Improved numerical solutions of nonlinear oscillatory systems

SUMMARY We consider numerical solutions of nonlinear oscillatory systems where closed‐form solutions do not exist. Such systems occur in buckling of columns, electrical oscillations of circuits containing inductance with an iron core, and vibration of mechanical systems with nonlinear restoring forc...

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Bibliographic Details
Published in:International journal for numerical methods in engineering 2020-05, Vol.121 (9), p.1898-1914
Main Author: Kaunda, Modify A. E.
Format: Article
Language:English
Subjects:
Algorithms
Asymptotic properties
Degrees of freedom
generalized Newmark scheme
Inductance
Initial conditions
Laplace transforms
Liapunov stability
Mechanical systems
Newmark trapezoidal scheme
nonlinear oscillatory systems
Nonlinear systems
one‐step multiple‐value methods
Stability
Truncation errors
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Summary:SUMMARY We consider numerical solutions of nonlinear oscillatory systems where closed‐form solutions do not exist. Such systems occur in buckling of columns, electrical oscillations of circuits containing inductance with an iron core, and vibration of mechanical systems with nonlinear restoring forces. We have improved the accuracy, stability, and speed of the generalized Newmark scheme of Zienkiewicz and Taylor using one‐step multiple‐value algorithms. Previously, a Laplace transform‐based method was developed to determine initial conditions of higher order derivatives of predictor algorithms together with the corrector algorithms cast in form of higher order Newton‐Raphson schemes, which are considered as consistent tangent operators that preserve quadratic rate of asymptotic convergence characteristics. For convenience, algorithms are applied to the solution of the van der Pol and Duffing's equations to show the concepts, but the procedures can also be applied to other systems such as the Bouc's, Coulomb's, and Mathieu equations. Comparisons are carried out regarding speed, accuracy, and stability, of results from the Runge‐Kutta and one‐step multiple‐value methods with remainder (truncation error) terms, previously not considered. Results compare favorably. A system of two‐degrees of freedom is also covered to illustrate how to extend the methods to deal with multiple‐degree of freedom systems.
ISSN:0029-5981
1097-0207
DOI:10.1002/nme.6292