Non-Linear Dynamics of Geared Systems Using the Asymptotic Method, Part 2: Computation of the Phase Angle
The increased interest for improved gear design has led to extensive research into the field of non- linear dynamics of such systems. The paper reveals a complex dynamic model to study the behavior in a gear- pair system taking into consideration backlash and time-dependent mesh stiffness and mesh damping. In many applications including turbo machinery, machine tools and diesel engines non-linearities are present due to tooth stiffness, damping and backlash that induced micro-vibrations of non-linear parametric type. In the mean time the input link of the driver ax and the output link of the driven ax induce non-linearities. The paper presents the use of asymptotic method in order to compute the amplitude of steady state motion. By this way the paper reveals the phenomena's characteristics of multiple jumps specific to the non-linear dynamic behavior of gear-pair due to: non-linearities of the input-output linkages, backlash and self-induced parametric excitations, caused by the tooth stiffness and damping. It was highlighted the interaction between fundamental resonance and the principal parametric resonance.
Schwingungs- und Stabilitätsprobleme bei mechanischen Systemen mit harmonischer Erregung†
Die Frage, wie ein mechanisches System resonanzartig auf eine harmonische Erregung antworten kann, ist eine der klassischen Problemstellungen der Schwingungslehre. Zu ihr soll im folgenden ein Beitrag geliefert werden. Er mus sich zwar der Kurze halber in mehrfacher Hinsicht beschranken, namlich bezuglich der behandelten mechanischen Systeme, der verwendeten mathematischen Methode und der abgeleiteten Ergebnisse, soll aber doch von einem neuen Standpunkt aus einen gewissen Uberblick uber die wichtigsten, in der Literatur heute z. T. noch weit verstreuten Tatsachen geben.
On the dynamics of oscillators with bi-linear damping and stiffness☆
Abstract The long time response of a class of harmonically excited, single degree-of-freedom, strongly non-linear oscillators is analysed. The non-linearity appears in both the damping and restoring forces, which are bi-linear functions of the system velocity and displacement, respectively, The existence and stability of single-crossing, periodic solutions is analysed, using a formulation which is appropriate for general piecewise linear systems. The oscillator examined is found to exhibit a regular behavior for some combinations of its parameters. However, there are sets of parameters for which the system undergoes bifurcations which may lead to loss of stability of periodic solutions and the appearance of chaotic response. Results obtained by the present methodology are compared with findings of previous studies on similar systems. Phenomena characterizing the response of the system are investigated in detail.