The periodic boundary value problem for a linear O.D.E. \(\ddot x + \alpha \dot x + ({2 \pi \over T})^2x = \lambda f(t) x\) \((0 \leq t \leq T)\), \(x(0) = x(T)\), \(\dot x(0) = \dot x(T)\), with spectral parameter \(\lambda\) is studied. Here a damping coefficient \(\alpha \neq 0\) and a smooth \(T\)-periodic function \(f \neq 0\) are given, the latter being allowed to vanish on some part of \([0,T]\). Under suitable assumptions on the Fourier coefficients of \(f\) the authors prove the existence of characteristic values \(\lambda \in \mathbb{R}\) and corresponding nontrivial solutions \(x(t)\). Their approach uses certain nonlinear branching equations and topological degree theory. Applications to Hill's equation are discussed. In a final section the results are employed to investigate the stability of a cylindrical shell which is subject to the action of compression or tension forces.