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.

对于一个实际的多项式p（小号）=一个_ñ小号^Ñ +⋯+一个₁小号+一个₀，其特性比通过定义{\阿尔法_i} \（：= {{A_I ^ 2} \ mathord {\左/ {\ vphantom {{a_i ^ 2} {{a_ {i-1}} {a_ {i + 1}}}}}} \ right。\ kern- \ nulldelimiterspace} {{a_ {i-1}} {a_ { I + 1}}}} \）为我= 1，2，⋯，ñ -1，和广义时间常数由下式定义τ ≔一个₁ /一个₀。相反，多项式p（s的每个系数）可以在以下方面来表示α_我和τ。我们目前命名为多项式的新家族特征比对称（CRS），其中一个多项式p（小号）被说成是CRS如果α_我= α _{ñ -我}1≤我≤ ñ - 1与任何τ。这种与根和{之间的关系纸优惠α_我，τ一个CRS多项式的}。结果表明，一些多项式的CRS的根的是在特定的半径的圆周ω _Ç而其余部分以四元组\（\ 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 {{\ Ω_c ^ 2} {\ lambda _i ^ *}}} \ right。\ kern- \ nulldelimiterspace} {\ lambda _i ^ *}}} \ right \}。\）。对于第五或较低阶的CRS多项式，我们推导出这些多项式的每根的阻尼比和自然频率可以在仅而言唯一表示{ α ₁，α ₂，τ }或{ α ₁，τ }为小于三阶。还显示了一个名为K多项式的特殊多项式是CRS多项式和的阻尼Ñ阶K-多项式可以通过仅选择一个单一的参数来调整α ₁。