System Eigenfrequencies Research Articles

Numerical Estimation of Torsional Dynamic Coefficients of a Hydraulic Turbine

The rotordynamic behavior of a hydraulic turbine is influenced by fluid‐rotor interactions at the turbine runner. In this paper computational fluid dynamics (CFDs) are used to numerically predict the torsional dynamic coefficients due to added polar inertia, damping, and stiffness of a Kaplan turbine runner. The simulations are carried out for three operating conditions, one at about 35% load, one at about 60% load (near best efficiency), and one at about 70% load. The runner rotational speed is perturbed with a sinusoidal function with different frequencies in order to estimate the coefficients of added polar inertia and damping. It is shown that the added coefficients are dependent of the load and the oscillation frequency of the runner. This affect the system′s eigenfrequencies and damping. The eigenfrequency is reduced with up to 65% compared to the eigenfrequency of the mechanical system without the fluid interaction. The contribution to the damping ratio varies between 30–80% depending on the load. Hence, it is important to consider these added coefficients while carrying out dynamic analysis of the mechanical system.

Numerical predictions of tuned liquid tank structural systems

A fully nonlinear 2-D σ -transformed finite difference solver has been developed based on inviscid flow equations in rectangular tanks. The fluid equations are coupled to a linear elastic support structure. Nonoverturning sloshing motions are simulated during structural vibration cycles at and outside resonance. The wave tank acts as a tuned liquid damper (TLD). The TLD response is highly nonlinear when large liquid sloshing occurs. The solver is valid at any water depth except for small depth when shallow water waves and viscous effects would become important. Results of liquid sloshing induced by horizontal base excitations are presented for small to steep nonbreaking waves at tank aspect ratios, depth to length, h/b of 0.5, 0.25 and 0.125, representing deep to near shallow water cases. The effectiveness of the TLD is discussed through predictions of coupling frequencies and response of the tank-structural system for different tank sizes and mass ratios between fluid and structure. An effective tank-structural system typically displays two distinct frequencies with reduced structural response (e.g., h / b = 0.5 ). These eigenfrequencies differ considerably from their noninteracting values. Hardening or softening spring behavior of the fluid, known to be present in solutions of pure sloshing motion in tanks, does not exists in the coupled system response. Strongest interactions occur with only one dominating sloshing mode when the nth sloshing frequency is close to the natural frequency of the structure, as the mass ratio between fluid and structure μ → 0 . Inclusion of higher modes reduces the efficiency of the TLD. Good agreement is achieved between the numerical model and a first-order potential theory approximation outside the resonance region when the unsteady sloshing motions remain small. When the free-surface amplitudes become large in the coupled system, the numerical peaks are larger and the troughs become lower as time evolves (typical nonlinear effects) compared to the linear solution. Nonlinearities were found to reduce the system displacement significantly, e.g., system resonance shifted to beating response, compared to linear predictions. It was also found that the system response is extremely sensitive to small changes in forcing frequency. In conclusion, if strong interaction exists, the coupled system exhibits nonlinearity in structural and free-surface response, but the coupled eigenfrequencies compare well with the linear predictions. Furthermore, the solver removes the need for free-surface smoothing for the cases considered herein (maximum wave steepness of 1.2). The numerical model provides a quick and accurate way of determining system eigenfrequencies which can be hard to identify and interpret in physical experiments.

Longitudinally Vibrating Elastic Rods with Locally and Non‐Locally Reacting Viscous Dampers

Eigencharacteristics of a longitudinally vibrating elastic rod with locally and non‐locally reacting damping are analyzed. The rod is considered as a continuous system and complex eigenfrequencies are determined as solution of a characteristic equation. The variation of the damping ratios with respect to damper locations and damping coefficients for the first four eigenfrequencies are obtained. It is shown that at any mode of locally or non‐locally damped elastic rod, the variation of damping ratio with damper location is linearly proportional to absolute value of the mode shape of undamped system. It is seen that the increasing damping coefficient does not always increase the damping ratio and there are optimal values for the damping ratio. Optimal values for external damping coefficients of viscous dampers and locations of the dampers are presented.

ON THE EIGENFREQUENCIES OF A FLEXIBLE ARM DRIVEN BY A FLEXIBLE SHAFT

The simulations of multibody dynamic systems with flexible components are generally based on solving the equations of motion by using approximate methods. This approach is taken because these systems' closed-form solutions are often not directly available. These methods often assume a solution as a finite series in terms of modal functions with time-varying coefficients. The eigenmodes of the system under study are preferable as the set of the basis functions used in these series because such expansions provide greater accuracy with fewer terms. As a consequence, accurate estimation of system eigenfrequencies and eigenmodes is extremely useful (potentially necessary) in the effective modelling and simulation of these systems. In this paper, a new general model consisting of rotor, shaft, hub, beam, and payload, as might be encountered in certain industrial robots, is presented and investigated. This model is similar in nature to those studied previously by a number of researchers, but it is more general in form. The authors believe that this model contains a more realistic (and higher fidelity) representation of the rotor–shaft–hub assembly of this system and its interaction with a flexible beam (arm) and associated payload. Through this model the relative influence of seven key dimensionless mass, stiffness and geometric parameters (ratios) on system eigenfrequencies and modes may be qualitatively and quantitatively investigated. These investigations may include many special cases such as flexible shaft+rigid beam, rigid shaft+flexible beam, cantilever–free beam, pinned–free beam, fixed–free shaft, etc. Given the volume of numerical studies which may be performed to this end, this paper concentrates on the effect of the two parameters representing the mass and stiffness ratios of the system manipulator on its driveline.

The representation of AC machine dynamics by complex signal flow graphs

Induction motors are modeled by nonlinear higher-order dynamic systems of considerable complexity. The dynamic analysis based on the complex notation exhibits a formal correspondence to the description using matrices of axes-oriented components; yet differences exist. The complex notation appears superior in that it allows the distinguishing between the system eigenfrequencies and the angular velocity of a reference frame which serves as the observation platform. The approach leads to the definition of single complex eigenvalues that do not have conjugate values associated with them. The use of complex state variables further permits the visualization of AC machine dynamics by complex signal flow graphs. These simple structures assist to form an understanding of the internal dynamic processes of a machine and their interactions with external controls. >

Fluid Coupling Characteristics and Vibration of Cylinder Cluster in Axial Flow. Part II: Experiments

The vibration characteristics of cylinder clusters in turbulent axial flow are studied experimentally, and the measurements are compared predictions by the theoretical model in Part I of this study. Two-, four- and 28-cylinder clusters were tested, with a maximum of four flexible instrumented cylinders in the cluster, the rest being rigid. The vibration is displayed in terms of the PSD, coherence and CSD phase of the vibration signals from one or two neighbouring cylinders, in two orthogonal directions; also, in terms of r.m.s vibration amplitude and the coupled intercylinder patterns of motion. Agreement between theory and experiment varies from reasonably good to excellent, which provides support for the validation of the theoretical model. Certain general characteristics of the vibration, applicable to any cluster in axial flow were established, e.g., that the dominant frequencies of the turbulence-excited vibration are not associated with pure modal patterns (corresponding to the system eigenfrequencies): the dominant frequencies are not discrete, and the modal patterns evolve continuously with frequency. Furthermore, by (i) studying mixed flexible/rigid four-cylinder clusters, (ii) varying the intercylinder distances and (iii) comparing vibration of one-, two- and four-cylinder cluster, insight into the modal and spectral characteristics of such systems has been gained.

Dynamics of a Cluster of Flexibly Interconnected Cylinders, Part 1: In Vacuum

This paper presents an analytical model for the dynamics of a cluster of flexible cylinders, the extremities of which are structurally interconnected, e.g., through “end plates” or other kinds of structural connectors. These connectors are modeled by sets of translational and rotational springs, such that resistance to both in-plane and out-of-plane deformation of the connectors is fully taken into account. The dynamics of the system in vacuum is examined here, and especially the effect of varying the different spring stiffnesses on the system eigenfrequencies. This provides a reference base and the necessary analytical tools for the study of the system in axial flow, which is presented as Part 2 of this work.

Dynamics of a Cluster of Flexibly Interconnected Cylinders, Part 2: In Axial Flow

In this paper the dynamical characteristics are discussed of a cluster of cylinders, the extremities of which are interconnected by beam-like connectors, modeled here by sets of linear springs as formulated in Part 1 of this study; resistance to both in-plane and out-of-plane deformation of the connectors is taken into account. The cylinders are subjected to axial flow, and hydrodynamic coupling in cylinder motions is taken into account. The system eigenfrequencies and modal shapes are calculated for different conditions of structural and hydrodynamic coupling. The effect of increasing flow on the dynamics of the system is also studied, up to and beyond the threshold of fluid-elastic instabilities.

Dynamics of a Flexible Cylinder in Subsonic Axial Flow

This paper examines the dynamics of a flexible cylinder with pinned ends immersed in axial subsonic flow, either bounded or unconfined. The problem proves to be surprisingly resistant to exact solution, as compared to the incompressible flow case, because of difficulties in determining precisely the inviscid aerodynamic forces. This paper presents a number of distinct formulations of these forces, involving different approximations: (1) a slender-body approximation; (2) an approximate three-dimensional formulation where, in the determination of the aerodynamic forces, the axial shape is prescribed in advance; and (3) an exact integral formulation of the generalized aerodynamic forces. In each case, Galerkin-type solutions yield the system eigenfrequencies which describe the dynamical behavior of the system. It is found that for sufficiently high flow velocities, divergence and flutter are possible. The different methods yield similar, but not quantitatively identical results. Interestingly, dependence of the dynamical characteristics on Mach number is shown to be weak for slender cylinders; for nonslender ones, it is stronger. Finally, a brief discussion of wave propagation in an unconstrained cylinder indicates the existence of a cutoff flow velocity for backward propagating waves, followed by wave amplification at higher flow, which is closely related to loss of stability inmore » the constrained system.« less