This paper presents vibration analysis of an autoparametric pendulum-like mechanism subjected to harmonic excitation. To improve dynamics and control motions, a new suspension composed of a semiactive magnetorheological damper and a nonlinear spring is applied. The influence of essential parameters such as the nonlinear damping or stiffness on vibration, near the main parametric resonance region, are carried out numerically and next verified experimentally in a special experimental rig. Results show that the magnetorheological damper, together with the nonlinear spring can be efficiently used to change the dynamic behaviour of the system. Furthermore, the nonlinear elements applied in the suspension of the autoparametric system allow to reduce the unstable areas and chaotic or rotating motion of the pendulum.

Dynamic mechanical systems possessing the pendulum arise in many practical application including special dynamical dampers [

A condition of the existence of an autoparametric vibrations is that, the structure has to consists at least of two constituting subsystems. The first one is a primary system that usually is excited by eternal force. The second subsystem, called as secondary, is coupled to the primary system by inertia terms. A classical example of an autoparametric system is the pendulum mounted to the oscillator, where the pendulum can both, oscillate or rotate [

In this paper we propose the concept of the use of combination of the magnetorheological damper together with the nonlinear spring applied in the autoparametric system suspension. Changing magnetorheological damping or nonlinear stiffness of the supporting spring, dangerous regions can be eliminated or moved away. This solution with MR damper gives reliable control possibilities and can help to react properly in critical situations. This kind of semiactive isolator can change dynamic behaviour to prevent undesired vibration or react properly according to varied initial conditions. The results are compared with a system with a classical viscous damper and a linear spring. The proposed idea can be used to design control strategy of an autoparametric system with an attached pendulum.

A model of the vibrating autoparametric system consists of a nonlinear oscillator (the primary system) with mass

Model of an autoparametric system with a pendulum and semiactive suspension composed of MR damper and nonlinear spring.

Determining the kinetic, potential energies, dissipation function of the system, and then applying Lagrange’s equations of the second kind we receive governing equations of motion:

Damping of the oscillator is studied in two variants, as linear viscous and nonlinear magnetorheological. Our concept of nonlinear damping is realized by application of the magnetorheological (MR) damper. The first step is to describe the nonlinear behaviour of the MR damper and to propose a proper mathematical model which characterizes its real behaviour. The characteristics are found by taking the sampled restoring force and input velocity based on the experimental data and curve fittings of a real MR damper. We propose to use a smooth function of modified Bingham’s model suggested in paper [

The considered MR model consists of a combination of viscous damping (

The Bingham model of MR damper (a) and exemplary characteristics for varied

Analytical solutions near the main parametric resonance received by the harmonic balance method for the system with typical linear viscous damping are presented in paper [

The experiment was performed on an autoparametrically two degree of freedom system presented in Figure

Photo of experimental rig (a), magnetorheological damper RD 1097-01 (b), and nonlinear oscillator springs (c, d).

A semiactive control system typically requires a small external power source for operation. Moreover, the motion of the structure can be used to develop the control forces. Therefore, semiactive control systems do not have the potential to destabilize the structural system, in contrast to active systems. Many studies have indicated that semiactive systems perform significantly better than passive devices [

In our experimental system we use magnetorheological damper

The magnetorheological damper

Such parameters allow very low damping force if the damper has to be switched off. In Figure ^{3}) and soft characteristic as no. 3 (^{3}). Six different types of linear and six nonlinear springs are used in the equipment.

Exemplary characteristics of different springs (a) and nonlinear damping force of MR versus velocity of the damper piston (b).

In this type of the systems near the parametric resonances the unstable areas may occur, and moreover, for certain parameters chaotic motion may be observed [

The bifurcation diagrams are calculated to investigate the effects of the influence of MR damping on dynamics for typical resonance range. For each value of the varied parameter, the same initial conditions are used so that the comparison could be made between different system's parameter values.

Figure

Two parameter space plot for different settings of MR damping

We propose to change the dynamic behaviour of the autoparametric system (for example transfer chaotic motion into periodic) by using MR damper which could allow for a change of the system motion online. The MR damper is a semiactive device, this means that the damping force can only be commanded by the input voltage adjusted to the MR damper. Therefore, the dynamic mechanism of the system with installed MR damper in the suspension can be updated in an online adaptive manner, according to the required conditions. In spite of the fact MR damper cannot activate positive force, the advantage is that the restoring force can be modified online without stopping the system.

Introducing MR damping we observe, that chaotic resonance tongues move towards the axis of amplitude of excitation

To have better insight into the parameters space plot, the bifurcation diagrams (crosschecks) have been done. Figures

Bifurcation diagrams

The rotation of pendulum is defined as a case when the motion amplitude exceeds

Interesting example is observed for frequency

Time history of pendulum motion for frequency

Introduction of MR damping does not change stability of the inverted pendulum for analyzed range of parameter

Another proposal to change the system dynamics is modification of nonlinear stiffness of a spring mounted in the suspension. Warminski and Kecik [

Parameter space plot of a pendulum for different values of hard stiffness

The bifurcation diagrams for varying

Bifurcation diagrams

Figure

Bifurcation diagrams

In real dynamical autoparametric structure, it is very important to keep the pendulum at a given, wanted attractor. The nonlinear systems are sensitive for initial conditions, environmental or working conditions, therefore monitoring of suitable parameters is required to improve dynamic. The existence of two or more solutions for the same parameter values in a nonlinear system indicates that the initial conditions play a critical role in determining the system's overall response. Therefore, slight disturbance may cause a jump from the oscillating pendulum to the rotation or chaotic motion.

Figure

Basins of attraction of a pendulum for

Time history of pendulum motion, for the case of chaotic swinging (attractor no. 3) obtained from numerical simulation is plotted in Figure

Time histories of a pendulum for

Time histories of a pendulum for

Figure

In Figure

Numerical and experimental time histories of a pendulum motion for

Therefore, after proper tuning of the system the response can be modified from chaotic to periodic motion and vice versa. It has been confirmed experimentally that the simple open loop technique, allows for an easy control of the system response.

This paper focuses on numerical and experimental investigations of an autoparametric system with a pendulum subjected to kinematic excitation. The dynamic response has been examined by constructing parameter plots which determinate the regular or irregular motion. The bifurcation diagram, Lyapunov exponents, time histories, and basins of attractions have been used to check nature of motion in those regions.

The results presented in the paper show that MR damping together with nonlinear spring included in the pendulum-like absorber structure can be an effective tool for reduction of dangerous unstable regions without a loss of the dynamical absorption effect. After proper tuning, the system can be maintained on a regular or a chaotic attractor. Moreover, by applying simple open loop control, it is possible to fit the structure response to the frequency of external excitation. Obtained results show that our semiactive suspension of the autoparametric system allows to freely move, both up and down, or left and right, the chaotic regions. Application of a closed loop control technique, leading to a smart dynamic absorber is a next step of our investigations.