The Normal Modes of Nonlinear n-Degree-of-Freedom Systems

R. M. ROSENBERG

Professor of Engineering .M~antcs, University of Callfo.rnla, Berkeley; Caiif.

A system of n masses, eqttal or not, intercomrec;ted ~Y 1101rl.~near "symmetrit;'' springs, and haui1rg " degrees of freedom is examitzed. The concept of tiormal modes is rigormesly defimd and the problem offi.11ding them is reduced to a geometrical maximum-mi11imum problem i1r an n-space of known metric. The solution of the geometrical problem redttr;es the conplcd equations of motion to 11 uncottPled eqteations whose n.atural frequettcies can alwa,•s be jou11il by a single gteadrature. An infinite class of systetns, of which the li1Jear system is a memher, has beett isol. ated for wMch the freqttency ampliltule can be}tnmd in closed form.

In some earlier papers {1, 2, 3) 1 the "normal modes" of certain nonlinea.r two-degre.e-of-!reedom systems were discussed, These systems consisted of masses and springs, a restriction being that the forces with-which the springs resist being. deflected are odd functions of these deflections.

Evidently, these papers have dealt with a heretofore undeJined subject matter since the concept,Qf normal modes is generally defined for linear systems only. In them, "normal solutions" is the term associated with a fundamental set of solutions, and it is a. well-known property of these, that a linear combination of the normal solutions yields all solutions of the system. Even if "it should be possible to isolate, for nonlinea.r systems, solutiol!s which are equivalent to the normal Yibrations of ,linear systems, such solutions cannot be used to construct new aolutions, sinc. e th~ superposition principle is ipso factQ inll.dmissible fu nonlinear systems.

The earlier method [2) consisted in identifying the search for the eigeovectors with the solution of a maximum-minimum problem in two-sp~~oce. Once the eigenvectors had been found in this manner they could be u' tilized to transform the two-degree-offreedom problem into two separate problems, each in a single degree of freedom. This approach suggested itself readily, and it was relatively simple to exploit it, because the geometrical problem (which is the equivalent of the dynamical one) is oue of geometry in two dimensions. It is natural to inquire whether it is even poasible, and if so, whether it is useful to extend this theory to n-"dimensions~ In fact, doubt as to the feasibility of this extension has formed the substance of a discu~ion of an earlier paper [START_REF] Rosenberg | On Normal Vibrations of a General ClaSl! of Nonlinear Dual-l\iode Systems[END_REF].t It is the purpose of this paper to extend the ea.rliEU" fElBUlts tO 1;ystems having n-degl'ees Of f reed01,11.

Equations of Motion

Conside. r a • system of ?Hnnsses (either equal t o or diffel'ent from each other ) interconnected by nonlinear springs, .and t he first r NumberS' in brackets designate Ueferences at end of paper. • This public discussion was offered as •a resua of the. presentation of reference [START_REF] Rosenberg | On Normal Vibrations of a General ClaSl! of Nonlinear Dual-l\iode Systems[END_REF] and last of the masses connected by nonlinear springs to fixed points (or to infinitely large masses). The springs also may be equal to or different from each other. The system has n degrees of freedom and is illustrated in Fig . . 1. The spring force of each spring is an Odd function of the spring deflection. Thus, if a ~pring S, is deflected by an amount u 1 , it rllSil!ts this deflection with a force S 1 (u;), and

S,( -t~) = -S;(u;)

There exists (by definition of admissible systems) an equilibrium position of the system, and the displacement of the ith mnss from its equilibrium position is denoted by x 1 , so that the x;, (i =(1, 2, ... , n) are the co-ordinates of the problem. Then, the equations of motion of the system are

In what follows, we shall only admit springs whose forces are everywhere analytic in the deflections and, moreover, we shall assume that these forces are represented with a. l!9.tisfactory degree of accuracy by finite Taylor expansions. In the technical sense, these restrictions represent no limitations on the springs.

Under them, the equations oi motion become

Yl m,x, = E a;.;(x;-1 -x 1 )i j=I, 3, ••• ~ i = 1, 2, ... , n} L-t a;H, ;(x 1 -Xc+t)l, _ (2)
j -1,3,. .• Xo"" Xn+1=0
Finally, it is convenient to normalize the. co-ordinates b:t means of the transformations

The equations of motion then take on their final form

Fig. 1 i = 1, 2, ... , n (4)
Since the system is conservative, the right-hand sides of the equations of motion must be derivable from a potential function.

In fact, one finds readily

~~ = :~, i = 1, 2, .•. , n (5) U-- ~ ~ {~ ( ~H -~)/+1} LJ LJ • + 1 l/1 1/t I ; • -t, 2., ... j-1, 3, ... J 'llli-1 'll~t . mo = 'llln+t = !X> (6)
Evidently, U i~ . symmetric with respect to the• .origin; in other words, U remain!! unchanged if any one, or some, or all of the ~H are replaced by -~H provided that, at the same time, the corresponding~. are replaced by -~,. This important property is a direct consequence of the fact that all spring forces are odd functions of the spring displncements.

The transformations (3} have normalized the system so that, in the transformed co-ordinates, the kinetic energy is

n 2T(~;) = L: ts • while, in the physical co-ordinates, n 2T(x;) = Y: m,:i:, 1 i
However, this transformation also admits a physical interpreta,tion. The nontransformed equations (2) are those of the physical system of Fig. 1. However, the trllollsformed equations of motion may be regarded as those of a unit ma&s which moves in an n-dimensional space under forces which are derivable from the poteritialfunction U defined in [START_REF] Klotter | Steady State Vibrations in Systems Having Arbitrary Restoring and Arbitrary Damping Forces[END_REF]; the~~ are, then, an orthogonal co-ordinate system in the n-space. Finally, it should be observed that the system ( 4), or (5) and ( 6), reduces to the linear problem when j = 1.

•

Normal Modes of Linear System

When dealing with the linear system, the normal modes, or the eigeove. ctors; are usually determined after the eigenvalues are found. In fact, one usually substitutes sinusoidal functions• of time

~1(1) = b, cos wt, i = 1, 2, . •. , n (7)
into the equations of motion and finds in this manner a characteristic function whose zeros li.re the eigenvalues. Then, the rth eigenvector v,(w,) is found by attaching to each eigenvalue w, the associated eigenvector ri,.. The b;, are then the n.-components of v,. It is evident that, in the very first step, thi11 n:tethod makes use of a property which is unique to the linear problem; namely, that the displacemeots are_ sinusoi4al functions of time. Consequently, this method can never lead to a generalization of the concept of normal modes which comprises the nonlinear problem, and which reduces to the familiar meaning of the eigenvectot when the problem reduces to the linear one.

Here, we wish to find a. new way of defining normal mQdes of the lin6llor system which must be such that (a) the normal modes of the linear system are uniquely and correctly defined by it and (b) the definition must be capable of a simple extension to the nonlinear system. This definition begins with the observation easily deducible from [START_REF] Courant | [END_REF], that

~.(1) = Mt + T) l • i-1, 2, ..• , n ~Ht(t) _ bm 1- _ 0 (8) --= -= Ci+l <;"+I= ~,(t) b,
The first of these states that the normal solutions of the linear system are all periodic of the Bame period, while the second states that the ratio of the displacement of any o.ne mass t o that of any other is identica.lly equal to a constant for all time. The first is a property whic:h muat be ret-ained in the nonlinear case while the second may either be retained in the form or, it is capable of the obvious generalization

i -1, 2, ... , n ~HI ~ ~i+lat), 1: <;11+1 = 0
where the ~i+t are single-valued functions of ~,.

(9)•

The second of (8) harbors a well-known property of normal vibrations. Suppose there exists o. time lo such that ~1(41) = 0.

Then, it must be t rue, in view of the second of (8), t hat ~., ~~ •... ~n all vanish at the same instant. In other words, all roasst>s of the system of Fig. 1 pass through their equilibrium position at the same instant if the system vibrat• es in normal modes. The two properties of vibrations in normal modes, (a) that all mnsses vibrate at the same frequency and, (b) that they pnss through their equilibrium position at the same time is, irl fact, a description of normal vibrations. If these properties are to be retained in the nonlinear system as generalized in (10), the boundary conditions

t;+t(O) = O, i -1, 2, ... , n -1 (11)
m\Jst be added to (10). If, however, (9) is to be retained, the condition (11) is .automatically satisfied.

One additional property of the normal modes of linear systems must be exhibited before the array is sufficient for actu{l.]ly finding the eigenvectors; this property is connected with the manner in which the system is set into motion. Let us suppose that we start the motion without initial velocity, so that the initial conditions are

~,(O) = b,, f,(O) .. 0, i -1, 2, , .• , n (12)
{We note here that the b; are not arbitrary. In fact, with arbitrary values ofb, the system would not vibrate in normal modes; instead, energy exchange between the mlloSSes (or in the transformed system Lissajoux figures of the unit mass) would result.

In fact, the b, are the ~,components of an eigenvector; finding these is the essential problem in the l!Olution of the eigenvalue problem. However, we assume here a priori. the well-known fapt that initial conditions like (9), and resulting in normal vibrations do exist. If it shoul(). be found in the generalization to nonlinear systems that such initial conditione do not exist, we shall say that the nonlinear system ia not capable of vibrating in normal modes.]

It follows from a.n application of L'Hopita.l's rule that

tH1(0) ~l+t(O) CIU /b~HI i -1, 2, •• . 1 n ~1(0) = ~,(0) -bU /b~l I en+1 =:!! 0
or, at the time t = 0, d~.

... = ()Uf?J~R

The meaning of this important property will be discussed a little further on. All these propertiE!I! of normal modes of linear systems are easily interpreted in a geometrical manner.

Consider the (n + I)-dimensional space whose orthogono.l coordinates are~., f2, ... , ~ •• U. When the system is released at the time t = 0, it occupies a position of mrucimum potential

U = -Uo (14)
because it is released without initial velocity. But U = -Uo defineS an n-dimensional equipotentialsurface or, more precisely, it definea the projection of the equipotential surface U = -Uo on the (~,, ~2, We are now in a position to discuss the meaning of (13). At the time t = 0, the point moving along the straight line defined previously, and termed a "modal relation" [START_REF] Roaenberg | Normal Modes of Nonlinear Dual-Mode Systems[END_REF], lies on the bounding surface, or in the domain defined by (14). At the !lame in~ stant, the relations (13) apply. But, since from ( 13)

n dU = L: (15) i ..... 1. 2, ...
It is evident that (13) are merely the well-known expreesions for the conditions that the modal line intersects the bounding surface defined by U = -Uo orthogonally.

We have now collected the properties, necessary to define a normal mode of the linear system having n degrees of freedom.

where u is given by ( 6) with j = l, it is a straight line in the ah ~~ •... , E.,)-space defined by the (n -1) equations (17) which are 11atisfied for all t by periodic solutions Ut) ~ Ut + T) of ( 16) in the closed domain u = -Uo of the celE!,' .. , En)-space, which passes through the origin of that space and which intersects the bounding surface defined by U = -Uo orthogonally.

An Example. AB an example of the foregoing definition, consider the three-dimensional linear problem, and let e. = e. ~~ = t], ea = r

The equations of motion of the physicAl system are nJt:i! -a,x -a2(xy)

Cit(x -y) -aa(Y -z) 1113i = aa(Y -z) -a.z
The transformed equations are f= as(TI r)

+ n~<~'/• 'fnt'/• -ma'l•
We may also write the system a8 where the potential function is defined as

u = ~ Cn~~.r -•~ C~~•;, -,:J;.r -~~ (nl!~;, -,~~.r -~ (m:.~.r
It is evident that U = -Uo defines an ellipsoid in the ce, Tf, .n-space which is symmetric with respect to the origin.

Let the normal solutions be

Tt(t) t{t) l ~(t) = c~, ~(t) = er ~.rare constants or ., = c~~. r = er~
The first of these defines B plane in the ( ~. Tf, n-space which contains the r~axis, and which intersects the(~, t])-plnne on the line 11 = c~~• The second defines another plane which contains the Tt-axis, and which intersects the (~. n-plane on the line r = er~• The intersection of these two planes is the line (OP), as shown in Fig. 2, and that line is the modal relation. Moreover,

( ( B<lUNOING ElllPSOIO
Fig. 2 thll.t line intersects the ellipsoi~ U ~ -Uo a~ two points P and P' and, at these points, the modal line is normal to the surface of the ellipsoid.

The model of the motion in hand is that of the unit mass traveling along the modal line in simple harmonic motion between the points P and P' where the modal line intersects the surface of the ellipsoid. At the extremities, the velocity vanishes, Jl,nd the co-ordinates of the point P are the initial displacement components.

Generafization of Normal Modes

In a purely verbal manner we shall say that the nonlinear system in Fig. 1 vibrates in normal modes when all masses execute periodic motions of the sam~ period, when all of them pass through the equilibrium position at the same instant, and when, at any time t, the position of all the masses is uniq1tely defined by the position of any one of them, In view of these properties and of the pre::ceding section, we give for normal modes, in genetal, the following:

Deflnlllon. The mth normo1 mode of the system

.. bU ~j = ac,' i = 1, 2, .. •i n ( 18
)
where U is given by ( 6) is a line {curved or straight) in the (~1, ~2, An illustration corresponding to that of Fig. 2 but applied to the nonline!J,r system of three degrees of freedom is shown in Fig. 3. It should be abserved that, in the linear case, the bounding surface is an ellipsoid while, in the nonlinear case it is an ovaloid which is, in general, not au ellipsoid.

While only one modal relation is referred to in th~ definitions and only one has been illustrated in Figs. 2 and3, it is known that, in the linear case a.t least, as many such modal lines exist as there are degrees offreedom. For instance, in the example of the three::degree-of-freedom system shown in Fig. 2, three modal lines e>dst and they are 1 in fact, the principal axes of the ellipsoid.

The Decoupled Equations

For the present, we leave open the question as to whether, or how, the modal lines can be found. Supposing simply that they ' -BOUIIOJNG OV~LOJ D fig, 3 are known, they ca,n be used in an obvious way to reduce the n-degree-of-freedom system of n-equa.tions, each in a single degree of freedom. For instance, the rth equation of motion is (20) in accordance with [START_REF] Den | Mechanical Vibrations[END_REF]. In (20} 1 the expression on the right-baud side d~notes the fact that, in general, the derivative oU/o~, depends on the three variables g,-,, The application of this procedure to the linear three-degree-offreedom system may be instructive. • The first equation of motion

(n~t)'/• ]} ~ = -m, 1 -~ 1 -1!h C~r . t r = 1, 2, 3
From these, the three natural freqtiencies, or eigenvalues, can now be found without difficulty.

It should be noticed .that knowledge of the eigenvectors has been assumed without any indication as to how they can be found. We shall now discus~ the questipn of finding the eigeuvectors withant prior knowledge of the eigenvalues.

Determination of the Eigenvectors

The dynamical problem in hand is to find the normal solutions of the system .. bU

~~ = o~,, i = 1, 2, ... , n (25)
Any solution of this system (including the normal solutions) is of the form

~~ = Mt), i = 1, 2, .•. , n (26)
Elimination of time between these functions and regarding ~I (for instance) as the independent variable, transforms (26) into

~~ "" ~~(td, i = 2, 3, ••. , n (27)
These functions constitute the trajectory of the system in the (~t, ~~ •... , ~ .. )-space. In a physical sense, they represent the trajectol'Y of the unit mass of the system (4), or ( 5) and ( 6).

It is well known [START_REF] Da | Le9ons sur la Th~orie Gberale des Surfaces[END_REF] that this trajectory is the geodesic in an n-dimensiona.l space whose metric is du~ = (Uo + U(~r, ~21 i, j = 2, 3, ... , n

While every solution of (30} represents a trajectory of the dynamical system in the (~1, ~2, , , ., ~n)-space, not all are modal lines. The modal lines are those solutions of (28) which satisfy the boundary conditions listed in the definition of normal modes. Since (30) is a system of (n -1) second-order equations, it gives rise to 2(n -1) constants of integration. But the trajectory is defined by the (n -1) equations ( 27), and the satisfaction of the required boundary conditions (one being the value a.t the origin, the other the value of the slope on the bounding surface) requires the determination of 2(n -1) constants. Therefore, it is possible, in principle a.t least, to solve the problem. N evertilelesa, the system (30) is such a. complicated system of highly nonlinear differential equations that the prospect of finding the general solution may be confidently regarded as hopeless.

Straight Modal Relations

Let us assume that, for certain forms of U, the modal relations are st,•aight lines. If tiley a,re, the second derivatives ~/, (j = Z, 3, ... , n) vanish. Then, since

bt;

i, j = 2, 3, ... , n (31)

Writing (31) in terms of differentials (instead of derivatives) and adding (G>U /b~;)d~; 2 to both sides of the equation, it is found that r.l~. d~.

?JU/?J~. = oU/b~2 =

[Actually, proceeding in the manner indicated yields (32) with dtt e>u;o~, absent. However, ~.is not intrinsically a. preterred co-ordinate;

it has assumed a apecia.l position only because we have assigned to it the role of the independent variable, Tbus it is clear that the first term in (32) may be added to the set. One can also obtain this result in a fom1al manner by considering all ~~ to depend on a parameter a, and by treating the variational problem J:

(Uo. + U)' 1 • (._ 1 ~ ••• ~r)''' da =stationary
where primes denote differentiation with respect to a.] But (32) i3 similar to (13), the only difference being that (13) was valid only at t = 0 while (32) applies at every value of t. Therefore (32) implies that tile modal line under discussion is normal to

all equipotential surfaces U ~ -H ~ -Uo
This leads to tile following: Theorem. Every straight line in tile(~.., ~a, ... , t.)-space which intersects all equipotential surfaces orthogonally is a modal line.

Conversely, every modal line which is straight intersects all equipotential surfaces orthogonally.

Homogeneous Systems

Among the systems illustrated in Fig. 1 whose equations of motion are given by ( 4), we select the cla~s who.se equations of motion are

i = 1, 2, ... n } mo = 1nn+t = oo • (33) k = odd integer
This class is called the homogeneous system of degree le because the right-hand sides are homogeneous functions. in the~~ of degree k. We shall show that, for this class, the modal relations are straight.

The system (33) has considerable interest, both from the mathematical and from the physical point of view. lts mathema tical interest resides in the fact tilat it represents a broad generalization of the linear problem since tile linear system is one of its members. Moreover, as we shall show later, in homogene-Olll> systems, the eigeuvalues as functions of the amplitudes can be found in terms of tabulated functions and without the approximations that must usually be made in the treatment of nonlinear problems.

Physically, the homogeneous system is such that the springs resist being deflected with a force that. is proportional to the kth power of the deflection. Thus the system is of interest wherever springs are used which have this property.

For our purposes, it is more convenient to write (33) in the form

•• "()U t = "()~,, i = 1, 2, .•. , n
We uow make u~ of the following J>roperty or surfaces in Euclidian n-space 1 : Consider a smooth surface in this space.

Then, the straight line between the origin (of that space) and tha.t point on the surface which is nearest to, or farthest from, the origin intersects the surface orthogonally.

We consider the equipotentialsurfaces

U = -H = const 0 > -H ~ -Ua
and we seek those points on each of them whose distanee from the origin is stationary. If the locus of these points is a straight line, its equation will not contain the distance of these points from the origin. Therefore the procedure is to write the equations of the equipotentialsurfaces in generalized pola.r co-ordinates and, then, to apply the• condition which singles out those points on the surfaces whose distance from the origin is stationary. A neeessary condition for the existence of straight modal relations is that the modules of the position vectors of these etationary points do not appear in the equations of the locus of these points. If the equations of the locus have real zeros, the conditions which are, both, necessary and suffident. They are such that, when r = 1 and the 0 1 take on all possible values between 0 < 8; ~ 211", all points on a hypersphere of radius Now, the locus of the stationary points in question is defined by

dr = 0 which implies oU "()(}; = 0, i = 1, 2, ..• , n -1
In view of (36), the last equation is equivalent to 1 This iti. not the I'H!paee whose metric is given in (26), The nspace under consideration here has the metric • oe ?18, = 0, i = 1, 2, ... , n -1

(37) and, thus, the condition which is neeessary for the existence of straight modal relations is sntisfied since e is a function of the angles 8; only. Whether the relations (37) have real roots 8 1 "

(r = 1, 2, ... ) cannot be answered without further calculation beca.use the f 1 have not been expressed explicitly as func.tions of the angles. We shall examine this question further for the rase of the nonlinea.r three-degree-of-freedom system. It is instructive to reduce these equations to the linear case and equal springs. In that case, the foregoing equations become cos 1 tp -sin 1 tp = -cos tp cot 0

Homogeneous System With Three Degrees of Freedom

, ( ~ •)k+l a 2 ( ~ fJ )4+1 k + 1 m1'!1 + k + 1 mt'/• -n~.t'l• a, ( fJ r )•+1 "' ( t )•+1 +---,---,- +---,- =H k + 1 1»2 I• m 1 I• k + 1 ~ /i
2 cos tp = (1/cot 0} -cot 8
This pair of equations has the following roots:

2 mode: tp = 0, 0 3 mode: lP !::: 305° 15', 8 60°

The eigenvectors corresponding to these roots are shown in Fig. 4.

Fig, 4

The roots of the transcendental equation yield the following wellknown results [START_REF] Den | Mechanical Vibrations[END_REF] (as is evident from Fig. 4):

1 mode: 1J 2'/t~, t ~.
2 mode: 11

3 mode: fJ

In the nonlinear homogeneous case, the solution of the transcendental equations will yield the values of the constants c in Fig. 5 properties stand out. For instance, in the linear problem, ( k = 1 ), the frequency of free vibrations is independent of the amplitude, while, in the nonlinear case, the frequency of free vibrations does depend on the amplitude.

When the spring is hard' (k > 1), the amplitude increases with the frequency; when the spring is soft' (0 < k < 1), the amplitude decreases when the frequency increases, These a.nd other properties of the backbone curves, easily deducible from (43), are illustrated in Fig. 5.

t] = c~,~• r -er,~. r -1, 2, ••.
< 41 ) Maximum-Minimum Properties and the largest value of r is equal to the number of modes. In general, one may expect as many sets of constants as there are degrees of freedom (i.e., here three), but the existence of superabundant modes [START_REF] Rosenberg | On Normal Vibrations of a General ClaSl! of Nonlinear Dual-l\iode Systems[END_REF] is not excluded. In all that follows, it will be assumed that the transcendental equations (40) have been solved, and that the constants in (41) are known.

The Eigenvalues

If the first of ( 41) is substituted in the first equation of motion, one finds [START_REF] Roaenberg | Normal Modes of Nonlinear Dual-Mode Systems[END_REF] (in view of the assumption that. the three masses are equal) ~ = -[a1 + a,(l -c~,)k)~; r = I, 2,...

(42)

If the eigenvalues are denoted by w, (r -1, 2, ..• ), (42) is integrated over one period tmder the initial conditions ~(0) = x, ~(0) .., 0, and use is made of the nomenclature w•/ar = P 1 , at!a2 = a,. it has been shown [START_REF] Rosenberg | On Normal Vibrations of a General ClaSl! of Nonlinear Dual-l\iode Systems[END_REF] that Before discussing thl11 equation, it should be remarked thn.t our entire discussion of the homogeneous system remains valid for noninteger, positive values of kif provisions are made for the odd characte~ of the spring forces. Tlus ca.n be done, for instance, by replacing terms of the sort ~1 in the equations of motion by ~~~~~-1 and terms like (~i-1 -~1)1+ 1 in the potential functions by 1~•-t -~~IH•. The reason for the extended validity of our discussion is that, with these provisioll8, the equipotential surfaces rema.in symmetric with respect to the origin. Equation ( 43) has been diiiCussed heretofore. [START_REF] Rosenberg | On Normal Vibrations of a General ClaSl! of Nonlinear Dual-l\iode Systems[END_REF]; it is the equation of the frequency-amplitude relations of free normal vibrations, called by Klotter [6) the backbones. A number of familiar It is well known [START_REF] Courant | [END_REF] that the eigenvalues of linear problems possess maximum-minimum properties. Moreover, it has bee.u shown [2) that the modal relations of the (linear or nonlinear) two-degree-of-freedom problem also have such properties. We would like now to show that this property also holds for n degree!! of freedom.

To do this, we focus attention on the geodesics in the space whose metric is Clearly, all are eolutions of the dynamical problem. Suppose we seek the geodesics between the origin of this n-space and a point which is mov~~oble on the bounding surfnce U ~ -Uo. This movable-end-point problem produces not only the Euler equa.-tiollB (30) (and thus solutions to the dynamical problem), but also tra.nsversality conditione. But these transversality conditions state (8] that the geodesics must intersect the bounding surface orthogone.lly. Therefore they are the modal relations.

In other words, the modal relations are the shortest and the longest of the lines of minimum length (geodesics) in then-space between any point on the bounding surface and the origin, where the space is defined by the metric (28).

  the relationsThe first of these, i.e., ~ = Ct~, robs the n-space of one dimension (because one of the co-ordinates is e.'Cpressible in terms of another). In fact, it defines an (n -I)-dimensional hyperspace which intersects the ~~~rplane on the line ~~ = C2~l• In this hyperspace, the equation~~ = c 3 E 1 defines an (n -2)-dimensional hyperspace which intersects the ~~EJ-plane on the line ~~ = c 1 E 1 •A continuation of this argument finally defines a. straight line passing through the origin of the n-space and intersecting at least twice the bounding surface within which the solutions must lie. The component of this straight line along the ~ra..xis, say, is b;~;. We may picture the normal vibration as a periodic (and in the linear problem a simple harmonic) motion of the unitmass traveling along this line between the. points where tbnt line pierces the surface defined by U = -U 0 • The projections of that motion on the ~;-axes, (i = 1, 2, ... , n) are the normal vibrations of the system.

  Definition. A normal mode of the linear system ): <>U ~,;; = ~~ i = 1, 2, .•. , n <>,.;

+

  T) of (18) in the closed domain U ~ -Uo of the(~~~ ~t, .•• , tn)space and which satisfies the boundary conditions that: (a) All ~;.,(0) = O, or the line passes through the origin of the (~1, ~2, •.. , ~,.)-space. (b) The line intersects the bounding smfa.ce defined by U = -Uo orthogonally.

  IS and the eqlmtions of the modal relations are supposed known. But this implies knowledge of the three const~nts c~, in ,. = 1, 2, 3 Intruducing this relation into the first equation of mo. tion yields the three equations . . a1 { a2 [

n 1 +

 1 2:: ~ / 2 > o, i these straight modal relations must satisfy the.eystem of equations I ?JU ~ I I C>U !• ~ + 4. tt ~-

Consider u homogeneous system

  like that of Fig. 1 but having only three degrees of freedom, and let a1.• = a,, at.t = a:, aa.t = aa, ~.~ = a, ~. = ~. ~2 = fJ, ~~ = !, 81 = 0, 82 = "'' H = B The equation of the equipotential surface is a

  n du' = (Uo + U) L dt' i-1, 2, ...

  • at the Tenth International CongFess• of Applied Mechanics in Stresa, Italy, 1960.Cpntributed by the Applied MedUioics Divis.iorr a.nd presented at the Wint.l!r Annual Meeting, New Yo-rk, N. Y., November 26-December 1, 1961, of TBB AMEIUCAI'i" SociETY oF l\b:cHA!'i"IOAL EwGlNEERS.

Di8cussion of tliis paper should be addressed to the Editorial De-p_ artment, ASME, United Engineering Center, 345 East 47th Street, New York 17, N. Y., and will be accepted until April 10, 1962. Dis-_ eus8ion received after the closing date will he returned. Manuscript received by ASME Applied 1\• Iechanics Division, November 1, 1960. Paper No. 61-WA-22.

  ••• , ~n}-space. In fact, (14) defines a domain in the 1Hl]}nce .surrounding the origin, and eJl solutions must lie in this domain (for, if they did not, the potential would have to exceed its maximum value-Uo). We shall, therefore, call it the "bounding surface."Now, in virtue of the second of (8), the normal modes satisfy

  ~" ~•H• If the equations (19) in one of the modes are known, one knows in particular the functions

	~r-1	~.---1(~1)
	~.	~.(~!)
	Introducing them into (20) gives
	where prin1es denote differentiation with respect. to ~1, and dots
	differentiation with respect to time t. (22)
	Knowledge of the equntions defining the modal relation includes
	the equation
		(23)
	Then, (19) becomes simply
		(24)

It follows that (21) is au equation in the single dependent variable ~~• If there are n modes r = 1, 2, ....• n, the system has now been reduced to n equations, each in the variable~~• A simplification of (21) can be. ~chieved if we choose r = 1 (which we are free to do). In that case, (17) is .. CJU b = ~ c~~~ e2> which is, obvipusly, an equation in ~1, only. If there are n modes, there are n equations (23) and, introducing each in turn into (22), there are n equations of the form (24), all in ~1•

  ••• , ~.)) t d~; 2 (28)

	j=l, 2, -..
	Consequently, the differential.equations of this geodesic are the
	Eulerequations of the variational problem

r~ ( n )~ J£, (Uo + U)'l• 1 + ._ L ~/~ d~1 =stationary e J-2,3, ... (29) These Euler equations are 2(Uo + U) {~;'' ( 1 + ~ ~/')-~ ~/~/~/'} + (1 + t t/') {tl' ?JU + t ~/~/ ?JU j Cl~; i*' ?Jt; -(r + f: ~/•) ?JU} = o (30) j:j::& bt;

  foa• the existence of str(l.ight modal relations are fulfilled Let the generalized polar co-ordinates be ~{ ... rj,(81, o~ . ... , 0,.-.), i ~ 1, 2, ...• n (34)

  1 in then-space are defined. If these co-ordinates are introduced into the equations of the equipotential surfaces -U = I; ~ ( ~~-~ . -+)Hl = 11 < Uo

		i=l k + 1 7nH/• rn.l•	-
	one finds
	_ U ~ ri"H	n+l 2::.:; i-1, 2, ... k + 1 a;.k	(35)
	or, more generally
	U(r, 81, 8~, ... , 8,.-1) = R(r)S(81, 82, •. , 8,.-1) = H (36)

Acknowledgments

Financial aid from the National Scien.ce Foundation which has supported this investigation is gratefully acknowledged. The author wishes particularly to expre81l his appreciation to the Director of the I118titut de Chronom6trie et de Microm6canique of the Faculte des Sciences at Besa.n~ton, Mr. P. Mesnage, who has kindly put the facilities of the Institute at his disposal for the completion of this study.

• In hard sprin,gs, the spring stiffness (i.e., the slope of the spring force) inorea.sss with increasing deflection, in soft springs, it decreases with increasing deflection.


R M Rosenberg

J K Kuo

Nonsimilar Normal Mode Vibrations of Nonlinear Systems Having Two Degrees of Freedom

When a non1inear system /l(wing several masses vibrates • in normal modes, the t• ime histories of the mot-ion of these masses are, in general, d~fferent in wave shape (although in certain specialnonl-inear systems they differ at most in amplitude, but not in shape). Wh en the 1uave shapes d• i:(fer, the normal mode vibration is called nonsimilar. In this pa.per, 11011similar itormalmode vibrations are analyzed with respect to wave shape and stabihty. The systems considered are those lying close to systems having similar normal mode vibrations. An example is worked out in detail, and a compar• ison with an experimental stu. dy i s reportea.

IN several earlier• papers [J, 2, 3, 4), 1 t he normal mode vibrations of certain nonlinear systems, having many degrees of freedom, have been studied. These systems consist of a chain of masses, each mass having a single degree of freedom of translaLion in the direction of the chain. Each is connected to others by nonlinear springs, and every spring force is an odd function of t he length change of that spring.

The term "normnl mode vibration" describes a vibration in wu•son of nil elements of the system, and "vibrnLion in unison" means that all masses execute equipcriodic motions, all pass through equilibrium at the same instant, all attain maximum displacement at t he same instant, and the position of any one mass at an .Y given instant of time defin es uniquely that of every other mass at the same instant.

\V hen the system is linear, the motion of every mass duriug a uormal mode vibration is a simple harmonic function of time havi ng t he same frequency aucl t he same zeros as every other, or

x,(t)/l' ;(l) = A, cos wi/A; cos wt == C;; where C;; is a constant.

Yl'hen the system is nonlinea r, the normal mode vibrations are no longer simple harmonic. Nevertheless, it is quite possible that, eveu then, the rntios of the displacement.s in normal mode vibrations are identically equal to constants. In fact, it wns shown t hat symmetric two-degree-of-freedom systems [START_REF] Rosenberg | On t he Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems[END_REF] and homogeneous n-degree-of-freedom systems [START_REF] Rosenberg | On Normal Vibrations of a General Class of Nonlinear Dual-Mode Systems[END_REF][START_REF] Roscnberg | The Normal Modes of Nonl incar n-Degreeof-Frccdom Systems[END_REF] are such t hat t heir normal mode vibrations satisfy

x,(t)/:t';(l) == const [START_REF] Rosenberg | On t he Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems[END_REF] no matter how strongly nonlinear the systems or how large the displacements. T he physical interpretation of [START_REF] Rosenberg | On t he Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems[END_REF] is that the motions, i.e., the wave shapes in the time-displacement plane, are similm• when the system vibrntes in normal modes.

While it has also been shown [START_REF] Rosenberg | Normal Modes of Nonlinear Dual-Mode Systems[END_REF] that normal mode vibrations of arbitrary spring-mass systems do not, in general, satisfy [START_REF] Rosenberg | On t he Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems[END_REF], all detailed studies have so far been restricted to cases of similar normal mode vibrations.

When t he displucements of any two masses m, aud m; during normal mode vibrations are such that x;(l)/x;(l) ~ const [START_REF] Rosenberg | Normal Modes of Nonlinear Dual-Mode Systems[END_REF] we shall sny that the normal mode vibration is nonsimilar. In this paper, normal mode vibrations of two-degree-of-freedom systems nre examined. However, the results can be extended in nn obvious wny to systems having many degrees of freedom. For t his reason, we shall describe a system of many degrees of freedom and specialize it Inter to the case of two degrees of freedom.

The System

As a physical model of admissible systems, consider a chain of n mass points P,, U = 1, . ., n), the mass of P, being m, . Since an infiuitely large mass eau only be accelerated by an infinitely large force, Lhe motion of a mass 111 0 = oo is u. = 0, or P 0 corresponds to a fixed point. T herefore we may, without increasing the number of degrees of freedom of the system, include a mass mo = oo in it; such an addition to the system prevents the degree of freedom of "rigid-body translation." Each mass m., (i = I, . . . , 11) has n single degree of freedom of translation 11; in the direction of Lhe chain. Each is connected to one, several, or all others by spri11gs that may be equal or uuequal, linear, nonlinenr, or nonlinearizable.! Thus, a system comprising n masses may have as many as tfn(n -1 )) springs. T his system is illustrated in Fig. 1. The corresponding system having two degrees of freedom is shown in Fig. 2.

We assume that there exists at least one configuration in which t he system can remain at rest indefini tely. This is the equilibrium configuration and, in the equilibrium configuration, every 11; == 0.

The spring forces are the only forces acting on the system. These are derivable from a potential functiou U where u = -lT [START_REF] Rosenberg | On Normal Vibrations of a General Class of Nonlinear Dual-Mode Systems[END_REF] and l' is the potenLial energy. The potential energy stored in any spring is a function of the absol-ute vcc/11e of the length change of that spring. In physical terms, this implies that a. spring resists (or aids) in the same degree being deflected by a. given amount, whether that deflection is a compression or extension.

If we put u, = u, 112 = v, the equations of motion of the twodegree-of-freedom system are into (4) one has () (X y)

0 mzv = -U('u, u) ou (4)
x = u:r; u m2'/t' m2'/t ' () ( X y) y = -U ---- o y m 1 ' /t' m/ /t (6)
and these may be regarded as t he equations of mot ion of a unit mass that moves in the :ry-plane under forces uU /ox, oU /uy.

That system of forces and unit mass will be called t he p seudosystem. To each motion u, v of the physical system there belongs uniquely a motion of the pseudo-system in the xy-plane, a nd the trajectory of the unit mass satisfies the E uler equat ions of f " "

o (U + h)'h (1 + y' 2 )'1•dx = 0 %1 (7)
T hat equat ion is [START_REF] Rosenberg | Normal Modes of Nonlinear Dual-Mode Systems[END_REF][START_REF] Knuderer | Nichtlincarc Mechanik, Springcr-Vcrlag[END_REF] 2(U + h)y" + (1 + y' 2 )(y'U: -U~) = 0 [START_REF] Unng | Harmonic Oscillations of Nonlinenr Two-Degree-of Freedom Systems[END_REF] where primes denote d ifl'erentiation with respect to x.

Inasmuch us the system is conservative, t he energy integral

T -U= h (9)
exists where his a nonnegalive constant equal to t he tot al energy and, for the physical and pseudo-systems, respectively,

U = U(u, v) } ( X y ) u = u --.-1 ' ---;-; 111! , 1n2 ' ( 10)
We admit only systems in which U is negative, definite.

Trajectories of Norm al Modes

One sees from [START_REF] Cournnt | cthodcn dcr J1lathcmatischen Physik[END_REF]

U + h = O ( Ll )
That r is a closed curve surrounding Lhe origin follows d irectly from the fact t hat U is negative definite, and h is nonnegative. The r -curve will be referred to as t he bounding curve. It is also the maximum equipotentialline because [START_REF] Su | On a Restricted Class of Cou]>lcd Hi ll's Equat-ions and Some Applications[END_REF] coincides with (9) when T = 0; i.e., when the velocities vanish, and at such an instant the total energy is potential. Finally, r is symmetric with respect to the origin [START_REF] Rosenberg | Normal Modes of Nonlinear Dual-Mode Systems[END_REF], and every traject ory which actually int ercepts r does so orthogonally [START_REF] Rosenberg | Normal Modes of Nonlinear Dual-Mode Systems[END_REF][START_REF] Roscnberg | The Normal Modes of Nonl incar n-Degreeof-Frccdom Systems[END_REF][START_REF] Knuderer | Nichtlincarc Mechanik, Springcr-Vcrlag[END_REF]. Solutions of (8) which correspond to vibrations in normal modes are called modal lines.

We shall use perturbation t heory t o determine the modal lines of systems which lie in the neighborhood of a " parent system" having known modal lines. In t lus respect, this paper has the same objective as Duffing's memoire [START_REF] Hang | Osdl/atory ,){otions, English translation[END_REF] had with respect to slightly nonlinear single-degree-of-freed om systems. However, our study is neither restricted to single-degree-of-freedom systems nor to nearly linear ones because the exact modal lines of many highly non linear multi-degree-of-freedom systems arc known [START_REF] Rosenberg | On t he Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems[END_REF][START_REF] Rosenberg | On Normal Vibrations of a General Class of Nonlinear Dual-Mode Systems[END_REF][START_REF] Roscnberg | The Normal Modes of Nonl incar n-Degreeof-Frccdom Systems[END_REF], aud t he theory to be presented here will hold for the modal lines of any systems in the neighborhood of these non linear parent systems.

The Perturbed System

Let U = U(x, y) be the given potential fun ction of a parent system fo r which a modal line

Y = y*(x) (12) is known. Further, let O(x, y) = U(x, y) + ew(x, y) ( 1 3)
be the potential function of a perturbed system. In ( 13), E is a given constant and w(x, y) is called the perturbatio1t potential; it is supposedly known. If lel is small, and if

ew(x, y) = O(e)
everywhere in the closed domain Jj bounded by the curve r,

where t he f'-curve is defined by

() + h = U(x, y) + EW(x, y) + h = 0 ( 14)
then t he perturbed system lies in the neighborhood of the "parent system" whose modal line is (12). We may, then, use first-order perturbation theory to find the modal line

y(x) = y*(x) + t1J(x) + . . . (15)
everywhere where the equation which (15) must satisfy is regular in 15. In view of ( 8) and ( 13), that equation is

• ( 1 + y' 2 )[y'(Uz + EW:)-(V~+ ew.) ] 11 = - 2(U + EW +h ) ( 16
)
I t is evident that ( 16) is singular on the f'-curve and, hence, does not possess a uniformly convergent power series in e about any point on f'. Nevertheless, it is always possible to use t he perturbation series (15) for constructing solutions t hat are valid everywhere in the open d omain D, and to continue these solutions analytically unt il they reach the f'-curve. The complications arising from the fact that ( 16) is singular on r will be di~cussed more fully later on, but first we should like to show that y• is bounded OU r even t hough the denominator of ( 16) vanishes OD it.

Let us suppose t hat a solution (15) exists which intercepts t he f'-c urve at a point (X, Y) where Y = y(X ). Then, ou f',

x(X, Y ) = y(X , Y) = 0 because t he f'-curvc is defined by ( 13), and Lhnt equation is the energy in tegra.! when the kinetic energy vanishes. Then, by I 'Hospi tal's rule,

y(X, Y) y(X, Y) V. (X, Y) y'( X) = ---= ---= -_L'--'----' x(X, Y) x( X, Y ) Ux(X , Y)
where use has been made of [START_REF] Hang | Osdl/atory ,){otions, English translation[END_REF]. But ( 16) is indeterminate when the foregoing value of y' is substituted in it; hence, it may also be evaluated by !'Hospital's rule. l' vfaki ng use of

d Tr 0 - dx v(x, y) = • + y'Uv
one finds, aft er some computations,

y "(X, Y) = _!. [ O•v + y'(Ovv -0 .. -y'O • • > J 3 0, ,., -X
By definition of V,(x, y), that quantity cannot vanish at (X, Y )

[because (X, Y) lies on r, and r does not pass through the origin).

Therefore, t he denominator of t he foregoing equation is not zero. Similarly, the above expression for y'(X) shows that that derivative is bounded when O,(X, Y) ;e 0. It follows that y"(X, Y) is bounded, and its value is given in the foregoing equation.

We shall now use first.-order perturbation theory to construct a solution of the form ( 15) which satisfies the equation ( 16) everywhere inside but. not on r. Using the notation

Q(x, y*(x) ) = Q*(x )
for U and w and for their partial derivatives, standard perturbation technique yields the equation which 1J(x ) must satisfy. It t urns out to be

2(U* + h)?J" + [2y*'(y*'U,*-U.*) +( I+ y*'•)U,*] ?J' + [2y*"Uv * + (1 + y*' 2 )(y*'U•v * -Uvv *)) 1/ + 2y*"w * + ( 1 + y*' 2 )( y*'w, * -w. *) = 0 (17)
But, we seek modal lines lying in t he neighborhood of ~traight ones. Therefore,

y*(x ) = ex ( 18)
For straight modal lines, y*" = 0, and one sees from (7) that, then, everywhere in D,

v*' == u.;u. = c ( 19)
Substituting ( I S) and ( 19) in ( 17), t hat equation simplifies considerably and becomes

2(U * + h)TJ " + (1 + c 2 )Ur*1J' + (1 + c 2 )(cU./-Uv/)1J = ( 1 + c 2 )(wv * -cw, *) (20)
The integration of t his equation is fncilitated by the recognition t hat nearly all coefficients are derivntives of U*(x) = U( x, y*(x)). For instance,

.!!_ U* = U* = ~ U(x, y*) + ~ U(:~, y*)y*' dx ox oy*
and in view of (19),

d -U* = (1 + c~) U, * dx One finds, similarly, that c/2 1 + c' -U* = (I + c 2 )U * + --U * dx2 vv c ••
It. follows t hat, i(one defines,

U*(x) + h = G(x)
the perturbation equation ( 20) becomes

(21) ( 22
) (23) = (1 + c 2 )(wv * -cw, *) (24)
It is this equation on which all further development is based.

In certain cases, it may be advantageous to introduce a transformation of the independent variable given by (25) This reduces (24) to the form

2 :~-[ G" -( 1 ~ c•)• U.v*] 1J = (1 + c 2 )(wy* -cw,*) (26 )
The t ransformation (25) has an interesting significance. Suppose we wish to determine the time t, required for the unit mass of the unperturbed pseudo-system to traverse a given arc of any trajectory in the xy-plane. If we denote the velocity of the unit mass by w, we find from the energy integral !w 2 = U + h that the transfer time along any trajectory is

1 f (dx 2 + dy 2 )'1• y2 [U( x, y) + h]'lt
If t hat t rajectory is a straight modal line, so that (18) holds, the transfer time becomes

( 1 + c 2 ) '/•f dx (1 + c2) '/tf dx t = -2- ( U* + h)'lt = -2 - G(x)'l •
But this is, except for a constant, t he transformation (25). Let us denote by T = (2/(1 + c 2 ) ] 'l•t the pseudo-time which t he unit m!U !S of the unperturbed pseudo-system requires to move from the origin along a straight modal line to a point whose absci~sa is x . Then the relation between ~ and x is the same as that between T and x. The times T and t coincide when c = ± 1; i.e., for modal lines whi ch are inclined by 45 deg to the x-axis.

Equation ( 26) is simpler in appearance than (24). However, the coefficients in (26) are still functions of x, not ~. It is necessary to replace x by ~ in them, before the equation can be integrated; this step is accomplished through the inversion of (25);

i.e., by x = ,p-•w.

Equations ( 24) or (26) are second-order differential equations having solutions of the form

1J = 1J(x, a, {J)
The constants of integration a and {J must be determined such that

y(x) = y*(x) + E1J(x, a, {J)
is a modal line. Since U(x, y) and w(x, y) are given functions, and h, c, and E are known constants, the integration of (24) could always be done on a computer.

Integrable Cases

There exist two cases of considerable physical interest where the perturbation equation can be integrated in terms of quadratures. One of these is the case of "weak coupling" while the second does not require t hat coupling be weak. We shall discuss these in order.

When the force in the coupling spring is of O(E) for any finite deflections of that spring, we shall say that the coupling is weak. Such weak coupling has been the subject of many investigations in the theory of nonlinear differential equations [START_REF] Hang | Osdl/atory ,){otions, English translation[END_REF].

The potential energy of the system arises from t he potentials U01, U02, U12 stored, respectively, in t he anchor s prings S 0 ,, So 2 , and the coupling spring s,. (Fig. 2). Clearly, Uoo is a fun ction of x only, a nd Uo~ is a function of y only . Hence, t heir mixed derivatives U, v vanish; in other words, the mixed derivative of the entire system is represented by the mixed derivative o 2 /oxoy-( u,.) of the coupling spring.

When coupling is weak, Uzv can only appear in a second-order theory and, hence, will not be present in (24 ). That equation, then, reduces to

2GTJ" + G'TJ' -G"TJ
A first integral of ( 27) is

(1 + c 2 )(wu * -CWz *) 1 '1) 1 = - [G'TJ + a -Sl(x)] 2G
where we have used the notation (27) (28) and , since G'(O) = r2(0) = 0, the initial slope is, from (28),

TJ'(O) = a/(2h) (31 )
These constants must be so determined that y y*(x) + ETJ(:t, a,/1) is a modal line; i.e., y(x) must pass through the origin, and it must intercept the f-cmve. From the fi rst condition one has 11 = 0.

As stated earlier, the perturbation t heory used here does not furnish a solution which is valid on r ; however, t he solutions may be continued analy tically tor.

Let the points whose locus is the r -curve be denoted by (X, Y) .

Then, the equation of the f -curve is

U(X,Y) + ew(X,Y) + h = 0 (32)
The condition that t his curve be intercepted by the analytical cont inuation of y(x) is satisfied by substituting into (32) the relation

Y = y*(X) + HJ(X,a) rl(x) = (1 + c 2
) fo"' (cwz *wu *)dx and the general solut ion of ( 27) is ( 2 9) where rx a-r2( 11) A second integrable case is that of the in-phase mode of almost symmetric systems, provided the coupling spring is nonlinearizeable or a lmost nonlinearizable. 3 H owever, it is not required t hat coupling be weak. A system of potent ial function . ' Thus, if A , = 0 or of O(e ), the term u •• * will be absem from (26). Then, the equation in 1J is (27) with c = 1, and its general integral is (30 ), again with c = 1.

TJ(X,a) = }[G(X)]'h Jo G(n)'h dn (33) ( "' a -rl(u) , 1 TJ(x) = ![G(x)]'h Jo G(u)'h du + {1[G(x)] ' (

Determination of Constants

The arbitrary constants a and 11 in (30) define the values of TJ U(X,y*(X) + ETJ(X,a)) + Ew(X,y*(X)

+ ETJ(X,a))

+ h = 0 (34)
It is readily seen t hat this equation will yield a relation of the form a= a(X) (

i.e., an equation connecting the init ial slope of '1 with the v!Liue X at which the f-curve is intercepted, and X is the x-amplitude of the motion. In other words, the analy tical continuation of every integral curve that lies sufficiently close to the straight modal line of the unperturbed system does intersect the f-curve at some point whose abscissa is X. Clearly, it is not possible to determine t he constant a uniquely from that result . Nevertheless, for future developments, it is necessary to determine the explicit relation between X and a. Let t he value of X be given by

X= X o +eA ( 36
)
where Xo is the (known ) amplitude of the unperturbed system in the mode whose modal line is y* = ex. Let it be assumed further that both the unperturbed and perturbed systems move at the same energy level h. T hen, the point a t which the straight modal line of the unperturbed system intercepts the l'-curve is found from

U(Xo,y*(Xo)) + h = 0 (37)
To find X , one must substitute (36) in (34) and expand the resulting equation in powers of €, retaining only terms up to the first power. In d oing this, it is necessary to evaluate

{ rx• du rx• Sl(u) } TJ(X o) = tlG(Xo)]'h a Jo G(u)'h -J o G(u)'h du Now, G(x)
is an even polynomial of at least second degree in x (when t he unperturbed problem is linear, otherwise it is of higher degree), and it vanishes at X o. The function r2(x) is also at least of second d egree in x ; it takes on its maximum v alue at Xo and it vanishes a t the origin. Therefore, bot h integrals in the foregoing equation always diverge; however, their products with [G(Xo)J'I• are bou nded, and one can readily fi nd, either from a limi t analysis or from l'Hospit.al's rule,

_a_ rl(Xo) TJ(X o) = -G'(Xo) + G'(Xo) (38)
When (38) is substituted in Lhe expansion of (3•1) a nd (37) is introduced in that relation, one find s after sonHl compu tat.ions

1 + c~ a = --[AG'(Xo) + w*(Xo)] + !l(Xo) c ( 30
)
Sin<:e a is a measure of the change in initial slope bet.wecn unperturbed and perturbed problems, and 1i is a measure of the change in x-a mplit.ude beLween the unperturbed problem and the analytical cont.inualion of t he integral curves of t.he perturbed problem, equations ( 30) and (36) jointly consLit.utc the explicit form of (35).

\Ve shall now show that there exists at m ost a single integral curve of t he pert,urbed problem which actually inter cepts the fi-curve ; in other words, there exist. s at most one value of init.ial slope (i.e., one v alue of a) for which an in tegral curve of the pert.tu•bed problem is a. modal line.

Let us write, in accordance with (30), .

.

-o

Hence, the rate of change of the end slope 1J'(X 0 ) 1Yith respect to a is given by o '(K) 2~ = J'(Xo) oa and a limit analysis (whose details are not r eproduced here) yields

J'(Xo) = lim _ r -•H-G' (Xo)J'I• _ _ u_ { 1 < i Xo-6 d .,-o v Eo - o G(tt)'h + 1 V~ ( -G'(Xo))>}
where lol > 0 is a small parameter and G'(Xo) < 0. In consequence, we find 101'/~~Yo)l = 00

Assume now that an integral exists which is a modal line; i.e., which intercepts the fi-curve. Then, the last equation states that no other integral curve passing through the origin and having an initial slope neighboring on that of the modal line can also reach the bounding curve b ecause the end slope 1J'(X 0 , a) is not a continuous funct.ion of a.

This result is not in contradiction with (30) because that equation holds only for the analytical continuations of t he integral curves. Nevertheless, (39) must be satisfied by the actual modal line because that line is t he only integral curve which coincides with its analyt.ical continuation.

The condition that 1J'(Xo) be bounded, together with the observation that J'(X 0 ) is unbounded might lead one to conjecture that a = 0; however, that conjecture contains the implicit a~s umpti on tha t l '( Xo ) is bounded, and such an assumption cannot be verified because w, and hence !1, are largely arbitrary.

To determine t he initial slope of the modal line of the perturbed system we observe that the left-hand side of (27) becomes self-adjoined when that equation is multiplied by G(x) -'h; it then becomes and the correspond ing self-adjoined homogeneous equation is One secs from (30), that tlie general solu tion of ( 41) is

1 { "' dn {3 , 1 1J = 1'/o(:r, a, {3) = } G(x ) h a J 0 G(u.)'h + G(x) ' (42)
Inasmuch as 1J must pass through t he origin, t he only case of interest here is t hat of {3 = 0, as sh own earlier.

Now, it is well known t hat for existen ce a nd uniqueness of solutions of cer tain inhomogeneous equations, these mus t satisfy a well-known orthogonality relation [START_REF] Cournnt | cthodcn dcr J1lathcmatischen Physik[END_REF]; however, because of t he unusual "boundary condition" that y(x) intercept a cert.ain curve (rather than t hat it pass through a given point on a given curve), one must investigate the question of applicability of that orthogonality condition. This is clone by multiplying t he inhomogeneous equation (40)

by l)o( x , a:) and integrat.ing it over 0

~ x ~ Xo, or rx• fx, rx• 2 Jo 1/o(G'/t71 ')'rlx - 0 1JoG-'I•G"1)dx = Jo l)o<pdx (43)
If one integrates t he first integral twice by parts, ( 43

) becomes rx• -a-'hG"1Jo) )dx = Jo 1Jo<Pdx (4-!)
The first term in equ ation ( 4< 1) vanishes in the lower limit because 1] 0 (0) = 1](0) = 0, a nd in the upper limit because G'h(Xo) = 0 a nd we require that the derivatives of 1J and l)o be bounded at the uppe r limit. The integral on t he left-hand side of ( 44) also vanishes because the quantity in the square brackets of t.hc integral is (•ll ), a nd that quantity is identically zero since 1' /o satisfies ( < !1 ). Hence, the ort.h ogonality condition [START_REF] Cournnt | cthodcn dcr J1lathcmatischen Physik[END_REF] is applicable; it is or, explicitly We, then, have the result that, to the first order in E, the modal line of an integrable, perturbed syste:m is tangent at the origin to that of the unperturbed system on which it ne:ighbors.

f x , •}a 0 rx• J 0 1]0rpdx = 0 Since G(u) ?-0 in 0 ~ 1t ~ Xo, the integral { "' du J 0 G(1t)'h (
In view of this result we have from ( 30

The solutions of unperturbed and perturbed systems are illustrated in Fig. 3. From this diagram, one sees that the perturbation has first-order effects on the x and y-amplitudes of the normal mode vibrations.

A meaaure of the change in y-amplit.ude is found from y(X) = y = y*(X) + e?)(X)

= y*(Xo +eA )+ e?)(Xo +eA) used standard perturbation techniques on the equations of motion, not on the geometrical equation ( 8) used here, and concluded that the vibration amplitudes of the perturbed system are in the same ratio to each other aa those of the unperturbed problem. That Huang's results are incorrect was shown recently by Szemplinska-Stupnicka [9] who showed that, in fact, certain first-order efl"ects are neglected in Huang's analysis, and that these can only be recovered by a second-order analysis (that one in turn losing certain second-order effects which must be recovered through a third-order analysis, and so forth). However, when using the geometrical methods applied here, all first-order effects are accounted for by a first-order analysis.

Fig. 1

 1 Fig.1

Fig. 2 Substituting

 2 Fig.2

  30 ) This results in an equation of the form where a and 11 are arbit rary constants of integration.

U( 1 : 1 11

 11 ,y) = Uo,(x ,y) + Uo.(x,y) + U, .(x,y ) is said to be symmetric if Ulo * == Uo•*• This requires, in physical terms, that the masses a re equal, or m, = m2 and t hat t he anchor springs are equal, or S01 = So•• The system is said t o be almost symmetric if IUoo*-Uoz*l = O(E). Itwillnowbeshown why the in-phase mode of an a lmost symmetric system is an integrable case when the coupling spring is a lmost, or altogether, nonlinearizable. As stated, t he mixed d erivative o•O / oxoy is equal to o•V,./ oxoy. Now, by definit ion of admissible systems, t he potenLial 0, 2 is a fun ction of t he absolute value lx -vi of the lengt h change. Hence, its Taylor expansion must be of the form "\' A m u,. = -L..t ---(x -y)"'+' m = l,3, . .. m+ 1 and I: mA,..(xy)"'-But for the in-phase mode of symmetric systems, y*(x)

and 1 ]

 1 1 a t t he origin. In fact, since G(O) = h in virtue of (23), one has from (30) 7](0) = /1h'l• s A spring is almost nonlincarizable if the linear component of the spring force is of 0 (<) .

rz

  21]= aJ(:c) -l (x) Sl(1l)l (x ) = [G(x)) /2 Jo G(u)'h duand where we have already put {3 = 0. Then, the end slopey'(X) = c + €1J'(Xo + EA) = c + €1J'(X0 )within O(e), and 21]'(Xo) = lim [aJ'(Xo -Eo) -l '(Xo -Eo))

  sign in t his interval, and ( 4G) may be written1 in the form rx'f "' dn~( 1 + c2)(wv * + cw, *)a J o o G(u)'l• dx = 0 where wv • -cw. * = wv *(x) -cw, *(x)(47) a nd :\: is some value of x in [0, Xo]. \Ve cannot impose the con• straint w. * -cw •• = 0 'By the mean-value theol'y of the integral calcnlus. on w, because w is largely an arbitrary function. For instance when the perturbation consista in perturbing one of the anchor springs only, either Wv * = 0, w. * .= 0, or vice versa; and the nonvanishing quantity never changes sign in [0, Xo]. Hence, the orthogonality condition becomes, finally, rx• rz du a Jo Jo G(u)'l• dx = 0 (48) But, as s(ated earlier, G(x) is nonnegative; hence, (48) can only be satisfied by a = 0.

  x) = -t !G(x)] / • J o G(u)'!s du (49)and, from (39), the change in x-amplitude between unperturbed and perturbed systems is defined throughA = -G'(~o) [ w*(Xo) + 1 ~ c 2 fl(Xo)]

A spring is said to be "nonlineariznble" if the Taylor expansion of the spring force lacks n linear term.

Acknowledgment

Financial aid from the National Science Foundation which has . supported tbis work is gratefully acknowledged.

The M otion and Its Stability

Once the modal line y(x ) of the perturbed system has been found by the methods described here, the motion of the perturbed system in the normal modes can be found by a simple quadrature. The substitution of the modal line where l' 0 = y' (Xo) is they-amplitude of the unperturbed system. If we write, similar to (3G ),

we have

where A is given in (50), nod ?](Xo) in (38) (with a = 0). l:Vfaking use of t hese, one finds

The result that the perturbation has first-order efJects on the amplitudes is at odds with those of Huang [START_REF] Unng | Harmonic Oscillations of Nonlinenr Two-Degree-of Freedom Systems[END_REF] who examined the nearly linear dynamic vibration absorber with a slightly non linear coupling spring. In our terminology, Huang's problem lies in the neighborhood of a homogeneous system of degree 1. Huang y Fig. 3 x(O) = Xo + eA, :t(O) = 0 where A is given in (50). The solution may be found by one quadrature and is of the form

It is expressly noted that (55), and hence (56), is free of '1• The y-motion is

and (57) does contain?). Since (57) depends on ?J and (56) does not, the normal mode vibration is, in general, nonsimilar.

Next we show that the ~/ability of the normal mode vibration of the 71erturbed system is the same as that of the unperturbed system.

The solutions whose stability is to be examined are

and the equations, identically satisfied by (58), are

Let the solutions of the perturbed system (59), but with slightly ditTerent initial values than (58), be

Then the solutions (58) are said to be stable if p(t) and q(t) are bounded for all t.

The equations of the first variation of (59) with respect to (58) are readily computed ; they are

But these are precisely the variational equations of the unperturbed system. Hence, the stability of t he normal mode solut ions of the two systems is the same. The stability problem is now solved because the stability of the unperturbed problem is deducible by known methods [START_REF] Rosenberg | On t he Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems[END_REF][START_REF] Rosenberg | On Normal Vibrations of a General Class of Nonlinear Dual-Mode Systems[END_REF][START_REF] Roscnberg | On t.lie Geometrizntiou of Normal Vibrations of No1tlinenr Systems Having Many Degrees of Freedom[END_REF][START_REF] Su | On a Restricted Class of Cou]>lcd Hi ll's Equat-ions and Some Applications[END_REF].

Example

An an example of the foregoing theory, we shall consider an almost symmetric system consisting of two masses and three springs, as shown in Fig. 2. The symmetry is disturbed by a slight change in the spring So2 which connects the mass m2 to t he fixed point. This change is to be such that the force of the per-. turbed spring remains an odd function of the spring deflection. Consequently,

The change in the bounding curve resulting from this perturbation is readily discussed. Because of t he third of (62), the perturbed bounding curve is tangent to that of the unperturbed problem where y = 0, i.e. , where both cross the x-axis. If the perturbation strengthens the spring S 0 2, the r-curve lies inside the r-curve for every y >= 0; if the perturbation weakens s 42, r lies outside r for every y >= 0.

We shall examine the in-phase mode of the perturbed system. Then, since the parent system is symmetric, y* = x or c = 1 [START_REF] Rosenberg | On t he Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems[END_REF], and

Substituting these quantities in (50 ), one has (because of c = 1) A = 0. Therefore, under the proposed perturbation, the X-amplit ude remains unchanged .

The change in y-amplitude is found by substituting the foregoing quantities in (52). The result is

~ow, G'(X 0 ) < 0 for all 0 < x ~ Xo. H ence, B has t he same sign as j (Xo ). If the perturbation is such as to strengthen the spring So2,J(x) < 0 for a ll 0 < x ~ Xo; therefore, they-amplitude will be diminished. If the perturbation weakens S o2, f(x) > 0, and the y-ampli tude will be increased. Both cases are illustrated in Fig. 4.

If it is the spring So 1 which is pertmbed rather than So2, one has w, = w. * = f'(x) and w. = 0. Then, in view of (29), Q(Xo) = 2j(Xo), and one finds as expected 2j(Xo) A = -G'( Xo)'

B = O

In that case, t he rand r-curves are tangent to each other OD the y-axis, and r lies inside I' for f(x) ~ 0, and outside for f(x) ;::: 0.

T he first of t hese results has also been checked experimentally on the analog computer. The unperturbed system chosen in the experiment was a symmetric, homogeneous system of degree 3 and with potential function The perturbation of the spring S 0 2 consisted of the addition of a small linear term which perturbed the homogeneity all well as the symmetry. In the case of a strengthened spring, the perturbation potential was EW(x, y) = -ty 2 and, in the case of a weakened S 0 2, the negative of the foregoing function was used. The resulting curves in the xy-plane are shown in Fig. 5 for both the unperturbed and the perturbed systems. Thus, these curves are the experimental counterpartB of Fig. 4 . All lines in Fig. 5, including the axis system, were recorded by an xy-plotter and only the lettering was added by hand. A comparison between Figs. 4 and5 shows that the agreement is excellent even though the perturbation was much larger than contemplated in a first-order theory. Quantitatively, the dif!'erence between the predicted and observed amplitude changes is hardly mensurable.


Joffrey Decourselle

Fabien Duchateau

Trond Aalberg
email: trondaal@idi.ntnu.no

Naimdjon Takhirov

Nicolas Lumineau

BIB-R: a Benchmark for the Interpretation of Bibliographic Records

Keywords: benchmark, migration, record interpretation, FRBRization, LRM, FRBR, MARC, dataset, evaluation metric

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Cultural institutions are responsible for cataloging and offering access to a large number of cultural items. The most popular format for libraries, MAchine Readable Cataloguing (MARC), available in different implementations such as MARC21 or UNIMARC, has shown some limitations in terms of interoperability, reuse, or information disambiguation [START_REF] Denton | FRBR and the History of Cataloging. Understanding FRBR: What It Is and How It Will Affect Our Retrieval Tools[END_REF]. The Functional Requirements for Bibliographic Records (FRBR) and its updated version Library Reference Model (LRM) [START_REF] Riva | Introducing the FRBR Library Reference Model[END_REF] have been designed to provide a sound and more explicit semantics which will enable new enhancements such as improved navigation and enrichment features [START_REF] Alemu | Linked Data for libraries: Benefits of a conceptual shift from library-specific record structures to RDF-based data models[END_REF][START_REF] Buchanan | FRBR: Enriching and Integrating Digital Libraries[END_REF][START_REF] Dickey | FRBRization of a Library Catalog: Better Collocation of Records, Leading to Enhanced Search, Retrieval, and Display[END_REF]. However, more than twenty years after the original specifications of FRBR, the model is still not widely used in libraries [START_REF] Coyle | FRBR, Twenty Years On[END_REF]. A major obstacle to the adoption of FRBR is the interpretation of records (e.g., FRBRization), which consists of migrating cultural heritage data from legacy formats (e.g., MARC) to models based on the FRBR semantics 4 .

In the last decades, the proliferation of FRBRization tools has demonstrated the need for specific enhancements (e.g., clustered deduplication, exploitation of added entries) to improve the process [START_REF] Decourselle | A Survey of FRBRization Techniques[END_REF]. Despite this progress, it is still very complicated to evaluate and compare FRBRization tools for several reasons. First, the experiments described in papers are rarely reproducible, mainly because the tools and the datasets are not available. A few catalog excerpts are provided, but they do not reflect the reality and the challenges of library catalogs because they are mainly used for illustrating specific cases [START_REF] Aalberg | The Value of MARC Data, or, Challenges of FRBRisation[END_REF]. Last but not least, the current metrics are not sufficient to evaluate all possible cases that might occur during FRBRization. In addition, these metrics imply that the user has to wait until the end of the FRBRization process before obtaining any insight about the resulting quality. To summarize, we advocate that the lack of a common FRBRization benchmark is an obstacle to the adoption of FRBR.

In this paper, we propose BIB-R5 , the first benchmark for evaluating FRBRization. It is composed of two datasets and a set of evaluation metrics. The goal of the first dataset T42 is to identify the weak and strong points of a tool by testing all possible issues that libraries may face during FRBRization. The second dataset BIB-RCAT is extracted from catalogs of three different cultural institutions and can be used for comparing or experimenting with the data quality that is typically found in real world catalogs. The assessment of the process relies on a set of metrics to predict the quality of the output and to evaluate the quality of a FRBRized catalog. An experimental study with three recent FRBRization solutions shows the benefits of our benchmark.

In the rest of the paper, we present related work in Section 2 and an overview of our benchmark BIB-R in Section 3. Evaluation metrics for pre-FRBRization and post-FRBRization are presented respectively in Sections 4 and 5 and the datasets are described in Section 6. The experimental study is detailed in Section 7. Section 8 concludes and outlines future work.

Related work

Issues related to FRBRization and identification of challenging bibliographic patterns are described in recent surveys [START_REF] Aalberg | The Value of MARC Data, or, Challenges of FRBRisation[END_REF][START_REF] Riva | Mapping MARC 21 Linking Entry Fields to FRBR and Tilletts Taxonomy of Bibliographic Relationships[END_REF][START_REF] Zhang | Implementing FRBR in libraries: key issues and future directions[END_REF]. In this section, we present rule-based FRBRization tools, and we focus on the evaluation of this process.

Tools. Due to page limitation, we refer to a recent survey for an exhaustive list of FRBRization tools [START_REF] Decourselle | A Survey of FRBRization Techniques[END_REF]. The last decade has seen the emergence of rule-based FRBRization tools, since grouping-records tools are not able to process complex structures [START_REF] Aalberg | The Value of MARC Data, or, Challenges of FRBRisation[END_REF][START_REF] Hickey | FRBRizing OCLC's WorldCat[END_REF]. We focus on three rule-based tools that are publicly available for experiments. The tool VFRBR, developed in the context of the Variations project, aims at FRBRizing catalogs with a focus on the music domain [START_REF] Notess | Scherzo: A FRBR-Based Music Discovery System[END_REF][START_REF] Riley | Enhancing Interoperability of FRBR-Based Metadata[END_REF]. Since music items are often described using added entries, VFRBR's strategy is to interpret these added entries as separate entities. The online catalog Sherzo is the proof of concept that lets users explore musical works, composers and related entities issued from 185,000 MARC records. Extensible Catalog (XC) is an open-source project for a complete Integrated Library System, which includes the FRBRization tool Metadata Service Toolkit [START_REF] Bowen | Moving Library Metadata Toward Linked Data: Opportunities Provided by the eXtensible Catalog[END_REF]. XC exploits added entries and is therefore able to detect complementary works for instance. The third tool FRBR-ML [START_REF] Takhirov | FRBR-ML: A FRBR-based framework for semantic interoperability[END_REF] is based on Aalberg's approach [START_REF] Aalberg | A Process and Tool for the Conversion of MARC Records to a Normalized FRBR Implementation[END_REF]. The authors discuss the possible structures of the FRBR output catalog, and the tool provides enhancements to disambiguate some complex cases by exploiting other catalogs or Linked Open Data knowledge bases.

Evaluation. Only the output of the FRBRization process is evaluated, under various forms: the most frequent option requires a ground truth or gold standard, i.e., an expert FRBRized catalog [START_REF] Manguinhas | FRBRization of MARC Records in Multiple Catalogs[END_REF]. The comparison between the expert FRBRized catalog and the FRBRized catalog produced by a tool indicates whether the tool is able to perform an acceptable transformation. One of the main issues is the manual construction of the expert FRBRized catalog. The FRBR-ML approach avoids the tiresome construction of a gold standard by converting the FRBRized catalog back to a MARC catalog [START_REF] Takhirov | FRBR-ML: A FRBR-based framework for semantic interoperability[END_REF]. The evaluation is performed between the initial MARC catalog and the converted one. With this type of evaluation, the drawback is that the last transformation (into MARC) may have a negative impact on the quality of the catalog. Besides, the rules that enable this last transformation have to be written too. To evaluate a process, metrics are required. In TelPlus, an aggregation metric is proposed to measure the percentage of aggregated content (e.g., Works, Persons, Places). FRBR-ML is evaluated with three metrics: redundancy, completeness and extension respectively measures duplicate data, loss of data and amount of enrichment.

Discussion. The digital library community has successfully identified the bibliographic patterns, as well as a few FRBRization issues. But there is no collection publicly available for testing each of these challenges. Thus, most FR-BRization tools have been tested against private datasets, whose characteristics are not clearly defined. Available metrics either assess the deduplication (aggregation) or compare two MARC collections (redundancy, completeness and extension). Thus, there is no metric which compares the quality of a generated FRBR collection, especially in terms of bibliographic patterns. And the whole FRBRization process is currently not evaluated (e.g., the tuning task). Contrary to other research domains, there is no benchmark for one of the most crucial task in the digital library community. Yet, we advocate that understanding the weak and strong points of the FRBRization process tends to promote novelty and enhancements in the future implementations. In addition, common datasets and evaluation metrics enable a fair comparison between the tools. In the next section, we describe our benchmark for FRBRization.

Overview of the benchmark

The FRBrization process has been described and enhanced in the last decade [START_REF] Aalberg | A Process and Tool for the Conversion of MARC Records to a Normalized FRBR Implementation[END_REF][START_REF] Decourselle | A Survey of FRBRization Techniques[END_REF]. Figure 1 depicts an overview of the FRBRization process. It is composed of three main steps (pre-FRBRization, FRBRization and post-FRBRization).

During pre-FRBRization, librarians are in charge of preparing the input catalog (traditionally in MARC) and tuning the tool. The optional preparation allows to clean fields, to delete empty records, etc. For the tuning task, librarians can configure some parameters (e.g., setting a decision threshold for deduplication), but the main challenge is to add, modify or delete rules. Next, the FRBRization starts using a clean catalog and a customized set of rules. The transformation of each record produces a set of entities and relationships according to the rules applied. A deduplication task is necessary to detect and merge entities that represent the same concept. Finally, the last step is post-FRBRization, during which optional tasks are performed on the raw FRBR collection [START_REF] Decourselle | A Survey of FRBRization Techniques[END_REF]. We only mention validation and enrichment. The former enables expert to verify and correct the generated FRBR catalog while the latter refers to the task of adding information from external sources. Most FRBRization approaches only evaluate the last step (post-FRBRization, using the FRBRized catalog), but the initial step has a strong impact on the final result in terms of quality and performance. Our benchmark BIB-R provides metrics and datasets to evaluate pre-FRBRization and post-FRBRization. It focuses on the foundations of FRBR, but it does not take into account the specificities and the complexity of the different implementations (e.g., FRBR-OO). The pre-FRBRization aims at preparing the catalog to be FRBRized and the set of rules. The records of a catalog include bibliographic patterns [START_REF] Aalberg | The Value of MARC Data, or, Challenges of FRBRisation[END_REF], which complicate FRBRization. The different ways of cataloging and the set of rules are subject to potential issues. Since FRBRization may last hours or days, it seems interesting to be able to detect and solve potential problems prior to running the process (e.g., updating the set of rules or cleaning fields and values). To the best of our knowledge, this detection of problems is a novel contribution for digital libraries. Our pre-FRBRization metrics analyze a set of rules according to a catalog for estimating the records that will not be FRBRized correctly. Each metric computes a percentage defined as the number of records concerned with the given pattern/issue divided by the total number of records.

Library practice allows for the description of different entities and relationships according to the nature of the item described. These structures can be generalized into common patterns, but because the structure mainly is implicit such patterns are difficult to detect and FRBRize correctly [START_REF] Aalberg | The Value of MARC Data, or, Challenges of FRBRisation[END_REF][START_REF] Riva | Mapping MARC 21 Linking Entry Fields to FRBR and Tilletts Taxonomy of Bibliographic Relationships[END_REF]. The most frequent core pattern includes a Work, an Expression, a Manifestation and (mostly) the Agent creator of the Work. The augmentation pattern is defined as an additional content to an existing Work, with the assumption that the new content does not alter the main Work (e.g., illustrations, forewords). Several scenarios occur to FRBRize this pattern, for instance the creation of a new Work or a note for the original Work. The derivation pattern means that one Work is the modification of another Work (e.g., translations, imitations), and it usually implies the creation of Expression(s) under the same Work or relationships between Works. The aggregation pattern is commonly described as a whole-parts relationship (e.g., ensemble, aggregative work). The FRBRization of aggregations mainly results in the creation of relationships between Works and optionally new Agents. The complementary works pattern aims at modelling a relationship between Works which have the same importance (e.g., sequels, accompanying works). The FRBRization of complementary works mainly results in the creation of relationships between Works. The metrics COR, AUG, AGG, DER and COW respectively compute the percentage of records with a core pattern, an augmentation, an aggregation, a derivation and a complementary work.

In addition to bibliographic patterns, records may include cataloging errors or quality problems. Authors of the TelPlus project have established six requirements for FRBRization [START_REF] Manguinhas | FRBRization of MARC Records in Multiple Catalogs[END_REF], that can be seen as errors in the initial records. We define the metrics MID, MPD, MUT, MOT, MRC and MAR which respectively compute the number of records that include the issues missing record identifier, missing publication date, missing uniform title, missing original title, missing relator code, and missing authoritative responsibility. We propose four new metrics related to cataloging issues. The metric MTF deals with missing type and form of material, which has an impact for correctly identifying Expressions (and sometimes Works). The metrics TLE and RLE relate to title linkage error and responsibility linkage error, which means that the unavailable related record (mainly in UNIMARC) has a negative impact in terms of completeness when FRBRizing. Finally, libraries make use of standards such as the International Standard Bibliographic Description (ISBD), widespread normalization of values (e.g., country codes) or codes specific to individual libraries (e.g., for a book category, value "r" corresponds to a roman). The metric CPN deals with these inconsistent cataloging practices and norms, which may contain useful information.

The set of rules has not been widely studied and mainly regarded as an artifact that needs to be tuned by librarians. Yet, this tuning has a crucial impact and its analysis can be exploited to improve FRBRization. In case of missing rule, a field cannot be processed by a rule, thus causing loss of information. Detecting these fields prior to FRBRization enables librarians to update the set of rules accordingly. A not used rule indicates that it is not useful for a given catalog.The conflicting rules issue occurs because the set of rules can be built using various techniques (e.g., written by librarians, merged from collected sets).

Actions associated to such rules can either be complementary or conflicting, which may degrade performance or quality during FRBRization. The metrics MR, UR and CR respectively compute the percentage of missing rules, not used rules and conflicting rules. The metric MR can be decomposed into more detailed metrics for a given pattern or issue, for instance MR-AUG to calculate the percentage of missing rules for detecting all augmentations. A formal notation of the pre-FRBRization metrics can be found in an online appendix [START_REF] Decourselle | Appendix: BIB-R: a Benchmark for the Interpretation of Bibliographic Records[END_REF].

Evaluating post-FRBRization

When the process of FRBRization is finished, librarians typically need to check the FRBRized catalog produced by the tool. This evaluation is the most studied in the literature, because the resulting quality is currently an obstacle to the adoption of FRBR and because most FRBRization tools have demonstrated their capabilities through an experimental validation based on the analysis of the produced FRBR catalog [START_REF] Manguinhas | FRBRization of MARC Records in Multiple Catalogs[END_REF][START_REF] Takhirov | FRBR-ML: A FRBR-based framework for semantic interoperability[END_REF]. In our context, we have chosen an evaluation based on expert FRBRized catalog. This is the most reliable evaluation, specifically because it directly assesses the quality of the FRBR catalog, including its complex relationships between entities. We have identified seven metrics to compare the FRBR catalog generated by a tool T and the expert FRBR catalog E. These metrics are useful to understand the weak and strong points of a FRBRization tool, to estimate the manual effort which is needed to complete the FRBRization, or to provide an insight about the rules that should be added to improve the quality.

The first four metrics deal with data (entities, relationships and properties from the FRBR model). The metric MD is related to the missing data issue, i.e., data which appears in the expert catalog E is missing in the tool's catalog T . This metric computes the ratio between the number of missing data and the total number of data in the expert collection. It can be redefined for each type of data, i.e., MD-E for entities, MD-R for relationships and MD-P for properties. The metric IAD deals with incorrectly added data, i.e., duplicate data (e.g., a property which appears twice in an entity, because of a bad deduplication for instance) and incorrect data (e.g., an entity that should not have been created or a property with an unexpected value). It is defined as the number of incorrect data in T (which is not in E) divided by the total number of data in T . Similarly to MD, the metric IAD can be redefined according to the data type. The metric DLE relates to errors in external link (e.g., to a referential or to the Linked Open Data). Either the link does not exist in E or it has a different value for the same external source. The metric calculates precision, i.e., the number of erroneous links in T divided by the total number of links in T . The metric SMD aims at computing semantic mismatch data, i.e., data which have a different semantics in both catalogs (e.g., a relationship translated by which appears as contributed to). The metric computes the amount of semantic mismatch data in T (compared to data in E) with regards to the total number of data in T .

The next metrics deal with patterns. The detection of the pattern in a MARC record is crucial because it provides the FRBR structure. Yet, only part of a pattern may be incorrect and the evaluation should reflect this . Note that it is not possible to verify information about bibliographic patterns without annotations in the expert collection. The metric MEND (main entity not detected) relates to the detection of the main entity of a pattern (e.g., an Expression in the case of a translation). It measures the percentage of main entities from E that have been correctly detected in T among all main entities from E. The metric MRND (main relationship not detected) checks whether the relationship associated to the main entity is correctly identified or not. For instance, an Expression is correctly identified but linked with a "is a revision" relationship rather than with a "is a translation" relationship. The metric MRND computes the percentage of main relationships from E that have been correctly detected in T among all main relationships from E. Finally, the metric ESE deals with errors in secondary element(s) of the pattern, which means that the main entity and its relationship have been correctly detected, but other elements (e.g., the translator) are missing or incorrect. The metric ESE computes the percentage of correct secondary elements in T among all secondary elements from E. A formal notation of the post-FRBRization metrics is given in an online appendix [START_REF] Decourselle | Appendix: BIB-R: a Benchmark for the Interpretation of Bibliographic Records[END_REF]. To use these metrics, it is necessary to have datasets with appropriate features.

Datasets

In BIB-R, two datasets allow the assessment of FRBRization tools. In our context, a dataset is a set of collections. Each collection, which contains records, is available in two input formats (MARC21 and UNIMARC) and it is associated with an expert FRBR collection. This expert collection has been manually created and verified by a librarian and three digital library researchers. All collections included in these datasets are based on the MARCXML and raw MARC formats. The records have been extracted from real-world catalogs, and modified when needed. The datasets are detailed in a report [START_REF] Decourselle | Open Datasets for Evaluating the Interpretation of Bibliographic Records[END_REF] and publicly available at http://bib-r.github.io/.

The first dataset T426 can be used for testing specific cases. In Section 4, we explained that a record has an inherent bibliographic pattern (e.g., core, augmentation) and it may include any number of issues (e.g., missing relator code, title linkage error). The objective of the dataset T42 is to check whether a FRBRization tool is able to handle each possible case. We define a unit test as the combination of a pattern and an optional issue. This dataset currently contains 42 meaningful tests which are crucial for testing specific aspects of FRBRization (a full list of combinations and statistics are available online).

The second dataset BIB-RCAT 7 simulates a real-world catalog in which various bibliographic patterns and issues may be found. It currently contains three collections (MARC21 and UNIMARC formats, and the expert FRBR). It is mainly composed of records from various catalogs (e.g., a public French library, a public Swiss hospital). The size of this catalog (560 records) is smaller than catalogs found in cultural institutions, since the expert FRBR collection requires a tiresome effort to be manually produced and verified.

Experiments

In this section, we demonstrate the benefits of our benchmark BIB-R for the evaluation of FRBRization. Three tools, which are publicly available 8 , have been used in these experiments: FRBR-ML, Extensible Catalog (XC) and Variations VFRBR. These tools are detailed in the Related Work (Section 2). The rest of this section describes three experiments using our benchmark: how to evaluate the strengths and weaknesses of FRBRization tools, how to compare tools in a real-world FRBRization scenario and how to facilitate the tuning of a tool. Due to page limit, only a few interesting results are presented, but all plots are publicly available in an online appendix [START_REF] Decourselle | Appendix: BIB-R: a Benchmark for the Interpretation of Bibliographic Records[END_REF].

Assessing strengths and weaknesses

This first experiment aims at demonstrating the benefit of the dataset T42 when it comes to evaluating the strengths and weaknesses of FRBRization tools. For the three tools, we have run each test from the dataset T42 and the evaluation is performed using post-FRBRization metrics. Note that we have not tuned the rules and rely on the set or rules provided with the tools, although they have been developed for different purposes. The first finding is about missing data. None of the tools completely FRBRize the data contained in the MARC records. Figure 2 illustrates this trend by showing the missings in terms of entities (MD-E), relationships (MD-R) and properties (MD-P) for various tests. With the core pattern (test 1.0), the tools may miss entities such as Concepts. Tools may also be implemented to merge some properties. For instance, XC merges the subtitle into the title, thus missing the subtitle property. The scores of VFRBR for missing data are strongly impacted by the fact that it does not create Work entities. The more complex the record becomes (tests 3.2 and 5.5), the more losses in the FRBRization. Secondly, only XC generates incorrectly added data, mainly in terms of properties and relationships (see online appendix). These additional data are in fact misplaced data, i.e., which should have been put in another entity or which should have linked other entities (e.g., the abstract is placed in the Work entity rather than in the Expression). Another study deals with the detection of patterns. Figure 3 depicts the scores obtained by the three tools for correctly detecting the bibliographic patterns without any cataloging issue (i.e., tests x.0). FRBR-ML obtains good results for detecting the core pattern (test 1.0). For other patterns (tests 3.0 and 5.0), it discovers half of the main entities (metric MEND) but it fails for the main relationship (metric MRND). VFRBR is not able to detect most patterns with its basic set of rules, even for the core pattern. This is mainly because this tool does not always create Works and Expressions. XC achieves good results with some patterns (core, complementary works) but not for derivations. Next, we note that all tools produce semantic mismatch, but only for the relationships (metric SMD-R). This issue occurs in 36 tests for FRBR-ML, 24 tests for VFRBR and 21 tests for XC (out of 42 tests), but the scores of the metric SMD-R are mostly below 10%, thus indicating that less than 10% of the relationships have a different semantics than in the collection annotated by experts. Since the more complex relationships are usually found in patterns, these results are also dependent on the ability of the tool to detect patterns. To summarize, our dataset T42 and post-FRBRization metrics are useful for understanding the failures of a tool.

Comparing tools in real-world context

The objective of this second experiment is to compare FRBRization tools in a real-world context using post-FRBRization metrics. The post-FRBRization metric DLE is not presented, since the expert FRBRized collection cannot include a link for each existing authority files or knowledge bases. All tools rely on their basic set of rules (no tuning). Fig. 4: Applying predictive metrics on BIB-RCAT for FRBR-ML basic rules and tuned rules

Facilitating the tuning

In this last experiment, we show how our pre-FRBRization metrics can help updating the set of rules. Only the FRBR-ML tool was used in this experiment, but the scenario could be applied to any tool. As shown in Table 1, the results of FRBR-ML for dataset BIB-RCAT could be improved. To provide insight to the expert, we compute predictive scores for the basic set of rules on the dataset BIB-RCAT. A subset of these scores is detailed in Figure 4. The white bar stands for the results with the basic set of rules. For instance, we note that 37% of the records contain cataloging practices (metric CPN). The basic set of rules contains many not used rules for the dataset BIB-RCAT (score of UR equal to 85%) and it lacks 24% of rules to take into account all fields from the dataset BIB-RCAT. Finally, the metrics for specific patterns indicate that 100% of the rules are missing to tackle derivations (metric MR-DER) and aggregations (metric MR-AGG). Based on these predictive scores, an expert has enhanced the basic set of rules of FRBR-ML. This update took 4 hours mainly for correcting minor changes (e.g., add rules for missing subfields) and implementing new templates to handle relator codes and missing concepts (e.g., augmentations, parent works).

The enhanced set of rules has been tested with the prediction metrics (black bars in Figure 4). Now, only 7% of the rules are missing to process all fields, and a few not used rules have been deleted (metric UR). The most significant enhancement deals with the pattern detection: all rules to identify derivations have been added, but the set still misses 67% of rules to process aggregations. Finally, FRBR-ML tuned with this enhanced set of rules was used to FRBRize the BIB-RCAT dataset. The results of this new FRBRization is shown in Table 1 (column FRBR-ML tuned ). As expected, the quality of this enhanced FRBRization is better than with the basic set of rules, especially for the patterns. Adding relevant new rules enables us to reduce the amount of missing data, but 29% of relationships and 21% of secondary elements in the patterns are still missing. This experiment demonstrates how the predictive metrics help librarians update the set of rules and thus improve the quality of the FRBRization.

Conclusion

In this paper, we described BIB-R, the first benchmark for evaluating the interpretation of bibliographic records. It includes a set of metrics and two datasets (T42 and BIB-RCAT). Extensive experiments with our dataset T42 have been performed with three recent tools (FRBR-ML, Variations VFRBR and Extensible Catalog) to demonstrate the possibility to identify strengths and weaknesses. Our experimental validation is also the first to compare FRBRization tools with the same datasets and metrics. Finally, we showed how the pre-FRBRization metrics can be useful to help librarians update the set of rules. The release of this benchmark brings different perspectives. We plan to add more records in the real-world dataset BIB-RCAT. The main challenge is to update the FRBR expert collection. Next, we could enhance the benchmark to enable evaluation of ergonomics (quality of graphical user interfaces), performance (execution time) and quality of the semantic enrichment (for instance based on the Knowledge Base Population challenge9 ).

Fig. 1 :

 1 Fig. 1: Overview of the FRBRization process

8Fig. 2 :Fig. 3 :

 23 Fig. 2: Experiment results for evaluating missing data

  Table1provides the results for the three tools. We note that they are able to identify only a few patterns (scores above 90% for the metrics MEND and MRND). VFRBR is the only tool to FRBRize half of the secondary elements of the patterns (ESE value equal to 55%). All tools successfully manage not to add incorrect data or produce different semantics (metrics IAD and SMD). However, they do not FRBRize almost half of the data (metric MD), mainly because of the incorrectly detected patterns. These average results for the three tools are understandable for several reasons: contrary to dataset T42, these real-world records from the dataset BIB-RCAT can combine several bibliographic patterns and issues. In addition, almost half of them include cataloging practices, which complicate the interpretation of the records. Finally, some additional entities (e.g., Concept) are not processed and created. The basic set of rules are not sufficient for achieving an acceptable quality. To conclude, this experiment showed that our dataset BIB-RCAT and associated metrics are useful to compare tools in a real-world context.

		FRBR-ML VFRBR XC FRBR-ML tuned
	MEND	94%	98%	94%	1%
	MRND	100%	100% 100%	29%
	ESE	99%	55%	100%	21%
	MD	44%	45%	45%	13%
	IAD	0%	0%	0%	0%
	SMD	0%	0%	0%	0%

Table 1 :

 1 Results of FRBR-ML, VFRBR and XC for the dataset BIB-RCAT

		100	FRBR-ML_tuned FRBR-ML		100	100
					85
		80			77
						67
	Percentage	60 40	37 37
				24
		20
				7
						0
		0
			CPN	MR	UR	MR-DER	MR-AGG
				Pre-FRBRization metrics

For instance, RDA, the LD4L project or BIBFRAME

http://bib-r.github.io/

T42 is a reference to the novel Hitchhikers Guide to the Galaxy

BIB-RCAT is a recursive acronym that stands for "BIB-RCAT Is Basically a Realworld CATalog"

http://www.nist.gov/tac/2016/KBP/

Acknowledgments

This work has been partially supported by the French Agency ANRT (www.anrt. asso.fr), the company PROGILONE (www.progilone.com/), a PHC Aurora funding (#34047VH) and a CNRS PICS funding (#PICS06945).