scholarly journals Lagrangians for Damped Linear Multi-Degree-of-Freedom Systems

2013 ◽  
Vol 80 (4) ◽  
Author(s):  
Firdaus E. Udwadia ◽  
Hancheol Cho
Keyword(s):  
Inverse Problem ◽  
Conservation Laws ◽  
Degree Of Freedom ◽  
Linear Vibration ◽  
Gyroscopic System ◽  
Simple Manner ◽  
Vibration Theory ◽  
Stiffness Matrices ◽  
Two Degree Of Freedom ◽  
Special Case

This paper deals with finding Lagrangians for damped, linear multi-degree-of-freedom systems. New results for such systems are obtained using extensions of the results for single and two degree-of-freedom systems. The solution to the inverse problem for an n-degree-of-freedom linear gyroscopic system is obtained as a special case. Multi-degree-of-freedom systems that commonly arise in linear vibration theory with symmetric mass, damping, and stiffness matrices are similarly handled in a simple manner. Conservation laws for these damped multi-degree-of-freedom systems are found using the Lagrangians obtained and several examples are provided.

scholarly journals A two degree-of-freedom linear vibration energy harvester for tram applications

2020 ◽  
Vol 140 ◽  
pp. 106657
Author(s):  
M. Perez ◽  
S. Chesné ◽  
C. Jean-Mistral ◽  
K. Billon ◽  
R. Augez ◽  
...  
Keyword(s):  
Energy Harvester ◽  
Degree Of Freedom ◽  
Vibration Energy ◽  
Linear Vibration ◽  
Vibration Energy Harvester ◽  
Two Degree Of Freedom

Inverse Problem for Lagrangian Dynamics for Multi-Degree-of-Freedom Systems With Linear Damping

2012 ◽  
Author(s):  
Hancheol Cho ◽  
Firdaus E. Udwadia
Keyword(s):  
Inverse Problem ◽  
Simple Procedure ◽  
Degree Of Freedom ◽  
Lagrangian Dynamics ◽  
Mass Matrices ◽  
Helmholtz Conditions ◽  
Linear Damping ◽  
Damped Systems ◽  
Stiffness And Damping ◽  
Two Degree Of Freedom

This paper deals with the inverse problem for Lagrangian dynamics for linear multi-degree-of-freedom systems. New results for linearly damped systems are obtained using extensions of results for single-degree-of-freedom systems. First, for a two-degree-of-freedom linear system with linear damping, the conditions for the existence of a Lagrangian are explicitly obtained by solving the Helmholtz conditions. Next, since the Helmholtz conditions are near-impossible to solve for general n-degree-of-freedom systems, a new simple procedure that does not require the use of the Helmholtz conditions and that is easily extended to n-degree-of-freedom linear systems, is developed. The emphasis is on obtaining the Lagrangians for these multi-degree-of-freedom systems in a simple manner, using insights obtained from our understanding of the inverse problem for single- and two-degree-of-freedom systems. Specifically we include systems that commonly arise in linear vibration theory with positive definite mass matrices, and symmetric stiffness and damping matrices. This method yields several new Lagrangians for linear multi-degree-of-freedom systems. Finally, conservation laws for these damped multi-degree-of-freedom systems are found using the Lagrangians obtained.

Analysis of an Offset Unsymmetric Gyroscope With Oblique Rotor Using (3×3) Matrices With Dual-Number Elements

1969 ◽  
Vol 91 (3) ◽  
pp. 535-541 ◽  
Author(s):  
An Tzu Yang
Keyword(s):  
Equations Of Motion ◽  
Degree Of Freedom ◽  
Spin Axis ◽  
Dynamic Equations ◽  
Gyroscopic System ◽  
Dual Number ◽  
System A ◽  
Six Degree Of Freedom ◽  
Special Case

Using 3 × 3 matrices of dual-number elements, dynamic equations are obtained for an offset unsymmetric gyroscope with obliquely placed rotor, a generalized six-degree-of-freedom gyroscopic system (shown schematically in Fig. 3). Equations of motion for a special case of the system, a two-frame symmetric gyroscope, conventional in all aspects except the rotor is inclined relative to its spin axis, are deduced; these equations are applied to the study of the effects of a slightly inclined rotor on (a) a two-frame symmetric gyroscope in steady precession and (b) a Faucualt’s gyrocompass.

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