  Consider the problem of finding the maximal nonpositive solvent $\Phi$ of the
quadratic matrix equation (QME) $X^2 + BX + C =0$ with $B$ being a nonsingular
$M$-matrix and $C$ an $M$-matrix such that $B^{-1}C\ge 0$, and $B - C - I$ a
nonsingular $M$-matrix. Such QME arises from an overdamped vibrating system.
Recently, Yu et al. ({\em Appl. Math. Comput.}, 218: 3303--3310, 2011) proved
that $\rho(\Phi)\le 1$ for this QME. In this paper, we slightly improve their
result and prove $\rho(\Phi)< 1$, which is important for the quadratic
convergence of the structure-preserving doubling algorithm. Then, a new
globally monotonically and quadratically convergent structure-preserving
doubling algorithm to solve the QME is developed. Numerical examples are
presented to demonstrate the feasibility and effectiveness of our method.
