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.

Complex Modal Analysis of a Nonmodally Damped Continuous Beam

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
Xing Xing ◽  
Brian F. Feeny
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Normal Modes ◽  
Damping Coefficient ◽  
Mode Analysis ◽  
Element Analysis ◽  
Assumed Mode Method ◽  
State Variable ◽  
Complex Modes ◽  
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. Assumed modes are applied to discretize the eigenvalue problem in state-variable form and then to obtain estimates of the true complex normal modes and frequencies. The finite element method (FEM) is also used to get the mass, stiffness, and damping matrices and further to solve a state-variable eigenvalue problem. A comparison between the complex modes and eigenvalues obtained from the assumed-mode analysis and the finite element analysis shows that the methods produce consistent results. The convergence behavior when using different assumed mode functions is investigated. The assumed-mode method is then used to study the effects of the end-damping coefficient on the estimated normal modes and modal damping. Most modes remain underdamped regardless of the end-damping coefficient. There is an optimal end-damping coefficient for vibration decay, which correlates with the maximum modal nonsynchronicity.

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.

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.

Simplified dynamic analysis of stepped thickness rectangular plate structures by the assumed mode method

2016 ◽  
Vol 231 (1) ◽  
pp. 177-187
Author(s):  
Dae-Seung Cho ◽  
Byung Hee Kim ◽  
Jin-Hyeong Kim ◽  
Nikola Vladimir ◽  
Tae-Muk Choi
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Dynamic Analysis ◽  
Rectangular Plate ◽  
Forced Response ◽  
Plate Structures ◽  
Assumed Mode Method ◽  
Frequency Responses ◽  
Mode Method ◽  
Assumed Mode

In this article, the assumed mode method is applied to simplified dynamic analysis of stepped thickness rectangular Mindlin plates and stiffened panels with arbitrary boundary conditions. The natural and frequency responses of stepped thickness plate structures subjected to harmonic point excitation force and enforced acceleration at boundaries, respectively, are considered. Potential and kinetic energies of the system are formulated and used to derive eigenvalue problem utilizing Lagrange’s equation of motion, and mode superposition method is further used for forced response assessment. Characteristic orthogonal polynomials having the property of Timoshenko beam functions are used for the assumed modes. Numerical examples analysing vibration of stepped thickness plate structures with different topologies and various sets of boundary conditions are provided. Numerical results are compared with the results from the relevant literature and finite element solutions obtained by a general finite element tool, and a very good agreement is achieved. Hence, it is expected that stepped rectangular plate structures satisfying the prescribed criteria regarding natural and frequency responses can be efficiently designed based on the proposed method.

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.

The Discretization Methods of a Rotating Flexible Cantilever Beam

2012 ◽  
Vol 226-228 ◽  
pp. 697-707
Author(s):  
Ji Hua Fan ◽  
Ding Guo Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Cantilever Beam ◽  
Interpolation Method ◽  
Flexible Beam ◽  
Transverse Deformation ◽  
Longitudinal Deformation ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Assumed Mode

The dynamics of a rotating flexible cantilever beam is investigated by using assumed mode method, finite element method and Bezier curve interpolation method in this paper. The longitudinal deformation and the transverse deformation of the flexible beam are considered and the coupling term of the deformation which is caused by transverse deformation is included in the expression of longitudinal deformation. The assumed mode, finite element and Bezier interpolation are used to discretize the deformation of the flexible beam, and then the dynamics equations are built by Lagrange equation, and a software package for the dynamic simulation of the rotating cantilever beam is developed. Two cases are considered in the simulations. One is the dynamics study which the external rotating torque is known, another one is that a rotating fixed axis flexible beam falls with gravity. According to the simulation results, assumed mode, finite element and Bezier interpolation method can be well used for discretizing the deformation of the flexible beam; the computational efficiency of finite element method is the lowest, and Bezier interpolation method is the highest; the calculation accuracy of the assumed mode method is lower than the Bezier interpolation method.

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.

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 An investigation into the free vibrations of carbon nanotubes using analytical and finite element methods

2021 ◽  
Author(s):  
Ishan Ali Khan
Keyword(s):  
Carbon Nanotubes ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Natural Frequencies ◽  
Nonlinear Eigenvalue Problem ◽  
Dynamic Stiffness ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Wide Range ◽  
Walled Cnts

Since their discovery, immense attention has been given to carbon nanotubes (CNTs), due to their exceptional thermal, electronic and mechanical properties and, therefore, the wide range of applications in which they are, or can be potentially, employed. Hence, it is important that all the properties of carbon nanotubes are studied extensively. This thesis studies the vibrational frequencies of double-walled and triple-walled CNTs, with and without an elastic medium surrounding them, by using Finite Element Method (FEM) and Dynamic Stiffness Matrix (DSM) formulations, considering them as Euler-Bernoulli beams coupled with van der Waals interaction forces. For FEM modelling, the linear eigenvalue problem is obtained using Galerkin weighted residual approach. The natural frequencies and mode shapes are derived from eigenvalues and eigenvectors, respectively. For DSM formulation of double-walled CNTs, a nonlinear eigenvalue problem is obtained by enforcing displacement and load end conditions to the exact solution of single equation achieved by combining the coupled governing equations. The natural frequencies are obtained using Wittrick-Williams algorithm. FEM formulation is also applied to both double and triple-walled CNTs modelled as nonlocal Euler-Bernoulli beam. The natural frequencies obtained for all the cases, are in agreement with the values provided in literature.

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