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TECHNICAL PAPERS

# Stability of a Time-Delayed System With Parametric Excitation

[+] Author and Article Information
Nitin K. Garg

Nonlinear Dynamics Lab, Department of Mechanical & Aerospace Engineering,  University of Florida, Gainesville, FL 32611nitingar@ufl.edu

Brian P. Mann1

Dynamical Systems Lab, Department of Mechanical & Aerospace Engineering,  University of Missouri, Columbia, MO 65211mannbr@missouri.edu

Nam H. Kim

Structural and Multi-disciplinary Optimization Lab, Department of Mechanical & Aerospace Engineering,  University of Florida, Gainesville, FL 32611nkim@ufl.edu

Machine Tool Research Center, Department of Mechanical & Aerospace Engineering,  University of Florida, Gainesville, FL 32611mhkurdi@ufl.edu

1

Corresponding author.

J. Dyn. Sys., Meas., Control 129(2), 125-135 (May 11, 2006) (11 pages) doi:10.1115/1.2432357 History: Received September 10, 2004; Revised May 11, 2006

## Abstract

This paper investigates two different temporal finite element techniques, a multiple element (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.

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## Figures

Figure 1

Multiple element stability predictions (graph A) from characteristic multiplier (μ) trajectories in the complex plane (shown by graphs B and C). The following parameters were used to generate these graphics ω=1, τ=2π, κ=0.1, and b=0.04.

Figure 2

Delayed Mathieu equation stability chart for two damping coefficient values: solid line indicates κ=0.1; dotted line indicates κ=0.2. The other parameters used to generate this graph are ω=1, τ=2π, and ϵ=0.

Figure 3

Three-dimensional plot depicting stability regions for the DDME with respect to parameters δ, b, and ϵ. Shaded regions are stable, and unstable regions are transparent. Predictions are for κ=0.2, τ=2π, and ω=1.

Figure 4

Fifth-order interpolated polynomials plotted as a function of the normalized local time

Figure 5

Stability boundaries are computed when evaluating the Floquet matrix over a period of 2π (dotted line) and a period of 2 (solid line). Stable simulated results are shown by +, and unstable simulation results are shown by ▿. Other parameters used for the computations are κ=0.2, b=0.01, τ=2π, and ω=π.

Figure 6

Stability boundaries are computed when evaluating the Floquet matrix over T=2π (dotted line) and T=4π (solid line). Stable simulated results are shown by + and unstable simulation results are shown by ▿. Other parameters used for the computations are κ=0.2, b=0.01, τ=2π, and ω=1∕2.

Figure 7

Stability boundaries of damped Mathieu equation. Predictions are for κ=0.1 (solid line) and κ=0.2 (dotted line).

Figure 8

Gradient plot of eigenvalues (μ) in δ versus ϵ parameter space

Figure 9

Convergence of eigenvalue (μ) to true eigenvalue with an increase in polynomial order

## Errata

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