Angular region: a measure of underdamped behavior

The natural modes of an underdamped dynamical system are given by the characteristic numbers of the quadratic operator pencil

$\text{P(s)=s}{}^2\text{I+sB+A,}$

where the operator A depends on the dissipative and reactive elements of the system, while B depends solely on the reactive elements. The operator P(s) for every applied stimulus vector signal x must satisfy:

$\text{(Bx,x)}{}^2\text{<4(Ax,x).}$

A measure of underdamped behaviour is suggested by predetermining an angular region |φ| containing all natural modes of the system,

$\text{|}\text{tan}\text{φ|?}\text{[4(Ax,x)?(Bx,x)}{}^2\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(Bx,x)}\text{.}$

When a comparison between positive operators A and B is available, say B²=KA, then

$\text{|}\text{tan}\text{φ|?}\text{√(4?K}{}^2\text{)}\text{K}\text{.}$

The paper is motivated by Duffin-Krein-Gohberg's earlier mathematical contributions.