MODAL ANALYSIS OF MECHANICAL SYSTEMS WITH IMPACT NON-LINEARITIES: LIMITATIONS TO A MODAL SUPERPOSITION

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INTRODUCTION
Analysis of the response of structures is convenient if a linear model can fully describe the structure. Within this framework, it is useful to introduce in either the "nite or in"nite dimension (Hilbertian case) the notion of eigenmodes of the structure. They are either normal modes (de"ned by adding conservative conditions to the model) or complex modes (taking into account viscous damping for example) [1,2]. The linear theory of di!erential systems provides the response of the structure to an external elementary sinusomK dal excitation in an interesting form: the full response is simply the superposition of the responses of each mode to the excitation. Such a formula is well known; this is the superposition formula which is the basis of modal synthesis [2,3]. The notion of modal synthesis can be extended to the case of sub-structures by using linear operator theory [3,4].
In the non-linear case, the notion of non-linear modes had been considered "rst. In the case of mathematically smooth non-linearities and for a "nite number of degrees of freedom (d.o.f.) with particular polynomial non-linearities, Rosenberg and others "rst introduced natural modes [5] and then non-linear normal modes [6}9], and investigated their stability. Since then, many methods have been used to introduce modes (natural, non-linear, non-linear normal, minimal normal, non-linear similar normal, etc.) in the case of non-linear structures; methods derived from the works of Rosenberg [10}13]; a stroboscopic method [14]; methods based on averaging and modal truncation [15}17]; direct or geometrical methods for conservative systems [18}20] or in the Hamiltonian frame [21}25]; PadeH approximation [26,27]; multi-spectral, Volterra series and HFRFs [28,29]; integral transforms [30]; Lie series [31]; methods based on normal forms in the Hamiltonian [32}34] or general frame [35}40]; or methods using centre manifold theory [41}47] or amplitude equations [48]. JeH zeH quel and Lamarque extended the modal superposition starting from non-linear modes built via normal forms, for systems with a few d.o.f. and smooth non-linearities [49}51], even for the case of complex modes [52]; this method is valid for su$ciently small non-linear oscillations. The non-linear modes and generalized masses obtained depend on the amplitudes of the normal co-ordinates. They are built up in order to agree at best with a resonance equation. The modes do not always verify the reciprocity condition which exists in the linear case.
The question of non-linear modes is obviously related to the search for periodic solutions to non-linear dynamical systems and the study of their stability [52,53]. It therefore involves a huge amount of literature dealing with numerous analytical methods, perturbation methods, and methods for bifurcation analysis for example references [32, 54}66].
In the "eld of stochastic behaviour, the question of non-linear modes has been examined already [67] and tools are available [68]. In the case of non-smooth non-linearities, only particular cases have been investigated; that is, normal modes for piecewise linear systems [69,70]. Sometimes, in order to deal with localized or weak non-linear non-smooth phenomena using linear methods, one can introduce a modi"ed dissipation or sti!ness matrix.
From the point of view of dynamics, vibro-impact systems have been thoroughly studied in the literature: global behaviours and periodic solutions have been investigated in the single-d.o.f. case (both analytically and numerically in references [71}74] or by means of a change of variables in reference [75]) and in the two-d.o.f. case (double impact oscillator in references [76,77] or impact damper in references [78,79]). Singularities in the dynamics of such systems have been pointed out in references in [80}82], and some authors have examined the e!ect of dry friction on mechanical systems with impacts in [83}86]. General results for vibro-impact systems can also be found in references [87,88]. Moreover, a modal approach has been introduced in reference [89] to deal with direct and inverse problems in discrete systems with impacts, based on the theory of non-linear normal modes (see reference [27]).
Nevertheless, to our knowledge, no attempt to build a modal superposition similar to the linear case exists in the case of hard non-smooth non-linearities such as friction or impact. The main aim of this work is to seek some answers to the question about possibility of building a modal superposition and a modal synthesis in the case of structures exhibiting a non-linearity of impact type. The problem is considered for the case of simple systems with one and two d.o.f.
In section 2, a single d.o.f. system is considered. Using a piecewise exact integration (2.1), the periodic responses under sinusomK dal excitation are studied (2.2). The building of a modal superposition in equation (2.3) is then tested by introducing successively a generalized eigenfrequency from free vibrations, a generalized mode and a generalized mass associated with forced oscillations.
In section 3, two-d.o.f. systems are considered. First the case of weak coupling and impact of a mass against a rigid stop is dealt with (3.1). Then the case of strong coupling is examined, again with impacts against a external rigid stop (3.2). The results obtained are applied to the case of direct impacts between two rigid solids (3.3). Finally, in section 4 conclusions are drawn on the relevance of the modal superposition formula obtained.

SINGLE-DEGREE-OF-FREEDOM SYSTEM
The system studied consists of a single-d.o.f. damped harmonic oscillator with a unilateral constraint, for which an impact law is de"ned (see Figure 1). The impact process Figure 1. Single-degree-of-freedom system with k" m, c"am, g"fm.
is considered to be instantaneous and the behaviour of the system at the time of impact is described using the coe$cient of restitution e3[0,1], characterizing the energy loss during impact. The equations governing the dynamics of the system are then If the velocity immediately before the impact at tN is zero, several cases can occur: if at tN the acceleration is negative, then the system is still described by equation (1) after the impact and the trajectory is tangent to the stop at tN . If, on the other hand, the acceleration is positive, then the system remains in contact with the stop for a non-zero time interval. It can be shown that sticking never occurs if f( x K?V . In the following, it is assumed that the system's parameters satisfy this condition.
Since this system is linear between two consecutive impacts, it is possible to determine a piecewise analytical form of the solution on 1>. Setting "( !a/4 and "a/2 . ∀k3-*, the solution on [t I\ , t I ] can be written in the form where t "0 and From equations (3), the following recursive relation gives the values of the constants A I and B I :

PERIODIC SOLUTIONS
Owing to the analytical form of the solution given by equations (3) and (4) it is possible to seek analytically a periodic solution, similar to that achieved in reference [72] or [71] for example. A solution of period n¹ with k impacts per cycle is here called (n, k)-periodic, where ¹"2 / is the period of the external excitation.

(n, 0)-periodic solutions
The simplest case consists of looking for n¹-periodic solutions which never impact against the stop. Such a case implies n"1 and the initial conditions leading to a (1,0)-periodic solution are x "f , Figure 2. (1, 1)-periodic solution to the system (2) for "2)6, x "13)32968 and xR "19)36619. and as it is assumed that f( x K?V , it can be shown that (n, 0)-periodic solutions exist if and only if ) \ or * > . The stability of these periodic solutions can be determined using a PoincareH map de"ned by a constant phase plane Z"¹ in the co-ordinates (X, >, Z)"(x, x , t mod ¹). As pointed out in reference [80], such a mapping is not everywhere continuous nor di!erentiable. Therefore, for each periodic solution corresponding to a "xed point of the PoincareH map where it is continuously di!erentiable, the stability can be investigated. It can be shown that (1, 0)-periodic solutions are always stable.

(n, 1)-periodic solutions
Similar to section 2.2.1, n¹-periodic solutions with one impact per cycle can be sought by using equations (3) and (4). In this case, the impact time can be analytically determined, and the initial conditions leading to (n, 1)-periodic solutions are then given by where A (t ) and B (t ) depend analytically on the system's parameters. An example of (1, 1)-periodic solution is shown in Figure 2.
As in section 2.2.1, the PoincareH map can be used (when it is de"ned and of class C locally) in order to determine the type of the periodic solutions obtained (see in Figure 3). The Jacobian matrix can be calculated analytically from the analytical form of the PoincareH map by taking into account the in#uence of the partial derivatives of the impact time t with respect to the initial conditions.
It can be shown that such a stability study is valid only if O > and O \ for in that case the PoincareH map is not di!erentiable at the "xed point considered. The method for seeking n¹-periodic solutions with two impacts per cycle is identical to the case with one impact per cycle, with a new unknown due to the second impact time t . By characterizing the periodicity of the solution and by using the analytical form (3), (4) a system of two non-linear equations with two unknowns (t , t ) can be obtained which can for example be solved using Newton's method. Once the value of t and t are known, the initial conditions of the system are given by show three examples of (n, 2)-periodic solutions that can be obtained analytically.
As in the case of (n, 1)-periodic solutions, the stability of the (n, 2)-periodic solutions can be studied by using the PoincareH map: when , + > , \ ,, it is possible to calculate analytically the Jacobian matrix of the PoincareH map, and its eigenvalues determine the stability or instability of the periodic solution.

Free oscillations
The free oscillations of the system are described by the following equation: x (0)"x , xR (0)"xR .
The solution of this equation can also be piecewise analytically written: the expression is identical to equation (3), the constants of integration being given by equation (4), with f "f "0.  If the equilibrium is a position that the system can physically reach, namely if x K?V '0, and if aO0, it can be shown that the system (6) has a "nite number K of impacts. This result is interesting from the modal point of view; it means that the free oscillations of the system are governed by the frequency , except for a bounded time span. Hence, it will be considered later on that the natural frequency of the system is .
Remark. When a"0, the number of impacts is in"nite and lim I> t there also is an in"nite number of impacts and t

Generalized mass and modal superposition
In this section, it is intended to establish a modal superposition formula for the irregular non-linear model previously introduced. The case x K?V '0 is considered for which the Figure 9. Existence of (4, 2)-periodic solutions: stable (solid curve) and unstable solutions (dotted curve): (a) initial displacements; (b) initial velocities. natural frequency is , and an attempt is made to obtain a modal superposition formula similar to the linear case, connecting the nth harmonic amplitude of the response to the amplitude of the forcing.
If a (n, k)-periodic solution is considered, i.e. a solution with period n¹ and k impacts per cycle. Due to the n¹-periodicity of the response, the Fourier coe$cients can be de"ned; for all j39 we have x (t) is known to be piecewise via equation (3). These k#1 integrals can then be calculated analytically. If kO0 de"ne The nth Fourier coe$cient is then given by Let the modal mass m LI be: Because the free damped steady state oscillations are linear oscillations, the generalized mode is here represented by the scalar 1. The nth harmonic amplitude can then be written in the form This is a modal superposition formula connecting the nth harmonic amplitude of the forced response to the amplitude of the forcing via the free response. This formula is similar to the formula that obtained in the linear case, but the mass (which should be equal to 1) is replaced by a modal mass (11).
Due to this de"nition, the mass is a complex: to give it a more physical meaning, it is necessary to consider its module. Thus, the unitary mass system with impact is modelled and, subjected to a sinusomK dal forcing of frequency , like a linear system without impact of mass "m LI ", subjected to the same forcing. This modelling holds in terms of spectral amplitude for a (n, k)-periodic response: the spectral amplitude is the same one for both systems.

Remark.
* In the case of a n¹-periodic response without any impacts, it was seen that only the case n"1 was possible. Thus, m "1 which is coherent since in this case the classical modal superposition formula applies and gives A ( )"" f/ I ( )".
* There is no longer a unique modal superposition formula as in the linear case, but an in"nity a priori; it depends on the period and number of impacts per cycle of the periodic solution.
* Preceding calculations give access to the whole Fourier series of a (n, k)-periodic response.
* An analytical expression of the module of the modal mass is obtained if an only if k"1, i.e., when the periodic solution has only one impact by period. Setting for the modal mass: Since t is analytically known in the case of a (n, 1)-periodic response, the modal mass "m L " can be expressed analytically as a function of the parameters of the system.

Examples of modal superposition
It is now possible to show some applications of the modal superposition formula previously established, using the analytical search for periodic solutions carried out in section 2.2. First consider the following set of parameters for the system: "2)5, a"0)05, x K?V "14, e"0)9 and f"20. Note that for these values of parameters x K?V 'f and e'0, so that sticking never occurs.
Figures 10 outlines an important di!erence compared to the linear case: the peak of amplitude corresponding to resonance does no longer exist around . Consequently, for close to the natural frequency of the system, the modal mass is the largest because there is no resonance between the external forcing and the system. The maximum amplitude occurs at a frequency higher than the natural frequency, but the maximum reached is much weaker than in the linear case; there is not a true peak of amplitude in the usual sense.
In addition, the modal superposition formula previously established enables the amplitude of nth harmonic of a solution, whose period is n¹ to be computed. In the linear case (without impact), this is su$cient to know the whole spectral response of the system: only n"1 occurs and the Fourier coe$cients are all zero except the "rst one. In this case, it is no longer su$cient, for the occurrence of impacts leads to periodic solutions with many harmonics, and the nth harmonic amplitude can be a poor approximation to the spectral amplitude of the response. For example, Figure 12 shows that the second-harmonic amplitude for a ¹-periodic solution can be the largest one. In the same way, the Fourier coe$cient c can become large, can be seen in Figure 10. Table 1 summarizes the di!erence  between the spectral amplitude and the nth harmonic amplitude for various periodic solutions.
For this example, a modal superposition formula can be built and is worthwhile as long as remains close to the &&primary resonance''. Nevertheless, this frequency area does not always correspond to the largest amplitudes.  An example showing a spectral behaviour di!erent from the preceding one can now be considered. In this case "1, a"0)02, x K?V "1, e"0)9 and f"20 (sticking to the stop may then occur, but only responses without sticking will be considered).
In Figure 18 two important characteristics can be seen. First of all, and contrary to the previous example, the response exhibits a true peak of spectral amplitude similar to the  resonance peaks observed in linear systems. However, the peak does not occur at the natural frequency , but at +2 . Furthermore, the approximation to the whole amplitude by the amplitude of "rst harmonic is very rough: resonance occurs through the Fourier coe$cient c of the solution, and the remainder of the harmonics are negligible in the neighbourhood of the peak.   may not be su$cient any more; it would be necessary to include at least the four "rst harmonics in the formula in order to get a closer approximation in the examples shown.

TWO-DEGREE-OF-FREEDOM SYSTEMS
This section deals with a system with two d.o.f., one of them being constrained by a stop: y3C(1>, 1).
This system can be written in the form with , F" f f and the term &&# impact'' represents the constraint induced on x by the occurrence of impacts.
The model just introduced is a general mathematical model. It will be studied in three stages: "rst of all, consider the case k "0 (which will be referred to as &&weak coupling'') for which the study is very close to the single-degree-of-freedom case in term of periodic solutions: it will give an initial outline of modal superposition with two d.o.f.
The general case where k O0 (referred to as &&strong coupling'') will then be studied, for which the modal superposition is similar to the case k "0, except that the search for periodic solutions becomes more complex. Finally it can be seen how the modal superposition can be written in the case of two rigid bodies colliding.
Note, initially, that the "rst case is a mathematical model which cannot easily be expressed in mechanical terms. Indeed, the action}reaction principle prevents x from acting on y without y acting on x. Therefore, it is necessary to consider carefully the physical conclusions that could be drawn from this model.

WEAK COUPLING
System (14), the special case where k "0, which is close to the single-d.o.f. case previously studied can be dealt with. and pre-multiply the system 14 without impacts by P\, a system in the new co-ordinates x x "P\X is obtained: where Written in that form, the system is easy to solve and as long as there is no impact: where "( !a/4, J "( !a/4 and Moreover, if "a/2 J and "a/2 J , the solution is given in the initial co-ordinate system by From this equation it can be inferred that t I is solution of the equation Moreover, by assumptions on x and y, at impact time which yields for the new variables: Thus, obtain the following relationships provide the constants of integration where u is given by

Search for periodic solutions
The search for periodic solutions (x, y) is equivalent to the search for periodic solutions (x , x ). According to equation (17), x "x, therefore the search for periodic solutions for x is similar to the one carried out in the case of a single-d.o.f. system. Thus, it is possible to determine x and x so that x is (n, k)-periodic where k3+0, 1, 2,. As regards x , the method is identical; only the recursion that gives A and B being di!erent. Thus 2;2 linear systems are obtained in (A (t ), B (t )). When the determinant is non-zero, x and xR are obtained so that x is (n, k)-periodic where k3+0, 1, 2,. The initial conditions for the original system co-ordinates (x, y) are given by equation (17):

Modal superposition
3.1.3.1. Free oscillations of the system. It was seen in Section 2.3.1 that, in the case of a single-d.o.f. system without external forcing, the number of impacts is "nite and the steady state response is periodic with frequency . Hence, in the case of weak coupling, the free response also exhibits a "nite number of impacts, and the steady state response for x is periodic with frequency . As for x , since there are no more impacts once the steady state response is reached, the equation of its movement is given by equation (15): This is the equation of a classical damped oscillator without external forcing, whose steady state response has frequency . In order to establish a modal superposition formula, it is necessary to start from the natural frequencies and . Moreover, in that case again, the generalized modes correspond to linear modes. Generalized masses and modal superposition. Consider a (n, k)-periodic solution. The Fourier coe$cients of the functions x and x can be calculated. These are quite similar to the coe$cients found for the single-d.o.f. system. Setting

3.1.3.2.
with H "j /n J and H "j /n J , the Fourier coe$cients are given by The Fourier coe$cient corresponding to the nth harmonic are inferred by using the coordinate transformation (17) cV where, I " ! #ai , I " ! #ai and 21 The nth Fourier coe$cients are then in the original basis: The contribution of the nth harmonic to the Fourier spectrum is given (as seen in equation (12)) by which can also be written as we can express A in the form TR T , and A in the form TR T where T , T , T and T denote the generalized modes of the system. If T "(a , b ) and T "(a , b ): then , three equations with four unknowns are obtained It is thus possible to arbitrarily set one of the unknowns; for example, as in the case of linear modes, if a "1, then The same type of calculations and assumptions for A lead to T "(0, 1), Using equations (23) and (24), relation (22) can then be written in the form This has just established a modal superposition formula holding for a two-d.o.f. system with weak coupling. This formula links the nth harmonic amplitude to the free response and the forcing by means of a modal mass. The vectors T , T , T and T represent the modes (left and right respectively). Note that T and T are merely the eigenvectors associated with and . Finally, as in the case of the single-d.o.f. system, there is no unique modal superposition formula which holds in all cases, but according to the period of the response and the number of impacts per cycle, it is necessary to choose the right formula.
Remark. The modal superposition formula seems to exhibit a reciprocity breaking compared to the linear case: ¹ and ¹ are di!erent whatever choice is made when setting one of the unknowns. Indeed, the second component of ¹ is always zero whereas that of ¹ is always non-zero. This is in fact due to the choice of the model, for which k "0 and k O0.

3.1.3.3.
Examples of modal superposition. For the "rst example, the following parameters are chosen: "2)5, "3)8, a"0)05, e"0)9, x K?V "14, f "20, f "18 and k "1. The second example is derived from the single-d.o.f. case and the chosen parameters are the following: "1, "3)8, a"0)02, e"0)9, x K?V "1, f "20, f "18 and k "1. The spectral amplitude for x is the same one as that obtained in the single-d.o.f. case. The continuous d.o.f. y exhibits a resonance for K similar to the linear case, but several additional secondary resonances occur for K /2 and K /3. Furthermore, for the frequency associated with a spectral amplitude peak for x, a small peak for y is found, which can become large if k is su$ciently large. Lastly, if is close to /2 and k is rather large, then the main peak of resonance can occur in the neighbourhood of /2.

STRONG COUPLING
In this section, the system (14) with k O0 is studied.

Analytical solution of the system
3.2.1.1. Decoupling equations. System (14) will be decoupled in order to be able to write the solutions x and y explicitly. The matrix K has the following characteristic polynomial: The system parameters will be chosen so that '0. Then the matrix K admits two distinct real eigenvalues:

Assume that
, then two eigenvectors associated with and are respectively ( v ) and ( v \ ), which de"ne the matrix The assumption '0 implies that its determinant is nonzero, and P is invertible with Then, if X"PY, the system (14) is multiplied by P\ in order to obtain The new system obtained using the new basis is given by as long as there is no impact. Thus these equations can be solved easily; with J "( !a/4, J "( !a/4 and The solution in the original basis is "nally given by 3.2.1.2. Gluing solutions at impact times. An impact occurs when x(t)"x K?V , namely when , the decoupled solution is given by From these equations it can be inferred that the equation veri"ed by t I is , Yet by assumptions on x and y, at the impact time: x (t> I )"x (t\ I ), x (t> I )"!exR (t\ I ), which yields, according to (28), For i3+1, 2,, then:

Search for periodic solutions
The search for periodic solutions (x, y) is equivalent to the search for periodic solutions (x , x ). Only (n, 0) and (n, 1)-periodic solutions will be sought. The following calculations are similar to those found in reference [76], with the addition of damping.
3.2.2.1. (n, 0)-periodic solutions. As for the single d.o.f system, it is easy to prove that the system admits a (n, 0)-periodic solution if, and only if, x and x are given by According to equation (28): It is thus possible to get results similar to those obtained for a single-d.o.f. system in section 2.2.1. This gives a condition that a (n, 0)-periodic solution exists; it requires From this inequality, a fourth degree polynomial in can be obtained, allowing the values of for which the system admits a (n, 0)-periodic response to be determined. The remark made in Section 2.2.1 still holds; the existence of (n, 0)-periodic solutions requires x K?V '0.
Using equation (29) , Initial conditions of the system leading to a (n, 1)-periodic solution are then given by The nth Fourier coe$cients of the system by the basis change (28) are then where I " ! #ai , I " ! #ai and In the original basis, The contribution of the nth harmonic is (as for the weak coupling in equation (21)) given by Let Three equations with four unknowns are obtained As in the case of weak coupling, it is possible to "x one of the unknows; for example, in order to respect a reciprocity condition as long as possible, let a "1/((1#v v ). Then 1#v v '0, for the parameters of the system verify ( In the same way, for A : Using equations (34) and (35), the relation (33) then becomes Thus a modal superposition formula for the two-d.o.f. system with strong coupling has been established. Remark.
* ¹ and ¹ thus de"ned are in fact eigenvectors associated with and .
* There is no reciprocity when k Ok , but when the matrix K is symmetrical, reciprocity occurs since then ¹ "¹ and ¹ "¹ .

TWO RIGID BODIES COLLIDING
In this section, a mechanical system consisting of two oscillating rigid bodies which can collide during their movement is studied.
It will be assumed that the equilibrium position x and x of the two solids are such that x !x "x K?V '0 (system without preload). The equations of the system in relative displacements are then Let a "c /m , " /m , f "g /m and similarly a "c /m , " /m , f "g /m , and assume hereafter that a "a "a. The system becomes x !x *x K?V .
When x (t)!x (t)"x K?V , an impact occurs at t and the restitution law leads the relative velocity between the two solids to be multiplied by a factor !e at the impact time:
Moreover, the conservation of the momentum provides the second equation to be able to determine post-impact velocities: A simple change of co-ordinates can be applied in order to express the system (14) in the form previously studied; let 32 x !x *0, where Thus the system is of the type (14); the results of the preceding study are then applicable. First of all note that k and k are proportional, so that there is always the case of the strong coupling studied in section 3. As regards the solutions of the system between two impacts, there is no work to do in order to decouple the system, since the initial system is already written in decoupled form. Thus the eigenvalues of the system are always real, and are exactly and . In this case v "1 and v "m /m . According to equations (34) and (35), the modes of the system are given by where "1/(1#m /m . Figure 23. Existence of (1, 0)-periodic solutions: (a) initial displacements for x (circle) and x (x-mark); (b) initial velocities for x (circle) and x (x-mark).
Using these modes, the modal superposition formula is as in equation (36), the expression of the modal masses being given by equation (32). It is interesting to note that in general there is no reciprocity, except if m "m . Finally, the periodic responses of the system can be dealt with. First of all, according to section 3.2.2, there are (n, 0)-periodic solutions because it was assumed that x K?V '0. In the same way, calculations of section 3.2.2 can search for (n, 1)-periodic solutions. Diagrams of existence of periodic solutions versus the frequency of the external excitation are obtained. From these periodic solutions, the modal superposition formula can then be tested, plotting the di!erence between the spectral amplitude of the system's response and the nth harmonic amplitude given by equation (36). For the "rst example presented, let m "1, m "0)7, "5, "13, a"0)05, e"0)9, f "20, f "18, and x K?V "14. As it was the case for the single-d.o.f. system, note in Figure 25 that the usual resonance of linear systems no longer occurs. Indeed, when the frequency of the external excitation is equal to the natural frequency of the system, the modal mass corresponding to the excited mode goes through a local maximum, so that the associated spectral amplitude does not present a particular peak. Nevertheless, spectral amplitude peaks appear for values of far away from the natural frequencies. The main peak is located in the neighbourhood of (which is one of the natural frequencies of the system after the change of variables), but it is di$cult to show a resonance in , for the peak is appreciably shifted from this frequency. From the modal point of view, the occurrence of such peaks does not correspond to a classical resonance (where the generalized natural frequency of the system is close to the frequency of external excitation); resonance can be interpreted as the locus of frequency where the generalized modal mass is minimum with respect to . As regards the closeness of the approximation to the spectral amplitude by the nth harmonic amplitude, note, as in the single-d.o.f. case, that the "rst harmonic is not always the leading term in the amplitude of a (1, 1)-periodic response (see Figure 26). Thus, the   constant coe$cient c overrides all the others for some values of . Yet if is considered to be close to the peaks in the spectral response, the "rst harmonic gives a close approximation to the total amplitude (see Table 2).
For the last example, a set of parameters close to the second example in the single-d.o.f. system is chosen: m "1, m "0)7, "1, "13, a"0)02, e"0)9, f "20, f "18, and x K?V "1. Looking at Figures 27 and 28 "rst note that amplitude peaks appear at the same frequencies for x and x , which is due to energy transmission between the two bodies through impacts. Moreover, the "rst harmonic amplitude can once again be far lower than the spectral amplitude; most of amplitude peaks in x come from A , and for x the second harmonic is often overriding.
Only the (1, 1)-periodic responses of the system have been studied. Nevertheless, it must be kept in mind that many other types of periodic solutions are possible, as was seen for the single-d.o.f. system, depending on the time period and the number of impacts per cycle. Theoretically, nothing can prevent a modal superposition formula for any (n, k)-periodic response from being written, but in practice such a response is hard to "nd by analytical means when k'1.

CONCLUSION
The feasibility of building a modal superposition formula for systems with irregular non-linearities of impact type has been investigated, imitating the procedure used in the smooth non-linear case [49,51]. The formula has been built for simple single-and two-d.o.f. systems with unilateral constraint and restitution law. The generalized modes and frequencies obtained turn out to be identical to the linear case, the non-linearity of the system being concentrated into the modal masses.
The examples considered show that the formula is valid in the case of a primary resonance for which the spectral amplitude is given by the Fourier coe$cient corresponding to the periodicity of the forced solution obtained. Nevertheless, these examples have above all illustrated the limitations to such a formulation. Firstly, the multiplicity of periodic solutions, with di!erent periodicity or number of impacts per cycle, compels several potential formulas to be built, and it is not possible to know a prori which one has to be used. Furthermore, main amplitude peaks may appear away from any a priori clearly identi"able resonance, which may be overidden by some unusual harmonics and consequently cause the modal superposition formula to fail. Therefore, it is not possible, in a general case, to build a modal superposition formula using only the usual sequence de"nition of generalized frequencies, de"nition of generalized modes and then de"nition of generalized modal masses; the non-linearities of impact type produce a limitation on the formulation of a general formula following the usual procedure.