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Damping Matrix. Classical Damping Matrix. Nonclassical Damping Matrix. 12. Dynamic Analysis and Response of Linear Systems.Two-Degree-of-Freedom Systems. Analysis of Two-DOF Systems without Damping. Vibration Absorber or Tuned Mass Damper. Modal Analysis. Modal Equations for Undamped Systems. Modal Equations for Damped Systems. Displacement ...
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nating the modal contribution factors. This normali-zation of the modal contributions suggests that the more modes shapes included in the analysis, the more accurate the algorithm will become. 2.2 Absolute Difference PSD Method After normalization by Equation 2, the ADPSD and the CDPSD have divergent calculation paths. The
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3 Matrices and the Free Response 3.1 Introduction 3.2 Eigenvalues and Eigenvectors 3.3 Natural 8 Modal Testing 8.1 Introduction 8.2 Measurement Hardware 8.3 Digital Signal Processing 8.4 1.2 SPRING-MASS SYSTEM. Simple harmonic motion, or oscillation, is exhibited by structures that have...
This modal matrix is then used to define the modal transformation equation with a new coordinate with 'p' as the principal coordinate. Dr. Peter Avitabile Modal Analysis & Controls Laboratory. Modal Mass, Modal Damping, Modal Stiffness. The mass becomes.9.9. MATRICES 9 (Modal & spectral matrices). by. A.J.Hobson. 9.9.1 Assumptions and denitions 9.9.2 Diagonalisation of a matrix 9.9.3 Exercises 9.9.4 Answers to exercises. When using normalised eigenvectors, the modal matrix may be denoted by N and, for an n×n matrix, A, there are 2n...
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Second, only by utilizing the first order modal data orthogonal normalization conditions and eigen equations , we derive the updated matrices. Third, the method can maintain matrices symmetry and sparsity and preserve system connectivity. The merit of the proposed method is simple and accurate in a practical engineering computation.
Modal analysis is a powerful tool to help us understand the vibration characteristics of mechanical structures. It is the process of determining the modal parameters (modal frequency, modal damping and modal shapes). It simplifies the vibration response of a complex structure by reducing the data to a set of modal parameters that can be analyzed.
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The modal matrix (each column of the modal matrix corresponds to a mode shape) is orthogonal to the mass and the stiffness matrix. The undamped normal modes in conjunction with proportional damping models (damping is not discussed in this write-up) simplify vibration analysis considerably.into a matrix Φ allows the matrices M and K to be diagonalized. Specifically, the eigenvectors are mass-normalized such that: ΦT MΦ = I and ΦT KΦ = Ω2 (3) The matrix Φ can be intuitively described as a matrix which trans-forms between object space and mode space: each column of Φ contains the shape of a normal mode, while ΦT f ...
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Therefore the modal data are identified by an experimental modal analysis. The transformation of the experimentally evaluated mode shapes into a closed analytical formulation and the extraction of modal input and output factors for sensors and actuators mass normalized mode shape of the r th mode.
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This method scales each eigenvector to result in a unit value of generalized mass. Equation 4-18. Numerically this method results in a modal mass matrix that is an identity matrix. This normalization approach is appropriate for modal dynamic response calculations because it simplifies both computational and data storage requirements. When mass normalization is used with a model of a heavy, massive structure, the magnitude of each of the terms of the eigenvectors is very small. diagonalize the damping matrix in a similar way, namely (17) where (18) In this equation, represents the percent of critical damping on mode . Combining the results of these equations, and examining the case of mass normalization where , Equation (3) can be uncoupled as (19)
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in which = modal mass matrix; and = modal stiffness matrix. The square of the modal fre-quency can be expressed in a matrix sense as (3) Combining Eqs. (2) and (3) leads to (4) Therefore, the stiffness matrix can be obtained by (5) where = diagonal matrix with the mass-normalized indices on the main diagonal given by (6) In Eq.
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