Summary: Rotating machinery support excitations can occur if a machine is installed on a base prone to ground motions or on-board moving systems such as ships and aircraft. This paper presents a formulation for the dynamic analysis of rigid rotors subject to base excitations plus mass imbalance. The formulation allows for six motions at the machine base and takes into account the linear/nonlinear spring characteristics of the supporting bearings. Equations of motion are derived using Lagrange’s equations. For rotor-linear bearing systems subject to mass imbalance plus harmonic excitations along or around lateral directions, analytical solutions for equations of motion are derived and analytical results in the time domain are compared with their counterparts obtained by numerical integration using the Runge-Kutta method and typical agreement is obtained. The system natural frequencies as affected by rotor speed are obtained using the QR algorithm using the DAMRO-1 program and compared with those obtained by MATLAB and excellent agreement is obtained. The frequency response (maximum amplitude of vibrations against the base excitation frequency) is characterized by peaks at natural frequencies of the rotating gyroscopic system. This necessitates extreme precaution when we design such rotating systems that are prone to base motions and mass imbalance. For systems with bearing cubic nonlinearity, results are obtained by numerical integration and discussed with regards to the time domain, fast Fourier transform (FFT) and Poincaré map. Periodic and quasi-periodic disk/bearings motions are observed. For systems with support cubic nonlinearity and subject to mass imbalance and base excitation, the FFT of disk horizontal and vertical vibrations is marked with sum and difference tones, \(\pm nf_{b} \pm f_{s}\) (\(n + m\) is always odd) where \(f_{s}\) is the rotating unbalance frequency and \(f_{b}\) is base excitation frequency.