A linearized formulation for fluid-structure-interaction problems exhibiting rigid-body-motion is presented. The linearized equations are evaluated on a stationary grid and require no mesh motion. Numerical tests are performed to validate the derived formulation by comparing the time evolution between the linear and non-linear equations. In all cases both the growth rate and angular frequency of the instability matches to within 0.1% accuracy. The derived formulation is used to predict the phenomenon of symmetry breaking for a rotating cylinder with an attached splitter-plate, the onset of aeroelastic pitch-oscillations for a NACA0012 airfoil at transitional Reynolds numbers, and investigate the structural sensitivity of the least stable eigenvalue for an oscillating cylinder.