 2p, region, (b) 2p4 region, (c) 2p3 region, (d) 2p2 region, (e) 2p, region.

0,

experimental

Computed regions using equation (24) in conjunction with Table 2 are shown along with experimental values in Figure 7. The basis of non-dimensionalization to the form @/(pi + pj)) is as indicated above for the 2pi regions. The experimental work on the five-storey structure used an earlier form of excitation equipment which limited base accelerations to the small values indicated (about 4g). The two-storey structure was, however, tested on the later apparatus and much higher acceleration levels were obtainable.

PARAMETRIC INSTABILITY IN STRUCTURES

505

Figure 7. (P, + p,) instability regions for five-storey model. -, Computed boundaries; 0, experimental boundary points. (a)p., + ps region, (b)P3 + p4 region, (c)p2 + p3 region, (dIpI + PZregion.

Figure 8. Computed region.

instability

regions for two-storey

model. Solution unstable within wedge-shaped

Figure 8 shows computed values for the two-storey structure of the instability regions of the various types against the parameter (c/Z). The values of the natural frequencies in this case were i=l Experimental pi Computed pi

88 84.89

2 241 222.74

506

A. D. S. BARR AND D. C. MCWHANNELL

310

300

298

___-_-_-~_______-_--_-_-+~4+ __

____

___

_.____----_-

-____________

--++++ ____

____ ___________---__296

_--me+++ ---++

_____-_---~--------------C ____________________-----

I

1

0

1

1

002

1

1

O,C4

c/L

Figure 9. Computer print-out facsimile for (PI + p2) region of two-storey model. -, Stable points; +, unstable points; -, approximation of equation (24).

4

6

Figure 10. Instability regions for two-storey model. -, points. (a) 2~~ region, (b)pI -t pz region, (c) 2p, region.

12

16

20

Computed boundaries; o , experimental boundary

PARAMETRIC

INSTABILITY

IN STRUCTURES

507

while the coefficient ml,* was computed as -0.9049. The combination region was obtained using the approach outlined in section 3.3. A more detailed picture of this region is given by the computer output shown in Figure 9, a printed cross indicating an unstable point. The two lines drawn on this figure are those obtained from the approximation given in equation (24) using the computed value of 1i2. Comparative plots of the 2p,, 2pZ and (p, +p2) regions against base acceleration are shown in Figure 10, the basis of the non-dimensionalization being as described above. 4.3.

COMPARISON

AND DISCUSSION

Figures 6,7 and 10 show that in all cases the experimental points lie reasonably near to the computed regions. The regions of Figures 6 and 7 are all narrow because of the low excitation involved; the region at 2pi could not be taken beyond l+g because of travel limitations in the vibrator. In both the principal and combination regions for the five-storey model, the unstable regions appear to be underestimated in the lower frequency modes and overestimated in the high frequency modes. It is difficult to account for this convincingly, although it is thought to be partly a result of the reliance experimentally on a visual indication of instability. It is generally difficult to be precise about the establishing of points marking a boundary between stability and instability. The entering of an unstable region is not necessarily accompanied by a rapid growth in amplitude to a large value ; non-linear inertial and elasticity effects cause the shape and position of the unstable zone to be a function of amplitude, and an unstable point just within one of the boundaries of the zones shown may simply appear experimentally as a small oscillation of fixed amplitude, and these are more easily detected in the low modes. For the same reasons, it is not usually satisfactory to try to define a stability boundary from observations on the decay rate of an initially deflected state. The 2pz region of Figure 6 for the five-storey model was complicated by the overlap with the p4 region, while within the lower edge of the 2p, region for (c&P/g) greater than 33, the mode shape became that of (pz fp,), one of the relatively unimportant combination regions. Similarly, the (pf + ps) mode was found within the lower boundary of the 2p4 region. The two-storey model results, shown in Figure 10, are very similar to those for the fivestorey case. However, due to higher damping ratios the experimental regions did not begin until much higher excitations had been reached. Combination instabilities seemed to begin abruptly for accelerations above 7g but could be obtained down to much lower accelerations (less than 2g) if the structure was released from an initially deflected state. Just below the 2p, region of Figure 10 are shown some boundary points of combination instabilities corresponding to excitation at -+(pi + ~2). This is generally expected to be a very narrow region of order c2. For both models, the instabilities at pt were also obtained although these were generally very narrow. Also, particularly for the lower modes, the model had to be carefully constructed and set up if forced oscillations due to initial imperfections or non-axial excitation were to be avoided, as these tended to dominate the instability and distort the apparent region of instability. The beating input type of instability discussed in sections 3.4.1 and 3.4.2 has been only briefly examined experimentally. With the structures and equipment described above, the damping and the extremely low values of E obtainable at high frequency excluded any poSSibility of finding the region. However, instability of this type [2Q = (k - l)] was detected when the multi-storey structure was replaced by a long slender cantilever beam having very low damping. The input was obtained by passing the outputs of two identical oscillators through a high gain amplifier arranged as a summer. The instability was difficult to obtain and needed very precise frequency adjustment. No trace of a 2Q = (k -!- 1) instability could be found.

508

A. D. S. BARR AND

D. C. MCWHANNELL

5. REMARKS ON THE AUTOPARAMETRIC PROBLEM The equations (7) for the lumped portal frame of Figure 2 constitute a simple example of parametric and autoparametric action in a vibrating structure. The parametric term in the equations is (-Aii)w, which occurs in the first equation and is brought about by the (given) support motion u. The support motion also acts as a direct forcing term on the cross-beam deflection u. If the base motion is taken to be sinusoidal, u = cco&V, then (7) takes the form ti + {pf + c.Q2_4 cos L?t + (A/R) ij> w + 3B(vi; + ti2) = 0, ti + {p:

-I-

$BRii} v + A(wti + 62) = -cL?2cos SZt.

(42)

The terms referred to as autoparametric are (A/R) i;w in the first of (42) and +BRih in the second. Although they are really just non-linear terms they are called autoparametric because of the way in which they appear in the equations. In fact the similarity goes further than this because the instability phenomena discussed above for normal parametric loadings can now take place within the structure, one mode or form of motion, providing the parametric excitation for another. As has already been seen, however, parametric action is significant only near certain excitation frequencies. If, for instance, in the above equations the acceleration B (assumed periodic) was to influence v parametrically, then the frequency of w would need to be near to 2p2 (for primary resonance) or to p2 (for secondary resonance). However, the frequency of w will generally be near top, so the condition for autoparametric action on v would bep, = 2p2 or alternativelyp, =p2. These are special cases of “internal resonance” within the structure, the general situation arising when any relation of the form mp, + np, = 0 holds, where m, n are integers. The situation described by equations (42) includes both parametric forcing (on w) and autoparametric forcing (on u and w), as well as the ordinary forcing term on v. An interesting case involving all forms of forcing arises if the relation Q = 2p, = 4p2 holds, or is at least very closely approximated. From the first equation the condition .Q = 2p, gives instability in w; the response w is at frequency pI and from the second equation this gives instability in u with frequencyp, because of the second (internal resonance) relationp, = 2p2. The displacements w and v do not grow indefinitely but are limited by non-linearities in inertia and elasticity not included in equations (42). The steady-state situation is thus one of a vertical support motion having frequency 9 at twice one of the normal mode frequencies of the structure, while the structure responds in two modes at frequencies p1 and P2 c=:p*>*

This type of behaviour has been produced experimentally and Plate 1, reproduced from a photograph, shows the appearance of the vibration under periodic illumination. The short uprights carrying end masses are fixed at their bases to the rigid cross-member which is fixed to the vibrator head and oscillated vertically at frequency ZJ = 28 Hz. The uprights are also joined at the top by a flexible cross-member which in this case was a uniform beam. The “sway” frequency of the frame was arranged to be 14 Hz while the symmetric mode involving, principally, bending of the cross-beam was arranged to be 7 Hz. The figure shows that both modes are involved in the response; in fact, the second mode of the cross-beam, forced by the 28 Hz support motion, is also in evidence. Further situations involving similar types of behaviour are reported in reference [7]. The analysis of equation systems of the type (42) under internal and external resonance conditions is given by Sethna [8] using the averaging method. Alternatively the method of Struble [5] used above in section 3.4 can be used and gives the same results.

Plate 1. Multiple flash exposure of autoparametric effect in a portal frame. Base vertical excitation frequency 28 Hz, lateral (sway) frequency 14 Hz, vertical motion of cross-beam frequency 7 Hr.

6 CONCLWSLONS It has been shown that support motion of a structure can act parametrically on its deflection and that in such instances it can exert a considerable influence on the response. For periodic support movement, the instability regions predicted theoretically agree reasonably well with experimental observation for the simple form of structure considered. Quasi-periodic support motion, due to simuhaneous application of two oscillatory input components of differing frequency, has also been considered theoretically for a single-degreeof-freedom system. A substantial instability zone is introduced if the input frequencies are close but high relative to the natural frequency of the system. Autoparametric action within a vibrating structure due to non-linear coupling terms has been considered briefly and has been shown qualitatively to modify the response of the structure if certain internal resonance conditions are met. If such conditions obtain, the behaviour of the structure is altered in that the modes related by the internal resonance conditions can interact in such a way that, for example, ordinary forced excitation of one mode will result in exponential growth of another. Such interactions would require to be taken into account in a proper response analysis. REFERENCES 1. E. MAS~A 1967 ~ecc~~~c~ 2, 243. On the i~tability of p~arne~~c~ly excited two degrees of freedom ~brat~g systems with viscous damping. Qmznic %zWfy ofElastic Systems. San Francisco : Holden-Day. 2. V. V. BOLOTKN 1964 !i'%e 3. C. S. Hsu 1963 f. a&. Me&. 30,367. On the parametric excitation of a dynamic system having multiple degrees of freedom. 4. E. METILER 1967Proc. iat, ConA llynumic Stability of Structures (1965), 169. StabiIity and vibration probIems of mechanical systems under harmonic excitation. 5. R. A. STRUBLE 1962 AQmli?tearDifirentialEquations. New York: McGraw-Hill. 6. C. S. Hsv 1965 J. appl. Me&. 32, 373. Further results on parametric excitation of a dynamic system. 7. A. D. S. BARR 1969 Proc. 2nd ht. Cmgr. Theory o~~achi~es and ~ecb~~srns, Zakopane, Poland, Vol. 1,365. Dynamic instabilities in moving beams and beam systems. 8. P. R. SETHNA 1965J. appl. Mech. 32,576. Vibrations of dynamical systems with quadratic nonlinearities.