Closure to “Discussion of ‘Classical Normal Modes in Damped Linear Dynamic Systems’” (1961, ASME J. Appl. Mech., 28, p. 152)

1961 ◽  
Vol 28 (1) ◽  
pp. 152-153
Author(s):  
T. K. Caughey
Keyword(s):  
Dynamic Systems ◽  
Normal Modes ◽  
Linear Dynamic Systems ◽  
Linear Dynamic

Normal and Quasi-Normal Modes in Damped Linear Dynamic Systems

1966 ◽  
Vol 33 (2) ◽  
pp. 413-416 ◽  
Author(s):  
J. S. Maybee
Keyword(s):  
Linear Systems ◽  
Dynamic Systems ◽  
Normal Modes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linear Dynamic Systems ◽  
Linear Dynamic ◽  
Necessary And Sufficient

A generalization of the concept of classical normal modes in damped linear systems is presented. It is then shown that a necessary and sufficient condition that such quasi-normal modes exist is that certain matrices associated with the system commute. Necessary and sufficient conditions of the same type are also obtained for the classical normal modes, but under more restrictive conditions.

Classical Normal Modes in Damped Linear Dynamic Systems

1965 ◽  
Vol 32 (3) ◽  
pp. 583-588 ◽  
Author(s):  
T. K. Caughey ◽  
M. E. J. O’Kelly
Keyword(s):  
Dynamic Systems ◽  
Normal Modes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Dynamic Systems ◽  
Linear Dynamic ◽  
Necessary And Sufficient

The purpose of this paper is to determine necessary and sufficient conditions under which both discrete and continuous damped linear dynamic systems possess classical normal modes.

Classical Normal Modes in Damped Linear Dynamic Systems

1960 ◽  
Vol 27 (2) ◽  
pp. 269-271 ◽  
Author(s):  
T. K. Caughey
Keyword(s):  
Dynamic Systems ◽  
Necessary Conditions ◽  
Normal Modes ◽  
Present Theory ◽  
Necessary And Sufficient Condition ◽  
Linear Dynamic Systems ◽  
Linear Dynamic ◽  
Damping Matrix ◽  
Necessary And Sufficient ◽  
Special Case

An analysis of the conditions under which a damped linear system possesses classical normal modes is presented. It is shown that a necessary and sufficient condition for the existence of classical normal modes is that the damping matrix be diagonalized by the same transformation that uncouples the undamped systems. Sufficient though not necessary conditions on the damping matrix are developed, and it is shown that Rayleigh’s solution is a special case of the present theory.

scholarly journals On the Uncertainty Identification for Linear Dynamic Systems Using Stochastic Embedding Approach with Gaussian Mixture Models

Sensors ◽  
2021 ◽  
Vol 21 (11) ◽  
pp. 3837
Author(s):  
Rafael Orellana ◽  
Rodrigo Carvajal ◽  
Pedro Escárate ◽  
Juan C. Agüero
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Dynamic Systems ◽  
Gaussian Mixture ◽  
Error Model ◽  
Fault Detection And Diagnosis ◽  
Deterministic Models ◽  
Linear Dynamic Systems ◽  
Linear Dynamic

In control and monitoring of manufacturing processes, it is key to understand model uncertainty in order to achieve the required levels of consistency, quality, and economy, among others. In aerospace applications, models need to be very precise and able to describe the entire dynamics of an aircraft. In addition, the complexity of modern real systems has turned deterministic models impractical, since they cannot adequately represent the behavior of disturbances in sensors and actuators, and tool and machine wear, to name a few. Thus, it is necessary to deal with model uncertainties in the dynamics of the plant by incorporating a stochastic behavior. These uncertainties could also affect the effectiveness of fault diagnosis methodologies used to increment the safety and reliability in real-world systems. Determining suitable dynamic system models of real processes is essential to obtain effective process control strategies and accurate fault detection and diagnosis methodologies that deliver good performance. In this paper, a maximum likelihood estimation algorithm for the uncertainty modeling in linear dynamic systems is developed utilizing a stochastic embedding approach. In this approach, system uncertainties are accounted for as a stochastic error term in a transfer function. In this paper, we model the error-model probability density function as a finite Gaussian mixture model. For the estimation of the nominal model and the probability density function of the parameters of the error-model, we develop an iterative algorithm based on the Expectation-Maximization algorithm using the data from independent experiments. The benefits of our proposal are illustrated via numerical simulations.

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