Dynamical stability of the response of oscillators with discontinuous or steep first derivative of restoring characteristic

• Published in: European journal of mechanics, A, Solids Vol. 23; no. 6; pp. 1041 - 1050
• Main Authors: Wolf, Hinko, Terze, Zdravko, Sušić, Aleksandar
• Format: Journal Article
• Language: English
• Published: Elsevier Masson SAS 01-11-2004
• Subjects: Dynamical stability; Floquet–Liapounov theorem; Non-linear oscillator; Floquet–Liapounov theorem; Non-linear oscillator; Dynamical stability
• ISSN: 0997-7538; 1873-7285
• DOI: 10.1016/j.euromechsol.2004.08.001

The influence of factors which can lead to incorrect prediction of dynamical stability of the periodic response of oscillators which contain a non-linear restoring characteristic with discontinuous or steep first derivative is considered in this paper. For that purpose, a simple one degree-of-freedom system with a piecewise-linear force-displacement relationship subjected to a harmonic excitation is analysed. Stability of the periodic response obtained in the frequency domain by the incremental harmonic balance method is determined by using the Floquet–Liapounov theorem. Responses in the time domain are obtained by digital simulation. The accuracy of determining the eigenvalues of the monodromy matrix (in the considered example) significantly depend on the corrective vector norm
‖
{
r
}
‖
, the accuracy
ɛ of numerical determination of the times when the system undergoes a stiffness change, and on the number of step functions
M (used in the Hsu's procedure), only for
‖
{
r
}
‖
>
1
×
10
−
5
,
ɛ
>
1
×
10
−
5
and
M
<
2000
. Otherwise, except if the maximum modulus of the eigenvalues of the monodromy matrix is very close to unity, their influence on estimation of dynamical stability is minor. On the contrary, neglecting very small harmonic terms of the actual time domain response can cause a very large error in the evaluation of the eigenvalues of the monodromy matrix, and so they can lead to incorrect prediction of the dynamical stability of the solution, regardless of whether the maximum modulus of the eigenvalues of the monodromy matrix is close to unity or not.