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Properties of Vibrating Systems
Published in William T. Thomson, Theory of Vibration with Applications, 2018
The orthogonal property of normal modes is one of the most important concepts in vibration analysis. The orthogonality of normal modes forms the basis of many of the more efficient methods for the calculation of the natural frequencies and mode shapes. Associated with these methods is the concept of the modal matrix, which is essential in the matrix development of equations.
Bilinear observer-based robust adaptive fault estimation for multizone building VAV terminal units
Published in Journal of Building Performance Simulation, 2023
Mona Subramaniam A., Tushar Jain, Joseph J. Yamé
For fault estimation, we use the error dynamics resulting from a similarity transformation Γ applied on (10), with , and where matrix Γ is the modal matrix of , i.e. is a diagonal matrix (Chen 1995). Consequently, (10) is transformed to where , , , and . Next, we shall introduce a matrix satisfying certain conditions, which would be helpful in designing the fault estimation filter in Proposition 3.4. The following proposition provides a sufficient condition for the existence of the matrix G.
Control for stability improvement of high-speed train bogie with a balanced truncation reduced order model
Published in Vehicle System Dynamics, 2022
Recall the transformation of the BT method in Equation (21). We have Apply the modal decomposition to the matrix . We have where is the modal matrix of and is a diagonal matrix consisting of eigenvalues of . Hence, the matrix reads where . The columns of the matrix are the modal eigenvectors of , while the diagonal matrix consists of the eigenvalues of . By examining the matrix , we can identify the two stable dominant modes in the transformed system. The modal participation of the physical degrees of freedom in the two dominant modes is shown in Figure 4. It is clear that the two dominant modes consist of special linear combinations of the physical degrees of freedom.
Numerical evaluation of multi-metric data fusion based structural health monitoring of long span bridge structures
Published in Structure and Infrastructure Engineering, 2018
Rohan Soman, Marios Kyriakides, Toula Onoufriou, Wieslaw Ostachowicz
where, is the strain modal matrix with dimensions is the diagonal matrix with inverse square of the natural frequencies with dimensions , is the transpose of the displacement mode shape with dimensions , is the inverse of the displacement flexibility matrix with dimensions and m is the number of modes of interest.