  We prove new enclosures for the spectrum of non-selfadjoint operator matrices
associated with second order linear differential equations $\ddot{z}(t) + D
\dot{z} (t) + A_0 z(t) = 0$ in a Hilbert space. Our main tool is the quadratic
numerical range for which we establish the spectral inclusion property under
weak assumptions on the operators involved; in particular, the damping operator
only needs to be accretive and may have the same strength as $A_0$. By means of
the quadratic numerical range, we establish tight spectral estimates in terms
of the unbounded operator coefficients $A_0$ and $D$ which improve earlier
results for sectorial and selfadjoint $D$; in contrast to numerical range
bounds, our enclosures may even provide bounded imaginary part of the spectrum
or a spectral free vertical strip. An application to small transverse
oscillations of a horizontal pipe carrying a steady-state flow of an ideal
incompressible fluid illustrates that our new bounds are explicit.
