Spinning diskΒ±spindle systems consisting of an elastic disk mounted on an elastic spindle by means of a three-dimensional, rigid clamp extend the rich literature on spinning disks and spinning shafts that are decoupled from each other. This work presents an exact, closed-form solution for the eigensolutions of such systems. The complex eigenfunctions have the classical properties of a gyroscopic system when the individual disk, spindle and clamp deΒ―ections for a given eigenfunction are collected in terms of an extended eigenfunction. Eigenvalue perturbations are calculated to determine the sensitivity of the zero speed eigenvalues and the critical speeds to system parameters. Additionally, critical speeds analogous to those of a rigidly supported (classical) spinning disk are examined for the coupled system. Whereas the rigidly supported disk does not experience critical speed instability in the one-nodal diameter eigenfunctions, the coupled system does. The exact solution admits a closedform modal analysis for the forced response to disk, spindle and clamp excitations. Response is calculated for two examples that demonstrate the strong diskΒ±spindle modal coupling that can exist and the potentially damaging transmission of excitation energy between the disk and spindle.