This paper focuses on the primary resonance of the time delayed Duffing oscillator solved by means of the multiple scales method. The second order approximation is used because the first approximation applied in the literature does not exhibit specific phenomena of the studied system. The Duffing system is investigated in two aspects. First of all, how delay displacement feedback influences the primary resonance of the classical Duffing oscillator in order to control the system and, on the other hand, how the external harmonic force influences vibrations of the system with time delay which is essential in applications, e.g. in machining. Stability of solutions and their bifurcations caused by the system parameters are shown. Especially, coexistence of possible stable and unstable solutions is investigated versus time delay and external excitation parameters. The selected analytical results are compared with numerical simulations. The model presented in the paper is considered as a general problem of non-linear oscillator with time delay but specific attention is paid to cutting process control.

Dynamics of a time delayed Duffing oscillator

We show that a "simple" differential equation modeling a garden-variety damped forced pendulum can exhibit extraordinarily complicated and unstable behavior. While instability and control might at first glance appear contradictory, we can use the pendulum's instability to control it. Such results are vital in robotics: the forced pendulum is a basic subsystem of any robot. Most of the mathematical methods used in this paper were initially developed in celestial mechanics, largely by Poincare. The literature of the field tends to be quite advanced indeed (see [1] and [11]); one object of this paper is to show that computer programs, properly used, can make these advanced topics transparent. All the computer-generated pictures in this paper were produced by the programs Planar Systems and Planar Iterations [6], both written by Ben Hinkle (now at Maple). 1. SOME PARALLELS IN CELESTIAL MECHANICS. When I was a graduate

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The Forced Damped Pendulum: Chaos, Complication and Control

: The primary goal of this work is to use mathematical modelling to assist in defining the operational limits of the Australian Army CH-47D Chinook when carrying mixed density slung loads. This report presents the first phase in the program: the development of a simple helicopter slung-load model for simulation and analysis of the system dynamics. General system equations of motion are obtained from the Newton-Euler equations in terms of generalized coordinates and velocities. The system is partitioned into coordinates such that the motion due to cable stretching is separated from that due to rigid-body, coupled dynamics. In the formulation used, the constraint forces appear explicitly and a solution to the resultant generalized accelerations can be determined by modelling the cable as a simple spring. An inelastic solution is also possible by nulling the stretching coordinates to obtain a relation for the suspension forces. The system equations are also extended for the multiple load case. The model is verified by imposing certain constraints in order to approximate a simple pendulum system and then comparing its behaviour with analytical results. Various configurations of the complete helicopter slung-load system, based on the CH-47B Chinook carrying standard military containers, are then examined in an investigation of the open-loop characteristics. In the investigation, several parameters such as the helicopter-load mass ratio, suspension configuration, and number of loads are varied and the resulting system modes examined. A number of simulations are also presented which demonstrate the characteristic behaviour of such systems.

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Mathematical Modelling of Helicopter Slung-Load Systems

Many technical and scientific objects are far too complex to be properly understood through pictures. 3D representations that can be touched and physically manipulated by the observer convey information not obtainable from 2D projections. Thanks to some emerging affordable rapid prototyping (RP) technologies, such models are beginning to find a role in design, science, and manufacturing. RP is already firmly established in the automotive industry and among designers of consumer products, including household appliances, toys, and electronics. With this personal case study from the arts and mathematics, I hope to encourage designers from other application domains to use RP technologies as a truly 3D physical visualization tool.

Rapid prototyping: a 3d visualization tool takes on sculpture and mathematical forms

The paper analyzes the concepts of uncertainty and surprise as key variables of a socio-ecological system's behavior in the context of the theory of complexity. Experiences from the past have shown that living with uncertainty is part of our daily life and surprises are only surprising because our perspective of system trajectories is basically linear and non-dynamic. The future of humanity is dependent on the understanding of the system's behavior and needs a change in perspective of linearity to non-linearity and from the planning imperative to a management hedging uncertainty and surprise. In the context of humanity's future, the theory of complexity offers a new perspective on system trajectories and their understanding of surprises and uncertainty. There is a need for a Gestaltwechsel—a change in perception—which helps to see things differently and fosters the search for new answers to emerging questions at the human-nature interface. Drawing on the case study of hazard management the paper will explain the necessity of analysis system's behavior and the taking into account of multi-agent behavior on the micro level which led to emergent behavior on the macro-level of the system. Regional geohazards are explained as the regional impact of an uncontrolled risk based on a state of a natural feature that has a direct impact on a regional population being affected by the appearance of a hazard and its development into damage. By acting in space, time and connectivity, people construct hazardscapes and change risk into regional geohazards. This concept shows relevance for future mitigation and adaptation measures. The theory of complexity can help in engendering the necessary shift in perspective. What is non-linear dynamic thinking as suggested by the theory of complexity? Why is the consideration of the system's behavior crucial and not just the number of system's elements? What is the role of agents in these systems? In addition, there are practical implications too: What does this shift in perspective mean for future hazard management and the future of humanity?

Surprise and Uncertainty—Framing Regional Geohazards in the Theory of Complexity

The Duffing Equation

A Double Pendulum

This title presents new ideas on the visualization of differential equations with user-configurable tools. The authors use the widely-used computer algebra system, Mathematica, to provide an integrated environment for programming, visualizing graphics, and running commentary for learning and working with differential equations.

VisualDSolve: Visualizing Differential Equations with Mathematica®

Invisible handshake

At the 13th International Snow Sculpting Championship in Breckenridge, CO, Jan 28. - Feb.l, 2003, Team Minnesota, USA, consisting of the above five authors, carved a 12-foot tall, twisted Scherk­ Collins Toroid celebrating a 7-year collaboration between Brent Collins and Carlo Sequin. This paper describes the concept, the detailed design, and the implementation of this very temporary piece of art. Figure 1. "Whirled White Web, " International Snow Sculpting Championships, Breckenridge, 2003.

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Dan Schwalbe

Papers

The Duffing Equation

A Double Pendulum

VisualDSolve: Visualizing Differential Equations with Mathematica®

Invisible handshake

Whirled White Web: Art and Math in Snow