Reliability of Mechanical Systems
During their operation mechnical systems are, in general, subjected to internal and external disturbances that may cause them to operate under non-optimal conditions, or even to fail. Failure can occur either as a sudden collapse of the system, or as the consequence of a gradual aging process. In this project mechanical systems are modeled as differential equations, and perturbating forces as stochastic processes. Both modes of failure are then described as exists from a safe operating region. The problem is, on the one hand, to determine failure probabilties, failure time, and critical components for such a system. On the other hand, the problem is to design a controller that prohibibts failure or increases the time to failure. The mathematical analysis requires techniques from complex dynamical systems, stochastic systems, and control of dynamical systems. Explicit computation of failure probabilities and times is done via numerical simulation. The figures above show the roll motion of a ship due to wave disturbances and a simulation of failure (=capsizing) times, with the best fitting 3-parameter Weibull distribution. (Supported by ONR)
- Collaborators
- Dr. Fritz Colonius, University of Augsburg, Germany
- Dr. Gerhard Häckl, Germany
- Dr. Ruey-Gang Lai
- Dr. Helmut Pradlwarter, University of Innsbruck, Austria
- Recent Publications
- Books
- Kliemann, W. and N. S. Namachchivaya (eds.), Nonlinear Dynamics and Stochastic Mechanics,
CRC Press, Boca Raton, 1995, 530 pp.
- Kliemann, W., W. F. Lanford and N. S. Namachchivaya (eds.) Nonlinear Dynamics and
Stochastic Mechanics, AMS Press, Fields Institute Communication Vol. 9, 1996, 238 pp.
- Papers, refereed
- Colonius, F. and W. Kliemann, Stability radii and Lyapunov exponents, in: Control of
Uncertain Systems, D. Hinrichsen and B. Martensson (eds.), Birkhäuser (1990), 19-56.
- Colonius, F. and W. Kliemann, Stabilization of Linear Uncertain Systems, in: Modeling,
Estimation and Control of Systems with Uncertainty (A. B. DiMasi, A. Gombani, A. B.
Kurzhansky, eds.), Birkhäuser (1991), 76-90.
- Colonius, F. and W. Kliemann, Minimal and maximal Lyapunov exponents of bilinear control
systems, J. Diff. Equations 101 (1993), 232-275.
- Colonius, F. and W. Kliemann, Stabilization of Uncertain Linear Systems via Lyapunov
Exponents, in Proceedings of IEEE Conference on Decision and Control 1991, Brighton, England
(1991), Vol. I. 887-893.
- Colonius, F. and W. Kliemann, Controlling the Dynamics of a Random System, in: Nonlinear
Stochastic Mechanics (N. Bellomo, F. Casciati, eds.), Springer (1992), 333-346.
- Colonius, F. and W. Kliemann, Random perturbations of bifurcation diagrams, Nonlinear
Dynamics 5 (1994), 353-373.
- Colonius, F. and W. Kliemann, Reliability assessment of dynamical systems with random
excitation, Proceedings of 32nd IEEE Conference on Decision and Control 1993, San Antonio
(1993), 3879-3884.
- Colonius, F. and W. Kliemann, Local robust stabilization of nonlinear oscillators under
parametric excitation, in: Stochastic Dynamics and Reliability of Nonlinear Ocean Systems (R.
A. Ibrahim, Y. K. Lin, eds.), ASME DE-Vol. 77, (1994), 1-5.
- Colonius, F. and W. Kliemann, A stability radius for nonlinear differential equations subject to
time varying perturbations, in: Proceedings of IFAC NOLCOS '95, (1995) 44-46.
- Colonius, F., F. J. de la Rubia, and W. Kliemann, Stochastic models with multistability and
extinction levels, SIAM J. Applied Mathematics 56(1996), 919-945.
- Colonius, F., G. Häckl, and W. Kliemann, Dynamic reliability of nonlinear systems under
random excitation, in: Vibrations of Nonlinear, Random, and Time-Varying Systems, ASME
DE-Vol. 84-1, (1995), 1007-1024.
- Colonius, F. and W. Kliemann, Stability of time varying systems, in: Vibrations of Nonlinear,
Random, and Time-Varying Systems, ASME DE-Vol. 84-1, (1995), 365-373.
- Kliemann, W., Nonlinear time series - bifurcation, chaos, and stationarity, Modelling and
Prediction Honoring Seymour Geisser, J. C. Lee, W. O. Johnson, and A. Zellner (eds),
Springer (1996), 389-401.
- Papers, not refereed
- Colonius, F., W. Kliemann, and S. Krull, Stability and stabilization of linear, uncertain systems
-A Lyapunov exponents approach, Report No. 372 of the Schwerpunktprogramm der Deutschen
Forschungsgemeinschaft 'Anwendungsbezogene Optimierung und Steuerung' (1992), 38 pp.
- Colonius, F. and W. Kliemann, Lyapunov exponents in nonlinear stochastic dynamics,
Proceedings of 32nd Annual Technical Meeting of the Society of Engineering Science, New
Orleans, Oct. 1995, 273-275.
- Pradlwarter, H. J., and W. Kliemann, First exit times in nonlinear dynamical systems by
advanced Monte Carlo simulation, Proceedings of the 11th Conference on Engineering
Mechanics, ASCE, Fort Lauderdale, May 1996, Vol. I, 523-526.