The mathematical principles of vibration reductor
The vibration reductor is modelled in this paper by the system of equation: \[ \ddot x_ 1+\phi (\dot x_ 1)+k_ 1(x_ 1-x_ 2)=0,\quad \ddot x_ 2+c\dot x_ 1+k_ 2(x_ 2-x_ 1)=0. \] \(x_ 1\) is the coordinate of the (one-dimensional) motion of the machine, \(x_ 2\) is that of the coupled vibration reductor. \(k_ 1\), c, \(k_ 2\) are constants. No particular statement is made about the function \(\phi\) (ẋ\({}_ 1)\). The aim of the paper is to choose the above constants so that the zero solution of the given equations is globally asymptotically stable. Four cases of certain inequalities must be studied. To this end some lemmas of Plies are used.
scientific article; zbMATH DE number 3976444