Normal Modal Vibrations for Some Damped n-Degree-of-Freedom Nonlinear Systems

1966 ◽  
Vol 33 (4) ◽  
pp. 877-880 ◽  
Author(s):  
George W. Morgenthaler
Keyword(s):  
Nonlinear Systems ◽  
Normal Mode ◽  
Nonlinear Vibrations ◽  
Normal Modes ◽  
Degree Of Freedom ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Damping Matrix ◽  
Necessary And Sufficient ◽  
Do So

T. K. Caughey1 has shown that a necessary and sufficient condition that a damped, linear, n-degree-of-freedom system possess classical linear normal modes is that the damping matrix be diagonalized by the same transformation which uncouples the undamped system. Rosenberg2 has defined normal modes for nonlinear n-degree-of-freedom undamped systems and has shown the existence of such modes for various classes of nonlinear systems. In linear systems, the frequency is independent of the amplitude and, if a set of masses is vibrating in unison, it is not surprising that in some cases they continue to do so as the motion damps out. In nonlinear vibrations, however, frequency depends upon amplitude so that a series of masses vibrating at different amplitudes in a Rosenberg normal mode might generally be expected to lose synchronization as their amplitudes damp out. Two classes of systems are discussed here in which normal modes are preserved under damping, and several examples are given.

scholarly journals Reachable Set Bounding for Homogeneous Nonlinear Systems with Delay and Disturbance

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6 ◽  
Author(s):  
Xingao Zhu ◽  
Yuangong Sun
Keyword(s):  
Stability Analysis ◽  
Time Delay ◽  
Nonlinear Systems ◽  
Reachable Set ◽  
Delay Systems ◽  
Sufficient Condition ◽  
Time Delay Systems ◽  
Positive Systems ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Reachable set bounding for homogeneous nonlinear systems with delay and disturbance is studied. By the usage of a new method for stability analysis of positive systems, an explicit necessary and sufficient condition is first derived to guarantee that all the states of positive homogeneous time-delay systems with degree p>1 converge asymptotically within a specific ball. Furthermore, the main result is extended to a class of nonlinear time variant systems. A numerical example is given to demonstrate the effectiveness of the obtained results.

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.

Analysis on Degree of Freedom and Singularity of Mechanism Based on Topological Graph Theory

2011 ◽  
Vol 393-395 ◽  
pp. 20-23
Author(s):  
Jian Guo Luo ◽  
Mao Yan He
Keyword(s):  
Graph Theory ◽  
Degree Of Freedom ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Topological Graph ◽  
Topological Graph Theory ◽  
Hybrid Mechanism ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Case Number

Based on the analysis of current developing state of graph theory, define the description of spacial moving capability of common couples and translation base and rotation base of mechanism, based on the new description method in topological graph theory. DOF(degree of freedom) of hybrid mechanism analysised with example based on the definition of dimensionity of branch spacial moving capability and mechanism spacial moving capability, necessary and sufficient condition of nonsingularity of mechanism presented, as well as the necessary and sufficient condition of singularity of mechanism deduced , in-phase and assimilation condition and in-phase and dissimilarity condition and asynchronism condition of limitation of input base of branch adopted, case number of position singularity and pose singularity and position and pose singularity obtained then, still the way of founding the combination and case number of common serial mechanism and parallel mechanism and hybrid mechanism mentioned.

Detection of Improper General Spatial Kinematic Chains With Different Types of Pairs

1994 ◽  
Author(s):  
Xian-Wen Kong ◽  
Ting-Li Yang
Keyword(s):  
Unsolved Problem ◽  
Degree Of Freedom ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Single Degree Of Freedom ◽  
Kinematic Chains ◽  
Single Degree ◽  
Different Types ◽  
Necessary And Sufficient ◽  
Topological Synthesis

Abstract Improper general spatial kinematic chains (GSKCs) due to the effect of pair types may be generated during the process of topological synthesis of GSKCs with different types of pairs. Thus, detection of improper GSKCs is necessary in topological synthesis of GSKCs with different types of pairs. Unfortunately, it is still an unsolved problem. In this paper, a method for detecting improper GSKCs is presented. Both a necessary and sufficient condition and a sufficient condition for proper GSKCs with R, P, H, T and C pairs are introduced at first. Based on these two conditions, an algorithm to detect improper GSKCs is then developed which is very efficient and suitable for topological synthesis of GSKCs with R, P, H, T and C pairs. The proposed algorithm has been applied to topological synthesis of 1- and 2-loop, single degree of freedom GSKCs with R, P, H, T and C pairs and the corresponding atlas is obtained.

Active Absorption of Viscously Damped System With Time Delay

2008 ◽  
Vol 75 (5) ◽  
Author(s):  
Yitshak M. Ram ◽  
Kumar Vikram Singh
Keyword(s):  
Time Delay ◽  
Passive Control ◽  
Degree Of Freedom ◽  
Sufficient Condition ◽  
Total Absorption ◽  
Necessary And Sufficient Condition ◽  
Controlled System ◽  
Damped System ◽  
Necessary And Sufficient ◽  
System With Time Delay

In general, it is not possible to obtain total motion absorption of a certain degree of freedom in a harmonically excited damped system by passive control. This paper presents a method of obtaining total absorption in viscously damped system by active control, including time delay, which is unavoidable in digital controlled system. The control is applied on one degree of freedom and the absorption is achieved at another point. This study is carried out by both complex and real analyses. The necessary and sufficient condition for obtaining total absorption is given. Examples demonstrate the various results.

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