Global dynamics of an autoparametric spring–mass–pendulum system

Abstract

In this paper, a study of the global dynamics of an autoparametric four degree-of-freedom (DOF) spring-mass-pendulum system with a rigid body mode is presented. Following a modal decoupling procedure, typical approximate periodic solutions are obtained for the autoparametrically coupled modes in 1:2 internal resonance. A novel technique based on forward-time solutions for finite-time Lyapunov exponent is used to establish global convergence and domains of attraction of different solutions. The results are compared to numerically constructed domains of attraction in the plane of initial position and initial velocity for the pendulum. Simulations are also provided for a few interesting cases of interest near critical values of parameters. Results also shed some light on the role played by other modes present in a multi-DOF system in shaping the overall system response.

Control of Unsteady Separation: A Dynamical Systems Approach

“…We have also explored the detection of Lagrangian Coherent Structures or LCS (developed for fluid flows under prior AFOSR award) fom problems outside the realm of fluid mechanics. We used our LCS techniques th evolutionhe phase space geometry of nonlinear vibration absorbers [4], and find a condition for cell-death in a model of biological signaling networks [2] (see Fig. 6).…”

Harmonic motions of a weakly forced autoparametric vibrating system

“…Equation (1.1) models an auto-parametric absorber by using a pendulum as an auxiliary system. Since the effect of vibration attenuation does not depend on external disturbances and the control effect is obviously better than passive control, this model has been widely applied in engineering, see [11,12,13].…”

“…In [15], the method of averaging is used to obtain first-order approximations of system to the response of the system, where a complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. System (1.1) also has been extended to consider for the nonlinear spring [11,16,17], the 3 degree-of-freedom (DOF) linear mass-spring-damper systems [13] and the single-degree-of-freedom (SDOF) systems with two pendulums [12]. In these cases, autoparametric vibrating system (1.1) with 0 0  F was studied for resonant excitations, where the stability of system is the focus of the research, see [18] Therefore, it yields that system (1.1) has a state of dynamic balance, namely, one 2π-periodic solution (a harmonic motion)…”

An Electromechanical Pendulum Robot Arm in Action: Dynamics and Control

“…Pendulum motion-driven systems have been intensively studied recently by both industries and research institutes because of their applications in different fields [1][2][3][4][5][6][7][8][9][10]. Some of these studies concern the analysis of the dynamical states and the development of control strategies to stabilize the dynamical state to a prescribed state.…”