Singularities in vibro-impact dynamics

Journal of Sound and Vibration (1992) 152(3), 427-460

SINGULARITIES

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DYNAMICS

G. S. WHISTON National Power plc, Technology and Environmental Centre, Kelvin Avenue, Leatherhead, Surrey KT22 7SE, England (Received 15 November 1989, and in revised form 17 December 1990)

The non-differentiable nature of vibro-impact dynamics can lead to breakdown of the global stable manifold theorem applicable to smooth dynamical systems. In one dimensional periodically excited system, the breakdown leads to the “shredding” of the stable manifolds, and this process is analyzed in detail. Bodily translation of filaments of stable manifolds can lead to homoclinic intersections, and this topic is discussed in the context of a new bifurcation due to “re-entrance”. Shredding leads to a strong similarity between the geometries of the singular subspaces and the geometries of homoclinic tangles and strange attracting sets.

1.

INTRODUCTION

Vibro-impact, the forced clashing together of closely spaced components of a flexible mechanical structure, is a poorly understood but industrially important phenomenon. Its occurrence ranges in importance from the annoying “rattles” to be found in cheap motor cars to the vastly expensive outages of nuclear power stations due to heat exchanger tube wear damage. Vibro-impact systems are beautiful examples of non-linear dynamical systems, displaying all of the effects of more usual systems plus some interesting extra ones. In a previous paper [ 11, an attempt was made to obtain a qualitative overview of the global dynamics of an undamped, linear, one-dimensional oscillator with a single sided amplitude constraint vibro-impacting under a harmonic excitation possibly in conjunction with a constant, positive pre-load. This paper represented a continuation of work on similar systems by many previous authors. Notable amongst these are S. Shaw and J. M. T. Thompson and their co-workers. Papers written by members of their two schools are contained in references [2-41 and [5,6] respectively. The study reported in reference [l] relied upon a combination of local analysis and computer simulation of a mapping P : C -2 sending an impact, represented as a point of the phase space Z, into its successor. Z is a two-dimensional (singular) manifold co-ordinatized by velocity and excitation phase at impact. A preliminary discussion in reference [l] was concerned with the topological structure of Z which is complicated by the possibility of trapping singular points. However, after having noted some of the complications, they were largely ignored in favour of the simulations. The purpose of the present study is to focus attention on the singularity structure. The occurrence of discontinuities in the impact successor map is generic, and it is shown to lead to breakdown of the global stable manifold theorem via a shredding process. An analysis of this phenomenon leads to an understanding of the geometry of strange attracting sets and homoclinic tangles in vibro-impact systems. Indeed, local analysis of an unstable, periodic, single impact response close to singularity enables one to gain an insight into the homoclinic bifurcation. 427 @ 1992Academic Press Limited 0022-460X/92,‘030427 + 34 %03.00/O

428

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S. WHISTON

Vibro-impact phase spaces contain one-codimensional singularity subspaces S, on which the impact successor map P is non-differentiable. This subspace comprises the pre-image, P-‘(Z,), of the graze-impact subspace Z, and it will be shown that its geometry is generically a “pleated surface”, the singularities of which arise as cross-sections of versa1 unfoldings of the elementary singularities of catastrophe theory. Such surfaces occur in both forced and autonomous systems with a single sided amplitude constraint. In the onedimensional systems discussed in this paper, S, will resemble part of the cusped caustic formed by an irregular focussing system. The purpose of this paper is to investigate the relationship between the existence and geometric structure of S, and the global dynamics of vibro-impact. Attention will be restricted to one-dimensional, undamped, linear oscillators with arbitrary periodic excitation and a single sided amplitude constraint. In such systems, the singularity subspace S, is locally one-dimensional and its characteristic singularities are parts of a cross-section of a two-surface with cusp ridges, swallowtail singularities and self-intersections. The impact successor map is differentiable except on S, and is generically not continuous on S,. It follows that the stable and unstable manifolds of saddle points will be “shredded” if they cross S,, and the nature of the shredding process is discussed in section 3. Section 2 is comprised of a mathematical review of vibro-impact dynamics and a brief discussion of the geometric structure of S, from the point of view of catastrophe theory [7,8]. A more detailed analysis of the latter topic is in preparation and will appear elsewhere. It is interesting to note that the discontinuity of P across S, leads to an effect similar to the generation of chaos. One of the characteristics of dynamical chaos is the loss of information: that is, two arbitrarily close initial conditions separate exponentially fast under iteration. With discontinuity, arbitrarily close initial conditions separate in discrete steps. For example, if xi is close to x0, P(x,) may be close to Pk(xo) for k > 1, and repeated crossing of S, leads to iteration of the separation process. One of the most interesting ways in which chaos arises in dynamical systems is via the intermediary of the generation of homoclinic tangles. The role of singularity structure in generating homoclinic tangles is discussed in section 4. The model to bear in mind is the bodily translation of segments of stable or unstable manifolds across each other due to the shredding discontinuity. This phenomenon is discussed in the context of the “re-entrance” bifurcation, involving the appearance or disappearance of a saddle point, the control-phase trajectory of which approaches the surface generated by S, as a parameter varies. The bifurcation involves transition to a period 1, double impact orbit with a graze component. In section 5 the theoretical tools developed in previous sections are applied to the analysis of the global structure of homoclinic tangles and strange attracting sets as revealed by numerical simulations. The geometrical structure of the latter objects can bear a very strong resemblance to the structure of S, and its time-reversed dual, WC= P(Z,), and the geometric imprint indicates the global significance of singularity on vibro-impact dynamics. The structure of S, is strongly influenced by the geometry of the set of periodic grazing impacts. Calculations characterizing the geometry are presented in the Appendix.

2. VIBRO-IMPACT

2.1.

DYNAMICS

GENERALITIES

The vibro-impact dynamical system is constructed from the free response of a onedimensional, linear, undamped oscillator with an arbitrary (smooth) periodic excitation : j;+y=pg(zt), where /I is the reduced amplitude and g with Jg(zr)(< 1 is the periodic excitation. It is assumed that r = mot, where w. is the oscillator frequency and t denotes

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time. The frequency ratio z is defined by z = 27r/oJ, where T is the period of excitation. Thus g(z( r +p)) =g(zr) for p = 2n/z. This system is non-generic in the sense that it lacks a damping term. The main effect of including damping is to add the complication of finite length impact orbits and analytical complexity. Otherwise, there is little effect on the geometry of singularity. One can convert the second order oscillator equation into a first order non-linear differential equation on R3 in the usual way. Thus, replacing y by Y: t-(y(s), j(r), 5) one obtains a smooth dynamical system: forX:(a,

d Y/dt =X( Y)

b, c)+(b, -a+g(zc),

1).

Phase space R3 will be co-ordinatized by yl , y2, y, denoting displacement, time. Note that because g is periodic, one can replace R3 by R2 x S’, where denotes time reduced modulo the reduced period. It will be assumed that vibro-impact is generated by constraining motion c>O the clearance parameter. In this case, it is useful to define the following phase space : L=

{y~R31y~ Gc},

E,’ = (Y~J%IY~~O, (+) oryz
J%= {.wLlYl

velocity and the circle S’ to y,
=c>,

E,o= {y~E,ly,=O}.

(-I)},

A point x in L, is said to lead to an impact if the free response trajectory from x crosses E,’ and the impact time r1 is the time of the first such crossing. The impact event is represented as the crossing point and the impact process is represented as an instantaneous rebound: that is, the process is defined by a map, P, : E,+-+E,-,

PI :(c, v, z)-r(c,

-rv, z),

where the parameter r, with 0 < r < 1, is the coefficient of restitution. It can be demonstrated that “all” points of EC- lead to impact and thus the flow F: (x, 5)+ Y(x, 5) (for Y(x, r) the trajectory of X with initial condition X) defines a mapping Pz : E;+E,t,

P2 :xdF(x,

A(x)),

where J_(x) is the first solution of Y,(x, r) = c. Note that while PI is a diffeomorphism, P2 is not yet well defined. However, modulo “loose ends”, one can now define an impact successor map : P: EC+-+EC+,

P :x-+P2(Pl(x))

or P= P2 0 PI.

The “loose ends” alluded to above are of paramount importance to the present analysis, for P2 is not yet defined on the zero velocity impacts. A point x of Ej is an extremum with value c of the free response displacement-time trajectory Y,(x. r). Writing Y,(x, r) = YJr), one obtains a family of functions from R to R parameterized by initial conditions. Such a family is referred to as an unfolding and these will be discussed later in the context of catastrophe theory.

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G. S. WHISTON

Figure 1. Singularities of the acceleration.

A smooth function from R to R, is said to have an Ak(&) [7,8] singularity for ka0 r1 if (dJ~/dr’)(rl)=O

forO
at

(dk+lfldrk+’ )(rl) >O, (+) or CO, (-).

For example, at an Ao(*) singularity, f( ~1) = 0 but f( rl) #O and f(r) crosses zero with positive or negative slope. If ZI is an Al(f) singularity,f(rl)=O,f(rl)=O butf(rl)#O. Thus an Al (+) singularity is a minimum and an A ,( - ) singularity is a maximum. Similarly, an A*( &) singularity is an ordinary infection point which crosses from negative to positive values off at an AZ(+) or from positive to negative at an AZ(-). Upon defining Yl(x, r) = Y,(x, r) -c for x in E,-, the non-degenerate impacts correspond to A,,(+) singularities at r > rO(x) and if Q,(X) ~0, rO(x) is an A,,(-). The points of E:c EC- are A p I(f) singularities, the nature of which is determined by the acceleration QG

50(x))

=

P&~o(X>)

-c.

At a maximum, A,(-) singularity, ?i(x, 7o(x)) < 0 and the free displacement-time trajectory immediately enters yl cc. However, at an A,(+) singularity or minimum, the trajectory immediately enters vl.? c. Such a point cannot represent an impact. AZ(f) singularities occur for Y,(x, ro(x)) = 0 and it is clear that if they exist they divide E,”into intervals of maxima and minima. The A*( +) singularities represent crossings from y1 < c into yl > c and lead to trapping behaviour. In the vibro-impact interpretation, the oscillator is trapped in E,”until the acceleration changes sign from positive to negative at an &( -) singularity of Y1. For this reason, A2(+) singularities will be called a-points and A*(-) singularities w-points, corresponding to the beginning and end of trapping. The Ak+2 singularities of Y, coincide with the Ak( &) singularities of the acceleration. Thus the simple zeros of the acceleration give rise to ordinary inflections. At the Al(f) singularities of the acceleration, Y1has AJ(~) singularities and these can be interpreted as bifurcation points in trapping behaviour. Consider Figure 1, which represents a single period of some forcing function. At clearance co, there is an A3( h) singularity of Yl at ro(x) = t3. If c is varied to CI, a region of maxima bounded by a- and w-points is created in the neighbourhood. As cl varies to c2 through co the a- and o-points merge at the A3 singularity and the “island” ]w, a[ of maxima disappears. At r = r4, the acceleration has an Az( -) and Yl has an A4( -) which corresponds to a degenerate w-point for the a-point a, . Similar interpretations exist for A , 4 singularities.

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The above considerations lead to the following definition of the vibro-impact dynamical system. Define the phase space by deleting generalized minima: C,‘= E=*\A+, where A*-uk,,, A(%+ 1, A) and A(2k-t 1, *) is the set of AZk+,(f) singularities in E,“. One can define P on generalized maxima in the same way as for finite velocity impacts. This also holds for the generalized w-points: A,

=kv,AOk +I,

A,=

u

A(2k,

-).

k>l

P is defined by the trapping convention on the generalized a-points A,. Thus, if a E A,, P(a) is the first w-point succeeding a. In this way, one can define a global dynamical system P : I&+-Xc+ with P a l-l and surjective map. It is important to note that some authors do not excise the “higher minima”, A+, from E. in defining impact phase space. By definition, a point of A’ cannot represent an impact :

that is, a free trajectory arriving from yl < c. Physically, however, it can represent an initial condition which, in the impact convention, remains trapped on y, = c until the sign of the acceleration changes at an w-point. In the phase portrait, all such points drift towards the nearest o-point and have the same image under P as the w-point and, in this model, P cannot be injective. Excision of A+ is equivalent to working in a phase space where points of A+ are collapsed onto their adjacent future bounding o-point. Although no point of A+ can be reached from the positive velocity impacts, they do act as limit points of chatter orbits. Both phase space models have their adherents, but the present paper in the excised model will be used, because of the advantage of a bijective impact successor map.

2.2.

UNFOLDINGS

AND

THE

GEOMETRY

OF

s,

In this section a brief discussion of the geometry of S, is presented from the point of view of catastrophe theory. Further details will be published elsewhere [9]. Recall that in section 2.1, the one-dimensional flow of y, , denoted Y, , allowed one to consider a family of solution trajectories parameterized by initial conditions. It was noted that such a family is known as an “unfolding” of an individual trajectory, the delineation of the nearby trajectories. A subfamily of all unfoldings are called “versal” and have well understood geometric structure. If the chosen unfoldings are versal, one can construct a local geometrical model of a singularity. The versatility conditions generalize the transversality conditions of the implicit function theorem. It is useful to consider the hypersurface generated by all time developed solutions of the free trajectories. To this end, define a map i:R’XR+R’=Rx(R’XR),

i :(x, z)-+(Ydx,

r), (4 t))

and let Im (i) = r. r is an injectively immersed hypersurface of R5. An equivalent way to visualize r is to define a related family Y, of functions by adding displacement as a parameter. Thus define r, :RS=Rx(R3~

R)+R,

r, : (a, (x, r))--r Y,(x, 5) -a.

The function Y, is a submersion so that the fibre Y,-‘(O) is a smooth, closed, co-dimension 1 submanifold of R’. Clearly, r= Y;‘(O).

432

G. S. WHISTON

If the space R3 of initial conditions is replaced by a one-dimensional subspace (for example, the initial time axis), R5 is replaced by R3 and r is replaced by a 2-surface, the “wrinkles” of which in the displacement direction correspond to the oscillations in the family of solutions in Figure 2. The locus of the values of the extrema of the family Y, is represented as a hypergraph or as the visible contour of the hypersurface r (in the direction of time increasing) by projection of r into the hyperplane G = r - ’(0) :

n:RS=Rx(R3~R)+R4=Rx(R3),

n:(a, (x,~))+(a, xl.

The singularity set S of Fi is the set of points y = (Yi(x, r), (x, r)) of r having 9(x, r) = 0. Thus S coincides with the fold set of the map p = alT of points y with Ker (T,(p)) # 0. S comprises all A a I singularities and away from the bifurcation set B of A,, singularities, it is a local 3-manifold. Interest will focus on the projection z(S) of S into G where there will be self-intersections as well as singularities a(@. The singularity subspace SC of the corresponding vibro-impact system is part of a locally two-dimensional cross-section of @). After having obtained at least an implicit knowledge of the singularities of Y via the singularities of the excitation, it is useful to decide if they are versally unfolded by F. The standard method to decide upon versatility of an n-parameter unfolding of an Ak singularity y of a family F of functions is to compute the rank of the matrix of coefficients of the (k - l)-jets of the parameter gradient VF( y). The latter matrix, itJ,_, ( y), has elements (Mk-~)~=(l/i!)(a’+‘F/ar’~Xj)(y)

for O
Figure 2. The wrinkled hypersurface.

1
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Explicit expressions for the parameter gradient VF at a general point are required. In the case of an undamped linear oscillator with periodic excitation the general solution is: Y(X, r) = (XCI - Y,(ro)) cos (5 - r0) + (zIO- YP(rO)) sin (5 - rO) + Yp( t), where Yp(t) is a periodic particular (x0, u. , zo>. This implies that (a Y/8x,)(x,

integral independent

r) = cos (5 - To(X)),

(8 Y/dso)(x,

(8 Y/&,)(x,

of initial conditions

x=

r) = sin (5 - to(x)>,

z) = A(x) sin (T - s0tx)) - 00(x) cm CT- ro(x)),

where A(x) is the initial deceleration (x0--pg(zro)). Also, because VY(x, r) is harmonic for k>O. It in r, there is a periodicity, (@+‘V Y/c??+~) =V(8+’ Y/fP+2)= -(~TY/cP) follows that the family Y can never versally unfold A > 2 singularities, and versality of A 1 and A2 singularities is decided by respectively non-vanishing of V Y or linear independence of V ? and V Y. With A = (r - ro(x)), M2 is given by cos (A) Il42=

-sin (A) (A)/2

-cos

A sin (A) - u. cos (A)

sin (A) cos (A)

-sin (A)/2

A (cos) (A) + v. sin (A) -(A

sin (A) - u. cos (A))/2

1 .

(1) The submatrix MO of O-jets is rank 1 (cos (A) and sin (A) are not simultaneously zero). Equivalently, VY#O. Thus, Y is versa1 for Al singularities which unfold into 2surfaces of maxima or minima. (2) The submatrix M, of l-jets is rank 2 (or VY and VY are linearly independent) because M, always has a unit 2 x 2 minor. Thus Y is versa1 for A2 singularities which unfold into cusp ridges. (3) As pointed out above, M2 has rank 2 because its first and third rows are linearly dependent. Thus Y does not unfold A3 singularities into swallowtails. (4) Y does not have sufficient parameters to versally unfold A > 3 singularities. The lack of versality for A3 singularities is a direct consequence of harmonicity of the oscillator “transient” terms. If damping is introduced, harmonicity is destroyed. One can include system parameters such as excitation amplitude and frequency ratio as further unfolding parameters. The addition of these parameters to the initial conditions leads to a rich family of versa1 and non-versa1 unfoldings. For example, the most immediately relevant unfolding is derived from initial velocity and time. Its “versatility matrix” is the submatrix of Mz above comprising the last two columns. Using this matrix, one comes to the following conclusions concerning the Xc,’unfolding. (1) All Al singularities with u. # 0 are versally unfolded into curves of minima. An A, with u. = 0 is not versally unfolded. Higher order methods have to be invoked to calculate the local geometry of So. (2) All A2 singularities with uo#O are versally unfolded into cusps. Higher order methods are required to deal with A2s having o. = 0. (3) A3 singularities can never be versally unfolded by any two-parameter unfolding. It has already been noted that adding clearance as a parameter cannot unfold A3 singularities into swallowtails. However, using excitation amplitude together with the Z,’ unfolding one does obtain a versa1 unfolding, illustrated in Figure 3(a). In order to understand

434

G. S. WHISTON (a)

(b)

+=u

‘5

T.sc

.?

Figure 3. Variation in the geometry of W with excitation amplitude in a harmonically excited system z= 1.23, r=0.98) with j? varying in steps of 0.01 between 0.06 and 0.17. The velocity scale is 0 $u,,
cc=O.l,

Figure 3(a) it is necessary to understand the process of construction of SCfrom the graph of extrema. The graph essentially comprises curves of maxima and minima with cuspoidal singularities and self-intersections. SC is derived from the graph by firstly deleting curves of minima and then deleting curves of maxima which do not correspond to the first maximum at the clearance level preceding the initial condition, given that SC= P- ‘(2,). For example, consider Figure 4(a) showing part of a transversal cross-section of a swallowtail singularity. The point x lies on the intersection of two curves of maxima and it is supposed that curve (1) represents maxima occurring earlier than those on curve (2). The two curves divide any small neighbourhood of x into four quadrants in which the amplitudes of the corresponding maxima are greater or less than the clearance. The segment of the curve of later maxima lying in the segment where the earlier maximum exceeds the cJearance cannot represent a grazing impact, while that lying in the complementary segment does. Thus the former segment must be deleted to obtain SCas in Figure 4(b). A self-intersection has led to a branch of SCand this corresponds to a repeated graze impact. A cusp singularity has also led to an “end”, labelled AZ in Figure 4(b). There is a l-l correspondence between self-intersections of SC and repeated graze impacts, the latter topic being discussed in the Appendix.

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zG!~~j,ter~c+lan - .;&E;z2 /’ (b)

(a)

Figure 4. Construction of .S, from singularities of the wrinkled hypersurface

The geometric features discussed above all occur in the computer simulation depicted in Figure 3(a), which represents a versa1 unfolding of a swallowtail obtained by adding excitation amplitude as an unfolding parameter in the case of harmonic excitation. In Figure 3(b) are depicted S, and WC in a system with harmonic forcing and c =O. 1, z= 1.48, p = 0.06 and r = 0.98. Of particular interest in this example is the mesh of intersections in the neighbourhood of the periodic graze impacts F,. Lack of space prevents further details of the singularity-theoretic aspects of the geometry of SCbeing presented here. A more detailed investigation will be published by Chillingworth [91. 2.3. DIFFERENTIABILITY OF P The set of points of E,” on which PF’ is defined will be denoted by Z, and P2-‘(ZC) = S’. Z, is the subspace of zero velocity impacts, representing events lying on trajectories from y
u,+c [ -s/v,

AoC+voS(voC-AoS)A,/v, (VOC- AoS)/v,

1’

where C and S denote cos (2, - ro) and sin ( rl - ro) and A0 and A, denote the decelerations at times r. and tl . It is clear from the presence of the terms v;’ in the matrix elements of JP,(x) that P2 cannot be differentiable on SC. In general, Z, will consist almost entirely of pre-images of maxima. More precisely, the set A( 1, -) will be “dense” in Z, with Z,\A( 1, -) comprising a finite number of higher singularities. Suppose, then, that x~P;l(A(l, -)). Then Y,(x, t,) = c and Y,(x, r,) >O, where rI = ro(P2(x)). In this case, the implicit function theorem can be applied to construct a local function ;1: V+R solving Y,((x!, r’) = 0 for X’E V by r’= d (x’). Given A-,one can also define a local maximum value function : y :x + Y,(x, A (x)). More generally, let ;lk be the time of the kth maximum of the displacement--time trajectory Y,(x, r) for r < to(x)

436

G.

S. WHISTON

and let yk be the corresponding lOCal maximum value fUr&OnS: x+ Yl(x, ak(x)). Expressions for vyk are required, and implicit differentiation of the exact solutions yields (dyk/drO)(x)=AO(x)

sin (~k(x)-~O(x))-uO(x)

(ayk/auO)(x)=sin

cm

@k(x)-TO(x)),

@k(x)-TO(X)).

Knowledge of vyk(x) enables one to consider the continuity of Pz at a point x of P;‘(A(l, -)). Suppose that x~P;‘(&l, -)) is modelled by yk(X) = c. Moreover, suppose that P;(x) is not a graze: uo(P~(x)) >O. Then if Vyk(X) ~0, yk : V+R is regular at x so that the fibre yk’ {c} of points in V with maxima at level c is a smooth one-dimensional submanifold of V. Indeed, V n yk 1{c} models 5” n V. Thus, with V chosen as an open disc about x, y;‘(c) divides V into two subspaces (see Figure 5) v+ = {X’EI+&‘)

ac}

I/- 3 (X’E V\y&‘) cc}.

and

If X’E V+, yk(x’) >c and, because Y,(x, r) Cc for ZO(X)< r<&(x), by choosing V small enough, one can always suppose that the crossing of y = c by Yi(x’, r) represents P2(x’). Thus if X’E V+, Pz(x’) is close to P2(x) =F(x, &(x)). However, if X’EV-, i.e., yk(X’) Cc, even though one can assume that Y,(x’, r) < c for ro(x’) < r C&(x), Yi(x’, r) need not cross y = c until a subsequent crossing for r >&k(x), because Yi(x’, r) < c for r in some neighbourhood of &(x). Thus if X’EV- , P2(x’) is close to P:(x) for some k > 1. Which k applies depends on the number of maxima of Yi(x, r) at level c and the position of x’ in V. This can be illustrated by assuming that Yi(x, r) has two maxima yk(x) = c, yl (x) = c and Yi(x, r) &-k(x).In this case, x lies in an intersection of fibres: x~yk’ {c] n y;' { c]. Assume that the intersection is transversal: vyk(x) and Vy, (x) are not co-linear (see Figure 6). In this case, y;’ {c} and y; I {c} divide V into four sectors. As above, let V+ and V- denote {yk>c} and {yk < c). If x’ lies in V +, one may

Vy*(x) ,/--/

Y v

I

(\v-

x

\\

v+

‘\__A

u;’

Figure

Figure

6. Local branching

-%

\\

\ \

/ ,/I

(C)

5. Local separation

in S,. via self-intersection

by S,

of singularities

of the wrinkled

hypersurface.

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assume that P2(x’) is close to P*(x). If x’ lies in V-, either yI (x’) 6 c, in which case P*(x) is close to P;(x) = F(x, ill(x)), or y( (x’) > c. In the latter case, if P,‘(x) has u. > 0, Pi is close to P:(x). Thus there are three possibilities: P2(x’) close to Z’*(x), P,‘(x) or P:(x). In the case of a single level c maximum, the possibilities are P2(x’) close to P2(x) or to P:(x). Generalizing to a train of r successive grazes, one sees that there are (r+ 1) possibilities. Note that in the case of harmonic excitation, there is a maximum of two successive grazes (see the Appendix). Similar considerations apply when Vyk(x) = 0 or when Vyk(x) and Vy, (x) are co-linear.

3. THE SHREDDING

PROCESS

This section is concerned with local curvesf: Z+C,‘where Z is an open interval about 0 in R and their translates P’ 0f around X,‘. If Pi-’ 0f does not intersect S, then P’ 0f is a is generically smooth curve. However, if P’-’ 0fmeets S,, P’ 0f need not be continuous-it sliced into two. Further intersections of its components with S, lead to further slicing, and this process will be called “shredding”. In the analysis the effect of intersections of stable manifolds with S, and the global geometry of related structures such as homoclinic tangles and strange attracting sets is borne in mind. The following analysis establishes some basic properties of the shredding of a curve transversal to S, by iteration of P. It will be established that, generically, the process is described in Figure 7. The local curvef(Z) is divided into two segments, I’ containing f(0) and I-, identified with the curvilinear segments [A, B] and [B, C], with B=f(O) in S,. Application of P cuts the curve into two disjoint sectors, P(Z+) transversal to Z, at P(B) and P(Z-) which branches off WC, the dual P(Z,), of S, at p(B). Further iteration of I+ yields p(Z’) tangentially branching off WCat p(B).

Figure

7. “Shredding”

of a linear submanifold

by transversal

intersection

with .S,

Let yk, /lk : CY+R be the usual local maximum value and maximum time functions. For analysis in ET, these have to be replaced by Yk, & : E:,‘-+ R defined by j& = yk QPI and & = &o P, , where PI is the inelastic rebound map. The gradient V&(x) at a point x of EC?is obtained from Vyk(x) and the Jacobian of PI :

[$kl~~Ol(X) = [~YkI~roI(PI(-~)) =Ao(P,(x)>sin(~k(x)-

~~(x))+ru~(x)cos (X(x)- ro(x)),

438

G. S. WHISTON

P

of P is required. ZI>, then

Jacobian

The :(uo,

ZO)+(UI,

If the component

functions

of P are defined by

Ac_Tz, s+hoc +AOSY 1

0

0

VI

-(ruoC +AoS) VI

where S = sin ( r1 - ro), C = cos (r l - zo) and A ](x) = Ao( P(x)) is the deceleration event P(x). It is important to note that JP is degenerate on the grazes 2, in XT On Z,,

at the

It follows that JP((d/i&)) = -(r/Ao)JP((d/dzo)). Equivalently, JP maps the normal bundle of Z, onto the tangent bundle of W,. After having noted the above properties of JP, let I be an open interval about 0 and suppose that :

f :I-a::,

fVWm

(f)

is an immersion (i.e., f’ # 0) mapping 0 to an isolated point x of intersection of the curve Im (f)and S, . The translated curve P ofmay be discontinuous at 0 and if it is continuous at 0, it need not be differentiable. Suppose that P(x) E Z, is modelled by yk(x) = c for yk a local maximum function defined on a neighbourhood V of x. (Note that yk now replaces jk defined above.) Then, if x does not lead to multiple grazes, S, n V is modelled by yk’ {c>. It follows that the intersection Im (f)n S, is transversal at x if (Vyk(X), f(0)) #O or tangential if (v&(x), f(0)) = 0. Equivalently, (Vyk(x), f(0)) = ( yk 0f)implies that the intersection is transversal if yk Of is non-stationary or tangential if yk Of is stationary at 0. If the intersection is tangential, either yk 0f is maximal, minimal or inflective. 3.1.

PROPOSITION

1

(f)crosses S, at x, P 0f is continuous at 0. Proof. Suppose first that x is not a branch point of S,. Then, for arbitrarily neighbourhoods V of x, S, n V divides V into two sectors V,, defined by If Im

~+={X’~~~y&‘)~:C}

and

small

U- = (X’E VIyk(x) cc}.

Define Im (f) n V* = J+ . The curve Im (P 0s) consists of the two segments P(J+) and P(J-) and one may assume that P(J+) n P(J_) is empty because P maps points in V+ close to P(x) and points in V- close to p(x). Thus P(J+) is attached to Z, at P(x) while P(J-) is close to p(x). Both P(J+) and P(J-) are continuous curves because P is smooth off S,. and one may assume that x is an isolated point of intersection. Moreover, P(J-) branches off W, at p(x) because points in J- arbitrarily close to x map arbitrarily close to P(x). Suppose that x is an n-fold branch point of S,. due to P(x), . . . , P”(x) lying in 2,. In this case, one can find arbitrarily small neighbourhoods V of x such that S, n V comprises fibres y,-’{c) for local maximum functions y,. One may assume that s(x) c&(x) if j< k and that the intersection {x> = n jy,:’ {c} is transversal. Define V, as above in terms of

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yl . Points in U+ arbitrarily close to x are mapped arbitrarily close to P(x). However, Uis divided by the branches of S, into n sectors U’_ and points in UC arbitrarily close to x are mapped arbitrarily close to P j”(x). Therefore, upon defining J* as above P(J+) is attached to 2, at P(x). Since Im (f) crosses S, at x, J_ can be assumed to lie entirely within a single sector, U!. In this case, P(J_) will branch off WC at the point pk+‘(x). These considerations are illustrated in Figure 8(a, b) for the case of n = 2 and of transversal intersection. In case (a), J- lies in {yl> c} and P(J-) will branch off WCat p(x). However, in case (b), J- lies in {y/c c} and it follows that P(J-) must branch off WCat P’(x). Similar considerations hold for Im (f) n S, a tangential crossing and x a higher branch point

Of&.

@)

I:(@;

Im (f) Figure

8. Transversal

intersections

at a branch

point.

3.2. PROPOSITION 2 The methods used above also establish the following proposition. If yk is the first maximum at x and yk 0f is a (higher) minimum at 0 then P 0f is continuous at 0. If yk is the first maximum at x and yk 0f is a (higher) maximum at 0, then P 0f is not continuous at x. Propositions 1 and 2 have yielded a picture of the discontinuities of the translates P o$ In order to obtain a complete geometrical picture it is necessary to analyze the singularities of (P 0f )’ via the slopes of the segments of Im (P 0f) at their discontinuities Assume the usual scenario in a neighbourhood U of x in S, n Im (f) and suppose that the intersection is not a (higher) maximum of yk 0f so that J, is a non-degenerate curve. Consider a sequence of points (ti) in f -‘( U+), ti#O and (ti)+O and Ui(ti)+O. The limit of the vectors JP( f(ti))( f ‘(ti)) IS re q uired. Upon defining Vzl(f(t))=Z(f(t))/u,(f(r)), it is clear that unless (Z(f(t)), f ‘(t)) converges to zero at a higher or equal rate than Qf(Q),

IIJP(f(O)(f’(OII willapproach 00.

Note that because rl( f(t))

approaches d,(x) as t approaches 0 and

z= rs(a/av,)

- (rvoC+ &s)(a/ar,).

Z( f (t)) converges to Vyk(x) and therefore the numerator converges to (vyk(x), f ‘(0)). Hence if the intersection is transversal, (vyk(x), f ‘(0)) ~0, the vector JP( f(t))(f’(t)) must “blow up” as t approaches zero. However, note that o,( f(t))JP( f (t))( f ‘(t)) is parallel to JP( f(t))( f ‘(t)), and it follows that as t approaches zero, the directions of the vectors JP( f (t))( f ‘( t)) approach the direction

T,’= (vyk(x), These considerations

f ‘(0)) y(x) =

(vyk(x),

f ‘(0))

‘2;) [

1 .

have therefore established the following proposition.

3.3. PROPOSITION 3 If Im (f) n S,. is transversal at x, P(J+) is transversal to Z, with slope -A,(x).

440

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Next, suppose that (Vyk(x),f'(O)) = 0.Then it is not obvious that IIJP(f(t))( f’(t)) II blows up as t approaches zero. To analyze this situation, one can apply LWipital’s rule to the quotient a(t)/P(t), where a(t) = (Z(f(t)),f’(t)) and B(t) = ui(f(t)). Suppose that Tdenotesf’(t). Then a(t)=T(Z, T) andP’(t)=T(v,). Using T(v~)=du,(T) yields of/p’= ulT(Z, T)I(-Ai
T>+ VI(X, T),

where T(Z, T) = (V& T) + (Z, VTT) and, typically, V,(Eizi(a/axi) (VZi, T). The components Zi of Z are given by zI = Ysin (r, - r,) Partial differentiation

rXiT(zi)(a/dxi)

=

z~=(w,,cos(~,-rO)+AOsin(t,-r,,)).

and

therefore yields

Vz,=rCVr,-rC(a/h,), Vzz=(-rur$+AC)Vz~+J', for V=~C(~/~%I,J+(~~OS-A&+(~AO/~~O)S)(~/~~~). It follows that a’(t)/B’(t) is of the form (P’+u,Q’)/(P”+uiQ”) independent of Q . Therefore, as t approaches zero, a’(t)/P’(t) PI/J”‘. The function P is given by

for P’, P”, Q’ and Q” approaches the limit of

and the function PI’is given by

-A,@, T)+Therefore, as t approaches zero, a’(t)//?(t) approaches - Y(x) = (-Al(x), 1). 3.4.

PROPOSITION

< VWc>,f’(O))Adx). approaches - 1 and the vector JP( f( t))( f ‘( t))

4

If Im (f) A SC is tangential at x but not maximal, P(J+) is transversal to Z, at P(x) with slope -A,(x). After having thus obtained useful information on the segment P(J+), it is necessary to consider the segment P(J-) which branches off W, at P*(x) when x is not a branch point of S,. The transversality of the branch is dictated by the following results. 3.5. PROPOSITION 5 Suppose that x is not a branch or end point of SC. Then P(L) is tangent to W, if and only if (Vyk(x),f'(O))=O. Proof. Let (ti) be a sequence of points in f- ‘(U_) which converges to 0 and consider the sequence of vectors (JP(f(ti))(f’(ti))). Let f’2(~)=(~2r 72), A~=A~(F(x)), C2= cos ( r2 - ro) and S2 = sin ( z2 - ro). Then the sequence of vectors converges to the vector -r--+C2 S2A2

T; = i

[

-

02

rS2

02 L

1

A 0c 2 _ rzl0s 2 + (ruoC2 + AoS2M2

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where the vector (a, /?) denotesf’(0). The tangent vector to WC at p(x) is JP(P(x))((B/ &,)(P(x))) and therefore if P(x) = (0, r,), A,(x) =&(P(x)), Cl = cos (ri - 50) and Si = sin ( r1 - to),

The above two vectors are co-linear if and only if

Expanding out this expression and gathering terms in a and /J, one obtains the following equation :

Given that x is not an end point of S,, A,(x) #O. Therefore the two vectors are co-linear if and only if (Vyk(x),f'(O))=O. Suppose that Im (f) crosses S, at x and thus both J+ are proper curvilinear segments. Then if x is not a branch point of S,, P(J-) branches transversally off WC at p(x) by proposition (5) and P(L) is transversal to 2, at P(x) by proposition (3). But p(J+) is also attached to WC at p(x). The relationship between the segments P(J-) and p(J+) at p(x) needs to be examined. Firstly, suppose that (Vyk(x), f'(0)) ~0. Then P(J+) is transversal to 2, at P(x) and has limiting tangent

T, =
. 1 [“1(1;’

The tangent to p(3+) at P2(x) is JP(P(x))( TL). Recall that the Jacobian JP is degenerate on Z,; that is, JP(y) maps both the normal and tangent onto the tangent to WC. Therefore, p( J+) is tangent to WCat p(x). But it has just been demonstrated that P( J_) is transversal to WC at p(x), and it therefore follows that p(J+) and P(J-) are transversal. 3.6. PROPOSITION 6 Suppose that x is not a branch or end point of S,. Then p(J+) is tangent to WC and P(J_) is transversal to WC at p(x) if (Vy,(x),f'(O))#O. Next, suppose that x is a 2-fold transversal branch point of S, and that Im (f) crosses S, at x. The analysis of P(J+) is unchanged, but the fate of P(J_) depends on the angle of intersection, Suppose that the branch is modelled by {x} =y; ’{c} n y; ’{c} with &k(x) c} or in {y,
442

G.

S. WHISTON

along a sequence of points converging to zero inf-‘(U-). Writingf(t) = (~0, 70)~ P(f(t))=(~12, 7z), A2=Ao(f(t)), C2=cos (72- 70) and S2=sin (72- zo) one obtains

JfYf(t))(f’(t))

JP(f(t))(f’(t))=

y+w,_r>[$ 1 .

- y

0

As t converges to zero, u2 converges to zero and z2 converges to a,(x). It follows that (Z, f ‘) converges to (Vy,(x),f'(O)) and if the latter is non-zero, v,(JP(f(t)))(f’(t)) converges to

Tm= -
[

1

AV’ +>) _1

.

Thus P(J-) is transversal to A- at P2(x) if (Vy,(x),f'(O)) #O.In order to compare P2(J+) and P(J-) at p(x)), one has to compare T, with a limit of tangents, JP(P( f(t))) 0 JP( f(t))( f’(t)), on a sequence of “times” t converging to zero in f- ‘( V+). Note that JP( f(t)) and JP(P( f(t)) are undefined at time zero because both P(x) and P(x) lie in Z,. Suppose therefore, thatf(t)=(uo, to), P(f(t))=(v,, zl) and p(f(t))=(s, z2). Also, let A1 and A2 denote Ao(f(t)), Ao(P2(f(t))) and C, , S, denote cos, sin of (7, - 70) and C2, S2 denote cos, sin of (72 - 7,). The two Jacobians are given by

-r

Cl

AoCl - ruoSl +

(rvoC1+ AoSJAI 01

01

70) =

JP(vo,

WI

-+

i

- r(u&

-6

+A,&)

01

VI

L

A

c

I 2

_

JP(u, , 7,) =

rzI

s

1

2

+

(wC2 + A&b4 u2

- (rulCl+ AIS) u2

1 1.

Split the Jacobian into terms multiplying powers of ul and v2: .P( &+J ) 70) = tr;‘M, + MO

JP(v, ) 7,) = u;‘(M2+ VlM3) + (M4+

and

Upon using these decompositions,

UlM,).

the matrix product is given by

(u,u2)-‘[M2 0 M, + u,M2 0 MO+ u2M4 0 Ml + V,V2(M3+ MS) 0 M, + &2(M3 + MS) 0 Mol. Therefore in the limit as t approaches zero and u1 and v2 approach zero, the vector z),u2JP(P( f(t))) 0 JP( f(t))( f’(0)) approaches the vector M2 0 M,( f’(O)), where M2 and Ml are now the limiting matrices M2=

-rS2A2 rS2

AIS2A2 -A,S2

1

and

M,=

-rSIA, rS1

(rooC, + AoSI)AI -WoG

+ AoSI)

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where P(x) = (0, r,), p(x) = ( u2, r2) and all quantities A, C and S refer to either P(x) or to p(x). Carrying out the matrix multiplication, one obtains -ww> A42

‘I M,(f’(0))

=

-AP!w

+~Kv&),f’(O))

~

T 2.

_1 [

3

It therefore follows that if (Vk(x),f'(0)) and (Vy,(x),f'(O)) are non-zero, Z-?5+) and P(L) are tangent at p(x). This establishes the following proposition. 3.7. PROPOSITION 7 If Im (f) crosses S, transversally at a 2-fold branch point of S, and J- lies in {Y,> c}, p(J+) and P(J_) are co-linear at P’(x). Finally, suppose that J- lies in {Yl
4. HOMOCLINICITY

AND SINGULARITY

4.1. SADDLE POINTS OF P Suppose that x is a subharmonic order k fixed point of P and Q,(X)E V’kand ro(x) = rk . Both vk and rk can be calculated by the imposition of cyclic boundary conditions on the free response displacement-time trajectory Y(Pl(x), r) from P,(x). Thus the four boundary conditions y(P1

y(P,(x),

(x),

TO(X)

50(x))

+

=

2W4

c,

=

P(Pl(X)t

c,

JYPI

To(X))

=

70(x)

(4,

-ruo(xh

+ 2Wz)

= uoh),

lead to a pair of linear equations in Y,(ro) and Yp(ro), where Y,(r) is the (periodic) particular integral part of the free response solution Y(x, r) = (c-Yp(rO)) cos tr - r0) + (UO-Ye)

sin (r - r0) +Yp(r),

Y(x, 5)=-(~-Y~(~0))sin(~-z0)+(v~-)‘,(~~))cos(z-s~)+~~(~). Applying the boundary conditions at time ro(x) + 2nk/z, one obtains c=(c-Yp(ro))

cos (2nk/z)-(ru0+_&,(z~))

vo=-(c-yp(z~))

sin (2nk/z)+y,(7,),

sin(2zk/z)-(ruo+Y~(ro))cos(2nk/z)+Y~(ro),

where periodicity of Y, has been used: Y,,(r + 2ak/z) equivalent to the linear equations c(l+C)+ruoS

i cS+v,(l

+rC)

I[ =

(1-C)

S

=y,(r).

-s

(1 -C)

These two equations

are

IL I Yp(ro) ‘p(70)

’

where C and S denote cos (2ak/z) and sin (2rkl.z). The determinant of the linear transformation is 2( 1 - C), and therefore if C # 1 one can solve for the vector ( yp( ro), _fp(70)). For certain special cases, the norm of the solution vector is independent of r. and this leads to a quadratic equation for u. (or vk), the coefficients of which are independent of

444

G. S. WHISTON

TO. This is true for harmonic excitation, and this will be assumed for the remainder of this section. In this case, the quadratic equation is

V,(( 1 - r)*S + z2C2( 1 + r)*) + 4cz2CS( 1 + r) Vk + 4z2Sz(c2- p2r2) = 0, where C and S now denote cos (r&/z) and sin (r&/z) 1One also has an equation expressing rk in terms of Vk: tan (~5~) = S( 1 - r) Vk/z(2Sc - (1-t I) V&Z). It is important to note that the solutions ( Vk , zk) of these algebraic equations can represent fixed points of P only if the following three conditions are also satisfied: (1) Vk are real; (2) Vk are positive; (3) y(Pl(x), z) is non-re-entrant in the interval ]ro(x), ro(x) + 2ak/z[: that is, y(P1 (x), z) < c for r,,(x) < r < r,,(x) +27&/z. Violation of any of these conditions

leads to a bifurcation where the corresponding fixed point disappears. The occurrence of the bifurcations to or from conditions (1) and (2) can be studied by examination of the governing quadratic equation. Both roots of the quadratic are complex if the discriminant is negative. With z and r regarded as fixed, the discriminant becomes zero at a critical value, cr*, of the parameter cr=c/P]r]. Note that it is assumed that the excitation is harmonic, p cos (zr), in which Casey,(t) is given by pycos(zr) for y=(l-z*)-‘. If the linear term in the quadratic is negative, CT*is a bifurcation value for a simple saddle-node bifurcation. If condition (3) holds for cr slightly less than CT *, the root Vkc corresponds to a stable (spiral) node and VL to a saddle. As o decreases further, the two fixed points move apart and, with condition (3) assumed to be continuing to hold, the saddle point (VF, 5;) must cross the time axis 2, (u. = 0) as o decreases to 1. At CT= 1, the constant term of the quadratic vanishes, yielding VL = 0. Also, z; = r/z mod (27r/z) for z > 1, and therefore (VL, zk) coincides with the graze fixed point (0, r/z). The latter fixed point corresponds to the exceptional trajectory ~((0, n/z), r) = by cos (zr) with PI y/ = c. The crossing of Z, by a saddle point at (T= 1 is the bifurcation representing the violation of condition (2). Note that in this case o = 1 is actually a crossing of S, n W, by the fixed point because (0, n/z) ES, n WC--being fixed on Z, . The bifurcation representing the violation of condition (3) can also be understood as a crossing of the subspace S, n W, by a saddle trajectory in control-phase space. In what follows, this will be called the “re-entrance bifurcation”. Note that if y(P,(x), 7) is nonre-entrant in ]ro(x), ro(x) +215/c/& then for all maxima yj(X) of y(P,(x), r) with jlj(x) in the interval, yi(x) < c. If cr+x(o) is the trajectory of a saddle point in C’ as CTvaries, then it is possible that one of the curves yj 0 x crosses y= c. At the first crossing, (T= (TI , condition (3) is violated. However, if V,,- is real and positive and xl =x(0,), then P(xJ zyl lies in Z, and P(y,) = xl : that is, xl is a period (2) point and x1 ES, n W,. For o > crl , yj 0 x( CT)> c so that x(o) is not a fixed point, although ( Vc ((T), s; (0)) satisfy the appropriate equations. The crossing of 2, discussed in the previous paragraph can be regarded as a degenerate example of the re-entrance bifurcation, occurring on 2, rather than in the interior of ET. With this interpretation, the re-entrance bifurcation is inevitable as o decreases from CT*to 1. As o decreases to 1 from cr*, a saddle trajectory must approach Z, and hence S, and W,. In so doing, its stable and unstable manifolds must interact with the singularity subspaces. In section 4.2 below, it is noted that a re-entrance bifurcation in E,‘\Z, occurs whenever the saddle curve x(o) approaches Zc faster than the highest velocity point of S, n W,. In section 4.3, it is conjectured that as X(D) approaches a point of S, n W, arbitrarily closely, prior to crossing, a homoclinic tangle will probably be present in its stable and unstable manifolds. In this way, one may regard the presence of homoclinicity

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as being generated by the inverse re-entrance bifurcation as cr increases towards o* from cr< 1. The intimate connection between homoclinicity and singularity may be illustrated by the fact that the crossing of 2, by a saddle trajectory is independent of subharmonic number. In some parameter ranges, there can be several co-existing saddles with different subharmonic numbers. Therefore as o decreases to 1, they must simultaneously approach (0, z/z) and hence each other, arbitrarily closely. In this case, heteroclinic and therefore homoclinic tangles are inevitable if the saddles all persist. Note that saddle points can exist in o < 1. These are produced when a stable node, represented by the root ( Vk+, z:) of the quadratic, destabilizes in a flip bifurcation to produce the saddle and a stable period 2 orbit. 4.2. LOCAL GEOMETRY OF S, AND WC If o is just above 1, the geometry of SC and WCin the neighbourhood of (0, B/Z) can be deduced by simple observations concerning the free displacement-time trajectory from x0 = (c, 0, a/z). From now on in this section, it will be assumed that z is rational and z > 1. (If Z-C1, (0, n/z) is replaced by (0, O).) The trajectory from x0 can be written as Y(X0, r) = (c-PI rl) cos (r - ro(x0)) +PI Ylcos (z(r - ro(x0))). The graze at r = ro(xo) is the sum of amplitudes and is therefore a global maximum yo(xo), and the other maxima I+ are arranged symmetrically around x0 at times ,$(x0) (see Figure 9). The maxima with dj(xo) < ro(xo) are labelled L-,(x0) with k> 0, and the maxima with &(X0)> ro(x) are labelled &(x0) with k>O. By symmetry, y_+(xo) =yk(xo) and IL(x,) - ro(xo)l = (&(x0) - ro(xo)l. The Lk, Y+ correspond to time running backwards from ro(xo) and are related to WC, and &, yk are related to points of SCas detailed below. Note that because vo(xo) = 0, one obtains

The curve sin (r - ro(x)) is plotted in Figure 9, and it follows that for the & maxima the slope is initially positive and increasing, while for the Lk maxima the slope is initially negative and decreasing with k. The variation of the J+(X) with x in 2, close to x0 is sketched in Figure 10 for c just above /?Iyl. The intersections of the curves yk with the line y = c correspond to points of SCn Z, if yj(X)
A-, Figure

A-, 9. Local

A-,

g

maxima

A,

A,

A,

in the neighbourhood

of n/z.

G. S. WHISTON

Figure 10. Trajectories of local maxima in the neighbourhood

of n/z.

note that there are clusters of points of SCn 2, to the right of J and clusters of points of WCn 2, to the left of J. These clusters give rise to a local “comb” structure of S, and WC close to x0 in C,‘. There also exist points of SCn 2, to the left of the WCn 2, cluster and points of WCn 2, to the right of the SCn 2, cluster. These correspond to maxima yk with negative slope and maxima y-k with positive slope. The slope of SCat x&Z, is &(x)/r and the slope of WCat XE2, is -&(x). Thus, because &(x) is maximal at x0, A(x) is approximately constant in a neighbourhood of x0 and the picture of SCand WC shown in Figure 11 is obtained in the neighbourhood of xo for PlyI slightly less than c. Note that by symmetry, there may be several points of SCn WC in X,‘\Z, on the line r~‘{n/z). Also, the gradient of Y,+at a point x of Z, is given by c7yk(x) = sin (&k(x) - TO(X))[--r[dldu0l(x> + 4x)[~l~~0l(x)l. It follows that for the segments of SC to the left of J and the segments of WCto the right of J, Vyk and Vyk are as depicted in Figure 11. Let x : R-*X+ be the trajectory of a saddle point as cr approaches 1 from above. Recall that x can cross SConly at a point of SCn WC with the creation of a period (2) point, and

Figure 11. Corresponding structure of S, in the neighbourhood

of n/z.

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that there is a re-entrance bifurcation if x(a) crosses the trajectory q : R+C+ of the highest point of S, n W, at some o = (T,. One can show that the trajectories cross if /?I71tan (~k/z) < cosec (2rrm/z). If the condition is true, there is a re-entrance bifurcation at or> 1 in C,‘\Z,. Otherwise, the bifurcation is degenerate, occurring on Z, at o = 1. 4.3. THE HOMOCLINICITY OF RE-ENTRANCE In this section, it will be supposed that a saddle trajectory x(o) is arbitrarily close to transversal intersection with the trajectory q(o) of the highest point of W, n S, when o is close to, and decreasing towards 1. Let x(a) =x0 and q(a) = qo. It is necessary to investigate the conditions for W’(XO) to intersect W”(x,). In the present dynamical system, the stable and unstable “manifolds” w”(x,) and W”(x,) need not be smooth. This follows from earlier considerations concerning the shredding of local 1-manifolds which cross S, and the conjugate properties of W, . The global “manifolds” are constructed by translation of local stable manifolds by using P and P-‘: that is, the global manifolds are the limits as II approaches infinity of W%) = koOF( WE)

and

V(n) = & p-Y W”,),

where W,” and Wi denote local stable manifolds. If P is differentiable, the global stable manifold theorem states that w” and W” are injectively immersed sub-manifolds of phase space. However, if P and P-’ are non-differentiable, W” and W” need not be differentiable. In the present type of system, given that x0 is assumed to be a hyperbolic fixed point, it may be supposed that there exists a first n (perhaps by re-defining Wj and WF), such that W”(k) n S, = fa and W”(k) n W, = j3 for k < n but W”(n) n S, and W(n) n W, are non-empty. Otherwise, W” and W” are smooth. It will be shown that if x and y are the first intersection points of W(n) n S, and of Ws(n) n W,, in the sense that the arc length of the segments [x0, x], [x0, y] along W” and W” are minimal, x and y must be transversal intersection points if x0 is close to qo. 4.3.1. Proposition 8 If cr is just above 1 and x0 is just above qo, W,“(x,) is approximately parallel to W, and Wl(xo) is approximately parallel to S, at qo. Proof. Suppose that A is an eigenvalue of JP(x,). Then A is one of d* = (Tr (JP) f (Tr (JP)* - 4 det (JP))‘/*)/2.

If (V, T) denotes an eigenvector, the slopes (V/T), by (V/T)+=

of the eigenvectors of A, are given

~(~J~(A~C-~~~S)+(~~J~C+A~S)A~) (-SA,(l

-r)f(2ro,C+A,S(l

+r))2-4r2u~)“2’

In the limit U,=O, one obtains (V/T)+=A,/r and (V/T)-=-AI. But (V/T)+=A,/r is the slope of S, on Z, at ro(xo) and (V/T)_ = -A _ is the slope of W, on Z, at 7o(xo). Therefore, because A(x) is maximal on Z, at (0, a/z), if cr is just above 1, it follows that WL(xo) is approximately parallel to W, and WL(xO) is approximately parallel to S,.

448

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It is interesting to note that for small finite uI , the eigenvalues A* are given by i/2 2: (2VlC+A;S(l +#4r2 1*= -‘(2rv,C+A,S(l +F))* [ VI [ 01 I II i.e., A.‘-0 and X - -A$( 1 + I)/v, -cc. Thus motion along w” becomes arbitrarily fast as u1 approaches zero. By proposition 8 it follows that for x(a) close to q(o) and u&(o)) small, W,U(x,,)is transversal to S, and WL(x,) is transversal to WC (see Figure 12). Note that the gradient

Figure 12. A saddle point close to contact with IV, n SC.

of y,,,(x) at XE W”(n) n SCis “towards” x0: that is, x0 lies in {ym< c>. This corresponds to the re-entrance bifurcation when the intermediate maximum v,(x(a)) = c. Recall that the segment SCn U of SCdivides a neighbourhood U of x into sectors U, : u, = {XE Ulym(x’) a}

and

u- = {X’EUIJ$JX’) < c}.

Let Z be an interval on W” about x and let I, = Z n U, . By the above remarks I_ is part of the interval [x0, x[ from x0 to x along W” (see Figure 13). Note that P maps points in U- close to p(x). But X~EU_ and x0 is fixed. Thus p(xo) =x0 is close to p(x). Equivalently, p(x) is close to x0 on WC-because p maps SConto WC. If K is a similar interval about y on W”, one can split K into K* depending upon some y-,,, 2 c or CC. By proposition 6, P(Z_) is transversal to WCat p(x) and similarly, P-‘(K-) is transversal to SCat P-‘(y)

Figure 13. Local intervals on w” and W”.

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Figure 14. First iterate of the local intervals.

(see Figure 14). The segments P(Z+) and P-‘(K,) are attached to Z, at P(x) and P-‘(y). The complete structure of W”(n + 1) and W”(n + 1) can now be deduced from the fact that P is smooth on [x0, x[ and P-’ is smooth on [x0, y[. Thus the segments P([xo, x[) and P-‘([x0, y[) are smooth curves. But P(Z-) cP([x,,, x[) and P-‘(K)- t P-‘([x0, y[). One can therefore write P([xo, x[ = [x0, p(x)[ and P-‘([XC,, y[) = [XO,P-‘(y)[ as connected segments along W” and w” which branch transversally off WCat p(x) and off S,, at Pe2(y). It is important to note that [x0, p(x)[ cannot intersect WC and [x0, P-‘(y)[ cannot intersect 2%. For [x0, P”(x)[ n WC= P([xo, x[) n P(Z,) = P([xo, x[ n Z,) = @ Similarly, [x0, P-*(y)[ n S,=P-‘([x0, y[) n P-‘(Z,) = because 1x0, x[nZ,=IZI. P-‘([x,,,y[ nZ,)=@ because [xO,y[ nZ,=fa. This observation implies that [x0, p(x)[ smoothly joins xo to p(x) without crossing WC and [x0, P-‘(y)[ smoothly joins x0 to P-‘(y) without crossing S,. It is therefore necessary for [x,,, p(x)[ to re-cross S, and [XO, P-‘(y)[ to re-cross WC. But in this case [x0, P(x)[ will probably cross w” and [x0, P-‘(y)[ will probably cross W”. To see this, note that there exist four possible permutations of the propositions P’(x) is (above, below) y on WC and P-‘(y) is (above, below) x on S,. These situations are depicted in Figure 15. It is clear that cases (a), (c) and (d) must lead to transversal intersections of w” and W” at homoclinic points marked H in the figures. However, case (b), where P:(x) is below y on WCand P-*(y) is below x on S, need not lead to homoclinic intersection, at least at the first iteration stage. However, it seems inevitable that, as x0 approaches a re-entrance bifurcation, homoclinic intersections should occur even in this case, perhaps in higher iterates of Wi and w”,. It is therefore conjectured that homoclinic tangles are inevitable in re-entrance bifurcations occurring at sufficiently low velocity.

5. SINGULARITY AND GLOBAL DYNAMICS In this section, the analysis of vibro-impact. These are the attracting sets. Both analyses stepwise limiting construction

of section 3 is applied to two topics in the global dynamics geometry of homoclinic tangles and the geometry of strange rely heavily on the concept of shredding applied to the of the global stable manifolds of a saddle point from local

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(b)

(a)

Cd)

(c)

Figure 15. Possible generation of homoclinic points. See text for descriptions of (a)-(d).

stable manifolds. By using a combination of numerical simulations explained in terms of analytical results, it will be shown that the geometry of the above objects can be largely explained in terms of the geometry of shredding. In some cases, the geometry of a homoclinic tangle defines the geometry of a strange attracting set. This topic is discussed below.

5. I.

THE

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According to the discussion of section 4, one can consider the existence of saddle points in the phase space of a vibro-impact system as arising from inverse re-entrance bifurcations or through flip bifurcations of stable nodes. In the former case, a saddle can be born with a ready made homoclinic tangle due to its proximity to the singularity subspaces SC and WC. Simulations indicate that for a broad class of systems, the tangle can persist as o increases towards cr* until homoclinicity is destroyed at a homoclinic tangency at cr, < o*. For o1 < cr < o*, the former tangle’s loops writhe into a strange attracting set which is born in an inverse “blue sky” catastrophe at tangency. The connection between the geometry of a tangle and the geometry of the singularity sets may be considered to arise as follows. Firstly, in any symmetric or antisymmetrically periodically forced vibro-impact system, it is easy to show that WC and SCare homeomorphic. The explicit mapping arises from the homeomorphism Z : (o. , TO)+( uo/r, - 20) which maps WC into SC. The global unstable manifold W”(x,) of a saddle point x0 is constructed

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by translation of a local unstable manifold WLu(xO)around z: under iteration by P. Thus W”(xO) is the limit as n approaches infinity of

Either W”(x0) is smooth, or there exists an integer n such that W”(xO, n) n SC is empty but w”(xO, n + 1) n SC is non-empty. Let x be a point of transversal intersection of WU(xO,n + 1) and S, and let Z be an interval on W’(x,,, n + 1) containing x. Then if W”(x,) tangles with W”(X& the &lemma implies that the iterate Z, = u,& Pk(Z) contains a sequence (Jk) of intervals which C’ accumulates on I: that is, Jk approach Z in parallel (see Figure 16). Because Z is transversal to S,, the sequence ( Jk) accumulates on I transversally with respect to S, . This generates a Cantor set C of points xk = Jk n SCof transversal intersection. Each of the points xk has p(xk) E w, and P(Jk-) branches off WCat p(xk). In this way, it becomes apparent that the geometry of W” is determined by that of WC. Note that because almost all xk lie in any given neighbourhood U of x and these xk lie in U+, where S, n U is modelled by a fibre yk ’{cl, the points p(xk) lie close to p(x) (see Figure 16). The segments P(Jk+) are all transversal to 2, at P(.z$, and accumulate on P(Z+). This follows from the fact that the points P(&) accumulate on P(x) and the segments P(Jk) have slope -Ao(P(xk)). Also note that p(Jk’) are attached tangentially to WC at P*(xk). Clearly, dual considerations apply to the interaction of W” and WC, the singularity subspace of P- ‘.

Figure 16. Cantor sets of branches of W” off W,..

It is important to consider the intersections of W” with WCand W” with S, and duality implies one only need consider W” n WC. If XE W” n WC, then P-‘(x) = ye W” n SC and it follows that the segment of W” through x must correspond to P(Z_) u p(Z+) for Z an interval on IV’ through y. Although P(Z-) is transversal to WC if Z is transversal to S,, p(Z+) is always tangent to WC. Therefore W” is not transversal to WC at a point of W” n WC. This implies that filaments of W” in the neighbourhood of WC will either be “parallel” to WC or correspond to the intersection points just described. This property further imprints the geometry of S, onto W”. However, note that it is to be expected that the imprinting will be most powerful for saddles close to S, and WCthan for high-velocity saddles. The imprinting process is strikingly illustrated by the simulation depicted in Figure 17. The system is harmonically excited and has parameters c= O-1, z= 3.7, /3 =O. 1 and r = 0.9; the plot has horizontal time axis and vertical velocity axis, where O
452

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I

Figure 17. Similarity between the geometry of a tangled unstable manifold and the geometries of WCand S,.

S, and WCare represented corresponding to-iterating 5000 points backwards (respectively) forwards off the time axis. The plot also depicts part of W” for a saddle point (represented by an asterisk), showing the orbits of 2000 points on the lower velocity part of a local unstable manifold under five iterations of P. Points to be noted are the Cantor sets of transversal intersection points of S, n W” mirrored by Cantor sets of segments of transversal branch points of W” and S, . Also, note the parallelism of the filaments of W” and of WC away from the branch points. The corresponding Cantor sets of transversal intersections of W” and Z, do not show up well. This is explained by the fact that JP(x) is dominated by u,- ’terms and thus one would expect low-velocity regions of W” to be sparsely represented in a finite representation of some W& Finally, note that the plots of WC and S, contain some transversal self-intersections, rather than branch points. This is due to numerical inaccuracy and disappears for more accurate, but much more time-consuming simulation. The tangle described above arose in the stable manifolds of a saddle point close to S, and WC. As mentioned above, the imprint of the geometry of S, and WC on a tangle is somewhat weaker for saddle points remote from the singularity sets. Consider Figure 18(a), which shows part of the tangled unstable manifold for a system with parameters x=0.1, =3*7, /3 =O-5 and r=0.98. Note that the left- and right-hand edges of the plot are identified and that the plot is for 0 < ug< 0.6 and depicts the orbits of 10 000 points in a local unstable manifold iterated forwards seven times. Upon comparing this plot with Figure 18(b), it is apparent that there is only a weak imprint of the geometry of S, most pronounced in the region of WC. However, there are also several apparent singularities appearing as branches of W” with itself. The generation of these discontinuities is represented in the sequence of sub-figures of Figure 18. In Figure 18(b) are depicted 5000 points on S, and WCtogether with W”(3) comprising three iterates of 750 points on a W,“.Note that W”(3) is continuous but has four transversal intersections xl , . . . , x4 with S,‘. In Figure 18(c) are depicted S,, WCand W”(4) represented by four iterates of 1000 initial conditions on Wfl. As predicted, W”(4) is disconnected. The disconnection can be explained as follows. Let Jk be small segments of W” about & . Then J: lies in [x0, x] and J; lies in 3x1, XZ[. Also, 5; lies in 1x2, xi] and J; lies in [x2, x3[. Similarly, J3+ lies in [x3, xz[, J3- lies in ]x~, x4[, J4- lies in 1x4, x3[ and J4+ lies in 1x4,).

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r

Figure 18. (a) The singular, tangled unstable manifold of a saddle point; (b) first intersection points of an iterate of a local unstable manifold; (c) a further iteration showing branches off Wand segments transversal to the time axis; (d) a further iteration of (c) which reveals much of the structure of (a).

P([x,,, xi]) is a continuous segment attached to Z, at P(xJ and P(]xi , xz[) must be a continuous segment transversally branching off WC at P’(xi) and p(x2). The segment P( [x2, x3]) is transversally attached to Z, at P(xz) and P(x~) forming the loop, L, depicted in the figure. Similarly, the segment P(]x3, xq[) transversally branches off WC at F(x~) and P(x,). The segment P&Q)) is transversally attached to Z, at P(A$. Note that the information obtained at this early stage of iteration explains a good deal of the coarser features of Figure 18(a). Figure 18(c) also depicts many new points of intersection of W”(4) and SC, These points, together with a further iteration to WU(5) depicted in Figure 18(d), generate many new d&continuities. Most of the definable structure of Figure 18(a) is shown in Figure 18(d). 5.2. GEOMETRY OF SINGULAR ATTRACTING SETS In two-dimensional, smooth, discrete dynamical systems, the best understood sets are the hyperbolic attractors. For example, if A is an indecomposable attractor in such a system, it is either a stable node or a subspace of topological 1. In the latter case, A coincides with the closure of the union of the unstable of its points, each unstable manifold being an immersed l-manifold.

attracting hyperbolic dimension manifolds

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The most detailed description of one-dimensional hyperbolic attractors is provided by the theory of expanding maps on branched l-manifolds which generalize Smale’s classic solenoidal expanding attractor. The latter object connects very neatly with the occurrence of fractal microstructure in such attractors because it has a direct representation as a Cantor l-manifold, locally R x C, where C is the classic Cantor set of excluded middles of the unit interval. The above models allow one to visualize a hyperbolic attractor as locally a complicated collection of parallel linear segments packed into a compact subspace of the plane. Another useful way to visualize the geometry of attractors occurs if there exists a nearby saddle point and its unstable manifold is drawn into the attractor. If this is the case, the one-dimensional submanifold wraps itself arbitrarily tightly about the attractor and thus delinates its geometry. In this section, the above connection between the geometry of a strange attracting set and the geometry of unstable manifolds of saddle points, together with the shredding process, will be utilized to discuss the geometry in vibro-impact dynamics. Simulations indicate that there are two main types of strange attracting sets in vibroimpact dynamics. The first type has some connections with the re-entrance process discussed earlier. Recall that subharmonic fixed points may be born in pairs of saddle and a stable node in an ordinary saddle node bifurcation. Several different pairs corresponding to different subharmonic numbers may co-exist in phase space. With attention restricted to the lowest velocity pair, the high-velocity component WY of the unstable manifold of the saddle is drawn into the node, while the low-velocity component W!! enters a region of the cylindrical phase space bounded by itself, WT and 2,. There are two possibilities. Either W!! is trapped within this region or it crosses its boundary. In the second case, the only possibility is that Wt! crosses WL , which causes a homoclinic tangle. In the first case, W!! has a limit set on some attractor within the region. Such attractors will be called lowvelocity attractors and because S, and W, inhabit low-velocity regions of XT, one would expect a strong imprint of the geometry of S, and WC on their geometry (see Figure 19). The imprint can be very marked. Consider Figure 19(a), which depicts WE, WT, S, and W, for a system with c = 0.1, z = 1.78, /? = 0.1 and r = 0.9. The saddle is subharmonic of order 1 and lies near velocity uo= 0.3. Note that the structure of the attracting set around which W!! wraps itself is very similar to that of the homoclinic tangle depicted in Figure 17, except that its loops are, of course, constrained within the region bounded by WT, WY’ and 2,. The geometry of the attractor is revealed in Figures 19(b)-(d). In Figure 19(b) are depicted S,, W, and W”(4) generated by four iterations of a local stable manifold, and three transversal intersection points x , , x2 and x3 with S, are shown. In Figure 19(c), w”(4) is replaced by W’(5) and one notes the disconnections, branch points with W, and transversal branches with Z, due to these points. Many new transversal intersections of W” and S, are generated by filaments of W” approximately parallel to W,. In Figure 19(d), much of the coarse structure of the attractor is revealed by the replacement of W”(5) by W”(6). Note the almost exponential growth of the number of points of S, n W’(n) with n. The second type of strange attracting set most often observed in a vibro-impact system is connected with the stable node of the conjugate pair of subharmonic fixed points mentioned above. In some parameter ranges with CT< 1, only a single globally attracting node can exist, and the node can be destabilized in a flip bifurcation creating a saddle point and a stable period (2) point. The flip bifurcations can cascade but simulations indicate that the cascade may be only partial, being interrupted by a homoclinic tangency in the stable manifolds of the saddle. Although what actually occurs is not clear, a strange

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(d)

L Figure 19. (a) Similarity between the geometry of a strange attracting set (as dehneated by the unstable manifold of an adjacent saddle) and the geometries of S, and W,; (b) first intersection points of an iterate of a local unstable manifold with S,; (c) a further iteration of(b) showing branches with Wand segments transversal to the time axis; (d) a further iteration of (c) which reveals much of the structure of (a).

attracting set is created and the tangled unstable manifold of the saddle point has asymptotic limit set on the attractor. Even in this type of system the geometry of the attractor can be strongly influenced by the geometry of the singularity subspaces. This is well illustrated by the attractor plotted in Figure 20(a), which also shows W, and S, for a system with parameters c = 0.1, z = 2.8, j3 = 5.0, r = 0.7 with 0
456

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(0)

.’

S. WHISTON

(b)

.-SC

(cl

(d)

/

P

SC

f----Y4

SC3

,pT \ wc \, 4 *i I,

x2

/

WY(4)

/.// 4

1

SP

\ “.* P(w) \

\ d

L--

SC -I\

4

W”(4)

‘.(, w ‘_

P(x, )

P%,)i

\

I J

%\, ‘._ w

a

Ph,)

P(x,l

P(x,)

a

Figure 20. (a) A global strange attracting set and its interaction with S and W. (b) first intersection points of an iterated local unstable manifold with S,; (c) a further iteration of (b) showing branches with Wand the time axis; (d) a further iteration of(c) revealing most of the structure of (a) WC at p(xl). The alternating nature of the saddle means that P(J-) is part of a smooth segment from the saddle to p(x,). Note that JV” cannot transversally cross WC and is confined beneath it in the left-hand part of the plot. Two new transversal intersections of W” and S, are created, labelled x2 and x3. In Figure 20(d), which shows W’(S), much of the geometry of the attractor is revealed. There are now two new disconnected components of W” branching transversally off WC at p(x2) and p(x3). A second new disconnected component appears to be attached to WCat its“end” P(a), but theory dictates that this is part of the filament P(J+) tangent to WCat p(x). A new transversal intersection point of S, and W” is created close to x1.

ACKNOWLEDGMENT The author would like to thank David Chillingworth for many useful discussions about singularities. REFERENCES 1. G. S. WHISTON 1987 Journal of Sound and Vibration 118, 395-429. Global dynamics of vibroimpacting linear oscillator. 2. S. W. SHAW and P. J. HOLMES 1983 Journal of Sound and Vibration 90, 129-155. A periodically forced piecewise linear oscillator.

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W. SHAW 1985 American Society of Mechanical Engineers, Journal of Applied Mechanics 85 APM14. Dynamics of a harmonically excited system having rigid amplitude constraints, part I : subharmonic motions and local bifurcations. 4. S. W. SHAW 1985 American Society of Mechanical Engineers, Journal of Applied Mechanics. 85 APMlS. Dynamics of a harmonically excited system having rigid amplitude constraints, part II : chaotic motions and global bifurcations. 5. J.M. T.THOMPSON and H. B. STEWART 1986Non-linear Dynamics and Chaos. New York: John Wiley. See Chapter 15. 6. J.M. T. THOMPSON, A. R. BOKAIAN and R. GHAFFARI 1984Journal ofEnergy Resources Technology 106,191-198. Subharmonic and chaotic motions of compliant offshore structures and articulated mooring towers. 7. J. W. BRUCE and P.J.GIBLIN1987Curves and Singularities. Cambridge University Press, second edition. 8 V. I. ARNOLD 1986Catastrophe Theory. New York: Springer-Verlag, second edition. 3. S.

APPENDIX : PERIODIC GRAZE ORBITS Let F” be the set of periodic points of Pz mod (2x/z) the orbirts of which remain on the time axis. Clearly, F” cS, n A- and to understand the geometry of S,, one has to generate information about F”. If x is any point of F”, the free response displacementtime trajectory Y(x, r) is periodic, and if z is irrational, this is only possible if either the oscillator component or the particular integral are identically zero. In the first case, one obtains only a periodic point if the global maximum of the particular integral equals the system clearance. The period of the corresponding point of F” is then the number of different times mod (2w/z) that the global maximum occurs. The particular integral can be identically zero only if the excitation is identically zero. If z is rational, the structure of F” can be deduced only if the excitation is known explicitly. In any case, the problem of calculating F" for the case of harmonic excitation is involved enough, and the remainder of this Appendix dedicated to that end. Using the considerations set out above and the results of proposition (Al), one can immediately write down proposition A2. Proposition Al. If XER~ is any initial condition for a free response trajectory under harmonic excitation, the displacement-time trajectory Yx(z) can have only the following types of multiple equal displacement extrema (coincidences) : (1) four coincidences comprising two maxima and two minima, each pair of coincidence times rI , z2 satisfying rr + ~2= 0 mod (2x/z) ;(2)three coincidences comprising two maxima and a minimum or vice versa; the pairs occur with r I+ r2 = 0 mod (2a/z) ; (3) two coincidences comprising two maxima, two minima or two inflections (AZ(+) and AZ(-)), each pair occurring at rI + r2 ~0 mod 28/z. Proof: The free response can be represented in the (y, 3)-plane as the locus of a point on a circle of radius r(x) moving around an ellipse generated by the particular integral. r(x) is given by r(x) = (((T - /IT cos (zT~(x)))~ + z2j3zy2sin2 (zr~(x)))“~,

where x is an initial condition (o., 0, ro). Writing co, s,, for cos (zro(x)) and sin (zro(x)) one obtains r dr/dro = z/? ysoAo

and, at dr/dro = 0,

r d*r/dri = z/3y(zc0A0 + SOdAo/dro),

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where A,,, the deceleration, is (o - j? cos (zz~)). In what follows, it will be assumed that 0 u. This case is as complicated as possible. The extrema of r occur when so= 0 (or r. = 0, n/z or 2x/z) or at A0 = 0. The latter zeros occur at the unique (IIand o points (mod (2~/z)) solving p cos (zr,,) = CJ.Consider the extrema at to = 0,2x/z. In this case, c,, = 1 and /I > cr implies A0 < 0. Also, 0 < z < 1 implies 1 < y < 00. Thus r is a maximum at 0,2x/z with value j3lrj - Q. At ro= X/Z, co= -1 so that Ao= Q + /I > 0 and ro= n/z is therefore a maximum with value /3Iy( + cr. At the zeros A0= 0, r d*r/dri = z*j?*& > 0 and thus a and w are local minima of r. It follows that the graph of r(ro) has the form shown in Figure Al. Examination of the number of solutions r” to r( ro) = r yields the following : (i) there are no solutions for Y< r”; (ii) at r=ro there are two solutions a and &, both inflection points; (iii) for r. < r < rl there are four solutions, two maxima (A0 > 0) and two minima (A0 < 0); (iv) for r = rl there are three solutions, two maxima and a minimum; (v) for rl
A<0 0

A<0

A>0 I

I

I

w

lrl.2

Q

z-50 2lr/‘?

Figure Al. The schematic graph of r(rO).

Note that the symmetry of the graph about n/z yields the symmetry rI + r2 = 0 mod (211/z) of the similar pairs of extrema. Proposition A2. Let FE denote the set of period k points of P2, the orbits of which lie entirely within A- in a clearance system with harmonic excitation. Then one has the following: (1) if z is irrational, F! = 0 if k > 1; also, Fy = 0 unless j?Iyj = c when F? = { (0, T,)}, where rl = 0 mod ~A/Z if z < 1 or tl = R/Z mod (2x/z) if z> 1; (2) if z is rational, FE = 0 if k > 2, but F? and Fi need not be empty. A direct approach to the calculation of F” makes use of a non-differentiable function defined as follows. Suppose that z is rational. Then there are clearly only a finite number of different maximum values Ok(x)). Define a global maximum, ,LL :A-+R, p : x+max (yk(x) [k > 1 >, where y1(x) = c corresponds to the initial value x0 = yl (x) = c, n,(x) = ro(x). /J(X) is the global maximum of Y,(r). Note that ,U is continuous but need not be differentiable. To see this, note that p is locally yk for some k except at intersection points yk(x) = ye(x) when ~1might switch from yk to y,, giving rise to a slope discontinuity. p cannot “end” with yk at a cuspoid “end’ point because an inflection is always dominated by a superior maximum. The importance of ~1 is summarized by the following two propositions. Proposition A3. Suppose that z is rational, z> 1 and PlyI fc. Then one has the following: (a) p(x)>ciff Ix~EP~-‘(E~\A-) u (P;‘(A-) n Pze2(E~\A-)); (b) p(x)=ciff lx(eF: (periodic order 2) ; (c) p(x) -CCiff 1x1EFy (periodic order 1). Proof. (a) Suppose that p(x) > c. Then there exists a first maximum yk(x) > c and Y,(r) must cross y = c at some time ro(x) < r1 < &(x). If Y,(r) -Kc for ro(x) < r < rI then P*(x) = (YJr,), r,) for YJr,) ~0 implying IxIEP;‘(E~\A-), where 1x1denotes x mod (2w/z). If Y(x, r2) = c for ro(x) < r2 < rl then P*(x) = (0, r2) (by the two maximum rule) and P%(x) = In the latter case lxl~P;‘(A-) n PC2(E,‘\A). Conversely, if (ircx, r,), z,).

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]xl~P;i(Cf\A-) or ]x]EP;‘(A-) n P;‘(X:C+\A-), YX(7) must cross y=c with YJzi)>O. Therefore p(x) > c. (b) If p (x) = c then 1x1E1;;. Because p(x) = c there exists a maximum yk(x) = c. The two Y,(7) c is impossible, by (a). (c) If p(x) 1. Then one has the following: (a) ~(x)=~(Z(x)) for Z:A-+A-, Z:(O, 7)+(0,-7) mod (21r/z); (b) P]r]>c implies p(x) > c for all x in A-; (c) /?Iy] = c implies p(x) > c if 70(x) #X/Z mod (2x/z) and p(x) = c if to(x) 3 B/Z mod (2x/z) ; (d) /3]y] < c implies there exists a point x* in A- at which p(x) =pUW) =c and P(X)>C if x#[x,, &Jl. Proof. (a) follows from Y& 7) = Yx(2x/z - 7). Suppose that PI yJ > c and define X=(7(pycos

(zQ=c}

and

Y(x)={7Jr(x)cos(7-t,(x)+~(x))>O}.

Clearly, X # @ and if X n Y(x) # 0 then Y,( 7) = T(X) cos (7 - 70(x) + 4 (x)) + Pr cos (~7) implies p(x) > c. But X n Y(x) = 0 if and only if Y(X)cos (7 - 7o(x) + 4 (x)) < 0 whenever Pr cos (~7) = c. This can be true only if both harmonic terms have constant relative phase: i.e., z = 1. Thus /3]y] > c and z # 1 imply ~1(x) > c. If PI y( = c then the above argument holds except for 7o(x) 3 a/z mod (2n/z) when I(X) = 0. In this case, Y,(7) is the exceptional trajectory J?r cos (~7) which has J+(X) = c for all k. Suppose that PlyI c on the complement of [x, , ~K/Z - x*] and that the assertion is true for all PI y] < c. However, it is clear that if the sign of (dp/dzo) is constant (where defined) either side ofx then statement (c) will be true. Note that ~1is piecewise smooth and the following considerations : (1) (dp/d70)>0 is preserved through a discontinuity of (dp/dz,); suppose that ,u is not C’ at y; then there is an interval ]y- E, y + E[ and integers k andj such that ~1=yk on ]y- E, y] and p =yj on [y, y+ E[; if (dp/dso) > 0 on ]v- E, ~1, the discontinuity occurs because (dyi/dzo) > (dyJd70) at y; thus (dyj/d70) > (dyk/dso) > 0 and the sign of (dp/ dzo) is preserved through y. (2) Next consider the possibility that (dp/dzo) changes sign when p =yk on some interval 1~- E, y + E[. Note that if (dyk/dz,) > 0 at y - E then (a) if y is an inflection point of yk then (dyk/dto) >O at y+ E so that the sign of (dp/dro) is preserved, and (b) if y is

460

G. S. WHISTON

a minimum of yk then (dp/dro) a maximum. Now and

< 0 at y - E so the minimum must have been preceded

(dd$)=A(y)C(y)~(~~‘)~y)-

1)

by

at (g)=O,

where 4~) = -Y(Y, TO(Y)), WY) = -Y(Y, MY)), C(y) = cos (&LY) - SO(Y)) and S(y) = sin (&( y) - ro(y)). Clearly, (dyk/dro) = 0 if and only if&(y) - ro( y) 3 0 mod (A). Suppose first that &( y) - ZO(y) = z mod (2~) when C(y) = - 1. Then (!$$)(y)=A(y)&+l)>O,

i.e.,yisamaximum.

If dk(y) - ro(y) ~0 mod (2~) then C(y) = 1 and

It therefore follows that (i) A(y) > B(y) implies y is a minimum of yk; (ii) A(y) = B(y) implies y is an inflection of yk; (iii) A(y)
[

@

b 7

sin

cos

(zkk(Y))

@kc

Y))

I[ =

c(y)

-S(y)

s(y)

c-

C(Y) I[

@

cos

@O(x))

zfir sin (zro(x))

I ’

Therefore C(y) = 1 and S(y) = 0 yield the following relationships : c-yk(y)=Py(cos(ZZO(y))-cos(Zak;G(Y))),

sin (zrO( y)) = sin (z&(y)).

By definition, A(y) = c - p cos (zzo( y)) and B(y) = yk( y) - fi cos (z&(y)) ; therefore, using the above equation for c- yk(y), one obtains A(Y)-B(Y)=-z’Pk](cos

(zro(y))-Cos

wkwh

The condition sin (zro( y)) = sin (z&(y)) implies that k,(y) - ro( y) z 0 mod (27r/z) or &(y) + 7o(y) E n/z mod (2x/z). In the first case, A(y) = B(y) : that is, y is a point of inflection. In the second case, A(y) - B(y) = -2z2j?l y( cos (zro( y)). This implies that (i) y is a maximum of ro(y)~]3a/2z, 2a/z[, (ii) g is a minimum of ro(y)~]n/z, 3n/2z[, and (iii) y is a point of inflection if zo(y) = 3~/2z. This exhaustive analysis has established that maxima of yk can occur only in ]3a/2z, 27r/ z[. It therefore follows that (dp/dzo)(x’) >O for x’ close to x in ]z/z, 21r/z[ must imply that, when defined, (dp/dro)(x’) ~0 in IX/Z, 3~/2z[. However, recall that r(x)) is given by r(x’) = ((c - j?r cos (zr~(x’)))~ + z2p2y2 sin’ (zro(x’)))“‘. Therefore for ro(x’) = 3~/2z, r(x’) = (c2 + z2p2y2)‘j2 > c. By using arguments analogous to those used in proving statements (b) and (c) above, it is easy to show that r(x’) > c implies p(x’)>O. But then c(y)>c for ro(y)=3~/2zmod (2a/z), and (dp/dro)>O in IX/Z, 3~/ 2z[ implies that p(x)) crosses y = c once and only once in IX/Z, 3~/2z[. Because r(x)) > c for X’E ]3n/2z, 2a/z[, I > c on the complement of [x*, 21c/z - x*1, where x* is the crossing point.