A new time integration scheme is presented for solving the differential equation of motion with nonlinear stiffness. In this new implicit method, it is assumed that the acceleration varies quadratically within each time step. By increasing the order of acceleration, more terms of the Taylor series are used, which are expected to have responses with better accuracy than the classical methods. By considering this assumption and employing two parameters δ and α, a new family of unconditionally stable schemes is obtained. The order of accuracy, numerical dissipation, and numerical dispersion are used to measure the accuracy of the proposed method. Second order accuracy is achieved for all values of δ and α. The proposed method presents less dissipation at the lower modes in comparison with Newmark's average acceleration, Wilson-θ, and generalized-α methods. Moreover, this second order accurate method can control numerical damping in the higher modes. The numerical dispersion of the proposed method is compared with three unconditionally stable methods, namely, Newmark's average acceleration, Wilson-θ, and generalized-α methods. Furthermore, the overshooting effect of the proposed method is compared with these methods. By evaluating the computational time for analysis with similar time step duration, the proposed method is shown to be faster in comparison with the other methods.

References

References
1.
Chopra
,
A.
,
2007
,
Dynamics of Structures: Theory and Applications to Earthquake Engineering
,
3rd ed.
,
Prentice-Hall
,
Upper Saddle River, NJ
.
2.
Paz
,
M.
, and
Leigh
,
W.
,
2003
,
Structural Dynamics: Theory and Computation
,
5th ed.
,
Springer
,
Netherlands
.
3.
Park
,
K. C.
,
1977
, “
Practical Aspects of Numerical Time Integration
,”
Comput. Struct.
,
7
(
3
), pp.
343
353
.10.1016/0045-7949(77)90072-4
4.
Bathe
,
K. J.
, and
Wilson
,
E. L.
,
1973
, “
Stability and Accuracy Analysis of Direct Time Integration Methods
,”
Earthquake Eng. Struct. Dyn.
,
1
, pp.
283
291
.10.1002/eqe.4290010308
5.
Keierleber
,
C. W.
, and
Rosson
,
B. T.
,
2005
, “
Higher-Order Implicit Dynamic Time Integration Method
,”
J. Struct. Eng.
,
131
(
8
), pp.
1267
1276
.10.1061/(ASCE)0733-9445(2005)131:8(1267)
6.
Chung
,
J.
, and
Hulbert
,
G. M.
,
1993
, “
A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method
,”
ASME J. Appl. Mech.
,
60
, pp.
371
375
.10.1115/1.2900803
7.
Kontoe
,
S.
,
Zdravkovic
,
L.
, and
Potts
,
D. M.
,
2008
, “
An Assessment of Time Integration Schemes for Dynamic Geotechnical Problems
,”
Comput. Geotech.
,
35
(
2
), pp.
253
264
.10.1016/j.compgeo.2007.05.001
8.
Hughes
,
T. J. R.
,
1987
,
The Finite Element Method: Linear Static and Dynamic Finite Element Analysis
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
9.
Dokainish
,
M. A.
, and
Subbaraj
,
K.
,
1989
, “
A Survey of Direct Time Integration Methods in Computational Structural Dynamics. I. Explicit Methods
,”
Comput. Struct.
,
32
(
6
), pp.
1371
1386
.10.1016/0045-7949(89)90314-3
10.
Subbaraj
,
K.
, and
Dokainish
,
M. A.
,
1989
, “
A Survey of Direct Time Integration Methods in Computational Structural Dynamics. II. Implicit Methods
,”
Comput. Struct.
,
32
(
6
), pp.
1387
1401
.10.1016/0045-7949(89)90315-5
11.
Chen
,
C.
, and
Ricles
,
J. M.
,
2008
, “
Stability Analysis of Direct Integration Algorithms Applied to Nonlinear Structural Dynamics
,”
ASME J. Eng. Mech.
,
134
(
9
), pp.
703
711
.10.1061/(ASCE)0733-9399(2008)134:9(703)
12.
Bathe
,
K. J.
,
1996
,
Finite Element Procedures
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
13.
Hughes
,
T. J. R.
, and
Belytschko
,
T.
,
1983
, “
A Precis of Developments in Computational Methods for Transient Analysis
,”
ASME J. Appl. Mech.
,
50
(
4b
), pp.
1033
1041
.10.1115/1.3167186
14.
Belytschko
,
T.
, and
Lu
,
Y.
,
1993
, “
Explicit Multi-Time Step Integration for First and Second Order Finite Element Semi-Discretizations
,”
Comput. Methods Appl. Mech. Eng.
,
108
(
3–4
), pp.
353
383
.10.1016/0045-7825(93)90010-U
15.
Hahn
,
G. D.
,
1991
, “
A Modified Euler Method for Dynamic Analysis
,”
Int. J. Numer. Methods Eng.
,
32
(
5
), pp.
943
955
.10.1002/nme.1620320502
16.
Rezaiee-Pajand
,
M.
, and
Alamatian
,
J.
,
2008
, “
Implicit Higher-Order Accuracy Method for Numerical Integration in Dynamic Analysis
,”
J. Struct. Eng.
,
134
(
6
), pp.
973
985
.10.1061/(ASCE)0733-9445(2008)134:6(973)
17.
Razavi
,
S. H.
,
Abolmaali
,
A.
, and
Ghassemieh
,
M.
,
2007
, “
A Weighted Residual Parabolic Acceleration Time Integration Method for Problems in Structural Dynamics
,”
Comput. Methods Appl. Math.
,
7
(
3
), pp.
227
238
.10.2478/cmam-2007-0014
18.
Chang
,
S. Y.
,
2007
, “
Improved Explicit Method for Structural Dynamics
,”
J. Eng. Mech.
,
133
(
7
), pp.
748
760
.10.1061/(ASCE)0733-9399(2007)133:7(748)
19.
Hilber
,
H.
, and
Hughes
,
T. J. R.
,
1978
, “
Collocation, Dissipation and Overshoot for Time Integration Schemes in Structural Dynamics
,”
Earthquake Eng. Struct. Dyn.
,
6
(
1
), pp.
99
117
.10.1002/eqe.4290060111
20.
Kavetski
,
D.
,
Binning
,
P.
, and
Sloan
,
S. W.
,
2004
, “
Truncation Error and Stability Analysis of Iterative and Non-Iterative Thomas-Gladwell Methods for First-Order Non-Linear Differential Equations
,”
Int. J. Numer. Methods Eng.
,
60
(
12
), pp.
2031
2043
.10.1002/nme.1035
21.
Goudreau
,
G. L.
, and
Taylor
,
R. L.
,
1972
, “
Evaluation of Numerical Integration Methods in Elastodynamics
,”
Comput. Methods Appl. Mech. Eng.
,
2
, pp.
69
97
.10.1016/0045-7825(73)90023-6
22.
Chang
,
S. Y.
,
2009
, “
Accurate Integration of Nonlinear Systems Using Newmark Explicit Method
,”
J. Mech.
,
25
(
3
), pp.
289
297
.10.1017/S1727719100002744
You do not currently have access to this content.