Analytical investigation of the dynamics of a nonlinear structure with two degrees of freedom

A two degree of freedom oscillator with a colliding component is considered. The aim of the study is to investigate the dynamic behavior of the system when the stiffness obstacle changes from a finite value to an infinite value. Several cases are considered. First, in the case of rigid impact and without external excitation, a family of periodic solutions are found in analytical form. In case of soft impact, with a finite time duration of the shock, and no external excitation, the existence of periodic solutions, with an arbitrary value of the period , is proved. Periodic motions are also obtained when the system is submitted to harmonic excitation , in both cases of rigid or soft impact. The stability of these periodic motions is investigated for these four cases.


!.INTRODUCTION
Vibrating systems with clearance between the moving parts are frequently encountered in technical applications. These systems with impacts are strongly nonlinear; they are usually modeled as a spring-mass system with amplitude constraint. Such systems have been the subject of several investigations, mainly in the simplest case of a one degree of freedom system [1-4] and more seldom for multi degree of freedom systems [5][6][7][8][9]. The system behavior during the contact between the moving parts can be described as rigid impact, usually associated with a restitution coefficient, or modeled as soft impact, with a finite time duration of the shock. Several other parameters such as damping, external excitation, influence the behavior of the system. The work is the continuation of a previous paper [10], in which a two degree of freedom oscillator is considered. The nonlinearity in this case comes from the Laboratoire Systemes Complexes, Universite d'Evry Val d'Essonne et CNRS FRE 2494,40 rue du Pelvoux, 91020 Evry cedex, France. mpascal@aramis.iup.univ-evry.fr nonlinearity in this case comes from the presence of two fixed stops limiting the motion of one mass. Assuming no damping and no external excitation, the behavior of the system is investigated when the obstacles stiffness changes from a finite value to an infinite one. In both cases, a family of symmetrical periodic solutions, with two impacts per period, is obtained in analytical form. In the present paper, a two degree of freedom system in presence of one fixed obstacle is considered. Assuming that no damping occurs, we investigate four cases: unforced system with rigid impact, unforced system with soft impact, forced system ( with harmonic excitation) with rigid impact and, at last, forced system with soft impact. In all cases, periodic solutions are found and stability results of these particular motions are obtained .

PROBLEM FORMULATION
The system under considemtion ( Fig.!) is a genemlization of the double oscillator investigated in the paper [10]. It consists of two masses rn 1 and m 2 connected by linear springs of stiffness k 1 and k 2 • The displacement z 1 of the mass m 1 is limited by the presence of a fixed stop. When z, is greater than the clearance, a contact of the first mass with the stop occurs; this contact gives rise to a restoring force associated to a spring stiffness k,. The mathematical model of the system is given by: ,K= , m~ -k 1 k 1 + k~ P=(~) and rp are the amplitude and the phase angle of the harmonic excitation.

UNFORCED SYSTEM
Let us consider the system without external excitation ( P = 0 ).

RIGID IMPACT
When the stiffness of the obstacle k 3 tends to infinity, a rigid After the impact, the system performs a free motion defined by: (:)=ccr{::} c 1 =(1,(t) 1 2(r)) Cl 1,(!) 1,(t)' 1 1 =AB,(t)A·' (i=l,2,3) In these formulas, ( m, ,m,) are the roots of the characteristic equation: The following properties for the r, matrices hold: r; Ctl-1, (t)1, (t) =I, for ij=l,2,3 (5) r,(r)r, (r) = 1/t)1,(tl Moreover, the coefficients C,/1) of the 4 by 4 satisfy the property: matrix C(t) c, (t) = c,.,,,., (t), (i, j = 3, 4) It results for the determination of the four scalar parameters (y, u, w, T) the four scalar equations: (-H,+C(-T))Z,=O, Z,=(l,y,u,w)' (8) or equivalently: (1,-I)z,-r,z, =0 Taking into account the properties (5) of the 1 1 matrices, the system (8) leads to: z, =C1,-Ir'1,z, The last equation of (9) reduces to w=O. From the first one, y and u are obtained in terms of the period: In case of rigid impact, a family of periodic solutions is obtained for which the initial conditions are defined in terms of the period and the initial velocity w of the non impacting mass is zero. For these particular motions, the conditions giving the positions and the velocities after the shock can be formulated by (II) These results are similar to the results obtained in [!OJ. The system considered in this previous paper is a symmetrical system with respect to the position z 1 of the first mass and it can be expected that the obtained results are due to this property. But it is not the real explanation because the system investigated now is not symmetrical. Remark In more general cases, the impact is described by a restitution coefficient r ( O<r<l). In this case, the new positions and the new velocities after the shock are given by: The initial conditions corresponding to a periodic orbit of period Tare obtained from the relation: lending to the solution .::q = ±n = 0 . In conclusion, if the restitution coefficient r is not equal to I. no periodic solution with one impact per period exists. This fact, of course , is due to the non conservation of the total energy in this case: it is not possible for the system to perform a periodic orbit if no external excitation occurs.

SOFT IMPACT
let us assume that the stiffness obstacle is bounded. The mathematical model of the system is given by : For z 1 :S:l Mi+Kz=O For z 1 ;:: I Let us assume that the initial conditions are A periodic solution is defined in two steps: given -For 0 $!::; 'l", z 1 >I, the system is defined by the motion equations ( 15). The time duration r of this constraint motion is defined by the condition: z 1 (r)=l (16) Let us denotes by Zc "'Z(r)"" (l,y, ,u 0 w J' the value of the parameters at the end of shock, with the condition u<: < 0. -For 1: ~ I s r + f, a free motion obtained from equations (14) and initial conditions Zc occurs. This motion finishes when Let us denotes by Z 1 ::::Z(r+T)=(l,y 1 ,u 1 ,w 1 Ythe value of the parameters at the end of free motion (u 1 > 0) The condition to obtain a periodic orbit of period r + f is given by the condition: The piecewise linear systems (14) and (15) give the two parts of the motion in analytical form.
-For 0$1::; r, the coristraint motion is deduced from a modal analysis of system (IS): , In these formulas, o-Po-1, C!>1 = (~J· <!>" = (~") define the characteristic frequencies and the eigenvectors ofthe constraint system (15). For the H, matrices, the properties (5) obtained for the r, matrices hold, together with the property -For • ~ t :S: 1: + T, the free motion is obtained from where the matrix C is defined by formulas z.
The condition (21) is equivalent to From the properties of f, and 11;, we deduce: The condition (22) leads to (25) (26) Two possible cases of periodic solutions can be deduced from (26), namely: Let us discuss the first conditions (27). In this case, from (23), we deduce: The condition (16) is fulfilled and the initial conditions are obtained from the equations: This system provides four scalar equations for the determination of the five parameters (y, u, w, r,T ). It results that, as in the case of rigid impact, f and hence the period can be chosen arbitrarily. Moreover, the conditions (28) and (II) obtained at the end of the shock are the same for both rigid and soft impacts. From (29), we deduce: The last equation (30), after the elimination of y, provides a relation F( r, f )=0 between the time duration r of the shock and the time duration f of the free motion. The other case X 1 = X 2 , det(f(-~) = 0 leads to no solution [10].
In both cases (soft or rigid impact), a family of periodic motions is obtained , with an arbitrary value of the period. Moreover, the conditions (28) obtained at the end of the shock for soft impact are consistent with Newton rules of rigid impact (11) with a restitution coefficient equal to one, i.e. with assumption of ideal elastic impact. This rather remarkable result have been already obtained for the symmetrical system of [10].

FORCED SYSTEM
Let us assume that the two masses are subjected to harmonic external excitations of period 21r I cu ,constant amplitudes ~,P 2 and constant phase angle rp . From the results obtained in the previous paragraph, where a family of periodic orbits is found with an arbitrary value of the period, it can be expected that for the forced system, periodic solutions of period 2:r f cu exist.

RIGID IMPACT
Let us investigate the case of rigid impact, with a restitution coefficient r = I . Starting from the initial conditions Z 0 =(l,y,u,w)' (u>O), the conditions Z< =(l,yc,uc,wJ' after the shock are obtained from (2) and the free motion performed by the system is given by: : where R = (R 1 , R 2 )' is the amplitude of the response: The free motion finishes at time t= T when z 1 (T) = 1, z 1 (T) > 0.
Let us denote by Z 1 = (z 1 , i 1 )' the positions and the velocities reached by the system at this time. The condition to obtain a periodic motion of period T is : Let us assume that T = 2:r I cu, rp = 0, z" = -z 0 • A periodic motion of period 21r f cu is obtained if the initial conditions Z 0 = (l,y,u,O)' are defined by the system: ri =r 1 (27r/cu), (i=l,2,3) Taking into account the properties (5) of the f 1 matrices, this system reduces to

Remark
In more general cases, the impact is described by a restitution coefficient r ( O<r< 1 ). The initial conditions and the phase angle related to a periodic solution of period 27r I OJ can also be obtained in analytical form. A similar solution has been studied in paper [6].

SOFT IMPACT
When the stiffness obstacle is bounded, the motion equations of the system are given by: The time duration r of this motion is obtained from the condition z 1 (r) = 1. Let us denote Zc = (l,yc,uc, we)' the value of the parameters at t= ' ( uc < 0 ).

UNFORCED SYSTEM
Let us consider a periodic motion of period T related to initial conditions z 00 = ( ~ 0 ) , z 00 = ( ~0 ) , where z 00 , z 00 are defined by: (r 1 -f)z 00 -r 2 z 00 = 0 r;::;:: r;(T) -rJzoo +(rr -£)zoo = 0 (46) Let us consider the perturbed motion defined by a set of new initial conditions: where This motion is defined for t>O, by: z 1 = r 1 (T +dT)z 00 + r 2 (T +dT)EZ 00 z 1 = r 3 (T + dT)z 00 + r 1 (T + dT)Ez 00 Assuming small perturbations dz 0 ,dz 0 of the initial conditions, Let us consider a perturbed motion related to initial conditions ( 4 7) and phase angle q; = rp 0 + d q; . The corresponding free motion performed by the system for t>O, is obtained from (29), with z< = Ez 0 • This motion ends at t= 2 7l + dT , when (J) z 1 ( Z;r + dT) = I and ± 1 ( Z;r + dT) > 0 . Let us denote by In this case, it is impossible that all the eigenvalues of the matrix N 0 lie strictly inside the unit circle. The periodic solution is unstable except if all these eigenvalues lie on the unit circle.

UNFORCED SYSTEM
When the stiffness of the obstacle is bounded and when there is no external excitation, the mathematical model of the system is given by (14) for the free motion and (15) for the constraint motion.
Let us consider a periodic motion of period r 0 +To = 2;r I w , where w is an arbitrary positive value in this case. This periodic solution is related to the initial conditions and the condition r 0 + T 0 = 2rc I m . (54) Let us consider the perturbed motion defined by a set of new initial conditions (47). This motion is defined in two steps: -For 0 ~I~ r = r 0 + dr: This motion ends when z 1 (r) =I and .± 1 (r) < 0. Let us denotes From systems (61) and (63),after the elimination of dr and dT, we deduce the correspondence between the initial perturbations and the final ones : (64) .
In Figures 2 and 3, the behavior of the periodic solutions is compared for unforced (ZiF) and forced system(ZiNF), in the rigid impact case. Figures 4 and 5 are related to soft impact: the bold part of the curves shows the free motion, the other part (ZicF or ZicNF) the constraint motion. Other results about the stability conditions when ru varies, will be presented at the Conference.

CONCLUSIONS
The main objective of this paper is to compare the behavior of the system when the stiffness of the obstacle changes from a finite value ( soft impact) to an infinite value (rigid impact). The results obtained for unforced systems about the existence of periodic solutions show that there is a smooth transition between the two cases. For both cases, the period T can be chosen arbitrarily, the initial conditions related to the periodic motion are obtained in terms of T. Moreover, in case of rigid impact, the initial velocity of the non impacting mass being zero, we can formulate the conditions giving the positions and the velocities after the shock by formula (I 1). In case of soft impact, we showed analytically that the conditions obtained at the end of the constraint motion ( when the contact of the first mass with the stop finishes), that the positions and the velocities at this time are obtained by the same rule. The case of forced systems is a more standard one. The investigations performed in this paper show how the results obtained in the case of unforced systems can lead very simply to obtain the periodic solutions in case of forced systems. The stability of periodic motions is based on mapping. In all cases, ( rigid or soft impact, unforced or forced systems), the matrix of the linear correspondence between the initial perturbations and the final ones, is obtained in close form. In several cases, some interesting properties of the corresponding eigenvalues are also obtained.