Nonlinear dynamics in multi-physics systems

The nonlinear dynamics of an electromechanical system is considered. The system consists of a linear oscillator nonlinearly coupled through an electromagnet to a linear electric circuit.  As it is known, electromechanical systems are characterized by the interaction among inertial, electric, and magnetic circuits, and they are nowadays widely used for devices that monitor and control machine and structural systems. Depending on the specific engineering application, electromechanical devices range from macro to micro and nano scales. 

When the oscillator is forced via harmonic voltage excitation of the electric circuit the long term mechanical dynamics evolves in a purely slow timescale, as shown in [1]. In this work, a slow invariant manifold is analytically derived and numerically validated by means of simulations on both full- and reduced-order systems. The slow time scale forced frequency–amplitude response of the full order system is computed by means of Poincaré mappings. The latter were analyzed in depth by using approximate slow invariant manifolds and various order reduced slow systems.

Three interesting non-linear phenomena were observed. The non-linear resonance related to the current quadratic nonlinearity, which imposes a natural linear resonance at half the frequency of the linear oscillator. The pull-in phenomenon, denoted by a jump after which the mass of the linear mechanical oscillator is pulled at a large distance and it is forced to oscillate about it. A peculiar irregular dynamics for high excitation amplitude involving a number of bifurcations characterized by dramatic qualitative changes of both the mechanical and electrical responses.

In [2], several multistable regions, with low amplitude solutions coexisting with oscillations of very high amplitude, able to induce the so-called pull-in phenomenon, were observed in the same system considered in [1]. Confined regions of chaos, also persisting in weakly nonlinear coupling regime, were also described. In particular, an isle of disconnected solutions was identified below the main frequency-response curve, displaying a rich bifurcation behavior responsible for the arise of several high amplitude period-1 responses alternated to quasiperiodic and chaotic regions. The concurrent realization of behavior charts in the most significant parameter planes pointed out the high sensitivity of the system response to slight variations of the coupling strength, capable to merge the isle with the main resonance branch, and in general to drastically modify the dynamical qualitative behavior. From an operational viewpoint, the obtained results furnished useful hints to properly design the system in order to avoid, or alternatively induce, the variety of dynamical responses encountered.

The weakly nonlinear dynamics of the linear oscillator nonlinearly coupled with a linear electric circuit is also analytically studied in conditions of primary electrical resonance in [3]. Willing to investigate the stability of the frequency-amplitude responses, the multiple scale analysis up to the third order is applied to the system of 2 degrees-of-freedom ODEs to obtain Cartesian Amplitude Modulation Equations (AMEs) enabling to reconstruct the particular mechanical and electrical solutions at each scaling order. The validation, carried out by comparison with the numerical integration, confirms the accuracy of the asymptotic approach in qualitatively and quantitatively grasping the mechanical and electrical responses of the coupled system, and their stability regions. Moreover, the analytical expression of the critical stability threshold allows to verify the influence of the main parameters on the system dynamical response; stability charts are also derived to properly tune the system parameters in order to predict and control the onset of instability. The post-critical behavior is also investigated by enriching the asymptotic approach with two further orders. The ensuing fifth order AMEs allow to catch rather accurately the numerical amplitudes of the quasiperiodic solutions originating from the torus bifurcations of the ODEs system, as well as to improve further the third order results related to the description of the frequency-response curves.

[1] Georgiou I.T., Romeo F., Multi-Physics Dynamics of a Mechanical Oscillator Coupled to an Electro-Magnetic Circuit, Int. Journal of Nonlinear Mechanics, doi: 10.1016/j.ijnonlinmec.2014.08.007, 2014.

[2] Settimi V., Romeo F., Dynamic regimes of a nonlinearly coupled electromechanical system, Int. Journal of Non-Linear Mechanics, 103, pp 68–81, 2018.

[3] Settimi V., Romeo F., High order asymptotic dynamics of a nonlinearly coupled electromechanical system, Journal of Sound and Vibrations, 432, pp 470–483, 2018.