In this work, a numerical framework for global stability analysis of rigid-body-motion fluid– structure-interaction problems is presented. The Jacobian matrices which arise in the linearization procedure are derived numerically via the first-order finite difference scheme. The linearized fluid–structure coupled equations are solved using an immersed boundary method. The linear stability solver is first tested on two canonical cases, i.e., the flow past a stationary cylinder and the flow past an isolated elastically mounted cylinder. An excellent agreement between the results obtained here and those from available published research is achieved. The solver is then used to study the linear stability of the flow past two elastically mounted cylinders in tandem arrangement. The variations in growth rate and frequency of two leading modes with reduced velocity are examined. The mechanisms of lock-in and galloping phenomena observed in nonlinear simulation are elucidated from the perspective of linear instabilities in the leading modes.