(L.L.F.) CURVES & PSD DERIVATION METHOD FOR EQUIPMENT MOUNTED ON SATELLITE PANELS

Transcription

1 (..F.) UV & D DVN MHD F QUMN MUND N N ndrea eresetti (*), hijs Van Der aan (#) (*) N Z - orino, taly; (#) / - Noordwijk, he Netherlands B his paper presents, for all those cases of equipment mounted on structural panels of a satellite, the approach of derivation of the imit oad Factor (..F.) curves related to the equipment mechanical design as well as the cceleration ower pectral Density (D) input spectra related to the equipment qualification vibration tests. Keywords imit load, effective/theoretical amplification factor, (..F.) curve, random vibration spectra, notching, hardmounted test, power spectral density, vibroacoustic. Nomenclature area of an impinged structural element (m 2 ) peak sound induced pressure peak (N/m 2 ) BW frequency bandwidth (Hz) D acceleration power spectral density (g 2 /Hz) c viscous damping of an oscillator (N s/m) D g ground acceleration D (g 2 /Hz) k stiffness of an oscillator (N/m) Q quality factor [/(2ξ)] f n, F n natural frequency of an oscillator (Hz) /3- oct. band pressure noise (db) relative to 0 f i,f f,f c band frequencies (initial, final, center) (Hz) ü acceleration (g) or (m/s 2 ) f ource uncoupled natural frequency [w /(2π)] ü g base excitation acceleration (g) or (m/s 2 ) f 2 oad uncoupled natural frequency [w 2 /(2π)] W pressure power spectral density (N/m 2 ) 2 /(Hz) g gravity acceleration factor g= 9.8 (m/s 2 ) w ource uncoupled circular frequency [k /m ] H acceleration transfer function H=H(w) w 2 oad uncoupled circular frequency [k 2 /m 2 ] m mass of an oscillator (kg) ξ ource critical damping ratio [c /(2 (k m ))] M mass of an acoustically impinged structure (kg) ξ 2 oad critical damping ratio [c 2 /(2 (k 2 m 2 ))] oop,ip out of plane, in plane directions µ oad to ource mass ratio [m 2 /m ] 0 reference pressure noise 0=2*-5 (N/m 2 ) φ oad to ource frequency ratio [f 2 /f ]=[w 2 /w ] r.m.s sound induced pressure r.m.s. (N/m 2 ) Γ n modal mass % (for n th mode). NDUN Dedicated (..F.) curves can be produced for equipment structural dimensioning purpose, tailored to each equipment-panel of a satellite. Usually, a set of three curves is generated, one for each principal direction of the panel, by considering the time consistent superposition of the main mechanical environmental contributions to the limit load, namely: quasi static, low frequency transients, random vibration and acoustic noise. s a general approach for high frequency loads, both pressure and acceleration power spectral densities can be treated in a similar manner to derive the acceleration responses, by using an equivalent random-acoustic D response spectra method, applicable to those panels classified as relevant acoustic receivers. nce defined the random vibration spectra at equipment mounting interface, and considering the dynamic coupling characteristics, then typical (..F.) curves are derived whose sloped behaviour is a function of the equipment mass. Moreover, considering that so far, based on the most important requirements sources, the equipment D spectra definition method is simply based on rules of inverse proportionality vs. the equipment mass, this paper also puts in evidence which is the expected influence of each dynamic parameter, like: geometrical characteristics, mass and surface density of full/empty panel and equipment, minimum required stiffness, equipment mounting position on panel, acoustic profile characteristics. he approach of generating equipment D spectra and (..F.) curves, has recently been implemented in the space industry for both the HH & NK ervice Modules (VM) during the design development phase, in progress nowadays. n practice, the approach explained here might become a general-purpose method for reducing both design/test

2 acceleration overloads. n fact, traditional methods sometimes can easily introduce some artificial criticalities and consequent risk of overdesign/overtesting as far as the equipment structural aspects. 2. DN n current satellite technology, equipment are normally grouped and mounted on dedicated platforms very often constituted by honeycombs (e.g. aluminium core with aluminium or carbon face sheets). ach panel assembly (platform + all mounted equipment) can be defined as an equipment-panel. For example Fig. shows a satellite weighting about 3.5 tons. comprising a.0 ton. VM, that is constituted by eight equipment-panels, as shown in Fig. 2 exploded view. VM Fig. 3.5 tons. atellite (verall View) Fig. 2.0 ton. VM tructure (xploded View) bove equipment-panel layout is also termed as satellite box-type, being the equipment mounted against the sidewalls (attached each other in a box shape) and resulting in direct contact with the vibroacoustic environment of the pacecraft (/) fairing. hus, both the panels and their mounted equipment are exposed, especially during the first phases of the flight, to low and high frequency environments that will induce, at a certain instant of the mission, the limit acceleration load. By definition, the limit load represents the maximum load expected to be acting on a structure during its design service life. n general, for satellite equipment-panels, the acceleration limit load is reached during the lift-off (/) phase, from the following main acceleration loads: - Quasi tatic (Q) + ow ransient () oads: (Q) - coustic oads: () - andom Vibration oads: (V) 3. N D MX Usually, a typical 3 x 6 loads combination matrix is considered for / structures, see ab. i. oad ets inear ccelerations (g) ngular ccelerations (rad/sec 2 ) ±F x ±Q y ±Q z ± Q x ± Q y ± Q z 2 ±Q x ±F y ±Q z ± Q x ± Q y ± Q z 3 ±Q x ±Q y ±F z ± Q x ± Q y ± Q z ab. i cceleration oads Matrix n particular, three (3) load sets result from the accelerations combination, where for each load set, the F is generated by considering simultaneous Q loads with all possible sign (±) combinations, whilst the V and (due to the low probability of peaks occurrence) are considered acting just one axis by a time. For simplicity a local co-ordinate system (r,t,l) is more suitable for panels geometry: r = normal to the panel surface (out-of-plane) t = parallel to the panel short side (in-plane ) l = parallel to the panel long side (in-plane 2)

3 o, for each load set/local direction j=(r,t,l) a number of eight (8) load cases is established, where the F value is the term in the matrix main diagonal. everal methods are available in literature to derive the F from individual load contributions; however, the most commonly used method is to consider the quare oot of the um of the quares () for Q,, V as defined in eq. []. (F) j = [(Q) 2 j +( ) 2 j + (V) 2 j] /2 [] nce the acceleration limit loads being evaluated, a Design Factor (DF) can be applied to get the design loads. Depending on project requirements, typical DF can be in the range of [.0 to...2.0]. 3. cceleration oads Matrix for quipment-panels he equipment-panels acceleration matrix is of the same type of ab. i. However, in this particular case, rotational accelerations can be disregarded due to the limited dimensions of the structures. he individual loads qualitatively can be considered as : (Q) low-medium; () medium-high; (V) ~ 3.2 cceleration oads Matrix for quipment imilarly to above, the same kind of loads already described can be identified. he individual load contributions qualitatively can be considered as: (Q) medium; () ~; (V) medium-high. However for an equipment case, proper acceleration amplifications due to dynamic coupling with the equipment-panel must be considered. n particular, due to the equipment typical high stiffness, the amplifications are particularly relevant for V, whilst are very limited for Q. n practice, the equipment load matrix can be conceived to contain in the main diagonal the effects of the V amplification. o, for each equipment-panel, an equipment load matrix can be defined as per ab. ii, where the F is represented by a curve correlating the equipment limit acceleration (g) versus its mass (kg). Details about the (..F.) curve philosophy are provided in {ef.}, {ef.2}, {ef.3}. oad ets ut-of-plane n-plane (-2) ( r ) ( t ) (l ) ± (..F.) urve(r) ± Q (t) ± Q (l) 2 ± Q (r) ± (..F.) urve(t) ± Q (l) 3 ± Q (r) ± Q (t) ± (..F.) urve(l) ab. ii quipment cceleration oads Matrix For equipment, the V acceleration is mainly due to random vibrations acting at the equipment-panel interface (/F), that was originally produced by acoustic loads. Usually, a strong difference can be noticed between hard mounted condition and flight configuration. he reason is related to the effects of the so-called ffective mplification Factor (Q eff ) that is always lower than equipment Quality Factor (Q). he Q eff can be defined as the nput/utput (/) accelerations ratio, see eq. [2]. (Q eff ) = (ü output /ü input ) < (Q) [2] For a wo-degree of Freedom ystem (DF) ource + oad, subjected to a random vibration, the Qeff formulations are given in {ef.2}. However, the Q eff worst case is always obtained in tuned conditions (φ=), being Q eff reaching the maximum expected value (Q eff *). he expression of Q eff * for a DF case is given by eq. [3], whose related dynamic parameters are defined in eq. [4] and eq. [5]. n particular, in eq. [4] the ü 2 represent the mean square accelerations for the DF masses ource m and oad m 2 (being the random vibrations of probabilistic nature). Q eff *={[(ξ (+µ) +ξ 2 (+3µ+µ 2 )+4[ξ 3 + ξ 2 ξ 2 (2+µ)+ξ ξ 2 2 (2+µ)+ξ 2 3 (+µ)]+6 ξ 2 ξ 2 2 (ξ +ξ 2 )] / [µ ξ +ξ 2 (µ 2 +µ)+4[ξ 3 µ + ξ 2 ξ 2 (+µ) +ξ ξ 2 2 (2+µ) +ξ 2 3 (+µ)]+6 ξ 2 ξ 2 [(ξ 2 + ξ 2 2 ) + 2 ξ ξ 2 ]]} /2 [3] Q eff = (< ü 2 2 > / < ü 2 >) [4] Q eff * = Q eff (φ=) [5]

4 t is evidenced here that the phenomenon of the effective amplification factor (applied for example to define notching in vibration testing, see {ef.4}) does not exclusively occur for random or acoustic vibrations, but is a very general phenomenon, valid for all vibration types (e.g. dynamic absorber theory is related to sine loads). he trend of eq. [3] shows that Q eff * is an inverse function of (µ). ll above facts represent the analytical basis for equipment (..F.) curve derivation approach. 4. VBU D s previously discussed, the equipment-panels represents the only significant high frequency load contribution to the equipment (..F.) curve. o the is the main source of equipment V. herefore, acoustically induced random vibrations on panels have been investigated and compared, as far as the following aspects are concerned: 4.) quipment-panels theoretical acoustic loads 4.2) quipment parametric D requirements 4.3) quipment-panels vibroacoustic test data 4.4) D criteria comparison 4. quipment-panels heoretical coustic oads When a structure is subjected to an acoustic field, the eak ressure ( peak ) can be assessed by eq. [7], where the ound ower pectral Density (W) is treated similarly to the case of an cceleration ower pectral Density (D) in random vibration. he acoustic Miles equation is applied, see eq. [6], where the W is calculated by eq. [2]. he formulas eq. [8] thru eq. [2] are also introduced. r.m.s. = [(π/2) Q F n W] /2 [6] peak = 3 r.m.s. [7] F n f c = (f i f f ) /2 [8] For a typical noise pressure level () in db defined at /3 or at octave band spectrum, the resulting bandwidth is defined by eq. [9]; so for a certain frequency (Fn= f c ) the bandwidth will be: For /3 oct. spectrum: BW=(f f -f i )= 0.23 f c = 0.23 F n [9-] For oct. spectrum: BW=(f f -f i )= (/2) /2 f c = F n [9-2] From the general definition of Noise pectral ressure () in (db), the W results: = 20 log(/0) (db) [0] =0 0 (/20) [] W=( 2 )/BW =(0 2 0 (/0) )/BW [2] he application of eq. [6] is a nearly exact solution for a structure having a dynamic behaviour ingle Degree of Freedom ystem (DF) like. n the other hand, for all cases of structures having several global modes of vibration, the modal responses must be weighted on the square of the modal masses (Γ n ) 2. However, for typical cases of noise impingement over relevant acoustic receivers (e.g. structures with low mass density M/ < 00 (kg/m 2 ) as meteoroids shields, equipment-panels, etc..) the dynamic behaviour of these structures is typically quite near to a DF, so that Miles formula associated to the global mode is deemed adequate and in general conservative with respect to exact solution. nce obtained for a certain condition the peak (N/m 2 ), the corresponding coustic eak cceleration ( peak ) in (g) units is given by eq. [3]. peak= [/(M g)] peak [3]

5 For example, when considering a structure as an ideal DF acoustic receiver with (Q=0), the acoustic noise environment of the pace huttle argo Bay /3 ct-4db () should induce some W and peak response spectra as shown in Fig. 3. V U i ] z H / N2 ^ a [ y t i HU pectrum /3 oct-4 db() N FQUNY [Hz) Fig. 3 pectral ressures for pace huttle argo Bay Noise (/3 oct-4db ) for Q=0 hus, for a generic case of a structure of a certain (M 2 / 2 ) ratio, the peak2 can be analytically derived by simply scaling the results available from a previously known case (M / ) and peak, being applicable from eq. [3] the following proportionality law: ( peak / peak2 )=( peak / peak2 ) ( /M )/( 2 /M 2 ) [4] Moreover, in order to derive the global accelerations acting at a structure entre of Gravity (c.o.g.), an equivalence between & V accelerations can be introduced as per eq. [5]. peak V peak [5] n other words, it is assumed that the peak acceleration produced by a DF under acoustic impingement being equivalent to the peak acceleration V produced on same DF by a Base xcitation andom Vibration D g, see Fig. 4 schematics. onsequently, the equivalence between and V, in (g) units, is established as follows: peak V peak = 3 ü 2 [6] From Fig. 4, the cceleration ransfer Function (Hü ) can be defined as the ratio between / accelerations. he main formulations from DF random vibration theory are reported in eq. [7] thru eq. [2], where the applicable integration field is between [-...+ ]. Hü = ü / ü g [7] Hü =[(2iwξ w n +w n 2 )/(-w 2 +2i wξ w n +w n 2 )] [8] D = D g Hü 2 [9] Q 2 =( D dw / D g dw)= ü 2 / ü g 2 [20] s n e D r e w o U ] a [ W i 0 3 U ower pectral Density N FQUNY (Hz) K U for [Q=0] 0 4 U peak i 0 3 K N FQUNY (Hz) ü 2 = D dw= Hü 2 D g dw [2] Moreover, in case of D g =constant, the D g can be taken out from the integrals, and the theoretical solution of eq. [2] in terms of DF mean square response is given by eq. [22]. D g D Fig. 4 DF- Base xcitation Model

6 ü 2 = [ (D g w n /4) (+(2 ξ) 2 )/(2 ξ ) ] [22] bove equation slightly differs from the so called Miles formula by the factor (Ψ = +(2ξ) 2 ); anyway for most / structural materials ξ<<, Ψ. n this case eq. [22] reduces to the Miles formula eq. [23]. ü 2 =[(D g w n /4) Q]=[(π/2) Q F n D g ] [23] For a DF case, the major contribution to above integrals is within the bell-shaped transfer function zone; thereby some approximate formulations like eq. [24] and eq. [25] can be introduced. Y N D 0. D i 0.0 D i 0 3 W ] g [ N K peak i quivalent D / spectra[m/=50;q=0] FQUNY (Hz) D -peak (g) for [M/=50][Q=0] peak D eak D g Q 2 [24] ü 2 / ü g 2 ( D eak / D g ) [25] Fig.5 esponse pectrum for huttle (M/=50) Finally, due to the D g equivalent excitation, the expected D plateau producing the peak (g) can be obtained by eq. [26], where the D profile corresponds to the D eak spectrum maxima, see Fig. 5. o, eq. [26] can be used as baseline for the derivation of equipment /F input D plateau. D D eak = W (Q/g) 2 (/M) 2 (D plateau in g 2 /Hz) [26] onsidering again the example of the pace huttle argo Bay noise of Fig. 3, for a structure having Q=0 and (M/)=50 (Kg/m 2 ); in this case the equivalence for V- in terms of D response spectra leads to the results shown in Fig quipment arametric D equirements From the experience gained on designing and testing past / projects, the main launch uthorities have established some parametric criteria for the definition of equipment random vibration qualification spectra. n particular from {ef.5} and {ef.6} the task of derivation of the equipment qualification random vibration test input profiles is simply based on a rule of inverse proportionality with respect to the equipment mass. hese criteria are investigated in the following. N-GF riteria ccording to{ef.5}for a huttle / mounted equipment of mass (m) located anywhere, the qual. test spectrum is given by ab. iii and eq. [27]; the slope (db/oct.) is fixed for all mass figures. For m > 22.7 Kg: D(m) = K [22.7/m] [27-a] For m 22.7 Kg: D(m) = K (K = 0.5 for all directions) [27-b] Frequency (Hz) evel D (g 2 /Hz) lope (db/oct.) y- D(m) D(m) D(m) - y2-4.5 Qual. est Duration 2.5 min./ axis ab. iii N-GF - quipment Qualification est riteria FQUNY (Hz) - riteria ccording to{ef.6}, for all equipment cases of mass (m) < 50 (kg) located on external panels or unknown / location, the qualification test spectrum is given by ab. iv and eq. [28].

7 For m < 50 Kg: D(m) = K o,i [m+20]/[m+] [28] (K o = 0.2 for out of plane direction (oop); K i = 0.05 for in plane directions (ip)) Frequency (Hz) evel D (g 2 /Hz) lope (db/oct) x - D(m) D(m) D(m) - x2-5 Qual. est Duration 2.5 min./ axis ab. iv - - quipment Qualification est riteria N- riteria omparison For an equipment of parametric mass [ ] (kg), the above D criteria comparison is shown in ab. v. he expected peak acceleration response (g) from Miles formula is shown in ab. vi. D(m) (g 2 /Hz) vs Mass (kg)! --oop ip N- x,y,z ab. v N- riteria omparison for quipment arametric D(m) lateau V peak (g) vs. Mass (kg)! Fn=00/200/300 Hz; Q=0 --oop 76/249/ /25/ /89/ 23 /58/ 93 84/9/46 68/97/8 57/ 8/ 99 48/67/82 - ip 3/60/ 96 98/ 39/ 70 87/ 22/ 50 72/ 02/ 25 54/ 77/ 94 44/63/77 38/53/ 65 3/ 44/54 N- x,y,z 46 / 65 / / 65 / / 65 / / 65 / / 65 / 80 46/65/80 46/65/80 3/44/54 ab. vi xpected V peak (g) from arametric D riteria for 00/200/300 (Hz) quipment (Q=0) 4.3 quipment-panels Vibroacoustic est Data From the vibroacoustic tests campaigns of some recently developed satellites, a set of measurements relevant to the max. D clipped plateau at equipment mounting /F of VM panels has been gathered, see Fig. 6 thru Fig.. n particular the D data from three different spacecraft acoustic qualification tests: XMM,, NG have been analysed, whose acoustic profiles are shown in ab. vii. Besides, for comparison purpose ab. vii also shows the acoustic qualification test requirement for HH satellite, which will be taken as a calculation example. 0 0 D 0, D 0, 0,0 0,0 (oop) axis (i-p) axes (oop) axis 0, QUMN-N Mass Density [Kg/m^2] 0, QUMN-N Mass Density [Kg/m^2] Fig. 6 NG-(M) est 44 db () VM D vs. Mass Density (kg/m 2 ) Fig. 7 XMM-(FM) est scaled to 47 db () VM D vs. Mass Density (-source)

8 0 0 D 0, D 0, 0,0 0,0 (oop) axis (i-p) axes (oop) axis 0, QUMN-N Mass Density [Kg/m^2] 0, QUMN-N Mass Density [Kg/m^2] Fig. 8 XMM-(M)- est at 47 db () Fig. 9 (FM)-est scaled 44 db () VM D vs. Mass Density (-source) VM D vs. Mass Density (-source) 0 0 D 0, D 0, 0,0 0,0 (oop) axis (oop) axis 0, QUMN Mass [Kg] Fig. 0 NG- Qual. est 44 db () VM D (oop) vs. quipment mass (kg) 0, QUMN Mass [Kg] Fig. XMM-Qual.est 47 db () VM D (oop) vs. quipment mass (kg) ct. Band entre Frequency (Hz) XMM [db] [db] NG [db] HH [db] [db] ab. vii pacecrafts - Vibroacoustic Qualification est equirements (0=2*0-5 (a)) 4.4 D riteria omparison D comparison has been performed, considering some equivalent acoustic loading conditions. n practice, no scaling of data shown in Fig. 6 thru Fig. has been applied, since qualification profiles of ab. vii for XMM, NG, have different but the expected theoretical D (oop) from eq. [26] result to be rather similar in all cases, see Fig. 2 at least in the range [50 50] Hz (which is typical for equipment-panels). o, in order to define an equipment limit analytical D requirement, the max. spectral D of Fig.2 can be considered (e.g for XMM and

9 HH the D results at 250 Hz while for / NG the results at 50 Hz). Finally, Fig. 3 and Fig. 4 respectively compare the theoretical, parametric and D test data. ^2 g [ Y N Deq D i Deq2 i Deq3 i Deq4 i W quivalent D utput [M/=50] ; [Q=0]. XMM NG 4. HH-NK ] g [ N K peak i peak2 i peak3 i peak4 i 0. XMM NG 4. HH-NK K cceleration (g) for [M/=50][Q=0] FQUNY (Hz) FQUNY (Hz) Fig. 2 omparison of heoretical D plateau and peak (oop) esponse pectra 0 0 D (oop) eq. [ 26 ] arametric: D (oop) eq. [28] D 0, D 0, 0,0 0,0 0,00 D (ip) eq.[26] (K i /K o ) QUMN-N Mass Density [Kg/m^2] 0,00 arametric: N D (oop) eq. [27] QUMN Mass [Kg] Fig. 3 quipment /F D nalytical vs.est Data D vs. quipment-panel mass density(kg/m 2 ) From Fig. 3 and Fig. 4, the following considerations are drawn: Fig. 4 quipment /F D arametric vs.est Data D vs. quipment mass (kg) - he D (oop) levels very often appear to be one order of magnitude higher than the (ip) ones. - omparing the D (oop) from various / VM the correlation degree of D plateau vs. panel mass density (Fig. 6 thru Fig. 9) is more significant than vs. equipment mass (Fig. 0, Fig. ). - trend of inverse proportionality exists between D (oop) and equipment-panel mass density. - n apparent trend of inverse proportionality exists between D (oop) and equipment mass. When comparing the D measurements vs. the equipment mass, the co-variance of data results ~ 0. n other words, it can be said: heavy equipment are always mounted on heavy equipment-panels whilst, heavy equipment cannot be mounted on light equipment-panels. o, it is pointed out that due to the effect of equipment-panels acoustic receiver characteristics, the D (oop) plateau response seems to be driven by the equipment-panel mass density rather than by the equipment mass. ummarising the results reported in the above graphs, it is underlined that: ) From / test data, the D (oop) plateau is always found in the range [ ] Hz, which is in good correlation with theoretical expectations of Fig. 2 and eq. [26]. n practice, the first two bending modes of / equipment-panels, normally lie below 250 Hz, and drive the max. D.

10 2) he D plateau when considered as a function of the equipment-panel mass density (Fig. 3) seems to be in a better correlation than when considered function of equipment mass (Fig. 4). 3) he D (ip) response is low; usually a D factor ratio (K o / K i ) = 2.4 is deemed conservative. 4) For equipment-panels, due to typical geometry, the D max. is generally found at their (c.o.g.). 5) For any kind of acoustic spectrum applied, a theoretical peak is expected at equipment-panel (c.o.g.) and can be analytically assessed as a function of and structure dynamics. Moreover, the peak value can be thought as acceleration input at the equipment mounting /F. 5. QUMN (..F.) UV & D NU 5. General riteria From previous considerations, the eq. [7] and eq. [3] enable the derivation of peak (g) at equipmentpanel (c.o.g.) and eq. [26] defines the D (g 2 /Hz) plateau. o, conservatively associating the Qeff* of eq. [3], the equipment response (g) can be derived; similarly to Qeff*, such a response will be a curve. 5.2 (..F.) urve - pplication ase n example of derivation of (..F.) curve has been considered for HH-VM panel (-y) nput Data for (..F.) urves Derivation onsider the equipment-panel(-y) layout shown in Fig. 2. he physical properties budget is given in ab. viii. he eigenvalues FM result of the fully loaded panel (see Fig. 5) is given in ab. ix. he qualification spectrum of ab. vii has been considered. n assy frequency of 00 Hz has been conservatively assumed for peak, W and the corresponding equipment D (oop) requirement has been evaluated in such a case. From frequency transient run of HH-VM FM (being the not performed yet) it results: Q 6 (g) all axes of equipment-panel(-y); Q 25(g) all axes for equipment. he V is negligible for equipment-panel, while 6.6 (g) (oop axis). he is negligible for equipment. Finally, for all structures a critical damping ξ=5% was assumed. quipment NN as B3 FM Grid quip. Mass (kg) FHU 383 FHH 392 FHHH FHHH FHU ssy (cog) anel rea (m 2 ) otal Mass (kg) Mass Density (kg/m 2 ) ~ 39 ab. viii VM anel(-y) hysical roperties Fully oaded VM anel(-y) Y FM=(oop) Mass Frequency (Hz) (kg) (oop) axis Modal Mass % ab. ix VM anel(-y) FM Modal esults (oop) Fig. 5 HH-VM anel(-y) FM (c.o.g.) 480 Dummy quipment at cog

11 quipment (..F.) curves & D he D results shown in ab. x have been derived from eq. [26] on the basis of the - trapezium profile of ab. iv. he corresponding equipment (..F.) curves are depicted in Fig. 6.. ( Freq. (Hz) D (oop) (g 2 /Hz) D (ip) (g 2 /Hz) lope (db/oct) time min / axis 3.62 (g-rms) 8.83 (g-rms) ab. x quipment Qualification est pectra for the quipment-panel(-y) of HH-VM 00 QUMN (..F.) urve (o-o-p) axis.. ( 50 QUMN M (m2) [Kg] QUMN (..F.) urve (i-p) axes F D Fo( µ( inc) ) M ] g [ F D M Fi( µ( inc) ) ). F m2( µ ) QUMN M (m2) [Kg] m2( µ ) QUMN M (m2) [Kg] Fig. 6 quipment (..F.) urves for the quipment-panel(-y) of HH-VM FM nalysis FM of the equipment-panel(-y) has been run taking into account a D equivalent plateau corresponding to the response at about 72.5 (Hz), in accordance to FM modal result of ab. ix and theoretical expectations of eq. [26]. Moreover, in order to assess equipment effective amplifications, a DF (m,c,k) dummy equipment was considered, mounted at panel(-y) (c.o.g.), with parametric mass of [m=0.,.0, 0] (kg) and a F n =00 (Hz) (e.g. quipment-anel (-y) - FHU.6 grms - FH.4 grms - FHHH.2 grms - FHU.4 grms - N(cog).4 grms the worst case corresponding to the HH Fig. 7 FM quipment-panel(-y) D (oop) equipment minimum stiffness requirement).he D / results are shown in Fig. 7. he results of two extreme conditions of dummy equipment (µ=0.2 and µ=0.002 ) are shown in Fig. 8 and Fig. 9. quipment-anel (-y) (m2/m=_ = 0.2) -Dummy qp. 2.4 grms -N (cog).4 grms quipment-anel (-y) (m2/m=_ = 0.002) -Dummy qp. 6.8 grms -N (cog).4 grms Fig. 8 FM / D (oop)-0 (kg) quipment Fig. 9 FM / D (oop)-0. (kg) quipment

12 5.2.4 oads omparison From above, the FM results are always encompassed by (..F.) curve. o, eventhough the (F) curve formulation is conservative, especially for heavy equipment cases the related (..F.) values are by far lower than those derived by using parametric approaches, see ab. vi. comparison of the response (g) for (oop) axis for a 00/200/300 (Hz) equipment with (Q=0) is shown in ab. xi. N (g) for quipment (oop) axis VM anel (-y) quip. Mass (kg) (..F.) urve (Q=25 g) (see Fig. 6) N arametric riteria Miles formula to eq. [27] quipment: 00/200/300 Hz arametric riteria Miles formula to eq. [28] quipment: 00/200/300 Hz FHU / 65 / / 94 / 5 FHH / 65 / / 94 / 5 FHHH / 65 / / 7 / 42 FHU / 65 / / 9 / 2 Dummy quipment located at anel (c.o.g.) / 65 / / 65/ / 65 / / 97 / 8 33 / 89 / / 249 / 305 ab. xi anel(-y) - quipment esponse (g) omparison for Fn=00/200/300 (Hz) (oop) (Q=0) 6. NUN For equipment mounted on satellite panels, analytical methods based on the dynamic coupling impedance can be used to derive more realistic (..F.) design curves and D test criteria. 7. FN {} M. rubert -N-J-D5882 Mass cceleration urve for / tructural Design (Nov. 89). {2}. eresetti - N pazio imit oad Factor urve (..F.) for reliminary Mechanical Design of omponents for pace Missions n ttempt of Mathematical Justification uropean onference on pacecraft tructures, Materials and Mechanical est, Braunschweig (D) (Nov. 98) --428/pg {3}. eresetti - N pazio imit oad Factor (..F.) urves for pacecraft lements uropean onference on /, Materials and Mechanical esting, Noordwijk (N)-(Nov. 2000) --468/pg {4} N-HDBK-7004 Force imited Vibration esting Handbook - (May 2000). {5} N, GV- ev. General nvironmental Verification pecification for & V ayloads, ubsystems, omponents GF (June 996). {6}, Draft 0H, uropean ooperation for pace tandardization pace ngineering esting (ct. 999). GNUMMY N UMMY