Does the command DMPRAT and the material property DMPR produce a constant damping ratio in a full harmonic analysis?
No, DMPRAT and MP,DMPR don't produce a constant ratio to critical (eta) in a full harmonic. This issue has been clarified in the damping documentation at 13.0. In Theory Manual Section 15.3.2 the damping contributed by dmpr is (2*dmpr/Omega)*K. Omega is the excitation frequency. So (2*dmpr/Omega) is a coefficient times the stiffness matrix like Beta damping. Beta = 2*eta/omega (omega is natural freq)2*dmpr/Omega = 2*eta/omega
eta (ratio to critical) = dmpr(omega/Omega).
In a single degree of freedom test you will see the ratio to critical be higher at low excitation frequencies and lower at high excitation frequencies.
A 1 dof test is provided below. Note that you can't measure the damping ratio by dividing the imaginary solution by the real solution. For example, at resonance, that is infinity, not any measure of damping. The only method I know is to compare the dynamic response to the static response. At resonance, the "dynamic load factor" will be 1/2*eta, where eta is the ratio to critical. The formula for the dynamic load factor at other frequencies is:
Away from resonance, the harmonic response is not a strong function of damping, so the deviation from ratio to critical is small.
ux2=ux(2) ! static displacement = 0.253