On Set-oriented Numerical Methods for Global Analysis of Non-smooth Mechanical Systems

Abstract

A large number of engineering systems include non-linear or even non-smooth behaviour. During development and optimisation of mechanical systems, a comprehensive dynamical analysis may be essential. Set-oriented Numerical Methods offer a new and efficient approach to global dynamical analysis without the requirement for a detailed a priori knowledge of the system dynamics. This paper shows the basic concepts behind this method and presents its application to a simple model of a vibro-impact device for ultrasonic/sonic drilling. In this drilling device an impacting mass transmits vibrational energy from a piezoelectric actuator as shock waves into a drill stem. Relative global attractors will be derived which show statistical distributions for the occurrence of all possible states. Periodic solutions can be identified parallel to the evaluation of chaotic behaviour. Thus a global understanding of the system performance can be obtained, offering new ways for model verification and system design.

Review on research and development of planetary penetrator

“…The nonlinear dynamic characteristics of USDC were studied at the University of Paderborn in Germany, and it was found that there is a wobble phenomenon when using a disk‐shaped mass motion. The final design replaces the disk‐shaped free mass with a free spherical mass, which solves the wobble phenomenon (Neumann & Sattel, 2007; Neumann, Sattel, & Wallaschek, 2007; Potthast et al, 2007). In Figure 8h, Honeybee Robotics developed an ultrasonic penetrator Auto‐Gopher II.…”

Control of a class of nonsmooth dynamical systems

“…For instance, the set-oriented method is applied to a simple nonsmooth mechanical model of an ultrasonic/sonic drilling device. Due to the complex dynamical behavior of the system, it is rather difficult to find periodic solutions by integrating the model, or to extract information from only a few long-term integrations (Neumann et al, 2007).…”

A numerical approach for nonsmooth ordinary differential equations

“…However, there are cases where such approaches are difficult to apply or even fail. This could be the case for systems with small or complicated attractors, making it difficult to find fixed points or periodic solutions (Neumann et al, 2007).…”