This paper deals with a forced, damped Boussinesq equation on the unit circle with quadratic nonlinearity, which represents small nonlinear oscillations of an elastic membrane in the presence of viscosity. We use the eigenfunction expansion method in order to construct global-in-time solutions with well-established convergence properties in Sobolev spaces. Existence and uniqueness follow from the construction. Numerical examples with the adaptation of the theoretical method show the viability of the method.