Integrability and Jacobi last multipliers of cubic Liénard differential equations with quadratic damping

2020 ◽  
Vol 9 (4) ◽  
pp. 499-507
Author(s):  
Maria V. Demina
Keyword(s):  
Differential Equations ◽  
Quadratic Damping

Free Oscillations of Systems Having Quadratic Damping and Arbitrary Restoring Forces

1955 ◽  
Vol 22 (4) ◽  
pp. 493-499
Author(s):  
Karl Klotter
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
First Integral ◽  
Linear Differential Equations ◽  
Free Oscillations ◽  
First Order ◽  
Restoring Forces ◽  
Linear Differential ◽  
Quadratic Damping

Abstract In this paper, systems are treated which are subjected to quadratic damping forces (of any magnitude) and to restoring forces of any type. The differential equations of motion for such systems can be transformed into linear differential equations of first order for the velocity squared, whatever the restoring forces may be. A first integral can be obtained readily. From it the exact relationships between any two consecutive maximum displacements (“amplitudes”) are derived. These relationships are discussed in detail for various types of restoring forces. Examples are worked out numerically and illustrated by graphs.

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