A numerical method to determine the stability of delay differential equations (DDEs) with time periodic coefficients is proposed. The DDE is converted into an equivalent partial differential equation (PDE) with a time periodic boundary condition (BC). The PDE, along with its BC, is then converted into a system of ordinary differential equations (ODEs) with time periodic coefficients using the Galerkin least squares approach. In the Galerkin approach, shifted Legendre polynomials are used as basis functions, allowing us to obtain explicit expressions for the approximate system of ODEs. We analyze the stability of the discretized ODEs, which represent an approximate model of the DDEs, using Floquet theory. We use numerical examples to show that the stability charts obtained with our method are in excellent agreement with those existing in the literature and those obtained from direct numerical simulation.
Skip Nav Destination
Article navigation
March 2015
Research-Article
Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients
Anwar Sadath,
Anwar Sadath
Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Ordnance Factory Estate
,Andhra Pradesh 502205
, India
Search for other works by this author on:
C. P. Vyasarayani
C. P. Vyasarayani
1
Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
e-mail: vcprakash@iith.ac.in
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Ordnance Factory Estate
,Andhra Pradesh 502205
, India
e-mail: vcprakash@iith.ac.in
1Corresponding author.
Search for other works by this author on:
Anwar Sadath
Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Ordnance Factory Estate
,Andhra Pradesh 502205
, India
C. P. Vyasarayani
Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
e-mail: vcprakash@iith.ac.in
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Ordnance Factory Estate
,Andhra Pradesh 502205
, India
e-mail: vcprakash@iith.ac.in
1Corresponding author.
Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received November 25, 2013; final manuscript received March 3, 2014; published online January 12, 2015. Assoc. Editor: Hiroshi Yabuno.
J. Comput. Nonlinear Dynam. Mar 2015, 10(2): 021011 (7 pages)
Published Online: March 1, 2015
Article history
Received:
November 25, 2013
Revision Received:
March 3, 2014
Online:
January 12, 2015
Citation
Sadath, A., and Vyasarayani, C. P. (March 1, 2015). "Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients." ASME. J. Comput. Nonlinear Dynam. March 2015; 10(2): 021011. https://doi.org/10.1115/1.4026989
Download citation file:
Get Email Alerts
Cited By
A Nodal-Lie-Group Beam Element for Absolute Nodal Coordinate Formulations
J. Comput. Nonlinear Dynam (March 2025)
Modal Analysis for Localization in Multiple Nonlinear Tuned Mass Dampers Installed on a Structure
J. Comput. Nonlinear Dynam (March 2025)
Free wave propagation in pretensioned 2D textile metamaterials
J. Comput. Nonlinear Dynam
Reduced-Order Modeling and Optimization of a Flapping-Wing Flight System
J. Comput. Nonlinear Dynam
Related Articles
Pole Placement for Delay Differential Equations With Time-Periodic Delays Using Galerkin Approximations
J. Comput. Nonlinear Dynam (September,2021)
Analytic Bounds for Instability Regions in Periodic Systems With Delay via Meissner’s Equation
J. Comput. Nonlinear Dynam (January,2012)
Galerkin Approximations for Stability of Delay Differential Equations With Distributed Delays
J. Comput. Nonlinear Dynam (November,2015)
Pole Placement for Time-Delayed Systems Using Galerkin Approximations
J. Dyn. Sys., Meas., Control (May,2019)
Related Proceedings Papers
Related Chapters
Dynamic Behavior in a Singular Delayed Bioeconomic Model
International Conference on Instrumentation, Measurement, Circuits and Systems (ICIMCS 2011)
Pre-Accidental Situations Highlighted by RECUPERARE Method and Data (PSAM-0029)
Proceedings of the Eighth International Conference on Probabilistic Safety Assessment & Management (PSAM)
A Design Method for Modified Smith Predictors for Non-Minimum-Phase Time-Delay Plants with Feedback Connected Multiple Time-Delays
Intelligent Engineering Systems through Artificial Neural Networks Volume 18