Accuracy and reliability of the numerical simulations for nonlinear dynamical systems are investigated with fourth-order Runge–Kutta method and a newly developed piecewise-constant (P-T) method. Nonlinear dynamic systems with external excitations are studied and compared with the two numerical approaches. Semianalytical solutions for the dynamic systems are developed by the P-T approach. With employment of a periodicity-ratio (PR) method, the regions of regular and irregular motions are determined and graphically presented corresponding to the system parameters, for the comparison of accuracy and reliability of the numerical methods considered. Central processing unit (CPU) time executed in the numerical calculations with the two numerical methods are quantitatively investigated and compared under the same computational conditions. Due to its inherent drawbacks, as found in the research, Runge–Kutta method may cause information missing and lead to incorrect conclusions in comparing with the P-T method.

References

References
1.
Nakamura
,
S.
,
1991
,
Applied Numerical Methods With Software
,
Prentice Hall
,
New Jersey
.
2.
Zingg
,
D. W.
, and
Chisholm
,
T. T.
,
1999
, “
Runge–Kutta Methods for Linear Ordinary Differential Equations
,”
Appl. Numer. Math.
,
31
, pp.
227
238
.10.1016/S0168-9274(98)00129-9
3.
Abukhaled
,
M. I.
, and
Allen
,
E. J.
,
1998
, “
A Class of Second-Order Runge–Kutta Methods for Numerical Solution of Stochastic Differential Equations
,”
Stochastic Anal. App.
,
16
, pp.
977
992
.10.1080/07362999808809575
4.
Zhang
,
S.
, and
Li
,
J.
,
2011
, “
Explicit Numerical Methods for Solving Stiff Dynamical Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
6
(
4
), p.
041008
.10.1115/1.4003706
5.
Wang
,
J.
,
Rodriguez
,
J.
, and
Keribar
,
R.
,
2010
, “
Integration of Flexible Multibody Systems Using Radau IIA Algorithms
,”
ASME J. Comput. Nonlinear Dyn.
,
5
(
4
), p.
041008
.10.1115/1.4001907
6.
Kuinian
,
L.
, and
Antony
,
P.
,
2009
, “
A High Precision Direct Integration Scheme for Nonlinear Dynamic Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
4
(
4
), p.
041008
.10.1115/1.3192129
7.
Shampine
,
L. F.
, and
Watts
,
H. A.
,
1971
, “
Comparing Error Estimators for Runge–Kutta Methods
,”
Math. Comput.
,
25
(
115
), pp.
445
455
.10.1090/S0025-5718-1971-0297138-9
8.
Hull
,
T. E.
,
Enright
,
B. M.
,
Fellen
,
B. M.
, and
Sedgwick
,
A. E.
,
1972
, “
Comparing Numerical Methods for Ordinary Differential Equations
,”
SIAM J. Numer. Anal.
,
9
(
4
), pp.
603
637
.10.1137/0709052
9.
Enright
,
W. H.
, and
Hayes
,
W. B.
,
2007
, “
Robust and Reliable Defect Control for Runge–Kutta Methods
,”
ACM Trans. Math. Software
,
33
(
1
), pp.
1
19
.10.1145/1206040.1206041
10.
Dai
,
L.
, and
Singh
,
M. C.
,
1997
, “
An Analytical and Numerical Method for Solving Linear and Nonlinear Vibration Problems
,”
Int. J. Solid Struct.
,
34
, pp.
2709
2731
.10.1016/S0020-7683(96)00169-2
11.
Dai
,
L.
, and
Singh
,
M. C.
,
2003
, “
A New Approach With Piecewise-Constant Arguments to Approximate and Numerical Solutions of Oscillatory Problems
,”
J. Sound Vib.
,
263
(
3
), pp.
535
548
.10.1016/S0022-460X(02)01065-9
12.
Abramson
,
H. N.
, and
Silverman
,
S.
,
1966
, “
The Dynamic Behavior of Liquids in Moving Containers
,” Paper No. NASA SP-106.
13.
Dodge
,
F. T.
,
2000
, “
 The New Dynamic Behavior of Liquids in Moving Containers
,” Southwest Research Institute, San Antonio, TX.
14.
Berlot
,
R. R.
,
1959
, “
Production of Rotation in a Confined Liquid Through Translational Motion of the Boundaries
,”
ASME J. Appl. Mech.
,
26
, pp.
513
516
.10.1115/1.3643980
15.
Dai
,
L.
, and
Singh
,
M. C.
,
1997b
, “
Diagnosis of Periodic and Chaotic Responses in Vibratory Systems
,”
J. Acoust. Soc. Am.
,
102
(
6
), pp.
3361
3371
.10.1121/1.420393
16.
Dai
,
L.
,
Xu
,
L.
, and
Han
,
Q.
,
2006
, “
Semi-Analytical and Numerical Solutions of Multi-Degree-of-Freedom Nonlinear Oscillation Systems With Linear Coupling
,”
Commun. Nonlinear Sci. Numer. Sci.
,
11
, pp.
831
844
.10.1016/j.cnsns.2004.12.009
17.
Dai
,
L.
, and
Wang
,
G.
,
2007
, “
Implementation of Periodicity-Ratio in Analyzing Nonlinear Dynamic Systems—A Comparison With Lyapunov-Exponent
,”
ASME J. Comput. Nonlinear Dyn.
,
3
(
1
), p.
011006
.10.1115/1.2802581
18.
Dai
,
L.
, and
Wang
,
X.
,
2013
, “
Diagnosis of Nonlinear Oscillatory Behavior of a Fluttering Plate With a Periodicity Ratio Approach
,”
Nonlinear Eng.
,
1
(
3–4
), pp.
67
76
.10.1515/nleng-2013-0001
19.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1979
,
Nonlinear Oscillations
,
Wiley
,
New York
.
20.
Brennan
,
M. J.
,
Kovacic
,
I.
,
Carrella
,
A.
, and
Waters
,
T. P.
,
2008
, “
On the Jump-up and Jump-Down Frequencies of the Duffing Oscillator
,”
J. Sound Vib.
,
318
, pp.
1250
1261
.10.1016/j.jsv.2008.04.032
21.
Junyi
,
C.
,
Chengbin
,
M.
,
Hang
,
X.
, and
Zhuangde
,
J.
, “
Nonlinear Dynamics of Duffing System With Fractional Order Damping
,”
ASME: J. Comput. Nonlinear Dyn.
,
5
(
4
), p.
041012
.10.1115/1.4002092
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