scholarly journals Parametric Resonances of a Spinning Cyclic Symmetric Rotor Assembled to a Flexible Stationary Housing Via Multiple Bearings

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
W. C. Tai ◽  
I. Y. Shen
Keyword(s):  
Equations Of Motion ◽  
Housing System ◽  
Combination Resonance ◽  
Linear Elastic ◽  
Model Free ◽  
Parametric Resonances ◽  
Symmetric Rotor ◽  
Cyclic Symmetric ◽  
Central Shaft ◽  
Multiple Bearings

This paper is meant to model free vibration of a coupled rotor-bearing-housing system. In particular, the rotor is cyclic symmetric and spins at constant speed while the housing is stationary and flexible. The rotor and housing are assembled via multiple, linear, elastic bearings. A set of equations of motion is derived using component mode synthesis, in which the rotor and the housing each are treated as a component. The equations of motion take 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. The rotor and the housing are connected via two linear elastic bearings. For the second example, the rotor is cyclic symmetric in the form of a disk with four evenly spaced radial slots. The housing and bearings remain the same. In both examples, instability appears as a combination resonance of the sum type between a rotor mode and an elastic housing mode. The cyclic symmetric rotor, however, has more instability zones. Finally, effects of damping are studied. Damping of the housing widens the instability zones, whereas the damping of the rotor does the opposite.

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.

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

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
W. C. Tai ◽  
I. Y. Shen
Keyword(s):  
Dominant Mode ◽  
Housing System ◽  
Cyclic Symmetry ◽  
Test Results ◽  
Benchmark Model ◽  
Rotor Bearing ◽  
Symmetric Rotor ◽  
Cyclic Symmetric ◽  
Square Plate ◽  
Multiple Bearings

This paper is to study ground-based response of a spinning, cyclic symmetric rotor assembled to a flexible housing via multiple bearings. In particular, interaction of the spinning rotor and the flexible housing is manifested theoretically, numerically, and experimentally. In the theoretical analysis, we show that the interaction primarily appears in coupled rotor–bearing–housing modes whose response is dominated by the housing. Specifically, let a housing-dominant mode have natural frequency ω(H) and the spin speed of the rotor to be ω3. In rotor-based coordinates, response of the spinning rotor for the housing-dominant mode will possess frequency splits ω(H)±ω3. In ground-based coordinates, response of the spinning rotor will possess alternative frequency splits ω(H)-(k+1)ω3 and ω(H)-(k-1)ω3, where k is an integer determined by the cyclic symmetry of the rotor and the housing-dominant mode of interest. In the numerical analysis, we study a benchmark model consisting of a spinning slotted disk mounted on a stationary square plate via two ball bearings. The numerical model successfully confirms the frequency splits both in the rotor-based and ground-based coordinates. In the experimental analysis, we conduct vibration testing on a rotor–bearing–housing system that mimics the numerical benchmark model. Test results reveal two housing-dominant modes. As the rotor spins at various speed, measured waterfall plots confirm that the housing-dominant modes split according to ω(H)-(k+1)ω3 and ω(H)-(k-1)ω3 as predicted.

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].

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.

Non-Linear Vibrations of Plates Under Thermal and Mechanical Loads

2005 ◽  
Author(s):  
P. Ribeiro
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Implicit Time ◽  
Linear Elastic ◽  
Time Integration Method ◽  
Thermal Fields ◽  
Non Linear ◽  
Linear Vibrations ◽  
The Time Domain ◽  
Constitutive Material

The geometrically non-linear vibrations of plates under the combined effect of thermal fields and mechanical excitations are analyzed. With this purpose, an accurate model based on a p-version, hierarchical, first-order shear deformation finite element is employed. The constitutive material of the plates is linear elastic and isotropic. The equations of motion are solved in the time domain by an implicit time integration method. The temperature and the amplitude of the mechanical excitation are varied, and transitions from periodic to non-periodic motions are found.

scholarly journals Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/hp Finite Element Model

2020 ◽  
Vol 10 (15) ◽  
pp. 5102
Author(s):  
Carlos Valencia Murillo ◽  
Miguel Gutierrez Rivera ◽  
Junuthula N. Reddy
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Variable Thickness ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Element Model ◽  
Linear Elastic ◽  
Commercial Code ◽  
The Finite Element Model ◽  
Hp Finite Element

In this paper, a seven-parameter spectral/hp finite element model to obtain natural frequencies in shell type structures is presented. This model accounts for constant and variable thickness of shell structures. The finite element model is based on a Higher-order Shear Deformation Theory, and the equations of motion are obtained by means of Hamilton’s principle. Analysis is performed for isotropic linear elastic shells. A validation of the formulation is made by comparing the present results with those reported in the literature and with simulations in the commercial code ANSYS. Finally, results for shell like structures with variable thickness are presented, and their behavior for different ratios r/h and L/r is studied.

Propagation of a nonlinear seismic pulse in an anelastic homogeneous medium

Geophysics ◽  
1993 ◽  
Vol 58 (7) ◽  
pp. 949-963 ◽  
Author(s):  
Dan W. Kosik
Keyword(s):  
Wave Propagation ◽  
Equations Of Motion ◽  
Surrounding Medium ◽  
Homogeneous Medium ◽  
Seismic Surveys ◽  
Linear Elastic ◽  
Cylindrically Symmetric ◽  
Unconsolidated Material ◽  
Linear Pulse ◽  
Implicit Finite Difference

Seismic surveys are often conducted using dynamite charges buried near the surface in unconsolidated material. In such material a large zone near the source should exist wherein nonlinear anelastic wave propagation, can be expected to take place, and have a significant impact on the way in which a seismic pulse forms and how its energy gets distributed into the surrounding medium. To obtain a solution for a propagating pulse in this zone, the equations of motion for nonlinear anelastic wave propagation, good to second order in the displacements, are solved numerically for the problem of a Gaussian pressure pulse acting on the interior cavity of a cylindrically symmetric hole in the medium. An implicit finite‐difference algorithm is used for the solution to the equations of motion for this problem. The anelastic medium is characterized by multivalued stress‐strain relations that exhibit hysteresis, and therefore a loss of energy per cycle, corresponding to a medium with a constant Q factor. Several numerical examples are calculated contrasting the nonlinear anelastic, linear anelastic, and linear elastic propagating pulses to one another. The nonlinear anelastic propagating pulse is found to have an amplitude that is several times larger than would be expected for a pulse in a linear medium and has a peak propagation velocity that is slightly less than that for a linear pulse. Dispersive effects are also evident for the nonlinear pulse.

Unilateral Impact and Contact of Elastic Structures Using Lagrangian Multipliers

2011 ◽  
Author(s):  
Sebastian Tatzko
Keyword(s):  
Equations Of Motion ◽  
Structural Equations ◽  
Contact Forces ◽  
Integration Scheme ◽  
Lagrangian Multiplier ◽  
Elastic Structures ◽  
Lagrangian Multipliers ◽  
Linear Elastic ◽  
Time Stepping ◽  
Numerical Effort

This paper deals with linear elastic structures exposed to impact and contact phenomena. Within a time stepping integration scheme contact forces are computed with a Lagrangian multiplier approach. The main focus is turned on a simplified solving method of the linear complementarity problem for the frictionless contact. Numerical effort is reduced by applying a Craig-Bampton transformation to the structural equations of motion.

The Use of Normal Modes in Problems of Forced Vibration and Impact

1957 ◽  
Vol 61 (560) ◽  
pp. 552-559
Author(s):  
R. P. N. Jones
Keyword(s):  
Forced Vibration ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Very High Frequency ◽  
Linear Elastic ◽  
Modes Of Vibration ◽  
Fundamental Equations ◽  
Very High ◽  
Simple Exposition

SummaryA simple exposition, using d'Alembert's principle and methods of virtual work, is given of the properties and applications of the normal modes of vibration of a linear elastic system. The use of the normal modes in problems of free and forced vibration and dynamic loading is discussed with the aid of simple examples, and it is shown that by these methods dynamical problems for any linear system may be solved without the use of the fundamental equations of motion, provided the natural frequencies and modes of the system are known. In most problems the solutions converge rapidly, so that only the first few modes of vibration need be considered, and in these cases the solution may be modified to give further improvement in convergence. Unsatisfactory convergence may be obtained, however, in problems where there is an exciting force of very high frequency, or an impact of short duration. An approximate allowance may be made for damping, provided this is small.

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.

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