scholarly journals On The Vibration of a Cracked Rotating Blade

1998 ◽  
Vol 5 (5-6) ◽  
pp. 317-323 ◽  
Author(s):  
Ming-Chuan Wu ◽  
Shyh-Chin Huang
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Crack Depth ◽  
Weighted Residual Method ◽  
Energy Principle ◽  
Discrete Equations ◽  
Rotating Blade ◽  
On Line ◽  
Released Energy ◽  
Blade Frequency

The dynamic behavior of a rotating blade containing a transverse crack was investigated. First, the local flexibility of the cracked blade was obtained by using the method of the released energy. An energy principle, in conjunction with a weighted residual method, was then applied to yield the discrete equations of motion. The equations of motion were further utilized to study the influences of the crack depth and location on the bending natural frequencies under various of rotation speeds. The numerical calculation showed that the crack effects the natural frequencies and the response appreciably only if it is relatively deep and locates near the root of the blade. However, the effects increase exponentially with the depth increases. In addition to the natural frequencies, the displacement responses of the blade with a crack under a constant lateral forces were discussed as well. This was done by calculating the deflections at the tip of the blade for various crack depths and locations. Similar to the rotation speed of the blade frequency, the deflection was offset by the increase of the rotation. However, the centrifugal effects increased significantly such that the crack’s effects became relatively insignificant. Nevertheless, the study showed that the changes on the natural frequency and the tip-deflection of the blade due to a crack may be used as indices for on-line detection of cracks.

Modal Analysis of a Rotating Shaft-Disk-Blades System With a Cracked Blade

1997 ◽  
Author(s):  
Ming-Chuan Wu ◽  
Shyh-Chin Huang
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Rotating Shaft ◽  
Coupled Modes ◽  
Energy Principle ◽  
Assumed Mode Method ◽  
Discrete Equations ◽  
Mode Method ◽  
Released Energy ◽  
Assumed Mode

Abstract The dynamic behavior of a rotating shaft-disk-blades system containing a cracked blade is investigated. With the crack released energy, the flexibility due to crack is evaluated. An energy principle in conjunction with the assumed-mode method is applied to yield the discrete equations of motion. Numerical examples are given for cases with between two and five symmetrically arrayed blades. The results show that there exist both torsion-bending coupled modes and blade-coupling modes, which occur at repeated frequencies. When there is a cracked blade, the frequencies of torsion-bending coupled modes decrease due to the crack, and blade-coupling modes have the phenomena of frequency bifurcation. Finally, the effects of shaft speed on the natural frequencies are illustrated.

scholarly journals Parametric Vibrations of Axially Moving Beams with Multiple Edge Cracks

2019 ◽  
Vol 24 (2) ◽  
pp. 241-252 ◽  
Author(s):  
Murat Sarıgül
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Crack Depth ◽  
Multiple Cracks ◽  
Mean Velocity ◽  
Beam System ◽  
Crack Tips ◽  
Axially Moving Beams ◽  
Edge Cracks ◽  
Axially Moving

Nonlinear transverse vibrations of axially moving beams with multiple cracks is handled studied. Assuming that the beam moves with mean velocity having harmonically variation, influence of the edge crack on the moving continua are investigated in this study. Due to existence of the crack in the transverse direction, the healthily beam is divided into parts. The translational and rotational springs are replaced between these parts so that high stressed regions around the crack tips are redefined with the springs' energies. Thus, the problem is converted to an axially moving spring-beam system. The equations of motion and its corresponding conditions are obtained by means of the Hamilton Principle. In numerical analysis, the natural frequencies and responses of the spring-beam system are investigated for principal parametric resonance in detail. Some important results are obtained; the natural frequencies decreases with increasing crack depth. In case of the beam travelling with high velocities, the effects of crack's depth on natural frequencies seems to be vanished.

Mode Localization of a Cracked Blade Disk

1999 ◽  
Vol 121 (2) ◽  
pp. 335-341 ◽  
Author(s):  
J. H. Kuang ◽  
B. W. Huang
Keyword(s):  
Equations Of Motion ◽  
Crack Depth ◽  
Numerical Results ◽  
Rotating System ◽  
Mode Localization ◽  
Rotating Speed ◽  
Rotating Blade ◽  
Localization Phenomenon ◽  
Blade Crack ◽  
Plane Mode

In this paper, the effect of blade crack on the mode localization of a rotating blade disk is studied. Pretwisted taper beams are used to simulate blades of a blade disk. The crack on the blade can be regarded as a local disorder of this periodically coupled blades system. An application of Hamilton’s principle and Galerkin’s method is used to formulate the equations of motion of the mistuned system. Effects of pretwisted angle, rotating speed, and crack depth of the blade on the in-plane and off-plane mode localizations of a rotating system are investigated. Numerical results indicate that the increase of rotating speed, pretwisted angle, and crack depth could enhance the localization phenomenon significantly.

scholarly journals Triply Coupled Bending-Bending-Torsion Vibrational Analysis of Rotating Beams Using Fem and Dynamic Finite Element Formulation

2021 ◽  
Author(s):  
Mohammad Shavezipur
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Convergence Rates ◽  
Equations Of Motion ◽  
Vibrational Analysis ◽  
Element Formulation ◽  
Shape Functions ◽  
Weighted Residual Method ◽  
Torsion Beam ◽  
Dynamic Finite Element

This research presents the numerical analysis of the triply coupled flap-wise, cord-wise and torsional vibrations of flexible rotating blades. Euler-Bernoulli bending and St. Venant torsion beam theories are considered to derive the governing differential equations of motion. Based on Finite Element Methodology (FEM), the cubic "Hermite" shape functions are implemented where the solution of the equations results in a linear engine problem. Then, the Dynamic (frequency dependent) Trigonometric Shape Functions (DISF's) for beam's uncoupled displacements are derived. The application of the Dynamic Finite Element (DFE) approach to the solution of the governing equations is then presented. The DFE formulation, based on the weighted residual method and the DTSF's results in a nonlinear engine problem representing eigenvalues and engine modes of the system. The applicability of the DFE method is then demonstrated by illustrative examples, where a Wittrick-Williams root counting technique is used to find the system's natural frequencies. The DFE approach, an intermediate method between FEM and "Exact" formulation, is characterized by higher convergence rates, and can be advantageously used when multiple natural frequencies and/or higher modes of beam-like structures are to be evaluated.

scholarly journals Dynamic Stability in a Cracked Turbo Blade-Disk

1999 ◽  
Author(s):  
J. H. Kuang ◽  
B. W. Huang
Keyword(s):  
Multiple Scales ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
Crack Depth ◽  
Disk System ◽  
Rotating Blade ◽  
Blade System ◽  
Instability Regions ◽  
The Stability ◽  
Multiple Scales Perturbation Method

Analysis of the stability in a cracked blade-disk system is proposed. The effect of modal localization on the stability in a rotating blade-disk was studied. A crack near the root of a blade is regarded as a local disorder in this periodically coupled blade system. Hamilton’s principle and Galerkin’s method were used to formulate the equations of motion for the cracked blade-disk. The instability regions of this cracked blade-disk system were specified by employing the multiple scales perturbation method. Numerical results indicate that the rotation speed, shroud stiffness and crack depth in the blades affect the stability regions of this mistuned system significantly.

scholarly journals Triply Coupled Bending-Bending-Torsion Vibrational Analysis of Rotating Beams Using Fem and Dynamic Finite Element Formulation

2021 ◽  
Author(s):  
Mohammad Shavezipur
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Convergence Rates ◽  
Equations Of Motion ◽  
Vibrational Analysis ◽  
Element Formulation ◽  
Shape Functions ◽  
Weighted Residual Method ◽  
Torsion Beam ◽  
Dynamic Finite Element

This research presents the numerical analysis of the triply coupled flap-wise, cord-wise and torsional vibrations of flexible rotating blades. Euler-Bernoulli bending and St. Venant torsion beam theories are considered to derive the governing differential equations of motion. Based on Finite Element Methodology (FEM), the cubic "Hermite" shape functions are implemented where the solution of the equations results in a linear engine problem. Then, the Dynamic (frequency dependent) Trigonometric Shape Functions (DISF's) for beam's uncoupled displacements are derived. The application of the Dynamic Finite Element (DFE) approach to the solution of the governing equations is then presented. The DFE formulation, based on the weighted residual method and the DTSF's results in a nonlinear engine problem representing eigenvalues and engine modes of the system. The applicability of the DFE method is then demonstrated by illustrative examples, where a Wittrick-Williams root counting technique is used to find the system's natural frequencies. The DFE approach, an intermediate method between FEM and "Exact" formulation, is characterized by higher convergence rates, and can be advantageously used when multiple natural frequencies and/or higher modes of beam-like structures are to be evaluated.

scholarly journals Mode Localization of a Cracked Blade-Disks

1998 ◽  
Author(s):  
J. H. Kuang ◽  
B. W. Huang
Keyword(s):  
Equations Of Motion ◽  
Crack Depth ◽  
Numerical Results ◽  
Rotating System ◽  
Mode Localization ◽  
Rotating Speed ◽  
Rotating Blade ◽  
Localization Phenomenon ◽  
Blade Crack ◽  
Plane Mode

In this paper, the effect of blade crack on the mode localization of a rotating blade-disk is studied. Pretwisted taper beams are used to simulate blades of a blade-disk. The crack on the blade can be regarded as a local disorder of this periodically coupled blades system. An application of Hamilton’s principle and Galerkin’s method is used to formulate the equations of motion of the mistuned system. Effects of pretwisted angle, rotating speed and crack depth of the blade on the in-plane and off-plane mode localizations of a rotating system are investigated. Numerical results indicate that the increase of rotating speed, pretwisted angle and crack depth could enhance the localization phenomenon significantly.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

Free Vibration Analysis of a Carbon Nanotube Reinforced Composite Beam

2014 ◽  
Vol 592-594 ◽  
pp. 2041-2045 ◽  
Author(s):  
B. Naresh ◽  
A. Ananda Babu ◽  
P. Edwin Sudhagar ◽  
A. Anisa Thaslim ◽  
R. Vasudevan
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Composite Beam ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Element Formulation ◽  
Reinforced Composite ◽  
Vibration Responses

In this study, free vibration responses of a carbon nanotube reinforced composite beam are investigated. The governing differential equations of motion of a carbon nanotube (CNT) reinforced composite beam are presented in finite element formulation. The validity of the developed formulation is demonstrated by comparing the natural frequencies evaluated using present FEM with those of available literature. Various parametric studies are also performed to investigate the effect of aspect ratio and percentage of CNT content and boundary conditions on natural frequencies and mode shapes of a carbon nanotube reinforced composite beam. It is shown that the addition of carbon nanotube in fiber reinforced composite beam increases the stiffness of the structure and consequently increases the natural frequencies and alter the mode shapes.

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