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At least 325 records · Page 18

Design and implementation of scalable tape archiver

In order to reduce costs, computer manufacturers try to use commodity parts as much as possible. Mainframes using proprietary processors are being replaced by high performance RISC microprocessor-based workstations, which are further being replaced by the commodity microprocessor used in personal computers. Highly reliable disks for mainframes are also being replaced by disk arrays, which are complexes of disk drives. In this paper we try to clarify the feasibility of a large scale tertiary storage system composed of 8-mm tape archivers utilizing robotics. In the near future, the 8-mm tape archiver will be widely used and become a commodity part, since recent rapid growth of multimedia applications requires much larger storage than disk drives can provide. We designed a scalable tape archiver which connects as many 8-mm tape archivers (element archivers) as possible. In the scalable archiver, robotics can exchange a cassette tape between two adjacent element archivers mechanically. Thus, we can build a large scalable archiver inexpensively. In addition, a sophisticated migration mechanism distributes frequently accessed tapes (hot tapes) evenly among all of the element archivers, which improves the throughput considerably. Even with the failures of some tape drives, the system dynamically redistributes hot tapes to the other element archivers which have live tape drives. Several kinds of specially tailored huge archivers are on the market, however, the 8-mm tape scalable archiver could replace them. To maintain high performance in spite of high access locality when a large number of archivers are attached to the scalable archiver, it is necessary to scatter frequently accessed cassettes among the element archivers and to use the tape drives efficiently. For this purpose, we introduce two cassette migration algorithms, foreground migration and background migration. Background migration transfers cassettes between element archivers to redistribute frequently accessed cassettes, thus balancing the load of each archiver. Background migration occurs the robotics are idle. Both migration algorithms are based on access frequency and space utility of each element archiver. To normalize these parameters according to the number of drives in each element archiver, it is possible to maintain high performance even if some tape drives fail. We found that the foreground migration is efficient at reducing access response time. Beside the foreground migration, the background migration makes it possible to track the transition of spatial access locality quickly.

Nemoto, Toshihiro↗

Wind Turbine Contingency Control Through Generator De-Rating

Maximizing turbine up-time and reducing maintenance costs are key technology drivers for wind turbine operators. Components within wind turbines are subject to considerable stresses due to unpredictable environmental conditions resulting from rapidly changing local dynamics. In that context, systems health management has the aim to assess the state-of-health of components within a wind turbine, to estimate remaining life, and to aid in autonomous decision-making to minimize damage to the turbine. Advanced contingency control is one way to enable autonomous decision-making by providing the mechanism to enable safe and efficient turbine operation. The work reported herein explores the integration of condition monitoring of wind turbines with contingency control to balance the trade-offs between maintaining system health and energy capture. The contingency control involves de-rating the generator operating point to achieve reduced loads on the wind turbine. Results are demonstrated using a high fidelity simulator of a utility-scale wind turbine.

condition monitoring↗

A fourth-order phase-field fracture model: Formulation and numerical solution using a continuous/discontinuous Galerkin method

Modeling crack initiation and propagation in brittle materials is of great importance to be able to predict sudden loss of load-carrying capacity and prevent catastrophic failure under severe dynamic loading conditions. Second-order phase-field fracture models have gained wide adoption given their ability to capture the formation of complex fracture patterns, e.g. via crack merging and branching, and their suitability for implementation within the context of the conventional finite element method. Higher-order phase-field models have also been proposed to increase the regularity of the exact solution and thus increase the spatial convergence rate of its numerical approximation. However, they require special numerical techniques to enforce the necessary continuity of the phase field solution. In this paper, we derive a fourth-order phase-field model of fracture in two independent ways; namely, from Hamilton’s principle and from a higher-order micromechanics-based approach. The latter approach is novel, and provides a physical interpretation of the higher-order terms in the model. In addition, we propose a continuous/discontinuous Galerkin (C/DG) method for use in computing the approximate phase-field solution. This method employs Lagrange polynomial shape functions to guarantee -continuity of the solution at inter-element boundaries, and enforces the required regularity with the aid of additional variational and interior penalty terms in the weak form. Finally, the phase-field equation is coupled with the momentum balance equation to model dynamic fracture problems in hyper-elastic materials. Two benchmark problems are presented to compare the numerical behavior of the C/DG method with mixed finite element methods.

42 ENGINEERING↗

Parallel Computing Strategies for Irregular Algorithms

Parallel computing promises several orders of magnitude increase in our ability to solve realistic computationally-intensive problems, but relies on their efficient mapping and execution on large-scale multiprocessor architectures. Unfortunately, many important applications are irregular and dynamic in nature, making their effective parallel implementation a daunting task. Moreover, with the proliferation of parallel architectures and programming paradigms, the typical scientist is faced with a plethora of questions that must be answered in order to obtain an acceptable parallel implementation of the solution algorithm. In this paper, we consider three representative irregular applications: unstructured remeshing, sparse matrix computations, and N-body problems, and parallelize them using various popular programming paradigms on a wide spectrum of computer platforms ranging from state-of-the-art supercomputers to PC clusters. We present the underlying problems, the solution algorithms, and the parallel implementation strategies. Smart load-balancing, partitioning, and ordering techniques are used to enhance parallel performance. Overall results demonstrate the complexity of efficiently parallelizing irregular algorithms.

Biswas, Rupak↗

Passive Rocket Diffuser Theory: A Re-Examination of Minimum Second Throat Size

Second-throat diffusers serve to isolate rocket engines from the effects of ambient back pressure during testing without using active control systems. Among the most critical design parameters is the relative area of the diffuser throat to that of the nozzle throat. A smaller second throat is generally desirable because it decreases the stagnation-to-ambient pressure ratio the diffuser requires for nominal operation. There is a limit, however. Below a certain size, the second throat can cause pressure buildup within the diffuser and prevent it from reaching the start condition that protects the nozzle from side-load damage. This paper presents a method for improved estimation of the minimum second throat area which enables diffuser start. The new 3-zone model uses traditional quasi-one-dimensional compressible flow theory to approximate the structure of two distinct diffuser flow fields observed in Computational Fluid Dynamics (CFD) simulations and combines them to provide a less-conservative estimate of the second throat size limit. It is unique among second throat sizing methods in that it accounts for all major conical nozzle and second throat diffuser design parameters within its limits of application. The performance of the 3-zone method is compared to the historical normal shock and force balance methods, and verified against a large number of CFD simulations at specific heat ratios of 1.4 and 1.25. Validation is left as future work, and the model is currently intended to function only as a first-order design tool.

Jones, Daniel R.↗

Evaluating The Relationships Between Supine Propriorception Assessments with Upright Functional Mobility and Balance Tests

INTRODUCTION Upon return to Earth, spaceflight sensorimotor adaptations can result in impaired posture and locomotion [1], [2], and current exercise countermeasures in the International Space Station are not sufficient to maintain sensorimotor function. In-flight countermeasures and assessment tools for sensorimotor function are needed to mitigate risks associated with mission-critical task performance upon return to Earth or arrival to 0.38 G on Mars. One proposed countermeasure for proprioceptive deconditioning includes training on a tilt board device [3], which will be tested in an upcoming bed rest study. In preparation for the said study, the objective of this pilot study is to compare performance in supine proprioceptive assessments to performance in functional upright activities. The results are intended to guide the selection of supine assessment tools for the upcoming bedrest study and the development of future inflight proprioceptive training and assessment tools for exploration missions. METHODS Seventeen healthy participants (8 males and 9 females, 27.9 ± 8.5 years) provided informed consent as approved by the Institutional Review Board (IRB) at NASA. A horizontal air-bearing sled was used to provide a proprioceptive challenge in a supine body orientation, allowing for mediolateral motion with minimal friction during supine stance [4]. Participants were loaded axially (30 to 60% body weight) with their feet on a vertically oriented tilt board and instructed to perform supine assessment activities using custom software that displays a cursor controlled by tilting the tilt board. In a single-leg static activity on the tilt board (TB-St), performance was measured as the percent time spent on a center target during a 30-second trial. During a two-feet dynamic activity on the tilt board (TB-Dy), performance was measured as the number of targets captured in 30 seconds. These supine tasks were compared to performance during two upright activities used to represent upright performance: completion time for a functional mobility task (FMT) [2],[5] and a performance score from force plate sway data during 30-second single-leg upright standing (USL). RESULTS AND DISCUSSION A Spearman's rank-order correlation was run to assess the relationship between upright and supine assessments. There were statistically significant, strong correlations between scores for the FMT and each TB activity, including TB-St, rs(17) = 0.50, p<0.05, TB-Dy3, rs(17) = 0.73, p<0.001, and TB-Dy9, rs(15) = 0.53, p<0.05. There were no statistically significant correlations between USL and each TB activity. Despite a lack of correlation with upright single-leg balance, the ability of the TB to indicate upright performance in the FMT is promising, as this task is a standard post-flight measure designed to characterize locomotor dysfunction [2], [5]. CONCLUSION These efforts will inform the selection of an appropriate sensorimotor assessment method for the upcoming bed rest study and similar future studies. While current work uses a bed rest analog to develop these technologies, future work aims to prepare capabilities for future in-flight sensorimotor training and assessment to mitigate the risks of proprioception impairment after spaceflight. ACKNOWLEDGEMENTS This work was supported by the University Space Research Association Internship Program. REFERENCES [1] Mulavara A.P. et al (2018) Med Sci Sports Exerc, 50(9), 1961-1980. [2] Mulavara A.P. et al (2010) Exp Brain Res, 202(3), 649-659. [3] Macaulay T.R. et al (2021) Front. Syst. Neurosci., 15, 658-985. [4] Goel R. et al (2017) Front. Syst. Neurosci, 11. [5] Koppelmans V. et al (2013) BMC Neurology, 12.

R Bellisle↗

Black Start of Unbalanced Microgrids Harmonizing Single- and Three-Phase Grid-Forming Inverters

As power systems are transforming with increasing penetrations of inverter-based resources (IBRs), system restoration using IBRs has drawn attention. Using distributed grid-forming (GFM) assets located near critical loads, either three-phase or single-phase, to establish microgrid voltages in the absence of a bulk grid, a distribution system could obtain high system survivability. For swift and secure recovery of a critical load in a single-phase lateral, local single-phase GFM inverters can form a microgrid, and then it can be combined with a neighbouring grid with the inverters remaining in GFM mode for voltage and frequency regulation until the bulk grid comes online. It leads to dynamic interoperation of single-phase GFM inverters with three-phase ones in the black start process. This paper studies the novel approach with electromagnetic transient (EMT) simulations. To evaluate the potential and the technical challenges of the heterogeneous IBR-driven black start, three-phase and single-phase GFM inverter models are developed, including negative-sequence control for voltage balance and a phase-by-phase current limiter (three-phase) and current magnitude limiter (single-phase). To examine dynamic aspects of the black-start process, the EMT simulation also models transformer and motor dynamics emulating their inrush and startup behavior as well as network dynamics. Involvement of grid-following assets to facilitate the black-start process is also modeled. By allowing multiple GFM inverters to collectively black start without leader-follower coordination, regardless of phases, a system can achieve extreme resilience. An inverter-driven black start of a heavily unbalanced 2-MVA distribution feeder using 1 three-phase and 3 single-phase GFM inverters is demonstrated. The simulation shows the heterogeneous system can maintain stability with the single-phase GFM dynamics coupled with the three-phase one.

grid-forming inverter↗

Black Start of Unbalanced Microgrids Harmonizing Single- and Three-Phase Grid-Forming Inverters: Preprint

As power systems are transforming with increasing penetrations of inverter-based resources (IBRs), system restoration using IBRs has drawn attention. Using distributed grid-forming (GFM) assets located near critical loads, either three-phase or single-phase, to establish microgrid voltages in the absence of a bulk grid, a distribution system could obtain high system survivability. For swift and secure recovery of a critical load in a single-phase lateral, local single-phase GFM inverters can form a microgrid, and then it can be combined with a neighbouring grid with the inverters remaining in GFM mode for voltage and frequency regulation until the bulk grid comes online. It leads to dynamic interoperation of single-phase GFM inverters with three-phase ones in the black start process. This paper studies the novel approach with electromagnetic transient (EMT) simulations. To evaluate the potential and the technical challenges of the heterogeneous IBR-driven black start, three-phase and single-phase GFM inverter models are developed, including negative-sequence control for voltage balance and a phase-by-phase current limiter (three-phase) and current magnitude limiter (single-phase). To examine dynamic aspects of the black-start process, the EMT simulation also models transformer and motor dynamics emulating their inrush and startup behavior as well as network dynamics. Involvement of grid-following assets to facilitate the black-start process is also modeled. By allowing multiple GFM inverters to collectively black start without leader-follower coordination, regardless of phases, a system can achieve extreme resilience. An inverter-driven black start of a heavily unbalanced 2-MVA distribution feeder using 1 three-phase and 3 single-phase GFM inverters is demonstrated. The simulation shows the heterogeneous system can maintain stability with the single-phase GFM dynamics coupled with the three-phase one.

grid-forming inverter↗

DCFC + Hydrogen Station Design Optimization [Slides]

Both direct current fast charging (DCFC) and Hydrogen stations are working to create successful long-term business models; however, analyses for DCFC and hydrogen fueling infrastructure are almost always performed separately. This work provides a detailed exploration of the benefit of integrating DCFC and hydrogen stations to lower the total system cost from load balancing and equipment cost sharing. To achieve this we have adapted the REopt optimization framework to simultaneously optimize the design and operation of integrated DCFC and H2 fueling station. Results indicate that 1) combining hydrogen fueling and DCFC stations can significantly reduce lifetime costs compared to separated stations. 2) Co-location with additional site load reduces DCFC costs, however, integration of DCFC with hydrogen provides an even greater cost reduction. 3) Adding PV to combined stations further reduces the lifetime station cost. 4) Capital investments in station combination today can help reduce the cost of operating DCFC tomorrow and 5) product diversification acts as a hedge against variability and enables a more dynamic response to market changes.

08 HYDROGEN↗

Validation and Demonstration of Control System Functional Capabilities within the IES Plug-and-Play Simulation Environment

The concept of an integrated energy system (IES) is meant to combine different energy technologies in synergistic ways to achieve a more secure and economical energy supply. The RAVEN-based HYBRID framework is used to find the optimal installed capacity and the optimal economical dispatch of each component of the IES. The new RAVEN plugin for grid and capacity optimization (HERON) only addresses the limits that affect the production variables and the corresponding rates of variation (explicit constraints). However, other variables are subject to constraints, and the associated limits should be accounted for (implicit constraints). In particular, for the power dispatch problem, the optimization algorithm takes into account the limits on the electrical power output and the corresponding hourly power variations but does not consider other constraints on process variables whose response affects the service life of the IES. This report describes a scheme that allows accounting for implicit constraints without increasing the size of the optimization problem. To obtain a more accurate approximation of the nonlinear dynamic behavior, a parametric version of the dynamic mode decomposition with control (DMDc) algorithm was developed to derive the state-space representation matrices of the IES components at different scheduling parameter. Thanks to this approach, a more accurate approximation of the system response can be obtained, the limits imposed by thermal mechanical implicit constraints can be translated into power dispatch limits, and the feedbacks to HERON power dispatcher can be provided. To assess the developed methodology, a power dispatching test case composed of three power generating and storage units (Balance of Plant, Secondary Energy Source, Thermal Energy Storage) was developed. The power output of each one of the three units was optimized to meet the imposed time-dependent load demand trajectory and to maximize the IES profitability by meeting both the explicit and implicit constraints.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Design of a Low Speed Fan Stage for Noise Suppression

This report describes the design of a low tip speed, moderate pressure rise fan stage for demonstration of noise reduction concepts. The fan rotor is a fixed-pitch configuration delivering a design pressure ratio of 1.378 at a specific flow of 43.1 lbm/sec/sq ft. Four exit stator configurations were provided to demonstrate the effectiveness of circumferential and axial sweep in reducing rotor-stator interaction tone noise. The fan stage design was combined with an axisymmetric inlet, conical convergent nozzle, and nacelle to form a powered fan-nacelle subscale model. This model has a 22-inch cylindrical flow path and employs a rotor with a 0.30 hub-to-tip radius ratio. The design is fully compatible with an existing NASA force balance and rig drive system. The stage aerodynamic and structural design is described in detail. Three-dimensional (3-D) computational fluid dynamics (CFD) tools were used to define optimum airfoil sections for both the rotor and stators. A fan noise predictive system developed by Pratt & Whitney under contract to NASA was used to determine the acoustic characteristics of the various stator configurations. Parameters varied included rotor-to-stator spacing and vane leading edge sweep. The structural analysis of the rotor and stator are described herein. An integral blade and disk configuration was selected for the rotor. Analysis confirmed adequate low cycle fatigue life, vibratory endurance strength, and aeroelastic suitability. A unique load carrying stator arrangement was selected to minimize generation of tonal noise due to sources other than rotor-stator interaction. Analysis of all static structural components demonstrated adequate strength, fatigue life, and vibratory characteristics.

Dalton, W. N.↗

Dynamic Force Reconstruction of Transient Flap Control Force Experiments in a Hypersonic Wind Tunnel

This work details the design of a sliced-cone model with a flap embedded within the slice. The flap is controlled by a fast acting servo to simulate a control force of a hypersonic vehicle. Design considerations such as sensor placement, expected loading, servo arm selection, and cavity temperatures are detailed in this work. Such a design imparts a dynamic response of the test article and as such, dynamic force measurement methodologies are required to reconstruct the control forcing. Three methods are of particular interest: the Sum of Weighted Accelerations Technique, Time Domain Deconvolution Method, and Frequency Domain Inverse Method. Novel versions of each method have been applied to this dataset to improve the reconstruction accuracy.

dynamic balance↗

Closed-form Static Analysis with Inertia Relief and Displacement-Dependent Loads Using a MSC/NASTRAN DMAP Alter

Solving for the displacements of free-free coupled systems acted upon by static loads is commonly performed throughout the aerospace industry. Many times, these problems are solved using static analysis with inertia relief. This solution technique allows for a free-free static analysis by balancing the applied loads with inertia loads generated by the applied loads. For some engineering applications, the displacements of the free-free coupled system induce additional static loads. Hence, the applied loads are equal to the original loads plus displacement-dependent loads. Solving for the final displacements of such systems is commonly performed using iterative solution techniques. Unfortunately, these techniques can be time-consuming and labor-intensive. Since the coupled system equations for free-free systems with displacement-dependent loads can be written in closed-form, it is advantageous to solve for the displacements in this manner. Implementing closed-form equations in static analysis with inertia relief is analogous to implementing transfer functions in dynamic analysis. Using a MSC/NASTRAN DMAP Alter, displacement-dependent loads have been included in static analysis with inertia relief. Such an Alter has been used successfully to solve efficiently a common aerospace problem typically solved using an iterative technique.

Barnett, Alan R.↗

Addressing Load Imbalance in Bioinformatics and Biomedical Applications: Efficient Scheduling across Multiple GPUs

Computational bioinformatics and biomedical applications frequently contain heterogeneously sized units of work or tasks, for instance due to variability in the sizes of biological sequences and molecules. Variable-sized workloads lead to load imbalances in parallel implementations which detract from efficiency and performance. Many modern computing resources now have multiple graphics processing units(GPUs) per computer for acceleration. These multiple GPU resources need to be used efficiently through balancing of workloads across the GPUs. OpenMP is a portable directive-based parallel programming API used ubiquitously in bioscience applications to program CPUs; recently, the use of OpenMP directives for GPU acceleration has become possible. Here, motivated by experiences with imbalanced loads in GPU-accelerated bioinformatics applications, we address the load balancing problem using OpenMP task-to-GPU scheduling combined with OpenMP GPU offloading for multiply heterogeneous workloads – loads with both variable input sizes, and simultaneously, variable convergence rates for algorithms with a stochastic component – scheduled across multiple GPUs. We aim to develop strategies which are both easy to use and have lower overheads, and may be incorporated incrementally in existing programs which already make use of OpenMP for CPU-based threading in order to make use of multi-GPU computers. We test different combinations of input size variability and convergence rate variability, and characterize the effects of these different scenarios on the performance of scheduling strategies across multiple GPUs with OpenMP. We present several dynamic scheduling solutions for different parallel patterns, explore optimizations, and provide publicly available example computational kernels to make these strategies easy to use in programs. This work will enable application developers to efficiently and easily use multiple GPUs for imbalanced workloads found in bioinformatics and biomedical applications.

Thavappiragasam, Mathialakan↗

An Analysis of Performance Enhancement Techniques for Overset Grid Applications

The overset grid methodology has significantly reduced time-to-solution of high-fidelity computational fluid dynamics (CFD) simulations about complex aerospace configurations. The solution process resolves the geometrical complexity of the problem domain by using separately generated but overlapping structured discretization grids that periodically exchange information through interpolation. However, high performance computations of such large-scale realistic applications must be handled efficiently on state-of-the-art parallel supercomputers. This paper analyzes the effects of various performance enhancement techniques on the parallel efficiency of an overset grid Navier-Stokes CFD application running on an SGI Origin2000 machine. Specifically, the role of asynchronous communication, grid splitting, and grid grouping strategies are presented and discussed. Results indicate that performance depends critically on the level of latency hiding and the quality of load balancing across the processors.

Djomehri, J. J.↗

Dynamic Force Reconstruction of Transient Flap Control Force Experiments in a Hypersonic Wind Tunnel

This work details the design of a sliced-cone model with a flap embedded within the slice. The flap is controlled by a fast acting servo to simulate a control force of a hypersonic vehicle. Design considerations such as sensor placement, expected loading, servo arm selection, and cavity temperatures are detailed in this work. Such a design imparts a dynamic response of the test article and as such, dynamic force measurement methodologies are required to reconstruct the control forcing. Three methods are of particular interest: the Sum of Weighted Accelerations Technique, Time Domain Deconvolution Method, and Frequency Domain Inverse Method. Novel versions of each method have been applied to this dataset to improve the reconstruction accuracy.

dynamic balance↗

Efficient Parallelization of Irregular Applications on GPU Architectures

With the enlarging computation capacity of general Graphics Processing Units (GPUs), leveraging GPUs to accelerate parallel applications has become a critical topic in academia and industry. However, a wide range of irregular applications with the computation-/memory-intensive nature cannot easily achieve high GPU utilization. The challenges mainly involve the following aspects: first, data dependence leads to coarse-grained kernel and inefficient parallelism; second, heavy GPU memory usage may cause frequent memory evictions and extra overhead of I/O; third, specific computation patterns produce memory redundancies; last, workload balance and data reusability conjunctly benefit the overall performance, but there may exist a dynamic trade-off between them. Targeting these challenges, this dissertation proposes multiple optimizations to accelerate two real-world applications: many-body correlation functions to simulate nuclear physics in a large-scale scientific system; the other is the eALS-based matrix factorization recommendation system. To accelerate the calculations of many-body correlation functions, this dissertation presents three frameworks in GPU memory management and multi-GPU scheduling. Firstly, an optimized systematic GPU memory management framework, MemHC, utilizes a series of new memory reduction designs in GPU memory allocation, CPU/GPU communications, and GPU memory oversubscription. Secondly, an enhanced multi-GPU scheduling framework, MICCO, particularly by taking both data dimension (e.g., data reuse and data eviction) and computation dimension into account. MICCO designs a heuristic scheduling algorithm and a machine learning-based regression model to generate the optimal settings of a proposed new concept to manage the trade-off. Thirdly, a locality-aware multi-GPU scheduling framework. This scheduler leverages pipeline batch generation with a looking-ahead strategy by building local dependency graphs for memory transfer reduction and better data reuse, achieving up to 79.92% memory cost reduction and 1.67x speedup. To parallelize the eALS-based recommendation system, this dissertation proposes an efficient CPU/GPU heterogeneous recommendation system, HEALS. HEALS employs newly designed architecture-adaptive data formats to achieve load balance and good data locality on CPU and GPU. To mitigate the data dependence, HEALS presents a CPU/GPU collaboration model for both task parallelism and data parallelism with multiple kernel computation optimizations. In summary, this dissertation efficiently accelerates two typical irregular applications on GPUs by building four frameworks, including CPU/GPU collaboration, GPU memory management, and multi-GPU scheduling.

Wang, Qihan↗

Hybrid methods for rotordynamic analysis

Effective procedures are presented for the response analysis of the Space Shuttle Main Engine turbopumps under transient loading conditions. Of particular concern is the determination of the nonlinear response of the systems to rotor imbalance in presence of bearing clearances. The proposed procedures take advantage of the nonlinearities involved being localized at only a few rotor/housing coupling joints. The methods include those based on integral formulations for the incremental solutions involving the transition matrices of the rotor and housing. Alternatively, a convolutional representation of the housing displacements at the coupling points is proposed which would allow performing the transient analysis on a reduced model of the housing. The integral approach is applied to small dynamical models to demonstrate the efficiency of the approach. For purposes of assessing the numerical integration results for the nonlinear rotor/housing systems, a numerical harmonic balance procedure is developed to enable determining all possible harmonic, subharmonic, and nonperiodic solutions of the systems. A brief account of the Fourier approach is presented as applied to a two degree of freedon rotor-support system.

Noah, Sherif T.↗