zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Friction induced hunting limit cycles: a comparison between the LuGre and switch friction model. (English) Zbl 1254.74085
Summary: Friction induced limit cycles are predicted for a simple motion system consisting of a motor-driven inertia subjected to friction and a PID-controlled regulator task. The two friction models used, i.e., (i) the dynamic LuGre friction model and (ii) the static switch friction model, are compared with respect to the so-called hunting phenomenon. Analysis tools originating from the field of nonlinear dynamics will be used to investigate the friction induced limit cycles. For a varying controller gain, stable and unstable periodic solutions are computed numerically which, together with the stability analysis of the closed-loop equilibrium points, result in a bifurcation diagram. Bifurcation analysis for both friction models indicates the disappearance of the hunting behavior for controller gains larger than the gain corresponding to the cyclic fold bifurcation point.

74M10Friction (solid mechanics)
34C05Location of integral curves, singular points, limit cycles (ODE)
Full Text: DOI
[1] Armstrong-Hélouvry, B.: Stick-slip and control in low-speed motion. IEEE transactions on automatic control 38, No. 10, 1483-1496 (1993)
[2] Armstrong-Hélouvry, B.; Amin, B.: PID control in the presence of static frictiona comparison of algebraic and describing function analysis. Automatica 32, 679-692 (1996) · Zbl 0862.93027
[3] Armstrong-Hélouvry, B.; Dupont, P.; De Wit, C. Canudas: A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica 30, 1083-1138 (1994) · Zbl 0800.93424
[4] De Wit, C. Canudas; Olsson, H.; Ström, K. J. A. \mathring{}; Lischinsky, P.: A new model for control of systems with friction. IEEE transactions on automatic control 40, 419-425 (1995) · Zbl 0821.93007
[5] Clarke, F. H., Ledyaev, Yu. S., Stern, R. J., & Wolenski, P. R. (1998). Nonsmooth analysis and control theory. Graduate texts in mathematics. New York: Springer-Verlag. · Zbl 1047.49500
[6] Dahl, P. (1968). A solid friction model. Aerospace Corp., El Segundo, CA, Technical Report TOR-0158(3107-18)-1.
[7] Dankowicz, H.; Nordmark, A.: On the origin and bifurcations of stick-slip oscillations. Physica D 136, 280-302 (2000) · Zbl 0963.70016
[8] Fey, R. H. B. (1992). Steady-state behavior of reduced dynamic systems with local nonlinearities. Ph.D. thesis, Eindhoven University of Technology, Netherlands.
[9] Gäfvert, M. (1997). Comparisons of two dynamic friction models. Proceedings of the sixth IEEE conference on control applications, Hartford.
[10] Galvanetto, U.; Knudsen, C.: Event maps in a stick-slip system. Nonlinear dynamics 13, 99-115 (1997) · Zbl 0898.70014
[11] Guckenheimer, J., & Holmes, P. (1983). Nonlinear oscillations, dynamic systems, and bifurcations of vector fields. Applied mathematical sciences. Vol. 42. New York: Springer-Verlag. · Zbl 0515.34001
[12] Hensen, R. H. A.; Van De, M. J. G. Molengraft; Steinbuch, M.: Frequency domain identification of dynamic friction model parameters. IEEE transactions on control systems technology 2, No. 10, 191-196 (2002)
[13] Karnopp, D.: Computer simulation of stick-slip friction in mechanical dynamic systems. ASME journal of dynamic systems, measurement and control 107, 100-103 (1985)
[14] Leine, R. I. (2000). Bifurcations in discontinuous mechanical systems of Filippov-type. Ph.D. thesis, Eindhoven University of Technology, Netherlands.
[15] Olsson, H. (1996). Control systems with friction. Ph.D. thesis, Lund Institute of Technology, Sweden.
[16] Parker, T. S.; Chua, L. O.: Practical numerical algorithms for chaotic systems. (1989) · Zbl 0692.58001
[17] Radcliffe, C. J., & Southward, S. C. (1990). A property of stick-slip friction models which promotes limit cycle generation. Proceedings of the 1990 American control conference, ACC, San Diego, CA (pp. 1198-1203).