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Qualitative and quantitative estimates for the fundamental frequency of uniform and optimized tetrahedral truss platforms are determined. A semiempirical equation is developed for the frequency of free-free uniform trusses as a function of member material properties, truss dimensions, and parasitic (nonstructural) mass fraction Mp/Mt. Optimized trusses with frequencies approximately two times those of uniform trusses are determined by varying the cross-sectional areas of member groups. Trusses with 3 to 8 rings, no parasitic mass, and member areas up to 25 times the minimum area are optimized. Frequencies computed for ranges of both Mp/Mt and the ratio of maximum area to minimum area are normalized to the frequency of a uniform truss with no parasitic mass. The normalized frequency increases with the number of rings, and both frequency and the ratio of maximum area to minimum area decrease with increasing Mp/Mt. Frequency improvements that are achievable with a limited number of member areas are estimated for a 3-ring truss by using Taguchi methods. Joint stiffness knockdown effects are also considered. Comparison of optimized and baseline uniform truss frequencies indicates that tailoring can significantly increase structural frequency; maximum gains occur for trusses with low values of Mp/Mt. This study examines frequency trends for ranges of structural parameters and may be used as a preliminary design guide.
A correlation is presented of experimental measurement and analytical prediction of free, planar oscillations of a tank of mercury hung on a pendulum. Measured frequencies are presented for the slosh mode and the pendulum-tank mode, and time-history wave forms are included for qualitative comparisons. The sloshing is represented analytically by a single pendulum or a spring-bob, and the size and location of their inertias are determined by matching forces and moments on the tank surface with linearized hydrodynamic theory for inviscid fluids. It is concluded that such analytical models are identical for small displacements in the present application (as they are in simpler applications) and that they provide excellent simulation of the stiffness and in ertia terms for fuel sloshing.
The exploration of the processes accompanying engine combustion demands quick-responding pressure-recording instruments, among which the piezoelectric type has found widespread use because of its especially propitious properties as vibration-recording instruments for high frequencies. Lacking appropriate test methods, the potential errors of piezoelectric recorders in dynamic measurements could only be estimated up to now. In the present report a test method is described by means of which the resonance curves of the piezoelectric pickup can be determined; hence an instrumental appraisal of the vibration characteristics of piezoelectric recorders is obtainable.
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Frequencies of cell division stages in suspensions of Prochloron cells, expressed at regular intervals throughout a natural day-night cycle from several colonies of four species of host didemnid, are given. The proportion of dividing cells of Prochloron living symbiotically in colonies of a didemnid, Diplosoma virens, rises from about 4% during the night (20.00-04.00 hrs.) to about 13% in the morning (0,.00-12.00 hrs.), and then falls again in the afternoon. Similiar, though less pronounced, changes were observed among Prochloron cells in two other symbiotic didemnids, Lissoclinum patella and L. voeltzkowi.
Natural vibration frequencies of orthotropic and anisotropic simply supported right circular cylinders are predicted using a higher-order transverse-shear deformation theory. A comparison of natural vibration frequencies predicted by first-order transverse-shear deformation theory and the higher-order theory shows that an additional allowance for transverse shear deformation has a negligible effect on the lowest predicted natural vibration frequencies of laminated cylinders but significantly reduces the higher natural vibration frequencies. A parametric study of the effects of ply orientation on the natural vibration frequencies of laminated cylinders indicates that while stacking sequence affects natural vibration frequencies, cylinder geometry is more important in predicting transverse-shear deformation effects. Interaction curves for cylinders subjected to axial compressive loadings and low natural vibration frequencies indicate that transverse shearing effects are less important in predicting low natural vibration frequencies than in predicting axial compressive buckling loads. The effects of anisotropy are more important than the effects of transverse shear deformation for most strongly anisotropic laminated cylinders in predicting natural vibration frequencies. However, transverse-shear deformation effects are important in predicting high natural vibration frequencies of thick-walled laminated cylinders. Neglecting either anisotropic effects or transverse-shear deformation effects leads to non-conservative errors in predicted natural vibration frequencies.
Natural vibration frequencies of orthotropic and anisotropic simply supported right circular cylinders are predicted using a higher-order transverse-shear deformation theory. A comparison of natural vibration frequencies predicted by first-order transverse-shear deformation theory and the higher-order theory shows that an additional allowance for transverse shear deformation has a negligible effect on the lowest predicted natural vibration frequencies of laminated cylinders but significantly reduces the higher natural vibration frequencies. A parametric study of the effects of ply orientation on the natural vibration frequencies of laminated cylinders indicates that while stacking sequence affects natural vibration frequencies, cylinder geometry is more important in predicting transverse-shear deformation effects. Interaction curves for cylinders subjected to axial compressive loadings and low natural vibration frequencies indicate that transverse shearing effects are less important in predicting low natural vibration frequencies than in predicting axial compressive buckling loads. The effects of anisotropy are more important than the effects of transverse shear deformation for most strongly anisotropic laminated cylinders in predicting natural vibration frequencies. However, transverse-shear deformation effects are important in predicting high natural vibration frequencies of thick-walled laminated cylinders. Neglecting either anisotropic effects or transverse-shear deformation effects leads to non-conservative errors in predicted natural vibration frequencies.
The scaling law for the m/n=3/2 error field (EF) penetration threshold is predicted numerically based on nonlinear single-fluid and two-fluid modeling using the TM1 code. The simulated penetration threshold of radial magnetic field b r at the plasma edge is scaled to the electron density n e , temperature T e , viscous time τ μ , toroidal field B t and the natural frequency ω in the form of b r /B t ∝n e αn T e αT τ μ αμ B t αB ω αω by scanning these parameters separately. Here, α n , α T , α μ , α B and α ω are the scaling coefficients on n e , T e , τ μ , B t and ω, respectively. Single-fluid modeling shows that the 3/2 EF threshold scales as b r /B t ∝n e 0.56 T e 0.6 τ μ -0.59 B t -1.15 ω, which is similar with the analytical scaling law in both the Rutherford and visco-resistive regimes. Yet, two-fluid modeling shows that the scaling law differs significantly in particular regarding the dependence on plasma rotation. In detail, the scaling coefficient α n on density decreases from 0.67 to 0.56 and α T on temperature decreases from 0.67 to 0.32, while α μ on viscous time is around -0.45 and α B on toroidal field decreases slightly from -1.15 to -1, when the ratio |ω E /ω *e | between plasma rotation frequency ω E and diamagnetic drift frequency ω *e varies from 0 to 10. Scans of the plasma rotation reveals that the penetration threshold linearly depends on the perpendicular rotation frequency (or natural frequency) ω ⊥e =ω E +ω *e , and there is a minimum in the required field amplitude when ω ⊥e 0. In addition, the enduring mystery of non-zero penetration threshold at zero plasma natural frequency in EF experiments is resolved by two-fluid simulations. We report that the very small island and smooth bifurcation in EF penetration near zero frequency is hard to detect in the experiment, leading to a finite penetration threshold within the capability of the experimental measurements.
While analyzing a rocket engine component, it is often necessary to account for any effects that adjacent fluids (e.g., liquid fuels or oxidizers) might have on the structural dynamics of the component. To better characterize the fully coupled fluid-structure system responses, an analytical approach that models the system as a coupled expansion of rigid wall acoustic modes and in vacuo structural modes has been proposed. The present work seeks to experimentally validate this approach. To experimentally observe well-coupled system modes, the test article and fluid cavities are designed such that the uncoupled structural frequencies are comparable to the uncoupled acoustic frequencies. The test measures the natural frequencies, mode shapes, and forced response of cylindrical test articles in contact with fluid-filled cylindrical and/or annular cavities. The test article is excited with a stinger and the fluid-loaded response is acquired using a laser-doppler vibrometer. The experimentally determined fluid-loaded natural frequencies are compared directly to the results of the analytical model. Due to the geometric configuration of the test article, the analytical model is found to be valid for natural modes with circumferential wave numbers greater than four. In the case of these modes, the natural frequencies predicted by the analytical model demonstrate excellent agreement with the experimentally determined natural frequencies.
The present report describes the process of redesigning a new rotor for the SSME Fuel Flowmeter. The new design addresses the specific requirement of a lower rotor speed which would allow the SSME operation at 1 15% rated power level without reaching a blade excitation by the wakes behind the hexagonal flow straightener upstream at frequencies close to the blade natural frequency. A series of calculations combining fleet flowmeters test data, airfoil fluid dynamics and CFD simulations of flow patterns behind the flowmeter's hexagonal straightener has led to a blade twist design alpha = alpha (radius) targeting a kf constant of 0.8256. The kf constant relates the fuel volume flow to the flowmeter rotor speed, for this particular value 17685 GPM at 3650 RPM. Based on this angle distribution, two actual blade designs were developed. A first design using the same blade airfoil as the original design targeted the new kf value only. A second design using a variable blade chord length and airfoil relative thickness targeted simultaneously the new kf value and an optimum blade design destined to provide smooth and stable operation and a significant increase in the blade natural frequency associated with the first bending mode, such that a comfortable margin could be obtained at 115% RPL. The second design is a result of a concurrent engineering process, during which several iterations were made in order to achieve a targeted blade natural frequency associated with the first bending mode of 1300 Hz. Water flow tests preliminary results indicate a kf value of 0.8179 for the f-irst design, which is within 1% of the target value. The second design rotor shows a natural frequency associated with the first bending mode of 1308 Hz, and a water-flow calibration constant of kf 0.8169.
Frequencies of natural vibrations in built-up shells determined by dynamic rigidities method
Hertzian contact theory is applied to a butt joint with specially mismatched bearing surfaces to devise a simple mathematical model of nonlinear axial force-displacement behavior in jointed members. Normalized tangent stiffness-force plots, for several values of a joint imperfection parameter, are presented for the sample case of solid structural members of circular section. The results illustrate the potential problem of high joint compliance at low axial-force levels, as well as the generally desirable stiffening and linearizing effects of preload. A nonlinear oscillator problem based on the static model is also formulated and solved to illustrate the effect of amplitude on natural frequency. As expected, natural frequency is low when amplitude is small. The results call attention to the important roles that tight tolerances and preload are expected to play in the design and fabrication of deployable and erectable truss-type space structures.
The natural frequencies of the electromagnetic resonances of conducting bodies with attached wires were determined. The bodies included twp cylinders and an approximate scale model of the NASA F-106B aircraft. All were three feet in length. Time domain waveforms of B-dot and D-dot were obtained from a sampling oscilloscope, and Prony analysis was used to extract the natural frequencies. The first four natural frequencies of the cylinders (and wires) were determined, and a comparison with calculated results of other investigators shows reasonable agreement. Seven natural frequencies were determined for the F-106B model (with wires), and these were compared with results obtained by NASA in 1982 during direct lightning strikes to the aircraft. The agreement between the corresponding natural frequencies of the model and the aircraft is fairly good and is better than that obtained in the previous work using wires with less resistance. The frequencies lie between 6.5 MHz and 41 MHz, and all of the normalized damping rates are between 0.14 and 0.27.
A computer model is reported of a PLL (phase-lock loop), preceded by a bandpass filter, which is valid when the bandwidth of the bandpass filter is of the same order of magnitude as the natural frequency of the PLL. New results for the PLL natural frequency equal to the bandpass filter bandwidth are presented for a second order PLL operating with carrier plus noise as the input. However, it is shown that extensions to higher order loops, and to the case of a modulated carrier are straightforward. The new results presented give the cycle skipping rate of the PLL as a function of the input carrier to noise ratio when the PLL natural frequency is equal to the bandpass filter bandwidth. Preliminary results showing the variation of the output noise power and cycle skipping rates of the PLL as a function of the loop damping ratio for the PLL natural frequency equal to the bandpass filter bandwidth are also included.
Torsional vibration systems, with finite numbers of degrees of freedom are investigated. In designing such systems, it is important to select parameters, so that the frequency spectrum of the natural vibrations is beyond the limits of the resonance danger region. In the case when the system has a large number of degrees of freedom, calculation of its natural frequencies, as well as tuning out the resonance danger zones is laborious. When the system being investigated has one group symmetry or another, solution of the problem mentioned above is facilitated, as a consequence of breaking down the frequency equations into several, which are smaller in size. An algorithm is given for tuning out the natural frequency spectrum from the forbidden region, by means of varying the rigidity of systems having group symmetries or group quasi-symmetries.
The special nonlinearities of the Jeffcott equations in rotordynamics are examined. The immediate application of this analysis is directed toward understanding the excessive vibrations recorded in the LOX pump of the SSME during hot firing ground testing. Deadband, side force and rubbing are three possible sources of inducing nonlinearity in the Jeffcott equations. The present analysis initially reduces these problems to the same mathematical description. A special frequency, named the nonlinear natural frequency is defined and used to develop the solutions of the nonlinear Jeffcott equations as asympotic expansions. This nonlinear natural frequency which is the ratio of the cross-stiffness and the damping, plays a major role in determining response frequencies. Numerical solutions are included for comparison with the analysis. Also, nonlinear frequency-response tables are made for a typical range of values.