On the non-linear dynamics of a forced plate with boundary conditions correction in subsonic flow

Abstract

• A complete modeling of a periodically driven plate with boundary conditions correction in subsonic ow is developed. • A transformation upon the Legendre polynomials is applied for converting the boundary conditions with correction. • The systems with or without correction show the similar global bifurcation structure and scaling property. • The correction can cause reversals which increase the diversity and complexity of the local bifurcation structure. When a plate is subjected to long-term vibration the original boundary conditions are no longer ideal and necessary correction should be considered during its dynamical modeling. This paper aims at the modeling of the boundary conditions correction and its influence on the nonlinear dynamics of a harmonic excited plate in subsonic flow . The plate boundary conditions with correction are non-autonomous and converted into autonomous by using a coordinate transformation upon the orthogonal Legendre polynomials. The fluid force is separated into the reactive and resistive parts. The reactive fluid force, due to plate motion, is derived from the bound and wake vorticity in Glauert’s expansion; the resistive force , which is independent on plate motion, is evaluated in drag coefficient . The governing nonlinear partial differential equation is discretized in space and time domains by using the Galerkin method. Results show that the present fluid model is in good agreement with other theories archived and is reliable. The unforced system loses its stability by divergence following a pitchfork-like bifurcation featured by the appearance of new bifurcated stable equilibrium points due to nonlinearity after instability. The boundary conditions correction can stabilize the system and does not change the bifurcation structure and scaling property. The symmetry-breaking/restoring bifurcations play an important role in the change of different period-1 motions, and the nature of such bifurcations is associated with the collision or division of attractors . Subharmonic motions and chaos appear alternatively, and the transition between chaos and periodic motions is generally in accompany with period-doubling bifurcations, explosion and condensation of attractors.