Mathematical Insights Into Linear Mode Localization in Nearly Cyclic Symmetric Rotors With Mistune

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
Y. F. Chen ◽  
I. Y. Shen
Keyword(s):  
Natural Frequencies ◽  
Necessary Condition ◽  
Localized Modes ◽  
Mode Localization ◽  
Linear Mode ◽  
Blade System ◽  
Cyclic Symmetric ◽  
Complete Set ◽  
Curve Veering ◽  
Cyclic Symmetric Rotors

In this paper, we develop a mathematical analysis to gain insights of mode localization often encountered in nearly cyclic symmetric rotors that contain slight mistune. First, we conduct a Fourier analysis in the spatial domain to show that mode localization can appear only when a group of tuned rotor modes form a complete set in the circumferential direction. In light of perturbation theories, these tuned rotor modes must also have very similar natural frequencies, so that they can be linearly combined to form localized modes when the mistune is present. Second, the natural frequency of these tuned rotor modes can further be represented in terms of a mean frequency and a deviatoric component. A Rayleigh–Ritz formulation then shows that mode localization occurs only when the deviatoric component and the rotor mistune are about the same order. As a result, we develop an effective visual method—through use of the deviatoric component and the rotor mistune—to precisely identify those modes needed to form localized modes. Finally, we show that curve veering is not a necessary condition for mode localization to occur in the context of free vibration. Not all curve veering leads to mode localization, and not all modes in curve veering contribute to mode localization. Numerical examples on a disk–blade system with mistune confirm all the findings above.

Mathematical Insights of Mode Localization in Nearly Cyclic Symmetric Rotors With Mistune

2013 ◽  
Author(s):  
Y. F. Chen ◽  
I. Y. Shen
Keyword(s):  
Natural Frequencies ◽  
Necessary Condition ◽  
Localized Modes ◽  
Mode Localization ◽  
Visual Method ◽  
Mean Frequency ◽  
Cyclic Symmetric ◽  
Complete Set ◽  
Curve Veering ◽  
Cyclic Symmetric Rotors

In this paper, we develop a mathematical analysis to gain insights of mode localization often encountered in nearly cyclic symmetric rotors with mistune. First, we conduct a Fourier analysis in the spatial domain to show that mode localization can appear only when a group of tuned rotor modes form a complete set in the circumferential direction. In light of perturbation theories, these tuned rotor modes must also have very similar natural frequencies, so that they can be easily reoriented when the mistune is present to form localized modes. Second, the natural frequency of these tuned rotor modes can further be represented in terms of a mean frequency and a deviatoric component. A Rayleigh-Ritz formulation then shows that mode localization occurs only when the deviatoric component and the rotor mistune are about the same order. As a result, we can develop an effective visual method — through use of the deviatoric component and the rotor mistune — to precisely identify those modes needed to form localized modes. Finally, we show that curve veering with respect to engine orders is not a necessary condition for mode localization to occur in the context of free vibration. Not all curve veering leads to mode localization, and not all modes in curve veering contribute to mode localization. A numerical example confirms the findings above.

scholarly journals Effects of Bearings and Housing on Mode Localization of a Nearly Cyclic Symmetric System

2016 ◽  
Vol 138 (5) ◽  
Author(s):  
Y. F. Chen ◽  
I. Y. Shen
Keyword(s):  
Finite Element ◽  
Reference System ◽  
Deductive Reasoning ◽  
Mode Shapes ◽  
Symmetric System ◽  
Finite Element Analyses ◽  
Localized Modes ◽  
Mode Localization ◽  
Bladed Disk ◽  
Cyclic Symmetric

This technical brief is to study how flexible bearings and housing affect mode localization of a nearly cyclic symmetric system with mistuning. This study is conducted via finite-element analyses and deductive reasoning. A reference system studied is a bladed disk with two groups of 24 localized modes. When bearings and housing are introduced into the reference system, their presence changes natural frequencies, mode shapes, and the number of the localized modes. Moreover, the mistuning causes bearing forces to surge for all the localized modes. A deductive reasoning based on the existing literature supports the observation from the finite-element analyses.

Modal Analysis of Mistuned Turbine Blade Packet Due to Combined Blade and Lacing Wire Damage

2019 ◽  
Vol 24 (3) ◽  
pp. 546-557
Author(s):  
Mangesh S. Kotambkar
Keyword(s):  
Material Properties ◽  
Natural Frequencies ◽  
Cyclic Symmetry ◽  
Vibration Localization ◽  
Symmetric Structure ◽  
Blade System ◽  
Cyclic Symmetric ◽  
Drastic Effect ◽  
Grouped Blades ◽  
Geometric Variation

The turbine disk blade system is a cyclic symmetric structure, initially tuned with all its blades perfectly identical in geometry and material properties; similarly interconnecting lacing wires are of equal stiffness. The cyclic symmetry of the bladed disks gets destroyed due to small differences in material properties or geometric variation between individual blades or lacing wires causing mistuning. Although mistuning is typically small, it can have a drastic effect on the dynamic response of the system. In particular, mistuning can also cause vibration localization for a few blades and the associated concentration of vibration energy can lead to an increase in blade amplitude and stress levels. Numerical simulations are performed with the characteristic equations of the simplified continuum model. Two different damage severity indices are included in the model to study the combined effect of cracked blades and damaged lacing wires on the natural frequencies of grouped blades. This study highlights the characteristic changes in the sub modal frequencies under combined damage in a stand still position. Although the major cause of mistuning is blade damage, lacing wire damage is more frequent and often acts as a precursor to blade damage and thus the present study focuses on mistuning due to combined damage.

Effects of Bearings and Housing on Mode Localization of a Nearly Cyclic Symmetric System

2014 ◽  
Author(s):  
Y. F. Chen ◽  
I. Y. Shen
Keyword(s):  
Boundary Condition ◽  
Reference System ◽  
Mode Shapes ◽  
Symmetric System ◽  
Localized Modes ◽  
Disk System ◽  
Element Analysis ◽  
Mode Localization ◽  
Bladed Disk ◽  
Cyclic Symmetric

This paper is to study how flexible bearings and housing affect mode localization of a nearly cyclic symmetric system with mistune. A finite element analysis is first conducted on a reference system that consists of a circular disk and 24 blades with mistune. The disk is annular with an inner rim and an outer rim. A fixed boundary condition is imposed at the inner rim, while the 24 blades with mistune are evenly attached to the outer rim and subjected to a free boundary condition. As a result of the mistune, the reference system presents 26 localized torsional modes as well as 24 localized in-plane modes in its blade vibration. When the fixed inner rim is replaced by a bearing support (i.e., an elastic boundary condition), not only the localized torsional modes can change their natural frequencies and mode shapes but also the number of the localized torsional modes may be increased to 28 in some range of bearing stiffness. Similarly, when the bladed-disk reference system is mounted on a stationary housing via a bearing support, the number of the localized in-plane modes can change from 24 to 33 modes. Moreover, localized mode shapes change significantly, and some of them involve significant housing deformation. To understand this phenomenon theoretically, we first demonstrate that the presence of bearing and housing provides additional degrees of freedom, which, in turn, allow the bladed-disk system to have additional disk modes. When the bearing and housing stiffness is properly tuned, some of these additional disk modes may possess significant torsional or in-plane displacement components in the blades. If these additional modes happen to have a natural frequency that is close to those of the localized modes of the reference system, these additional modes will join the localized modes to form new localized modes. As a result, the number of localized modes increases and the mode shapes change significantly.

The Influence of a Cracked Blade on Rotor’s Free Vibration

2008 ◽  
Vol 130 (5) ◽  
Author(s):  
Yi-Jui Chiu ◽  
Shyh-Chin Huang
Keyword(s):  
Free Vibration ◽  
Natural Frequency ◽  
Natural Frequencies ◽  
Bending Moment ◽  
Rotor System ◽  
Coupled Modes ◽  
Mode Localization ◽  
Coupled Vibrations ◽  
Crack Location ◽  
Blade System

The influence on the coupled vibrations among shaft, disk, and blades of a rotor system due to a cracked blade was investigated analytically. On the shaft-disk-blade system, without a cracked blade, there exist three types of coupled modes, shaft disk blade (SDB), disk blade (DB), and blade-blade. A cracked blade was found to change not only the natural frequencies but also the types of coupled modes. First, the DB modes disappeared and were replaced by SDB modes. Second, in some modes the disk experienced mode localization due to the cracked blade. Numerical results showed that natural frequencies varied with the blade’s crack location and depth. At no bending moment places the crack imposed no effect. The crack affected the natural frequency slightly until its depth reached as deep as 70% then the frequency dropped abruptly.

Localized and Non-Localized Normal Modes in a Strongly Nonlinear Discrete System

1991 ◽  
Author(s):  
Alexander F. Vakakis
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Linear Vibration ◽  
Mode Localization ◽  
Linear Mode ◽  
Strongly Nonlinear ◽  
System A ◽  
Coupling Stiffness ◽  
Nonlinear Normal ◽  
Nonlinear Discrete System

Abstract The free oscillations of a strongly nonlinear, discrete oscillator are examined by computing its “nonsimilar nonlinear normal modes.” These are motions represented by curves in the configuration space of the system, and they are not encountered in classical, linear vibration theory or in existing nonlinear perturbation techniques. For an oscillator with weak coupling stiffness and “mistiming,” both localized and nonlocalized modes are detected, occurring in small neighborhoods of “degenerate” and “global” similar modes of the “tuned” system. When strong coupling is considered, only nonlocalized modes are found to exist. An interesting result of this work is the detection of mode localization in the “tuned” periodic system, a result with no counterpart in existing theories on linear mode localization.

Mode Localization and Delocalization in Constrained Strings and Beams

1997 ◽  
Author(s):  
Chris H. Riedel ◽  
Chin An Tan
Keyword(s):  
Bernoulli Beam ◽  
Constrained System ◽  
Reflection And Transmission ◽  
Mode Localization ◽  
High Frequencies ◽  
Transmission Coefficients ◽  
Curve Veering ◽  
Elastic Constraints ◽  
Euler Bernoulli Beam ◽  
Free Vibration Response

Abstract The free vibration response of a string and a Euler-Bernoulli beam supported by intermediate elastic constraints is studied and analyzed by the transfer function method. The constrained system consists of three subsystems coupled by constraints imposed at the subsystem interfaces. For both the string and beam systems, curve veering and mode localization are observed in the lower modes when the distance between the elastic constraints is varied. As the mode number increases, the modes of the system become extended indicating that the coupling springs have little effect on the system at higher modes. A wave analysis is employed to further investigate the behavior of the systems at high frequencies. Reflection and transmission coefficients are formulated to show the effects of the constraints on the coupling of the subsystems. The weakly bi-coupled beam produces an interesting phenomena where a particular mode experiences no localization while neighboring modes are localized. The frequency at which this occurs is termed the delocalization frequency. Only one delocalization frequency exists and it occurs where the reflection coefficient of the propagating wave becomes zero.

Sensor Egregium – An Atomic Force Microscope Sensor for Continuously Variable Resonance Amplification

2021 ◽  
pp. 1-23
Author(s):  
Rafiul Shihab ◽  
Tasmirul Jalil ◽  
Burak Gulsacan ◽  
Matteo Aureli ◽  
Ryan Tung
Keyword(s):  
Finite Element Analysis ◽  
Atomic Force Microscope ◽  
Natural Frequencies ◽  
Proof Of Concept ◽  
Element Analysis ◽  
Force Microscopy ◽  
Atomic Force ◽  
Wide Range ◽  
Curve Veering ◽  
Dynamic Phenomena

Abstract Numerous nanometrology techniques concerned with probing a wide range of frequency dependent properties would benefit from a cantilevered sensor with tunable natural frequencies. In this work, we propose a method to arbitrarily tune the stiffness and natural frequencies of a microplate sensor for atomic force microscope applications, thereby allowing resonance amplification at a broad range of frequencies. This method is predicated on the principle of curvature-based stiffening. A macroscale experiment is conducted to verify the feasibility of the method. Next, a microscale finite element analysis is conducted on a proof-of-concept device. We show that both the stiffness and various natural frequencies of the device can be highly controlled through applied transverse curvature. Dynamic phenomena encountered in the method, such as eigenvalue curve veering, are discussed and methods are presented to accommodate these phenomena. We believe that this study will facilitate the development of future curvature-based microscale sensors for atomic force microscopy applications.

A Partial Generalization of Mann's Theorem Concerning Orthogonal Latin Squares

1988 ◽  
Vol 31 (4) ◽  
pp. 409-413 ◽  
Author(s):  
E. T. Parker ◽  
Lawrence Somer
Keyword(s):  
Latin Square ◽  
Latin Squares ◽  
Necessary Condition ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Complete Set

AbstractLetn = 4t+- 2, where the integert ≧ 2. A necessary condition is given for a particular Latin squareLof ordernto have a complete set ofn — 2mutually orthogonal Latin squares, each orthogonal toL.This condition extends constraints due to Mann concerning the existence of a Latin square orthogonal to a given Latin square.

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