Determination of the first n modes of a tall building from a reduced eigenvalue problem

1964 ◽  
Vol 54 (4) ◽  
pp. 1233-1254
Author(s):  
Moshe F. Rubinstein
Keyword(s):  
Eigenvalue Problem ◽  
Natural Frequencies ◽  
Tall Building ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Story Building

Abstract The first n natural frequencies and mode shapes of an N degree of freedom structure (n < N) are derived from the solution of a reduced eigenvalue problem of order smaller than N. The reduced eigenvalue problem is formulated by using experience to select approximations to the first n modes desired. Accuracy is improved when more than n modes are selected. The method is illustrated by a study on an 18 story building.

scholarly journals Direct Method for the Determination of Coefficients of Characteristic Equation of a MDOF System

2019 ◽  
Vol 15 (2) ◽  
pp. 26-31
Author(s):  
Mahesh Chandra Luintel
Keyword(s):  
Characteristic Equation ◽  
Natural Frequencies ◽  
Direct Method ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Vibratory System ◽  
Single Degree Of Freedom ◽  
Dynamic Matrix ◽  
Eigen Values

Dynamic response of any single degree of freedom (SDOF) vibratory system is studied by evaluating its natural frequencies whereas that of any multi degree of freedom (MDOF) vibratory system is studied by evaluating its natural frequencies and corresponding mode shapes. Efficient method to determine the natural frequencies and mode shape of a MDOF system is to determine its dynamic matrix and to calculate its eigen-values and eigen-vectors. As the number of degree of freedom (DOF) of the system increases, the size of the dynamic matrix increases and the use of a computer program or package become essential. Hence this paper proposes a new method to directly calculate the coefficients of characteristic equation of any degree of freedom system from which eigen-values and then natural frequencies can be determined.  

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.

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.

scholarly journals Coupled flexural - torsional vibration and stability analysis of pre-loaded beams using conventional and dynamic finite element methods

2021 ◽  
Author(s):  
Heenkenda Jayasinghe
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Axial Force ◽  
Torsional Vibration ◽  
Natural Frequencies ◽  
Nonlinear Eigenvalue Problem ◽  
Compressive Force ◽  
Mode Shapes ◽  
Dynamic Finite Element ◽  
Starting Point

Dynamic Finite Element (DFE) and conventional finite element formulations are developed to study the flexural - torsional vibration and stability of an isotropic, homogeneous and linearly elastic pre-loaded beam subjected to an axial load and end-moment. Various classical boundary conditions are considered. Elementary Euler - Bernoulli bending and St. Venant torsion beam theories were used as a starting point to develop the governing equations and the finite element solutions. The nonlinear Eigenvalue problem resulted from the DFE method was solved using a program code written in MATLAB and the natural frequencies and mode shapes of the system were determined form the Eigenvalues and Eigenvectors, respectively. Similarly, a linear Eigenvalue problem was formulated and solved using a MATLAB code for the conventional FEM method. The conventional FEM results were validated against those available in the literature and ANSYS simulations and the DFE results were compared with the FEM results. The results confirmed that tensile forces increased the natural frequencies, which indicates beam stiffening. On the contrary, compressive forces reduced the natural frequencies, suggesting a reduction in beam stiffness. Similarly, when an end-moment was applied the stiffness of the beam and the natural frequencies diminished. More importantly, when a force and end-moment were acting in combination, the results depended on the direction and magnitude of the axial force. Nevertheless, the stiffness of the beam is more sensitive to the changes in the magnitude and direction of the axial force compared to the moment. A buckling analysis of the beam was also carried out to determine the critical buckling end-moment and axial compressive force.

An Extension of the Holzer Method for Redundant Drive Trains

1974 ◽  
Vol 96 (2) ◽  
pp. 697-698 ◽  
Author(s):  
M. S. Hundal
Keyword(s):  
Natural Frequencies ◽  
Mode Shapes ◽  
System Of Equations ◽  
Redundant System ◽  
Drive Trains ◽  
Redundant Drive

A method is described for the determination of natural frequencies and mode shapes of closed drive trains. It is an extension of the Holzer method to a redundant system. The “error” for a given value of frequency is computed by the solution of a tridiagonal system of equations.

Analysis Behavior of a Rig Shafting Vibration Set Changes Bearing Parameters

2013 ◽  
Vol 437 ◽  
pp. 98-101 ◽  
Author(s):  
Van Thanh Ngo ◽  
Dan Mei Xie
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Natural Frequencies ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Lateral Vibration ◽  
Eigenvalues And Eigenvectors ◽  
Fem Model ◽  
Critical Speeds ◽  
Bearing Stiffness

Frequently, in the design of machines, some of parameters that directly affect the rotordynamics of the machines are not accurately known. In particular, bearing stiffness support is one such parameter. Taking a rig shafting as an example, this paper studies the lateral vibration of the rig shafting with multi-degree-of-freedom by using finite element method (FEM). The FEM model is created and the eigenvalues and eigenvectors are calculated and analyzed to find natural frequencies, critical speeds, mode shapes. Then critical speeds and mode shapes are analyzed by set bearing stiffness changes. The model permitted to identify the critical speeds and bearings that have an important influence on the vibration behavior.

scholarly journals On the determination of natural frequencies and mode shapes of cable-stayed bridges

2001 ◽  
Vol 25 (12) ◽  
pp. 1099-1115 ◽  
Author(s):  
F.T.K. Au ◽  
Y.S. Cheng ◽  
Y.K. Cheung ◽  
D.Y. Zheng
Keyword(s):  
Natural Frequencies ◽  
Mode Shapes ◽  
Cable Stayed Bridges

A Normalized Modal Eigenvalue Approach for Resolving Modal Interaction

1997 ◽  
Vol 119 (3) ◽  
pp. 647-650 ◽  
Author(s):  
M.-T. Yang ◽  
J. H. Griffin
Keyword(s):  
Monte Carlo ◽  
Eigenvalue Problem ◽  
Perturbation Analysis ◽  
Material Properties ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Modal Parameters ◽  
Modal Interaction ◽  
Eigenvalue Approach ◽  
The Difference

Modal interaction refers to the way that the modes of a structure interact when its geometry and material properties are perturbed. The amount of interaction between the neighboring modes depends on the closeness of the natural frequencies, the mode shapes, and the magnitude and distribution of the perturbation. By formulating the structural eigenvalue problem as a normalized modal eigenvalue problem, it is shown that the amount of interaction in two modes can be simply characterized by six normalized modal parameters and the difference between the normalized frequencies. In this paper, the statistical behaviors of the normalized frequencies and modes are investigated based on a perturbation analysis. The results are independently verified by Monte Carlo simulations.

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