The dynamics of the pendulum suspended on the forced Duffing oscillator

A numerical study of an application of magnetorheological (MR) damper for semi-active control is presented in this paper. The damper is mounted in the suspension of a Duffing oscillator with an attached pendulum. The MR damper with properties modelled by a hysteretic loop, is applied in order to control of the system response. Two methods for the dynamics control in the closed-loop algorithm based on the amplitude and velocity of the pendulum and the impulse on–off activation of MR damper are proposed. These concepts allow the system maintaining on a desirable attractor or, if necessary, to change a position from one attractor to another. Additionally, the detailed bifurcation analysis of the influence of MR damping on the number of periodic solutions and their stability is shown by continuation method. The influence of MR damping on the chaotic behavior is studied, as well.

/pdf/magnetorheological-damping-and-semi-active-control-of-an-20o6f4gh03.pdf

Magnetorheological damping and semi-active control of an autoparametric vibration absorber

In this paper we propose a new method to detect and classify coexisting solutions in nonlinear systems. We focus on mechanical and structural systems where we usually avoid multistability for safety and reliability. We want to be sure that in the given range of parameters and initial conditions the expected solution is the only possible or at least has dominant basin of attraction. We propose an algorithm to estimate the probability of reaching the solution in given (accessible) ranges of initial conditions and parameters. We use a modified method of basin stability (Menck et al. in Nat Phys 9(2):89–92, 2013). In our investigation we examine three different systems: a Duffing oscillator with a tuned mass absorber, a bilinear impacting oscillator and a beam with attached rotating pendula. We present the results that prove the usefulness of the proposed algorithm and highlight its strengths in comparison with classical analysis of nonlinear systems (analytical solutions, path-following, basin of attraction ect.). We show that with relatively small computational effort (comparing to classical analysis) we can predict the behaviour of the system and select the ranges in parameter’s space where the system behaves in a presumed way. The method can be used in all types of nonlinear complex systems.

/pdf/basin-stability-approach-for-quantifying-responses-of-2dwhjuc8g3.pdf

Basin stability approach for quantifying responses of multistable systems with parameters mismatch

The work focuses on autoparametric vibrations of system composed of a non-linear oscillator with an attached pendulum. A combination of semi-active damper together with a non-linear spring mounted in the suspension of system is proposed. The spring is made from a shape memory alloy (SMA) while damping is realized through a magnetorheological (MR) damper. The MR damper is used for controlling damping of the system, SMA spring is used to change system׳s stiffness. The system is solved numerically and verified experimentally on a custom made experimental rig. Specifically, non-linear resonances are investigated, and their influence on the system dynamics and absorption effect.

Dynamics and control of an active pendulum system

Synchronous motion of two vertically excited planar elastic pendula

In this work we study the frequency and dynamic response of a damped Duffing system attached to a parametrically excited pendulum vibration absorber. The multiple scales method is applied to get the autoparametric resonance conditions and the results are compared with a similar application of a pendulum absorber for a linear primary system. The approximate frequency analysis reveals that the nonlinear dynamics of the externally excited system are suppressed by the pendulum absorber and, under this condition, the primary Duffing system yields a time response almost equivalent to that obtained for a linear primary system, although the absorber frequency response is drastically modified and affected by the cubic stiffness, thus modifying the jumps defined by the fixed points. In the absorber frequency response can be appreciated a good absorption capability for certain ranges of nonlinear stiffness and the internal coupling is maintained by the existing damping between the pendulum and the primary system. Moreover, the stability of the coupled system is also affected by some extra fixed points introduced by the cubic stiffness, which is illustrated with several amplitude-force responses. Some numerical simulations of the approximate frequency responses and dynamic behavior are performed to show the steady-state and transient responses.

https://downloads.hindawi.com/journals/sv/2008/827129.pdf

Evaluation of the Autoparametric Pendulum Vibration Absorber for a Duffing System

Vibration control in mechanical systems is an important tool by means of which undesirable vibrations are suppressed or at least attenuated. This problem appears in many applications with beams, frames, mechanisms, structures, cranes, etc. There are three fundamental methodologies which are normally denoted as passive, semi-active and active vibration control. Passive vibration control relies on the addition of mass, stiffness and/or damping to the system to reduce the primary response and can be used for a narrow band of excitation frequency and stable operating conditions, but it is not recommended for variable forcing frequencies and uncertain system parameters. Semi-active vibration control deals with adaptive spring or damper characteristics, which are tuned according to the operating conditions. Active vibration control achieves better performance by adding actuators or actuated degrees of freedom to the system and controlling them according to feedback and feedforward information obtained from sensors on the system. Most of the research on mechanical vibration absorbers is devoted to minimise the system response in the presence of vibrations. Sometimes this is performed by changing the structure, adding extra degrees of freedom, considering nonlinear stiffness or damping elements, using active/smart materials or applying actuators. Active vibration control has been rapidly gaining some well-deserved popularity in the field of mechatronics because of many technological advances, enabling the analysis and real time applications of passive and active vibration absorbers for complex mechanical systems. In general, the addition of extra degrees of freedom leads to highly nonlinear and coupled dynamics. In the case of nonlinear systems the application of appropriate nonlinear control schemes is then recommended to achieve satisfactory operation. This paper considers a mechanical structure, consisting of an oscillating rigid bar coupled to a classical passive absorber, which is able to attenuate harmonic vibrations only at a nominal excitation frequency. The study is motivated by applications to several engineering systems. For a variable excitation frequency a third degree of freedom is added to the original system, including a proper controller to automatically place the passive absorber along the rigid bar, for each measured frequency. The resulting dynamical system is underactuated and strongly nonlinear, hence, nonlinear control techniques are applied. Since the output to be controlled is not controllable about the set of equilibria then, an indirect controller based on nonlinear control laws combined with optimal attenuation, is proposed. Thus, the active vibration control methodology starts from the measurement of the excitation frequency, which serves to compute the position of the passive absorber. This leads to optimal attenuation and then, the desired behaviour is asymptotically achieved by a nonlinear controller. Two control strategies are analysed, 1) a partial feedback linearisation technique and 2) an output regulation technique. They both use feedback and feedforward information of the overall system to get a robust performance.

/pdf/active-vibration-absorber-for-an-oscillating-structure-using-23w017iv31.pdf

Gerardo Silva-Navarro

Papers

Evaluation of the Autoparametric Pendulum Vibration Absorber for a Duffing System

Active Vibration Absorber for an Oscillating Structure Using Nonlinear Control