Kiwamu, Terra

One thought for tracking PI mode frequency drift over a lock stretch (as seen here) is to utilize the drumhead modes: these are reliable in amplitude/presence and by monitoring their freq drift we could predictively change PI filters, etc. to handle drifting of higher modes.

Eigenfrequencies of the test masses have nominally been defined as proportional to square root of Young's modulus: f ~ E^1/2, where E := Young's modulus. The Young's modulus for fused silica has some temperature dependence ( measured to be (dE/dT)/E = 1.52e-4 / K ); as temperature increases, Young's modulus increases, so the mode frequency increases over a lock stretch. In this simple case we would expect all modes in a given test mass to have the same relative drift. Instead we see that del_f has mode dependence as well, i.e. the ~ 15007 Hz differential drumhead mode always drifts more than the ~ 8160 Hz drumhead mode of the same optic (relative to their frequency). I suspect this is due to different modes having more or less torsional versus longitudinal motion and so frequency dependence has both E (Young's) and G (shear) dependence. Peter also suggested nonuniformity of temperature in the test mass volume (as different modal motions interact with certain parts of that volume differently). To this end we define, for an arbitary mode m:

f_m = g(T) , where g is some mode-specific temperature dependent function involving E,G

     ==>  f_m + del_f_m ~= g₀ + g₁del_T + . . .  

     ==> del_f_m / f_m = g₁del_T / g₀ = A * del_T

Similarly, for another mode n: f_n + del_f_m ~= h₀ + h₁del_T  ==>  del_f_n / f_n = B * del_T where h is some function with some different temp dependence

Then the ratio of the change of frequency over frequency of those specific modes is constant: 

(del_f_m/f_m) / (del_f_n/f_n) = A / B = constant

I've quickly looked at three longer recent locks, specifically at change of drumhead modes compared to change of 15007 Hz modes of the ETMS. ETMY seems to agree on a constant while ETMX a bit wandery . I'll look more systematically tomorrow to see if I can narrow it up.

ETMY (del_Drumhead / Drumhead) / (del_15007 / 15007) = 1.38, 1.31, 1.31   |    ETMX (del_Drumhead / Drumhead) / (del_15007 / 15007) = 1.29, 1.45, 2.45