In this paper, we study the limit cycle oscillations of multiple double integrators with coupled dynamics, subject to a constant disturbance term and switching inputs. Such systems arise in a variety of control problems where the minimization of both fuel and number of input transitions is a key requirement. The problem of finding the minimum switching limit cycle, among all the fuel-optimal solutions satisfying given state constraints, is addressed. Starting from well known results available for a single double integrator, two suboptimal solutions are provided for the multivariable case. First, an analytic upper bound on the number of input switchings is derived. Then, a less conservative numerical solution exploiting the additional degrees of freedom provided by the phases of the limit cycles is presented. The proposed techniques are compared on two simulation examples.