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# Asymptotics for the Characteristic Roots of Delayed Dynamic Systems

[+] Author and Article Information
Pankaj Wahi1

Mechanical Engineering, Indian Institute of Science, Bangalore 560012, Indiapankaj@mecheng.iisc.ernet.in

Anindya Chatterjee

Mechanical Engineering, Indian Institute of Science, Bangalore 560012, Indiaanindya@mecheng.iisc.ernet.in

Here $f$ is the chip thickness, and $Fx$ is the $x$ component of the cutting force.

Our calculations were done using MAPLE 6 (Windows), which, for these irrational powers, needs a little patience. We found it useful to do the expansion one term at a time. For each term, we divided by the (known) largest surviving power of $N$, and then asked for the limit as $N→∞$.

Note that we have considered $β$ fixed as $ϵ→0$. An argument allowing $β$ to grow as $ϵ→0$ can be developed, but is trickier and avoided here.

1

Corresponding author.

J. Appl. Mech 72(4), 475-483 (Oct 29, 2004) (9 pages) doi:10.1115/1.1875492 History: Received December 01, 2003; Revised October 29, 2004

## Abstract

Delayed dynamical systems appear in many areas of science and engineering. Analysis of general nonlinear delayed systems often begins with the linearized delay differential equation (DDE). The study of these linearized constant coefficient DDEs involves transcendental characteristic equations, which have infinitely many complex roots not obtainable in closed form. Here, after motivating our study with a well-known delayed dynamical system model for tool vibrations in metal cutting, we obtain asymptotic expressions for the large characteristic roots of several delayed systems. These include first- and second-order DDEs with single delays, and a first-order DDE with distributed as well as multiple incommensurate delays. For reasonable magnitudes of the coefficients of the DDEs, the approximations in each case are very good. Subsequently, a fourth delayed system involving coefficients of disparate magnitude is analyzed using an alternative asymptotic strategy. Finally, the large root asymptotics are complemented with calculations using Padé approximants to find all the roots of these systems.

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## Figures

Figure 3

Regions considered in the complex plane

Figure 4

Roots of Eq. 4 with ψ=0.1 and p=2

Figure 1

Simple model for tool vibrations

Figure 2

Roots of Eq. 6 for a=1

Figure 5

Characteristic roots of Eq. 19 for a1=a2=a3=a4=1

Figure 6

Roots of Eqs. 39,40 for a=1 and ϵ=0.05. Plus signs: Newton-Raphson. Circles: asymptotic, Case (1,1). Triangles: asymptotic, Case (m,3). Rectangles: asymptotic, Case (2,2).

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