Abstract
The local instability of 2 degrees of freedom (DOF) weakly damped systems is thoroughly discussed using the Liénard‐Chipart stability criterion. The individual and coupling effect of mass and stiffness distribution on the dynamic asymptotic stability due to mainly infinitesimal damping is examined. These systems may be as follows: (a) unloaded (free motion) and (b) subjected to a suddenly applied load of constant magnitude and direction with infinite duration (forced motion). The aforementioned parameters combined with the algebraic structure of the damping matrix (being either positive semidefinite or indefinite) may have under certain conditions a tremendous effect on the Jacobian eigenvalues and then on the local stability of these autonomous systems. It was found that such systems when unloaded may exhibit periodic motions or a divergent motion, while when subjected to the above step load may experience either a degenerate Hopf bifurcation or periodic attractors due to a generic Hopf bifurcation. Conditions for the existence of purely imaginary eigenvalues leading to global asymptotic stability are fully assessed. The validity of the theoretical findings presented herein is verified via a nonlinear dynamic analysis.
Highlights
In previous studies of the 3rd author, based on 2-DOF and 3-DOF cantilevered models 1 under partial follower loading nonconservative systems, it was shown that in a small region of divergence instability, flutter dynamic instability may occur before divergence static instability, if very small damping is included 2, 3
The local dynamic stability of discrete systems under step conservative loading when small dissipative forces are included is governed by the matrix-vector differential equation 8–11 : Mq Cq Vq 0, 1.1 where the dot denotes a derivative with respect to time t; q t is an n-dimensional state vector with coordinates qi t i 1, . . . , n ; M, C and V are n × n real symmetric matrices
Matrix M associated with the total kinetic energy of the system is a function of the concentrated masses mi i 1, . . . , n, being always positive definite; matrix C the elements of which are the damping coefficients cij i, j 1, . . . , n may be positive definite, positive semidefinite as in the case of pervasive damping 12, 13, or indefinite 14–16 ; V is a generalized stiffness matrix with coefficients kij i, j 1, . . . , n whose elements Vij are linear functions of a suddenly applied external load λ with constant direction and infinite duration 17, that is, Vij Vij λ; kij
Summary
In previous studies of the 3rd author, based on 2-DOF and 3-DOF cantilevered models 1 under partial follower loading nonconservative systems , it was shown that in a small region of divergence instability, flutter dynamic instability may occur before divergence static instability , if very small damping is included 2, 3. Kounadis in two very recent publications 10, 11 has established the conditions under which the above autonomous dissipative systems under step conservative loading may exhibit dynamic bifurcational modes of instability before divergence static instability, that is, for λ < λc[1], when infinitesimal damping is included These bifurcational modes may occur through either a degenerate Hopf bifurcation leading to periodic motion around centers or a generic Hopf bifurcation leading to periodic attractors or to flutter. The question which arises is whether there are combinations of values of the abovementioned parameters mass and stiffness distribution which in connection with the algebraic structure of damping matrices may lead to dynamic bifurcational modes of instability when the system under discussion is unloaded Such local due to unforced motion dynamic instability will be sought through the set of asymptotic stability criteria of Lienard-Chipart 8, 18 which are elegant and more readily employed than the well-known Routh-Hurwitz stability criteria. The conditions of a double purely imaginary root leading to a new dynamic bifurcation, whose response is similar to that of a generic Hopf bifurcation, are properly established
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
