Characteristic Ratio Symmetric Polynomials and Their Root Characteristics

Abstract

For a real polynomial p(s) = a_nsⁿ + ⋯ + a₁s + a₀, its characteristic ratios are defined by ${\alpha _i}: = {{a_i^2} \mathord{\left/{\vphantom {{a_i^2} {{a_{i - 1}}{a_{i + 1}}}}} \right.\kern-\nulldelimiterspace} {{a_{i - 1}}{a_{i + 1}}}}$ for i = 1, 2, ⋯, n−1, and the generalized time constant is defined by τ ≔ a₁/a₀. In contrast, every coefficient of the polynomial p(s) can be represented in terms of α_i and τ. We present a novel family of polynomials named characteristic ratio symmetric (CRS), where a polynomial p(s) is said to be CRS if α_i = α_n−i for 1 ≤ i ≤ n − 1 with any τ. This paper deals with the relationships between the roots and {α_i, τ} of a CRS polynomial. It is shown that some of the roots of the CRS polynomial are on the circle of a specific radius ω_c while the rest appear in four-tuples $\left\{{{\lambda _i},{{\omega _c^2} \mathord{\left/ {\vphantom {{\omega _c^2} {{\lambda _i},}}} \right. \kern-\nulldelimiterspace} {{\lambda _i},}}\lambda _i^*,{{\omega _c^2} \mathord{\left/ {\vphantom {{\omega _c^2} {\lambda _i^*}}} \right. \kern-\nulldelimiterspace} {\lambda _i^ *}}} \right\}.$. For CRS polynomials of the fifth or lower order, we derive that the damping ratio and natural frequency of every root of these polynomials can be uniquely represented in terms of only {α₁, α₂, τ} or {α₁, τ} for less than third order. It is also shown that a special polynomial named K-polynomial is a CRS polynomial and the damping of an nth-order K-polynomial can be adjusted by just choosing a single parameter α₁.