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.
The transition to chaos in a simple mechanical system
Abstract A simple mechanical device and its response to periodic excitation is considered. The system consists of an inverted pendulum with rigid barriers which limit the amplitude variation from the unstable upright position. The static stable rest positions correspond to the pendulum leaning against one of the barriers. When subjected to periodic excitation the system response can be quite complicated and may include one or several stable subhannonics and/or chaotic motions. The analysis presented here is based on a piecewise linear model which allows explicit analytic expressions to be determined for many bifurcation conditions including: the appearance of certain types of subharmonics by saddle-node bifurcations, the secondary bifurcations of these subharmonics, and a global bifurcation which results in the creation of horseshoes.