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.

Wave-Induced Vibrations of an Axial-Loaded Immersed Timoshenko Beam Carrying an Eccentric Tip Mass with Rotary Inertia

2010 ◽  
Vol 54 (01) ◽  
pp. 15-33
Author(s):  
Jong-Shyong Wu ◽  
Chin-Tzu Chen
Keyword(s):  
Closed Form ◽  
Timoshenko Beam ◽  
Forced Vibration ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Form Solution ◽  
Rotary Inertia ◽  
Mode Shapes ◽  
Mode Superposition Method ◽  
Tip Mass

Under the specified assumptions for the equation of motion, the closed-form solution for the natural frequencies and associated mode shapes of an immersed "Euler-Bernoulli" beam carrying an eccentric tip mass possessing rotary inertia has been reported in the existing literature. However, this is not true for the immersed "Timoshenko" beam, particularly for the case with effect of axial load considered. Furthermore, the information concerning the forced vibration analysis of the foregoing Timoshenko beam caused by wave excitations is also rare. Therefore, the first purpose of this paper is to present a technique to obtain the closed-form solution for the natural frequencies and associated mode shapes of an axial-loaded immersed "Timoshenko" beam carrying eccentric tip mass with rotary inertia by using the continuous-mass model. The second purpose is to determine the forced vibration responses of the latter resulting from excitations of regular waves by using the mode superposition method incorporated with the last closed-form solution for the natural frequencies and associated mode shapes of the beam. Because the determination of normal mode shapes of the axial-loaded immersed "Timoshenko" beam is one of the main tasks for achieving the second purpose and the existing literature concerned is scarce, the details about the derivation of orthogonality conditions are also presented. Good agreements between the results obtained from the presented technique and those obtained from the existing literature or conventional finite element method (FEM) confirm the reliability of the presented theories and the developed computer programs for this paper.

Sensitivity Analysis of a Flexible Marine Riser Using Sobol’s Method

2017 ◽  
Author(s):  
Firooz Bakhtiari-Nejad ◽  
Arastou Azimi ◽  
Robert G. Parker
Keyword(s):  
Sensitivity Analysis ◽  
Structural Parameters ◽  
Unit Length ◽  
Mode Shapes ◽  
External Diameter ◽  
Marine Riser ◽  
Assumed Mode Method ◽  
Flexible System ◽  
Mode Method ◽  
Sobol’S Method

In this research, sensitivity analysis of upper and lower end angles in addition to the midpoint displacement of the flexible marine riser with respect to structural parameters such as: uniform mass per unit length, external diameter and the tension of the riser are conducted. Harsh environmental circumstances of ocean flow in addition to exerted tension on top of risers may lead to irreparable damages, so it is important to have a parametric study of dynamic response before it is controlled. The “Sobol” method is applied here as a reliable statistical method to sensitivity analysis of a flexible system. Motion equation of the system is developed based on Hamilton’s principle. The riser is modeled as a distributed parameter system. Moreover, simulations are carried out based on Assumed Mode Method (AMM) to solve PDE of the riser through mode shapes and generalized coordinates. Finally, the results of sensitivity analysis are presented.

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.

Complex Modal Analysis of a Non-Modally Damped Continuous Beam

2013 ◽  
Author(s):  
Xing Xing ◽  
Brian F. Feeny
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Separation Of Variables ◽  
Mode Shapes ◽  
Modal Assurance Criterion ◽  
Assumed Mode Method ◽  
State Variable ◽  
Mode Method ◽  
Cantilevered Beam ◽  
Assumed Mode

This work represents an investigation of the complex modes of continuous vibration systems with nonmodal damping. As an example, a cantilevered beam with damping at the free end is studied. Traditional separation of variables for this problem leads to a differential eigenvalue problem which requires a numerical solution. In this paper, assumed modes are applied to discretize the eigenvalue problem in state-variable form, to then obtain estimates of the frequencies and modes. The finite-element method (FEM) is also utilized to get the mass, stiffness, and damping matrices and further to solve a state-variable eigenproblem. A comparison between the assumed-mode and finite-element eigenvalues and modal vectors shows that the methods produce consistent results. The comparison of the modes was done visually and also by using the modal assurance criterion (MAC) on the modal vectors. The assumed-mode method is then used to study the effects of the damping coefficient on mode shapes and modal damping.

Free Vibration Analysis of Rectangular Plates With Multiple Rectangular Openings and Arbitrary Edge Constraints

2014 ◽  
Author(s):  
Dae-Seung Cho ◽  
Nikola Vladimir ◽  
Tae-Muk Choi
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Relevant Literature ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Ocean Engineering ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Assumed Mode

Free vibration analysis of plates with openings represents an important issue in naval architecture and ocean engineering applications. Namely, they are often primary design members of complex structures and knowledge about their dynamic behavior becomes a prerogative for the proper structural design. This paper deals with application of assumed mode method to free vibration analysis of rectangular plates with multiple rectangular openings at arbitrary defined locations. Developed method can be applied to both thin and thick plates as well as to classical and non-classical edge constraints. In the assumed mode method natural frequencies and mode shapes of a corresponding plate are determined by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange’s equations of motion. The developed procedure actually represents an extension of a method for the natural vibration analysis of rectangular plates without openings, which has been recently presented in the relevant literature. The effect of an opening is taken into account in a simple and intuitive way, i.e. by subtracting its energy from the total plate energy without opening. Illustrative numerical examples include dynamic analysis of rectangular plates with single and multiple rectangular openings with various thicknesses and different combinations of boundary conditions. Also, the influence of the rectangular opening area on the plate dynamic response is analyzed. The comparisons of the results with those obtained using the finite element method (FEM) is also provided, and very good agreement is achieved. Finally, related conclusions are drawn and recommendations for future investigations are presented.

Dynamic Characteristic Analysis of a Rotor-Bearing System Using Global Assumed Mode Method With Different Polynomial Function

2011 ◽  
Author(s):  
T. N. Shiau ◽  
J. R. Chang ◽  
C. H. Kang ◽  
C. Y. Liao
Keyword(s):  
High Speed ◽  
Polynomial Function ◽  
Mode Shapes ◽  
Characteristic Analysis ◽  
Assumed Mode Method ◽  
Higher Order Modes ◽  
Rotor Bearing ◽  
Mode Method ◽  
Assumed Mode ◽  
Bearing System

In this study, the dynamic analysis of a domestic high speed rotor bearing system in turbo machines by using global assumed mode with different polynomial is investigated. This system consists of rotating multi flexible shaft, rigid disks and stiffness bearing effects. The analysis includes the whirl speeds, critical speeds, and mode shapes. The Global Assumed Modes Method (GAMM) and Finite Element Method (FEM) are employed to model the rotor-bearing system, and the accuracy of the results is discussed. With the application of GAMM, similarity transformation of different types of polynomials and interval has been investigated. The results show that using different polynomial function in GAMM have similar results, and which are also be agreed with the FEM. The results also show that the number of polynomial can be increased as the interval of the assumed mode function is altered. Consequently, the convergence of higher order modes will be more accurate.

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.

scholarly journals High-Fidelity Sensitivity Analysis of Modal Properties of Mistuned Bladed Disks Regarding Material Anisotropy

2018 ◽  
Author(s):  
Adam Koscso ◽  
Guido Dhondt ◽  
E. P. Petrov
Keyword(s):  
Sensitivity Analysis ◽  
Finite Element ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Finite Element Models ◽  
Coordinate Systems ◽  
Bladed Disks ◽  
Bladed Disk ◽  
Material Orientation

A new method has been developed for sensitivity calculations of modal characteristics of bladed disks made of anisotropic materials. The method allows the determination of the sensitivity of the natural frequencies and mode shapes of mistuned bladed disks with respect to anisotropy angles that define the crystal orientation of the monocrystalline blades using full-scale finite element models. An enhanced method is proposed to provide high accuracy for the sensitivity analysis of mode shapes. An approach has also been developed for transforming the modal sensitivities to coordinate systems used in industry for description of the blade anisotropy orientations. The capabilities of the developed methods are demonstrated on examples of a single blade and a mistuned realistic bladed disk finite element models. The modal sensitivity of mistuned bladed disks to anisotropic material orientation is thoroughly studied.

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