A computer implemented method for mitigating drill tool assembly vibrations by adjusting drilling operations while drilling a wellbore, the method comprising: obtaining drilling operations parameters regarding a drilling operation; obtaining drill tool assembly data regarding a drill tool assembly design to be used in the drilling operation; calculating a baseline solution for the drill tool assembly design using the drilling operations parameters and the drill tool assembly data, wherein the baseline solution is a particular solution of the equations of motion that correspond to smooth drilling with no vibration; constructing one or more linear frequency-domain models around the baseline solution, wherein linearization of the equations of motion around the baseline solution allows independent linear harmonic analysis of the eigenstates at each vibration frequency and use of superposition to analyze the overall dynamic motion; wherein one or more of the frequency-domain models includes a velocity-dependent damping relationship; utilizing one or more of the linear frequency-domain models to transform the obtained drilling operations parameters and the obtained drill tool assembly data into one or more vibrational indices characterizing an excitation response of the drill tool assembly design under a range of available drilling operating conditions; determining preferred drilling operating conditions to mitigate vibrations based at least in part on one or more of the vibrational indices; monitoring drilling operating conditions while drilling a wellbore using the drill tool assembly; and adjusting drilling operations to maintain drilling operating conditions at least substantially within a range of the preferred drilling operating conditions. obtaining drilling operations parameters regarding a drilling operation; obtaining drill tool assembly data regarding a drill tool assembly design to be used in the drilling operation; calculating a baseline solution for the drill tool assembly design using the drilling operations parameters and the drill tool assembly data, wherein the baseline solution is a particular solution of the equations of motion that correspond to smooth drilling with no vibration; constructing one or more linear frequency-domain models around the baseline solution, wherein linearization of the equations of motion around the baseline solution allows independent linear harmonic analysis of the eigenstates at each vibration frequency and use of superposition to analyze the overall dynamic motion; wherein one or more of the frequency-domain models includes a velocity-dependent damping relationship; utilizing one or more of the linear frequency-domain models to transform the obtained drilling operations parameters and the obtained drill tool assembly data into one or more vibrational indices characterizing an excitation response of the drill tool assembly design under a range of available drilling operating conditions; determining preferred drilling operating conditions to mitigate vibrations based at least in part on one or more of the vibrational indices; monitoring drilling operating conditions while drilling a wellbore using the drill tool assembly; and adjusting drilling operations to maintain drilling operating conditions at least substantially within a range of the preferred drilling operating conditions.