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 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.

Squeal Analysis of a Disc Brake Considering Damping between a Disc and a Pad Lining

2012 ◽  
Vol 157-158 ◽  
pp. 1000-1003
Author(s):  
Ke Wei Zhou ◽  
Cheol Kim ◽  
Min Ok Yun ◽  
Ju Young Kim
Keyword(s):  
Finite Element ◽  
Equations Of Motion ◽  
Element Model ◽  
Disc Brake ◽  
Vibration Modes ◽  
Assumed Mode Method ◽  
Frictional Damping ◽  
System Components ◽  
Mode Method ◽  
Assumed Mode

The improved equations of motion for a friction-engaged brake system have been newly derived on the basis of the assumed mode method and frictional damping. The equations of motion with a finite element model were constructed by a set of vibration modes found from FE modal analysis on all system components. Consequently, the modal information of system components are combined with equations of motion derived from the analytical model. Numerical analysis showed the mode which was unstable in an undamped case became stable in a damped case.

Numerical Analysis of the Iced Transmission Line Galloping

2014 ◽  
Vol 607 ◽  
pp. 894-900
Author(s):  
Li Qin ◽  
Tian Yuan Xu
Keyword(s):  
Wind Speed ◽  
Transmission Line ◽  
Lagrange Equation ◽  
Equations Of Motion ◽  
Lyapunov Stability Theory ◽  
Assumed Mode Method ◽  
Torsional Response ◽  
Critical Wind Speed ◽  
Mode Method ◽  
Assumed Mode

By used the assumed mode method to simulate the iced transmission line galloping,with three generalized coordinates to represent the iced transmission line galloping. In order to avoid the complicated calculation of vector,used the Lagrange equation to build the nonlinear equations of iced transmission line from the perspective of energy,used the Runge-Kutta numerical calculation to solve the equations of motion and get the iced transmission line’s across-wind,along-wind and the torsional response. Based on Lyapunov stability theory to deduce the critical wind speed of the iced transmission line galloping. And had used a test iced transmission line to verify the feasibility of the numerical solution and the critical wind speed.

Dynamic Modeling of a Flexible Manipulator With Prismatic Links

1999 ◽  
Vol 121 (4) ◽  
pp. 691-696 ◽  
Author(s):  
B. J. Torby ◽  
I. Kimura
Keyword(s):  
Moving Boundary ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Flexible Manipulator ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Finite Element Node ◽  
Assumed Mode ◽  
Three Dimensional Space

In this paper the equations of motion for a flexible multi-link manipulator are derived. Each link of the manipulator, including those with prismatic motion, is represented by two finite elements in three-dimensional space. The prismatic links are treated as beams with moving boundary conditions, and the position of finite-element node points are not changed relative to the link. The equations are generated using Maple V, and the paper discusses a general approach for eliminating small terms. A sample calculation is performed for a RRP (Stanford arm) manipulator, and the shift of natural frequencies with time are plotted. Results are compared to those obtained by the assumed-mode method.

scholarly journals DYNAMIC ANALYSES OF A FLEXIBLE VEHICLE MOVING ALONG A FLEXIBLE GUIDEWAY CONSIDERING THE TIP-OFF EFFECT

2013 ◽  
Vol 13 (01) ◽  
pp. 1350009 ◽  
Author(s):  
S. N. CHOU ◽  
F. P. CHENG ◽  
C. S. HUANG
Keyword(s):  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Flexible Beam ◽  
Differentiation Formula ◽  
Assumed Mode Method ◽  
Dynamic Analyses ◽  
Mode Method ◽  
Assumed Mode ◽  
Bdf Method ◽  
Free Beam

A semi-analytical solution for the tip-off response of a vehicle moving along a guideway is obtained, considering the dynamic interaction between the two subsystems. The guideway is modeled as an inclined simply-supported uniform flexible beam, and the vehicle as a flexible free-free beam under a pre-specified thrust force. The equations of motion for the vehicle and guideway are developed using the Lagrangian approach and the assumed mode method based on the Euler–Bernoulli hypothesis. In the form of nonlinear differential equations, they are solved by the Petzold-Gear backward differentiation formula (BDF) method. The solutions obtained are validated by comparing them with the published results for the models with a rigid vehicle running over a rigid guideway or a flexible guideway. Comparisons of the present solutions with the existing ones for the vehicle and guideway reveal the advantages of the approach proposed herein. Other effects on the tip-off responses of the vehicle that are investigated include the length of the guideway, distance between the shoes of the vehicle, and mass and rigidity ratios of the vehicle to the guideway. The results presented herein provide valuable information for the design of the vehicle launch system.

Dynamics of a Rotating Shaft Subject to a Three-Directional Moving Load

2007 ◽  
Vol 129 (3) ◽  
pp. 386-389 ◽  
Author(s):  
Huajiang Ouyang ◽  
Minjie Wang
Keyword(s):  
Axial Force ◽  
Moving Load ◽  
Equations Of Motion ◽  
Bending Moment ◽  
Axial Direction ◽  
Force Component ◽  
Rotating Shaft ◽  
Assumed Mode Method ◽  
Rayleigh Beam ◽  
Mode Method

This paper presents a dynamic model for the vibration of a rotating Rayleigh beam subjected to a three-directional load acting on the surface of the beam and moving in the axial direction. The model takes into account the axial movement of the axial force component. More significantly, the bending moment produced by this force component is included in the model. Lagrange’s equations of motion for the modal coordinates are derived based on the assumed mode method and then solved by a fourth-order Runge-Kutta algorithm. It is found that the bending moment induced by the axial force component has a significant influence on the dynamic response of the shaft, even when the axial force and speed are low and, hence, must be considered in such problems as turning operations.

Parameter Sensitivity Analysis of Rotating Beams in Frequency Domain

2009 ◽  
Author(s):  
Masoud Ansari ◽  
Ebrahim Esmailzadeh ◽  
Nader Jalili
Keyword(s):  
Sensitivity Analysis ◽  
Closed Form ◽  
Characteristic Equation ◽  
Frequency Analysis ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Tip Mass ◽  
Assumed Mode

Many mechanical rotating systems can be modeled as a cantilever beam attached to a rotating substrate. Vibratory beam gyroscopes are good examples of such systems. They consist of a rotating beam with a tip mass, attached to a rotating base. Due to the base rotation, the governing partial differential equations of the system are coupled, and hence, the system undergoes coupled torsional-bending vibrations. The coupling effect complicates the frequency analysis of the system, especially in determining the system characteristic equation. Many investigators have chosen to use the assumed mode method in their analysis of such systems instead of extracting the exact mode shapes of the system. In spite of all these difficulties, this paper addresses the exact frequency analysis of such systems and presents a closed-form frequency characteristic equation and evaluates the accurate values of the natural frequencies. The application of the proposed method is not limited to the system at hand, as it can be utilized for analyzing general systems with coupled governing equations of motion. Having analyzed a closed-form frequency equation has two valuable advantages: a) it can serve as the basis for the subsequent time-domain analysis; and b) it can be very essential in developing control strategies. In this study a thorough sensitivity analysis is performed to determine the effects of different parameters on the natural frequencies of the coupled vibrating system. The proposed method reveals some interesting findings in the systems which were difficult, if not impossible, to be revealed by the assumed mode method commonly utilized in many research work reported recently in literature.

Dynamics of a Flexible Rod in a Quick Return Mechanism

1994 ◽  
Vol 116 (1) ◽  
pp. 70-74 ◽  
Author(s):  
Heow-Pueh Lee
Keyword(s):  
Numerical Simulations ◽  
Equations Of Motion ◽  
Angular Speed ◽  
Matrix Form ◽  
Euler Beam ◽  
Assumed Mode Method ◽  
Point Support ◽  
Mode Method ◽  
Stiff Spring ◽  
Assumed Mode

The equations of motion in matrix form are formulated for a flexible rod in a quick return mechanism using Hamilton’s principle and the assumed mode method. The rod is considered as an Euler beam. The crank is assumed to be rigid and rotating at a constant angular speed. The translating-rotating joint connecting the crank to the flexible rod is assumed to be a frictionless moving point support for the flexible rod. This support is regarded as a very stiff spring acting on the rotating flexible rod. Results of numerical simulations are presented for various prescribed crank positions, crank lengths, and crank speeds.

The Assumed Mode Method in Structural Dynamics of Bladed-Disk-Shaft Systems

1990 ◽  
Author(s):  
Naim Khader ◽  
Sanier Masoud
Keyword(s):  
Equations Of Motion ◽  
Analytical Investigation ◽  
Assumed Mode Method ◽  
Bladed Disk ◽  
Wide Range ◽  
Out Of Plane ◽  
Mode Method ◽  
Inertia Parameters ◽  
Inertia Properties ◽  
Assumed Mode

Analytical investigation into the effect of transverse bending of continuous flexible shafts is presented. While the blades are allowed both in-plane and out-of-plane deformations, the considered disk is rigid, and the shaft is allowed to bend in two planes. The assumed mode method is used to express flexible blade and shaft deformations, and the Lagrangian approach is used to derive the governing equations of motion for the considered structure. Stiffness and inertia properties of an actual experimental rotor, typical of a fan stage, are used in the analysis. Calculations are performed for three different disk-shaft configurations, and results are presented for different shaft stiffness and inertia parameters, as well as for a wide range of rotational speed.

A Method to Improve the Modal Convergence for Structures With External Forcing

1987 ◽  
Vol 54 (4) ◽  
pp. 904-909 ◽  
Author(s):  
Keqin Gu ◽  
Benson H. Tongue
Keyword(s):  
Spatial Distribution ◽  
Free Vibration ◽  
Convergence Rate ◽  
Traditional Approach ◽  
Vibration Modes ◽  
External Forcing ◽  
Slow Convergence ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Assumed Mode

The traditional approach of using free vibration modes in the assumed mode method often leads to an extremely slow convergence rate, especially when discete interactive forces are involved. By introducing a number of forced modes, significant improvements can be achieved. These forced modes are intrinsic to the structure and the spatial distribution of forces. The motion of the structure can be described exactly by these forced modes and a few free vibration modes provided that certain conditions are satisfied. The forced modes can be viewed as an extension of static modes. The development of a forced mode formulation is outlined and a numerical example is presented.

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