Vibration of Bending-Torsion Coupled Resonance in a Rotor

2013 ◽  
Author(s):  
Hiroyuki Fujiwara ◽  
Tadashi Tsuji ◽  
Osami Matsushita
Keyword(s):  
Numerical Simulations ◽  
Natural Frequency ◽  
Torsional Vibration ◽  
Rotational Speed ◽  
Coupling Effect ◽  
The Other ◽  
Resonance Phenomenon ◽  
Horizontal Direction ◽  
Bending Vibration ◽  
Rotor Systems

In certain rotor systems, bending-torsion coupled resonance occurs when the rotational speed Ω (= 2π Ωrps) is equal to the sum/difference of the bending natural frequency ωb (= 2π fb) and torsional natural frequency ωθ(= 2πfθ). This coupling effect is due to an unbalance in the rotor. In order to clarify this phenomenon, an equation was derived for the motion of the bending-torsion coupled 2 DOF system, and this coupled resonance was verified by numerical simulations. In stability analyses of an undamped model, unstable rotational speed ranges were found to exist at about Ωrps = fb + fθ. The conditions for stability were also derived from an analysis of a damped model. In rotational simulations, bending-torsion coupled resonance vibration was found to occur at Ωrps = fb − fθ and fb + fθ. In addition, confirmation of this resonance phenomenon was shown by an experiment. When the rotor was excited in the horizontal direction at bending natural frequency, large torsional vibration appeared. On the other hand, when the rotor was excited by torsion at torsional natural frequency, large bending vibration appeared. Therefore, bending-torsion coupled resonance was confirmed.

Experimental Study of Torsional-Bending Coupled Vibration of a Rotor System With a Bladed Disk

2013 ◽  
Author(s):  
Takeshi Kudo ◽  
Koki Shiohata ◽  
Osami Matsushita ◽  
Hiroyuki Fujiwara ◽  
Akira Okabe ◽  
...  
Keyword(s):  
Natural Frequency ◽  
Torsional Vibration ◽  
Magnetic Bearing ◽  
Rotor System ◽  
Active Magnetic Bearing ◽  
Coupled Vibration ◽  
Bending Vibration ◽  
Nodal Diameter ◽  
Rotor Systems ◽  
Bladed Disk

An experimental investigation was conducted to confirm the bending-torsion coupled vibration of a rotor system with a bladed disk. For a rotor with relatively long blades such as in the latest low-pressure steam turbines, coupled vibration with shaft torsional vibration represents the bladed disk natural frequency of a nodal diameter (k) of zero (umbrella mode). Today this well-known behavior is reflected in the design of steam turbine rotor systems to prevent the blade vibration resonance due to torque excitation caused by the electric power grid, a standard for which is proposed by ISO 22266-1. The bending-torsion coupled resonance of rotor systems occurs, however, under specific conditions due to rotor unbalance. When the rotor’s rotational speed (Ω) is equal to the sum/difference of the bending natural frequency (ωb) and torsional natural frequency (ωθ), namely, Ω = ωθ ± ωb, there is coupled resonance, which was experimentally observed with a rotor with a relatively simplified shape. In this study, the test apparatus for a flexible rotor system equipped with a shrouded bladed disk driven by an electric motor was constructed to confirm the vibration characteristics, by envisioning the bending-torsion coupled resonance as applied to actual rotor systems of turbo machinery. A radial active magnetic bearing (AMB) was employed to support the rotor by controlling bearing stiffness and damping, and applying lateral directional excitation of forward and backward whirl to the rotor. A servomotor was also equipped at the end of the rotor system to excite the torsional vibration. The resonance of a bladed disk with nodal diameter (k) of zero, which was coupled with the rotor’s torsional vibration, was observed under the above condition (Ω = ωθ − ωb) through AMB excitation of the rotor’s bending natural frequency. Conversely, the torsional excitation caused by the servomotor was confirmed as causing the coupled resonance of rotor bending vibration.

Torsional Vibration Analysis of Shafts With Sections of Tapered Diameter

1993 ◽  
Author(s):  
John R. Baker ◽  
Keith E. Rouch
Keyword(s):  
Torsional Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Displacement Function ◽  
The Other ◽  
Mode Shapes ◽  
Axial Distance ◽  
Rotational Degree ◽  
Rotor Systems ◽  
Constant Diameter

Abstract This paper presents the development of two tapered finite elements for use in torsional vibration analysis of rotor systems. These elements are particularly useful in analysis of systems that have shaft sections with linearly varying diameters. Both elements are defined by two end nodes, and inertia matrices are derived based on a consistent mass formulation. One element assumes a cubic displacement function and has two degrees of freedom at each node: rotation about the shaft’s axis and change in angle of rotation with respect to the axial distance along the shaft. The other element assumes a linear displacement function and has one rotational degree of freedom at each node. The elements are implemented in a computer program. Calculated natural frequencies and mode shapes are compared for both tapered shaft sections and constant diameter sections. These results are compared with results from an available constant diameter element. It is shown that the element derived assuming a cubic displacement function offers much better convergence characteristics in terms of calculated natural frequencies, both for tapered sections and constant diameter sections, than either of the other two elements. The finite element code that was developed for implementation of these elements is specifically designed for torsional vibration analysis of rotor systems. Lumped inertia, lumped stiffness, and gear connection elements necessary for rotor system analysis are also discussed, as well as calculation of natural frequencies, mode shapes, and amplitudes of response due to a harmonic torque input.

Vibration Diagnosis Featuring Blade-Shaft Coupling Effect of Turbine Rotor Models

2010 ◽  
Vol 133 (2) ◽  
Author(s):  
Norihisa Anegawa ◽  
Hiroyuki Fujiwara ◽  
Osami Matsushita
Keyword(s):  
Natural Frequency ◽  
Rotational Speed ◽  
Coupling Effect ◽  
Rotating Shaft ◽  
Cross Sectional ◽  
Additional Resonance ◽  
Exciting Frequency ◽  
Excited Vibration ◽  
Out Of Plane ◽  
Blade System

As is well known, zero and one nodal diameter (k=0 and k=1) modes of a blade system interact with the shaft system. The former couples with torsional and/or axial shaft vibrations, and the latter with bending shaft vibrations. This paper addresses the latter. With respect to k=1 modes, we discuss, from experimental and theoretical viewpoints, in-plane blades and out-of-plane blades attached radially to a rotating shaft. We found that when we excited the shaft at the rotational speed of Ω=|ωb−ωs| (where ωb is the blade natural frequency, ωs the shaft natural frequency, and Ω is the rotational speed), the exciting frequency ν=ωs induced shaft-blade coupling resonance. In addition, in the case of the in-plane blade system, we encountered an additional resonance attributed to deformation caused by gravity. In the case of the out-of-plane blade system, we experienced two types of abnormal vibrations. One is the additional resonance generated at Ω=ωb/2 due to the unbalanced shaft and the anisotropy of bearing stiffness. The other is a flow-induced, self-excited vibration caused by galloping due to the cross-sectional shape of the blade tip because this instability disappeared in the rotation test inside a vacuum chamber. The two types of abnormal vibrations occurred at the same time, and both led to the entrainment phenomenon, as identified by our own frequency analysis technique.

A Study of Wind Generator Set Blades Dynamic Analysis

2012 ◽  
Vol 522 ◽  
pp. 413-416
Author(s):  
Jian Jie Zhang ◽  
Wen Lei Sun
Keyword(s):  
Dynamic Analysis ◽  
Wind Turbine ◽  
Natural Frequency ◽  
Torsional Vibration ◽  
Second Order ◽  
Response Analysis ◽  
Stochastic Dynamic ◽  
Complex Eigenvalue ◽  
Bending Vibration ◽  
Wind Generator

This paper on wind generator set blades had a simple introduction, and the modal analysis was used on the blades dynamic analysis, the conclusion that blades the different stage in the direction of the force and deformation was drawn, it can be concluded: the first three-order natural frequency at, the blades are expressed as a pure bending vibration. It can be seen from the blade vibration figure: a frequency of blades waving in the direction of the first natural frequency, first natural frequency of the second-order frequency blades shimmy direction, three frequencies for the blades waving direction of the second-order natural frequency, torsional frequency of fourth order for the direction of the shimmy, five order of frequency of blades placed waving direction of the three natural frequencies. According to the theory of vibration, the vibration in the process of energy is concentrated in bands 1 and 2, so the first stage waving vibration (bending vibration) is the main vibration of the wind generator set blades; 4 and 5 vibration modes performance for bending dominated accompanied shimmy and torsional vibration. Visible, the shimmy vibration and torsional vibration in high-end part is still not the main vibration.It provides the theoretical basis so as to improve the quality of the design of wind turbine blades. Blade is one of the wind generator set key components, many aerodynamic problems are faced, the aerodynamic efficiency of the blades is to be considered an important factor in the blade design, and their job security is more important factor. The wind turbine power source is the natural randomness of strong wind, the blades often run in the stall condition, the system has a strong stochastic dynamic process, the transmission of abnormal is irregular power input, the main structural components to withstand higher than normal rotation mechanical fatigue loading several times, forming a unique wind turbine dynamics. Wind generator set blades as a flexible structure, the load acting on has a cross and random variability, which is an inevitable occurrence of vibration, the vibration characteristics of their study is very necessary. Blades the structure and strength of the wind generator set reliability plays an important role. With the emergence of high-power wind generators, finite element numerical analysis of the theories and methods have been applied to the structural design of wind generator set. Modern wind generator set blades shape and internal structure complex calculations,it is necessary to study the exact dynamics model and the analysis of blades structure [. The institutional dynamics is to constitute the institutional elements of the inertia and institutions in the rigid parts caused by the vibration of a subject. The kinetic analysis features include: regular modal complex eigenvalue analysis, frequency and transient response analysis, (noise) acoustic analysis, random response analysis, the response and the impact of spectrum analysis, power sensitivity analysis. Here only the blade to vibrate and modal to be analysed [.

Vibration Diagnosis Featuring Blade-Shaft Coupling Effect of Turbine Model Rotors

2010 ◽  
Author(s):  
Norihisa Anegawa ◽  
Hiroyuki Fujiwara ◽  
Osami Matsushita
Keyword(s):  
Natural Frequency ◽  
Rotational Speed ◽  
Coupling Effect ◽  
Rotating Shaft ◽  
Analysis Technique ◽  
Additional Resonance ◽  
Exciting Frequency ◽  
Out Of Plane ◽  
Blade System ◽  
Bearing Stiffness

It is well known that zero and one nodal diameter (k = 0 and k = 1) modes of a blade system interact with the shaft system. The former is coupling with torsional and/or axial shaft vibrations, and the latter with bending shaft vibrations. This paper deals with the latter. With respect to k = 1 modes, we discuss experimentally and theoretically in-plane blades and out-of-plane blades attached radially to a rotating shaft. We found that when we excited the shaft at the rotational speed of Ω = |ωb – ωs| (where ωb = blade natural frequency, ωs = shaft natural frequency and Ω = rotational speed), the exciting frequency ν = ωs induced shaft-blade coupling resonance. In addition, in the case of the in-plane blade system, we encountered an additional resonance attributed to deformation caused by gravity. In the case of the out-of-plane blade system, we experienced two types of abnormal vibrations. One is the additional resonance generated at Ω = ωb / 2 due to the unbalanced shaft and the anisotropy of bearing stiffness. The other is a flow-induced self-exited vibration caused by galloping due to the cross-section shape of the blade tip, because this instability disappeared at the rotation test inside a vacuum chamber. Both occurred at the same time, and both led to the entrainment phenomenon, which was identified by our own frequency analysis technique.

Influence of Centrifugal Force on Characteristics of Turbine Shaft for Micro-Spindle for Micro-Cutting

2012 ◽  
Vol 565 ◽  
pp. 582-587
Author(s):  
Wei Li ◽  
Zhi Xiong Zhou ◽  
Xiang Ming Huang ◽  
Chen Chen ◽  
Ling Yun Meng
Keyword(s):  
Finite Element ◽  
Natural Frequency ◽  
Centrifugal Force ◽  
Torsional Vibration ◽  
Rotational Speed ◽  
Fe Simulation ◽  
Great Influence ◽  
Vibrational Modes ◽  
Micro Cutting ◽  
Turbine Shaft

The centrifugal force has great influence on characteristics of the turbine shaft. The changes of stress, diameter and natural frequency of the turbine shaft for self-developed micro-spindle for micro-cutting following with the rotational speed were studied by finite element (FE) simulation, which showed that stress, diameter of the turbine shaft and frequency of torsional vibration in elastic shaft coupling integrated with the turbine shaft caused by the centrifugal force increases almost linearly with increase of rotational speed and the higher the speed, the larger the increase. The frequencies of other vibrational modes were influenced by centrifugal force. Materials with good properties can improve characteristics of turbine shaft caused by the centrifugal force well.

On the kinetics of Hadamard-type fractional differential systems

2020 ◽  
Vol 23 (2) ◽  
pp. 553-570 ◽  
Author(s):  
Li Ma
Keyword(s):  
Differential Operator ◽  
Numerical Simulations ◽  
Type Inequality ◽  
The Other ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Kinetics Of ◽  
Theoretical Results

AbstractThis paper is devoted to the investigation of the kinetics of Hadamard-type fractional differential systems (HTFDSs) in two aspects. On one hand, the nonexistence of non-trivial periodic solutions for general HTFDSs, which are considered in some functional spaces, is proved and the corresponding eigenfunction of Hadamard-type fractional differential operator is also discussed. On the other hand, by the generalized Gronwall-type inequality, we estimate the bound of the Lyapunov exponents for HTFDSs. In addition, numerical simulations are addressed to verify the obtained theoretical results.

The Analytical Imbalance Response of Jeffcott Rotor During Acceleration

2000 ◽  
Vol 123 (2) ◽  
pp. 299-302 ◽  
Author(s):  
Shiyu Zhou ◽  
Jianjun Shi
Keyword(s):  
Natural Frequency ◽  
Initial Condition ◽  
Constant Acceleration ◽  
Transient Vibration ◽  
Critical Speeds ◽  
Rotor Systems ◽  
Jeffcott Rotor ◽  
Physical Insight

Since many rotor systems normally operate above their critical speeds, the problem of accelerating the machine through its critical speeds without excessive vibration draws increasing attention. This paper provides an analytical imbalance response of the Jeffcott rotor under constant acceleration. The response consists of three parts: transient vibration due to the initial condition of the rotor, “synchronous” vibration, and suddenly occurring vibration at the damped natural frequency. This solution provides physical insight to the vibration of the rotor during acceleration.

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