Formulas for Vibrating Strings with Obstacles

Henri Cabannes

June 16, 2001

The string moves in the plane Oxu (orthonormed system of references) ; the extremities are fixed at the points (x = ±1/2, u = 0), and the shape of the string at time t is u(x,t). The string is initially at rest in the position u(x,0) = u0(x) ³ -h, with 0 < h < 1.

1  Fixed Point-Mass Obstacle(2)

The obstacle is located at the point x = 0, u = +h. The function u0(x) is an even function and u0(0) = 1. We introduce the two functions K(y) and f(y), periodic with period 1, and défined for -(1/2) < y < (1/2) by :
K(y) = ì
í
î
|y|-[|y|] -  1

2
ü
ý
þ
2

 
,
(1)

f(y)+y = sgn(y){ 1-u0(y)} /2.
(2)

[|y|] denotes the highest integer smaller than |y|, absolute value of y ; sgn(y) denotes the sign of y. The function K(y) is even, the function f(y) is odd. Then we introduce the new variables X and T, and the new functions K1 and K2.
X(x,t) = x+  f(x+t)+f(x-t)

2
,
(3)

T(x,t) = t+  f(x+t)-f(x-t)

2
.
(4)

K1 = K ì
í
î
2  X+T

3-h
ü
ý
þ
+K ì
í
î
2  X-T

3-h
ü
ý
þ
,
(5)

K2 = K ì
í
î
 2(X+T)+1-l

3-h
ü
ý
þ
+K ì
í
î
 2(X-T)+1-l

3-h
ü
ý
þ
.
(6)

The parameter l is equal to h when 0 < 4x < 1+h, and equal to 3 when 1+h < 4x < 2.  With all those auxiliary functions, the motion of the string is represented by the function :
u(x,t) =  1+h

3-h
{ 1-2|X|} +  3-h

2
{ K1-K2} .
(7)

The motion is an almost periodic function of the time, and a periodic function when h is rational.

2  Fixed Straight Line Obstacle(3)

The obstacle is located on the line u = -h. The initial position is ''unimodal'', which means that the derivative (du0/dx) is positive or zero when -(1/2) < x < a < (1/2), and negative or zero when -(1/2) < a < x < (1/2). We assume u0(a) = 1. We introduce the new function L(y) = {1-(y/2)} 2 when 0 < y < 1, and then L(y+1) = L(y). We introduce also the functions F(y), L1, L2 and L3 :
F(y) =  1

2
{ 1-u0(y)} sgn(y-a).
(8)
 
L1 = L ì
í
î
 F(x+t)

1+h
ü
ý
þ
+L ì
í
î
 F(x-t)

1+h
ü
ý
þ
,
(9)

L2 = L ì
í
î
 F(x+t)-F(x-t)

2(1+h)
+  1

2
, ü
ý
þ
,
(10)

L3 = L ì
í
î
 1-eF(x+t)

1+h
ü
ý
þ
-L ì
í
î
 1-eF(x-t)

1+h
ü
ý
þ
.
(11)

e = ±1 possesses the sign of F(x+t)+F(x-t).

The function u(x,t) which represents the motion of the string is given by formula (12) for the points which encounter obstacles : |F(x+t)+F(x-t)| < 1-h, and by formula (13) for points which do not encounter obstacles : 1-h < |F(x+t)+F(x-t)| < 1.
u(x,t) =  1-h

2
 { F(x+t)+F(x-t)} 2

2(1+h)
+(1+h){ L1-2L2} ,
(12)

u(x,t) =  1-h

1+h
{ 1-|F(x+t)+F(x-t)|}+(1+h){ L1-L3} .
(13)

The motion is an almost periodic function of the time, and a periodic function when h is rational.

3  Rebound on a Curvilinear Obstacle(4)

The obstacle is the curve u = lsin(2px), and the intial position of the string u(x,0) = u0(x) = cos(px). Before the first contact the motion is the free oscillation u(x,t) = w(x,t) = cos(px)cos(pt) ; the time of the fist contact is
t1 = t(x) = (1/p)Arccos{ 2lcos(px)} .
(14)

For |2l| < 1, the first contact corresponds to a rebound and, after this rebound, the motion of the string is represented by the function
u(x,t) = w(x,t)-  1-4l2

2l
G(x,t),
(15)

G(x,t) = px-Arctan  (1+4l2)sin(px)-4lcos(pt)

(1-4l2)cos(px)
.
(16)

For |2l| < (1/3), the time of the second contact is
t2 = 2-t(x),
(17)

and this second contact corresponds still to a rebound. After this second rebound the motion of the string is again the free oscillation, so that the motion is periodic with period 2.

4  Wrapping on a Curvilinear Obstacle(4)

Always considering an obstacle u = lsin(2px), and the string always in the initial position u(x,0) = u0(x) = cos(px), one obtains for the peculiar case l = -(1/2), the wrapping of the string on the convex part of the obstacle, and the rebound on the concave part.

For the new peculiar case corresponding to the obstacle u = (1/3)cos(3px) , the motion of the string is periodic with period T = (4/3), with successive wrapping and unwrapping on the convex part of the obstacle.

References

[1]. L. Amerio and G. Prouse, Study of  the motion of a string vibrating against an obstacle, Rendiconti di Matematica, Serie VI, 8, 563-584, (1975).

[2] . H. Cabannes , Motion of a vibrating string against a fixed point-mass obstacle. C.R. Acad. Sci. Paris, Série II 298, 613-616, (1984).

[3] . H. Cabannes, Motion of a string in the presence of a straight rectilinear obstacle. C.R. Acad. Sci. Paris, Série II 295, 637-640, (1982).

[4] . H. Cabannes, Motion of a vibrating string in the presence of a convex obstacle, C.R. Acad. Sci. Paris, Série II 301, 125-129, (1985).

[5] . A. Haraux and H. Cabannes, Almost periodic motion of a string vibrating against a straight fixed obstacle, Nonlinear Analysis, Theory, Methods and Applications 7, 129-141, (1983).

[6] . H. Cabannes, Periodic  motions of a string vibrating against a fixed point-mass obstacle, Mathematical Methods in Applied Sciences 6, 55-67, (1984).

[7] . H. Cabannes, Cordes vibrantes avec obstacles, Acustica 55, 14-20, (1984).

[8] . H. Cabannes and C. Citrini, Editors, Vibrations with unilateral constraints, In Proceedings of the Euromech Colloquium 209, Tecnoprint, Via del Legatore 3, Bologna, Italy, (1986).




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