Eigenvalues analysis of a diesel valve gear mathematical model

We consider the mathematical model for (non)linear oscillations of a diesel valve gear. We transform the original 2nd order system of ODEs resulting from Newton's law into a linear 1st order system of ODEs of dimension six with external forces. We analyse the eigenvalues of the corresponding matrix in terms of the original input parameters (i.e. masses and the coefficients of the springs involved in the corresponding gear system), vary the parameters within some naturally interesting intervals and consider the response of the system. This is a non-trivial task, since it involves a large number of variables. However at least empirically, it is easy to see that the coefficients k4, k2, k0 of the characteristic polynomial of the corresponding matrix are in the following relation k2=αk42 and k0=βk43 for some real values α, β. This simplifies the analysis of the position of eigenvalues of the corresponding matrix in the complex plane and allows to determine the critical region for which the diesel valve gear admits possibly non-oscillatory motion in the plane of new parameters α and β.