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1Introduction1.1GeneralThe various mechanical loads are not all equally important and depend on the type of the mechanical structure:i.e.does it concern a primary structure, the spacecraft structure or other secondary structures(such as solar panels, antennas,instruments and electronic boxes).Requirements are specified to cover loads encountered by handling,testing,during the launch phase and operations in transfer andfinal orbit,such as[167]:•natural frequencies•steady-state(semi-static)acceleration•sine excitation•random excitation•acoustic noise•transient loads•shock loads•temperaturesNatural frequencies:The location of natural frequencies is a primary design requirement for all parts of the spacecraft.This requirement is imposed in order to limit the dynamic coupling of the spacecraft with the launch vehicleSemi-static and low frequency sinusoidal loads:The design of the primary structure is determined to a large extent by the semi-static and low fre-quency sinusoidal loads(up to approximately50Hz)Sinusoidal and random loads:To a large extent,the sinusoidal and random loads determine the design of secondary structures(solar panels,antennas, electronic boxes).Acoustic loads:Light structural parts with relatively large surface areas(such as solar panels and spacecraft antennas)are more sensitive to acoustic loads than sinusoidal and random base excitation.J.Wijker,Random Vibrations in Spacecraft Structures Design,Solid Mechanics and Its Applications165,c Springer Science+Business Media B.V.200921IntroductionShock loads:Deployable structures experience high shock loads;for example during latch-up of hinges in the requiredfinal position of these mecha-nisms.This is especially the case when the deployment speeds are too high.Temperatures:Temperature variations usually cause high thermal stresses in structures.In general,the various coefficients of expansion are accounted for in the choice of the structural materials.Thermal deformations are taken into account when working with structures that must be aligned with each other.Random Loads:The design of instruments and electronic boxes is determined by the random base excitation.All these different types of load are described in detail in[224,225].In this book the random mechanical and acoustical vibrations of determin-istic and statistical dynamic systems,in the low and high frequency range, are considered,and the following topics will be discussed in great detail:•Vibrations of deterministic linear mechanical dynamic systems exposed to mechanical random loads and or enforced motion(acceleration)•Vibrations of deterministic linear mechanical dynamic systems exposed to random acoustic loads(sound pressures)•Random vibration of statistically defined mechanical systems and loads using Statistical Energy Analysis(SEA)•Non-linear structures excited to random(white noise)mechanical loads analyzed by using the Fokker-Planck-Kolmogorov(FPK)equationThe theory of random vibration is strongly related to the design of space-craft structures and will be illustrated with simple and more difficult worked examples;each section are ends with posed problems;usually answers are provided.Figure1.1shows a cross section of a typical spacecraft.This may be a com-munication,scientific or other spacecraft.For this spacecraft,the acoustic and the mechanical random vibration environment outside and inside the space-craft structure will be discussed.The spacecraft structure is an assembly of structural elements:shells of revolution,panels,shear panels,struts,etc.The spacecraft structure provides strength and stiffness properties to the space-craft in order to survive test and launch loads.Among other systems on the outside of the spacecraft there are the antenna reflector and both solar wings,constituting the spacecraft solar array.Both the antenna reflector and the solar array are in folded or stowed configuration, because:•the folded systemsfit better under the fairing of the launch vehicle,and •the folded systems can carry the launch loads better.The central structure of the spacecraft is called the primary structure of the structure,and forms the backbone(load path)of the structure.In general,1.1General3plete spacecraftfuel tanks are needed for propulsion.The attitude control systems and are fixed to the central body.These tanks are relatively heavy.Spacecraft quadrilateral sandwich platforms are supported by the central body and side panels used to mount payload and equipment boxes.In Fig.1.1 we see the top and lower platform.The antenna is mounted to the top plat-form.The payload and equipment is distributed so as to fulfill center of gravity requirements posed by the launch vehicle authority.The spacecraft side panels will close the structure box.Solar wings and part of the equipment are mounted to the side panels.For launch,the spacecraft is placed on the launch vehicle payload adapter structure.In the liftoffphase of the launch,the exhaust streams of the engines and solid rocket boosters will produce sound waves propagating to the launch vehicle,and will impinge on the launch vehicle structure and fairing.The sound pressures(acoustic load)will excite the launch vehicle structure,which will transfer the vibrations to the interface spacecraft launch vehicle.The acoustic loads are random in nature,hence the derived mechanical vibrations are random too.In the ECSS1standard[56],general“Qualification”acoustic loads are specified and given in Table1.1.The vibrating fairing transfers acoustic loads under the fairing.In general, these have a reverberant nature;they are denoted sound pressure level(SPL) and are given in decibels(dB)with a reference pressure p ref=2×10−5Pa, 1European Corporation of Space Standardization.41IntroductionTable1.1.Acoustic qualification test levels and duration[56]Octave band0dB=2×10−5Pa31.513063135.51251392501435001381000132200012840001248000120OASPL147Duration:2minFig.1.2.Acoustic loads converted into random mechanical loadswhich is ostensibly the audible limit of the human ear[133].The sound pres-sure is relatively low with respect to the static atmospheric pressure of105 Pa,1Bar,but large areal light weight structures are very sensitive to dynamic sound pressures.The sound pressure will excite the outside equipment and the outside spacecraft structure,especially the external panels.The random mechani-cal vibration of the external panels will(a)excite thefixed equipment(see Fig.1.2)and(b)will generate acoustic loads in the inside cavities of the space-craft,which in turn will excite the internal load-carrying structures like the central structure and the lower platform.Summarizing,it can be said that the sound pressures will cause random mechanical vibrations in the space-craft structure,creating a rather heavy random vibration environment for the spacecraft payload,tanks,equipment,etc.Even the direct transfer of vibro-acoustic energy to an unit(experiment,instrument,box,...)structure cannot be neglected due to the large unit surfaces[196].In the ECSS standard[56]“Qualification”mass dependent random en-forced accelerations are specified and given in Tables1.2and1.3.1.3Random Acoustic Vibration5 Table1.2.Random vibration test levels and duration for equipment with mass M≤50kg[56]Location Duration Frequency range(Hz)Levels(g2/Hz)Equipment located on external panel or with unknown location Vertical20–1003dB/oct.2.5100–3000.12M+20M+1 min/axis300–2000−5dB/oct. Lateral20–1003dB/oct.2.5100–3000.15M+20 min/axis300–2000−5dB/oct.Equipment not located on external panel All axes20–1003dB/oct.2.5100–3000.05M+20M+1 min/axis300–2000−5dB/oct.Table1.3.Random vibration test levels and duration for equipment with mass M>50kg[56]Frequency range(Hz)Levels(g2/Hz)Remark20–1003dB/oct.100–3000.0911.12G rms300–2000−3dB/oct.Duration:all axes 2.5min/axis1.2Random Mechanical VibrationIn part I we discuss the aspects of random vibrations of deterministic me-chanical structures.Predictions made about the random vibrations levels are limited to the low frequency domain because the vibration theory is based on simple single degree of freedom(SDOF)systems.These forms the basis for the modal displacement method(MDM),which is frequently used in the finite element applications.1.3Random Acoustic VibrationLarge-area light-weight mechanical structures are very sensitive to random acoustic loads.The same procedures as discussed in part I are applied,how-ever,now the applied loads are distributed over the surface of the mechanical structure.The distributed load application is discussed in part II.Plane waves, rain on the roof and reverberant(diffuse)soundfields are considered.In thefirst part of the chapter3fluid structure interaction(FSI)is ignored. The exposed pressurefield cause structural responses,but the influence of the vibrating structure on the pressurefield is ter on,the full FSI is discussed in detail,e.g.radiation,which will introduce radiation damping. Both analytical and approximate methods will be discussed.61Introduction1.4Statistical Energy AnalysisComplementary to low frequency mechanical vibrations,the statistical energy analysis(SEA)method is discussed in part III.The basis of the SEA method is the powerflow between oscillators or groups of oscillators.The structures and loads are described in a statistical manner that in contrast with the deterministic description of structures using thefinite element method.The number of modes per frequency dictates the application of the SEA method in the higher frequency bands.Both random mechanical and acoustical loads can be considered within the framework of the SEA method.1.5Fokker-Planck-Kolmogorov EquationPart IV is more or less based on Gaussian,white noise processes,leading to a Markoffprocess in which each event is dependent only on the event before it. The Fokker-Planck-Kolmogorov(FPK)partial differential diffusion equation is derived from the Markoffprocess.The unknown in this FPK equation is the transition probability density function,and after integration,the joint prob-ability function.Mean values and correlation functions(second moments), up-crossings andfirst passage statistics can be obtained from the FPK equa-tion.The stochastic differential equations(SDE),either in Itˆo or Stratonovich sense(definition of integration)are closely linked to the FPK equation.To solve nonlinear random vibrational problems we can use the FPK equa-tion;analytical and numerical methods are discussed.Huge computer power is needed to solve the FPK equation numerically. In general,the applications of the FPK equation is restricted to nonlinear dynamic systems with a few DOFs.。
