Vibration of a Spinning Rotationally Periodic Rotor

2007 ◽  
Author(s):  
Hyunchul Kim ◽  
I. Y. Shen
Keyword(s):  
Theoretical Analysis ◽  
High Speed ◽  
Experimental Studies ◽  
Mode Shapes ◽  
Integer Multiple ◽  
Numerical Studies ◽  
Theoretical Predictions ◽  
Symmetric Rotor ◽  
Unified Algorithm ◽  
Cyclic Symmetric

This paper studies the vibrations of a spinning, rotationally periodic (also known as cyclic symmetric) rotor through theoretical analysis and experimental studies. The theoretical analysis consists of two parts. The first part is Fourier analysis of mode shapes of a stationary rotor with periodicity N. A periodic mapping of the n-th mode shape shows that its k-th Fourier coefficient is generally zero, except when k ± n is an integer multiple of N. The second part is to apply the derived mode shapes through a unified algorithm developed by Shen and Kim [1] to predict primary and secondary resonances of spinning, rotationally periodic rotors. The experimental study focuses on vibration measurements of a spinning disk carrying 4 pairs of evenly spaced brackets mounted on a high-speed air-bearing spindle. Initially, experimentally measured waterfall plots do not agree well with those from theoretical predictions. Further numerical studies show that mistune of rotationally periodic rotors could substantially change their waterfall plots. After the mistune is modeled, experimental and theoretical results agree very well with a difference of only 0.8% in natural frequencies observed in the ground-based coordinates.

Ground-Based Vibration Response of a Spinning, Cyclic, Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Hyunchul Kim ◽  
I. Y. Shen
Keyword(s):  
Theoretical Analysis ◽  
Fourier Expansion ◽  
Vibration Response ◽  
Modal Testing ◽  
Mode Shapes ◽  
Stationary Mode ◽  
Spinning Disk ◽  
Phase Index ◽  
Symmetric Rotor ◽  
Cyclic Symmetric

This paper is to study ground-based vibration response of a spinning, cyclic, symmetric rotor through a theoretical analysis and an experimental study. The theoretical analysis consists of three steps. The first step is to analyze the vibration characteristics of a stationary, cyclic, symmetric rotor with N identical substructures. For each vibration mode, we identify a phase index n and derive a Fourier expansion of the mode shape in terms of the phase index n. The second step is to predict the rotor-based vibration response of the spinning, cyclic, symmetric rotor based on the Fourier expansion of the mode shapes and the phase indices. The rotor-based formulation includes gyroscopic and centrifugal softening terms. Moreover, rotor-based response of repeated modes and distinct modes is obtained analytically. The third step is to transform the rotor-based response to ground-based response using the Fourier expansion of the stationary mode shapes. The theoretical analysis leads to the following conclusions. First, gyroscopic effects have no significant effects on distinct modes. Second, the presence of gyroscopic and centrifugal softening effects causes the repeated modes to split into two modes with distinct frequencies ω1 and ω2 in the rotor-based coordinates. Third, the transformation to ground-based observers leads to primary and secondary frequency components. In general, the ground-based response presents frequency branches in the Campbell diagram at ω1±kω3 and ω2±kω3, where k is phase index n plus an integer multiple of cyclic symmetry N. When the gyroscopic effect is significantly greater than the centrifugal softening effect, two of the four frequency branches vanish. The remaining frequency branches take the form of either ω1+kω3 and ω2−kω3 or ω1−kω3 and ω2+kω3. To verify these predictions, we also conduct a modal testing on a spinning disk carrying four pairs of brackets evenly spaced in the circumferential direction with ground-based excitations and responses. The disk-bracket system is mounted on a high-speed, air-bearing spindle. An automatic hammer excites the spinning disk-bracket system and a laser Doppler vibrometer measures its vibration response. A spectrum analyzer processes the hammer excitation force and the vibrometer measurements to obtain waterfall plots at various spin speeds. The measured primary and secondary frequency branches from the waterfall plots agree well with those predicted analytically.

Ground-Based Vibration Response of a Spinning, Cyclic Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects

2008 ◽  
Author(s):  
I. Y. Shen ◽  
Hyunchul Kim
Keyword(s):  
Theoretical Analysis ◽  
Fourier Expansion ◽  
Vibration Response ◽  
Modal Testing ◽  
Mode Shapes ◽  
Secondary Resonances ◽  
Spinning Disk ◽  
Phase Index ◽  
Symmetric Rotor ◽  
Cyclic Symmetric

This paper is to study ground-based vibration response of a spinning, cyclic symmetric rotor through a theoretical analysis and an experimental study. The theoretical analysis consists of three steps. The first step is to analyze vibration characteristics of a stationary cyclic symmetric rotor with N identical substructures. For each vibration mode, we identify a phase index n and derive a Fourier expansion of the mode shape in terms of the phase index n. The second step is to predict rotor-based vibration response of the spinning, cyclic symmetric rotor based on the Fourier expansion of the mode shapes and the phase indices. The rotor-based formulation includes gyroscopic and centrifugal softening terms. Moreover, rotor-based response of repeated modes and distinct modes is obtained analytically. The third step is to transform the rotor-based response to ground-based response using the Fourier expansion of the stationary mode shapes. The theoretical analysis leads to the following conclusions. First, gyroscopic effects have no significant effects on distinct modes. Second, the presence of gyroscopic and centrifugal softening effects cause the repeated modes to split into two modes with distinct frequencies ω1 and ω2 in the rotor-based coordinates. Third, the transformation to ground-based observers leads to primary and secondary resonances. In general, the ground-based response presents resonance branches in the Campbell diagram at ω1 ± kω3 and ω2 ± kω3, where k is phase index n plus an integer multiple of cyclic symmetry N. When the gyroscopic effect is significantly greater than the centrifugal softening effect, two of the four resonance branches disappear. The remaining resonances take the form of either ω1 + kω3 and ω2 − kω3 or ω1 − kω3 and ω2 + kω3. To verify these predictions, we also conduct a modal testing on a spinning disk carrying 4 pairs of brackets evenly spaced in the circumferential direction with ground-based excitations and responses. The disk-bracket system is mounted on a high-speed, air-bearing spindle. An automatic hammer excites the spinning disk-bracket system and a laser Doppler vibrometer measures its vibration response. A spectrum analyzer processes the hammer excitation force and the vibrometer measurements to obtain waterfall plots at various spin speeds. The measured primary and secondary resonances from the waterfall plots agree well with those predicted analytically.

Experimental Verification of Ground-Based Response of a Spinning, Cyclic Symmetric, Rotor Assembled to a Flexible Stationary Housing via Multiple Bearings

2014 ◽  
Author(s):  
W. C. Tai ◽  
I. Y. Shen
Keyword(s):  
Laser Doppler Vibrometer ◽  
Nodal Line ◽  
Spindle Motor ◽  
Housing System ◽  
Mode Shapes ◽  
Coupled Modes ◽  
Spin Speed ◽  
Resonance Frequencies ◽  
Symmetric Rotor ◽  
Cyclic Symmetric

This paper is to present an experimental study that measures ground-based response of a spinning, cyclic, symmetric rotor-bearing-housing system. In particular, the study focuses on rotor-housing coupled modes that are significantly dominated by housing deformation. In the experiments, a ball-bearing spindle motor, carrying a disk with four evenly spaced slots (i.e., the rotor), is mounted onto a stationary housing. The housing is a square plate supported with steel spacers at four corners and fixed to the ground. Two different ways are used to excite the rotor-housing system to measure frequency response functions (FRFs). One is to use an automatic hammer tapping at the disk, and the other is to use a piezoelectric actuator attached to the housing. Vibration of the rotor and housing is measured via a laser Doppler vibrometer and a capacitance probe. The experiments consist of two parts. The first part is to obtain FRFs when the rotor is not spinning. The measured FRFs reveal two rotor-housing coupled modes dominated by the housing. Their mode shapes are characterized by one nodal line in housing and one nodal diameter in the rotor. The second part is to obtain waterfall plots when the rotor is spinning at various speeds. The waterfall plots show that the housing dominant modes split into primary branches and secondary branches as the spin speed varies. The primary branches almost do not change with respect to the spin speed. In contrast, the secondary branches evolve into forward and backward branches. Moreover, their resonance frequencies increase and decrease at four times of the spin speed. The measured results agree well with the predictions found in the authors’ previous theoretical study [1].

Modal Analysis of Ballooning Strings With Small Sag

1999 ◽  
Author(s):  
Rongjun Fan ◽  
Sushil K. Singh ◽  
Christopher D. Rahn
Keyword(s):  
Steady State ◽  
High Speed ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Theoretical Predictions ◽  
Nonlinear Equations Of Motion

Abstract During the manufacture and transport of textile products, yarns are rotated at high speed and form balloons. The dynamic response of the balloon to varying rotation speed, boundary excitation, and disturbance forces governs the quality of the associated process. Resonance, in particular, can cause large tension variations that reduce product quality and may cause yarn breakage. In this paper, the natural frequencies and mode shapes of a single loop balloon are calculated to predict resonance. The three dimensional nonlinear equations of motion are simplified via small steady state displacement (sag) and vibration assumptions. Axial vibration is assumed to propagate instantaneously or in a quasistatic manner. Galerkin’s method is used to calculate the mode shapes and natural frequencies of the linearized equations. Experimental measurements of the steady state balloon shape and the first two natural frequencies and mode shapes are compared with theoretical predictions.

Mode Evolution of Cyclic Symmetric Rotors Assembled to Flexible Bearings and Housing

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
Hyunchul Kim ◽  
Nick Theodore Khalid Colonnese ◽  
I. Y. Shen
Keyword(s):  
Vibration Mode ◽  
Fluid Dynamic ◽  
Travelling Wave ◽  
Modal Testing ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Phase Index ◽  
Symmetric Rotor ◽  
Cyclic Symmetric ◽  
Inertia Forces

This paper is to study how the vibration modes of a cyclic symmetric rotor evolve when it is assembled to a flexible housing via multiple bearing supports. Prior to assembly, the vibration modes of the rotor are classified as “balanced modes” and “unbalanced modes.” Balanced modes are those modes whose natural frequencies and mode shapes remain unchanged after the rotor is assembled to the housing via bearings. Otherwise, the vibration modes are classified as unbalanced modes. By applying fundamental theorems of continuum mechanics, we conclude that balanced modes will present vanishing inertia forces and moments as they vibrate. Since each vibration mode of a cyclic symmetric rotor can be characterized in terms of a phase index (Chang and Wickert, “Response of Modulated Doublet Modes to Travelling Wave Excitation,” J. Sound Vib., 242, pp. 69–83; Chang and Wickert, 2002, “Measurement and Analysis of Modulated Doublet Mode Response in Mock Bladed Disks,” J. Sound Vib., 250, pp. 379–400; Kim and Shen, 2009, “Ground-Based Vibration Response of a Spinning Cyclic Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects,” ASME J. Vibr. Acoust. (in press)), the criterion of vanishing inertia forces and moments implies that the phase index by itself can uniquely determine whether or not a vibration mode is a balanced mode as follows. Let N be the order of cyclic symmetry of the rotor and n be the phase index of a vibration mode. Vanishing inertia forces and moments indicate that a vibration mode will be a balanced mode if n≠1,N−1,N. When n=N, the vibration mode will be balanced if its leading Fourier coefficient vanishes. To validate the mathematical predictions, modal testing was conducted on a disk with four pairs of brackets mounted on an air-bearing spindle and a fluid-dynamic bearing spindle at various spin speeds. Measured Campbell diagrams agree well with the theoretical predictions.

Ground-Based Response of a Spinning, Cyclic Symmetric Rotor Assembled to a Flexible Stationary Housing via Multiple Bearings

2013 ◽  
Author(s):  
W. C. Tai ◽  
I. Y. Shen
Keyword(s):  
Theoretical Analysis ◽  
Coordinate System ◽  
Equation Of Motion ◽  
Periodic Coefficients ◽  
Numerical Example ◽  
Housing Systems ◽  
Rotor Bearing ◽  
Symmetric Rotor ◽  
Cyclic Symmetric ◽  
Multiple Bearings

This paper is to study free response of a spinning, cyclic symmetric rotor assembled to a flexible housing via multiple bearings. In particular, the rotor spins at a constant speed ω3, and the housing is excited via a set of initial displacements. The focus is to study ground-based response of the rotor through theoretical and numerical analyses. The paper consists of three parts. The first part is to briefly summarize an equation of motion of the coupled rotor-bearing-housing systems for the subsequent analyses. The equation of motion, obtained from prior research [1], employs a ground-based and a rotor-based coordinate system to the housing and the rotor, respectively. As a result, the equation of motion takes the form of a set of ordinary differential equations with periodic coefficients of frequency ω3. To better understand its solutions, a numerical model is introduced as an example. In this example, the rotor is a disk with four radial slots and the housing is a square plate with a central shaft. The rotor and housing are connected via two ball bearings. The second part of the paper is to analyze the rotor’s response in the rotor-based coordinate system theoretically. When the rotor is at rest, let ωH be the natural frequency of a coupled rotor-bearing-housing mode whose response is dominated by the housing. The theoretical analysis then indicates that response of the spinning rotor will possess frequency components ωH ± ω3 demonstrating the interaction of the spinning rotor and the housing. The theoretical analysis further shows that this splitting phenomenon results from the periodic coefficients in the equation of motion. The numerical example also confirms this splitting phenomenon. The last part of the paper is to analyze the rotor’s response in the ground-based coordinate system. A coordinate transformation shows that the ground-based response of the spinning rotor consists of two major frequency branches ωH − (k + 1) ω3 and ωH − (k − 1) ω3, where k is an integer determined by the cyclic symmetry and vibration modes of interest. The numerical example also confirms this derivation.

Modal Analysis of Ballooning Strings With Small Curvature

2000 ◽  
Vol 68 (2) ◽  
pp. 332-338 ◽  
Author(s):  
R. Fan ◽  
S. K. Singh ◽  
C. D. Rahn
Keyword(s):  
High Speed ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Small Displacement ◽  
Axial Motion ◽  
Theoretical Predictions ◽  
Nonlinear Equations Of Motion

During the manufacture and transport of textile products, yarns are rotated at high speed. The surface of revolution generated by the rotating yarn is called a balloon. The dynamic response of the balloon to varying rotation speed, boundary excitation, and aerodynamic disturbances affects the quality of the associated textile product. Resonance, in particular, can cause large tension variations that reduce product quality and may cause yarn breakage. In this paper, the natural frequencies and mode shapes of a single loop balloon are calculated to predict resonance. The three-dimensional nonlinear equations of motion are simplified under assumptions of small displacement and quasi-static axial motion. After linearization, Galerkin’s method is used to calculate the mode shapes and natural frequencies. Experimental measurements of the steady-state balloon shape and the first two natural frequencies and mode shapes are compared with theoretical predictions.

Parametric Resonances of a Spinning Cyclic Symmetric Rotor Assembled to a Flexible Stationary Housing via Multiple Bearings

2012 ◽  
Author(s):  
W. C. Tai ◽  
I. Y. Shen
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Housing System ◽  
Mode Shapes ◽  
Element Analysis ◽  
Linear Elastic ◽  
Rotor Bearing ◽  
Symmetric Rotor ◽  
Cyclic Symmetric ◽  
Inertia Forces

This paper is to present findings from a theoretical study on free vibration and stability of a rotor-bearing-housing system. The rotor is cyclic symmetric and spinning at constant speed, while the housing is stationary and flexible. Moreover, the rotor and housing are assembled via multiple, linear, elastic bearings. For the rotor and the housing, their mode shapes are first obtained in rotor-based and ground-based coordinate systems, respectively. By discretizing the kinetic and potential energies of the rotor-bearing-housing system through use of the mode shapes, a set of equations of motion appears in the form of ordinary differential equations with periodic coefficients. Analyses of the equations of motion indicate that instabilities could appear at certain spin speed in the form of combination resonances of the sum type. To demonstrate the validity of the formulation, two numerical examples are studied. For the first example, the spinning rotor is an axisymmetric disk and the housing is a square plate with a central shaft. Moreover, the rotor and the housing are connected via two linear elastic bearings. Instability appears in the form of coupled vibration between the stationary housing and spinning rotor through three different formats: rigid-body rotor translation, rigid-body rotor rocking, and elastic rotor modes that present unbalanced inertia forces or moments. For the second example, the rotor is cyclic symmetric in the form of a disk with four evenly spaced slots. The housing and bearings remain the same. When the rotor is stationary, natural frequencies and mode shapes predicted from the formulation agree well with those predicted from a finite element analysis, which further ensures the validity of the formulation. When the cyclic symmetric rotor spins, instability appears in the same three formats as in the case of axisymmetric rotor. Number of instability zones, however, increases because the cyclic symmetric rotor has more elastic rotor modes that present unbalanced inertia forces or moments.

Effects of Curvature on Slamming Loads

2016 ◽  
Author(s):  
John B. Weber ◽  
Raj Das ◽  
Mark Battley
Keyword(s):  
High Speed ◽  
High Performance ◽  
Experimental Studies ◽  
Rigid Bodies ◽  
Small Scale ◽  
Marine Vehicles ◽  
Numerical Studies ◽  
Impact Forces ◽  
Structural Responses ◽  
Peak Impact Force

Much research has been directed at understanding and predicting water slamming loads for a range of geometries of varying rigidity and size. Analytical and numerical studies focused on slamming of cylindrical rigid bodies are present in literature but there are relatively few experimental studies useful for validation purposes, none of which methodically investigate a range of curvatures. Despite the current understanding of slamming loads and structural responses, high speed marine vehicles still experience slamming related failures in operation. In this study, nominally rigid, singly curved prismatic specimens of varying curvature are subjected to constant velocity water impacts relevant to those encountered by high performance offshore racing yachts and other high-speed craft. Peak impact forces of 14 to 52 kN were recorded while testing specimens with radii ranging from 0.300 to 5.000 m. Experimental peak impact force and event impulse are found to be significantly lower than predicted by numerical and small scale empirically derived methods. A modification is introduced which improves the empirical model.

Linear Hydroelastic Analysis of High-Speed Catamarans and Monohulls

1999 ◽  
Vol 43 (01) ◽  
pp. 48-63
Author(s):  
Ole Andreas Hermundstad ◽  
Jan Vidar Aarsnes ◽  
Torgeir Moan
Keyword(s):  
High Speed ◽  
Hydrodynamic Interaction ◽  
Linear Method ◽  
Mode Shapes ◽  
Ship Motions ◽  
Regular Waves ◽  
Integral Theorem ◽  
Strip Theory ◽  
Theoretical Predictions ◽  
Hydroelastic Analysis

A linear method for hydroelastic analysis of high-speed vessels is presented. It is based on a modal approach and represents a generalization of the high-speed strip theory of Faltinsen & Zhao (1991a). Hydrodynamic interaction between the hulls of catamarans is properly accounted for by utilizing symmetry. It is demonstrated how an integral theorem can be utilized to find the hydrodynamic force for general mode shapes. Theoretical predictions of ship motions and sectional forces and moments are compared with experimental data for a flexible high-speed catamaran model in regular waves. The influences of hull interaction and the application of the integral theorem are demonstrated.

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