Active damping controllers with proof-mass electrodynamic actuators : design, analysis and experiments

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POLITECNICO DI MILANO

Corso di Laurea Magistrale in Ingegneria Aeronautica Scuola di Ingegneria Industriale e dell’Informazione

Active damping controllers with proof-mass

electrodynamic actuators: design, analysis and

experiments

Politecnico di Milano

Relatore: Prof. Lorenzo Dozio

Tesi di Laurea di: Davide di Girolamo Matr. N. 853938

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Abstract

This thesis work is a preliminary study for the development of a universal active device to be attached on demand to flat lightweight surfaces for low-frequency vibration and noise reduction. The device is based on a commercially available vibration speaker used as a proof-mass actuator. In this early prototype phase, the speaker is driven by a low-cost audio amplifier and the device is tested on an aluminum thin panel arranged in the laboratory. The feedback signal is provided by a small lightweight accelerometer and the controller is digitally implemented on a standard desktop computer running a real-time software. After deriving and implementing suitable modeling and simulation tools of the dynamics of the system under study, an experimental characterization of each component is carried out. In particular, a procedure is devised for the identification of the electro-mechanical parameters of the vibration speaker, which were not available from the manufacturer. The active unit applies a skyhook controller based on the direct feedback of the co-located transverse velocity of the vibrating surface. The active damping control is first evaluated on a simplified single-degree-of-freedom model of the plate to get insight into its physical behavior and stability properties. Then, it is numerically simulated on a full plate model and experimentally tested on the laboratory prototype plate. It is found that the skyhook controller can damp some low-frequency modes of the structure. However, the feedback gain must be limited due to the destabilizing coupling between the dynamics of the proof-mass actuator and the dynamics of the plate. In order to guarantee good stability properties, the actuator resonance should have a low natural frequency and it should be well damped. Two strategies are studied and implemented with the aim of improving the stability margins of the control system. The first approach is based on introducing a compensation filter in the loop such that it modifies in an appropriate manner the dynamics of the actuator and makes the system less prone to control spillover. The second strategy follows a similar approach by implementing a local feedback loop on the inertial device. Both methods are first studied on the dynamic models of the system and then tested on the prototype panel. Closed-loop performances are compared and some conclusions on their advantages and limitations are drawn.

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Sommario

Questo lavoro di tesi rappresenta uno studio preliminare per lo sviluppo di un dispos-itivo attivo montabile on-demand e posizionabile arbitrariamente su un qualsiasi pannello sottile per la riduzione delle sue vibrazioni e del rumore emesso.

Sempre più spesso l’inquinamento acustico inficia la qualità della nostra vita. Ad esempio, le finestre, o più generalmente i pannelli sottili, sono il mezzo con la quale il rumore esterno si propaga all’interno degli edifici. Per salvaguardare il benessere della persona sia in ambienti domestici che in ambienti lavorativi, recentemente la ricerca si è focalizzata sullo studio di soluzioni tecnologicamente all’avanguardia che garantiscano un elevato isolamento acustico. La soluzione classica consiste nell’applicazione di tratta-menti passivi, quali l’aggiunta di materiale smorzante su una vasta area del pannello, che però aumentano molto l’ingombro e il peso di quest’ultimo. In più i trattamenti passivi sono efficaci sono alle alte frequenze. Questo ha portato la ricerca a dar sempre maggior importanza ai metodi di controllo attivo per la riduzione delle vibrazioni e del rumore. Questo aspetto, già in fase di sviluppo nell’automotive e nel campo aeronautico, rimane però spesso sottovalutato in un mercato più globale. Questo lavoro si va ad inserire pro-prio in questo ambito nel quale l’aspetto economico, la compattezza e la flessibilità del dispositivo diventano imprescindibili.

Entrando nel dettaglio di questo lavoro, si sviluppa un’analisi teorica e sperimentale di un controllore attivo delle vibrazioni composto da un attuatore inerziale asservito da op-portuni compensatori tali da massimizzare la sua performance. Altro obiettivo principale di questa tesi è verificare la possibilità di utilizzare un “vibration speaker” come attuatore inerziale. Un vibration speaker è un dispositivo in grado di trasformare una superficie vibrante in un altoparlante. Questo accade poiché lo speaker trasmette la vibrazione al pannello su cui è montato che, vibrando, sposta le molecole d’aria attorno a se generando cos`ı un campo acustico. Sfruttando questo semplice principio fisico, durante questo lavoro è quindi stato possibile utilizzare un semplice vibration speaker come unità di controllo tale da creare un’interferenza distruttiva delle vibrazioni indotte da una sorgente di ru-more esterno casuale. La scelta degli attuatori inerziali è stata guidata da due ragioni fondamentali. La prima è legata al fatto che, confrontati con delle patch piezo-elettriche, riescono a fornire una forza normale al pannello e quindi ad agire in modo più efficace e diretto sulla vibrazione flessionale di quest’ultimo. La seconda ragione è che un attua-tore inerziale è attaccabile e rimovibile dal pannello a piacere e questo, da un punto di vista commerciale, offre la possibilità di un posizionamento del dispositivo solo quando necessario. In più posizionando un sensore in corrispondenza dell’attuatore è possibile realizzare un’unità di controllo collocata e compatta.

Pur essendo cos`ı versatili e reperibili a basso costo, gli attuatori elettro-meccanici pre-sentano delle importanti limitazioni. Accoppiando quest’ultimi con una struttura, essi introducono un ritardo di fase di 180o _{che, ad anello chiuso, limita il margine di}

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aumentare il guadagno in retroazione, e quindi incrementare lo smorzamento del pan-nello. Sebbene sia possibile aumentare il margine di stabilit`a mantenedo il contenuto armonico dell’attuatore e del sistema ben separati, questo presenta due limitazioni. La prima risiede nel fatto che risulta impossibile diminuire notevolmente la frequenza natu-rale dell’attuatore inerziale senza ricadere in problematiche di realizzazione causate dalla deflessione statica della massa sismica dovuta alla gravit`a, e la seconda legata al fatto che la realizzazione del nostro dispositivo prescinda dalla conoscenza del contenuto armonico del pannello.

Durante questo lavoro si è inizialmente modellizzata la dinamica dell’unità di con-trollo, dell’amplificatore (necessario al funzionamento del vibration speaker) e di un pan-nello sottile e omogeneo. Per la modellizzazione di quest’ultimo si è utilizzato un software (S-GUF) recentemente sviluppato da una collaborazione tra Politcenico di Milano e Uni-versité de Nanterre Paris X. Dopodiché è stata sviluppata una parte matematica nella quale si sono analizzati i vari accoppiamenti tra i diversi sottosistemi presentati. Dopo aver dato largo spazio alla parte di modellizzazione, è stata eseguita una caratterizzazione sperimentale dell’attuatore, dell’amplificatore e del pannello. Tutto ciò è stato utile ai fini di una validazione e identificazione dei parametri definiti nella modellizzazione matem-atica precedentemente eseguita. Si sono, quindi, analizzate e numericamente valutate le limitazioni già ampiamente descritte per il controllo diretto in velocià. Sia dalla simu-lazione che dalla prova sperimentale si è dimostrato come un controllo attivo in velocità aumenti lo smorzamento modale. Si conclude che maggiore è il guadagno d’anello e mag-giore è la riduzione dei picchi di risonanza confermando quanto già ci si aspettava dalla teoria. Inoltre, in questa prima parte, si è analizzato l’effetto che la dinamica elettrica, dello pseudo-integratore e del amplificatore hanno sul sistema ad anello chiuso. Successi-vamente si sono progettati analizzati e testati sperimentalmente due differenti metodi per aumentare il margine di stabilità del sistema. Il primo metodo analizzato consiste in un compensatore a monte dell’attuatore utile a modificarne appropriatamente la dinamica. Dopo aver quindi mostrato e validato gli effetti che i differenti parametri del filtro intro-ducono, si sono testati due differenti compensatori. Dai risultati risulta evidente come il margine di stabilità sia decisamente aumentato rispetto al caso di semplice retroazione in velocità. La principale limitazione di questo metodo è il design stesso del compensatore. Infatti i parametri del filtro vengono settati in funzione della dinamica dell’attuatore iso-lato che, quando viene accoppiato alla struttura, subisce il cosiddetto shift in frequenza. Quindi, senza la conoscenza del contenuto armonico del pannello risulta complicato pro-gettare un compensatore valido per ogni tipo di struttura sottile. Una via alternativa che cerca di oltrepassare queste limitazioni è l’utilizzo di Local feedback loop. L’obiettivo principale di questo metodo è quello di creare un loop locale sull’attuatore in modo tale da aumentare lo smorzamento attivo di quest’ultimo. Aumentare lo smorzamento attivo dell’attuatore si traduce in un aumento del margine di stabilità del sistema ad anello chiuso. Dopo aver simulato e riprodotto sperimentalmente questa strategia di controllo si può concludere che si riesce a raggiungere prestazioni molto simili a quelle del compen-satore evitando di dover settare i parametri relativi ad esso. Tuttavia questa soluzione richiede un sensore per il controllo locale e quindi un maggior costo di realizzazione.

Per entrambi i metodi di controllo appena descritti si deve evidenziare un effetto fon-damentale. All’aumentare dello smorzamento attivo, l’azione dell’unità di controllo risulta talmente forte da far si che la vibrazione del pannello in quel punto sia praticamente an-nullata. Questo implica che ad alti guadagni la dinamica del sistema risponda come una nuova dinamica vincolata anche nel punto dell’unità di controllo. Questo effetto, perciò, fa si che aumentare il guadagno al di sopra di un certo valore limite non porti a nessun

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vantaggio vibro-acustico aggiuntivo. Da qui è previsto lo sviluppo di un lavoro futuro nel quale venga presa in considerazione la posizione ottimale del dispositivo tale da garan-tire una minimizzazione dell’energia cinetica totale del pannello. Inoltre lo sviluppo di un tool che guidi l’utente all’auto-posizionamento del dispositivo porterebbe a un grandissimo passo in avanti verso la commercializzazione del dispositivo stesso. Nella stessa direzione, è possibile l’estensione del dispositivo a più unità di controllo o persino a più dispositivi con una configurazione prefissata (ragno, stella, ecc.) in modo tale da garantire un mag-gior vantaggio vibro-acustico evitando l’effetto precedentemente descritto. Un tool già in fase di sviluppo è il self-tuning, utile all’auto-settaggio del valore massimo di guadagno che garantisca la stabilità del sistema di controllo con un certo livello di robustezza.

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Acknowledgements

In primis ringrazio il mio professore Lorenzo Dozio per il costante sostegno fornitomi e per l’aiuto datomi durante lo sviluppo di tutto questo lavoro. Ringrazio, inoltre, la mia famiglia per avermi dato la possibilità di realizzare il mio sogno di laurearmi in Ingegneria Aeronautica. Uno speciale ringraziamento va alla mia nonna Anna che, pur essendo venuta a mancare, ha sempre promesso e desiderato essere al mio fianco durante tutto il mio percorso formativo. Uno speciale e sincero ringraziamento va alla mia ragazza Ilaria che mi ha sopportato e supportato durante tutti i momenti più difficili di questo percorso e durante lo svolgimento di questo lavoro. Infine vorrei ringraziare i miei più cari amici e colleghi Marco, Simone e Vanessa per aver condiviso con me questi anni.

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List of Figures

1.1 Vibration speaker . . . 2

1.2 Typical bode diagram (magnitude and phase) of the proof-mass acceleration response of a voltage-driven inertial actuator. . . 3

1.3 Fully centralized and fully decentralized active vibration control systems on a flat surface. . . 5

1.4 Active vibroacoustic control of a panel with an array of decentralised feed-back controllers driving secondary force actuators from collocated velocity sensors. . . 5

2.1 Schematic of a current-driven electromagnetic inertial actuator attached to the ground. . . 9

2.2 Lumped parameter model for the study of the blocked force response. . . . 10

2.3 Bode diagram (magnitude and phase) of the blocked force response of the current-driven inertial actuator in Table 2.1. . . 11

2.4 Bode diagram (magnitude and phase) of the blocked force response of the current-driven inertial actuator in Table 2.2. . . 12

2.5 Pole-zero map of the inertial actuator in Table 2.2. . . 12

2.6 Schematic of a voltage-driven electromagnetic inertial actuator attached to the ground. . . 13

2.7 Bode diagram (magnitude and phase) of the blocked force response of the voltage-driven inertial actuator in Table 2.1. . . 14

2.8 Bode diagram (magnitude and phase) of the blocked force response of the voltage-driven inertial actuator in Table 2.2. . . 14

2.9 Comparison between the blocked force response of the current-driven and voltage-driven inertial actuator in Table 2.1. . . 15

2.10 Comparison between the blocked force response of the current-driven and voltage-driven inertial actuator in Table 2.2. . . 15

2.11 Comparison of the pole-zero map of the current-driven (black) and voltage-driven (red) inertial actuator in Table 2.1. . . 16

2.12 Bode diagram (magnitude and phase) of the full (black) and residualized (red) blocked force response for the voltage-driven inertial actuator in Ta-ble 2.1. . . 17

2.13 Lumped parameter model for the study of the base impedance. . . 18

2.14 Base impedance of the inertial actuator in Table 2.1. . . 19

2.15 Base impedance of the inertial actuator in Table 2.2. . . 20

2.16 Typical frequency response (magnitude) of an audio amplifier. . . 21

2.17 Bode diagram (magnitude and phase) of a typical amplifier. . . 21

2.18 S-GUF: geometric description. . . 22

2.19 Kinematic description in sublaminate k: equivalent single-layer (ESL) and layerwise (LW) description. . . 23

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2.20 Schematic of a single-dof model of a vibrating structure equipped with a voltage-driven inertial actuator. . . 30 2.21 Schematic model of the force transmitted between the inertial actuator and

the plate. . . 36 2.22 Schematic of a voltage-driven inertial actuator attached to the plate. . . 38 2.23 Schematic of a voltage-driven inertial actuator and a co-located

accelerom-eter attached to the plate. . . 40 2.24 Bode plot of the sensor/actuator open-loop frequency response function. . . 47 2.25 Control point velocity FRF without and with control unit mounted at the

center of the panel described in the table 2.6 . . . 47 3.1 DAYTON DAEX25 electrodynamic exciters: manufacturer specifications. . 49 3.2 The DAYTON DAEX25VT-4 exciter fixed to the ground with an

accelerom-eter on top to measure the acceleration of the proof mass. . . 50 3.3 Experimental setup used to measure the proof mass acceleration response

of each vibration speaker. . . 51 3.4 Frequency response function (magnitude and phase) in terms of proof mass

acceleration. Experimental results for inertial actuators labelled 1 and 2. . . 51 3.5 Frequency response function (magnitude and phase) in terms of proof mass

acceleration. Experimental results for inertial actuators labelled 3 and 4. . . 52 3.6 Frequency response function (magnitude and phase) in terms of proof mass

acceleration. Experimental results for inertial actuators labelled 5 and 6. . . 53 3.7 Frequency response functions (magnitude and phase) in terms of proof mass

acceleration. Experimental results for the six inertial actuators. . . 53 3.8 Frequency response function ˜Hav(s) (magnitude and phase) for increasing

values of the ratio ms/ma. . . 54

3.9 Frequency response function ˜Hdv(s) (magnitude and phase) of the Dayton

3 exciter. . . 56 3.10 Comparison of the measured and identified frequency response functions

(magnitude and phase) for the Dayton exciters 1 and 2. . . 58 3.11 Comparison of the measured and identified frequency response functions

(magnitude and phase) for the Dayton exciters 3 and 4. . . 58 3.12 Comparison of the measured and identified frequency response functions

(magnitude and phase) for the Dayton exciters 5 and 6. . . 59 3.13 Pictures of Lepai LP-2020A+ amplifier . . . 59 3.14 Experimental setup used to measure the frequency response function of each

Lepai amplifier. . . 60 3.15 Measured frequency response functions (magnitude and phase) for the Lepai

2020A+ amplifiers labelled 1, 2 and 3. . . 60 3.16 Measured frequency response functions (magnitude and phase) for the Lepai

2020A+ amplifiers labelled 4 and 5. . . 61 3.17 Comparison of the measured and identified frequency response functions

(magnitude and phase) for the Lepai 2020A+ amplifiers labelled 1,2 and 3. 62 3.18 Comparison of the measured and identified frequency response functions

(magnitude and phase) for the Lepai 2020A+ amplifiers labelled 4 and 5. . 62 3.19 Picture of the clamped plate with the array of measuring microphones. . . . 63 3.20 First eight experimental modal shapes and natural frequencies. . . 64 3.21 First eight simulated modal shapes and natural frequencies. . . 64

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4.1 Equivalent SDOF model of the plate equipped with an ideal force actuator (fa) and subjected to a disturbance force fd. . . 68

4.2 Left: Scheme of the skyhook control approach. Right: Equivalent skyhook model . . . 69 4.3 Skyhook control: root locus when the equivalent SDOF model of the plate

is subjected to a control force provided by an ideal force actuator. . . 69 4.4 Skyhook control: root locus when the equivalent SDOF model of the plate

is subjected to a control force provided by the Dayton vibration speaker (Dayton 2). . . 70 4.5 Skyhook control: frequency response function (a) and root locus (b) when

the equivalent SDOF model of the plate is subjected to a control force provided by the Dayton vibration speaker. . . 71 4.6 Skyhook control: root locus when the equivalent SDOF model of the plate

is subjected to a control force provided by an ideal force actuator and the velocity signal is obtained with a pseudo-integration of the acceleration. . . 71 4.7 Skyhook control: frequency response function (a) and root locus (b) when

the equivalent SDOF model of the plate is subjected to a control force provided by the Dayton vibration speaker and the velocity signal is obtained with a pseudo-integration of the acceleration. . . 72 4.8 Skyhook control: frequency response function (a) and root locus (b) when

the equivalent SDOF model of the plate is subjected to a control force provided by the Dayton vibration speaker and amplified by the Lepai audio amplifier, and the velocity signal is obtained with a pseudo-integration of the acceleration. . . 73 4.9 Sketch of the prototype plate reporting disturbance position (D) and

sensor-actuator dual and collocated position (A). . . 74 4.10 Simulation of the vibration reduction at the central location of the full

model of the prototype plate using a skyhook control. . . 74 4.11 Picture of the prototype plate equipped with one central Dayton vibration

speaker as the control actuator and one off-center Dayton vibration speaker as disturbance source. . . 75 4.12 Another picture of the prototype plate equipped with one central Dayton

vi-bration speaker as the control actuator and one off-center Dayton vivi-bration speaker as disturbance source (view from above). . . 76 4.13 Experimental vibration reduction at the central location of the prototype

plate using a skyhook control. . . 77 5.1 Block diagram of the system: compensation filter + inertial actuator. . . . 79 5.2 Bode diagram (amplitude and phase) of the second-order filter in Eq. (5.1) 80 5.3 Bode diagrams of the second-order compensation filter for different values

of ωc. . . 82

5.4 Bode diagrams of the second-order compensation filter for different values of ξc. . . 82

5.5 Bode diagrams of the compensated frequency response of the Dayton device for different values of ωc. . . 83

5.6 Bode diagrams of the compensated frequency response of the Dayton device for different values of ξc. . . 83

5.7 Block diagram of the system: compensator + amplifier + inertial actuator. 84 5.8 Compensated actuator bode diagram . . . 84

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5.9 Bode diagrams of the compensated frequency response of the Dayton device driven by the Lepai amplifier for different values of ωc. . . 85

5.10 Bode diagrams of the compensated frequency response of the Dayton device driven by the Lepai amplifier for different values of ξc. . . 85

5.11 Bode diagram (amplitude and phase) of the experimental frequency re-sponse of the compensated control unit with different values of ωc (ξc= 1

fixed). . . 86 5.12 Experimental Bode diagram with ξcfixed (ξc= 1) . . . 87

5.13 Skyhook control with compensation filter C1: frequency response function (a) and root locus (b). . . 89 5.14 Skyhook control with compensation filter C2: frequency response function

(a) and root locus (b). . . 89 5.15 Simulation of the vibration reduction at the central location of the full model

of the prototype plate using a skyhook control with the compensation filter C1. . . 90 5.16 Simulation of the vibration reduction at the central location of the full model

of the prototype plate using a skyhook control with the compensation filter C2. . . 91 5.17 Experimental vibration reduction at the central location of the prototype

plate using a skyhook control with the compensation filter C1. . . 92 5.18 Experimental vibration reduction at the central location of the prototype

plate using a skyhook control with the compensation filter C2. . . 92 6.1 Local feedback loop strategy; V denotes the input voltage, a is the

accel-eration, I is the ideal integrator and gi is the inner gain loop. . . 93

6.2 Bode diagram (a) and root locus (b) of the system reported in Fig. 6.1b. . . 94 6.3 Block diagram of the local feedback control loop involving amplifier . . . . 95 6.4 Bode diagram (a) and root locus (b) of the system reported in Fig. 6.3. . . 95 6.5 Bode diagram (amplitude and phase) of the experimental frequency

re-sponse of the control unit with a local direct velocity feedback. . . 96 6.6 Comparison of the experimental ad simulated Bode diagrams when the

control unit is equipped with a local feedback control. . . 97 6.7 Block diagram of the double-loop control system; V is the input voltage,

d is the disturbance force, ft is the force transmitted by the actuator, a

is the acceleration of the plate, v is the velocity, ma is the proof mass of

the actuator, PI is the pseudo integrator, IAr is the ”coupling” transfer

function between actuator and structure as better explained in Appendix A, and finally g and gi are the outer and inner loop gains, respectively. . . . 97

6.8 FRF (6.8a) and root locus (6.8b) of the closed loop system with the local feedback loop gain value: gi = 5 . . . 98

6.12 Picture of the prototype plate with one central Dayton vibration speaker as the control actuator and one off-center Dayton vibration speaker as dis-turbance source. The control actuator is equipped with an accelerometer for the implementation of the local feedback loop. . . 100

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6.13 Experimental vibration reduction at the central location of the prototype plate using a skyhook control with a local feedback loop. . . 101 A.1 Block diagram: V is the input voltage, ft is the force transmitted by the

actuator, a is the acceleration of the plate, IAr is the ”coupling” transfer

function between actuator and structure as better explain in the rest of this appendix. . . 111 A.2 Schematic of a voltage-driven inertial actuator . . . 111

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List of Tables

2.1 Parameters of the inertial actuator in Ref. [37]. . . 11 2.2 Parameters of the inertial actuator in Ref. [13]. . . 11 2.3 First ten frequency parameters for a fully clamped square plate - FSDT

theory. . . 27 2.4 First ten frequency parameters for a fully clamped square plate - (3,3,2)

theory. . . 28 2.5 Parameters for the actuator’s electromechanical model. . . 46 2.6 Geometry and physical parameters of the panel. . . 46 3.1 Mechanical parameters of the Dayton exciters (R = 4.2 Ohm, kem = 3.86

N/A). . . 57 3.2 Geometry and physical parameters of the panel. . . 63 3.3 First eight natural frequencies of the clamped plate: experimental results,

numerical results and relative error. . . 65 3.4 1-DOF parameters. . . 65 5.1 Parameters of the compensation filters used in the simulation and

experi-ments of the skyhook vibration control. . . 88 5.2 Parameters of the pseudo-integrator used when the compensators are

intro-duced in the feedback loop. . . 88 6.1 Combinations of inner and outer loop gains tested on the prototype plate. . 99

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Chapter 1

Introduction

“Silence is a universal gift that few appreciate. Maybe because it can not be bought.

The rich buy noise.

The human soul delights in the silence of nature, which is revealed only to those who seek him.”

Charlie Chaplin

1.1 Motivation

Noise pollution has a deep detrimental effect on psychological health and quality of life, for example in home and working environments. The outer building envelope, in particular windows and window-like surfaces, often constitutes the primary path for external noise sources to travel into the building and disturb a person’s everyday life. An introduced noise can be very annoying and harmful, affecting the wellbeing of residents and workers. Another common form of noise pollution occurs in transport vehicles. Vibrations of cabin panels and shell structures generate a high level of interior noise inside aircrafts, tractors, cars and trains. In recent years, people are more and more aware of vibroacoustic comfort. Indeed, cabin noise quality and speech intelligibility have become increasingly important in the automotive field and they are the main factors driving the private market of VIP aircrafts and helicopters.

According to what outlined before, it is not surprising that a huge effort is being made to find innovative and effective solutions which would ensure higher acoustic insulation and comfort. This effort is also enforced by the increasing number of strict noise regulations appearing in all countries. The importance of these studies in a global market should not be overlooked. The increase in the population of large cities has led to increased noise pollution and some municipalities have already launched calls for tenders in order to develop specific equipments capable of reducing exposure to noise by their citizens [1, 2]. The classical approach to limit noise pollution arising from lightweight vibrating sur-faces is the application of passive treatments over a large area such as added mass or damping material [3, 4, 5]. Surface treatments are often effective at suppressing higher frequency vibrations. However, the reduction of low-frequency noise usually requires a lot of material leading to heavy and bulky solutions, which could violate some requirements in terms of maximum weight and occupied space. Active control systems [6] can help in achieving acoustic performances of panels without introducing extra mass.

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1.2 Scope of the thesis

The present thesis can be considered as a preliminary study in the development of a control system involving active devices to be arbitrarily placed on vibrating surfaces to damp vibrations and thus reduce noise radiation.

More specifically, this work presents a theoretical and experimental investigation of an active damping controller with an inertial or proof-mass electrodynamic actuator, which is driven by suitable compensators to maximize its performance [7]. The choice of a proof-mass actuator as a control device has two main reasons. First, compared to a piezoelectric patch actuator [8], it is capable to provide a transverse force normal to the vibrating surface, thus acting more directly and more efficiently on the flexural vibration of the panel, which is responsible for its sound radiation. If an accelerometer sensor is placed in the footprint of the inertial device, a collocated unit can be also realized. Second, the control unit should be easily attached to and removed from the vibrating panel several times, so that it can be applied and put into work only when needed.

Figure 1.1: Vibration speaker

Another principal aim of this work is to investigate the possibility of using commercial and low-cost voice-coil vibration speakers as proof-mass actuators. Vibration speakers are vibration transducers capable of turning surfaces into loudspeakers (see Fig. 1.1). A loudspeaker is an electroacoustic transducer having several parts such as the diaphragm and the voice coil. The voice coil is a movable electromagnet, i.e., passing current through the coil creates a magnetic field and reversing the current switches the polarity of the magnetic field. At the base of the speaker is a permanent magnet. When the polarity of the magnetic field of the coil matches that of the permanent magnet, the two like fields repel one another and the coil moves outward, pushing the diaphragm. When the magnetic forces are opposite one another, they attract each other. This pulls the coil inward, pulling on the diaphragm. Alternating the electricity flowing through the coil will cause the coil to move up and down quickly. This makes the diaphragm move, which in turn causes the air pressure to change and create sound. A vibration speaker is similar, except that there’s no diaphragm. Instead, the voice coil attaches to a solid surface so that it will vibrate against that surface. The solid surface will vibrate with the speaker, displacing air molecules around it. Because the speaker transfers vibrations to the surface it is mounted on, the surface itself will send out sound. According to this physical mechanism, the simple idea in this work is to use a vibration speaker as a control device instead of a sound device, i.e., introduce vibration waves into the attached surface which can suppress the vibration induced by external noise sources instead of generating sound.

While being very versatile and effective, inertial actuators also have some drawbacks when used as control units [9, 10]. Figure 1.2 shows the typical magnitude and phase

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di-Magnitude (dB) -90 0 90 180 Phase (deg) Bode Diagram Frequency (Hz)

Figure 1.2: Typical bode diagram (magnitude and phase) of the proof-mass acceleration response of a voltage-driven inertial actuator.

agrams of a proof-mass actuator in terms of acceleration of the seismic mass with respect to a voltage applied to the coil. Note that the same curve, after proper scaling, describes the amplitude of the force provided by the device. It can be observed that the actuator dynamics introduces in the generated force a phase shift of 180o degrees around some resonance frequency. Additional phase lag effects are also visible in the higher frequency range. This dynamics can significantly limit the maximum stability gain of a feedback control system [11]. It is possible to increase the stability of the closed-loop system by selecting actuators having internal dynamics decoupled from the structure under analy-sis [12]. An ideal configuration involves a proof-mass actuator with natural frequency well damped and approximately one decade lower than the fundamental mode of the vibrat-ing surface. However, the mechanical resonance of the actuator cannot be arbitrarily low since this can give rise to problems with the static deflection of the suspended mass due to gravity [13]. Therefore, other solutions aimed at properly compensating the actuator dynamics should be devised. The evaluation of two different compensators for the selected vibration speakers will be another important part of the present work.

1.3 Technical background

As already outlined, active control strategies applied on lightweight vibrating panels emerged in the last three decades as a viable way of reducing the structural vibration and, as a result, the sound transmission in the audio low frequency range. In general, the human audible frequency range goes from 20Hz to about 20kHz. Disturbances above 1 kHz can be typically reduced using passive techniques. Active technology has found potential use in the 50Hz to 1kHz range [14].

The development of active controllers for noise and vibration suppression has carried out through different strategies that can be classified into two principal groups.

The first one is based on the centralized feedforward control strategy to directly control the global sound field. This approach can be implemented as active noise control (ANC) and active noise and vibration control (ANVC). ANC systems consist in using loudspeakers as actuators to create directly a sound field that destructively interferes with the sound field of disturbance. ANVC systems, on the other hand, involve structural actuators to alter the vibration behavior of the radiating surface such that the acoustic level is reduced [15].

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In both cases, the acoustic error sensors are microphones and they are used to define a cost function that is minimized by the controller [16]. The main drawback of this approach is that the feedforward control strategy requires a reference signal well correlated with the disturbance to be controlled [17, 18]. For random disturbances this is very complicated and, according to the goal of this thesis, the feedforward approach will not be considered. The second group includes systems which implement feedback controllers employing structural sensor and actuator pairs only [7, 19]. This second group is especially suited for the vibration and radiated noise control of thin panels since no reference signal is required. Moreover, good performances can be achieved with broadband excitation spectra. As it is possible to express the total kinetic energy of the plate through classical vibration modes, by analogy, it is possible to express the sound radiation through some quantities called radiation modes [20]. Formulation of the problem of sound radiation in these terms leads to the so-called active structural acoustic control (ASAC) [21]. As such, ASAC formulations tend to minimize a cost function containing informations associated with radiation modes. On the other hand, a simpler and effective alternative approach to reduce the vibration and sound radiation, especially at low frequencies, is the active vibration control (AVC) strategy. Unlike ASAC, this formulation does not make use of any information related to radiation modes and is based on structural information only. This is possible because, at low frequencies, the panel response is typically dominated by well-separated resonant modes, and, controlling them means achieving a reduction in sound levels [7]. Due to a good balance between simplicity and effectiveness, AVC is the strategy selected in this thesis work.

The AVC strategy can be implemented using a single-input-single-output (SISO) con-figuration or a multiple-inputs-multiple-outputs (MIMO) arrangement when the vibrating surface is equipped with many structural actuators and sensors. MIMO configurations for AVC systems typically employ the same number of actuators and sensors, which are used in pairs and collocated. As a general classification, they can involve a fully centralized, a partially decentralized or a fully decentralized architecture [22, 23].

In a fully centralized arrangement (see Fig. 1.3), a single controller manages directly all the actuator/sensor pairs, which are fully coupled, i.e., the control signal provided to each actuator receives information from all the available sensors [24]. This solution can achieve the best performance in terms of vibration and sound reduction. However, as the number of transducers increases, the complexity of the controller rises up rapidly and the stability of the closed-loop system can be highly vulnerable to individual failures of control units.

Partially centralized architectures are based on grouping control transducers according to some clustering strategies. A separate controller is implemented for each cluster [25].

In a fully decentralized vibration control system [26, 27], the sensors and actuators are connected in individual, local loops, each control signal being dependent only on the signal from the collocated sensor. Even if the number of transducers could be high, the implementation of the controller remains simple, with a high degree of scalability. Moreover, this strategy is very robust against possible failures of the control units.

As shown in [23, 28], direct velocity feedback using multiple velocity sensors and col-located force actuators is well known for adding damping to a structure and thus, for a given disturbance excitation, reduce its vibration (see Fig. 1.4). This strategy, also called skyhook control since it is equivalent of having viscous dampers attached to a fixed point in the sky [27, 28, 29, 30], leads to an unconditionally stable closed-loop system when ideal collocated actuator/sensor pairs are used, i.e., the control gain of each local loop can be arbitrarily increased without inducing instability.

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Figure 1.3: Fully centralized and fully decentralized active vibration control systems on a flat surface.

Figure 1.4: Active vibroacoustic control of a panel with an array of decentralised feedback controllers driving secondary force actuators from collocated velocity sensors.

In practical realizations, the control force is provided by an actuator with internal dynamics such as a proof-mass device and the velocity signal comes from the integration of the acceleration measurement. Moreover, the actuator/sensor pair cannot be considered perfectly collocated at any frequency. In such cases, the skyhook control system is only conditionally stable and the control gain must be limited to avoid instability.

Since the higher is the gain the higher is the closed-loop damping, small control gains can give unsatisfactorily performance. One approach to overcome this limitation and let the control gain to reach high values consists in introducing in the loop a suitable compensator capable of altering the internal dynamics of the proof-mass actuator [6, 31, 32, 33]. An example is provided in [34] where a second-order compensator is proposed as a notch-like filter to cancel the mechanical resonance of the actuator. In this case, it is necessary to know exactly the influence that the structure has on the dynamics of the actuator, especially if the resonance of the actuator is not low enough to guarantee a clear separation between the two peaks. In our case this can be very complex because the aim is to design a device having characteristics independent of the panel. Another limitation introduced by this compensator is the possible saturation of the actuator, which would lead to nonlinear effects and instability of the closed loop system.

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applied on the inertial actuator. The aim is to make the actuator’s response as close as possible to the ideal one in order to reduce the phase lag as much as possible. This strategy has been proposed to try to overcome the problem of coupling between actuator and structure by operating directly on the actuator inner-loop control.

1.4 Structure of the thesis

As already outlined, the thesis originates from the idea of developing a universal AVC device to be attached on demand to any flat lightweight surfaces for vibration and noise suppression. The device should be based on commercially available vibration speakers as control actuators and small accelerometers as vibration sensors.

After a brief outline on the motivation of this effort and its technical background as discussed in the present chapter, the most important aspects related to the modeling of the dynamics of the control unit and its coupling with the flexible surface onto which it is mounted are presented in Chapter 2. The vibration speaker is modeled as a voltage-driven proof-mass device and its dynamics is presented in terms of blocked force response, frequency response of the suspended mass and base impedance. A section is dedicated to the mathematical model of the audio amplifier to be adopted to drive the vibration speaker. As shown later, the dynamics of the amplifier can affect the closed-loop response of the system. It is worth noting that the model of the flat vibrating surface is based on an original formulation recently developed through a collaboration between Politecnico di Milano and Universit´e of Paris X.

Chapter 3 is aimed at presenting the experimental characterization of the control unit and the prototype panel arranged in the laboratory, so that the models derived in the previous chapter can be validated and the unknown parameters can be identified.

The skyhook controller is described in Chapter 4 along with a set of simulation and experimental results. At first, it is analyzed theoretically on a single-degree-of-freedom (SDOF) spring-mass-damper system as an equivalent representation of the dynamics of the fundamental mode of the prototype plate. The effects of the mechanical dynamics of the vibration speaker and the electrical dynamics of the audio amplifier on the closed-loop system are clearly outlined. The results from this first part have been useful to understand the physics of the problem and were then extended to the full plate model. The skyhook controller was finally tested on the prototype panel by placing one control unit at the center of the plate.

Chapters 5 and 6 present two solutions for improving the stability margin of the sky-hook control system. The first solution is based on introducing a compensation filter in the control loop as proposed in [13]. The second solution involves the modification of the dynamics of the inertial actuator through a local velocity feedback loop. Both strategies are simulated on the SDOF equivalent model of the plate and tested experimentally in an open-loop arrangement on the isolated control unit. As a final step, the performances of the skyhook control with the compensation filter and the local feedback loop are tested experimentally on the prototype panel.

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Chapter 2

Modeling Aspects

This chapter presents the main aspects related to the mathematical modeling of the system under study. In particular, each component is first analyzed separately and then coupled models are properly derived. As outlined in the Introduction, the system selected as a prototype to test the concept involves a flat plate equipped with lightweight and low-cost vibration speakers to be used as inertial actuators, driven by low-cost audio amplifiers.

2.1 Vibration speakers

The vibration speakers adopted in this study can be considered as proof-mass electro-dynamic actuators (also called inertial actuators or vibration exciters in the following). A proof-mass actuator is a force actuator based on the principle that accelerating a suspended mass results in a reaction force on the supporting structure. A possible conceptual scheme of a real device is represented by a mass, elastically connected to a support structure by membranes which guide the linear motion of the moving mass. If the mass is subjected to a magnetic force arising from a voice coil transducer, the inertial device is called electro-magnetic actuator. A voice coil transducer is an energy transformer converting electrical energy into mechanical energy which consists of a permanent magnet producing a uniform magnetic flux density normal to the gap and a coil free to move axially within the gap. The current flowing into the coil can be provided by a current source or a voltage source and the actuator is said to be current-driven or voltage-driven, respectively.

The theoretical dynamic behavior of an inertial actuator could be represented by using: • the blocked force response

• the proof mass acceleration response

• the base impedance for both current-driven and voltage-driven units.

A simple lumped-parameter model is adopted here as a basic description of the mechanical and electrical dynamics of the device. As shown in the next chapter, this model fits well the experimental data.

2.1.1 Blocked force response

The blocked force response of an inertial actuator is defined as the transfer function between the electrical source (either current or voltage) and the force exerted by the device when it reacts against a perfectly rigid surface.

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Lagrange’s equations for electromechanical systems [36] are used to derive the blocked force response. The following nomenclature is adopted:

ma proof mass of the actuator (kg)

ca damping of the actuator (Ns m−1)

ka suspension stiffness of the actuator (N m−1)

z(t) absolute displacement of the moving mass (m) L coil inductance (H)

R coil resistance (Ω)

q(t) electric charge in the coil (F)

i(t) electric current flowing in the coil (A) kem electromagnetic coupling factor (NA−1)

v(t) electric voltage source

The energies involved in the system are: – complementary kinetic energy

T⋆ = 1 2ma˙z 2 – potential energy V = 1 2kaz 2

– complementary magnetic energy

W_m⋆ = 1 2L ˙q 2_{+ k} emz ˙q – dissipation function D = 1₂ca˙z2+ 1 2R ˙q 2

The virtual work done by non-conservative forces is given by: δWnc = δqv

The Lagrangian of the system is

L = T⋆− V + Wm⋆ (2.1)

where the degrees of freedom of the problem are the absolute displacement z(t) of the proof mass and the electric charge q(t) in the coil. Accordingly, the Lagrange’s equations are written as d dt ∂L ∂ ˙z −∂L_∂z +∂D ∂ ˙z = Qz d dt ∂L ∂ ˙q −∂L_∂q +∂D ∂ ˙q = Qq which lead to the following dynamic equations:

     maz(t) + c¨ a˙z(t) + kaz(t) − kem˙q(t) = 0 L¨q(t) + R ˙q(t) + kem˙z(t) = v(t) (2.2)

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Since the electric current is the rate of change of the electric charge with respect to time, i.e., i(t) = ˙q(t), the above equations can be also written as a function of z(t) and i(t) as follows      maz(t) + c¨ a˙z(t) + kaz(t) − kemi(t) = 0 L˙i(t) + Ri(t) + kem˙z(t) = v(t) (2.3)

The first equation describes the mechanical behavior, whereas the second is referred to the dynamics of the electrical part. Note that, from a mechanical point of view, the actuator is schematically equivalent to a spring-mass-damper system subjected to a control force fc(t) = kemi(t). The current i(t) is obtained as a result of the electrical equation if the

actuator has a voltage source (voltage-driven device). Otherwise, it is directly provided by the current source for a current-driven device and the electrical equation is discarded.

Current-driven device

In the case of a current-driven actuator, the current is provided by a current source and thus the electric dynamics given by the second equation in Eq. (2.3) is by-passed. As a result, the behavior of the inertial actuator is represented by the mechanical equation only as

maz(t) + c¨ a˙z(t) + kaz(t) = kemi(t) (2.4)

which corresponds to the scheme in Fig. 2.1, where fc(t) is the contribution to the overall

force of the actuator related to the external electrical command.

fc(t) fc(t) ma ca ka kem z(t) fc(t) = kemi(t)

Figure 2.1: Schematic of a current-driven electromagnetic inertial actuator attached to the ground. Taking the Laplace transform yields

z(s) = kem

ka+ sca+ s2ma

i(s) (2.5)

After calling f (t) = kaz(t) + ca˙z(t) the mechanical force transmitted by the device as

shown in Fig. 2.2, the blocked force exerted by the actuator is given by fB(t) = fc(t) − f (t) = kemi(t) − kaz(t) − ca˙z(t)

Using Eq. (2.4), it follows that

fB(t) = maz(t)¨

and in the Laplace domain we have

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fc(t) fc(t) fB(t) f (t) f (t) ma ca ka z(t) maz + f − f¨ c = 0 f = kaz + ca˙z fB= fc− f = maz¨ fc(s) − f (s) = s2maz(s) f (s) = (ka+ sca)z(s) fB(s) = s2maz(s) ⇒

Figure 2.2: Lumped parameter model for the study of the blocked force response.

Combining the above equation with Eq. (2.5) yields the blocked force response of a current-driven inertial actuator as

Hf i(s) = fB(s) i(s) = kems2 s2_{+ 2ξ} aωas + ω2a (2.6) where ωa= r ka ma

is called the actuator resonance frequency or suspension frequency of the inertial actuator and

ξa=

ca

2√kama

is the equivalent mechanical damping ratio.

Two representative examples are shown in the following to highlight the typical fre-quency behavior of the blocked force response.

The first example refers to a small-size electromagnetic actuator having parameters reported in Table 2.1 as given in Ref. [37]. The mechanical frequency and damping ratio are equal to fa = ωa/(2π) = 23.3 Hz and ξa = 0.284, respectively. The Bode diagram

(magnitude and phase) of the blocked force response Hf i(s) is shown in Fig. 2.3.

The second example involves an actuator with parameters in Table 2.2 as given in Ref. [13], having a mechanical frequency and damping ratio equal to fa = ωa/(2π) = 55

Hz and ξa= 0.04, respectively. The corresponding response is reported in Fig. 2.4.

It is clear from Figures 2.3 and 2.4 that, at low frequencies, the blocked force response Hf i(s) of a current-driven inertial actuator is controlled by the stiffness of the proof-mass

suspension and is out of phase with driving signal. The magnitude of the blocked force response function increases proportionally to the square of frequency and peaks at the

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Table 2.1: Parameters of the inertial actuator in Ref. [37].

Parameter Symbol Value Units

Proof mass ma 0.024 kg

Suspension stiffness ka 511 N/m

Suspension damping coefficient ca 1.99 Ns/m

Coil inductance L 0.6 mH

Coil resistance R 2.7 Ω

Electromagnetic factor kem 2.16 N/A

101 102 103 104

10−2 10−1 100

blocked force response of a current−driven inertial actuator

Frequency (Hz) Magnitude (N/A) 101 ₁₀2 ₁₀3 ₁₀4 −270 −180 −90 0 90 180 270 Frequency (Hz) Phase (deg)

Figure 2.3: Bode diagram (magnitude and phase) of the blocked force response of the current-driven inertial actuator in Table 2.1.

Table 2.2: Parameters of the inertial actuator in Ref. [13].

Parameter Symbol Value Units

Proof mass ma 0.012 kg

Suspension stiffness ka 1433 N/m

Suspension damping coefficient ca 0.332 Ns/m

Coil inductance L 0.1 mH

Coil resistance R 8 Ω

Electromagnetic factor kem 0.53 N/A

actuator mechanical resonance frequency. The peak value is dependent on the damping coefficient of the suspension. The actuator in Table 2.1 is more damped than the second case. Above the actuator resonance frequency, the blocked force response is in phase with the driving signal and show a flat response magnitude. Accordingly, the device behaves like a high-pass filter (it has no authority over rigid body modes) with an high-frequency asymptote given by the electromagnetic constant kem. As a result, above a frequency

approximately equal to 2ωa the actuator can be regarded as an ideal force generator.

The transfer function in Eq. (2.6) has two zeros in the origin z1,2 = 0 and two complex

conjugate poles at p1,2= −ξaωa± jωa

p 1 − ξ2

a. As an example, the pole-zero map of the

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101 102 103 104 −50 −40 −30 −20 −10 0 10 20

blocked force response of a current−driven inertial actuator

Frequency (Hz) Magnitude (N/A dB) 101 102 103 104 −270 −180 −90 0 90 180 270 Frequency (Hz) Phase (deg)

Figure 2.4: Bode diagram (magnitude and phase) of the blocked force response of the current-driven inertial actuator in Table 2.2.

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 −500 −400 −300 −200 −100 0 100 200 300 400 500 0.007 0.0115 0.017 0.024 0.034 0.055 0.12 100 200 300 400 500 100 200 300 400 500 0.0035 0.007 0.0115 0.017 0.024 0.034 0.055 0.12 0.0035 Pole−Zero Map

Real Axis (seconds−1

)

Imaginary Axis (seconds

−1)

Figure 2.5: Pole-zero map of the inertial actuator in Table 2.2.

Voltage-driven device

In the case of a voltage-driven actuator, the control current flowing into the coil results from the coupled electro-mechanical dynamics expressed in Eq. (2.3). The corresponding schematic is sketched in Fig. 2.20.

Taking the Laplace transform of the dynamic equations (2.3) yields      (ka+ sca+ s2ma)z(s) = kemi(s) (R + sL)i(s) + skemz(s) = v(s)

From the second equation,

i(s) = 1

R + sLv(s) − skem

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fc(t) fc(t) ma ca ka kem z(t) R L v(t) i(t)

Figure 2.6: Schematic of a voltage-driven electromagnetic inertial actuator attached to the ground.

and putting i(s) into the first equation gives

z(s) = kem

(R + sL)(ka+ sca+ s2ma) + sk2em

v(s)

Therefore, the blocked force fB(s) = s2maz(s) can be related to the input voltage by the

following transfer function

Hf v(s) = fB(s) v(s) = kems2 (R + sL)(s2_{+ 2ξ} aωas + ω2a) + sk 2 em ma (2.7)

or, more explicitly,

Hf v(s) = kems2 Ls3_{+ (2ξ} aωaL + R)s2+ (2ξaωaR + ωa2L + k 2 em ma)s + ω 2 aR (2.8)

The blocked force of a voltage-driven actuator can be also related to the electrical current flowing in the coil. Since v(s) = Ri(s), the following transfer function can be introduced H_{f i}v(s) = fB(s) i(s) = kemRs2 Ls3_{+ (2ξ} aωaL + R)s2+ (2ξaωaR + ω2aL + k 2 em ma)s + ω 2 aR (2.9)

Comparing Eq. (2.6) with Eq. (2.8), it is evident that the blocked force response of a voltage-driven actuator has an additional pole with respect to an actuator with a current source due to the first-order electrical dynamics.

The Bode diagram (magnitude and phase) of the blocked force responses Hf v(s) for the

actuators in Tables 2.1 and 2.2 are shown in Figures 2.7 and 2.8, respectively. However, the comparison with the current source case can be more appreciated by looking at the transfer function H_{f i}v(s) as in Figures 2.9 and 2.10. It is clear that, at low frequencies, the response of the voltage-driven actuator is very similar to the current-driven case. In the frequency region around the mechanical resonance of the actuator, the voltage source introduces an additional damping effect associated with the back-electromotive force, which could be significant (as for the device with parameters in Table 2.1) or negligible (as for the device of Table 2.2). Another difference relies in the high-frequency behavior. The current-driven case predicts a flat magnitude with an asymptote equal to the electromagnetic factor and a zero-phase response. Instead, the electrical pole associated with the electrical dynamics induced by the increasing inductance effects of the actuator gives a high-frequency slope

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101 102 103 104 10−2

10−1 100

blocked force response of a voltage−driven inertial actuator

Frequency (Hz) Magnitude (N/V) 101 102 103 104 −270 −180 −90 0 90 180 270 Frequency (Hz) Phase (deg)

Figure 2.7: Bode diagram (magnitude and phase) of the blocked force response of the voltage-driven inertial actuator in Table 2.1.

101 102 103 104 −50 −40 −30 −20 −10 0 10 20

blocked force response of a voltage−driven inertial actuator

Frequency (Hz) Magnitude (N/V) 101 102 103 104 −270 −180 −90 0 90 180 270 Frequency (Hz) Phase (deg)

Figure 2.8: Bode diagram (magnitude and phase) of the blocked force response of the voltage-driven inertial actuator in Table 2.2.

of the magnitude response of -20 dB/dec, which corresponds to a phase lag of 90 degrees. Note that the real electrical pole has a frequency approximately equal to R/L.

For the sake of completeness, Fig. 2.11 shows the pole-zero map of the actuator in Table 2.1 when it has a current and a voltage source.

Static residualization of the electrical dynamics

The blocked force response of voltage-driven actuators can be also approximated by performing a static residualization of the electric dynamics. From the second equation of the system (2.3), after neglecting the dynamics described by the derivative of the electric

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101 102 103 104 10−2

10−1 100

Bode diagram − blocked force response

Frequency (Hz) Magnitude (N/A) 101 102 103 104 −270 −180 −90 0 90 180 270 Frequency (Hz) Phase (deg) Hf i(s) Hv f i(s)

Figure 2.9: Comparison between the blocked force response of the current-driven and voltage-driven inertial actuator in Table 2.1.

101 102 103 104 −50 −40 −30 −20 −10 0 10 20

Frequency (Hz) Magnitude (N/A) 101 102 103 104 −270 −180 −90 0 90 180 270 Frequency (Hz) Phase (deg) Hf i(s) Hv f i(s)

Figure 2.10: Comparison between the blocked force response of the current-driven and voltage-driven inertial actuator in Table 2.2.

current, it follows that

i(t) = 1 Rv(t) −

kem

R ˙z(t) which is substituted into the mechanical equation leading to

maz(t) +¨ ca+ k2 em R ˙z(t) + kaz(t) = kem R v(t)

In the above representation it is clear the contribution to the damping given by the back-electromotive force. The Laplace transform of the displacement of the mass is thus

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ex-−4500 −4000 −3500 −3000 −2500 −2000 −1500 −1000 −500 0 −150 −100 −50 0 50 100 150 1 0.94 0.982 0.993 0.997 0.998 0.999 1 1 500 1e+03 1.5e+03 2e+03 2.5e+03 3e+03 3.5e+03 4e+03 4.5e+03 0.94 0.982 0.993 0.997 0.998 0.999 1 Pole−Zero Map

Real Axis (seconds−1)

Imaginary Axis (seconds

−1)

Figure 2.11: Comparison of the pole-zero map of the current-driven (black) and voltage-driven (red) inertial actuator in Table 2.1.

pressed as z(s) = kem R 1 s2_m a+ s ca+k 2 em R + ka v(s)

The corresponding blocked force response is given by

H_{f v}r (s) = fB(s) v(s) = kem R s2 s2₊_2ξ aωa+ k 2 em Rma s + ω2 a (2.10)

A comparison of the residualized and full response is represented in Fig. 2.12 for the inertial actuator in Table 2.1 (similar considerations can be made for the inertial actuator in Table 2.2). It is noted that the resonance region characterizing the voltage-driven case is correctly captured, in particular with reference to the overall damping (mechanical + electromagnetic) of the device. Instead, the high-frequency dynamics introduced by the electrical part is residualized and, as expected, the related response is flat.

2.1.2 Proof mass acceleration response

In this section, the dynamic behaviour of an electromagnetic inertial actuator is char-acterized in terms of its proof-mass acceleration response, defined as the transfer function between the electrical source (either current or voltage) and the acceleration of the moving mass.

Current-driven device

Since the acceleration of the proof mass is the second order derivative of its dis-placement, i.e., a(s) = s2z(s), and the blocked force is related to the acceleration by fB(s) = maa(s), we have Hf i(s) = fB(s) i(s) = ma a(s) i(s) = maHai(s) (2.11) so that Hai(s) = a(s) i(s) = 1 ma Hf i(s) = (kem/ma)s2 s2_{+ 2ξ} aωas + ω2a (2.12)

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101 102 103 104 10−2

10−1 100

Frequency (Hz) Magnitude (N/V) 101 102 103 104 −270 −180 −90 0 90 180 270 Frequency (Hz) Phase (deg) Hf v(s) Hr f v(s)

Figure 2.12: Bode diagram (magnitude and phase) of the full (black) and residualized (red) blocked force response for the voltage-driven inertial actuator in Table 2.1.

Therefore, the proof mass acceleration response is equal to the blocked force response translated by a quantity corresponding to the value of the moving mass.

Voltage-driven device

Following the same procedure used before, we can write

Hf v(s) = fB(s) v(s) = ma a(s) v(s) = maHav(s) (2.13) so that Hav(s) = a(s) v(s) = 1 ma Hf v(s) = (kem/ma)s2 (R + sL)(s2_{+ 2ξ} aωas + ωa2) + sk 2 em ma (2.14)

or, more explicitly,

Hav(s) = (kem/ma)s2 Ls3_{+ (2ξ} aωaL + R)s2+ (2ξaωaR + ωa2L + k 2 em ma)s + ω 2 aR (2.15)

Similarly, the acceleration response in terms of the current flowing in the coil is ex-pressed as H_aiv(s) = (Rkem/ma)s 2 Ls3_{+ (2ξ} aωaL + R)s2+ (2ξaωaR + ω2aL + k 2 em ma)s + ω 2 aR (2.16)

When a static residualization of electrical dynamics is performed, the acceleration response is simplified to the following function

H_avr (s) = kem Rma s2 s2₊_2ξ aωa+ k 2 em Rma s + ω2 a (2.17)

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2.1.3 Base impedance

Another way of characterizing the dynamic behaviour of a proof mass actuator is to study its base impedance. The base impedance of the device is defined as:

ZB(s) =

fB(s)

vB(s)

(2.18)

where fB and vB = szB(jω) are the blocked force and velocity at the base of the actuator,

respectively. fc(t) fc(t) fB(t) f (t) f (t) ma mB ca ka v(t) = ˙z(t) vB(t) = ˙zB(t) maz + f − f¨ c = 0 f = ka(z − zB) + ca( ˙z − ˙zB) fB = fc+ mBz¨B− f

Figure 2.13: Lumped parameter model for the study of the base impedance. Referring to the lumped parameter scheme in Fig. 2.13, we can write

maz(t) + f (t) − f¨ c(t) = 0 ⇒ smav(s) + f (s) − fc(s) = 0 f (t) = ka[z(t) − zB(t)] + ca[ ˙z(t) − ˙zB(t)] ⇒ f (s) = ka s + ca [v(s) − vB(s)] ⇓ ka s + ca+ sma v(s) = ka s + ca vB(s) + fc(s) fB(t) = fc(t) − f (t) + mBz¨B(t) ⇒ fB(s) = fc(s) − f (s) + smBvB(s)

Combining the previous equations, the blocked force is expressed as

fB(s) =    smB+ ka s + ca sma ka s + ca+ sma    vB(s) + sma ka s + ca+ sma fc(s)

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Introducing ZmB(s) = smB Zma(s) = sma Zs(s) = ka s + ca we have the following expression for the blocked force

fB(s) = ZmB + ZsZma Zs+ Zma vB(s) + Zma Zs+ Zma fc(s)

and thus the so-called open-loop impedance is given by

ZOL= ZmB + ZsZma Zs+ Zma 101 102 103 104 100 102 104

Bode diagram − base impedance

Frequency (Hz) Magnitude (N/m/s) 101 102 103 104 −50 0 50 100 150 200 250 Frequency (Hz) Phase (deg) ZmB+ma ZmB ZOL

Figure 2.14: Base impedance of the inertial actuator in Table 2.1.

Figures 2.14 and 2.15 show the open-loop base impedances (blue line) of the actuators having parameters in Tables 2.1 and 2.2, respectively. At low frequencies, the actuator base impedance has a phase of 90o degrees and the magnitude rises proportionally to frequency. It is therefore mass like, corresponding to the impedance of the total mass of the actuator ma + mB moving in phase (dashed black line). Around the actuator

resonance frequency, the magnitude of the base impedance shows a resonance peak and an antiresonance dip, and the phase drops. For higher frequencies the phase of the base impedance converges back to 90o degrees, corresponding to the impedance of the actuator base mass mB (dashed-dotted black line).

2.2 Audio amplifiers

To ensure that the vibration speaker works properly, a power amplifier must be used upstream. A power amplifier is a device that reproduces low-power signals at a level that is strong enough for driving (or powering) actuators. The amount of amplification provided

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101 102 103 104 −10 0 10 20 30 40 50 60

Bode diagram − base impedance

Frequency (Hz) Magnitude (N/m/s) 101 102 103 104 −100 0 100 200 Frequency (Hz) Phase (deg) ZmB+ma ZmB ZOL

Figure 2.15: Base impedance of the inertial actuator in Table 2.2.

by an amplifier is measured by its gain, which is the relationship that exists between the signal measured at the output with the signal measured at the input. Referring to voltage quantities, the gain (also called voltage gain) is defined as the ratio between output and input voltage. As discussed in the next chapter, the vibration speakers adopted in this study have a nominal impedance of approximately 4 Ohm. Therefore, commercially available audio amplifiers can be employed to drive them.

Broadly speaking, an audio amplifier can be modeled as a band pass filter, so that the characteristic transfer function can be expressed as follows

H(s) = ˆgamp

s

(s + a)(s + b) (2.19)

where ˆgamp is the gain, a is the low-frequency pole and b is the high-frequency pole. The

frequency range between the poles is called the bandwidth of the amplifier. A typical frequency response normalized with respect to the maximum gain is reported in Fig. 2.16. Note that there is a typical requirement on the minimum bandwidth of an audio amplifier since the range of human hearing is approximately from 20 Hz to 20 kHz [38], even if it reduces to a maximum frequency of about 15 kHz with age.

When the frequency range of interest is limited to the low-to-medium frequencies, the amplifier can be represented by an high-pass filter. The corresponding simplified model includes only a low-frequency pole, so that the transfer function can be expressed as

H(s) = gamp

s

s + a (2.20)

where gamp is the amplifier gain and a is the low-frequency pole.

Figure 2.17 shows an example of a high-pass filter model compared to a band-pass filter model, where it is clear the validity of the representation in the low-to-medium frequency range. In this example it is also worth noting the phase response of the amplifier within the bandwidth, where it is equal to -180o degrees. Indeed, this case refers to an inverting audio amplifier so that the output signal is amplified and inverted with respect to the input. Note that the audio output sounds the same whether the output signal is inverted or not.

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Figure 2.16: Typical frequency response (magnitude) of an audio amplifier. 0 1 2 3 4 5 Amplitude (V/V) 100 102 104 106 -270 -225 -180 -135 -90 Phase (degrees) Frequency (Hz)

Figure 2.17: Bode diagram (magnitude and phase) of a typical amplifier.

2.3 Flexible plate

As already outlined, the prototype system under investigation involves a flat plate as a representative vibrating surface. In this section, the mathematical formulation at the basis of the plate model used in the subsequent dynamic analysis is briefly presented. It is worth noting that the formulation described below, developed in the last years through a scientific collaboration between Politecnico di Milano and Universit´e of Paris X, is very general and represents a powerful and versatile modeling tool, in particular for multilayered composite and sandwich panels. Indeed, a thin isotropic monolithic plate, as the one of interest in this thesis work, could be modeled using classical kinematic theories without the need of