Asymptotic stability of a pendulum with quadratic damping
The equation considered in this paper is x ′ ′ + h ( t ) x ′ | x ′ | + ω 2 sin x = 0 , where h ( t ) is continuous and nonnegative for t ≥ 0 and ω is a positive real number. This may be regarded as an equation of motion of an underwater pendulum. The damping force is proportional to the square of th...
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| Published in | Zeitschrift für angewandte Mathematik und Physik Vol. 65; no. 5; pp. 865 - 884 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Basel
Springer Basel
01.10.2014
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0044-2275 1420-9039 |
| DOI | 10.1007/s00033-013-0361-x |
Cover
| Abstract | The equation considered in this paper is
x
′
′
+
h
(
t
)
x
′
|
x
′
|
+
ω
2
sin
x
=
0
,
where
h
(
t
) is continuous and nonnegative for
t
≥
0
and ω is a positive real number. This may be regarded as an equation of motion of an underwater pendulum. The damping force is proportional to the square of the velocity. The primary purpose is to establish necessary and sufficient conditions on the time-varying coefficient
h
(
t
) for the origin to be asymptotically stable. The phase plane analysis concerning the positive orbits of an equivalent planar system to the above-mentioned equation is used to obtain the main results. In addition, solutions of the system are compared with a particular solution of the first-order nonlinear differential equation
u
′
+
h
(
t
)
u
|
u
|
+
1
=
0
.
Some examples are also included to illustrate our results. Finally, the present results are extended to be applied to an equation with a nonnegative real-power damping force. |
|---|---|
| AbstractList | The equation considered in this paper is
x
′
′
+
h
(
t
)
x
′
|
x
′
|
+
ω
2
sin
x
=
0
,
where
h
(
t
) is continuous and nonnegative for
t
≥
0
and ω is a positive real number. This may be regarded as an equation of motion of an underwater pendulum. The damping force is proportional to the square of the velocity. The primary purpose is to establish necessary and sufficient conditions on the time-varying coefficient
h
(
t
) for the origin to be asymptotically stable. The phase plane analysis concerning the positive orbits of an equivalent planar system to the above-mentioned equation is used to obtain the main results. In addition, solutions of the system are compared with a particular solution of the first-order nonlinear differential equation
u
′
+
h
(
t
)
u
|
u
|
+
1
=
0
.
Some examples are also included to illustrate our results. Finally, the present results are extended to be applied to an equation with a nonnegative real-power damping force. |
| Author | Sugie Jitsuro |
| Author_xml | – sequence: 1 givenname: Jitsuro surname: Sugie fullname: Sugie, Jitsuro email: jsugie@riko.shimane-u.ac.jp organization: Department of Mathematics, Shimane University |
| BackLink | https://cir.nii.ac.jp/crid/1570009752741440000$$DView record in CiNii |
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| CitedBy_id | crossref_primary_10_1007_s00605_018_1192_9 crossref_primary_10_1088_1361_6404_ac1446 crossref_primary_10_1088_1742_6596_2090_1_012008 crossref_primary_10_1155_2022_7803798 crossref_primary_10_1063_5_0159852 |
| Cites_doi | 10.2307/20022396 10.1007/b139028 10.1007/BF01231091 10.1007/s00605-011-0297-1 10.1090/S0002-9939-2013-11615-1 10.1006/jdeq.1995.1087 10.1007/s10884-012-9256-3 10.1119/1.14703 10.1093/qmath/12.1.123 10.1016/j.jmaa.2010.04.035 10.1016/0022-460X(82)90491-6 10.1016/S0029-8018(99)00026-8 10.1016/S0029-8018(98)00023-7 10.1016/0362-546X(95)00093-B 10.1007/s11071-010-9861-9 10.1016/j.na.2011.07.028 10.1016/0029-8018(82)90012-9 10.2219/rtriqr.49.209 10.2298/PIM0999119C 10.1002/9780470549162 10.2307/2303961 10.1016/j.camwa.2009.07.014 10.1007/BF03184624 10.1007/s10440-013-9839-y 10.1007/978-1-4684-9362-7 10.3233/ISP-1991-3841303 10.1080/00423114.1999.12063109 10.5957/jsr.1978.22.3.178 10.1115/1.4011142 |
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| Keywords | Phase plane analysis Asymptotic stability 34D45 37C75 Secondary 34C10 Comparison of solutions Quadratic damping force Damped pendulum 70K05 Primary 34D23 |
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| References | Wylie (CR35) 1940; 47 Smith (CR27) 1961; 12 Richardson (CR23) 1963; 11 Citterio, Talamo (CR7) 2010; 59 Sugie (CR28) 2011; 74 Dalzell (CR9) 1978; 22 Kovacic, Rakaric (CR18) 2011; 64 CR36 CR13 CR12 CR33 Simmonds (CR26) 1982; 84 Sugie (CR29) 2013; 141 Neves, Pérez, Valerio (CR21) 1999; 26 Bacciotti, Rosier (CR2) 2005 Bass, Haddara (CR3) 1991; 38 Sugie, Hata (CR30) 2012; 166 Hatvani, Krisztin, Totik (CR15) 1995; 119 Sugie, Hata, Onitsuka (CR31) 2010; 371 Peirce (CR22) 1908; 44 Biringen, Chow (CR5) 2011 Eng, Lau, Low, Seet, Chin (CR10) 2008; 16 Michel, Hou, Liu (CR19) 2008 Sugie, Shimadu, Yamasaki (CR32) 2012; 24 Cvetićanin (CR8) 2009; 85 Ahmed, Tapley (CR1) 1984; 33 Nelson, Olsson (CR20) 1986; 54 Klotter (CR17) 1955; 22 CR24 Berg (CR4) 1999; 33 Shimozawa, Tohtake (CR25) 2008; 49 Hatvani (CR14) 1995; 25 Hatvani, Totik (CR16) 1993; 6 Cardo, Francescutto, Nabergoj (CR6) 1982; 9 Halanay (CR11) 1996 Taylan (CR34) 2000; 27 M. Berg (361_CR4) 1999; 33 M. Taylan (361_CR34) 2000; 27 J. Sugie (361_CR29) 2013; 141 J. Sugie (361_CR30) 2012; 166 361_CR24 A.H. Ahmed (361_CR1) 1984; 33 A. Halanay (361_CR11) 1996 B.O. Peirce (361_CR22) 1908; 44 C.R. Wylie Jr (361_CR35) 1940; 47 J. Sugie (361_CR32) 2012; 24 P.D. Richardson (361_CR23) 1963; 11 L. Hatvani (361_CR14) 1995; 25 A. Bacciotti (361_CR2) 2005 R.A. Smith (361_CR27) 1961; 12 J. Sugie (361_CR31) 2010; 371 S. Biringen (361_CR5) 2011 M. Citterio (361_CR7) 2010; 59 I. Kovacic (361_CR18) 2011; 64 D.W. Bass (361_CR3) 1991; 38 L. Cvetićanin (361_CR8) 2009; 85 M.A.S. Neves (361_CR21) 1999; 26 A.N. Michel (361_CR19) 2008 Y.H. Eng (361_CR10) 2008; 16 D.S. Simmonds (361_CR26) 1982; 84 J.F. Dalzell (361_CR9) 1978; 22 361_CR33 361_CR12 361_CR13 R.A. Nelson (361_CR20) 1986; 54 361_CR36 L. Hatvani (361_CR15) 1995; 119 K. Shimozawa (361_CR25) 2008; 49 J. Sugie (361_CR28) 2011; 74 A. Cardo (361_CR6) 1982; 9 K. Klotter (361_CR17) 1955; 22 L. Hatvani (361_CR16) 1993; 6 |
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Mech. doi: 10.1115/1.4011142 – volume: 84 start-page: 453 year: 1982 ident: 361_CR26 publication-title: J. Sound Vibr. doi: 10.1016/0022-460X(82)90491-6 – volume: 27 start-page: 921 year: 2000 ident: 361_CR34 publication-title: Ocean Eng. doi: 10.1016/S0029-8018(99)00026-8 |
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| Snippet | The equation considered in this paper is
x
′
′
+
h
(
t
)
x
′
|
x
′
|
+
ω
2
sin
x
=
0
,
where
h
(
t
) is continuous and nonnegative for
t
≥
0
and ω is a... |
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| StartPage | 865 |
| SubjectTerms | Asymptotic stability Comparison of solutions Damped pendulum Engineering Mathematical Methods in Physics Phase plane analysis Quadratic damping force Theoretical and Applied Mechanics |
| Title | Asymptotic stability of a pendulum with quadratic damping |
| URI | https://cir.nii.ac.jp/crid/1570009752741440000 https://link.springer.com/article/10.1007/s00033-013-0361-x |
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