H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 37
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Severalvariations of the pendular parallelogram (fig.5.18) have been adapted with general successand used in the studies reported in references5.15 through 5.18, and 5.23 through 5.26.Another technique widely used is the installation of tanks on rollers or bearings, as indicated by the sketch in figure 5.19. This technique was utilized in the studies reported inreferences 5.19 through 5.22, and will berpfprrpdi~ R S I J ~ S ~ I J P secti~r.I?~&a!icgwith slosh tests in full-scale tanks.The excitation technique employed for boththe pendular and platform support system isusually a pushrod actuated by an eccentric orcam. The driving mechanism shown in figure5.18(a)and described in reference 5.27 is unique as amechanical system in that it permits instantaneous variations in the amplitude of oscil-172THE DYNAMIC BEHAVIOR OF LIQUIDS(a) NASA, Langley Research Center.fSupport members from werheadcrossbeams to framePreloading bracket andtension compression--Tension compression load cellTest bedFrame'/Hydraulic cylinderStationary support(b) NASA, Lewis Renearch Center.FIGURE5.18.-Test apparatus for mechanically exciting liquid-propellant tank configurations.lation of the tank and also permits oscillationof the tank at any given frequency withoutinducing oscillations at other frequencies asin the case of speeding up a motor.
Similarconditions of excitation are also feasible throughthe use of electromagnetic or hydraulic shakersas the driving source.Two t,ank-support systems frequently employed (refs. 5.28 and 5.29) for damping studiesare shown in figure 5.20. The tripod supportlegs may be situated near a tank diameterso that the tank can be readily tipped to excitethe motions of the fluid, or the motions maybe excited by careful manipulations of a paddle.SIMULATION AND EXPERIMENTAL TECHNIQUES173Southwest Rwarch Institute.(d) Space Technology Laboratories, Inc.FIGURE5.18.-Concluded.The fluid motions may be measured by astrain gage bridge on a torque bar connectingthe tank bottom frame to the main frame, asin figure 5.20(a), or by installing a load cellas one of the tripod supports.Natural Frequencies and Mode ShapesThe response of a fluid to the excitation ofits container is of prime significance when themotion of the container is essentially periodicand the period coincides with the first naturalfrequency of the fluid.
Experimental evidence has shown that, though substantialfluid amplitudes may be attained a t higherfrequency excitations, such as those for thesecond and higher natural modes, the forcesassociated with these fluid motions are generallyof secondary importance. Thus the fundamental problem in analyzing propellant sloshingin a launch vehicle or spacecraft is the establishment of the characteristics of the firstlateral mode, that is, the frequency and modeshape. However, it is generally convenient touse the same apparatus to measure the frequencies and mode shapes for the first two orthree modes, and this is usually done as amatter of course.The references cited present data which document the natural frequencies and mode shapesfor the fluid motions in tanks of many of thegeometrical configurations of interest.
Reference 5.17 treats the case of the circular cylinderand presents frequency data in dimensionlessform for transverse and i o n g h u h a l modes forthe horizontal cylinder and for the transversemodes for the upright cylinder-theconfiguration of general concern to launch vehicles.Figure 5.21 shows typical mode shapes measured in a tank 30.5 centimeters in diameterand 61 centimeters long.Experimental data on frequencies and modeshapes for the sphere are given in references174DYNAMICITHETankBEHAVIOR--,FIGURE5.19.-Platform-roller tank support and excitation,-system.30 cm diam76 cm diamSupportringL/OF LIQUIDSences 5.16 and 5.21.
Figure 5.24 shows themode shapes measured for the first threemodes for a horizontal toroid.The general technique for measuring thenatural frequencies is to oscillate the tank atlow amplitudes and record the frequency atwhich the amplitude of the undistorted waveshape reaches a maximum without rotation.Above the natural frequency, the fluid surfaceand particles will exhibit a tendency to rotateand, during this rotation, a ~ronouncedincreasein wave amplitude is possible. However, therotary motions are associated with the problemof nonlinear sloshing treated in references 5.25and 5.26, and summarized in chapter 3.The photographs presented in figures 5.21through 5.24 demonstrate the effectiveness ofthe photographic technique for recording themode shape and emphasize the convenience ofclear plastic tanks for studies of fluid-sloshingcharacteristics.DampingI~ o a bcell 1Strain gages and torsion bars:iquid surfaceFIGURE 5.20.-Tripodsupport systems.
(a) 30-centimeter-diameter tanli; (b) 76-centimeter-diameter tank.5.17 through 5.19, and 5.25. The first twomode shapes are shown in figure 5.22.The normal oscillntions of fluids in oblatespheroids are presented in reference 5.18, andreference 5.15 presents n study of the Centauroxygen tank which is a "near" spheroid.Mode shapes for the first three natural modesfor fluids in a horizontally oriented spheroidtire shown in figure 5.23..
Results of studies on the natural frequenciesi~ndmode shapes of toroids are given in refer-The motions of the fluids in launch vehicletanks, if unchecked, are capable of exertinglarge upsetting forces and moments on thevehicle. The general procedure for limitingthese inputs, and thus minimizing the requirements of the vehicle control system to overcome them, is to install slosh baffles in thetanks. (See ch. 4.) Such baffles are ofvaried design such as rings, conical frustums,cans, and so forth, but in all cases the weightof the baffles is a burden which reduces payload ~apabilit~y.As a consequence, considerable research has been devoted to evaluatingthe damping of various types of baffles tomaximize their efficiency and minimize theirweight.
These studies involve the principalgeometric tank shapes of interest.Techniques for measuring the damping supplied by slosh baffles in small research tanks aregiven in the literatsure for cylinders (refs.5.28 and 5.30), spheres (refs. 5.20, and 5.22through 5.24), and oblate spheroids (refs.5.15 and 5.29). Two basic techniques areused: the logarithmic decay method or theforced response method.The logarithmic decay method involves themeasurement of the rnte of decay of t,he fluidoscillation in a given rnode-invariably the firstFIGURE5.21.-Mode shapes for circular cylinder.
(a) First two transverse modes for horizontal cylinder: (b) first threelongitudinal modes for horizontal cylinder; ( c ) first two transverse modes for an upright cylinder.mode, since higher modes are so heavilydamped that the oscillations are neither of---k-&--+:n ;-+nvn=t nnr --n r -____.n ~ n n_ h-.l pt o accuratemeasurement, especially if the tank is fittedwith even a minimum of sloshing baffles.The damping may be interpreted in terms ofthe natural logarithm of the ratio of the forcesimposed by the fluid on the tank duringconsecutive oscillations (e.g., ref. 5.24), or ofthe ratio of the amplitudes of the flnid motionsduring consecutive oscillations (e.g., ref.
5.25).I n either case, a trnnsducer is used to obtainan effective measure of the magnitude of theJ U U PIUIIVVA~v u wL---aA~I~!sloshing fluid and the signal is fed into a readoutsystem.At least four types of transducer systems havebeen used effectively to measure fuei-sioshingamplitudes. These are sketched in figure 5.25.The first is a capacitance wire system whichconsists of ttvo capacitance probes which aremounted parallel to, but offset slightly andinsulated from, the I\-alls of the tank at thelocation of the antinodes of the sloshing fluid.These probes yield electrical outputs proportional to the difference in height of the liquidsurface at these points and are self-compensat-176TEE DYNAMIC BEHAVIOR OF LIQUIDSFIGURE5.22.-First- and second-mode shapes for fluid in a sphere.ing so as to maintain a constant zero level asthe tank is drained.
This technique is described in detail in the appendix of reference5.14.The second technique, shown by the sketchof figure 5.25(b), involves the use of concentricfloats which are positioned in the tanks a tdesired radial locations by means of wires orcables. Accelerometers or velocity pickups aremounted on the floats and sense the motions ofthe floats as they rise and fall uith the fluidduring sloshing.A third technique utilizes pressure pickupswhich are mounted in the tank wall or insertedin the fluid to measure the fluid head duringsloshing, as shown in figure 5.25(c).The final, and perhaps most straightforwardtechnique, is to install the tank on some formof load cell arrangement designed to indicatethe forces or moments applied to the tank bythe sloshing fluid.
Load cells used may be ofthe standard commercial type (ref. 5.29) ormay consist of specially designed strain-gagebeam systems. Also, in this category, is theload cell mounted as a series element of thetank excitation system such as that utilized forthe damping studies reported in reference 5.22.Transducer outputs from any of the aforementioned systems can be readily recorded byan oscillograph to yield a permanent record ofthe decay of fluid amplitudes and associateddamping. However, the Dampometer hasproven to be a rapid and very useful devicefor measuring the damping of singledegree-offreedom systems such as that associated withthe decay of fluid in one of its lower naturalmodes.
The Dampometer is an electronic instrument which converts the frequency anddamping characteristics of the analog transducer signal (the input) into a digital count(the output). The input signal is first converted from a damped sinusoid to an Archimedes spiral which is then displayed on twooscilloscopes, one for observation and thesecond for further reduction of the data. Acover which contains a photocell is placed overthe second oscilloscope. In addition, a seriesof interchangeable calibrated disks are providedwith radial slits of different lengths \I-hich maybe individually placed over the second scope.SIMULATION AND EXPERIMENTAL TECHNIQUES177By proper adjustment of the gain, the spiralgenerated from the transducer signal appearson the perimeter of the scope and moves towardthe center as the magnitude of the fluid motiondiminishes and the output spiral decays.
Asthe signal moves across the slits, the photocelldetects the intermittent light and provides theinput for the counter. By using the numberand length of the slits for a given disk, and thecounter's reading, the frequency and dampingassociated with the transducer output arereadily obtained from a simple formula. Byadjusting the sensitivity of the instrument, thedamping is readily obtained for different magnitudes of the sloshing wave.The measurement of damping by the forcedresponse method is not as convenient orstraightforward as the logarithmic decrementmethod, but it may be advantageous if the testfacility and instrumentation are such that anaccurate velocity response curve is available.If such is the case, the damping may be determined by measuring the amplitude of thepeak response a t the natural frequency or byratioing the bandwidth a t the half-power pointto the natural frequency (ref.
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