On the stability of nonconservative systems with small dissipation
In the present work, we study the paradoxical influence of small dissipative and gyroscopic forces on the stability of linear nonconservative systems consisting of the nonpredictable (at first glance) behavior of a critical nonconservative loading. By studying bifurcations of multiple roots of the characteristic polynomial of the nonconservative system considered, the analytical description of this effect is obtained. The model of a disk brake describing the appearance of a creak in the braking of a car is considered as a mechanical example.
KeywordsCharacteristic Polynomial Simple Eigenvalue Disk Brake Dissipative Force Double Root
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- 4.V. V. Bilitin, Nonconservative Problems of Elastic Stability Theory [in Russian], Fizmatgiz, Moscow (1961).Google Scholar
- 7.V. I. Gugnina, “Extension of the D. K. Faddeev method to polynomial matrices,” Dokl. Akad. Nauk UzSSR, 1, 5–10 (1958).Google Scholar
- 8.O. N. Kirillov, “Destabilization paradox,” Dokl. Ross. Akad. Nauk, 395, No. 5, 614–620.Google Scholar
- 9.K. Popp, M. Rudolph, M. Kroger, and M. Lindner, “Mechanisms to generate and to avoid friction induced vibrations,” in: VDI-Berichte, No. 1736 (2002).Google Scholar
- 10.A. P. Seiranyan, “Destabilization paradox in the stability problems of nonconservative systems,” Usp. Mekh., 13, No. 2, 89–124 (1990).Google Scholar
- 11.A. P. Seiranyan, “On stabilization of nonconservative systems by dissipative forces and indeterminacy of the critical loading,” Dokl. Ross. Akad. Nauk, 348, No. 3, 323–326 (1996).Google Scholar
- 12.A. P. Seiranyan and O. N. Kirillov, “On the influence of small dissipative and gyroscopic forces on the stability of nonconservative systems,” Dokl. Ross Akad. Nauk, 393, No. 4, 483–488 (2003).Google Scholar
- 13.A. P. Seyranian and O. N. Kirillov, “Bifurcation diagrams and stability boundaries of circulatory systems,” Theor. Appl. Mech., 26, 135–168 (2001).Google Scholar
- 15.M. I. Vishik and L. A. Lyusternik, “Solution of some perturbation problems in the case of matrices and self-adjoint and non-self-adjoint differential equations, I,” Usp. Mat. Nauk, 15, No. 3, 3–80 (1960).Google Scholar
- 16.N. I. Zhinzher, “Influence of dissipative forces with incomplete dissipation on the stability of elastic systems,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 19, No. 1, 149–155, (1994).Google Scholar