Tunable electromagnetic resonant shunt using pulse-width modulation

https://doi.org/10.1016/j.jsv.2021.116018Get rights and content

Abstract

This article proposes a novel mean for tuning the natural frequency of an electromagnetic resonant shunt, using a pulse-width modulation (PWM) circuit. It is used to modulate the value of the capacitance of the shunt, and the electrical frequency is shown to be proportional to the command parameter of the PWM, the duty cycle. An easy and efficient strategy to tune the resonant shunt in real time is then proposed, thus obtaining a low powered and always stable vibration control device. The article proposes the theory of PWM, giving a robust method to predict the dynamics of the system. Then, an accurate multi-mode theoretical model of the tunable resonant shunt coupled to an elastic structure is proposed and experimentally validated on an elastic multi-mode structure, in the case of two different control strategies. The first one is a standard resonant shunt with both the electrical frequency and damping optimized to reduce a given resonance peak. The second one is based on a resonant shunt with the electrical damping as low as possible, which creates an antiresonance and a “notch” type mechanical response at the driving frequency. Both strategies are experimentally validated with real time variation and adaptation of the electrical frequency, obtaining an efficient vibration control device, able to reduce by a factor 40 the vibration level.

Introduction

The search for increasing performance, acoustic comfort and fatigue life of mechanical systems and structures often necessitates the design of vibration reduction systems. Apart from including special passive damping materials into the structure, such as viscoelastic polymers or foams [1], [2], [3], the use of mechanical vibration absorbers such as dynamic vibration absorbers (DVAs) or Lanchester dampers [4], [5] is common. Equivalent damping properties can be obtained using electromechanical shunts, in which a transducer converts the vibratory energy of the host structure into electrical energy in a dedicated electronic circuit, designed to dissipate it and/or to counteract the structure’s vibrations. Depending on the physics of the transducer, piezoelectric or electromagnetic shunts have been proposed in the pioneering works [6], [7], and have been addressed in a huge number of contribution since (see [8], [9], [10] and reference therein).

Using a DVA or a resonant shunt (its electromechanical equivalent), two practical issues are often addressed, depending on the type of the excitation signal. (i) If it has a broadband frequency content, one is often interested in attenuating as much as possible one or several resonances of the host structure. Depending on the figure of merit of the absorber (the mass ratio for the DVA; the electromechanical coupling factor for the shunts) and the damping ratio of the considered mechanical mode, substantial vibration reduction can be obtained on a single resonance [5], [9], [11], [12]. Several resonances can also be attenuated with more complex circuits [13]. (i) For a tonal excitation signal, DVAs and resonant shunts, damped as less as possible, can be efficiently used to almost cancel the vibration energy by creating an antiresonance at the absorber natural frequency [9]. In those two cases of broadband or tonal excitation signal, the fine tuning of the absorber natural frequency, either on a particular host structure resonance or on the excitation frequency, is mandatory. This has stimulated the interest in designing adaptive absorbers, which can self-tune by adaptively changing some of their parameters. This article addresses a novel strategy for tuning the natural frequency of an electromagnetic resonant shunt, using an electronic chopper with pulse-width modulation (PWM). This technology is well known and used in power electronics technologies to regulate the power flowing in a circuit, thanks to electronic components switched at a high frequency [14], [15]. In this paper, this principle is adapted to modify the characteristics of a resonant circuit.

In designing an adaptive shunt, one has to deal with two issues. First, one has to provide a technology able to change the shunt natural frequency to a desired value. Next, a suitable shunt tuning control strategy must be formulated. The pioneering works in this field were proposed for piezoelectric transduction in [16], [17]. The former proposed to change the electrical natural frequency using a motorized variable resistor included in the synthetic inductor of the shunt; the control law was based on the minimization of the root mean square (RMS) value of the vibration signal. In the latter, the authors use a simple capacitive shunt constituted of a ladder of discrete capacitor controlled by switches, to adapt the capacitance value and change the equivalent stiffness of a mechanical absorber. An additional step was proposed in [18], with a synthetic impedance based on a real-time digital signal processor (DSP), which can be easily tuned by changing the parameters of the block diagram in the DSP. The control law was based on the RMS value of the vibration signal. An important update was proposed in [19], with an analog tunable inductance including a voltage controlled resistor and a new control law based on the phase of the shunt impedance. This control law was adapted to multi-mode damping in [20], with a DSP-based shunt impedance. Those techniques, all based on piezoelectric transduction, were applied to an electromagnetic shunt for the first time in [21]. Since then, we can notice three interesting updates with analog electronic circuits, in the case of piezoelectric shunts [22], [23], [24] (with a fully analog circuit including junction gate field-effect transistors in the first, a photoresistive opto-isolator with a phase locked loop processor in the second and a digital potentiometer in the third). The so-called sweeping and switching technique, that uses an adaptive piezoelectric shunt impedance, was also proposed for broadband vibration damping (see [25] and reference therein). Recently, programmable digital shunts have also been proposed [26], [27]. For electromagnetic transduction, some tunable DVAs, with a variable capacitance to change their apparent stiffness, where proposed in [28], [29]. However, it seems that no contribution on adaptive resonant electromagnetic shunts has been proposed since the pioneering work [21].

It is worth to notice the so-called synchronized switch damping (SSD) techniques which are also intrinsically adaptive. They consist in switching the electromechanical transducer on two distinct shunt impedances, synchronously with the oscillations of the host structure. This idea was initially proposed in [30], [31] for piezoelectric transduction and developed in numerous contributions since then (see [32] for a recent review). SSD techniques in the case of electromagnetic transduction were first proposed in [33]. Those SSD techniques are very interesting since they appear adaptive, efficient, intrinsically stable and require low power. Their effect can be viewed as a resonance peak reduction, similarly to traditional resistive or resonant shunts, with higher performance for one degree of freedom host structure [34]. However, they cannot produce a “notch” type response, as a lightly damped resonant shunt does by creating an antiresonance. They are then not as efficient as an adaptive resonant shunt in the case of a tonal excitation filtering.

Recently, a new shunt technique was proposed, by considering that the shunt effect of the host structure could be chosen equivalent to a force in phase with the velocity of the structure, to be equivalent to a damping force [35], [36], [37]. To create this, a constant (DC) voltage source is connected to the piezoelectric transducer through a pulse width modulation (PWM) circuit, with the duty cycle modulated synchronously with the structure oscillations. It is shown that increasing the DC voltage can lead to a complete resonance peak cancellation, with possible unstabilities if the DC voltage is too large. To our knowledge, these contributions are the only one using PWM techniques for structural vibration reduction. Though using PWM circuits, our present contribution is based on a different point of view, since it is used to modulate the value of a given component of the shunt and not to fully synthetize a suitable shunt voltage signal.

A different approach, more related to active control, consists in using a feedback of a measurement of either displacement, velocity or acceleration, which is amplified to supply a magnetic or a piezoelectric transducer. It enables to change the apparent stiffness/inertia of the structure and thus its natural frequencies. Only a few recent references are given here. In [38], the principle is applied to a single degree of freedom oscillator with an acceleration feedback, to provide a tunable inertia effect to change the resonance frequency. In [39], a similar approach, with displacement feedback, is applied to a multi degree of freedom system (a clamped free piezoelectric beam), with unconditional stability. In [40], the vibration damping of a plate is addressed by coupling it to a mechanical resonator enhanced with an inerter (a flywheel) and combined with an electromagnetic actuator and a velocity feedback, with stability issues due to a non colocalized control. In all these approaches, and contrary to the shunt and SSD techniques discussed above and addressed in the present paper, a power is injected in the system, which can lead to stability issues.

In the context of resonant shunts, the present article proposes an original technique to tune the electrical natural frequency of the shunt by using a PWM circuit. It is here realized with an electromagnetic transduction, but it could equally be realized with a piezoelectric transduction. Indeed, we use the PWM circuit to modulate the value of the capacitance of the shunt. It is shown that the electrical natural frequency is directly proportional to the duty cycle of the PWM, thus enabling an efficient and easy mean of tuning the resonant shunt in real time. As it will be shown, as compared to other works of the literature, our strategy has the advantage of being semi-passive (contrary to active control strategies, no energy is injected in the circuit so that the system dynamics is unconditionally stable. Only a small amount of energy is here necessary to supply the PWM circuit switches and their command). It is thus completely equivalent to a passive RC resonant shunt for which the value of C can be changed in real time. It shares similarities with the SSD family of techniques because we include switches in the shunt circuit. However, here, the switching (PWM) frequency is chosen much higher than the dynamics of the mechanical system, whereas in the SSD techniques, the switching is synchronized with the oscillations of the system and thus in the same frequency range. Finally, while the SSD techniques are nonlinear (the switch commutation creates non-smooth oscillations and shocks), our technique is fully linear because of the separation of the frequency bands (the dynamics of the system and the PWM frequency).

The outline of the papers follows. Section 2 gives the basics of the PWM technique, showing that the low frequency dynamics of the system can be obtained by an average of its fast dynamics, at the PWM frequency. Section 3 presents a multi-mode model of an elastic structure coupled to the tunable resonant shunt. Then, Section 4 shows an experimental validation of the PWM electronic circuits. Finally, Section 5 presents an experimental validation of the tunable resonant shunt on a mechanical multi-mode elastic structure.

Section snippets

Pulse-Width Modulation theory

The basic component used in this article to obtain a tunable circuit is a chopper, that is placed between two electrical circuits Z1 and Z2. As shown in Fig. 1, it is composed of two switches A and B that are controlled by complementary periodic rectangular pulse signals to obtain a pulse-width modulation (PWM). If the PWM frequency fPWM=1/T (with T the period of the rectangular pulse signals) is chosen much larger with respect to the characteristic frequency content of the circuits Z1 and Z2,

Governing equations

In this section, we address the coupling of the tunable resonant electrical circuit of Fig. 3 to a mechanical structure, through an electromagnetic transducer, as depicted on Fig. 4. We gather the displacements of all the points of the structure in a vector u(t), which is expanded on a subset of NN* modes (ωi,Φi), i=1,N, of the natural mode basis of the structure with the electromagnetic transducer in open-circuit:u(t)=i=1NΦixi(t),where xi(t) is the ith modal coordinate. One obtains the

Practical implementation of the chopper

The schematic of electronic circuit and its practical implementation build for the experiments are depicted in Fig. 6. It essentially consists in an inverter leg realized using two MOSFETs (International Rectifier IRFP 4468Pbf) which were selected considering their low drain to source resistance (typically 2 mΩ) when switched on, their high switching frequency (up to 100 kHz) and their maximum voltage rating (100 V). Since the current changes of sign during operation, the switches must be

Experimental setup

An experimental mechanical flexible structure, used previously in [9], shown in Fig. 13, was designed to test the tunable resonant shunt. It is composed of three identical plates, the bottom one being clamped to a frame, sustained by four identical beams, all built in aluminium. The geometrical characteristics were chosen to obtain two bending modes as the first two modes (with f1=30.8Hz and f2=85.4Hz their eigenfrequencies, see Fig. 13(c)), with the first twisting mode shifted to higher

Conclusion

In this article, we presented a novel mean of realizing a tunable electromagnetic shunt absorber using a PWM circuit to continuously change the apparent capacitance – and consequently the electrical frequency – of the shunt. This technique has several advantages. First, it does not changes the intrinsic stability of the resonant shunt: it is just a mean of changing the apparent value of the shunt capacitance, without injecting external energy into the shunt. It also requires very low additional

CRediT authorship contribution statement

Michel Auleley: Formal analysis, Investigation, Validation, Writing - original draft. Christophe Giraud-Audine: Conceptualization, Formal analysis, Investigation, Validation. Hervé Mahé: Supervision, Funding acquisition. Olivier Thomas: Conceptualization, Formal analysis, Investigation, Validation, Writing - original draft, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (43)

  • M. Lallart et al.

    Self-powered circuit for broadband, multimodal piezoelectric vibration control

    Sens. Actuators A

    (2008)
  • S. Krenk

    Frequency analysis of the tuned mass damper

    J. Appl. Mech.

    (2005)
  • M. Vakilinejad et al.

    A comparison of robustness and performance of linear and nonlinear Lanchester dampers

    Nonlinear Dyn.

    (2020)
  • S. Behrens et al.

    Electromagnetic shunt damping

    Proceedings of the 2003 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM 2003)

    (2003)
  • J.A.B. Gripp et al.

    Vibration and noise control using shunted piezoelectric transducers: a review

    Mech. Syst. Signal Process.

    (2018)
  • M. Auleley et al.

    Enhancement of a dynamic vibration absorber by means of an electromagnetic shunt

    J. Intell. Mater. Syst. Struct.

    (2020)
  • M. Berardengo et al.

    Piezoelectric resonant shunt enhancement by negative capacitances: optimisation, performance and resonance cancellation

    J. Intell. Mater. Syst.Struct.

    (2019)
  • O. Thomas et al.

    Performance of piezoelectric shunts for vibration reduction

    Smart Mater. Struct.

    (2012)
  • R. Darleux et al.

    Broadband vibration damping of nonperiodic plates by piezoelectric coupling to their electrical analogues

    Smart Mater. Struct.

    (2020)
  • R.D. Middlebrook et al.

    A general unified approach to modelling switching-converter power stages

    1976 IEEE Power Electronics Specialists Conference

    (1976)
  • M.H. Rashid

    Power Electronics: Devices, Circuits, and Applications

    (2014)
  • Cited by (11)

    • Automated electromagnetic generator with self-adaptive structure by coil switching

      2022, Applied Energy
      Citation Excerpt :

      Up to date, only vibration EMGs with passive smart enhancing of power generation have been developed [32–34]. The widely used methodology to provide adaptability is based on impedance tuning for maximum power point tracking [33–37]. Another much-explored methodology is to insert additional components to the passive generator structures mainly for resonance frequency self-tuning [38–41].

    • On the dynamic stability and efficiency of centrifugal pendulum vibration absorbers with rotating pendulums

      2022, Journal of Sound and Vibration
      Citation Excerpt :

      Note that in the case of a real automotive driveline (a simple model of which consists in successive rotors linked through torsional springs [61]), the CPVA should be placed as close as possible from the source of excitation (i.e. the engine). Doing so, the antiresonance generated by the pendulums exists on every driveline components located after the CPVA [62], which allows to isolate the whole driveline from the torque fluctuations. The response of the first three harmonics of the rotor’s acceleration are obtained by substituting the pendulums’ solutions in Eq. (13a).

    • Hybrid electromagnetic shunt damper with Coulomb friction and negative impedance converter

      2022, International Journal of Mechanical Sciences
      Citation Excerpt :

      Both isolators showed good performance for nonlinear vibration isolation. Apart from structural parameter adjustment, Auleley et al. [19] showed that the natural frequency of electromagnetic shunt could also be varied by introducing a digital pulse width modulator to the EMSD. With careful design of shunt circuits and structural parameters, EMSDs have been widely applied in various ways to cope with very diversified applications, including micro-vibration mitigation on space crafts [12], locker at bridge cables [35,36], energy harvesting on railroad transportation [37], dual-function of energy harvesting and vibration control on automotive suspensions [2,38], semi-active control on seat suspensions [39], frame structures vibration control [40], and energy capturing from human motion [41,42], etc.

    View all citing articles on Scopus
    View full text