Mechanical Engineering Department
Analysis and Development of a Vibration Isolation
System using Low Cost Actuators and Active
Decentralized Control
Supervisor: Prof. Stefano Manzoni
Co-Supervisor: Ing. Dirk Mayer
Master Thesis of:
Rocco Libero GIOSSI
ID Number: 842411
“God ain’t nothing but the mind working overtime”
-Anonymous
The problem of vibration isolation is an important topic of study when isolation of precise equipment from environmental disturbances is considered. This thesis has the objective to create an isolation system made by low-cost instrumentation with the most simple and effective control logic. The present work is an experimental dissertation. Thus, it has the aim of verifying which are the reliable aspects and which ones must be further studied.
Firstly, the problem of vibration isolation is discussed following the work done in the last years for single and multiple degrees of freedom. Inertial mass actuators are chosen among others due to their simple functioning, affordability and availability. “Sky-hook” control logic is chosen according to various authors as starting point. The structure under study is then analysed and the isolation system is set-up accord-ing to the analysis done, positionaccord-ing the actuators chosen in Reactive configuration. Lastly, a way to implement a reliable centralized isolation control logic based on a modal “Sky-hook” is presented. Due to the simplicity requirement, a decentralization method is proposed and further simplified, for the specific analysed case, showing good simulation results.
The experimental results of the implemented system show good results despite ex-perimental implementation problems. Additional work has still to be done to achieve an isolation system completely made by low-cost instrumentation.
L’isolamento da vibrazioni `e un argomento di rilevante importanza quando l’isolamento di strumentazione di precisone da disturbi esterni `e preso in considerazione. L’obbiettivo che si prende in carico la presente tesi `e quello di sviluppare e implementare un sis-tema di isolamento costituito da componentistica di basso costo utilizzando la logica di controllo pi`u semplice e efficace possibile. Il lavoro qui presentato `e di tipo speri-mentale e quindi ha il compito di verificare quali siano gli aspetti che possono essere affrontati e quali necessitano di ulteriore studio.
In prima istanza, il problema dell’isolamento da vibrazioni `e analizzato seguendo quanto fatto da diversi autori negli ultimi anni. Tale problema `e analizzato sia per un singolo grado di libert`a che per multipli. Gli attuatori di massa inerziali sono scelti come attuatori utilizzati in questa tesi grazie alla loro semplicit`a di funziona-mento, economicit`a e reperibilit`a. La logica di controllo scelta `e lo “Sky-hook” che in concordanza con molti autori viene presa come punto di partenza.
La struttura che `e caso di studio di questa tesi `e dunque analizzata. Il sistema di iso-lamento viene quindi costruito in concordanza con l’analisi sui sistemi di isoiso-lamento precedentemente fatta. In tale sistema di isolamento gli attuatori scelti vengono messi in configurazione reattiva.
Come ultimo punto, viene presentato un modo per trovare un controllo di tipo cen-tralizzato basato su un approccio “Sky-hook” di tipo modale. Data la richiesta di semplicit`a della logica di controllo legata all’obbiettivo della presente tesi, un metodo di decentralizzazione `e qui proposto. Una ulteriore semplificazione di tale metodo viene derivata mostrando buoni risultati di simulazione. Essa `e per`o valente esclusi-vamente per il caso in esame.
Buoni risultati sperimentali sono ottenuti utilizzando il sistema di isolamento stu-diato e la logica di controllo derivante dall’ultima semplificazione del controllo de-centralizzato, nonostante si siano riscontrate difficolt`a nell’implementazione di tali esperimenti da un punto di vista pratico. Ulteriore lavoro `e comunque richiesto per raggiungere l’obbiettivo di poter costruire un sistema di isolamento interamente composto da strumentazione di basso costo.
Introduction 1
1 Vibration Isolation Problem 5
1.1 Single d.o.f. Problem . . . 5
1.1.1 Ideal “Sky-hook” Control . . . 7
1.1.1.1 Control Comparison . . . 11
1.1.2 Actuator Selection . . . 13
1.1.3 IMA Actuators Functioning . . . 14
1.1.3.1 Proof-mass Configuration – Stability Analysis . . . . 16
1.1.3.2 Reactive Configuration – Stability Analysis . . . 19
1.1.3.3 Configuration Choice . . . 20
1.2 Multi d.o.f. Problem . . . 21
1.2.1 Modal Formulation . . . 22
1.2.2 Structures Interaction . . . 23
1.2.2.1 Stability Limit . . . 23
1.2.2.2 Limits and Aproximation . . . 25
1.2.3 Ideal Control of Structures for Vibration Isolation . . . 26
2 Experiment Description 29 2.1 LEGO R Structure . . . 29
2.1.1 Laser Vibrometer Analysis . . . 30
2.1.2 Modal Analysis and Reconstructed Behaviour . . . 32
2.2 Mounting System Set-up . . . 35
2.2.1 Actuators and Sensors Location . . . 36
2.2.2 Actuators choice -Electrical and Mechanical Behaviour . . . 37
2.2.3 Physical Coupling . . . 38
2.3 Modal Analysis of System Set-up . . . 39
2.3.1 Experimental Results - Symmetric Behaviour . . . 40
3 Control Design 43
3.1 Modal Control . . . 43
3.1.1 Modes Controllability and Observability . . . 44
3.1.1.1 Modes Selection . . . 46
3.1.2 Modal Control Design . . . 48
3.1.3 Actuators Behaviour Compensation . . . 49
3.1.4 Simulation Results . . . 52
3.2 Decentralized Co-Located Control . . . 54
3.2.1 Advantages of Decentralized Control . . . 54
3.2.2 Decentralized Control Derivation from Centralized Control . . . 55
3.2.2.1 State Space Formulation . . . 56
3.2.2.2 Least-Square Method . . . 58
3.2.2.3 Example and Comparison . . . 62
3.2.3 Further Control Simplification for the Analysed Symmetric Case . . . 65
3.2.3.1 Simplification Dissertation . . . 66
3.2.4 Simulation Results . . . 75
3.3 Control Selection . . . 77
4 Experimental Results 81
Conclusions 87
1.1 Single d.o.f. Scheme . . . 6
1.2 Damping change Response . . . 7
1.3 Skyhook damper: a) possive configuration; b) equivalent active con-figuration . . . 8
1.4 Skyhook Effect . . . 9
1.5 Block Diagram . . . 10
1.6 Control Effect Comparison . . . 12
1.7 (a) Stewart platform; (b) Experimental results of [1] . . . 13
1.8 Sketch of Inertial Mass Actuator . . . 14
1.9 Inertial Mass Actuator Scheme . . . 15
1.10 I.M.A.: a) Reactive Configuration; b) Proof Mass Configuration . . . . 16
1.11 Blocked Forece Responce . . . 17
1.12 Maximum reduction achievable on ω0 decreasing ωa . . . 19
1.13 Reactive vs P.Mass Conf. Control Effect - Comparable Attenuation: (a) Bode diagram; (b) Impulse response . . . 20
1.14 Reactive vs P.Mass Conf. Control Effect - Maximum P.Mass Gain: (a) Bode diagram; (b) Impulse response . . . 21
1.15 Nyquist Plot: a)Unconditionally stable Sys; b)Conditionally stable Sys 24 1.16 Ideal reachable behaviour . . . 27
2.1 MyPhotonics project experiments . . . 29
2.2 LEGO R Plate . . . 30
2.3 Preliminary Measurements: a) Laser Vibrometer; b) Structure Set-up 31 2.4 Measurement points visualization . . . 31
2.5 Experimental F.R.F. of Co-Located point . . . 32
2.6 Coherence: a) Co-located point (6); b) Corner point (66) . . . 33
2.7 Prony Stability Diagram . . . 34
2.8 F.R.F. Reconstruction: a) Co-located point (6); b) Corner point (66) . 35 2.9 Mode Shapes: a) 1st Mode; b) 2nd Mode; c) 3rd Mode; d) 4th Mode; e) 5th Mode; f) 6th Mode . . . 36
2.10 Plate mounting system schceme . . . 37
2.11 Actuator-plate coupling . . . 38
2.13 F.R.F.s of the mounted system . . . 40
2.14 Prony Stability Diagram - Isolation system . . . 41
3.1 Solution of Controllability Gramian - Sum of kinetic and Potential energy . . . 47
3.2 Bode diagram comparison - Centralized vs Ideal Control . . . 52
3.3 Penalty gian ξ changes - Centralized . . . 53
3.4 Decentralized Control Strategies - Lynch [2] . . . 55
3.5 Example’s Root Locus: (a) Original vs Centralized Controlled; (b) Different Control Actions . . . 63
3.6 Penalty gain ξ change: (a) 1st Mode; (b) 2nd Mode; (c) 3rd Mode . . . 64
3.7 Gains Error . . . 72
3.8 Bode diagram comparison - Decentralized vs Centralized Control . . . 75
3.9 Penalty gian ξ changes - Decentralized . . . 76
3.10 Bode diagram comparison - Decentralized Controls vs Centralized One 77 3.11 Penalty gain ξ change comparison: (a) Centralized Control; (b) De-centralized Control on all Modes; (c) DeDe-centralized Control on the 1st Mode; (d) Decentralized Control by Chen . . . 78
4.1 Final isolation system configuration . . . 81
4.2 Integation . . . 82
4.3 Bode diagram comparison - Real vs Simulation . . . 83
4.4 Bode diagram comparison - 10 kHz vs 1 kHz . . . 84
Scalar Quantities
m = Clean Body mass r = Damping
k = Stiffness
x = Clean Body displacement xp= Disturbance displacement
X = Clean Body variable in Laplace domain Xp = Disturbance variable in Laplace domain
ξ = Adimensional damping ratio
ξj = Adimensional damping ratio of the mode j
ω0= Single degree of freedom eigenfrequency
ωj = Eigenfrequency of mode j
f0 = Eigenfrequency in [Hz]
ω = Frequency variable in [rad/s] s = Laplace variable
h = Feedback control gain for a s.d.f. system f = Force
d(s) = Transmissibility function G(s) = System transfer function H(s) = Controller transfer function S(s) = Sensitivity function
K = Constrant related to the number of windings B = Magnetic flux density
l = length of conductor
i(t) = Current flowing into the windings
v(t) = Relative velocity between coil and magnet V (t) = Voltage
L = Inductance R = Resistance C = Capacitance
ψ = Electrical-mechanical coupling coefficient ma= Actuator mass
ra= Actuator damping
ka= Actuator stiffness
ωa= Actuator eigenfrequency
Yik= Transfer mobility
φjf = Mode shape of mode j at point f Ajik = Modal residue of mode j
H1= 1 st
transfer function estimator
N = Number of frequency domain measured points ωh = Measured frequency
˜
Gik(ωh) = Transfer funcion reconstructed response at the frequency ωh
E2 = Error function
EEk = Expected value of Kinetic energy
EV = Expected value of Potential energy βik = Index of Ha´c formulation
p = Penalty gain
C1(s) and C2(s) = Chen compensators
hcab = Centralized control matrix [Hc] element of position (a; b) hPab = Decentralized control matrix [HP] element of position (a; b) hIab = Decentralized control matrix [HI] element of position (a; b) βab,f = Service matrix [βf] element of position (a; b)
rc,ab = Control action damping matrix [Rc] element of position (a; b)
kc,ab= Control action stiffness matrix [Kc] element of position (a; b)
σc,ab = Control action modal damping matrix [σc] element of position (a; b)
Indexes
f = Sensor-actuator couple index j = Mode index
n = number of sensor-actuator couples m = number of modes
Vectors
x = Displacement vector in physical coordinates
y = Displacement output vector in physical coordinates xp = Disturbance displacement vector
z = State vector
w = Output vector in State Space formulation f or fc= Controlling force vector
γ = Modal state vector q = Modal vector q
R= Reduced modal vector
g = Controlling modal force vector ˜
gR= Reduced controlling modal force vector i = Current vector V = Voltage vector
Matrices
[M] = Mass matrix [R] = Damping matrix [K] = Stiffness matrix[Rm] = Scaled damping matrix
[Km] = Scaled stiffness matrix
[B0] = Input matrix related to physical coordinates [C0] = Output matrix related to physical coordinates [A] = State Space matrix
[B] = Input matrix in State Space formulation [C] = Output matrix in State Space formulation [B1] = Consistent submatrix of [B]
[C1] = Consistent submatrix of [C]
[φ] = Mode shapes matrix [σ] = Modal damping matrix [Ω] = Eigenfrequency matrix [S(s)] = Sensitivity function matrix [G(s)] = Transfer function matrix [d(s)] = Transmissibility matrix [H(s)] = General control matrix [W (t)] = Controllability gramian [Q(t)] = Observability gramian [Km] = Modal gains matrix
[M ] = Reducing matrix on input [N ] = Reducing matrix on output
[φc] = Input matrix of the controllable reduced modes
[φo] = Output matrix of the observable reduced modes
[L] = Inductance matrix [R] = Resistance matrix
[ψ] = Electrical-mechanical coupling coefficient matrix [H] = Output voltage control matrix
[Hc] = Centralized control matrix
[HP] = Decentralized control matrix - Proportional part
[HI] = Decentralized control matrix - Integral part
[P ] = Proportional gain matrix [D] = Derivative gain matrix
[Rc] = Damping control action matrix
[Kc] = Stiffness control action matrix
[σc] = Modal damping control action matrix
[βf] = Least-Square service matrix related to sensor-actuator couple f
[Π] = Least-Square service matrix composed by the elements of [βf]
State of the Art
Isolation of components from environmental disturbances is an important topic to prevent possible damage of the component or to carrying on suitable measures re-ducing or neglecting the interference coming from it.
In the field of interest, it is shown by various authors that an active control is nec-essary to avoid the trade-off between low and high frequency isolation performances depending on the damping mount. The challenge in passive isolation is to have a mount as stiff as possible to better support the equipment, from the statically point of view, and as soft as possible to better isolate the equipment, from a dynamic point of view. This is described in deep by Crede in [3].
Active isolation from vibration can be divided in two classes. The first one deals with the isolation of a vibrating body from a receiving structure, the second one deals with the isolation of a body from vibration imposed by another source. The first one, as rotating machine, deals more with deterministic source of disturbances and can be solved using adaptive feedforward control approach as described by Fuller, Elliott and Nelson in [4] and mentioned by Preumont in [5]. The second type, isolation of precise instrumentation from unpredictable noise or isolation of vehicles from track irregularities, deals more with random external excitation and can be solved by a feedback approach.
A simple solution for the second type of problem, that is the case of study, is a skyhook control as shown by Fuller, Elliott and Nelson in [4] and Preumont in [5]. Preumont suggest the usage of absolute velocity feedback or force feedback. The body or structure that must be isolated will be referred as clean body.
Different authors studied the problem and good results are achieved by Beard et. al. in [6] and Bartel et. al. in [7] by means of the usage of piezoelectric materials, in the form of stack [6] or bending beams (bimorphs) [7]. Despite their effectiveness the available stroke is of the order of 10µm for a 1 cm long actuator (stack).
In the field of relatively high displacements, order of millimetres, Kaplow et. al. in [8] studied the isolation of a flexible telescope, using decentralized local feedback by means of the minimization of a cost function, and Karnopp in [9] studied isolation for automotive suspensions, showing good results with the usage of standard velocity feedback control. An in-depth study of the last theme was done by Li et. al. in [10],
where the usage of a linear and non-linear Kalman filter estimation is adopted. Another interesting system is the cubic hexapod, or Stewart platform, that is intro-duced by Preumont in [5] and studied more in details by Thayer et. al. in [1] in terms of comparison in different control approaches.
One more possible solution is the usage of proof-mass actuators (or inertial mass actuator (IMA)). These types of actuators, used in the field of vibration isola-tion, are widely studied by Institute of Sound and Vibration Research, University of Southampton. They can be used properly as inertial actuators or as reactive ac-tuators. Elliott et. al. in [11] analyse the stability limits of the two configurations in which the clean body is considered rigid and the system is subjected to constraint displacement imposed.
The unknown dynamic of the disturbance emitting source influence the system be-haviour and Elliott et. al. in [12] and Yan et. al. in [13] propose simple criteria to judge the conditional stability of the controlled system.
A pure theoretical study is carried out by Sciulli in [14] and [15] about interaction of rigid or not rigid structures for isolation purpose. The structures considered are Euler-Bernoulli beams and their interaction is found thanks to the theory developed by Yang in [16].
OpenAdaptronik Project
This thesis is part of OpenAdaptronik, a project which objective is to create reliable and easy to use vibration control systems. Researches are done to obtain guidelines that help to choose, for non-accustomed persons, actuators, sensors, amplifier, control board, control logics, open-source programs and implementing systems in a cheap and easy way, in order to obtain a “make by your own” portal. Of course, all the produced material is free to be consulted. OpenAdaptronik is a young project and it has the possibility to find a good way on its purpose. It belongs to Fraunhofer LBF – institute for system reliability and structural durability in Darmstadt that belongs to Fraunhofer institute. Institute widely spread on German territory and in research areas as production, energy, biotechnology and so on.
Objective and Organization of the Thesis
Active control is seen to be a good solution for the problem of vibration isolation as discussed in [5]. In the field of interest, as described by Fuller in [4], isolation from unpredictable random excitation, in which feedback control is preferred instead of adaptive feedforward control, is taken into account.
The objective of this thesis is to demonstrate, according to OpenAdaptronik purpose, that an isolation system can be implemented with low cost instrumentation and sim-ple control logic and that the system can be reproducible.
The studied structure belongs to the project MyPhotonics that on it, mounts inter-esting laser interferometer experiments. Both the structure and the experiments of
MyPhotonics are principally made by LEGO R.
In order to achieve the objective, the work is devided into four phases as described in the following.
In Chapter 1 the feedback approach to vibration isolation is studied for single and multiple degrees of freedom. In a first part actuators and control strategies are com-pared and chosen to have the most effective and easy to use system. Then in a second part interaction of structures is analysed in terms of control stability limits and an ideal control strategy is proposed.
In Chapter 2 the mentioned LEGO R structure is analysed to understand its be-haviour. The isolation system implementation is presented using low-cost actuators in Reactive configration. A modal analysis of the presented set-up is done to find the obtained structure parameters.
In Chapter 3, based on the approach proposed in 1, a modal control is presented. A decentralization method is proposed and compared with the one of Chen et. al. in [17] showing good simulation results. The method proposed, based on state space formu-lation of the modal system, leads to a simple proportional-integral output feedback on velocity. A further simplification of the last one, for the specific thesis purpose, is shown.
Lastly, in Chapter 4 experimental results of the implemented isolation system are shown.
Vibration Isolation Problem
Vibration isolation is an important task when vibration of precise equipment, due to external sources, must be neglected, or at least, attenuated. Active control can produce good results in damping natural frequencies achieving disturbances rejection, as shown by various authors.
Different solutions, in terms of control strategies and actuators solutions are here discussed. Due to OpenAdaptronik requirements, some solutions must be neglected, but they are not necessary only a restriction choice.
In the first part, a simplified single degree of freedom (d.o.f.) problem is considered to understand the necessity of active control in vibration isolation and to judge which is the most simple and effective solution for the thesis purpose.
In the second part, multi d.o.f. problem is considered and an ideal feedback control is proposed.
1.1
Single d.o.f. Problem
To understand the problems related to vibration isolation it is useful to start analysing a single d.o.f. system. This analysis will lead to the comprehension of why active con-trol is useful in avoiding the trade-off problem related to classical passive mounting systems.
With reference to Figure 1.1, as introduced by Preumont in [5], the system can be divided into three parts. The first one is the moving mass that represents the equipment to be isolated, defined as Clean Body. The second one is the mounting system represented by a stiffness and a damper in parallel. The third one is the source of disturbance that here is represented by a displacement of constraint imposed. Defining m, the mass of the clean body, k and r, the stiffness and damping of the mounting system and xp, the disturbance displacement, it is possible to write the
dynamic equation of the system represented in Figure 1.1 in time domain as expressed by eq. 1.1.
Figure 1.1: Single d.o.f. Scheme
m¨x + r( ˙x − ˙xp) + k(x − xp) = 0 (1.1)
Passing in the Laplace domain (eq. 1.2), it is possible to define the transmissibility function d(s) as expressed in eq. 1.3.
(ms2+ rs + k)X = (rs + k)Xp (1.2)
X Xp
= d(s) = rs + k
ms2+ rs + k (1.3)
Adimensional damping ratio and resonance frequency of the considered system are defined as in eq. 1.4. ξ = r 2mω0 ; ω0 = r k m (1.4)
To reduce the response of the system to incoming disturbance it is necessary to in-crease the damping in order to dein-crease the amplitude of oscillations in the resonance region. Considering the resonance frequency constant, two areas of importance can be distinguished. The first one is in the neighbourhood of the resonance and the second one is the one after that, at higher frequencies, called seismic region. In these areas, the contribution of adimensional damping ratio is critical in the transmissibil-ity function.
In fact, increasing the adimensional damping ratio ξ, the transmissibility function experiences a reduction in terms of magnitude of response in the neighbourhood of the resonance region, but, at the same time, experiences a decrease in the rate of decay in the seismic region in terms of decibels per decade of the magnitude. The best case for the resonance region and the worst for the seismic one is when ξ = 1.
In fact, as expressed in eq. 1.5, the transmissibility function in this case has a double pole located in ω0 but also a zero located in ω0/2 that leads to a rate of decay of -20
dB in the seismic region.
ξ = 1 → d(s) = 2ω0 s + ω0 2 (s + ω0)2 (1.5) 101 102 103 freq. [Hz] -100 -50 0 50 Magnitude dB ξ = 0.005 ξ = 0.09 ξ = 0.25 ξ = 1 101 102 103 freq. [Hz] -4 -3 -2 -1 0 Ψ [rad] -20 dB
Figure 1.2: Damping change Response
As shown by the Bode diagram of Figure 1.2, the trade-off problem in passive mounts is evident. That means that reducing the response in the resonance region increase the response in the seismic region. Some passive solutions to this problem are given by Crede in [3] and the solution called “Sky-hook”, widely used as a starting point in this type of problems, is discussed in the next section.
1.1.1 Ideal “Sky-hook” Control
One of the possible solution, that can be adopted to solve the above-mentioned trade-off problem, is the so called “Sky-hook” solution.
Figure 1.3: Skyhook damper: a) possive configuration; b) equivalent active configu-ration
This solution is provided by an additional damper h in series with the mounting one. The additional damper is grounded as shown in Figure 1.3 (a), from which the name “Sky”. Thus, the equation of motion in Laplace domain and the Transmissibility function d(s), can be write as expressed in eq. 1.6. With this solution, the influence of the second damper is only on the denominator of d(s).
(ms2+ (r + h)s + k)X = (rs + k)Xp
d(s) = rs + k
ms2+ (r + h)s + k (1.6)
If it is possible to impose the value of the damper as h = 2mω0− r, the denominator
of the transmissibility function d(s) become critical damped (ξ = 1) as shown by eq. 1.7. This solution ensures a -40 dB amplitude decrease after the resonance frequency. In Figure 1.4 the transmissibility functions of the original system, whit non-critical and critical damping, and the “Sky-hook” configuration with critical damping are compared, where it is possible to see the effectiveness of the additional damping.
d(s) = rs + k (s + ω0)2
(1.7) Being not always possible to have a fixed system to attach the additional damper, an equivalent “Sky-hook” solution can be adopted. This solution is based on absolute velocity feedback control, giving back to the system a force proportional to the absolute velocity of the clean body. The scheme is shown in Figure 1.3 (b). The equation of motion of the system in this case is given by eq. 1.8.
101 102 103 f [Hz] -100 -50 0 50 Magnitude dB
Passive sub-critical damping Passive critical damping Active critical damping
101 102 103 f [Hz] -4 -3 -2 -1 0 Ψ [rad]
Figure 1.4: Skyhook Effect
Introducing G(s), the transfer function of the system, eq. 1.8 can be organized as eq. 1.9, in which d(s) is expressed by eq. 1.3.
X = G(s)f + d(s)Xp ; G(s) =
1
ms2+ rs + k (1.9)
Imposing a classical negative feedback control approach the force can be expressed as eq. 1.10. In which h is the control gain.
f = −H(s)X ; H(s) = hs (1.10)
The controlled system can be represented as shown in the block diagram of Figure 1.5.
It is then possible to define the sensitivity function S(s) as the ratio between the clean body displacement (X) and the ground displacement (Xp) of the controlled
Figure 1.5: Block Diagram
S(s) = X Xp
= 1
1 + G(s)H(s)d(s) (1.11)
S(s) can be reduced for this case as shown by eq. 1.12.
S(s) = DG DG+ hs Nd Dd = Nd DG+ hs = rs + k ms2+ (r + h)s + k (1.12)
Eq. 1.12 is equal to eq. 1.6. Thus “Sky-hook” damper and absolute velocity feedback are equivalent and if the control gain is chosen to be h = 2mω0− r the effect is the
same shown in Figure 1.4.
While using a canonical negative feedback control approach whit h > 0, it is possible to demonstrate, using the Routh-Hurwitz criterion, that the controlled system is unconditionally stable, as shown below.
m k r + h 0 h1 h1 = − 1 r + h(−(r + h)k) = k > 0
Feedback approach is especially useful when the source of disturbance has an unpre-dictable behaviour. Otherwise, as discussed by Fuller et. al. in [4] and mentioned by [5], feedforward adaptive control could be more efficient.
1.1.1.1 Control Comparison
In feedback control, direct measure of the velocity could be a problem. In fact, it is necessary a signal derivation for displacement measurements or a signal integration for acceleration ones. This implication can create problems considering the necessity of a simple system.
Thus, two different control approaches are compared to the above-mentioned abso-lute velocity feedback. Those are pure Acceleration Feedback and Linear Quadratic Regulator (LQR) approaches.
1. Acceleration Feedback
Imposing again a canonical negative feedback where the force is expressed in eq. 1.13
f = −H(s)X ; H(s) = as2 (1.13)
The sensitivity function, S(s), can be wrote as expressed by eq. 1.14.
S(s) = rs + k
(m + a)s2+ rs + k (1.14)
Thus, using a strictly positive gain (a > 0), the controlled system is uncon-ditionally stable. The advantages of the pure acceleration feedback are the possibility of directly using the signal coming from the sensor (accelerometer) without manipulation and the unconditional stability of the controller. Unfor-tunately, as shown in Figure 1.6, the controlled system via pure acceleration feedback, at equal control effort to absolute velocity feedback, has no relevant positive effects.
2. LQR
The state space formulation and control logic for LQR control approache are expressed in eq. 1.15.
˙
z = [A] z + [B] u ; u = − [H] z (1.15)
At equal control effort LQR can produce the same effect of absolute veloc-ity feedback on the controlled system, as shown in Figure 1.6, but it carries essentially two drawbacks.
(a) Measurements
Being based on a state space formulation displacement and velocity of clean body must be known at the same time frustrating the first need of measurement simplification. This problem could be solved introducing a Deterministic Observer or Kalman Filter, that are the second drawback. (b) Deterministic Observer or Kalman Filter
The usage of the Observers introduces a degree of complexity that should be neglected for the thesis purpose. Moreover, for structures control Ob-servers could create problems of implementation.
101 102 103 freq. [Hz] -100 -50 0 50 Magnitude dB NC Sky-Hook LQR ACC 101 102 freq. [Hz] -4 -3 -2 -1 0 Ψ [rad]
Figure 1.6: Control Effect Comparison
Proceeding by exclusion, acceleration feedback is discharged for no relevant effects with equal control effort with respect to absolute velocity feedback and LQR is discharged for complexity of implementation. Thus, absolute velocity feedback, or equivalent Sky-hook control, is chosen as control logic to be implemented, being the most simple and effective between the compared ones.
1.1.2 Actuator Selection
For the vibration isolation purpose, different types of actuators can be used. Bartel et. al. ( [6] and [7]) analysed usefulness and applicability of Piezoelectric actuators, Thayer et. al. ( [1]) investigated the possibilities of Cubic Hexapod while Institute of Sound and Vibration Research, University of Southampton proposed the usage of Proof-Mass actuators (or inertial mass actuator (IMA)).
The Piezoelectric actuators solution is tried in two different configurations, Stack actuators in [6] and Bending Beams (Bimorphs) in [7]. This solution is really effective, but the available stroke is of the order of 10µm for a 1 cm long actuator (stack). This order of displacement is sufficient for seismic isolation systems, but for other applications in which the base motion is of the order of millimetres, actuators with a longer throw are required. Moreover, for the project purpose, piezoelectric actuators are too expensive and they must be discarded.
(a) (b)
Figure 1.7: (a) Stewart platform; (b) Experimental results of [1]
Cubic Hexapod solution, or Stewart platform, ( [5] and [1]) has higher displacement range and it is considered for high displacement vibration isolation systems and po-inting systems in aerospace applications. The solutions proposed by Thayer et. al. in [1] dampen well the low frequency region but introduce some overshoots in the high frequency one. In Figure 1.7 (a) the Hexapod and in Figure 1.7 (b) the experi-mental results of Thayer are shown.
For six-axis vibration isolation, Hexapod is a good and effective solution, but it is again discarded, for the thesis purpose, due to the high level of technology and com-plexity required.
Proof-mass actuators are thus considered due to their simplicity and easily under-standable functioning. Elliot et. al. studied their behaviour and possible adoption
for vibration isolation. This type of actuators can manage displacements of the or-der of millimetres and, whit a proper configuration, they can be used as three-axis isolators. Moreover, Proof-mass actuators have the advantage of being essentially loud speakers, thus cheap ones can be found on the market.
The choice of actuators is thus done by exclusion. Piezoelectric materials have too high cost and too low available displacement and Hexapod has a too high technologic complexity. Thus, for this thesis, Proof-mass actuators are chosen having a simple functioning, as it is discussed in the next section (Section 1.1.3), and being easily reachable. In Chapter 2 Section 2.2.2, the choice of the Proof-mass used in this thesis, is discussed.
1.1.3 IMA Actuators Functioning
Figure 1.8: Sketch of Inertial Mass Actuator
Inertial mass actuators are direct drive, limited motion devices that use a permanent magnet field and a coil winding to produce a force proportional to the current applied to the coil. These devices are useful in linear motion applications requiring high acceleration and high frequency actuation. The actuator can be sketched as shown in Figure 1.8.
As discussed in [18] and [19] the electro mechanical conversion mechanism is governed by the Lorentz Force Principle and Amp`ere’s law as sated by eq. 1.16.
f (t) = KBli(t) ; V (t) = KBlv(t) (1.16)
K = Constant related to the number of windings B = Magnetic Flux density
l = Length of the conductor
i(t) = Current flowing into the windings
v(t) = Relative velocity between coil and magnet f (t) = Generated force
V (t) = Generated voltage
Being the magnetic flux generated by a permanent magnet, B is constant, thus the constant KBl can be summarized in a new constant ψ, named electrical-mechanical coupling coefficient.
Eq. 1.17 express the general relationship between voltage and current in a coil. Generally, in an Inertial mass actuator the capacitance C is negligible.
V (t) = Ldi(t)
dt + Ri(t) + C Z
i(t)dt (1.17)
The resulting voltage needed to actuate the moving part is the sum of the coil voltage and the Lorentz one.
Figure 1.9: Inertial Mass Actuator Scheme
The electrical and mechanical system can be considered separated having, however, a mutual dependence expressed by the Lorentz law. The two systems are schematized in Figure 1.9. The mechanical part can be seen as a moving mass, magnet or coil, connected to a static part, coil or magnet, throughout a stiffness-damper system. Thus, taking as reference Figure 1.9, it is possible to express the relative velocity between coil and magnet in eq. 1.16 as v(t) = ˙x − ˙xp. Thus, the constitutive
equations of the Inertial mass actuator can be summarized in a system of three equations as expressed by eq. 1.18, in which the first equation is the time domain equivalent of eq. 1.8.
m¨x + r ˙x + kx = f + r ˙xp+ kxp Ldidt+ Ri + ψ ( ˙x − ˙xp) = V f = ψi (1.18)
Inertial mass actuators can be used in vibration isolation in two different tions, Reactive and Proof-mass configuration (Figure 1.10). In Reactive configura-tion, as shown in Figure 1.10 (a), the moving part of the actuator is rigidly connected to the clean body, while in the Proof-mass one, Figure 1.10 (b), the actuator is con-nected in series with the mounting system. In this last case the static part of the actuator is rigidly connected to the clean body and the moving part of it is free to move and it is used to introduce force into the system.
Figure 1.10: I.M.A.: a) Reactive Configuration; b) Proof Mass Configuration
It is now necessary to understand the stability limits of the two configurations in order to choose the best one for the thesis purpose. In the following analysis, for sake of simplicity, the electrical dynamic behaviour is neglected considering the force directly proportional to the voltage applied to the coil.
1.1.3.1 Proof-mass Configuration – Stability Analysis
A Proof-mass actuator can express force only after its resonance frequency. This condition is expressed by the blocked force response in eq. 1.19 and shown by the Bode diagram of Figure 1.11.
ft fa = mas 2 mas2+ ras + ka (1.19) Consequently, the lower the resonance frequency the larger the frequency range of action of the actuator is. Lowering the resonance frequency generally means reducing
100 101 freq. [Hz] -40 -20 0 20 40 Magnitude dB 100 101 freq. [Hz] 0 1 2 3 4 Ψ [rad]
Figure 1.11: Blocked Forece Responce
the stiffness of the actuator, that can cause a gravity-induced sag proportional to the inverse of the square of the actuator’s resonance frequency as stated by Beranek et. al. in [20]. Thus, the lower the resonance frequency the greater the sag. To overcome this problem Benassi et. al. proposes two methods in [21] and in [22].
However, introducing the resonance frequencies of the two separated systems, actu-ator and clean body ones, ωa and ω0 respectively (eq. 1.20), it is necessary, in order
to introduce force into the system and control it, that ωa< ω0.
ωa= r ka ma ; ω0 = r k m (1.20)
Despite these two preliminary conditions, with reference to Figure 1.10 (b), the sta-bility limit of the Proof-mass configuration can be judge starting from the sensitivity function of eq. 1.21 in which the control logic is still absolute velocity feedback of the clean body as expressed by eq. 1.10.
S(s) = (rs + k) mas 2+ r as + ka as4+ bs3+ cs2+ ds + e (1.21) Where: a = mma b = ma(r + ra− h) + mra c = ma(k + ka) + mka+ rra d = kra+ rka e = kka
Using the Routh-Hurwitz criterion shown below with the additional condition b > 0, it is possible to reach the four stability conditions of eq. 1.22.
a c e b d 0 h1 h2 0 k1 0 f1 h1 = − 1 b(ad − bc) > 0 h2 = − 1 b(−be) = e k1 = − 1 h1 (bh2− h1d) > 0 f1 = − 1 k1 (−k1h2) = h2= e > 0 1) h < (r + ra) + m ma ra 2) h < (r + ra) + m ma ra− ad mac 3) h < (r + ra) + m ma ra− b1 ma 4) h > (r + ra) + m ma ra− b2 ma (1.22) Where b1 and b2 are the solutions of the polynomial of eq. 1.23 deriving from the
Routh-Hurwitz condition k1> 0.
eb2− cdb + ad2 < 0 (1.23)
The first three conditions of eq. 1.22 increase the damping of the actuator itself, while the fourth one acts on the clean body. This condition, according to Elliot et. al. in [11], can be approximated to eq. 1.24 to express the maximum gain achievable.
Here it is possible to notice that, at equal stiffness, the higher the damping of the actuator the higher the absolute value of the maximum gain is. The negative sign is due to the convention choice represented in Figure 1.10 (b).
hmax ' − 2ξamω20 ωa = −rak ka (1.24) In Figure 1.12 controlled and uncontrolled response in frequency domain of the Proof-mass configuration are compared, maintaining constant the parameters of the mount-ing system, the mass of the clean body and the dampmount-ing of the actuator, changmount-ing the resonance frequency of the last one. The gain used for each resonance frequency considered is the maximum achievable one, thus the one that guarantee the maximum attenuation of the clean body resonance.
100 101 102 freq. [Hz] -40 -20 0 20 40 60 80 Magnitude dB Controlled Not Controlled 100 101 102 freq. [Hz] -4 -3 -2 -1 0 1 2 Ψ [rad]
Figure 1.12: Maximum reduction achievable on ω0 decreasing ωa
1.1.3.2 Reactive Configuration – Stability Analysis
For the case of Reactive configuration (Figure 1.10 (a)), maintaining the hypothesis of negligible dynamic influence of the electrical system, the stability limit reduces to
the one discussed in Section 1.1.1, where the total mass is the sum of the clean body and actuators ones and the same for the damping and stiffness. Under the conditions of:
• f = −HX • H = hs ; h > 0
The controlled system, in Reactive configuration, is unconditionally stable. 1.1.3.3 Configuration Choice
In order to understand which is the best configuration choice for the thesis purpose the two configurations, trying to control approximately the same system, are com-pared. In Figure 1.13 the two configuration gives the same attenuation of the clean body resonance frequency while in Figure 1.14 the Proof-mass has a near maximum control gain. 100 101 102 103 freq. [Hz] -100 -50 0 50 Magnitude dB Reac. NC IMA NC Reac. C IMA C 100 101 102 103 freq. [Hz] -4 -2 0 2 Ψ [rad] (a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time [s] -5 0 5 10 15 20 25 30 35 Reac. Control IMA control (b)
Figure 1.13: Reactive vs P.Mass Conf. Control Effect - Comparable Attenuation: (a) Bode diagram; (b) Impulse response
Observing the impulse response of the two cases, two considerations can be done on the Proof-mass configuration. The first one is that, increasing the gain, oscillations appears due to the actuator resonance frequency because the adimentional damping ratio of the actuator ξa is affected negatively by the control gain. The second one is
that a careful choice of the actuator must be done to guarantee that ωa < ω0 and
that the damping of the actuator is sufficiently high to ensure a good attenuation of the clean body resonance frequency.
These two consideration plus the stability limits discussed in Section 1.1.3.1 and farther investigated for a three d.o.f. system by Zimmerman in [23], make easy the choice of Reactive configuration instead of Proof-mass one. Thus, being the
100 101 102 103 freq. [Hz] -100 -50 0 50 Magnitude dB Reac. NC IMA NC Reac. C IMA C 100 101 102 103 freq. [Hz] -4 -2 0 2 Ψ [rad] (a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time [s] -10 -5 0 5 10 15 20 25 30 35 Reac. Control IMA control (b)
Figure 1.14: Reactive vs P.Mass Conf. Control Effect - Maximum P.Mass Gain: (a) Bode diagram; (b) Impulse response
most simple and effective system, Inertial Mass Actuators in Reactive configuration, controlled with a classical negative absolute velocity feedback control, is used in this thesis.
1.2
Multi d.o.f. Problem
It is now necessary to pass from the analysis of a single d.o.f. system to the multi d.o.f. one. Clearly, this introduces complexity to the problem of isolation. Maintain-ing the conclusions done in the previous Section on actuators type, actuators location and type of control, it is possible to write the equation of motion of a generic multi d.o.f. system as expressed by eq. 1.25 in which x is the displacement vector and f the controlling force vector. In this formulation, it is considered that the distur-bance comes from displacement of constraints, as discussed previously, thus, xp is the incoming displacement disturbance vector.
[M] ¨x + [R] ˙x + [K] x = [Bf] f + [Bp] [Rp] ˙xp+ [Kp] xp
(1.25) Considering the usage of reactive configuration, the input matrices, the controlling force one and the disturbance one, are equal, as expressed by eq. 1.26.
[Bf] = [Bp] =B0
(1.26) Generally, it is not possible to have an actuator for each degree of freedom thus the relation of eq. 1.27 holds.
B0 6= [I] (1.27)
Thus, in many cases, the number of actuators are less than the number of d.o.f. to be controlled.
1.2.1 Modal Formulation
Multi d.o.f. systems can be divided in lamped parameters systems and structures. In this thesis, the attention is on structures. For this types of problems, if light damped structures are considered, it is possible to write the equation of motion in modal form. Introducing the relation of eq. 1.28, where q is the modal vector and [φ] is the eigenvector matrix, it is possible to express eq. 1.25 as eq. 1.29 pre-multiplying by [φ]T. x = [φ] q (1.28) [φ]T[M] [φ] ¨q + [φ]T [R] [φ] ˙q + [φ]T[K] [φ] q = = [φ]T B0 f + [φ]T B0 [Rp] ˙xp+ [Kp] xp (1.29) Thanks to the orthogonality property of the eigenvector matrix it is possible to derive the relations of eq. 1.30.
◦ [φ]T [M] [φ] = [I]
◦ [φ]T [R] [φ] = [σ] = diag(σj)
◦ [φ]T [K] [φ] = [Ω] = diag(ωj2) (1.30)
Where σj = 2ξjωj, ωj is the eigenfrequency of the mode j and ξj is the adimensional
damping ratio of the mode j. With the relation of eq. 1.30 it is possible to write the equation of motion in modal form as in eq. 1.31.
[I] ¨q + [σ] ˙q + [Ω] q = [φ]T B0 f + [φ]T B0
[Rp] ˙xp+ [Kp] xp
(1.31) The presence of the choice of reactive configuration is explicit in the terms [Rp]
and [Kp] realted to disturbance while is implicit in the modal parameters and mode
1.2.2 Structures Interaction
Until now the source of disturbance is considered as it would come from displacement of constraint imposed, thus from an object of null mass. If this hypothesis is released it is possible that the dynamic behaviour of the emitting source has influence on the control strategy due to structures interactions being them interconnected by the presence of the actuators in reactive configuration.
A pure theoretical view of this problem is analysed by Sciulli in [14] and [15]. In his work Sciulli studied the interaction of Euler-Bernoulli Beams using the theory developed by Yang in [16] and some ways to isolate one structure from the other. He shows that it is still possible to isolate the clean one from the dirty one by means of absolute velocity feedback.
However, the problem complexity increases when structures other than Euler-Bernoulli Beams are considered, but interaction of structures still has influence of the control performances. Thus, it is necessary to understand in which way this interaction affects the control.
1.2.2.1 Stability Limit
Interaction of structures and possible base dynamics has negative influence on the control that can become conditionally stable. This possibility is studied by Elliot et. al. in [12] and by Yan et. al. in [13]. These two authors try to give criteria to judge the possible stability limits.
Yan bases his criterion on the modes shape of the interacting structures. His method is based on the hypothesis of using decentralized control. Decentralized control is a type of control logic used in structures control that, instead of using a complete network between sensors and actuators, uses each sensor signal as drive signal for only its co-located actuator. This means that it is necessary to have the same number of sensors and actuators. In Section 3.2.1 the advantages of decentralized control with respect to centralized one are discussed.
However, being the method of Yan based on decentralized control it is not possible to use it as a general criterion, but it helps in understanding the nature of the problem. Sensitivity function of eq. 1.11 can now be expressed in matrix form as in eq. 1.32
[S(s)] = ([I] + [G(s)] [H(s)])−1[d(s)] (1.32)
Where:
• [G(s)] = System Transfer Function Matrix • [H(s)] = Control Matrix
Following the Nyquist criterion, to have an unconditionally stable system it is neces-sary that no encirclements of the point (−1; 0) in the Real-Imaginary plane could ever be done by the functions [G(s)] [H(s)]. This implies that if the functions [G(s)] [H(s)] cross the negative real axis the system could become unstable.
As example, it is possible to take the two Nyquist plots of two systems of Figure 1.15. As shown by Figure 1.15 (a), there is no possibility that encirclements of point (−1; 0) could be done, thus the system is unconditionally stable. Instead, in Figure 1.15 (b) the negative real axis is crossed and encirclements could be done, thus the system is conditionally stable.
-5 0 5 -8 -6 -4 -2 0 2 4 6 8 (a) Real Axis Imaginary Axis -2 -1 0 1 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 (b) Real Axis Imaginary Axis
Figure 1.15: Nyquist Plot: a)Unconditionally stable Sys; b)Conditionally stable Sys
In the case studied by Yan the control logic can be expressed as shown by eq. 1.33.
[H(s)] = diag(hk)s ; hk > 0 (1.33)
Thus, being hk> 0, it is necessary that the product [G(s)] s doesn’t cross the negative
real axis to have an unconditionally stable system. Passing in the frequency domain, Yan expresses the mobility of a point q in which is acting an actuator as shon by eq. 1.34. Gq(jω)jω = ˙ xq fa = Yqq− Yqb (1.34)
Where Yqq and Yqb are respectively the input and transfer mobility. If Yqq is smaller
than Yqb the system is conditionally stable. According to Rao in [24], for a multi
Yik = jωGik = ∞ X j=1 jωφjiφjk ω2 j − ω2+ 2jξjωjω (1.35)
In the neighbourhood of a resonance frequency j the mobility simplifies to eq. 1.36.
Yik|ω=ωj ' jωjφjiφ j k 2jξjωj2 = φ j iφ j k 2ξjωj (1.36)
Thus, the two mobility of eq. 1.34 can be expressed as eq. 1.37, in which φjq is the
mode shape of the clean body structure in point q at the mode j and φjb is the mode shape of the disturbance emitting source structure at the mode j.
Yqq|ω=ωj ' φjq 2 2ξjωj ; Yqb|ω=ωj ' φjqφjb 2ξjωj (1.37) Eq. 1.34 can now be write, in the neighbourhood of a resonance frequency j, as eq. 1.38. Gq(jω)jω|ω=ωj ' φjq 2 1 −φ j b φjq 2ξjωj (1.38) This formulation shows that only if condition of eq. 1.39 is satisfy the product [G(s)] [H(s)] could cross the negative real axis and thus become conditionally stable.
φjb φjq
> 1 (1.39)
Condition 1.39 could be verified only if the two modes are in phase and φ j q < φ j b . This means that only if two light weight structures are considered and the disturbance emitting one has strong influence, the system could become unstable.
1.2.2.2 Limits and Aproximation
As discussed in the previous section, the problem rises if interaction of light weight structure is considered. For the thesis purpose, the system to be implemented must be usable in wide range of cases, thus the knowledge of the emitting source behaviour is generally unknown. This leads to the impossibility of the judgement of the stabil-ity limits above depicted.
This limits of knowledge force to consider the systems interaction negligible in a first approach. Moreover, if the disturbance emitting source has no light weight behaviour or has low mode shapes amplitude with respect to the clean body structure ones, displacement of constraint is a good approximation. Thus, in this thesis, displace-ment constraint approximation is chosen and the system is simplified to a system composed by actuators in reactive configuration and clean body structure only.
1.2.3 Ideal Control of Structures for Vibration Isolation
Thanks to the simplification done the system can still be formulated as expressed by eq. 1.31. If it would be possible to introduce into the system modal forces, expressed as a vector g, each mode of the clean body structure could be controlled independently. Thus, the system could be expressed as in eq. 1.40.
[I] ¨q + [σ] ˙q + [Ω] q = g + [φ]TB0 [Rp] ˙xp+ [Kp] xp
(1.40) Following the idea of “Sky-Hook” control with absolute velocity feedback shown in Section 1.1.1, under the hypothesis of modal forces, each mode could be controlled with the modal velocities. Here, if the control logic is expressed as in eq. 1.41, each mode could be critically damped and thus each mode will experience the same effect as absolute velocity feedback on a single d.o.f. shown in Section 1.1.1.
g = −2 (diag(ωj) − diag(ξjωj)) ˙q (1.41)
As example a system with ten modes is considered. In Figure 1.16 are compared the transmissibility function, thus the uncontrolled system response, and the sensitivity function, thus the controlled system response, in a disturbance input point. As it is possible to see a good attenuation is achieved on all modes.
In conclusion, the isolation problem is shown. Firstly, the problem of isolation is studied for a single d.o.f. problem and the necessity of an active control is shown to solve the trade-off problem related to passive isolation systems. The active control chosen is the so called “Sky-hook” control. This type of solution is analysed in terms of effectiveness and it is compared with two different possible approaches. Subsequently the choice of the appropriate type of actuator for the thesis purpose is discussed and different configurations are compared. for the single d.o.f. problem Inertial Mass Actuators are chosen and the most simple and effective configuration is found to be the Reactive one. In a second part the problem of multi d.o.f. isolation is considered. The modal formulation of the problem for a structure is shown and the limits on the control stability due to structures interaction in the field of interest is introduced. Due to forced lack of knowledge, approximation on the considered source of disturbance is done. Lastly, an ideal solution to the isolation problem is proposed.
100 101 102 freq. [Hz] -50 0 50 100 Magnitude dB
Not Controlled Sys Controlled Sys 100 101 102 freq. [Hz] -4 -3 -2 -1 0 Ψ [rad]
Experiment Description
Isolation methods shown in the previous chapter must be related to a specific system. The dynamic behaviour of the system under consideration must be known in order to judge how to counter the incoming disturbances.
In the first part the structure considered, LEGO R structure from MyPhotonics project, is analysed in an independent way from the isolation system due to the particularity of its construction.
In the second part the mounting system for the isolation purpose is defined and dis-cussed.
In the third part the system build in the second one is analysed in order to get the necessary knowledge to develop a control strategy.
2.1
LEGO
RStructure
(a) (b) (c)
Figure 2.1: MyPhotonics project experiments
MyPhotonics is a German project that builds laser interferometer experiments in which the structural parts are made by LEGO R. As OpenAdaptronik, its objective is to create guidelines for a “make by your own” portal that allow to create experiments without precise and expensive equipment. Some of the experiments done by
MyPho-tonics are shown in Figure 2.1.
Due to exposition necessity, the structure carrying the experiments should be isolated from environmental disturbances, otherwise exposition could be compromised. O-penAdaptronik takes the responsibility of implementing the isolation system with low cost instrumentation. The structure to be isolate is the support one, shown in Figure 2.2. For the sake of simplicity, the structure is free from the modular experiments of MyPhotonics.
Figure 2.2: LEGO R Plate
The structure is completely made by LEGO R bricks with a precise order that tries to confer structural elasticity to it. Due to the particularity of the considered structure, it is necessary to understand if linear theory can be adopted or if the structure shows strange behaviours. If linear behaviour could be considered verified, the theory depicted in Section 1.2 can be used.
2.1.1 Laser Vibrometer Analysis
The analysis of the structure is carried out with the usage of a laser vibrometer and the structure is suspended. The suspended structure and the laser vibrometer head pictures are shown in Figure 2.3.
The laser vibrometer used is a 2D type one. Thus, the structure is sampled in sixty-six points on the upper surface. The disposition of the measuring points is the one shown in Figure 2.4.
During the analysis, the structure is exited with a linear shaker that gives to the system a White-Noise Random excitation on the frequency bandwidth from 0 to 1000 Hz. With reference to Figure 2.4, the structure is excited in point 6 in z direction, through the lower surface of it (the side not visible in Figure 2.3 (b)). Point 6 is also a measurement point in order to achieve a co-located measured point. Each point is
(a) (b)
Figure 2.3: Preliminary Measurements: a) Laser Vibrometer; b) Structure Set-up
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 x [m] -0.3 -0.2 -0.1 0 0.1 0.2 0.3 y [m] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 measurement points excitation point plate boundaries
Figure 2.4: Measurement points visualization
measured over a time period of 3.2 seconds and averaged over 15 records. To each signal an Hanning window is applied.
The analysis shows good results in the frequency range investigated. In Figure 2.5 the Frequency Response Function (F.R.F.) is shown. The shown response is the 1st estimator H1 of the structure response of the co-located point (Point 6). As shown
a correct estimate of the F.R.F. in presence of measured noise on the output. 0 100 200 300 400 500 600 700 800 900 1000 freq. [Hz] -140 -120 -100 -80 -60 Magnitude dB 0 100 200 300 400 500 600 700 800 900 1000 freq. [Hz] -6 -4 -2 0 2 Ψ [rad]
Figure 2.5: Experimental F.R.F. of Co-Located point
In Figure 2.6 are reported the coherence functions of the co-located point and one of the other corner points, in this case point 66. As it is possible to notice the coherence function of the co-located point is close to 1 along all the analysed frequency range, while the corner one start decreasing after 400÷450 Hz. Thus, the collected data can be considered usable below this frequency. Moreover, being the excitement signal random, it is possible to judge the linearity of the system through the coherence, as shown by Schwarz et. al. in [27]. Thus, the analysed system has a linear behaviour below 450 Hz.
2.1.2 Modal Analysis and Reconstructed Behaviour
Thanks to the linearity of the structure in the frequency range between 0 and 450 Hz, it is possible to evaluate trough linear methods the parameters of the system. In order to find the modal parameters, eigenfrequencies and adimensional damping ratios, the Prony method is adopted (Weiss et. al. in [28] and Meunier et. al. in [29]). Prony method is a time domain analysis on discrete data achieved from the
0 100 200 300 400 500 600 700 800 900 1000 freq. [Hz] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ 2 (a) 0 100 200 300 400 500 600 700 800 900 1000 freq. [Hz] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ 2 (b)
Figure 2.6: Coherence: a) Co-located point (6); b) Corner point (66)
inverse Fourier transform of the receptance. Receptance can be expressed as in eq. 2.1. Gik(ω) = m X j=1 Ajik ωj2− ω2+ 2jξ jωjω (2.1) Where:
• m = Considered modes number
• Ajik= Modal residue at mode j of point i exited in point k
Prony method solves in a least square sense the characteristic polynomial of the analysed system. Being not possible to know the exact number of poles in the fre-quency range of interest the solution must be iterate increasing the number of poles suggested in the analysed frequency range. If a certain number of poles remains in the threshold level of ±1% the considered pole is stable. A stable pole is an eigen-frequency of the structure analysed. For the found stable poles, the adimensional damping ratio related to them must remain in the threshold level of ±10% otherwise the pole must be discharged.
The stability diagram of Figure 2.7 shows the results of the Prony method on the eigenfrequencies identification. It is possible to notice that the first eight eigenfre-quencies are clearly identified while for the remaining five the stability diagram is a bit confused.This could be caused by a lack of coherence of signals different from the co-located one in the frequency region identified in Figure 2.7 by the cyan bounded region.
100 150 200 250 300 350 400 450 Frequency [Hz] -20 -10 0 10 20 30 40 50 60 70 80 Number of Poles
Figure 2.7: Prony Stability Diagram
Ajik = φjiφjk (2.2)
Thanks to the co-located measure it is possible to define the mode shape of that point being valid the relation of eq. 2.3. Thus, all the other points mode shapes con be found. Ajii= φji 2 → φji = q Ajii (2.3)
Unfortunately, the formulation of the Prony method is ill-conditioned in determining the modal residues. Thus, it is necessary to find the modal residues with a least square method based on the quadratic error function of eq. 2.4 that tries to minimize the error between the measured data and a target function.
E2 = N X h=1 H1ik(ωh) − ˜Gik(ωh) 2 (2.4)
Where:
• N = Number of frequency domain measured points • ωh = Measured frequencies
• H1ik(ωh) = 1st estimator of the measured response function
• ˜Gik(ωh) = Reconstructed response
The reconstructed behaviour found with the analysis above depicted is reported in Figure 2.8 for the F.R.F. of the co-located point and one of the corner points.
0 50 100 150 200 250 300 350 400 freq. [Hz] -140 -120 -100 -80 -60 Magnitude dB [m/N] Measurement Data Reconstructed Signal 0 50 100 150 200 250 300 350 400 freq. [Hz] -4 -2 0 2 4 Ψ [rad] (a) 0 50 100 150 200 250 300 350 400 freq. [Hz] -180 -160 -140 -120 -100 -80 -60 Magnitude dB [m/N] Measurement Data Reconstructed Signal 0 50 100 150 200 250 300 350 400 freq. [Hz] -4 -2 0 2 4 Ψ [rad] (b)
Figure 2.8: F.R.F. Reconstruction: a) Co-located point (6); b) Corner point (66)
The modal reconstruction visualization is shown in Figure 2.9 for the first six modes to give a sample of the results achieved with the analysis done.
Despite the system analysed in this section is not the one that will be used for the isolation purpose, as discuss in the next section, the investigation done is useful to observe that, despite the unconventional nature of the LEGO R structure, a linear behaviour is detectable in the low frequency region and that modal parameters can be found through standard methods of identification.
2.2
Mounting System Set-up
As seen in the previous chapter, the structure can be considered having a linear behaviour, for the case of vibration isolation. Thus, it is now necessary to build the isolation system of the LEGO R plate.
This system must follow the considerations on actuators positioning done in Chapter 1. Moreover, it must fit with the OpenAdaptronik requirements of simple, intuitive, and reproducible system. The purpose of OpenAdaptronik is to create complete
0.6 0.5 0.4 y [m] 0.3 -0.05 0 0 0.2 0.05 0.1 x [m] 0.1 0.2 0.3 0 (a) 0.6 0.5 0.4 y [m] 0.3 -0.05 0 0 0.2 0.05 0.1 x [m] 0.1 0.2 0.3 0 (b) 0.6 0.5 0.4 y [m] 0.3 -0.05 0 0.2 0 0.1 0.05 x [m] 0.1 0.1 0.2 0 0.3 (c) 0.6 0.5 0.4 y [m] 0.3 -0.05 0 0 0.2 0.05 0.1 x [m] 0.1 0.2 0 0.3 (d) 0.6 0.5 0.4 y [m] 0.3 -0.05 0 0.2 0 0.05 0.1 x [m] 0.1 0.1 0.2 0 0.3 (e) 0.6 0.5 0.4 y [m] 0.3 -0.05 0 0 0.2 0.05 0.1 x [m] 0.1 0.1 0.2 0 0.3 (f)
Figure 2.9: Mode Shapes: a) 1st Mode; b) 2nd Mode; c) 3rd Mode; d) 4th Mode; e) 5th Mode; f) 6th Mode
systems composed by cheap and reliable components. It is not possible to have all the requirements at ones thus, here the attention is focused on the actuators selection and configuration choice.
2.2.1 Actuators and Sensors Location
The choice of the actuator positioning must be in reactive configuration as discussed in Chapter 1. This choice takes the additional advantage of leaving the surface of the structure free to be used for the experiments implementation of MyPhotonics. In a first approach, it is possible to consider the LEGO R structure as a plate. As discussed by various authors among which Kim et. al. in [30], one possible config-uration, for the vibration isolation of this type of problem, is to place the actuators in the four corners in a symmetric way, as shown in Figure 2.10.
With this configuration three rigid body motion are allowed at frequencies greater than 0 Hz. These are the vertical displacement, the pitch rotation and the roll one, while the two horizontal displacements and the yaw rotation are set to 0 Hz. The necessity of simplicity drive the choice of the sensors positioning to the co-located solution. In fact, with this configuration it is possible to get a decentralized formu-lation of the control logic. As briefly discussed in Section 1.2.2.1, a decentralized
Figure 2.10: Plate mounting system schceme
control needs the co-location of sensors and actuators, because the signal coming from one sensor is directly fed back to its actuator without having interactions with the other sensors signals. Thus, every sensor-actuator couple create a local subsys-tem independent from the other subsyssubsys-tems reducing the complexity of the control logic.
2.2.2 Actuators choice
-Electrical and Mechanical Behaviour
According to OpenAdaptronik requirements the actuator to be used must be cheap and easy to find. Other members of the project selected various types of Inertial Mass Actuators that can be used in vibration control and fits the requisite. Some of these are directly derived from laud speakers and all of them can be easily find on the web. On OpenAdaptronik site discussion and selection of the actuators are shown.
The selected actuator must be able to support the structure and control it. The one that fits the requests is the RockWood Bass-Shaker (100 W), that it is possible to find on the web at a price between 15 and 25 euro and it has a useful run of ±2 mm, run sufficient for the isolation purpose.
The mechanical and electrical nominal parameters of the chosen actuator are reported in Table 3.1.
The total weight of the LEGO R structure is approximately 5.9 kg. Each actuator must carry approximately a quarter of the total weight. The static deformation derived from it can be calculated and it is around 0.6 mm. The useful run in negative direction is reduced but it is still sufficient for the purpose.
The electrical behaviour is not a relevant problem, in fact it is possible to compensate it with the control logic, as it will be discussed in Chapter 3 Section 3.1.3.
Nominal Measurement value units m 0.2727 [kg] r 2 [N/(m/s)] k 22387 [N/m] ψ 1.6444 [N/A] [V /m] R 3.8994 [Ohm] L 0.5 [mH]
Table 2.1: Nominal values of RockWood actuator
2.2.3 Physical Coupling
The physical coupling is done by gluing two LEGO R bricks on the top of the moving part of the actuator. For RockWook actuators, and in general for IMA derived from loud speakers, the moving part is the magnet and not the coil. The coupling bricks are then connected to the plate as normally LEGO R brick does.
Figure 2.11: Actuator-plate coupling
This solution drives to two considerations. The first one is that the original LEGO R structure can be removed from the mounting system without compromise its integrity and can be easily attach again to the actuators. The second one derives from the analysis done in the previous section. As it was possible to notice the embed between the brick is sufficiently high to maintain a compact structure and permit a behaviour, at least at the analysed frequencies, similar to the one of a continuous structure. Also for this case it is possible to consider the embed of the bricks sufficiently strong to create a rigid connection between the LEGO R structure and the moving part of the actuator.
The second consideration fits with the conclusions done in Chapter 1 Section 1.2.1. Thus, from now on, the magnet, moving part of the actuator, is consider part of the entire system to be controlled and thus the modal parameters of it, identified in the modal analysis that follows, carries the information of the actuators’ moving part.
2.3
Modal Analysis of System Set-up
The knowledge of the modal parameters of the structure is needed to understand how to counter the incoming disturbances as discussed in Section 1.2.3.
In order to find the mode shapes, eigenfrequencies and adimentional damping ratios of the structure plus the isolation system the disturbance source should be neglected. Being the actuators used inertial type ones they are composed by two parts as dis-cussed in Section 1.1.3. The magnet is rigidly attached to the structure while the coil is the part that introduce the disturbance. In order to make a modal analysis not affected by the disturbance, the coils of the actuators are rigidly mounted on a service structure. The service structure is than fixed on the analysis’ table, table with a great mass that should take to zero the incoming disturbance.
In Figure 2.12 the measurement set-up is shown. The analysis in this case is done with the dynamometric hammer in the corner points where the actuators and sensors are positioned. In Figure 2.12 the sensors and actuators are numerated. This numeration is taken as reference from now on.
