Fr Edr Iksson A N D Ar

B No Rdm A Rk

Keywords: impact oscillations, norm al forms, non-sm o oth system s, discontinuity mapping, computer algebra

On normal form calculations in impact oscillators

Ma ts H

Introduction

Modern dynamical systems theory supplies tools for local analysis of the dynamics of ordinary di¬erential equations. Concepts such as Poincaré mappings, centre manifolds and normal forms are helpful in understanding motion and bifurcations. The underlying assumption for these techniques is that the vector eld is su¯ciently well behaved. However, in models of mechanical systems one can nd that assumptions that are natural from the modelling point of view give vector elds that lack smoothness across boundaries in state space. Oscillating systems where impacts are possible is one example where it is not straightforward to employ standard methods.

In impact oscillators one or more motion limiters are present. Typically, a wall acts as a one-sided amplitude constraint. This situation is not uncommon in engineering systems, where models that incorporate this behaviour have been used to understand gears [START_REF] Pfei® Er | Rattling models from deterministic to stochastic processes[END_REF][START_REF] Karagiannis | Theoretical and experimental investigations of gear-rattling[END_REF], railway wheelsets [START_REF] Nordstr¿ M Jensen | On a new route to chaos in railway dynamics[END_REF][START_REF] Knudsen | Bifurcations and chaos in a model of a rolling railway wheelset[END_REF], mooring towers [START_REF] Thompson | Complex dynamics of compliant o® -shore structures[END_REF], printing devices [START_REF] Tung | The dynamics of an impact print hammer[END_REF], the mechanics of bipedal walking [START_REF] Garcia | The simplest walking model: stability, complexity and scaling[END_REF][START_REF] Mcgeer | Dynamics and control of bipedal locomotion[END_REF]Adolfsson et al. 1999), and several other dynamical problems.

The motion during contact has a much shorter time-scale than the typical motion of the system; consequently, a very common model for the impact is to assume that velocities change at an instant. This is the model that will be used in this paper.

Many papers have dealt with this subject. Among the rst investigations using a dynamical systems perspective we nd [START_REF] Shaw | A periodically forced piecewise linear oscillator[END_REF], Shaw (1985a;b) and [START_REF] Thompson | Chaotic dynamics of an impact oscillator[END_REF]. Subsequent work has revealed dynamical features found in smooth systems, as well as special features of the model, such as the dynamics close to grazing impact (see [START_REF] Nordmark | Non-periodic motion caused by grazing incidence in an impact oscillator[END_REF], and the possibility of an in nite number of impacts in nite time (see [START_REF] Budd | Chattering and related behaviour in impact oscillators[END_REF]. The main focus has been on one-degree-of-freedom systems, but systems with several degrees of freedom have been considered in [START_REF] Shaw | The onset of chaos in a two-degree-of-freedom impacting system[END_REF], Aidanp a a & [START_REF] Gupta | Periodic and chaotic behaviour of a threshold-limited two-degree-of-freedom system[END_REF], [START_REF] Fredriksson | Grazing bifurcations in multibody systems[END_REF] and the thesis of [START_REF] Lee | The theoretical and numerical analysis of impact oscillators[END_REF]. The study of such systems are, in general, complicated by the abundance of parameters, thus special cases are usually considered in order to reduce the number of parameters. To the authors' knowledge no review paper has been written, but the theme issue edited by Bishop (1994) is dedicated to impact oscillations.

Previous investigations usually assume that the dynamics between impacts is given by linear equations of motion. As the impact law relates the velocity after impact to the velocity before impact, a coe¯cient of restitution is used. The advantage of this approach is that a closed-form solution for the motion between impacts can conveniently be written down, at least as long as the system has one degree of freedom. For impact oscillators little attention has been paid to more general approaches to local stability calculations, where one cannot rely on special features of the system. A scheme for the derivation of local expressions would be desirable, giving means to analyse systems where the interesting dynamics is impossible to capture using linearized equations of motion. This would also be of interest for the engineering community. The aim of the present paper is to propose such an idea. Combined with standard techniques it gives a way to obtain a local Poincaré mapping P on the form

P (z) = Lz + N (z); (1.1)
where z is a local coordinate in the Poincaré section, L is the matrix that gives the linear part of the mapping and N (z) are the higher-order terms. To obtain (1.1) is the starting point if one would like to use normal forms to study bifurcations. The paper is organized as follows. Firstly, we discuss the type of systems that we have in mind. Local mappings are introduced, and an idea on how to handle the impact part of the local dynamics is presented. This is done by de ning a discontinuity mapping. This mapping encapsulates the contribution to the dynamics coming from impacts. Three examples are studied. Finally, we discuss the results and how these can be extended.

Motion and impact

Our assumptions about the system are largely inspired by models based on connected rigid bodies. Such multibody models often o¬er su¯ciently accurate and computationally e¯cient models. They also relate well to the impact approximation, since a truly rigid body must change velocities discontinuously upon impact with a rigid surface. We assume that impact is due to a single body colliding with a surface. Thus, only one condition for contact needs to be checked. The motion of the impacting body is coupled to a larger system (in an arbitrary way) between impacts.

(a) State space and °ow

The system is assumed to have n degrees of freedom. We denote coordinates on the con guration manifold by q 1 ; q 2 ; : : : ; q n . Similarly, the velocities of the system are u 1 ; u 2 ; : : : ; u n . The state x is

x = q 1 u 1 . . . q n u n (2.1)
and the time derivative _ x of a motion is related to x by a vector eld F :

_ x = F (x): (2.2)
In many practical cases, time-periodic forcing is present. Indeed, all examples below have periodic forcing. In developing the general arguments, we do not need to treat this case separately. We can always rewrite the equations in the standard fashion, by extending x with a phase angle 2 S 1 , and add _ = 1 to the equations of motion.

We denote the state space where the motion limiter is removed (or thought to be penetrable) by X . This is partly a notational convenience. The main concern here is how to obtain locally valid expressions, thus we will not have much to say about global features of the state space. We write = (x; t) for the state space ®ow:

: X R ! X ; (2.3)
the mapping given by the solution of the di¬erential equation (2.2). When writing the arguments explicitly we sometimes use an indexed time t, t (x). Thus, t (x) is the state reached by following a trajectory from an initial point x during the time interval t.

(b) The impact law

We de ne the function H = H(x) to be equal to the distance from the impacting body to the motion limiting surface when H(x) > 0. The equation H(x) = 0 is the condition for contact. We also allow H(x) < 0, meaning a state which breaks the geometry imposed by the model. The contact condition can be interpreted as a surface in X with an equation H(x) = 0. We denote the set of points that ful ls this equation as :

= fx 2 X : H(x) = 0g:
(2.4) When a trajectory reaches , it is disrupted by the impact law. We view the impact law as a mapping G : X ! X , which is only of interest on . The impact law leaves coordinates on the con guration manifold unchanged. The velocities are changed, where we allow the new velocities to depend on both velocities and the con guration coordinates of the impact. We write

x a = G(x b ); (2.5)
where x a is the state immediately after impact, and x b is the state immediately before impact.

S x = t 1 (x) ~- x x' - F G ° t 1 (x) - F G ° t 1 (x) - F t 2 °F t 1 (x') F t 2 °F t 1 (x') F Figure 1.
Impacting and non-impacting motion.

The discontinuity mapping

We now develop the main idea of the paper. Related work can be found in [START_REF] Fredriksson | Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators[END_REF] and in Dankowicz & Nordmark (2000). For a derivation of the linearization, see also M uller (1995).

(a) Motivation

Away from , all of the dynamics is in the ®ow. If we wish to analyse non-impacting motion close to a point x 0 , it is natural to consider the mapping T , where T > 0 is the time-interval of interest (the period in the case of periodic motion). A trivial decomposition can be made,

T = t 2 t 1 ; (3.1)
where

t 1 + t 2 = T .
Including impacts, we assume that t 1 is also the time of ®ight for an orbit starting at a point •

x to reach a point x 2 :

x = t1 (• x): (3.2)
Using the impact law, and then t 2 , the image of • x is

t 2 G t 1 (• x): (3.3)
This situation is indicated in gure 1. However, one should note that the expression t 2 G t 1 is valid as a mapping only for points having a ®ight time equal to t 1 . In general the ®ight time for points in a neighbourhood of •

x will be di¬erent. A decomposition similar to that of ®ow maps would be convenient, which inspires the following idea: can we nd a mapping C such that the correct mapping for all points close to •

H < 0 H = 0 H > 0 G (x) x x ~G ° t C (x ) F t C (x ) F C (x)

S

x can be written as t 2 C t 1 ? This mapping C must then incorporate the jump in velocities implied by the mapping G as well as the di¬ering ®ight times. The mapping C will be referred to as the discontinuity mapping, since it takes care of all aspects of passing the discontinuity imposed by the impact law.

Using this viewpoint, the in®uence of impacts is in a natural way separated from the dynamics of the ®ow. Calculating the ®ow for a xed time is also natural in numerical computations, where can be locally expanded and the coe¯cients can be determined by integrating the variational equations for the xed times t 1 and t 2 .

If, rstly, we follow the ®ow for a xed time t 1 , the image of the starting point might have H < 0. This is not consistent with the model, and the discontinuity mapping C will have to address this. The de nition set for C is thus a neighbourhood N of x, where N X . The rst step in de ning C is to note that for x 2 N we can trace the ®owline passing through x to the intersection with . We use t C = t C (x) to denote the short time of ®ight from x to . If H(x) < 0 then t C (x) < 0, and if H(x) > 0 the time of ®ight to is positive. The latter situation is shown in gure 2.

In this way we obtain a mapping from N to , and the image of x is tC (x) (x). Using the impact law we get a point G tC (x) (x) close to G(x). We now use this as an initial condition and we follow the motion (disregarding the wall) for a time t C (x). The discontinuity mapping is then

C(x) = tC (x) (G tC (x) (x)): (3.4)
It follows that C is a mapping from a state space neighbourhood of x (disregarding the wall) to a state space neighbourhood of G(x) (again disregarding the wall). Choosing x = x we have t C (x) = 0; hence C(x) = G(x), as expected.

(b) Expansions

To write C in a useful form, we wish to express it as an expansion in y = x x:

C(x) = G(x) + Ly + N (y); (3.5)
where L is the matrix giving the linearized mapping and N denotes nonlinear terms. It is not hard to translate the ideas in the previous section into a step-by-step description of how to obtain this expression.

(i) Firstly we calculate the ®ow close to x. We use the tilde to stress the expansion point, thus writing ~. It is sought as

~= x + t ~(y; t); (3.6)
where ~is an expansion in y and t. We determine the unknown coe¯cients in ~(y; t) by using the fact that the ®ow satis es (3.10)

@ @t (x; t) = F ( (x; t)) (
where O(y) 2 denotes terms of order two and higher. We then have

x a = G(x b ); (3.11)
where x b (x) is known.

(v) The ®ow close to x = G(x) is now calculated. This is similar to the rst step, with only a change in the expansion point. For obvious reasons it is convenient to denote the initial condition with a subscript `a', and we use the circum®ex as we have previously used the tilde: ^= x a + t ^(x a x; t):

(3.12) (vi) Lastly, we insert t = t C in the expression for the ®ow:

C(x) = ^(x a ; t C ); (3.13)
where x a = x a (x) has been calculated in previous steps. Finding the linearization is straightforward. The zeroth-order term in ~is F = F (x), thus ~= x + t( F + O(y; t)):

(3.14)

By writing Hx = @H @x (x);

(3.15)

the function H(x) is

H(x) = Hx y + O(y) 2 ; (3.16)
where the gradient Hx is a row vector. We use an unknown row vector tCx to tentatively write t C as

t C = tCx y + O(y) 2 :
(3.17 Note that this form of the linearized map is uniquely speci ed by demanding that vectors v orthogonal to Hx should be mapped to Gx v (since C is the same as G on ) and that F should map to F (since C maps a ®owline through x to a ®owline through x).

If we wish to derive the mapping to higher order the computations rapidly get complex. A computer algebra system is very helpful in order to automate the calculations and to minimize errors. For error checking it is also helpful to note some of the characteristics of the mapping. If we tentatively use the identity mapping for G, then C is identity. If we take x 2 , then C = G. These observations can be used to check that complicated expressions evaluate as expected.

Examples

Let us investigate how the ideas above can be used in practical calculations. When writing out expansions we use brackets [i; j; : : : ; k] as indices to label coe¯cients. If A is a function of the real variables x 1 ; x 2 ; x 3 , we write

A = i;j;k 0 i+ j+ k<m Ã[i;j;k] i!j!k! (x 1 x1 ) i (x 2 x2 ) j (x 3 x3 ) k + O(x x) m : (4.1)
An expansion that includes linear terms is then

A = Ã[0;0;0] + Ã[1;0;0] (x 1 x1 ) + Ã[0;1;0] (x 2 x2 ) + Ã[0;0;1] (x 3 x3 ) + O(x x) 2 : (4.2) (a) A forced n-degree-of-freedom system
We introduce some assumptions to obtain a system with an arbitrary number of degrees of freedom, but with similarities to the familiar one-degree-of-freedom periodically forced case. The basis for the one-degree-of-freedom model is a particle moving along a line. This motion is now coupled to another system. Thus, the state space without motion limiter, X , has the structure

X = R 2 M S 1 ; (4.3)
where the factor R 2 is due to the position and velocity of the particle, M denotes the submanifold of the state space which describes the system that is coupled to the particle and the factor S 1 is included by assuming periodic forcing. Furthermore, we assume that the equations of motion are

_ q 1 _ u 1 . . . _ q n _ u n _ = u 1 A 1 (x)
. . .

u n A n (x) 1 : (4.4)
When the motion limiter is taken into the model we assume that the impact process only involves the particle model. The coordinate q 1 describes the position of the particle and the motion limiter is located at a critical coordinate q 1 = q c , hence

H(x) = q 1 q c : (4.5)
The impact law only changes the velocity of the particle, which we write as

u 1a = g(u 1 b ): (4.6)
All other velocities are una¬ected by the impact. Writing the impact law as

u 1a = û1 + g [1] (u 1b ũ1 ) + O(u 1b ũ1 ) 2 ; (4.7)
and denoting

A i [0;:::;0] = A i [0] ; (4.8)
we obtain from (3.23) the expression for the linearization L of C: for some choice of • . The Poincaré mapping using this section is sometimes called the stroboscopic mapping and denoted as P S . For impacting systems the situation is more complicated. The impact velocity is often of major interest. A natural idea is then to use a subset of (u 1 > 0 or u 1 < 0) as a Poincaré section. The mapping is then referred to as the impact mapping, P I . Another choice is to use (with q 1 > q c ) and use this set as a section. This choice is more in conformity with the current approach. Assume that we have a motion, making m impacts at the phases ~1; ~2; : : : ; ~m before returning to ; let P ! ~ 1 be the mapping, using only the ®ow, from a neighbourhood of the starting point in to the section of xed phase where the rst impact occurs. Similarly we write P ~ 1 ! ~ 2 ; P ~ 2 ! ~ 3 ; : : : ; P ~ m ! . We use Cj = ~ to denote the restriction of C by xing the phase angle to the impact phase. The stroboscopic mapping is then

L = û1 ũ1 0 0 0 0 0 0 Â1 [0] g [1] Ã1 [0] ũ1 g [1] 0 0 0 0 0 0 0 1 0 0 0 0 Â2 [0] Ã2 [0]
P S = P ~ m ! Cj = ~ m P ~ 1 ! ~ 2 Cj = ~ 1 P ! ~ 1 : (4.14)
To nd periodic motion, standard root-nding algorithms can be employed. The linearization of Cj = ~ is then useful. We obtain it simply by removing the last row and last column in (4.9). It is worth noting that the determinant is the same for the restricted mapping, thus the Poincaré section volume change due to impact is also given by (4.10).

(b) A general one-degree-of-freedom system

We now consider a one-degree-of-freedom system, and we calculate an expansion of C to order two. We use (4.4) with n = 1, where for simplicity we write q = q 1 , u = u 1 and A = A 1 . The special structure of the system can be used to nd shortcuts in the calculations. Using Q, U and for the components of the ®ow, we make the trial solution

Q = q + t(u + 1 2 t (x; t)); (4.15)
where we use ~for expansions at (x; 0), and similarly we write ^to stress that x is the point of expansion. By di¬erentiating Q with respect to time we obtain U :

U = @Q @t = u + t (x; t) + 1 2 t @ (x; t) @t : (4.16)
The unknown coe¯cients in can be determined by di¬erentiating U with respect to time and inserting all expressions in an expansion of A. We obtain

+ 2t @ @t + 1 2 t 2 @ 2 @t 2 = A(Q; U; ): (4.17)
Carrying out the rst step, we wish to obtain U to order two, hence we need ~to order one: ~= ~[0;0;0;0] + ~[1;0;0;0] (q q) + ~[0;1;0;0] (u ũ) + ~[0;0;1;0] ( ~) + ~[0;0;0;1] t + O(x x; t) 2 : (4.18)

Expanding A to order one and using (4.17 To nd t C we note that since t C = 0 when q = q, it is natural to seek t C of the form t C (x) = (q q) (x);

(4.20)

where is an expansion in x x. The unknown coe¯cients in are found by inserting t = t C into Q(x; t) = q, from which we obtain 1 + (u + (q q) 1 2 ) = 0:

u b = u 1 ũ Ã[0;0;0] (q q) + 1 2ũ Ã[1;0;0] + 1 ũ Ã[0;0;1] + 1 ũ Ã[0;0;0] Ã[0;1;0] 1 ũ2 Ã2 [0;0;0] (q q) 2 + 1 ũ Ã[0;1;0] + 1 ũ Ã[0;0;0] (q q)(u ũ) 1 ũ Ã[0;0;1] (q q)( ~) + O(x x) 3 : (4.23)
The expansion of the impact law is

u a = û + g [1] (u b ũ) + 1 2 g [2] (u b ũ) 2 + O(u b ũ) 3 : (4.24)
To get the expansion of the ®ow close to x we just change the tilde to a circum®ex. Inserting t = t C and the initial conditions, we nd that the position component q C of C(x) is q C = q + û ũ (q q) + 1 2ũ 2 Â[0;0;0] + û ũ 2g [1] Ã[0;0;0] (q q) 2 + 1 ũ g [1] û ũ (q q)(u ũ) + O(x x) 3 ; (4.25) and the velocity component u C is

u C = û+ 1 ũ f Â[0;0;0] g [1] Ã[0;0;0] g(q q)+g [1] (u ũ) + 1 2ũ û ũ Â[1;0;0] g [1] Ã[1;0;0] 1 2ũ 2 ( Â[0;0;1] g [1] Ã[0;0;1] ) + 1 2ũ 3 Ã[0;0;0] ( Â[0;0;0] g [1] Ã[0;0;0] ) + 1 2ũ 2 ( Â[0;0;0] Â[0;1;0] 2g [1] Â[0;1;0] Ã[0;0;0] +g [1] Ã[0;0;0] Ã[0;1;0] )+ g [2] 2ũ 2 Ã2 [0;0;0] (q q) 2 + g [1] ũ ( Â[0;1;0] Ã[0;1;0] ) 1 ũ2 ( Â[0;0;0] g [1] Ã[0;0;0] ) g [2]
ũ Ã[0;0;0] (q q)(u ũ)

+ 1 ũ f Â[0;0;1] g [1] Ã[0;0;1] g(q q)( ~)+ 1 2 g [2] (u ũ) 2 +O(x x) 3 : (4.26)
The phase-angle component C is simply where z is the Poincaré section coordinate and P ~ ! ~ is a linear mapping. This viewpoint can be practical both for (numerically) nding periodic orbits and for studying the e¬ects of the nonlinear terms. As an example we take the one-degree-of-freedom system with the acceleration given by A = ! 2 q 2 !u + af(! 2 1) cos( ) 2 ! sin( )g; (4.29)

and with the impact law

u a = ru b : (4.30)
Here the forcing is chosen so that if impacts are disregarded the motion will settle on q = a cos( ), u = a sin( ). In the following a is varied while the parameters are kept Since we work with the restriction Cj = ~ , the point of expansion is the triplet q, ũ, a c . We use the notation C [i;j;k] for the coe¯cients in the expansion of Cj = ~ . Hence C [1;1;1] is the coe¯cient in front of (q q)(u ũ)(a a c ).

We can obtain the coe¯cients C [i;j;k] by applying the result of the previous section. The Jacobian of (the restriction of ) I is the identity matrix and T is the period of interest.

C is L C , L C = r 0 (1 + r) ũ r ; ( 4
Choosing ! = 1=4:9, = 0:6, r = 1 and q c = 1, we can nd a saddle-node bifurcation involving 4-periodic motion with one impact when a = a c = 0:915 66. The impact velocity is ũ = 0:368 15 and the impact phase is ~= 2:727 95. To approximately nd the bifurcated motion we calculate the centre manifold and the reduced dynamics. Using À as a coordinate, we write an expansion for the centre manifold as

z = z + 1 À + M [0;1] (a a c ) + M [2;0] 1 2 À 2 + ; (4.42)
where 1 is the eigenvector corresponding to the eigenvalue

1 = 1 of L = L T L C .
There is an element of choice involved in normal form calculations. We have here taken where 1 is the left eigenvector of L with eigenvalue 1 , to make the expansion unique. A comparison of the approximate solution with a solution found numerically can be done in two ways. This is because a numerical routine for nding periodic orbits typically assumes that q > q c . Thus, either we nd the bifurcated solution numerically at a convenient phase and move it to the section ~, or else we move the approximate solution to the section used for the numerical solution. The latter approach is taken in gure 3, where a bifurcation diagram at = 0 is shown. The eventual fate of the branches is thoroughly investigated in [START_REF] Nordmark | Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators[END_REF]. Both the stable and the unstable branch are destroyed in grazing bifurcations as a is increased (the stable when a = 0:975 55, the unstable when a = 1).

Results and discussion

By using the discontinuity mapping, we have shown how to obtain series expansions of mappings in impact oscillators with a unilateral displacement limiter. This is helpful in analysing bifurcations. Some extensions of this work are straightforward. We can change the assumption that the impact law can be written explicitly, by allowing an implicit relation G(x a ; x b ) as long as it can be solved locally. Systems where only one impact can occur at a given time, but with many di¬erent contact possibilities, can be brought into the present framework by applying the di¬erent discontinuity mappings several times, once for each contact. Other aspects are more subtle and call for future studies. Models where spatially separated parts of one or several bodies can make contact simultaneously have not received much attention. In such models two contact conditions can be ful lled at the same time. One scenario that might be possible is that a parameter change will change the order in which the impact events occur for a periodic motion. Since linear stability is given by multiplying the Jacobians of the di¬erent mappings, which, when composed, gives the Poincaré mapping; this corresponds to switching order between the Jacobians of two discontinuity mappings. Since matrix multiplication in general does not commute, this might lead to non-di¬erentiable dynamics exactly when the order is switched.

Figure 2 .

 2 Figure 2. The discontinuity mapping. Flowlines are dashed when H < 0.

  3.7) for all x; t 2 X R. (ii) The next step is to nd the time t C (x) as an expansion in y. The unknown coe¯cients in t C are calculated by inserting the expression for the ®ow into an expansion of H: H( ~(x; t C )) = 0: (3.8) (iii) We can now calculate the point of impact, x b : x b = ~(x; t C (x)): (3.9) (iv) We now direct our attention towards the impact law G. Using the notation Gx for the Jacobian of G at x, we write the expansion as G(x) = G(x) + Gx y + O(y) 2 ;

)

  By inserting this into (3.8) we get Hx y + Hx F tCx y + O(y) 2 = 0the identity matrix. Using the impact law we nd x a = G(x) + Gx I F Hx Hx F y + O(y) 2 ; (3.21) and by substituting x a and t = t C into ^we have C(x) = G(x) + Gx I

  local state-space volume change caused by an impact is given by the determinant of L. Since L is triangular we immediately havedet L = g [1] û1 ũ1 : (4.10)This is independent of the dimension of the system. For the much-used model where the impact law is modelled by a coe¯cient of restitution r,u 1a = ru 1 b ; Poincar ¶ e mappingsFor periodically forced systems without impacts, the canonical candidate for a Poincaré section is := fx 2 X : = • g; (4.13)

  t C into U gives u b :

  A saddle-node bifurcationIn the case where the equations of motion between impacts are linear, all nonlinearities are in C. Assuming that we study an orbit with one impact, we can use the section with = ~to write the Poincaré mapping as P S (z) = P ~ ! ~ Cj = ~ (z);(4.28)

C

  [0;2;0] = C [0;0;2] = C [0;1;1] = 0 0 ; (4.36) where = a c f(! 2 1) cos( ~) 2 ! sin( ~)g ! 2 q (4.37) and = ( 4 ! ũ) + a c ũf(! 2 1) sin( ~) + 2 ! cos( ~)g: (4.38) The mapping P ~ ! ~ is P ~ ! ~ (z; a) = P ~ ! ~ (ẑ; a c ) + L T (z ẑ) + (I L T )

Figure 3 .

 3 Figure 3. Bifurcation diagram when = 0. The unstable branch is dashed.

M [0;1] = 0; 1 M [2;0] = 0;(4.43)

We thank Dr Hanno Ess ¶ en for comments. This work was supported by the Swedish Research Council for Engineering Sciences (TFR).


by proceeding along the lines of the analysis for similar single-degree-of-freedom oscillators [13,[START_REF] Natsiavas | Vibration of harmonically excited oscillators with asymmetric constraints[END_REF]. This type of analysis exploits the linearity of the equations of motion within each time interval where the system possesses constant characteristics. As expected [10], this extension is not a trivial task. A key feature that permits this generalization is the derivation of a suitable expression for the form of the solution of the linear problem.

The basic steps of the present analysis are briefly outlined in the following three sections. First, a suitable solution form is obtained for a general discrete linear system under harmonic excitation. This solution form then presents the basis for determining exact periodic motions of the non-linear system, as well as their stability. The analysis developed can be applied in a number of different areas. Here, the validity and effectiveness of the method is tested by applying it to two classes of important engineering problems. In this way, useful information is also obtained for the dynamics of the examples considered, which are vibration absorbers with elastic stops and other oscillators under internal resonance. The results of the numerical study reveal that some parameter combinations lead to a periodic or quasi-periodic response, while others lead to chaotically modulated motions. Comparison with the response of oscillators with continuous non-linearities is also made.

MECHANICAL MODEL AND SOLUTION FORM

The system examined is modeled as a general linear multiple-degree-of-freedom oscillator, including a component carrying elastic stops with damping and restoring forces, expressed by x>xs.

(

In equation (1), x is the relative displacement of the oscillator carrying the stops with respect to the component of the system these stops collide with. An example of such a system with c =eN, c' =eN+ Cs, k = kN and k' = kN+ ks is presented in Figure 1. In general, the position of the stops within the overall system can be arbitrary. The components of the system are subjected to forcing: /,(t) = F; cos (wt + l/J; + ljJ ),

where 4> is an unknown phase angle introduced for the purpose of facilitating the search of periodic solutions. Since the function F(x~ x) is piecewise linear, the exact form of the analytical solution of the system can be obtained for x ~ Xs and x < Xs. Before doing this, the displacements are normalized by a characteristic length, Xc, i.e., y 1 = xtf Xc. This length can be the gap length Xs, when Xs is non-zero. In addition, a normalized time is chosen according tor= rot. Then, the governing equations in the time intervals where the stiffness and damping remain constant can be cast in the linear form

My+ Cy+ Ky=f('r) + k, (2)
where M, C and K are the inertia, damping and stiffness matrices, respectively, while k is a vector including the constant terms of equation (1). Therefore, the general solution in the interval considered is

y=yh+Yp•

The homogeneous part, Yh• is expressed in terms of the eigenvalues s, = p 11 + iu 11 and the corresponding eigenvectors fn =ex,+ ip 11 , determined by

n= 1, ... , N,
where N is the total number of degrees of freedom. Note that the solution of the above eigenvalue problem is equivalent to that obtained by converting equation ( 2) to the classical first order 2N-space form. For oscillatory systems, the s,'s and y,'s appear in complex conjugate pairs. Making use of this, Yh is expressed in terms of real quantities as

N Yh( r) = L A,( r)c," n "" l
where the N x 2 matrix A, is defined by

A,( r) =e'"T[B, cos (G,r) + C, sin (G,r)],
while B, = [a, p,], C, = [ -p, a,] and the constants c, are determined by satisfying initial conditions.

Next, the forcing vector f( r) is split in the form : f c cos ( r + </>) + f s sin ( r + </> ), and the particular solution of equation ( 2) is determined in the form from

[ K -M C ](Yc )=( fc ) , C M-K Y s -fs Kp=k.
Then, the particular equation is rewritten in the form yp(r)=g(r) cos </>+h(r) sin t/J+p.

Introducing the vector z=

[y y]\ (3)
and combining all the above, the solution is finally expressed in the form

z( r) = tl>( r)y+ Jl( r) cos 4> + v( r) sin 4> + l;, (4)
where ~=(:).

PERIODIC STEADY STATE VIBRATIONS

Based on the solution form (4), a methodology is presented in this section, which leads to exact, n-periodic solutions of the system examined in a systematic and efficient way. These solutions are characterized by a single change of the damping and stiffness parameters within a response cycle. Since the system is non-linear, other vibrational modes are also possible. However, the motions considered are expected to be the most dominant. Then, the form of the solution sought is dictated by the form of u = y N, which is depicted in Figure 2 : namely, the solution is zl ( rl) in 0 ~ rl = r ~ rr and Zz( r z) in 0 ~ r 2 = r-rr ~ r!' where rr is the first contact time and r! = 2nn -rr . The determination of the desired solutions is based on the fact that they satisfy the following set of periodicity and matching conditions :

(5) (6) The above represent a set of 4N + 2 transcendental equations for the 4N + 2 unknowns of the problem (i.e., y 1 , y 2 , t ~ and <P ). However, generalizing a solution procedure developed for similar but single-degree-of-freedom oscillators [ 13,[START_REF] Natsiavas | Vibration of harmonically excited oscillators with asymmetric constraints[END_REF], the task of solving these equations can eventually be reduced to the solution of a single transcendental equation, as explained next.

First, application of equation ( 5) leads to [START_REF] Shaw | [END_REF] where w = [yl y 2 ]T. Note that for the solution form of Figure 2, ~1 =0. Next, for a given value of rr ' solution of equation [START_REF] Shaw | [END_REF] provides the vector w in the form w=wc cos q,+ws sin t/J+wo. [START_REF] Shaw | [END_REF] Then, application of equation ( 6) leads to two algebraic equations of the form bs sin q,+bc cos <P=bo.

(9, 10)

Solving equations ( 9) and (10) for sin lfJ and cos t/J and applying a trigonometric identity, the correct value of -rf will give rise to (11) The strategy leading to a complete numerical solution of the problem is based on the observation that all the matrices and vectors of equation [START_REF] Shaw | [END_REF], as well as the a's and b's in equations ( 9) and (10), are functions of the system parameters and -rf only. Moreover, 0 < -rt < 2Hn. Then, the solution procedure starts by assigning values to -rt within the range from 0 to 21rn. This permits the determination of w from equation [START_REF] Shaw | [END_REF] and the evaluation of the corresponding value of G from equation ( 11 ), independently of the value of t/J. In doing so, the bisection method is applied in locating a possible solution for rr, when a sign change of G is detected between subsequent trial values of rf. Determination of rf allows the calculation of t/J from equations ( 9) and (10), the evaluation of the w from equation ( 8) and therefore the solution histories of Zt and z2 from equation (4).

The above methodology can be modified and applied to other important cases, including systems with symmetric characteristics.

STABILITY OF PERIODIC RESPONSE

The information about the stability properties of a steady state motion is important because only stable solutions are realizable in practice. Moreover, the way in which a solution loses stability presents useful information in searching for other possible motions that may coexist. For piecewise linear systems, the classical methodologies are not applicable in performing the stability analysis of located periodic solutions. Here, a method employed before for other similar oscillators with one and two degrees of freedom will be generalized and applied.

Following reference [12], the essential information about the stability of a solution of a piecewise linear oscillator is obtained by just analyzing the propagation of arbitrary perturbations of the starting conditions within a single interval of two successive contact times. This actually implies that the steps of the analysis that follows are general and can be applied to solutions of piecewise linear systems of arbitrary form.

The stability procedure starts by letting .1 r 1 = .1 rt-.1 r3 , where .1 denotes a small perturbation of the variable it is applied on. Next, consider an exact periodic solution z 1 and a perturbed solution z 1 , obtained by perturbing the initial conditions of z 1 slightly (see Figure 2). Then, (12) where the vector Liz1 represents the deviation of Zt from Zt at time rt. Note that the corresponding variation of u1 is zero and, therefore, its position in .1z 1 is filled by the time variation .1 -rf instead. Application of equation ( 12) yields 4'.( rr + .1 r,)yt(Zo+ LiZo) + Jl( rr + .1 rt) cos 4> + v( rr + .1 tf} sin l/J = z.( rf) +Liz., (13) where the vector Li.zo involves the deviation from the correct initial conditions, Zo=z.(O), leading to z 1 • These initial conditions are determined by letting r 1 =0 in equation ( 4), once a periodic motion has been located by the analysis of the previous section. Next, expanding the left side of the last equation and keeping up to first order terms leads to tPt( rt)r.Lizo+ dJ.( rf)y.L1 '• +cit( rr)Li -rf =Liz., where q 1 ( r 1 ) = p., ( r 1 ) cos <P + v, ( r 1 ) sin <j), and the matrix r 1 is formed by adding the vector cl> ! 1 (0) q 1 (0) in the last column of the matrix cl> j" 1 (0). The zeroth order terms drop out because they satisfy equation (5). Then, the last equation can be put in the form with Q, = Rlt 1 R10. The matrix R 11 is formed by adding the vector -[ <i > 1 ( rf)y 1 +tit( rt)] in the last column of the identity matrix of order 2N x 2N, while R 10 is obtained by subtracting the vector <h1( rf)y, from the last column of the matrix cl>,( rt)r, .

Proceeding in a similar fashion in the interval 0 < r 2 < r~ yields where, again, the matrix Q 2 is found in terms of the system parameters only. Then, the relation between the perturbations at the beginning and at the end of the first response period is expressed by [START_REF] Natsiavas | Vibration of harmonically excited oscillators with asymmetric constraints[END_REF] where n = Q 2 Q 1 • Finally, after m response cycles, the perturbations from the correct initial conditions will be related to first order with the original perturbations by Therefore, if A. is the eigenvalue of ll with the largest modulus, the effect of the original error will vanish with time, provided that Ill < I. In such cases, the periodic solution examined is asymptotically stable. When IA.I > 1, the solution is unstable. Finally, when

IA-1 = 1, bifurcations occur, resulting in qualitative changes of the system dynamics [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields[END_REF].

According to the results presented in the next section, three types of bifurcations were encountered in this study. The first type is a saddle-node bifurcation: it occurs with A.= 1 and results in the classical jump phenomena. The second type happens with ).. = -I and is associated with period doubling of the response. The third type is the so-called Hopf bifurcation, it occurs when a pair of complex eigenvalues has modulus one and results in an amplitude modulated response of the system. Note that if the damping coefficients are non-negative, the Hopf bifurcation is excluded for single degree of freedom oscillators [START_REF] Shaw | [END_REF]12], but it occurs for systems with N';?!::2 .

APPLICATIONS

This section verifies the analysis developed by presenting results for systems with two degrees of freedom (the system shown in Figure 1 with N = 2). The mass m 1 is subjected to a forcing F 1 cos r , while where

The numerical results obtained for this system are presented in the following two subsections. In all cases considered, y 0 = -1. For other systems, the only difference in the analysis will be in the form of M, C, K, f and k.

VIBRATION ABSORBERS WITH ELASTIC STOPS

The idea of using stops in vibration absorption was presented in earlier studies [START_REF] 0rmondroyd | [END_REF]1]. The main interest there was to limit excessive amplitudes of vibration of the auxiliary mass of the absorber. Here, the analysis developed is applied in order to show some further enhancements and some limitations in the performance of the absorber with stops.

First, in Figure 3 is presented the response diagram of the main mass for three systems with parameters originally in pretension and loses partial contact at y 0 and, as a consequence, presents softening characteristics, while the second non-linear oscillator possesses hardening characteristics. The effectiveness of the stops is obvious, when comparing the response diagram of the linear absorber (thin line) with the softening/ hardening absorber in the lower/ higher forcing frequency ranges, respectively. These benefits were expected from previous work on absorbers with continuous non-linearities (e.g., [17][18][19]). In fact, the present setup provides a practically feasible way of designing non-linear absorbers. However, in analogy to the non-linear absorber with continuous characteristics, the non-linearities also introduce some potential dangers, as explained next.

J.l = 0• 1, f3 = 0•1, p = 1, s 1 = 0•
In Figure 3, the unstable branch of the softening oscillator is generated with ).. = 1 bifurcations, associated with jump phenomena of the response at m = 0• 789 and 0• 806, and coexists with two other stable periodic solutions. Similar bifurcations are captured for the hardening system at m = I• 244 and 1• 259. In addition, the latter system exhibits period doubling (l= -1) bifurcations (at 0•805 and 0•816) as well as Hopf bifurcations (at 1•034 and 1•061).

Response diagrams obtained by raising the non-linearity level are presented in Figure 4. This is done by increasing p' to 1• 732 and 4, which corresponds to stops with twice and 15 times the original stiffness of the absorber (approaching conditions of impact dampers), respectively, while maintaining 'i=0•02 and the values of the other parameters as before.

All the bifurcations of the lower resonance branches occur with ).. = -1, while ).. = I gives rise to the unstable motions of the higher resonance branches. Branch C is common to both diagrams. From the last plot, the most important observation is that increasing the non-linearity level causes a significant expansion of the zone of the unstable solutions originated by Hopf bifurcations near the original resonance ro = I. This is in accordance with results for absorbers with continuous non-linearities [ 18,19] and gives rise to rich but dangerous dynamics within that frequency range. For example, in Figure 5

0•6 0•4 0•2 0•0 -0•2 -0•4 0•8 (c) 0•6 0•4 r- . ~ 0•2 0•0 -0•2 -0•4 I 0•8 (e) 0•6 0•4 0•2 .:/ 0•0 f- ' •. '~ \ -0•2 .. : -0•4 -0•2 -0•1 ! ! •. ,..... ,, . ......_.., .. ~ ' .. ---•} -• .. -• •~ •-' - - 1-(b) T (d) ~ - ! ~ (f) •:' (:• --~- -• ••-. -•• . •:-. . . . ~~~L>:• . . -. 0•0 0•1 0•2 -0•2 -0•1 Y1 '• •'
...: .. , according to a scenario explained in reference [START_REF] Aronson | Bifurcations from an invariant circle for two parameter families of maps of the plane: a computer assisted study[END_REF]. Then, following a single torus doubling (Figure 5(e)), the system settles into chaotic response (Figure 5(f)) without any further observable torus doubling (see references [START_REF] Arneodo | [END_REF] and [22] for similar results and explanations).

To examine the effect of these motions on the performance of the absorber, in Figure 6 cycles. Clearly, the response involves beats, which is in agreement with previously reported experimental results [2,3]. In Figure 6 shows that caution should be exercised in choosing the absorber parameters properly. The present analysis can then be used as a guide in adjusting the parameters and eliminating these unstable responses.

OSCILLATORS WITH INTERNAL RESONANCE

The systems examined above have "equivalent" linearized counterparts with close natural frequencies. In this section, it is shown that some of the phenomena observed for the non-linear absorber can also occur in other oscillatory systems, when their parameters are chosen in ways that lead to internal resonance conditions [23][24][25][26]. A number of some new and interesting features also appear and are analyzed.

First, in Figure 7 is presented the response diagram for a system with parameters p = fJ = 0• 5, p = 0•9, p' = 0•65, '• = 0•01 and ' 2 = ' 2 = 0•02. This combination of parameters leads to 2: 1 internal resonance conditions. First, note that there exist two resonance peaks around ro = 0• 65 with very close frequencies, while the peak of the resonance branch around 1• 3 consists of unstable solutions, generated by A.= -1 bifurcations at 1• 203 and 1• 348. As expected, these bifurcations give rise to a periodic response with a period equal two times the forcing period. To illustrate this, n = 2 solutions arising in that frequency range are also determined and shown. In addition, there is a branch of n = 1 unstable solutions in the range from ro =0•963 to 0•990, originated by a Hopf bifurcation. Again, this branch is located around the middle of the primary resonance frequencies. However, there is a qualitative difference between the response of the present system and that of the oscillators analyzed in the previous section: namely, varying the last set of parameters may result in unstable ranges near the middle of the primary resonances, generated by period doubling and not by Hopf bifurcations. For other sets of parameters, several unstable intervals may appear in the same range, some generated by period doubling and the others generated by Hopf bifurcations. For example, decreasing the damping ratios of their amplitude is larger (by more than an order of magnitude near resonance) than the amplitude of the coexisting n = 1 motions. Moreover, the n = 2 solutions undergo saddlenode bifurcations, giving rise to a very narrow frequency range in which two stable and one unstable n = 2 solutions coexist. Finally. note the multiple peaks for both then= 1 and n = 2 branches. As for the 2: 1 case, changing the parameters may result in multiple frequency ranges involving unstable response generated by either A.= -1 or Hopf bifurcations. For In fact, in some cases the bifurcation takes place with a pair of complex conjugate eigenvalues equal to one, while a real eigenvalue has a value very close to -1, indicating conditions close to codimension two bifurcations [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields[END_REF]. To present a small but representative sample of some of the interesting dynamics encountered in that range, in Figure 10 is shown a sequence of Poincare plots (y 2 ,j 2 ) obtained by direct integration for the parameters of the last example. The behavior encountered in this section presents similarities and differences with the behavior of systems with asymmetric but continuous non-linearities [24]. Here, in addition to the unstable zones generated by the Hopf bifurcation between the primary resonant frequencies, there may exist zones that originated with A,= -1 bifurcations. This last type of bifurcation does not have an analog in the analysis of the steady state solutions captured in reference [24]. The reason is that the stability of those solutions is governed by linearized differential equations (flows), while here it is governed by difference equations (maps).

SYNOPSIS AND CONCLUSIONS

An investigation is carried out on the long-time response of a class of multiple-degreeof-freedom oscillators with strongly non-linear characteristics. The non-linearity of the system is modeled as a component with piecewise linear damping and stiffness properties. The solution procedure exploits the piecewise linear nature of the governing equations. Suitable solution forms are first developed for the response within the intervals where the system characteristics remain constant. Then, exact, n-periodic, single crossing steady state motions are determined by imposing a set of periodicity conditions. For an N-degree-offreedom system, this results in a system of 4N + 2 transcendental equations. However, generalizing previous work on similar single-degree-of-freedom systems, the numerical effort is reduced to finding the solution of sets of linear equations and eventually to a single transcendental equation for the crossing time. Finally, the analytical procedure is complemented by a stability analysis which is appropriate for solutions of systems with arbitrary piecewise linear characteristics. As in the case of single-degree-of-freedom oscillators, this task is reduced to examining the propagation of errors in starting conditions within a single linear motion interval.

The present analysis provides efficient means for performing parametric studies in a large number of systems encountered in engineering practice. Here, the applicability and effectiveness of this analysis is demonstrated by examining the response of vibration absorbers with elastic stops, as well as the response of other non-linear oscillators with parameters leading to 2:1 and 3:1 internal resonance conditions. At the same time, the results illustrate some interesting features of the dynamics of these systems: namely, the absorber with stops is shown to possess superior technical characteristics. It provides a way in which to design non-linear absorbers and to suppress vibration levels over broader forcing frequency ranges than the conventional absorber, on both sides of the original resonant frequency. However, stability considerations in conjunction with numerical integration reveal that high amplitude beating and chaotic response may arise near the original resonance, because the coexisting periodic response may become unstable due to Hopf bifurcations. Next, similar phenomena are observed for other piecewise linear oscillators exhibiting 2:1 and 3:1 internal resonances. In these cases, zones of unstable periodic solutions are also observed in the vicinity of the middle of the primary resonances of the system. The resulting motions are again of much higher amplitude than that of the coexist• ing unstable n = 1 response. Several unstable zones may exist, generated by either Hopf or period doubling bifurcations. It is shown that the Hopf bifurcation of subharmonic solutions results in multiple invariant circles. The coexistence of harmonic with large-period periodic and quasi-periodic responses is also presented. In all cases, similarities and differences in the response of systems possessing continuous non-linearities are noted.

  x)= c'x+k'x+(k-k')xs.

~+Figure 1 .

 1 Figure 1. A mechanical model of the system.

  u

Figure 2 .

 2 Figure 2. The solution form of u = y N.

  F 2 = 0. For convenience, introduce the additional dimensionless parameters n.= ,.,,~ ., ""'' , with similar relationships holding for the primed variables (see equation (1) for their definition) and Then, the inertia, damping and stiffness matrices and the forcing vectors are expressed by c=[8o, o J J.l82 ' K=[~' ~~J f=(~)'ll cos~.

Figure 3 .

 3 Figure 3. The response diagram of the main mass: p' -I, 0• 707 and 1• 414.

  are shown samples of the Poincare sections of y , vs. y 2 , for p ' =4. These plots were obtained by direct integration of the equations of motion at several values of m and by sampling the longtime response once every forcing cycle.

Figure 4 .

 4 Figure 4. The response diagram of the main mass : p' = 1• 732 and 4.

Figure 5 .

 5 Figure 5. Poincare sections: {a) ro= 1•0180; (b) ro= 1•0220; (c) ro= 1•0225; (d) ro= 1•0230; (e) m= 1•0233; (f) ciH •0250.

1 - 1 • 5 .Figure 6 .

 1156 Figure 6. (a) The response history at w = 1•018; (b) projection onto the (j 1 ,j 2 ) phase plane (w = 1•018); (c) response history at ro = 1•025.

  Figure 5(f) and is chaotically modulated. The amplitude of the corresponding unstable periodic solutions at ro = 1•018 and 1•025 is only 0•052 and 0•058, respectively. Comparison

Figure 7 .

 7 Figure 7. The response diagram of m 1 (2:1 internal resonance).

Figure 8 .

 8 Figure 8. The response diagram of m 1 (3 : I internal resonance).

2 wFigure 9 .

 29 Figure 9. The response diagram of m 1 (3 :1 internal resonance, p = 0•9).

1 Figure 10 .

 110 Figure 10. Poincare sections: (a) o1 = 1•130 and 1•129; (b) w = 1•127; (c) di = 1•127.

  at the same frequency. The interesting feature of this solution is that it consists of nine invariant circles, possibly originated by a Hopf bifurcation of an n = 9 solution. When m is varied, each of these invariant circles undergoes changes similar to those of the single circle of Figure5(a).


the problem of stabilizing steady motions.

1.

2.

3. 4. 5. 6.

7.

REFERENCES RUMYANTSEV V.V., On the control and stabilization of systems with cyclic coordinates. PMM, Vo1.36, No.6, 1972. LILOV L.K., On the stabilization of the steady-state motions of mechanical systems with respect to part of the variables. PMM, Vo1.36, No.6, 1972. SAMSONOV V.A., On the feasibility of stabilizing steady motions of systems with pseudocyclic coordinates. PMM, Vo1.45, No.3, 1981. MERKIN D.R., Introduction to the Theory of Motion Stability. Moscow, Nauka, 1976. RUMYANTSEV V.V., On the stability of a gyroscope in gimbals. PMM, Vo1.22, No.3, 1958. KONOSEVICH B-I., On the stability of the steady motions of an asynchronous gyroscope in gimbals. In: Mechanics of Solid. Iss. 9. Kiev, Naukova Dun&a, i977. __ WARLAMOV H.P., The phase topology of a problem of gyroscope motion. In: Mechanics of Solids, Iss. 13, Kiev. Naukova Dumka, 1981.

PMM U.S.S.R., Vo1.49,No.2,pp.153-159,1985 0021-8928/85 $1O.OO+O.oo Printed in Great Britain Pergamon Journals Ltd.

Translated by J.J.D.

NORMAL OSCILLATIONS OF A STRING WITH CONCENTRATED MASSES ON NON-LINEARLY ELASTIC SUPPORTS*

E.G. VEDENOVA, L.I. MANEVICH, and V.N. PILIPCHUK

The problem of constructing and classifying normal oscillations of a string with concentrated masses on non-linearly elastic supports is considered (special limit cases are linear and vibro-impact systems). It is shown that in the limit of intensive impact action, the non-linear system has supplementary properties of synnnetry which enables this problem to be solved effectively. On the other hand, the normal oscillations of a vibro-impact system can be used as the generating solutions for dynamic calculations of essentially non-linear systems that are close to them. The connection between localized normal oscillations and solutions of the soliton type are discussed.

The existence of normal oscillations as special particular SolUtiOnS of linear conservative system is due to the properties of symmetry inherent in it that can be partly retained in the non-linear case /l, 2/. The possibility of constructing in some strongly non-linear system synchronous motions that have a number of properties of normal linear oscillations /3, 4/ is related to this property. The normal "principal" oscillations have already been considered by Lyapunov /5/. Due to these properties a multidimensional non-linear system in the normal oscillation mode behaves as a non-linear oscillator.

1. Consider the transverse oscillations of a regular chain of masses connected to each other by a weightless string, and interacting with non-linearly elastic supports (Fig. 1) which in the limit become rigid limiters with some gap 2e. The reaction of the j-th support is qj = a (a,,'@-'

where Uj is the deflection of the string in the respective cross section, n is an integer, and a is the stiffness parameter. Among the various type of motions of non-linear systems are those of the simplest mode, such as normal oscillations. When n = 1 (a string with concentrated masses on linearly elastic supports) we have A' normal oscillations; the spectrum of respective natural frequencies is discrete and limited.

When n> i we distinguish the cases of strong and weak connection along the string. In the first case the non-linearity is small and the system belongs to the Lyapunov class of systems /5/. We thus have the problem of constructing non-linear normal oscillations which, as the amplitude decreases,becomenormal oscillations of a normalized system. This simple case is not considered further.

l Prikl. Matem.Mekhan.,49,2,203-211,1985 1 Fig. 1 In the second case, if we neglect the connection along the chain, we obtain a system of non-interacting homogeneous oscillators (the potential energy of each oscillator is a homogeneous function of the displacement). Exact solutions of that system are, however, difficult to use as the first approximation in any scheme of the theory of perturbation, since they are expressed in terms of special functions. At the same time, the properties of the oscillation process, including also the mode of normal oscillations, essentially depends now on the index n.

When II -1 (e.g. n = 2. 3) normal oscillations are still close to harmonic (in time) and, consequently they can be defined in terms of the quasiharmonic approximation in the standard scheme ofthemethod of averaging. If, however, n>> 1 the system behaviour proves to be closer to the other limit case (??+ X) that corresponds to the characteristic of the elastic medium shown in Fig. 2, i.e. to an impact interaction. The investigation of normal oscillations of systems of the class of the situation consideredinthese circumstances which are the least favourable from the point of view of the quasiharmonic approach, is the basic content of this paper.

2.

Let us first consider a string with two unit concentrated masses in the limit case of impact interaction, when the non-linearly elastic supports become rigid limiters. In that case the equations of motion are

(2.1)
where g is the stress impact interaction which cuts-in when iu,( =e (i= 1, 2).S is the tension of spring, and I is the distance between the supports (masses).

When ;' = cl (there is no connection) both masses move independently of each other, undergoing only absolutely elastic impact against the elastic supports. The respective coordinates u;(,i = 1. ?) are determined as follcws: (3.31

The system (2.X), unlike (2.1) does not contain functions with discontinuities of the second kind.

Relations (2.3) may be considered as the equations of motion of unit mass in the (q,.~~) plane in a periodic potential field of force: when y = 0 we have inertial motion.

A set of equipotential lines is shown in Fig. 3, where the trajectory in the plane (yl. (is) is mapped, using (2.2) into the configurational space of the initial system (ul. up). That image "turns" the plane (q,. (I?) along the lines into a square ii: {-n 2 ( vi .< .1 3, i = 1, 2) with subsequent stretching in the directions YI. 'i? e times. For fairly small energies the system moves without impacts so that the trajectory 2 Note that the pattern in square K (the equipotential curves of a linear system) has an axis of rotational symmetry of second order, and the gap (Fig. 3) obtained by multiple mirror reflection of that pattern (the motive) relative to the straight lines (2.4) has a fourthorder axis of symmetry. This is related to the substantial difference of the dynamic properties of the linear model (without impact) and modes of motion with impacts against a limiter. Thus, in the first instance we have two normal oscillations, i.e. the natural oscillations of a linear system with two degrees of freedom. With impacts, there are already four (Fig. 3), since besides a symmetric (inphase) and skew-symmetric (antiphase) forms, two supplementary spatially localized forms appear, which at the limit y-+0 become straight lines qI = 0, q2 = 0 and correspond to one of the masses being at rest.

3. The part played by normal oscillations is primarily dependent on the possibility of their use as providers of solutions when equations of motion are asymptotically integrated. Because of this, we shall obtain the approximate (averaged) equations which defines the behaviour of a system about forms enumerated above. When constructing such approximate equations it is advisable to utilize the intrinsic small parameter of the problem that reflects the basic property of the impact mode, i.e. the relatively small magnitude of the coupling parameter e? = 4e2y,(nzv2)

Physically the constraint e?<l is obviously connected with the condition of realizing impacts. The smallness of the parameter e predetermines the availability in the system of "rapid" and "slow" times, which enables us to use the averaging method in this or that modification. Below, in each case we present only shortened equations that correspond to the first approximation. Classification of normal forms by the type of behaviour in the neighbourhoods of these forms is carried out on this basis.

Inphase fom ~~ = 'pl. We set 'pl = T + T1 (rl),cp2 = T -I-q2 (tl)

, where T= ',Tt.L.e-'t is the rapid and t, = ET the slow time, respectively. The rapid time is, naturally, related to the rate of impact.

Carrying out the averaging in (2. where m(E) is a L-c-periodic function defined on the segment i-n, nj by the formula

The trivial solution of the averaged system corresponding to the inphase form is orbitally stable in the "small". In this sense we refer tc the inphase form as of the "stable" type.

Antiphase form 'PI = -92. Let (pl = r + V, (Q, t2 = --e + V2 (t&

In this case the equations relative to i&, iz may be reduced to the form dz (h -M __o. d'(BI 1. TA dt,' dl,z -2~(~~,~*)=o

The trivial solution of the averaged system corresponds to the antiphase form and is orbitally unstable. Hence we regard the antiphase form as being of the "unstable" type.

3

As previously noted, that straight line is a normal form only when y = 0. YfO, the curvilinear form is "on the average" However, when directed along the straight line (Fig. 3).

(p? = 0

Let us assume that the trajeCtOq in the course of motion remains in the band --;r'2 < v, < n/2. Sq.(2.3) then takes the form (2.1) Setting q, = T +-T,(tJ and averaging (3.1! over T. we obtain the equations which indicate orbital stability of the trajectory (p2 = 0 relative to the perturbations This means that a reasonably large energy imparted in the initial instant to one of the oscillators is subsequently maintained by it. The localized form belongs in the sense indicated above to the "stable" type. A similar conclusion also holds for the second localized form. 4. A model with two degrees of freedom reveals an important feature that is inherent in all discrete systems of the class considered here, i.e. the existence of a form of localized oscillations. Other types of elementary solutions that do not appear in linear systems emerge only when X> 3, as will be shown later. In this connection, we will now analyse a discrete model with three degrees of freedom.

In that case the equations that define the phases iri (i = 1,2,3) have the fom where 'F#Jz (Fa30. Introducing the dimensionless parameter of time 'I = ',',xue-lt, we write this system in the form (4.1) Equations (4.1) dimensional space in field form a spatial c(, .< n.2 (i = 1. 2. 3). may be considered as the equation of motion of a unit mass in threea periodic potential field of force. The equipotential surfaces of this pattern obtained by multiple mirror reflection of a cubic element -n,:! < Inside the element the field of force corresponds to the linear problem with three degrees of freedom, and ahs three axes of symmetry of a triaxial ellipsoid. The field in the whcle space of (ql.~?.q3) has 13 axes of symmetry of the cubic lattice, and hence, unlike the linear problem, 13 normal oscillations are possible here and not three. However the relations shown in Fig. 4, comprise all essentially different cases (the remaining forms are easily reestablished on considerations of symmetry). Here 1, = ET; E?= 4ye'i(n%'). It is assumedincasesl, 2, Sthat -rt2<yl<nl!(i=1,2,3). The masses shown in Fig. 4 as fixed, are at rest only in the "mean", executing high-frequency oscillations with small amplitudes.

These results show the existence of localized forms of oscillations of the "stable" type for systems with two degrees of freedom also (solutions 1, 2 and a similar form with localization on the third mass). At the same time a new effect appears in the system with three degrees of freedom: the possibility of cooperative motion of two adjacent masses (of the stable type) when there are connections undeformed on the average (solution 3). As in Sect.3, the in-phase form (solution 6) is of the stable type (denoted in Fig. 4 by the plus sign), and the antiphase form (solution 7) of the unstable type (denoted by the minus sign). Other forms of motion in which displacements of two adjacent masses have different signs (solutions 4, 8), and, also, the form in which the fixed mass (on the average) separates two deformed connections (solution 2) also belong to the unstable type.

These results (Fig. 4) enable us to represent the possible forms of normal oscillations for N>3 also.

4

We can similarly construct the solutions in subsequent approximations, with the amplitude A represented by the expansion A = A, + v+A1 -+ . . . 6. Since the classification of normal forms was carried out using averaged equations, a numerical integration of the initial system with the characteristic gj=Ujn was carried out, corresponding in practice to the impact process with initial conditions specified in the "neighbourhood" of "normal" forms of the "stable" and "unstable" types.

Below, we present the results obtained for a model with two degrees of freedom for p = 0.5; B = 1; e = 1 (Fig. 5).

Fig.5

When the normal form relates to the stable type, with initial conditions somewhat different from the "exact", the trajectory remains close to the form.

This occurs for the inphase (Fig. 5,a) and localized (Fig. 5,c assuming for every t a dependence of the functions iii @) on the index j that is smooth compared with "t(L). After substituting (7.3) into (7.1) and simple transformations, we obtain d*Cj dt2 7 y (lij+l + L'j_1 -2Gj) + 4yL;j $ aUi"-' = 0 (5.4) j =O, *I, *2, . . .

By virtue of the smooth dependence on j, the set of functions UI (1) can be approximated by one function of two variables (V (2. t) and the finite difference operator in (7.4) by the second partial derivative with respect to the space coordinate. We then have a single differential equation in partial derivatives with respect to L: (r, 1) CPC':ai + y12aw/ax= + 4yU + aU2"-1 = 0 (i-5) which corresponds to the use of continuous approximation for System (7.4). This non-linear equation has a localized solution of the soliton type with the provisional frequency w, that exceeds the maximum natural oscillations of system (7.1) after linearization %a1 = zl/u_ If e*= C&P-ok,, < 1, then in the first order with respect to e the soliton Solution may be expressed in explicit form. For instance, when n = 2 /7-g/ 6 (7.6) so that Uj = (-i)'U(Zj* t).

(7.i)

Solutions of the soliton type may be similarly obtained when n > 2.

In the region of applicability of the continuous approximation (7.5), when a<l and the function Vk,t) is in fact smoothly varying along the coordinate z, its localization is weakly expressed and the amplitude can be arbitrarily small. The linear and non-linear terms of system (7.5) are of the same order in E.

As E increases the amplitude increases, and the tendency to a pronounced localization becomes apparent. A soliton-type solution is virtually indistinguishable from a localized normal form. The continuous approximation and, consequently, the analytic representation (7.6) and (7.7) is no longer applicable here. In that case, the analysis carried out above in terms of normal oscillations is the most reasonable.

The analytic formulae of the form (7.6) and (7.7) and the localized normal forms investigated in this paper relate to one type of solution but correspond to different regions of amplitude variation. 1.

2.

3.

4.

5. 6.

7.

8.

  IJ) is a saw-tooth function of period '2 and amplitude 1, (Ij is phase of the i-th mass, and iJ is the absolute velocity, constant in time.Passing to the general case (';#(I). we shall consider (2.2) as the change of variables. Such rough transformations were first utilized in /6/. The equations of motion then reduce to the form d%, lr,i[~n~q,i-nly?)l="

  =2,...) (2.4)

  Fig.2Fig.3in the plane (v,,Q.) remains inside the square K. Passing into a new element of the plane corresponds to an impact on the limiter. Note that the pattern in square K (the equipotential curves of a linear system) has an axis of rotational symmetry of second order, and the gap (Fig.3) obtained by multiple mirror reflection of that pattern (the motive) relative to the straight lines (2.4) has a fourthorder axis of symmetry. This is related to the substantial difference of the dynamic properties of the linear model (without impact) and modes of motion with impacts against a limiter. Thus, in the first instance we have two normal oscillations, i.e. the natural oscillations of a linear system with two degrees of freedom. With impacts, there are already four (Fig.3), since besides a symmetric (inphase) and skew-symmetric (antiphase) forms, two supplementary spatially localized forms appear, which at the limit y-+0 become straight lines qI = 0, q2 = 0 and correspond to one of the masses being at rest.

  ) forms. If, however, the forms belong to the unstable type (Fig.S,b, the antiphase form), a rapid "drift" of the trajectory from the form is observed. 7. Let us consider the connnection between normal oscillations and the soliton type solutions for an infinite chain. If in this system N+ ca,e = 1, the equations of motion in a finite-difference approximations of elastic forces in the string have the form #ajl&: -1' (2U' -U _ -U _ ) +-au?c:(1) satisfies the non-linear differential equation d2C'/di2 + qyc + aUP-= 0.(5.2)A solution of the soliton type can be sought in the class of weakly modulated standing waves U, = (-l)jU, (1) (7.3)

9. MANEVICH L.I. and PINSKII M.A., On normal oscillations in non-linear systems with two degrees of freedom. Prikl. Mekhanika, Vol.8, No.9, 1972. MANEVICH L.I. and PINSKII M.A., On the use of symmetry when calculating non-linear oscillations. Izv. AN SSSR, MTT, No.2, 1972. ROSENBERG R.M., On the variation of systems with many degrees of freedom. In advances in Applied Mechanics. Vo1.9, New York-London Acad. Press, 1966. MANEVICH L.I. and MIKHLIN IU.V., On periodic solutions close to rectilinear vibration modes. PMM, Vo1.36, No.6, 1972. MALKIN I.G., Some Problems of the Theory of Non-linear Oscillations. Moscow, Gostekhizdat. 1956. ZHURAVLSV V.F., A method of analysing vibro-impact systems using special functions. Izv. AN SSSR. MTT, No.2, 1976. KOSEVICH A.M. and KOVALSV A.S., Selflocalization of oscillations in a one-dimensional anharmonic chain. ZHETF, Vo1.67, No.5, 1974. VEDENOVA E.G. and MANEVICH L.I., Periodic and localized waves in vibro-impact systems of regular structure. Mashinovedenie, N0.4, 1981. VEDENOVA E.G., Solitons in essentially non-linear system of impact type. Mashinovedenie, No.1, 1983.

Y1'-Y2=Ip3 s= w-Y3 Fig. 4 5. We shall now show how the results related to the limit case n+oJ can be used to construct oscillations when n>I.

As an example, we shall consider the system with two degrees of freedom

We shall seek the normal form of oscillations that corresponds to the localized form of the limit system (k+ 00).

We set U, = z10 + IPL' (CL"), lL* = L.-zL. (UJ (5.2) Ug = '4n (q), Cf = L't,;l (the period of the function n is provisionally taken as equal to 4). Substituting (5.2) into (5.1), we obtain the following system of equations in u (&I), v (110):

Setting L",=, = --LJ. 1" lU& = 0 (5.4)

we eliminate the terms in braces which correspond to a periodic system of pulses (6 functions in the variable CF.

The first of conditions (5.4) is satisfied by the appropriate selection of the amplitude