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Unsteady processes in machines.

Friedrich Pfeiffer¹

¹Lehrstuhl B fur Mechanik, Technische Universitat-Munchen Arcisstrasse 21, D-80333 Munchen, Germany.

Chaos (Woodbury, N.Y.)

|December 1, 1994

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This study presents a unified Lagrangian theory for analyzing unsteady multibody systems, addressing phenomena like stick-slip and impacts in machine couplings. The approach models complex dynamics, including potential chaos, for improved mechanical system design.

Area of Science:

Multibody Dynamics
Mechanical Engineering
Nonlinear Dynamics

Background:

Machine couplings exhibit play and friction, leading to complex behaviors like stick-slip and impacts under load.
These phenomena render mechanical system structures time-variant or unsteady, complicating traditional modeling approaches.
Existing models struggle to uniformly address the time-variant degrees of freedom and unsteady coupling characteristics caused by stick-slip and impacts.

Purpose of the Study:

To establish a uniform Lagrangian theory for modeling unsteady multibody systems with stick-slip and impact events.
To develop numerical algorithms for analyzing contact problems and identifying switching points in dynamic systems.
To apply the developed theory to practical engineering applications and investigate potential chaotic behaviors.

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Main Methods:

Utilized a Lagrangian approach, incorporating constraint equations and equations of motion via Lagrange multipliers.
Employed indicator functions to evaluate stick-slip and impact events, enabling specialized numerical algorithms.
Formulated contact problems as complementarity problems, solvable with efficient algorithms.

Main Results:

Developed a unified theoretical framework for analyzing systems with time-variant degrees of freedom and unsteady coupling characteristics.
Successfully applied the theory to diverse applications including rattling gears, impact drilling machines, turbine blade dampers, and a woodpecker toy.
Demonstrated the possibility of chaotic behavior arising from parameter variations in some analyzed systems.

Conclusions:

The proposed Lagrangian approach provides a robust method for modeling complex dynamics in machines and mechanisms with play and friction.
The integration of indicator functions and complementarity problems offers efficient solutions for analyzing stick-slip and impact events.
The theory's applicability to real-world systems highlights its potential for enhancing the design and understanding of mechanical systems, including those exhibiting chaotic dynamics.