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NASA NTRS1993

A finite-element-based perturbation model for the rotordynamic analysis of shrouded pump impellers: Part 2: User's guide

This report describes the computational steps involved in executing a finite-element-based perturbation model for computing the rotor dynamic coefficients of a shrouded pump impeller or a simple seal. These arise from the fluid/rotor interaction in the clearance gap. In addition to the sample cases, the computational procedure also applies to a separate category of problems referred to as the 'seal-like' category. The problem, in this case, concerns a shrouded impeller, with the exception that the secondary, or leakage, passage is totally isolated from the primary-flow passage. The difference between this and the pump problem is that the former is analytically of the simple 'seal-like' configuration, with two (inlet and exit) flow-permeable stations, while the latter constitutes a double-entry / double-discharge flow problem. In all cases, the problem is that of a rotor clearance gap. The problem here is that of a rotor excitation in the form of a cylindrical whirl around the housing centerline for a smooth annular seal. In its centered operation mode, the rotor is assumed to give rise to an axisymmetric flow field in the clearance gap. As a result, problems involving longitudinal or helical grooves, in the rotor or housing surfaces, go beyond the code capabilities. Discarding, for the moment, the pre- and post-processing phases, the bulk of the computational procedure consists of two main steps. The first is aimed at producing the axisymmetric 'zeroth-order' flow solution in the given flow domain. Detailed description of this problem, including the flow-governing equations, turbulence closure, boundary conditions, and the finite-element formulation, was covered by Baskharone and Hensel. The second main step is where the perturbation model is implemented, with the input being the centered-rotor 'zeroth-order' flow solution and a prescribed whirl frequency ratio (whirl frequency divided by the impeller speed). The computational domain, in the latter case, is treated as three dimensional, with the number of computational planes in the circumferential direction being specified a priori. The reader is reminded that the deformations in the finite elements are all infinitesimally small because the rotor eccentricity itself is a virtual displacement. This explains why we have generically termed the perturbation model the 'virtually' deformable finite-element category. The primary outcome of implementing the perturbation model is the tangential and radial components, F(sub theta)(sup *) and F(sub r)(sup *) of the fluid-exerted force on the rotor surface due to the whirling motion. Repetitive execution of the perturbation model subprogram over a sufficient range of whirl frequency ratios, and subsequent interpolation of these fluid forces, using the least-square method, finally enable the user to compute the impeller rotor dynamic coefficients of the fluid/rotor interaction. These are the direct and cross-coupled stiffness, damping, and inertia effects of the fluid/rotor interaction.

Baskharone, Erian A.↗

NASA NTRS1985

Structural parameter identification of distributed systems using finite element approximation

A system identification technique is developed for classes of distributed systems using finite element approximations. Vibrating systems represented by partial differential equations have physical parameters associated with mass, stiffness, and damping distributions which need to be known in order to properly control and design mathematical models of the system. In order to identify these parameters a weighted least-squares algorithm and modified Newton-Raphson method is used for the identification process. The theory and technique is demonstrated by estimating the system parameters of a vibrating cantilever beam made up of several different structural properties.

Lee, K. Y.↗

NASA NTRS1987

Recent developments in the experimental identification of the dynamics of a highly flexible grid

Control effectiveness tests of reaction wheel actuators attached to a highly flexible grid are reported. Analytic determination of actuator control effectiveness is accomplished with finite-element modeling. Experimental determination is done with a least-square parameter identification algorithm that identifies the control coefficients of the second-order difference equation model of each vibration mode. The algorithm assumes a model with frequency and damping predetermined from free-decay tests for each mode. Accounting for the difference in forced and resonant frequency was necessary to produce control effectiveness estimates that are in reasonable agreement with the analytic predictions. The average error for control effectiveness coefficients greater than 5/sq sec was 6.384 percent.

Montgomery, Raymond C.↗

NASA NTRS1976

Curve fitting of aeroelastic transient response data with exponential functions

The extraction of frequency, damping, amplitude, and phase information from unforced transient response data is considered. These quantities are obtained from the parameters determined by fitting the digitized time-history data in a least-squares sense with complex exponential functions. The highlights of the method are described, and the results of several test cases are presented. The effects of noise are considered both by using analytical examples with random noise and by estimating the standard deviation of the parameters from maximum-likelihood theory.

Bennett, R. M.↗

NASA NTRS1991

Vector magnetic fields in sunspots. I - Stokes profile analysis using the Marshall Space Flight Center magnetograph

The Marshall Space Flight Center (MSFC) vector magnetograph is a tunable filter magnetograph with a bandpass of 125 mA. Results are presented of the inversion of Stokes polarization profiles observed with the MSFC vector magnetograph centered on a sunspot to recover the vector magnetic field parameters and thermodynamic parameters of the spectral line forming region using the Fe I 5250.2 A spectral line using a nonlinear least-squares fitting technique. As a preliminary investigation, it is also shown that the recovered thermodynamic parameters could be better understood if the fitted parameters like Doppler width, opacity ratio, and damping constant were broken down into more basic quantities like temperature, microturbulent velocity, or density parameter.

Balasubramaniam, K. S.↗

NASA NTRS1993

Modal characterization of the ASCIE segmented optics testbed: New algorithms and experimental results

New frequency response measurement procedures, on-line modal tuning techniques, and off-line modal identification algorithms are developed and applied to the modal identification of the Advanced Structures/Controls Integrated Experiment (ASCIE), a generic segmented optics telescope test-bed representative of future complex space structures. The frequency response measurement procedure uses all the actuators simultaneously to excite the structure and all the sensors to measure the structural response so that all the transfer functions are measured simultaneously. Structural responses to sinusoidal excitations are measured and analyzed to calculate spectral responses. The spectral responses in turn are analyzed as the spectral data become available and, which is new, the results are used to maintain high quality measurements. Data acquisition, processing, and checking procedures are fully automated. As the acquisition of the frequency response progresses, an on-line algorithm keeps track of the actuator force distribution that maximizes the structural response to automatically tune to a structural mode when approaching a resonant frequency. This tuning is insensitive to delays, ill-conditioning, and nonproportional damping. Experimental results show that is useful for modal surveys even in high modal density regions. For thorough modeling, a constructive procedure is proposed to identify the dynamics of a complex system from its frequency response with the minimization of a least-squares cost function as a desirable objective. This procedure relies on off-line modal separation algorithms to extract modal information and on least-squares parameter subset optimization to combine the modal results and globally fit the modal parameters to the measured data. The modal separation algorithms resolved modal density of 5 modes/Hz in the ASCIE experiment. They promise to be useful in many challenging applications.

Carrier, Alain C.↗

NASA NTRS1989

Tau approximation techniques for identification of coefficients in parabolic PDE

A variant of the Tau method, called the weak Tau method, is developed on the basis of the weak form of the PDE for use in least-squares parameter estimation; also presented is a suitable abstract convergence framework. The emphasis is on the theoretical framework that allows treatment of the weak Tau method when it is applied to a wide class of inverse problems, including those for diffusion-advection equations, the Fokker-Planck model for population dynamics, and damped beam equations. Extensive numerical testing of the weak Tau method has demonstrated that it compares quite favorably with existing methods.

Banks, H. T.↗

NASA NTRS1976

Development of a solid propellant viscoelastic dynamic model

The results of a one year study to develop a dynamic response model for the Space Shuttle Solid Rocket Motor (SRM) propellant are presented. An extensive literature survey was conducted, from which it was concluded that the only significant variables affecting the dynamic response of the SRM propellant are temperature and frequency. Based on this study, and experimental data on propellants related to the SRM propellant, a dynamic constitutive model was developed in the form of a simple power law with temperature incorporated in the form of a modified power law. A computer program was generated which performs a least-squares curve-fit of laboratory data to determine the model parameters and it calculates dynamic moduli at any desired temperature and frequency. Additional studies investigated dynamic scaling laws and the extent of coupling between the SRM propellant and motor cases. It was found, in agreement with other investigations, that the propellant provides all of the mass and damping characteristics whereas the case provides all of the stiffness.

Hufferd, W. L.↗

NASA NTRS2011

Dynamic Shape Reconstruction of Three-Dimensional Frame Structures Using the Inverse Finite Element Method

A robust and efficient computational method for reconstructing the three-dimensional displacement field of truss, beam, and frame structures, using measured surface-strain data, is presented. Known as shape sensing , this inverse problem has important implications for real-time actuation and control of smart structures, and for monitoring of structural integrity. The present formulation, based on the inverse Finite Element Method (iFEM), uses a least-squares variational principle involving strain measures of Timoshenko theory for stretching, torsion, bending, and transverse shear. Two inverse-frame finite elements are derived using interdependent interpolations whose interior degrees-of-freedom are condensed out at the element level. In addition, relationships between the order of kinematic-element interpolations and the number of required strain gauges are established. As an example problem, a thin-walled, circular cross-section cantilevered beam subjected to harmonic excitations in the presence of structural damping is modeled using iFEM; where, to simulate strain-gauge values and to provide reference displacements, a high-fidelity MSC/NASTRAN shell finite element model is used. Examples of low and high-frequency dynamic motion are analyzed and the solution accuracy examined with respect to various levels of discretization and the number of strain gauges.

Gherlone, Marco↗

NASA NTRS1995

Actuator placement for active sound and vibration control of cylinders

Active structural acoustic control is a method in which the control inputs (used to reduce interior noise) are applied directly to a vibrating structural acoustic system. The control concept modeled in this work is the application of in-plane force inputs to piezoceramic patches bonded to the wall of a vibrating cylinder. The cylinder is excited by an exterior noise source -- an acoustic monopole -- located near the outside of the cylinder wall. The goal is to determine the force inputs and sites for the piezoelectric actuators so that (1) the interior noise is effectively damped; (2) the level of vibration of the cylinder shell is not increased; and (3) the power requirements needed to drive the actuators are not excessive. We studied external monopole excitations at two frequencies. A cylinder resonance of 100 Hz, where the interior acoustic field is driven in multiple, off-resonance cylinder cavity modes, and a cylinder resonance of 200 Hz are characterized by both near and off-resonance cylinder vibration modes which couple effectively with a single, dominant, low-order acoustic cavity mode at resonance. Previous work has focused almost exclusively on meeting objective (1) and solving a complex least-squares problem to arrive at an optimal force vector for a given set of actuator sites. In addition, it has been noted that when the cavity mode couples with cylinder vibration modes (our 200 Hz case) control spillover may occur in higher order cylinder shell vibrational modes. How to determine the best set of actuator sites to meet objectives (1)-(3) is the main contribution of our research effort. The selection of the best set of actuator sites from a set of potential sites is done via two metaheuristics -- simulated annealing and tabu search. Each of these metaheuristics partitions the set of potential actuator sites into two disjoint sets: those that are selected to control the noise (on) and those that are not (off). Next, each metaheuristic attempts to improve this initial solution by calculating the change in the objective value when one selected actuator site is turned off and one actuator site that previously was not selected is turned on. All such pairwise exchanges are performed and the exchange that improves the objective the most is made. Eventually the search is unable to improve the objective value and a local optimum (with respect to pairwise exchanges) is reached. Both simulated annealing and tabu search provide mechanisms to escape local optima and allow the search to continue until (hopefully) a global optimum is found. Our experiments with the 100 Hz and 200 Hz cases confirm that both metaheuristics are able to uncover better solutions than those selected based upon engineering judgement alone. In addition, the high quality solutions generated by these metaheuristics, when minimizing interior noise, do not further excite the cylinder shell. Thus, we are able to meet objective (2) without imposing an additional constraint or forming a multiobjective performance measure. An additional observation is that in many cases the amplitude and phase values for several chosen actuator sites were nearly identical. This natural grouping means that fewer control channels are needed and the resulting control system is simpler. Currently no power requirements have been set, so objective (3) cannot be addressed. A set of experiments is planned with a laboratory test article (a cylinder). For these experiments the transfer matrices will be generated experimentally. It is hoped that the predicted performance of the best actuator sites found by our metaheuristics will correlate well with the measured performance.

Kincaid, Rex K.↗

NASA NTRS1975

Experimental evaluation of multiplane-multispeed rotor balancing through multiple critical speeds

The experimentally proven range of application of the influence-coefficient balancing method, especially the least-squares procedure, is extended to include the case of multiple bending critical speeds within the operating range of a test rotor. Tests were conducted on a laboratory quality machine capable of speeds up to 18,000 rpm in order to investigate several distinct practical aspects of flexible-rotor balancing. These include: (1) balancing for operation through four bending critical speeds, (2) balancing of rotors mounted in both rigid and flexible bearing supports, (3) balancing of rotors with various amounts of measured vibration-response data and different numbers of correction planes, and (4) balancing of rotors with different arbitrary initial-unbalance configurations. The results show that a lightly damped, flexible rotor can be balanced systematically and efficiently through four bending critical speeds. It is concluded that the influence-coefficient method should be equally applicable to rotors or shafts having more than four bending critical speeds in their operating-speed ranges.

Tessarzik, J. M.↗

NASA NTRS1999

Optimized System Identification

In system identification, one usually cares most about finding a model whose outputs are as close as possible to the true system outputs when the same input is applied to both. However, most system identification algorithms do not minimize this output error. Often they minimize model equation error instead, as in typical least-squares fits using a finite-difference model, and it is seen here that this distinction is significant. Here, we develop a set of system identification algorithms that minimize output error for multi-input/multi-output and multi-input/single-output systems. This is done with sequential quadratic programming iterations on the nonlinear least-squares problems, with an eigendecomposition to handle indefinite second partials. This optimization minimizes a nonlinear function of many variables, and hence can converge to local minima. To handle this problem, we start the iterations from the OKID (Observer/Kalman Identification) algorithm result. Not only has OKID proved very effective in practice, it minimizes an output error of an observer which has the property that as the data set gets large, it converges to minimizing the criterion of interest here. Hence, it is a particularly good starting point for the nonlinear iterations here. Examples show that the methods developed here eliminate the bias that is often observed using any system identification methods of either over-estimating or under-estimating the damping of vibration modes in lightly damped structures.

Juang, Jer-Nan↗

NASA NTRS1996

Large-eddy simulation of a backward facing step flow using a least-squares spectral element method

We report preliminary results obtained from the large eddy simulation of a backward facing step at a Reynolds number of 5100. The numerical platform is based on a high order Legendre spectral element spatial discretization and a least squares time integration scheme. A non-reflective outflow boundary condition is in place to minimize the effect of downstream influence. Smagorinsky model with Van Driest near wall damping is used for sub-grid scale modeling. Comparisons of mean velocity profiles and wall pressure show good agreement with benchmark data. More studies are needed to evaluate the sensitivity of this method on numerical parameters before it is applied to complex engineering problems.

Chan, Daniel C.↗