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BOUNDED DISPLACEMENT OF A DAMPED NON-LINEAR STRING
Journal of Sound and Vibration (1996) 193(5), 1115–1121
LETTERS TO THE EDITOR BOUNDED DISPLACEMENT OF A DAMPED NON-LINEAR STRING S. M. S L. G. K Berkeley Engineering Research Institute, P.O. Box 9984, Berkeley, CA 94709, U.S.A. (Received 3 October 1995)
1. In this letter, we consider a damped elastic string of length l which is fixed at both ends, has zero initial displacement and velocity, and is under distributed external forces. The dynamics of such a string can be represented by the following nonlinear partial differential equation (see, e.g., references [1–3]):
0 g
1
l
utt (x, t) + 2dut (x, t) = a + b
ux2 (x, t) dx uxx (x, t) + f(x, t),
0
(1a)
for all x $ (0, l) and t e 0, with the boundary conditions u(0, t) = u(l, t) = 0,
(1b)
for all t e 0, and the initial conditions ut (x, 0) = 0,
u(x, 0) = 0,
(1c)
for all x $ (0, l), where u(·, ·) $ R denotes the transversal displacement of the string due to the distributed external force f(·, ·) $ R, the constant d q 0 represents the damping coefficient in the string, a q 0 and b q 0 are constant real numbers, and ut M1u/1t, utt M1 2u/1t 2, ux M1u/1x and uxx M1 2u/1x 2. The non-linear model of vibrating strings in equations (1) was originally derived by Kirchhoff [4], and later rederived by Carrier [1], Narasimha [2] and Oplinger [3]. Researchers have studied the system (1) from physical and mathematical points of view; see, e.g., references [1–25]. In this letter, we establish the boundedness of the displacement of the string represented by equations (1) when the external force is bounded. This result has not been established despite extensive study of the system (1). In order to define the boundedness precisely, we introduce two function spaces, as follows. (i) By X2 we denote the space of functions u(x, t), where u: (0, l) × R+:R, for which >u >X 2Msup te0
$g
l
=u(x, t)=2 dx
0
%
1/2
Q a.
We say that a u $ X2 is X2 -bounded. (ii) By Xa we denote the space of functions v(x, t), where v: (0, l) × R+:R, for which >v >X aMsup sup =v(x, t)= Q a. t e 0 x $ (0, l)
We say that a v $ Xa is Xa -bounded. Note that XaWX2 because La (0, l)WL2 (0, l). Thus, if a function is Xa -bounded, then it is X2 -bounded. However, the converse may not be true; for instance, u(x, t) = x−1/3 sin t, where x $ (0, l) and t e 0, is in X2 but not in Xa . 1115 0022–460X/96/251115 + 07 $18.00/0
7 1996 Academic Press Limited
1116
We now make the following assumption (A1) on the force f: the force f $ Xa . In this letter, our goal is to provide a proof for the Xa -boundedness of u: that is, to show that the system (1) is bounded-input bounded-output stable. We achieve this goal by taking an energy approach. 2. Our plan to establish the Xa -boundedness of the displacement of the string, u, is as follows. We define an energy-like (Lyapunov) function of time for the system (1) and denote it by V. We prove that due to the Xa -boundedness of the force f and the dissipative effect of 2dut , the function V is a bounded function of time. Then, from the boundedness of V, we conclude that u is Xa -bounded. It appears that due to the non-linearity in the model of the string, the energy approach is the only approach to prove the boundedness of the displacement. We define the scalar-valued function V as V(t)ME(t) +
g
l
[dut (x, t)u(x, t) + d 2u 2(x, t)] dx,
(2)
0
for all t e 0, where 1 2
E(t)M
g
l
[ut2 (x, t) + aux2 (x, t)] dx +
0
b 4
0g
l
1
2
ux2 (x, t) dx ,
0
(3)
and u(·, ·) satisfies equations (1). We can rewrite V as V(t) =
1 2
g
+
l
[(ut (x, t) + du(x, t))2 + d 2u 2(x, t) + aux2 (x, t)] dx
0
b 4
0g
1
l
2
ux2 (x, t) dx ,
0
(4)
for all t e 0. Now, we prove some of the properties of V. Lemma 2.1. The function V(t) e 0 for all t e 0, and V(0) = 0. Proof. Since u(x, 0) = 0 for all x $ [0, l], we have ux (x, 0) = 0 for all x $ [0, l]. Using u(x, 0) = 0, ux (x, 0) = 0 and ut (x, 0) = 0 for all x $ [0, l] in equation (4), we conclude that V(0) = 0. The non-negativeness of V(t) for all t e 0 is obvious from equation (4). q In the following, we will use the La -norm of V, defined as >V >aMsup V(t). te0
Lemma 2.2. If >V >a Q a, then ux is X2 -bounded. More precisely, >ux >X 2 E
0
2>V >a a
1
1/2
Q a.
(5)
1117
Proof. From equation (4), we have a 2
g
l
ux2 (x, t) dx E V(t) E >V >a Q a,
(6)
0
for all t e 0. Therefore, equation (5) follows. q Remark. It can be shown that >V >a Q a if and only if u, ux and ut are X2 -bounded. However, this result is of no need in this paper. Now, we show that V can be bounded by E defined in equation (3). Lemma 2.3. The function V satisfies V(t) E K(d)E(t),
(7)
for all t e 0, where K(d) = 1 +
6
7
dl 1 + 2dl/p max 1, . p a
(8)
Proof. By Scheeffer’s inequality, which is a Poincare´-type inequality (see, e.g., reference [26, p. 67]), we have
g
l
u 2(x, t) dx E
0
l2 p2
g
l
ux2 (x, t) dx,
(9)
0
for all t e 0. Furthermore, we have the inequality
g
l
ut (x, t)u(x, t) dx E
0
l 2p
g
p 2l
l
ut2 (x, t) dx +
0
g
l
u 2(x, t) dx,
(10)
ux2 (x, t) dx,
(11)
0
for all t e 0. Using equation (9) in equation (10), we obtain
g
l
ut (x, t)u(x, t) dx E
0
l 2p
g
l
ut2 (x, t) dx +
0
g
l 2p
l
0
for all t e 0. Substituting equations (11) and (9) into equation (2), we obtain V(t) E E(t) +
dl 2p
0g
l
ut2 (x, t) dx +
0
1 + 2dl/p a
g
l
1
aux2 (x, t) dx ,
0
(12)
for all t e 0. Therefore, V(t) E E(t) +
6
dl 1 + 2dl/p max 1, p a
70 g 1 2
1
l
[ut2 (x, t) + aux2 (x, t)] dx ,
0
(13)
and, finally,
0
V(t) E 1 +
6
dl 1 + 2dl/p max 1, p a
71
E(t),
for all t e 0. Thus, equation (7) holds with K(d) given in equation (8).
(14) q
1118
Up to this point, we have proved some of the properties of the function V. Next, we show that due to the Xa -boundedness of the force f and the dissipative effect of 2dut , the function V along the solution of the system (1) is bounded. Lemma 2.4. The function V along the solution of the system (1) is a uniformly bounded function of time. Proof. From equation (4), we obtain (the argument (x, t) of the functions is deleted) V (t) =
g
g
l
[(utt + dut )(ut + du) + d 2ut u + auxt ux ] dx + b
0
l
ux2 dx
0
g
l
uxt ux dx,
(15)
0
for all t e 0. Substituting utt from equation (1a) into equation (15), we obtain
g
V (t) = −d
l
ut2 dx + da
0
g
l
uxx u dx + db
0
0 g
l
ux2 dx
+ a+b
0
g
l
ux2 dx
0
1g
g
l
uxx u dx
0
l
(uxx ut + uxt ux ) dx +
0
g
l
(ut + du)f dx,
(16)
0
for all t e 0. From the boundary conditions u(0, t) = u(l, t) = 0, we have ut (0, t) = ut (l, t) = 0 for all t e 0. Thus, we obtain
g
l
uxx u dx =
0
g
l
(uux )x dx −
0
g
g
l
g
l
ux2 dx = −
0
l
(uxx ut + uxt ux ) dx =
0
g
l
ux2 dx,
(17a)
0
(ux ut )x dx = 0.
(17b)
0
Using equation (17) and E defined in equation (3) in equation (16), we can rewrite V as V (t) E −2dE(t) +
g
l
(ut + du)f dx,
(18)
0
for all t e 0. Furthermore, we have the inequality
g
l
0
(ut + du)f dx E
g 2
g
l
(ut + du)2 dx +
0
1 2g
g
l
f 2 dx E gV(t) +
0
1 2g
g
l
f 2 dx,
(19)
0
for all t e 0, where g q 0 is an arbitrary number. We choose g Q 2d/K(d), where K(d) is that given in equation (8). Using equations (7) and (19) in equation (18), we obtain the differential inequality V (t) E −(2d/K(d) − g)V(t) +
1 f(t), 2g
(20)
for all t e 0, with V(0) = 0, where
g
l
f(t)M
0
f 2(x, t) dx,
(21)
1119
for all t e 0. By assumption (A1), f is Xa -bounded; hence it is X2 -bounded, and 0 Q f(t) E >f >2X 2 Q a for all t e 0. By a comparison theorem given in references [27, p. 29] or [28, p. 30], we conclude that the function V in equation (20) satisfies V(t) E
1 2g
g
t
e−(2d/K(d) − g)(t − t)f(t) dt,
(22)
0
for all t e 0. Therefore, V(t) E
>f >2X 2 2g
g
t
e−(2d/K(d) − g)(t − t) dt,
(23)
0
for all t e 0. By making the change of variable t − t = u in equation (23), we obtain V(t) E
>f >2X 2 2g
g
t
e−(2d/K(d) − g)u du E
0
>f >2X 2 2g
g
a
e−(2d/K(d) − g)u du,
(24)
0
for all t e 0. Therefore, V(t) E >f >2X 2 /2g(2d/K(d) − g) for all t e 0, and hence >V >a E
>f >2X 2 Q a. 2g(2d/K(d) − g)
(25) q
That is, V is a uniformly bounded function of time.
Having proved the boundedness of V, we finally prove that the displacement of the string, u, is Xa -bounded. Theorem 2.5. Consider the string represented by equations (1) for which assumption (A1) holds. The displacement of the string, u, is Xa -bounded. Proof. Using the fundamental theorem of calculus and the fact that u(0, t) = 0 for all t e 0, we obtain =u(x, t)= =
bg
x
0
b g
x
uj (j, t) dj E
=uj (j, t)= dj E
0
g
0g
l
l
=ux (x, t)= dx E l 1/2
0
0
ux2 (x, t) dx
1
1/2
, (26)
for all x $ (0, l) and t e 0, where the last inequality follows from the Cauchy–Schwarz inequality. Thus, =u(x, t)= E l 1/2 >ux >X 2 E
0
2l >V >a a
1
1/2
Q a,
(27)
for all x $ (0, l) and t e 0, where the last inequality follows from equation (5). Using equation (25) in equation (27), we obtain >u >X a E
0
l ag(2d/K(d) − g)
1
1/2
>f >X 2 Q a.
(28) q
1120
Remark. Note that equation (28) relates the norm of the displacement of the string, u, to the norm of the force f applied to it. It is clear from equation (28) that if the amplitude of the force is small, then the displacement of the string is small. 3. In this letter, we have proved the boundedness of the displacement of a damped non-linear elastic string when it is under a bounded force. If the force applied to the string is Xa -bounded, then the displacement of the string is Xa -bounded. Our proof is achieved by showing that the energy-like function V of the string is bounded for all time. In our proof, having the damping coefficient d q 0 is of crucial importance, because only in this case the coefficient of V on the right-hand side of equation (20) can be made negative: a negative coefficient for V guarantees that V is bounded. 1. G. F. C 1945 Quarterly of Applied Mathematics 3, 157–165. On the non-linear vibration problem of the elastic string. 2. R. N 1968 Journal of Sound and Vibration 8, 134–146. Non-linear vibration of an elastic string. 3. D. W. O 1960 Journal of the Acoustical Society of America 32, 1529–1538. Frequency response of a nonlinear stretched string. 4. G. K 1877 Vorlesungen u¨ber Mathematische Physik: Mechanik. Leipzig: Druck and Verlag von B. G. Teubner. 5. G. V. A 1969 Journal of the Acoustical Society of America 45, 1089–1096. Large-amplitude damped free vibration of a stretched string. 6. G. V. A 1969 Journal of the Acoustical Society of America 46, 667–677. Stability of non-linear oscillations of stretched strings. 7. A. A and S. S 1984 in Nonlinear Partial Differential Equations and Their Applications (H. Brezis and J. L. Lions, editors) VI, 1–26. Marshfield, MA: Pitman. Global solutions to the Cauchy problem for a nonlinear hyperbolic equation. 8. R. W. D 1978 Proceedings of the Royal Society of Edinburgh 82A, 19–26. The initial value problem for a nonlinear semi-infinite string. 9. R. W. D 1980 Quarterly of Applied Mathematics 38, 253–259. Stability of periodic solutions of the nonlinear string. 10. A. I. E 1972 Journal of the Acoustical Society of America 51, 960–966. Driven nonlinear oscillation of a string. 11. C. G 1984 Journal of the Acoustical Society of America 75, 1770–1776. The nonlinear free vibration of a damped elastic string. 12. M. P. M 1991 Nonlinear Analysis, Theory, Methods & Applications 17, 1125–1137. Mathematical analysis of the nonlinear model for the vibrations of a string. 13. G. P. M 1979 Nonlinear Analysis, Theory, Methods & Applications 3, 613–627. On classical solutions of a quasilinear hyperbolic equation. 14. J. M 1984 Journal of the Acoustical Society of America 75, 1505–1510. Resonant, nonplanar motion of a stretched string. 15. W. G. N 1995 Journal of Mathematical Analysis and Applications 192, 689–704. Global solution of a nonlinear string equation. 16. T. N 1971 Memoirs of the Faculty of Engineering, Kyoto University 33, 329–341. A note on the nonlinear vibrations of the elastic string. 17. K. R and G. V. A 1983 Journal of Sound and Vibration 86, 85–98. Non-linear response in strings under narrow band random excitation, part I: planar response and stability. 18. G. T 1978 Journal of Sound and Vibration 58, 95–107. Analysis of a randomly excited non-linear stretched string. 19. G. T 1983 Journal of Sound and Vibration 87, 493–511. A parametrically driven harmonic analysis of a non-linear stretched string with time-varying length. 20. G. T 1989 Journal of Sound and Vibration 129, 215–235. Wave synthesis in a non-linear stretched string with time-varying length or tension.
1121
21. G. T 1989 Journal of Sound and Vibration 129, 316–384. Non-linear string random vibration. 22. G. T 1995 Journal of Sound and Vibration 185, 51–78. Parametric oscillations of a non-linear string. 23. Y. Y 1987 Nonlinear Analysis, Theory, Methods & Applications 11, 1155–1168. Some non-linear degenerate wave equations. 24. K. Y and T. T 1985 Bulletin of the Japan Society of Mechanical Engineers (JSME) 28, 2699–2709. Nonlinear forced oscillations of a string, 1st report: two types of responses near the second primary resonance point. 25. K. Y and T. T 1986 Bulletin of the Japan Society of Mechanical Engineers (JSME) 29, 1223–1260. Nonlinear forced oscillations of a string, 2nd report: various type of responses near the primary resonance points. 26. D. S. M´, J. E. P ´ and A. M. F 1991 Inequalities Involving Functions and Their Integrals and Derivatives. Dordrecht: Kluwer Academic. 27. G. B and G.-C. R 1989 Ordinary Differential Equations. New York: John Wiley; fourth edition. 28. V. L, S. L and A. A. M 1989 Stability analysis of Nonlinear Systems. New York: Marcel Dekker.
LETTERS TO THE EDITOR BOUNDED DISPLACEMENT OF A DAMPED NON-LINEAR STRING S. M. S L. G. K Berkeley Engineering Research Institute, P.O. Box 9984, Berkeley, CA 94709, U.S.A. (Received 3 October 1995)
1. In this letter, we consider a damped elastic string of length l which is fixed at both ends, has zero initial displacement and velocity, and is under distributed external forces. The dynamics of such a string can be represented by the following nonlinear partial differential equation (see, e.g., references [1–3]):
0 g
1
l
utt (x, t) + 2dut (x, t) = a + b
ux2 (x, t) dx uxx (x, t) + f(x, t),
0
(1a)
for all x $ (0, l) and t e 0, with the boundary conditions u(0, t) = u(l, t) = 0,
(1b)
for all t e 0, and the initial conditions ut (x, 0) = 0,
u(x, 0) = 0,
(1c)
for all x $ (0, l), where u(·, ·) $ R denotes the transversal displacement of the string due to the distributed external force f(·, ·) $ R, the constant d q 0 represents the damping coefficient in the string, a q 0 and b q 0 are constant real numbers, and ut M1u/1t, utt M1 2u/1t 2, ux M1u/1x and uxx M1 2u/1x 2. The non-linear model of vibrating strings in equations (1) was originally derived by Kirchhoff [4], and later rederived by Carrier [1], Narasimha [2] and Oplinger [3]. Researchers have studied the system (1) from physical and mathematical points of view; see, e.g., references [1–25]. In this letter, we establish the boundedness of the displacement of the string represented by equations (1) when the external force is bounded. This result has not been established despite extensive study of the system (1). In order to define the boundedness precisely, we introduce two function spaces, as follows. (i) By X2 we denote the space of functions u(x, t), where u: (0, l) × R+:R, for which >u >X 2Msup te0
$g
l
=u(x, t)=2 dx
0
%
1/2
Q a.
We say that a u $ X2 is X2 -bounded. (ii) By Xa we denote the space of functions v(x, t), where v: (0, l) × R+:R, for which >v >X aMsup sup =v(x, t)= Q a. t e 0 x $ (0, l)
We say that a v $ Xa is Xa -bounded. Note that XaWX2 because La (0, l)WL2 (0, l). Thus, if a function is Xa -bounded, then it is X2 -bounded. However, the converse may not be true; for instance, u(x, t) = x−1/3 sin t, where x $ (0, l) and t e 0, is in X2 but not in Xa . 1115 0022–460X/96/251115 + 07 $18.00/0
7 1996 Academic Press Limited
1116
We now make the following assumption (A1) on the force f: the force f $ Xa . In this letter, our goal is to provide a proof for the Xa -boundedness of u: that is, to show that the system (1) is bounded-input bounded-output stable. We achieve this goal by taking an energy approach. 2. Our plan to establish the Xa -boundedness of the displacement of the string, u, is as follows. We define an energy-like (Lyapunov) function of time for the system (1) and denote it by V. We prove that due to the Xa -boundedness of the force f and the dissipative effect of 2dut , the function V is a bounded function of time. Then, from the boundedness of V, we conclude that u is Xa -bounded. It appears that due to the non-linearity in the model of the string, the energy approach is the only approach to prove the boundedness of the displacement. We define the scalar-valued function V as V(t)ME(t) +
g
l
[dut (x, t)u(x, t) + d 2u 2(x, t)] dx,
(2)
0
for all t e 0, where 1 2
E(t)M
g
l
[ut2 (x, t) + aux2 (x, t)] dx +
0
b 4
0g
l
1
2
ux2 (x, t) dx ,
0
(3)
and u(·, ·) satisfies equations (1). We can rewrite V as V(t) =
1 2
g
+
l
[(ut (x, t) + du(x, t))2 + d 2u 2(x, t) + aux2 (x, t)] dx
0
b 4
0g
1
l
2
ux2 (x, t) dx ,
0
(4)
for all t e 0. Now, we prove some of the properties of V. Lemma 2.1. The function V(t) e 0 for all t e 0, and V(0) = 0. Proof. Since u(x, 0) = 0 for all x $ [0, l], we have ux (x, 0) = 0 for all x $ [0, l]. Using u(x, 0) = 0, ux (x, 0) = 0 and ut (x, 0) = 0 for all x $ [0, l] in equation (4), we conclude that V(0) = 0. The non-negativeness of V(t) for all t e 0 is obvious from equation (4). q In the following, we will use the La -norm of V, defined as >V >aMsup V(t). te0
Lemma 2.2. If >V >a Q a, then ux is X2 -bounded. More precisely, >ux >X 2 E
0
2>V >a a
1
1/2
Q a.
(5)
1117
Proof. From equation (4), we have a 2
g
l
ux2 (x, t) dx E V(t) E >V >a Q a,
(6)
0
for all t e 0. Therefore, equation (5) follows. q Remark. It can be shown that >V >a Q a if and only if u, ux and ut are X2 -bounded. However, this result is of no need in this paper. Now, we show that V can be bounded by E defined in equation (3). Lemma 2.3. The function V satisfies V(t) E K(d)E(t),
(7)
for all t e 0, where K(d) = 1 +
6
7
dl 1 + 2dl/p max 1, . p a
(8)
Proof. By Scheeffer’s inequality, which is a Poincare´-type inequality (see, e.g., reference [26, p. 67]), we have
g
l
u 2(x, t) dx E
0
l2 p2
g
l
ux2 (x, t) dx,
(9)
0
for all t e 0. Furthermore, we have the inequality
g
l
ut (x, t)u(x, t) dx E
0
l 2p
g
p 2l
l
ut2 (x, t) dx +
0
g
l
u 2(x, t) dx,
(10)
ux2 (x, t) dx,
(11)
0
for all t e 0. Using equation (9) in equation (10), we obtain
g
l
ut (x, t)u(x, t) dx E
0
l 2p
g
l
ut2 (x, t) dx +
0
g
l 2p
l
0
for all t e 0. Substituting equations (11) and (9) into equation (2), we obtain V(t) E E(t) +
dl 2p
0g
l
ut2 (x, t) dx +
0
1 + 2dl/p a
g
l
1
aux2 (x, t) dx ,
0
(12)
for all t e 0. Therefore, V(t) E E(t) +
6
dl 1 + 2dl/p max 1, p a
70 g 1 2
1
l
[ut2 (x, t) + aux2 (x, t)] dx ,
0
(13)
and, finally,
0
V(t) E 1 +
6
dl 1 + 2dl/p max 1, p a
71
E(t),
for all t e 0. Thus, equation (7) holds with K(d) given in equation (8).
(14) q
1118
Up to this point, we have proved some of the properties of the function V. Next, we show that due to the Xa -boundedness of the force f and the dissipative effect of 2dut , the function V along the solution of the system (1) is bounded. Lemma 2.4. The function V along the solution of the system (1) is a uniformly bounded function of time. Proof. From equation (4), we obtain (the argument (x, t) of the functions is deleted) V (t) =
g
g
l
[(utt + dut )(ut + du) + d 2ut u + auxt ux ] dx + b
0
l
ux2 dx
0
g
l
uxt ux dx,
(15)
0
for all t e 0. Substituting utt from equation (1a) into equation (15), we obtain
g
V (t) = −d
l
ut2 dx + da
0
g
l
uxx u dx + db
0
0 g
l
ux2 dx
+ a+b
0
g
l
ux2 dx
0
1g
g
l
uxx u dx
0
l
(uxx ut + uxt ux ) dx +
0
g
l
(ut + du)f dx,
(16)
0
for all t e 0. From the boundary conditions u(0, t) = u(l, t) = 0, we have ut (0, t) = ut (l, t) = 0 for all t e 0. Thus, we obtain
g
l
uxx u dx =
0
g
l
(uux )x dx −
0
g
g
l
g
l
ux2 dx = −
0
l
(uxx ut + uxt ux ) dx =
0
g
l
ux2 dx,
(17a)
0
(ux ut )x dx = 0.
(17b)
0
Using equation (17) and E defined in equation (3) in equation (16), we can rewrite V as V (t) E −2dE(t) +
g
l
(ut + du)f dx,
(18)
0
for all t e 0. Furthermore, we have the inequality
g
l
0
(ut + du)f dx E
g 2
g
l
(ut + du)2 dx +
0
1 2g
g
l
f 2 dx E gV(t) +
0
1 2g
g
l
f 2 dx,
(19)
0
for all t e 0, where g q 0 is an arbitrary number. We choose g Q 2d/K(d), where K(d) is that given in equation (8). Using equations (7) and (19) in equation (18), we obtain the differential inequality V (t) E −(2d/K(d) − g)V(t) +
1 f(t), 2g
(20)
for all t e 0, with V(0) = 0, where
g
l
f(t)M
0
f 2(x, t) dx,
(21)
1119
for all t e 0. By assumption (A1), f is Xa -bounded; hence it is X2 -bounded, and 0 Q f(t) E >f >2X 2 Q a for all t e 0. By a comparison theorem given in references [27, p. 29] or [28, p. 30], we conclude that the function V in equation (20) satisfies V(t) E
1 2g
g
t
e−(2d/K(d) − g)(t − t)f(t) dt,
(22)
0
for all t e 0. Therefore, V(t) E
>f >2X 2 2g
g
t
e−(2d/K(d) − g)(t − t) dt,
(23)
0
for all t e 0. By making the change of variable t − t = u in equation (23), we obtain V(t) E
>f >2X 2 2g
g
t
e−(2d/K(d) − g)u du E
0
>f >2X 2 2g
g
a
e−(2d/K(d) − g)u du,
(24)
0
for all t e 0. Therefore, V(t) E >f >2X 2 /2g(2d/K(d) − g) for all t e 0, and hence >V >a E
>f >2X 2 Q a. 2g(2d/K(d) − g)
(25) q
That is, V is a uniformly bounded function of time.
Having proved the boundedness of V, we finally prove that the displacement of the string, u, is Xa -bounded. Theorem 2.5. Consider the string represented by equations (1) for which assumption (A1) holds. The displacement of the string, u, is Xa -bounded. Proof. Using the fundamental theorem of calculus and the fact that u(0, t) = 0 for all t e 0, we obtain =u(x, t)= =
bg
x
0
b g
x
uj (j, t) dj E
=uj (j, t)= dj E
0
g
0g
l
l
=ux (x, t)= dx E l 1/2
0
0
ux2 (x, t) dx
1
1/2
, (26)
for all x $ (0, l) and t e 0, where the last inequality follows from the Cauchy–Schwarz inequality. Thus, =u(x, t)= E l 1/2 >ux >X 2 E
0
2l >V >a a
1
1/2
Q a,
(27)
for all x $ (0, l) and t e 0, where the last inequality follows from equation (5). Using equation (25) in equation (27), we obtain >u >X a E
0
l ag(2d/K(d) − g)
1
1/2
>f >X 2 Q a.
(28) q
1120
Remark. Note that equation (28) relates the norm of the displacement of the string, u, to the norm of the force f applied to it. It is clear from equation (28) that if the amplitude of the force is small, then the displacement of the string is small. 3. In this letter, we have proved the boundedness of the displacement of a damped non-linear elastic string when it is under a bounded force. If the force applied to the string is Xa -bounded, then the displacement of the string is Xa -bounded. Our proof is achieved by showing that the energy-like function V of the string is bounded for all time. In our proof, having the damping coefficient d q 0 is of crucial importance, because only in this case the coefficient of V on the right-hand side of equation (20) can be made negative: a negative coefficient for V guarantees that V is bounded. 1. G. F. C 1945 Quarterly of Applied Mathematics 3, 157–165. On the non-linear vibration problem of the elastic string. 2. R. N 1968 Journal of Sound and Vibration 8, 134–146. Non-linear vibration of an elastic string. 3. D. W. O 1960 Journal of the Acoustical Society of America 32, 1529–1538. Frequency response of a nonlinear stretched string. 4. G. K 1877 Vorlesungen u¨ber Mathematische Physik: Mechanik. Leipzig: Druck and Verlag von B. G. Teubner. 5. G. V. A 1969 Journal of the Acoustical Society of America 45, 1089–1096. Large-amplitude damped free vibration of a stretched string. 6. G. V. A 1969 Journal of the Acoustical Society of America 46, 667–677. Stability of non-linear oscillations of stretched strings. 7. A. A and S. S 1984 in Nonlinear Partial Differential Equations and Their Applications (H. Brezis and J. L. Lions, editors) VI, 1–26. Marshfield, MA: Pitman. Global solutions to the Cauchy problem for a nonlinear hyperbolic equation. 8. R. W. D 1978 Proceedings of the Royal Society of Edinburgh 82A, 19–26. The initial value problem for a nonlinear semi-infinite string. 9. R. W. D 1980 Quarterly of Applied Mathematics 38, 253–259. Stability of periodic solutions of the nonlinear string. 10. A. I. E 1972 Journal of the Acoustical Society of America 51, 960–966. Driven nonlinear oscillation of a string. 11. C. G 1984 Journal of the Acoustical Society of America 75, 1770–1776. The nonlinear free vibration of a damped elastic string. 12. M. P. M 1991 Nonlinear Analysis, Theory, Methods & Applications 17, 1125–1137. Mathematical analysis of the nonlinear model for the vibrations of a string. 13. G. P. M 1979 Nonlinear Analysis, Theory, Methods & Applications 3, 613–627. On classical solutions of a quasilinear hyperbolic equation. 14. J. M 1984 Journal of the Acoustical Society of America 75, 1505–1510. Resonant, nonplanar motion of a stretched string. 15. W. G. N 1995 Journal of Mathematical Analysis and Applications 192, 689–704. Global solution of a nonlinear string equation. 16. T. N 1971 Memoirs of the Faculty of Engineering, Kyoto University 33, 329–341. A note on the nonlinear vibrations of the elastic string. 17. K. R and G. V. A 1983 Journal of Sound and Vibration 86, 85–98. Non-linear response in strings under narrow band random excitation, part I: planar response and stability. 18. G. T 1978 Journal of Sound and Vibration 58, 95–107. Analysis of a randomly excited non-linear stretched string. 19. G. T 1983 Journal of Sound and Vibration 87, 493–511. A parametrically driven harmonic analysis of a non-linear stretched string with time-varying length. 20. G. T 1989 Journal of Sound and Vibration 129, 215–235. Wave synthesis in a non-linear stretched string with time-varying length or tension.
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21. G. T 1989 Journal of Sound and Vibration 129, 316–384. Non-linear string random vibration. 22. G. T 1995 Journal of Sound and Vibration 185, 51–78. Parametric oscillations of a non-linear string. 23. Y. Y 1987 Nonlinear Analysis, Theory, Methods & Applications 11, 1155–1168. Some non-linear degenerate wave equations. 24. K. Y and T. T 1985 Bulletin of the Japan Society of Mechanical Engineers (JSME) 28, 2699–2709. Nonlinear forced oscillations of a string, 1st report: two types of responses near the second primary resonance point. 25. K. Y and T. T 1986 Bulletin of the Japan Society of Mechanical Engineers (JSME) 29, 1223–1260. Nonlinear forced oscillations of a string, 2nd report: various type of responses near the primary resonance points. 26. D. S. M´, J. E. P ´ and A. M. F 1991 Inequalities Involving Functions and Their Integrals and Derivatives. Dordrecht: Kluwer Academic. 27. G. B and G.-C. R 1989 Ordinary Differential Equations. New York: John Wiley; fourth edition. 28. V. L, S. L and A. A. M 1989 Stability analysis of Nonlinear Systems. New York: Marcel Dekker.