Periodic motions of a string vibrating against a fixed point-mass obstacle: II

A string fixed at both ends A and B, can oscillate in a plane in which there is a fixed point obstacle, placed in the middle of the line AB. The string is initially at rest with a prescribed shape, symmetric with respect to the normal mid-plane of the segment AB. Using results established before [9] we find new periodic motions.


Introduction
During the past few years several works have been devoted to the motion of a string vibrating in the presence of obstacles. Amerio and Prouse [l] are the first to have considered the problem and they have proved, for a straight line obstacle, the global existence of the solution. Schatzman [2] and Bamberger [3] have studied the case of concave obstacles. Cabannes and Haraux [4, 51 have studied, for straight line obstacles, the periodic or almost periodic character of the solution. Betro and Gotusso [6] have made numerical computations; Citrini and d'Acunto [7] have considered the shock of two strings. The case of a pointmass obstacle has been studied by Reder [8] and Cabannes [9].
The present work is devoted to the investigation of periodic plane motions of a vibrating-string fixed at both ends A and B in the presence of a point mass obstacle fixed in the middle of AB. Initially, the string is at rest with a prescribed shape, which is symmetric about the median line normal to the segment AB: figure 1. The elongation (distance from the equilibrium position) possesses three extrema; one of there, located on the median of AB, is above the obstacle, the other two are below the obstacle at positions symmetric with respect to the median. We prove that, when the ratio of corresponding elongations is a rational number, the motion of the string is periodic and we compute the period; when the ratio is irrational the motion of the string is not periodic; it is probably almost periodic but this result still has to be proved.

Statement of the Problem
The equilibrium position of the string is along the segment A B of the x-axis, the points A and B having as abscissae k0.5; it can oscillate in a plane 1 xu above a fixed point mass obstacle, placed at the origin. The function u(x, I ) which represents the departure from the equilibrium position satisfies the following conditions: The contact of the string with the obstacle corresponds to zero values of the function u(0, t ) and persists during the intervals of time when the reaction of the obstacle on the string is positive, let (cf.

1-
The new initial shape u(x,O) is composed of straight line segments, all having the same slope k = 2(M + 2). We define an odd, increasing function F ( y ) , such that F( y )y is a periodic function with period 1, by the relation,

x 1 = u(x,O) = u(F(x),o).
If the motion of the string, initially at rest in the position u(x, 0), is periodic with whole number period N, then the motion of the string initially at rest in the position a ( x ) = u(x,O) = v(F(x),O} is periodic with the same period. Moreover, since the initial data are even functions, the necessary and sufficient condition for the motion of the string to be periodic, is that the motion of the middle of the string is periodic. The motion of the string and the motion of the mid-point therefore have the same periods (cf. [9], theorem 3). We adopt henceforth for the function a ( x ) the function represented in figure 2.

Motion of the Middle of the String
The motion of the middle of the string u(0,t) can be studied without the knowledge of the complete motion of the string. Before the first contact, we have the free oscillation The first contact begins at the time C,, which is the smallest positive root of the equationfo(t) = 0; we therefore have: This first contact ends at the time Do when the reaction vanishes; Do is the smallest root greater than C , of the equation (dfo/dt) = 0; we have therefore After time Do we have a new free oscillation. In general after the time D, which follows the end of a contact we have a new free oscillation in which the motion of the middle of the string f D n ( t ) is defined (cf.

191, theorem 5 ) by the relations:
(1 3) The contact which follows the detachement at the time D, appears at the time t = Cn+l which is the smallest root greater than D, of the equationfDn(t) = 0, and this contact ends at the time f = D,+ 1 , which is the smallest root greater than C,, of the equation (d/dt)fDn(t) = 0. As fDn(Dn) = 0, we have also f D n ( l + D,) = 0, and Cn+l is at most equal to 1 + D,.
The first complete free oscillation, that which occurs after time Do, is defined by the functions

('3
We deduce that in the interval Do < t < 1 + Do, the function fDO(t) has a first maximum Mf = 2, a minimum m0 = 1 -M and a second maximum M! = 1 :

Fig. 3.
We assume that the free oscillation, which starts at the time D,, has, for D, < f < 1 + D,, a first maximum MY, a minimum m" and a second maximum M ; .
The formulae (13) prove that the same situation is valid for the free oscillation which starts at the time D,, 1. As this is true for n = 0, this is always true and from formulae (13) we have: We note that the sum Mym" + M," is independent of the index n and so has the value M + 2.
Moreover it follows from formulae (16) and (17)  To prove this result we assume first that we have M < 2, and prove that the following inequalities are satisfied for all values of n : The property being true for n = 0, it is sufficient to deduce its validity for n + 1 from that for n. This  We assume then that we have M > 2, and we prove that the following inequalities are satisfied for all values of n The property is true for n = 0. We then have by induction   We put M = p / 4 , p and q being mutually prime integers, p positive or zero, q strictly positive. We first assume that p > 2q, and we put I) ') The notation [x] denotes the greatest integer smaller or equal to x, that means the integral part of x. If p is even, q is odd and the inequality (23-1) is always satisfied. If p is odd the equality (23-2) is satisfied for ( j + 1 ) p = (2m + 1 ) q ; m is an integer and 2m + 1 an oddmultipleofp. Wehavethereforem = ( p -1)/2andj = q -1, or m = ( 3 p -1)/2 and j = 3 q -1,. . . . Also when the index j varies from 0 to 2q the inequality (23-1) is always satisfied except if p is odd and i f j = q -1. The equality is then satisfied and we have We assume first that p is an odd integer. We denote by s), sj . . . the set of integers greater than uj-1 , less or equal to oj; sj = 1 + oj-1 , . . . The period of the motion is then DoD3p,2 or D3p/2D3p, that is, half of the value given by the formula (26).

Calculation of the Period for M = p / q < 2
When p is less than 29, we put: The number of groups of intervals of the first type is (ap-p ) + (2 qap-,) = 2 qp . The number of groups of intervals of the second type i s p -1 (from a. to ap-except a(p-1)/3. There is one interval of the third type. We have also 12q2 + 3 p q -2p2 P + 2 9

( 3 5 ) T = DOD6, =
When p is even the equality (30-2) is never satisfied and the table 11-a is valid by cancelling only the lines of rank 3oj. We have and for j = p / 2 the set sj contains the value sj = sPl2 = q. If M < 1 we have q = Sp/2 < if M > 1 we have q = sPl2 = up/2. So, using the tables 11-a, or the tables 11-b we obtain: The period of motion is D0D(3p/2) or D3p,2D3p, that means the half of the value given by the formula (35). The motion of the string is therefore not periodic when M is irrational. As we have said in the introduction it is probable that the motion is almost periodic. We hope that a reader will be able to prove this conjecture.

Conclusion
We have studied the case of a string initially at rest in a position represented by Fig. 2. The maxima displacements are reached for x = 0 and for x = f a ; their ratio M = u (O,O)/u(O,a) has been assumed rational: M = p / q . We have then proved that the motion is periodic and we have computed the periods: formulae (28) and (35). The transformation (6), which conserves the integral periods, allows us to pass from the case of figure 2, to the case of Fig. 1, the ratio of maxima displacements being the same. Also when the string is initially at rest in a position represented on the Fig. 1, with M = p/q ( p and q integers), the motion of the string is periodic and the period has as value: T = 1 2 q 2 + 3 p q -2p2 ifp < 2 q T = 3 p 2 + 3 p q -8 q 2 i f p 2 2 q .
When M is irrational, the motion is not periodic.