Evaluation of effet of blade internal modes on sensitivity of Advaned LIGO T0074-00-R Norna A Robertson 5 th Otober 00. Introdution The urrent model used to estimate the isolation ahieved by the quadruple suspension system for the most sensitive mirrors (ETM and ITM), the results of whih were presented in the suspension oneptual design, (LIGO T0003-00-D), does not inlude several features whih will affet the isolation at high frequenies. These features inlude: ) effets due to the finite mass of the blades ) the violin modes of the wires One of the ations raised in the report of the SUS design requirements review (LIGO L00-00-D) was to evaluate the effets of blade internal modes. This omes under ) above. ) is not diretly addressed in this doument. In this paper we present an estimate of the peak height of the lowest in frequeny of the blade internal modes, and ompare it to the expeted sensitivity of Advaned LIGO at that frequeny, whih is limited by sapphire internal losses. We onlude that given the assumptions presented here, the internal modes of the blades will not require to be passively damped.. Isolation The isolation urves presented in the oneptual design ome from the MATLAB model of the quadruple suspension. This model urrently onsists of 4 unoupled sets of dynamial equations, orresponding to vertial motion, yaw, longitudinal and pith (together) and transverse and roll (together). In this paper we will fous on the vertial isolation as being the limiting effet. The lowest internal mode of a blade involves vertial motion. It will ouple in to horizontal motion of the mirror through rossoupling from vertial to horizontal. Higher modes ould involve twisting, whih ould produe diret horizontal motion. However the horizontal isolation should be more than adequate at the frequenies of suh internal modes, as demonstrated below.. Diret Horizontal Isolation An estimate of the residual motion due to diret horizontal exitation an be made as follows. The horizontal transfer funtion of the quad falls off as f 7, taking into aount eddy urrent damping between the first stage and its support. (If ative ontrol is used, the transfer funtion would fall as f 8 at higher frequenies where the gain is rolled off,).
The transfer funtion should ontinue to fall as a steep funtion of frequeny at least up to the violin mode frequenies. By 75 Hz (the frequeny of the lowest of the blade internal modes) it is estimated to have a value of ~.5 x0-3, assuming f 7 behaviour. This value an be ombined with the residual noise from the isolation platform, speified to be not greater than 3x0-4 m/ Hz at these frequenies, giving an overall bakground level of horizontal displaement noise of ~4.5x0-7 m/ Hz. A blade internal mode might have a loss orresponding to a Q of 0 4 (the value we assume for maraging steel). Thus a well oupled mode might produe a peak at a level of ~4.5x0-3 m/ Hz. This is well below the expeted internal thermal noise level for sapphire at 75 Hz, ~8x0 - m/ Hz.. Vertial Isolation The MATLAB model for vertial motion assumes that the blades are ideal massless springs. The overall behaviour of a blade from whose tip a mass is suspended via a wire is treated as a simple mass/spring unit, where the spring onstant is found by adding in series the spring onstant of the blade and the wire. The behaviour is dominated by the spring onstant of the blade, whih is muh less than that of the wire. In reality a blade has finite mass. A more omplete treatment of the blade/wire/mass system an be arried out, where the mass and moment of inertia of the blade are inluded. This has been done for example by Husman (999), where he uses the Lagrangian tehnique for analysing the system. The potential energy of the system onsists of two terms, orresponding to the energy stored in the blade and the wire. The kineti energy terms orrespond to the translational energy of the mass and the blade and the rotational energy of the blade. From the resulting equations of motion, the magnitude of the transfer funtion (transmissibility) from the base of the blade to that of the suspended mass an be found. Examples, using Husman s formula (see Appendix A), of the transmissibility for the three different blades in the urrent baseline design for Advaned LIGO are shown in Figure. Note that there is no damping assumed in this model. Several features should be noted. ) There are two resonanes in eah urve. The lower resonane is the familiar one orresponding to a simple spring/mass system. The upper resonane orresponds to the mode of the ombined blade/wire/mass system where the blade tip and mass move out of phase as the wire strethes between them. These peaks are suffiiently high in frequeny that they should not ompromise the overall sensitivity (but note omment in onlusions below). ) The transmissibility falls off as f - above the first resonane as expeted. However the transmissibility flattens out around 00Hz for these blades. Note that the flattening out is not due to the presene of the seond resonane. It would still be seen if the mass were rigidly attahed to the end of the blade. It is due to the finite mass of the blade and the fat that the wire suspending the mass is attahed at a point that is not the entre of perussion. The blade/wire/mass system is behaving in a manner analogous to a ompound pendulum.
These urves an be used to give an estimate of the overall vertial transmissibility of the quadruple pendulum at a partiular frequeny. We note that above the oupled resonanes of a system onsisting of several stages, the overall transmissibility tends to the produt of the individual transmissibilities of the unoupled stages. Thus an estimate of the overall transmissibility for the quadruple pendulum onsisting of these three blade Figure : Vertial transmissibility of the three blade/wire/mass stages in the baseline Advaned LIGO quadruple suspension design. Eah urve represents an unoupled stage onsisting of a partiular blade, the wire or wires suspended from it and the mass it supports in that stage alone (sine there are two blades per stage, the mass involved is half of the total mass of that stage). The blades are assumed to be triangular. Red, solid = top stage, blue, dotted = seond stage, green, dashed = third stage. Note that the peak heights are limited by the number of data points displayed. stages plus a final stage of fused silia fibres suspending the mirror an be found by multiplying the individual transmissibilities from figure, and an estimated transmissibility of the final stage..3 Estimate of Vertial Transmissibility for Quadruple Pendulum We require to alulate the transmissibility of the final stage. Sine we want to onsider the unoupled behaviour, the vertial frequeny an be found from the spring onstant of a silia fibre supporting ¼ of the load (sine there are 4 fibres). For the partiular parameters used in the baseline design (fibre radius = 00 miron, length = 0. m and 3
sapphire mass = 40 kg) this yields a frequeny of. Hz. Note that this is less than the oupled frequeny of ~8 Hz whih is more familiar from noise urves. At this point we note that the estimated frequenies of the first internal mode of the three different blades in the baseline design are 75, 98 and 8 Hz (from top to bottom). So we shall onsider the situation at the lowest of these frequenies, namely 75 Hz, where in general the isolation is less. The produt of the transmissibilities is thus given approximately by 3x0-3 x 9x0-3 x.5x0 - x (./75) =.8x0-9 The residual noise on the isolation platform supporting the quadruple pendulum is taken as 3x0-4 m/ Hz. We further assume a ross-oupling of 0.% from vertial to horizontal, thus giving an overall horizontal noise level of 8.4x0 - m/ Hz. To put this into ontext with respet to any internal modes, suh peaks if fully oupled might appear at a level of Q above this bakground. For Q=0 4, this would give a peak height of 8.4x0 - m/ Hz. This is a fator of ~0 below the estimated sapphire internal thermal noise level..4 Consideration of Effet of Damping on Vertial Isolation. The above estimation has been made by multiplying the transmissibilities of 4 stages, assuming no damping. However if eddy urrent damping is used the uppermost stage will be damped, and this will effet its transmissibility. To test the signifiane of damping on the onlusions above, a model of a blade/wire/mass system with damping was required. A slightly simpler model was used for this one in whih the effet of the wire strething is not inluded. The resulting transfer funtion an be obtained by letting the spring onstant of the wire go to infinity in the formula derived by Husman. It an also be derived diretly from writing down the equation of motion for suh a system, and it was heked that these two methods yielded the same relationship. It was then straightforward to inorporate a damping term in the latter formulation (see Appendix A). Figure shows the results of this analysis arried out on the top blade i.e. in the uppermost stage, the stage in whih damping will be applied. Several features should be noted. ) On omparing the original urve inluding the finite wire spring onstant with the urve where the wire is assumed infinitely stiff, we see in the latter urve the flattening of transmissibility without the high frequeny peak. ) The effet of damping (here hosen to give a Q of the system of around 5) is to slightly modify the urve between the resonane and the region of flattening, but the additional effet of the damping by 75 Hz is not signifiant. Thus we an onlude that our original estimate of noise level is valid in the presene of damping. 4
Figure : Vertial transmissibility of the top blade/wire/mass stage in the baseline Advaned LIGO quadruple suspension design. Red dashed line shows the original, undamped ase as in figure. Blue dotted line shows the same system with an infinitely stiff wire. Blak solid line is damped version of the latter, with a Q value of ~5. 3. Conlusions From the arguments presented here, it appears that it should not be neessary to passively damp the internal modes of the blades. However one feature whih this analysis reminds us of is that there is another family of peaks orresponding to the upper modes of the blade/wire/mass systems. And we have also not inluded the violin modes in this analysis. An undesirable situation ould arise where there is overlap of mode frequenies e.g. of an internal blade resonane and a wire strething resonane. A hek should be made on suh a potential overlap one the baseline design parameters are more firmly hosen, and before ommitting to a final design. It should be noted that in the analysis presented here several parameter values have been used whih ould take different values in a future design. ) The so-alled shape fator, α, whih is a geometri fator dependent on blade shape, and is used in the alulation of spring onstant, has been taken to equal.38 here (α= for retangle, α=.5 for triangle). This value was experimentally found to 5
fit measurements of spring onstants for the blades designed for GEO 00. Measurements on blades more reently aquired have suggested that a value of around.5 fits the new experimental data better. Further investigation of blade design using finite element analysis will help to more fully explore their behaviour under load and should lead to a better estimation of this fator. For the purposes of this doument we have hosen to use the value whih leads to a larger spring onstant, and hene higher unoupled frequenies, whih gives a more onservative estimate of the isolation. ) The blade design has been arried out assuming that the maximum stress in the blades should not exeed ~800 MPA, a value whih is ~ half of the yield strength. Again this may be onservative, and one ould push up the stress to say ~ /3 of the yield strength, and still be within the linear stress/strain region. By doing this, one ould push up the internal mode frequenies of the blades, by a fator whih for the first internal mode is equal to the ratio of the stresses, assuming the spring onstant is not hanged. An example is given in the Appendix. A summary of the key blade equations an be found in Torrie (999). It should be noted that if better high frequeny vertial isolation were required, one method of doing this ould be to modify a blade suh that the mass is suspended effetively at the blade s entre of perussion. For example Husman analysed and experimentally investigated this effet for a GEO blade, by extending the tip and adding a suitably hosen small mass to the new tip to move the entre of perussion. In pratie however, adding more mass will redue the internal mode resonant frequeny, and so it is not simple to predit whether suh a modifiation gives any signifiant redution in the peak motion assoiated with the resonane. Referenes Husman, ME, 999, PhD Thesis Suspension and Control for Interferometri Gravitational Wave Detetors (University of Glasgow) Torrie, CIE, 999, PhD Thesis Development of Suspensions for the GEO 00 Gravitational Wave Detetor (University of Glasgow)
Appendix A: Summary of Equations and Parameters Used A Model Used by Husman (999) y o θ blade y y (looking from the side) The variables y o, y and y measure the vertial displaement of the base to whih the blade is rigidly attahed, the tip of the blade, and the suspended mass respetively. Note that the suspended mass is assumed point-like in this analysis. The transfer funtion / o is given by (.f. eqn. 4.43 in Husman) 0 k w ( ml 9I ant ) s + 9kl ) 4 ( 9I m + m l m ) s + ( 9I k + m l k + 9( k + k ) l m ) s + 9k k l = eqn. ant ant w where k w and k are the spring onstants of the wire and the antilever blade respetively, m and m are the masses of the blade and suspended mass respetively, I ant is the moment of inertia of the blade about a vertial axis through its entre and l is the length of the blade. For a triangular blade I = m l /8. ant A simplified model an be derived assuming y = y, whih is equivalent to setting k w. In that ase, and for a triangular blade, the transfer funtion beomes w w w 0 ml s + kl = eqn. ( m l + l m ) s + k l A Derivation of Simpler Model Inluding Damping. The relationship given in eqn. an be derived diretly from the equation of motion of a blade lamped at one end with a (point) mass rigidly attahed to its tip. 7
y 0 θ y Consider the rotational equation for the motion of the blade ( y y0 ) l ( m m ) y lm θ + eqn. 3 Itot, 0 = Γ = k 0 Here I tot,0 is the moment of inertia of the blade plus mass through a vertial axis at the wide end of the blade rigidly attahed to the ground. The first term on the right hand side is the restoring torque due to the blade when its tip is y y0 defleted by θ, where θ =. The seond term is the torque introdued by the l aeleration of the ground, y 0, where l m is the position of the entre of mass of the blade/mass assembly, measured from the wide end of the blade. I tot,0 is given by I blade,0 + I mass,0. Using the parallel axis theorem, the first term = I m ( l / 3) = m l / ant +, where we have assumed a triangular blade, for whih the entre of mass of the blade is at l /3 from the base. The seond term is m l. ( m + 3m ) It an be shown that l m is given by lm = l. 3 m + m ( ) Combining all of the above, and using Laplae transforms, we an derive eqn. To add damping, we inlude a fator in eqn. 3 on the right hand side of the form bθ. The resulting transfer funtion is 0 ( m l + l m ) s + bs + k l l l m s + bs + k = eqn. 4 The value of b an be suitably hosen to give a resonane with a ertain Q value. 8
A 3 Parameters Used in Figures and. Blade (top) k=4.9*0^3; kw=.4*0^5; m=.4; l=0.5; m=8.; for damping b=(m*l*l/q)*((k/m)^0.5) and Q=5 Blade Blade 3 k=4.5*0^3; kw=.*0^; m=0.8; l=0.48; m=8.; k=4.47*0^3; kw=.4*0^; m=0.54; l=0.4; m=3.; A 4 Example of Two Blade Designs: same k, different first internal mode frequeny Blade (as above) spring onstant 4.9*0^3 N/m length = 0.5 m, base = 0.4 m, thikness = 0.005 m max. stress = 880 MPa first internal mode = 75 Hz Blade (alternative) spring onstant 4.9*0^3 N/m length = 0.457 m, base = 0.095 m, thikness = 0.005 m max. stress = 050 MPa first internal mode = 90 Hz ------------------------------------------------------------------------------------------------------------ 9