Nitin K. Garg Nonlinear Dynamics Lab, Department of Mechanical & Aerospace Engineering, University of Florida, Gainesville, FL 32611 e-mail: nitingar@ufl.edu Brian P. Mann 1 Dynamical Systems Lab, Department of Mechanical & Aerospace Engineering, University of Missouri, Columbia, MO 65211 e-mail: mannbr@missouri.edu Nam H. Kim Structural and Multi-disciplinary Optimization Lab, Department of Mechanical & Aerospace Engineering, University of Florida, Gainesville, FL 32611 e-mail: nkim@ufl.edu Mohammad H. Kurdi Machine Tool Research Center, Department of Mechanical & Aerospace Engineering, University of Florida, Gainesville, FL 32611 e-mail: mhkurdi@ufl.edu Stability of a Time-Delayed System With Parametric Excitation This paper investigates two different temporal finite element techniques, a multiple ele- ment (h-version) and single element (p-version) method, to analyze the stability of a system with a time-periodic coefficient and a time delay. The representative problem, known as the delayed damped Mathieu equation, is chosen to illustrate the combined effect of a time delay and parametric excitation on stability. A discrete linear map is obtained by approximating the exact solution with a series expansion of orthogonal polynomials constrained at intermittent nodes. Characteristic multipliers of the map are used to determine the unstable parameter domains. Additionally, the described analysis provides a new approach to extract the Floquet transition matrix of time periodic systems without a delay. DOI: 10.1115/1.2432357 Keywords: bifurcation, delay differential equations, parametric excitation, Mathieu equation 1 Introduction The stability of systems governed by time-periodic differential equations is important to various fields of science and engineer- ing. For instance, recent literature has described applications in high-speed milling 1–5, quantum mechanics, structures under oscillating loads, and rotating helicopter blades 6. Some of the methods available for stability analysis are Hill’s method 7–9, Floquet theory 10–16, and perturbation 7,17,18. Sinha and Wu 19, Sinha 20, Sinha et al. 21–23, Butcher et al. 24, Ma et al. 25,26, Bueler et al. 27, and Szabo and Butcher 28 have used Chebyshev polynomials to analyze the stability and control of time-periodic systems. The effect of time delay on control stability has been examined by Yang and Wu 6, Horng and Chou 29, and Chung and Sun 30, who studied the effect of a time delay on structural dynamics. The delayed damped Mathieu equation DDME provides a representative system with both a time delay and parametric ex- citation. Mathieu 31 used this equation, without the time-delay and damping terms, to study the oscillations of an elliptic mem- brane. Bellman and Cooke 32 and Bhatt and Hsu 33 both made attempts to lay out the criteria for stability using the D-subdivision 34 method combined with the theorem of Pontryagin 35. In- sperger and Stépán have used an analytical and semi- discretization approach, which is applicable to a combination of problems with finite and functional time delays, to examine the stability of the DDME 36–38. The use of orthogonal polynomials to solve systems with para- metric excitation and no time delay has been adopted by many authors. For instance, Chang et al. 39 studied the response of linear dynamic systems. Sinha and Chou 40 and Sinha et al. 13 used orthogonal polynomials to investigate the behavior of time- periodic differential equations. Orthogonal polynomials are used because they decouple the successive solutions, as presented in Ref. 41, which will increase the rate of convergence, thereby making the process less computationally expensive. In this paper, interpolated orthogonal polynomials are used to determine the sta- bility of a system with both parametric excitation and time de- layed feedback. The present work describes two different temporal finite ele- ment analysis approaches that can be used to ascertain the stability behavior of both linear autonomous and nonautonomous systems with a single time delay see previous work in Refs. 4 and 42–46. In particular, this paper examines the stability of the DDME using temporal finite element analysis. A set of orthogonal polynomials, constrained for C 1 continuity, are used to obtain a discrete linear map that closely approximates the exact solution. Characteristic multipliers of the map, which are obtained from the finite-dimensional monodromy operator that closely approximates the actual infinite-dimensional system, are then used to determine the stable and unstable parameter domains. Two different approaches are used to formulate dynamic maps that describe the system evolution: i a multiple element method is described that divides the minimum time period of the system into a finite number of temporal elements. An approximate solu- tion is then obtained for a single period as a linear combination of interpolated polynomials. This technique employs cubic polyno- mials as trial functions to approximate the exact solution. Asymptotic convergence of the approximated solution to the exact solution is obtained by increasing the number of elements that discretize the time domain. This approach is called h-convergence as in spatial finite element analysis. ii A single-element method that utilizes a linear combination of higher-order orthogonal poly- nomials, coupled with C 1 continuity, is applied in the second ap- proach. These polynomials approximate the exact solution by uti- lizing a single temporal element. Asymptotic convergence of the approximated solution to the exact solution is obtained by increas- 1 Corresponding author. Contributed by the Dynamic Systems, Measurement, and Control Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS MEASUREMENT, AND CON- TROL. Manuscript received September 10, 2004; final manuscript received May 11, 2006. Assoc. Editor: Jordan Berg. Journal of Dynamic Systems, Measurement, and Control MARCH 2007, Vol. 129 / 125 Copyright © 2007 by ASME