Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 2, pp. 479-504, Warsaw 2010 FOURIER EXPANSION SOLUTION FOR A SWITCHED SHUNT CONTROL APPLIED TO A DUCT Salvatore Ameduri The Italian Aerospace Research Centre, Capua, Italy e-mail: s.ameduri@cira.it Monica Ciminello ”Federico II” University of Naples, Aerospace Engineering Department, Naples, Italy e-mail: monica.ciminello@gmail.com In the present work, a semi analytic approach aimed at estimating the effects on reduction of the pressure sound level by synchronised switched shunt logic is described. The displacement field within a 1D longitudinal air column through a Fourier series expansion has been formalised by assigning a sinusoidal perturbation and fluid-structure interface condi- tion on the left and right boundaries, respectively. To simulate the no control operative condition, the solution has been computed for the en- tire time domain, keeping invariant all circuitry properties; then for the switch working modality, the solution has been computed by splitting the entire time domain into partitions; for any partition, specific circu- itry properties (e.g. piezo voltage, electrical field...) have been selected. Based on the displacement information, the related sound pressure level has been compared for no controlled and controlled operative conditions, with and without signal amplification. Key words: synchronised switched shunt control, piezoelectric, pressure sound level Nomenclature Tel,Tmech – electrical circuit andmechanical period, respectively Ttot – total simulation time t0 – initial instant x,y,z – axis, fluid column length, depth and height, respectively U0 – excitation amplitude 480 S. Ameduri, M. Ciminello u0(t) – displacement at x=0 p(x,t) – pressure ω – excitation angular frequency c – sound speed ξ – fluid decay rate L – fluid column length Mp – piezo transmitted moment d – half beam span b – depth of the column and of the beam ζ – piezo extension along the beam span, ζ ∈ [0,1] D – bending stiffness ρ – fluid density E0 – boundary conditions constant term β1,β2 – boundary conditions coefficient, proportional to velocity and displa- cement, respectively f(x),g(x) – initial displacement and initial velocity law, respectively v(x,t) – function used for boundary conditions homogenisation u(x,t) – semi-analytic solution, displacement distribution P(x,t) – differential equation right hand term of the boundary conditions homegeneous problem Q(x,t),R(x,t) – displacement and velocity initial conditions right hand term of the boundary conditions homegeneous problem, respectively X(x),T(t) – space and time depending factor of the solution, respectively λn – n-th problem eigen value ϕn(x),bn(t) – n-th orthogonal space and n-th time dependent function used for Fourier series expansion, respectively pn(t),qn(t) –n-th coefficient ofFourier series expansionof P(x,t) and Q(x,t), respectively D1n,D2n – n-th general integral constants An,Bn – n-th particular integral constants µn,νn,rn(t) – n-th coefficient of Fourier series expansion of f(x), g(x) and R(x,t), respectively p̂ – pressure squared value, averaged with respect time and space d31,g31 – piezoelectric charge and voltage constant, respectively Yp,Ys – piezoelectric and structure Youngmodulus, respectively νp,νs – piezoelectric and structure Poisson modulus, respectively Fourier expansion solution for a switched shunt... 481 1. Introduction Sound and vibration control can be considered a real technological challenge because of the large amount of related problems and peculiar complexity. As a consequence, many efforts have been spent on defining, realising and cha- racterising different typologies of control techniques, tailored on the specific problem. Among the different strategies, due to their promising properties in terms of lightness, simple design and low cost, a wide amount of interest has been focused on the shunt circuits. Through the shunt control architectures, struc- tural vibrations are reduced by using time variant electric circuits integrated with electromechanical PZT suitably positioned on structural elements. The first three schemes illustrated on the left in Fig.1 represent the most commonpassive shunt circuits.A lot ofworks in literature (Erturk and Inman, 2008; Hagood and Von Flotow, 1991; Lesieutre, 1998; Park and Inman, 1999) have shown how the resistive shunt dissipates energy through Joule effect, the capacitive shuntchanges the local stiffnessof the structure,while shuntingwith inductive introduces an electrical resonance, which can be optimally tuned to the one of the system, analogously with a mechanical vibration absorber. Fig. 1. Configuration of four different shunted circuits It is well known that passive techniques are among the most commonly adopted because they never provide the structure with artificial energy and their functionality is essentially based on time invariant (fixed) modifications of the structural mass, damping and stiffnessmatrices. However, despite their easy implementation and lowcost, their performance is generally inadequate to face all the necessities, particularly concernedwith optimal strategies of smart structures solutions. In this case in fact, the control system characteristics have to satisfy some important requirements, among all, adaptability of the parameters. This is the reason why in recent years there has been a growing interest in the semi-active control. A semi-active device can be broadly defined as a passive device in which the properties (stiffness, damping, etc.) can be va- ried in real time with a low power input. Although they behave in a strongly 482 S. Ameduri, M. Ciminello nonlinear way, semi-active devices are inherently passive and cannot destabi- lise the system. They are also less vulnerable to power failure and have good thermal stability, particularly useful in aerospace applications (DeMarneffe et al., 2008; Preumont, 1997). These reasons, jointly with the good performance exhibited within the low frequency range, justified the large amount of the- oretical, numerical and experimental investigations. Among the shunt schemes sketched inFig.1, the last on the right represents the switch architecture. The principle of the shunt using a switch is to store the electric charge and use its effect in opposition to the structural movement within a very short time constant. One of the first concepts of commutation of a shunted circuitwas proposed in Clark (1999, 2000), where the author studied a case of resistive shunt with comparable openand closed circuit periods (this is the reasonwhy it is referred as ”state switching”). This kind of system can be assimilated to a variable stiffness mechanical system. InRichard et al. (1999, 2000), the authors proposed to close the circuit for a very short period and to add an inductor to augment the charge on thePZT device. This technique is called ”Synchronized Switch Damping”, that can be specialised according to the following shunt architecture: • Synchronized Switch Damping on Short, where the shunt is purely resi- stive; • Synchronized SwitchDamping on Inductance, with an inductive compo- nent; • Synchronized Switch Damping on Voltage source, where the shunt inc- ludes a voltage source (Lefeuvre et al., 2006)] (which place the system in the class of the active control, needing an external power supply) and involves a risk of instability. This problem can be mitigated with a slow variation of the voltage following the average amplitude of the vibrations, as proposed in Lallart et al. (2005). In the cited works dealing with this technique, the dampingwas estimated with a simple 1Dmodel.Thedevelopment and implementation ofmdofmodels have also been faced. The team of Clark (Corr andClark, 2001, 2002; Corr, 2001) compared the state switching and the synchronised switching showing that the last one is more effective. Further improvementswere presented inCorr andClark (2003) where the authors developed the switching synchronisation technique based on different modal filters. The technique required complex filters and power supply, but the performancewas good. This kind of work is at themoment an object of another American staff (Collinger andWickert, 2007). Fourier expansion solution for a switched shunt... 483 The team of Daniel Guyomar refined the inductive switched shunt and proposed an autonomous circuit (Petit et al., 2002, 2004). They showed that the detection of a local maximum is not optimal for the case of multimodal control. The authors proposed probabilistic criteria (detection of maxima si- gnificantly exceeding the average level) giving a good result (Guyomar and Badel, 2006). The necessity of extending benefits due to this technique to more realistic applications has led to numerical solving schemes, prevalently based on a FE approach. Some test cases have been carried out, including this time some examples also on elasto-acoustic systems. In their works, Ameduri, Ciminello et al. (2008a-c) studied the finite ele- ment formulation of a synchronised switched shunt applied to both isotropic and anisotropic structure with collocated PZT patches. Themultimodal con- trol was optimised by a genetic algorithm. Finally, an original circuit based on a tachometer component was presented. However, due to the complexity of real applications, despite the efficient reduction techniques employed, numerical computations result heavyand, con- sequently, time consuming. On the contrary, a semi-analytical solution would allow eliminating the time consumption due to the integration. Inhiswork,Ameduri et al. (2007) facedwith the implementation of a semi- analytical solution of a however complex structural system suitably reduced. Then, the analytical solution to the related system of differential equations, for a sinusoidal and constant excitation, was found out; finally, the theoreti- cal solution was fitted to the specific problem, i.e. the switch shunt control implementation. InDucarne et al. (2007), amulti degree-of-freedom (dof) electromechanical model of a structure with piezoelectric elements coupled to state switching and synchronous switching electric circuits was derived. By restricting the analysis to onemechanical dof only, the system free responsewas analytically obtained. A similar analysis was conducted to obtain the forced response of the structure subjected to harmonic forcing of any frequency, except that no analytical expressions were available. As a general conclusion, it was proved that the only parameter that influences the performances of the synchronous switching devices was the coupling coefficient, that had to be maximised in order to enhance the vibration attenuation. The interest in extending switched shunt control benefits also to acoustic applications is confirmed by the amount of numerical and experimental inve- stigations carried out as regards (Ducarne et al., 2007; Guyomar and badel, 2006; Petit et al., 2002). These works deal with the problem of controlling 484 S. Ameduri, M. Ciminello the sound transmission properties of structural elements used to insulate an internal noise source from the external environment. Another problem is the soundpressure level attenuationwithin enclosures. The fundamental idea is to implement a control acting on a PZTnetwork, suitably distributed on the bo- undary of the enclosure (Ciminello et al., 2008b). The effects originated from this solution can be appreciated by predicting and/or measuring the sound pressure level at different points of the air volume taken into consideration. In the work at hand, the attention is paid to this last problem: pressure sound level attenuation through a switched shunt architecture implemented on a metallic plate, acting on a finite length air column. A semi-analytical approach, consisting in solving the telegraph equation through the Fourier expansion series strategy, has been adopted to find out the time dependent displacement field along the 1D horizontal air column. The related boundary conditions have been assigned by imposing a sinusoidal perturbation on the left frontier andby formalising the fluid-structure interaction on the right side. Here, a couple of PZTpatches bonded on the two faces of an aluminiumplate and connected to an external switched shunt circuit, provide the control action on the air column. Before computing the solution, a preliminary validation process has been carried out by verifying the satisfaction of the assigned conditions and proving its convergence by estimating the series coefficients. Then the simulation of the displacement field within the fluid domain, in presence and absence of control, at different amplification levels, has been per- formed. For the no control condition, the solution has been computed within the considered time interval; on the other hand, due to the non-linearity of the switch architecture, the controlled solution has been estimated at a different time interval: any partition is limited by the instants at which the circuit is switched on and electrical properties (i.e. voltage, electrical field, charge on the leads) undergo variation. The results have been expressed in terms of the displacement and sound pressure level computed in different points of the spatial domain; finally, the global attenuation achieved has been evaluated by squared average values of the sound pressure level over spatial and time domains. 2. Switched shunt control system The adopted electrical network is an RLC resonant circuit having the PZT as capacitor. In Fig.2, the circuit is sketched. Fourier expansion solution for a switched shunt... 485 Fig. 2. Synchronised switched shunt circuit A couple of collocated PZTpatches are bonded to the structure. The idea is to generate the control force that opposes itself to the motion with the maximum amplitude and without tuning requirement. To this end, the ”on” state (switch closed) is synchronisedwith themaximumsignal detected by the sensors. This produces the maximum flowing charge into the inductor which sends it reversed to the actuator. Thismeans that the control force is in phase opposition with respect to the local displacement of the structure. Finally, the switching time, i.e. the period the circuit is switched on, is generally assumed 10 to 50 times lower than themechanical period to control. The inductive element of the circuit is thus chosen according to Eq. (2.1), relating such an element to the piezo capacitance and the electric angular frequency (Clark, 1999, 2000; Ciminello et al., 2008a,c; Corr and Clark, 2001, 2002, 2003;Corr, 2003;HagoodandvanFlotow, 1991; Lesieutre, 1998;Richard et al., 1999, 2000) L= 1 ω2C (2.1) Moreover, if the switchmechanism is set according to thehighest frequency of interest, the natural band range of the control system is naturally defined. To summarise circuit workingmodalities: Open Circuit State: in absence of any shunted configuration, that is to say, no connection to a passive electrical network, no current flows and the voltage is a function of the displacement; Shunted Circuit State: every time the piezo voltage reaches a maximum, the switch is closed.Theconnectionof thePZTelectrodes to the external circuit is realised.Thevoltage is givenby two contributions: the opencir- cuit signal (proportional to the deformations) added to the offset signal (proportional to the electrical charge). 486 S. Ameduri, M. Ciminello Fig. 3. Switch signal simulation Theswitchingmechanismproducesnaturally amplifiedvoltage (Fig.4) and the charge behaviour as shown in Fig.5. Fig. 4. Voltage signal simulation The collocated configuration of the sensor-actuator and the absence of an external power supply, that means no energy injected into the system, guarantee unconditional stability of the control. Some drawbacks can be found in the inner behaviour of the control. The effect of this kind of control is in fact the reduction of the vibration ampli- tude not by damping but subtracting a fraction of the mechanical energy of the system at resonance and giving it back, transferring energy to the high frequencies. Fourier expansion solution for a switched shunt... 487 Fig. 5. Charge signal simulation 3. Problem formulation and solving strategy The considered physical problem is sketched inFig.6.A1Dair columnexcited on the left boundary (x=0) by a signal u0(t) u0(t)=U0 sin(ωt) ∀t (3.1) and controlled on the right by twoPZTpatches bonded to an aluminiumalloy plate (see detail at the bottom of Fig.6). Fig. 6. Scheme of the problem 488 S. Ameduri, M. Ciminello The displacement field u(x,t) within the air column is described by the telegraph equation ∂2u ∂t2 − c2 ∂2u ∂x2 + ξ ∂u ∂t =0 (3.2) resulting from the application of themass equation and the balance of inertial (time 2nd derivative), elastic (spatial 2nd derivative) and dissipative (time 1st derivative) forces, acting on a fluid element. The boundary condition on the left side of the domain, coherently with (3.1), is given u(0, t) =u0(t)=U0 sin(ωt) ∀t (3.3) The other boundary condition on the right side of the domain (x=L) results from the formalisation of the fluid/structure interaction at the interface and from the semi-active nature of the switched shunt control. The structure, a plate very long in the z direction, behaves according to Timoshenko’s ”long rectangular plate” theory (Timoshenko andWoinowsky-Krieger, 1959). The acoustic pressure at the interface, p(L,t), jointly with the piezo ac- tion (moment Mp(t)), produces the displacement of the middle of the plate, up(y=0), described by the classical elastic beam theory up(t)= 3 8 p(L,t)b D d4− Mp(t) D ζ ( ζ− 3 2 ) d2 (3.4) The same theory provides also the deformation within the piezo, εp. Then, by considering the piezoelectric constitutive law (Preumont, 1997), the open circuit voltage V can be computed V = g31 Eptp 1−νp εp (3.5) with b and D being the plate depth and bending stiffness, respectively. This voltage, used by the logic to detect the instant at which the circu- it must be switched on, is modified according to what was explained in the previous Section (Fig.4, comparing the open state and circuit generated vol- tages). By exploiting the Crawley and De Luis transmission model (Crawley and Luis, 1987) that proposes a linear relation between transmitted actions and applied voltage, the moment Mp(t) can be computed. Since the fluid and plate displacements are the same at the interface u(L,t)=up(t) ∀t (3.6) Fourier expansion solution for a switched shunt... 489 and recalling that p(x,t)=−ρc2 ∂u ∂x (3.7) Eq. (3.4) can be finally rewritten to formalise the required boundary condition β1 ∂u ∂x ∣∣∣∣ L +β2u(L,t)=E0 ∀t (3.8) where β1 =1 β2 = 8 3 D ρc2bd4 E0 =− 8 3 Mp(t) ρc2bd2 ζ ( ζ− 3 2 ) (3.9) For sake of simplicity, no inertial and damping terms have been taken into account in condition (3.8). This, as alreadymentioned, restricts the validity of themodel to frequencies far fromthe structural resonance. Suchanassumption is however coherent with the low band range of interest for the switched shunt control. The initial conditions are represented by the displacement and velocity at the time t0 described by the assigned functions f(x) and g(x) u(x,t)= f(x) ∂u ∂t ∣∣∣∣ t0 = g(x) ∀x (3.10) To compute a semi-analytical solution, through a Fourier series expansion, boundaryconditions (3.3) and (3.8) have to becomehomogeneous (Haberman, 1987). It is possible to demonstrate that this can be achieved by assuming the function v(x,t)=u(x,t)+ ( β2x β1+β2L −1 ) U0 sin(ωt)− E0x β1+β2L (3.11) After introducing relation (3.11) into Eq. (3.2) and the related boundary con- ditions, (3.3) and (3.7), and initial conditions, (3.10), a new problem is found ∂2v ∂t2 − c2 ∂2v ∂x2 + ξ ∂v ∂t =P(x,t) (3.12) Boundary conditions read v(0, t) = 0 β1 ∂v ∂x ∣∣∣∣ L +β2v(L,t)= 0 ∀t (3.13) initial conditions read v(x,t0)= f(x)+Q(x,t0) ∂v ∂t ∣∣∣∣ t0 = g(x)+R(x,t0) ∀x (3.14) 490 S. Ameduri, M. Ciminello where P(x,t)=U0ω ( β2x β1+β2L −1 ) [ξcos(ωt)−ω sin(ωt)] Q(x,t)= ( β2x β1+β2L −1 ) U0 sin(ωt)− E0x β1+β2L (3.15) R(x,t)=U0ω ( β2x β1+β2L −1 ) cos(ωt) Equation (3.12) may be solved by separating space and time variables, i.e. assuming the solution v(x,t) as a product of the two unknown functions, X and T , the former depending on x, the latter on t v(x,t)=X(x)T(t) (3.16) As a result, Eq. (3.12) is split into two separated problems d2X dx2 +λ2X(x)= 0 d2T dt2 + ξ dT dt + c2λ2T(t)= 0 (3.17) The solution to (3.17)1 can be expressed as a sum of infinite orthogonal func- tions, ϕn(x), satisfying conditions (3.13) ϕn(x)= sin(λnx) (3.18) being λn the nth root of the transcendental eqauation (Boyce and DiPrima, 2008) β1λn+β2 tan(λnL)= 0 (3.19) By expressing the solution v(x,t) as a sum of the products of these functions with the corresponding time dependent ones, bn(t) v(x,t)= ∞∑ n=1 ϕn(x)bn(t) (3.20) Eq. (3.10)2 can be written into a new formalism ∞∑ n=1 ϕn(x) (d2bn dt2 +ξ dbn dt + c2λ2nbn(t) ) =P(x,t) (3.21) This relation, by introducing the Fourier expansion of P(x,t) P(x,t)= ∞∑ n=1 ϕn(x)pn(t) (3.22) Fourier expansion solution for a switched shunt... 491 with pn(t)= L∫ 0 P(x,t)ϕn(x) dx L∫ 0 ϕ2n(x) dx = 2U0ω[ξcos(ωt)−ω sin(ωt)] [cos(λnL)sin(λnL)−λnL] (3.23) reduces itself to a time dependent equation d2bn dt2 + ξ dbn dt +c2λ2nbn(t)= pn(t) (3.24) Equation (3.24) assumes the solution in the form bn(t)= [D1n sin(ωnt)+D2ncos(ωnt)]e − 1 2 ξt ︸ ︷︷ ︸ general integral +An sin(ωt)+Bncos(ωt)︸ ︷︷ ︸ particular integral (3.25) with ωn = √ c2λ2n− ξ2 4 (3.26) The constants An and Bn can be determined by substituting the particular integral into Eq. (3.24) An =2U0ω 2 ξ 2− (c2λ2n−ω 2) [ξ2ω2+(c2λ2n−ω 2)2][cos(λnL)sin(λnL)−λnL] (3.27) Bn =2U0ωξ c2λ2n [ξ2ω2+(c2λ2n−ω 2)2][cos(λnL)sin(λnL)−λnL] The remaining constants, D1n and D2n, can be computed by imposing that v(x,t), formalised as in (3.20) and including (3.25), satisfies initial conditions (3.14): — 1st initial condition D1n sin(ωnt0)+D2ncos(ωnt0)= [−An sin(ωt0)−Bncos(ωt0)+µn+qn(t0)]e ξ 2 t0 (3.28) — 2nd initial condition D1n [ ωncos(ωnt0)− ξ 2 sin(ωnt0) ] −D2n [ ωn sin(ωnt0)+ ξ 2 cos(ωnt0) ] = (3.29) = [−Anωcos(ωt0)+Bnω sin(ωt0)+νn+rn(t0)]e ξ 2 t0 492 S. Ameduri, M. Ciminello being µn, νn, qn and rn the terms of the Fourier series expansion of f(x), g(x), Q(x,t) and R(x,t), respectively µn = L∫ 0 f(x)ϕn(x) dx L∫ 0 ϕ2n(x) dx νn = L∫ 0 g(x)ϕn(x) dx L∫ 0 ϕ2n(x) dx qn(t)= L∫ 0 Q(x,t)ϕn(x) dx L∫ 0 ϕ2n(x) dx = 2E0 λn(β1+β2L) − 2U0 sin(ωt) λnL− sin(λnL)cos(λnL) (3.30) rn(t)= L∫ 0 R(x,t)ϕn(x) dx L∫ 0 ϕ2n(x) dx =− 2U0cos(ωt) λnL− sin(λnL)cos(λnL) Finally, by assembling (3.11), (3.18), (3.20) and (3.25), the required solution to problem (3.2) can be written u(x,t)= ∞∑ n=1 sin(λnx){[D1n sin(ωnt)+D2ncos(ωnt)]e − ξ 2 t+ (3.31) +An sin(ωt)+Bncos(ωt)}− ( β2x β1+β2L −1 ) U0 sin(ωt)+ E0x β1+β2L The corresponding pressure field, p(x,t), may be obtained by deriving with respect to x the above solution andmultiplying it by −ρc2 p(x,t)=− 1 ρc2 ∞∑ n=1 λncos(λnx){[D1n sin(ωnt)+D2ncos(ωnt)]e − ξ 2 t+ (3.32) +An sin(ωt)+Bncos(ωt)}− β2U0 sin(ωt)+E0 β1+β2L The corresponding pressure squared value averaged with respect to space and time has been computed p̂= 1 Ttot t0+Ttot∫ t0 1 L L∫ 0 p2(x,t) dxdt (3.33) Fourier expansion solution for a switched shunt... 493 For the no control case, simulation relations (3.31) and (3.32) can be exploited to compute the displacement and pressure field within the entire considered time interval. On the other hand, for the switched control simulation, due to the discontinuity of circuit parameters, like voltage, charge and hence E0, the use of (3.31) and (3.32) is restricted to the time intervals in which the circuit parameters keep constant, that is during the switch on and off stationary states. In practice, the solution at any time interval is computed by assuming as initial conditions the configuration (in terms of displacement and velocity) computed at the last instant of the previous time interval. 4. Numerical results The results herein presented have been obtained considering the parameters summarised in Table 1. Before computing the displacement field (in Fig.14) and corresponding pressure distribution, a preliminary validation process has been carried out on solution (3.31). At first, the validation of boundary conditions (3.3) and (3.7) has been proved, just expressing relation (3.28) at x = 0 and x = L. Then, the co- nvergence of coefficients An, Bn, D1n, D2n has been verified by exciting the system far and at the fluid resonance. In Figs.7-9, the mentioned coefficients vs. harmonic order n have been plotted at 50Hz and for the first two normal modes. Fig. 7. Fourier series coefficients vs. harmonic order at an excitation frequency of 50Hz 494 S. Ameduri, M. Ciminello Table 1. Simulation parameters Main simulation parameters Time interval 1st resonance frequency [Hz] 170 2nd resonance frequency [Hz] 340 Frequency considered for out 50 of resonance simulations [Hz] Fluid column properties Length [m],L 1 Depth [m], b 0.5 Height [m], 2d 0.06 Density [kg/m3], ρ 1.19 Sound speed [m/s], c 340 Decay rate [dB/s], ξ [28] 200 Beam properties In-plane (yz) dimensions [m] 0.06×0.5 Thickness [m], ts 5 ·10 −4 Youngmodulus [GPa], Ys 72 Poisson ratio, νs 0.33 Density [kg/m3], ρs 2700 1st resonance frequency [Hz], fs 737 Piezo properties In-plane (yz) dimensions [m] 0.03×0.5 Thickness [m], tp 5 ·10 −4 Youngmodulus [GPa], Yp 59 Poisson ratio, νp 0.32 d31 [m/V] −35 ·10 −11 g31 [Vm/N] 8 ·10 −3 The evident convergence allows for considering a small order for a good solution accuracy. After this preliminary validation, the results in terms of the displacement and soundpressure level have been computed in the time domain.At first, the lowest resonance frequency of the structure has been estimated and assumed as the upper threshold, coherently with the assumption boundary conditions (3.7) based on the absence of structural inertial and damping actions. Thenormaliseddisplacementvs. timeandcolumnaxis, at 50Hzand for the first two resonance frequencieswithout control has been plotted inFigs.10-12. Fourier expansion solution for a switched shunt... 495 Fig. 8. Fourier series coefficients vs. harmonic order at the 1st resonance frequency Fig. 9. Fourier series coefficients vs. harmonic order at the 2nd resonance frequency Fig. 10. Normalised displacement vs. time for x/L=0, 0.25, 0.50, 0.75, 1 at 50Hz 496 S. Ameduri, M. Ciminello Fig. 11. Normalised displacement vs. time for x/L=0, 0.25, 0.50, 0.75, 1 at the 1st resonance frequency Fig. 12. Normalised displacement vs. time for x/L=0, 0.25, 0.50, 0.75, 1 at the 2nd resonance frequency Thecorrespondingmodal shapesof thefluidcolumnhavebeenextracted at a fixed time in the steady state regime, from the last two figures (see Fig.13). Thenormalised displacement at x=L vs. time, for the first two resonance frequencies and gain amplification of 1 and 10 have been compared with no controlled response in Figs.14 and 15. In Fig.14, one can easily see the circuit ability of inducing also a phase shift (Petit et al., 2002, 2004; Richard et al., 1999, 2000). Finally, in Figs.16 and 17, the normalised displacement field and the dif- ference between no controlled and controlled (amplification = 1) case vs. time and column axis have been plotted. Fourier expansion solution for a switched shunt... 497 Fig. 13. 1st and 2ndmodal shapes Fig. 14. Normalised displacement vs. time and amplification level at x=L for the first resonance Fig. 15. Normalised displacement vs. time and amplification level at x=L for the second resonance 498 S. Ameduri, M. Ciminello Fig. 16. Normalised displacement vs. time and column axis for the first resonance Fig. 17. No controlled-controlled normalised displacement difference vs. time and column axis for the first resonance Anattenuation of 1.5 and 2.3dBwithout amplification has been estimated at the fluid-structure interface for the first and second resonance frequencies. The maximum reduction (7.9, 11.2dB at 170 and 340Hz, respectively) has been observed for the maximum amplification considered. Tohave an ideaof the energyattenuation, the squaredvalue of thepressure has been computed and reported for fixed locations along the column. The corresponding squaredmean energy, SME, estimated through (3.33), has been reported on the last column. Benefits are evident for different locations along the fluid column, even though zero attenuation values have been detected on themodal shape nodes. Fourier expansion solution for a switched shunt... 499 Table 2. Sound pressure reduction vs. spatial domain and amplification First resonance: 170Hz P P P P P P P PP Gain x/L 0 0.25 0.50 0.75 1.00 SME Sound pressure attenuation [dB] 1 2.1 1.9 0 1.8 1.5 1.7 5 5.7 5.1 0 5.1 5.4 9.7 10 12 10 0 9.8 9.5 17.3 Second resonance: 340Hz P P P P P P P PP Gain x/L 0 0.25 0.50 0.75 1.00 SME Sound pressure attenuation [dB] 1 2.8 0 2.3 0 2.1 3.8 5 8.5 0 8.3 0 8.2 15.2 10 13 0 14 0 14 24.6 Generally, the control proved to bemore effective for the 2nd frequency, by achieving a mean reduction of 24.6dB, 6.5 times higher than in the reference case without amplification. The minimum averaged attenuation (1.7dB) has been detected for the 1st resonance, without amplification. Finally, to have an idea of control authority for the entire frequency range (0-500Hz), the mean pressure level estimated with and without control (for Gain = 1) by exciting the system through a stepped sine signal has been compared in Fig.18. Fig. 18. No controlled-controlledmean pressure level vs. frequency The related attenuation, as shown inFig.19, has proved tomainly interest the peaks zones. 500 S. Ameduri, M. Ciminello Fig. 19. No controlled-controlled dBmean pressure level reduction vs. frequency 5. Conclusions and further steps In the present work, the possibility of exploiting the switched shunt control architecture to sound pressure attenuation within enclosures has been dealt with.Many applications have been already published on the vibration attenu- ation through piezo network controlled by this mentioned logic. The related benefits within the low frequency band led to development of numerical tools aimed at predicting related benefits, and addressing experimental campaigns to verify effectiveness and point out eventual drawbacks. Also, applications focused on acoustic problems have been carried out, generally oriented to interior noise radiation and sound pressure level within enclosures. The main advantage of the switch logic is fully described in the referred literature and summarised in the dedicated Sectio of the present paper. The idea is to generate a control force opportunely synchronised to the maximum amplitude and shortly temporised with the sensor output signal. These ope- rating conditions guarantee a pulse kind actuation force out of phase with respect to the local structural displacement and independent of the structural response in a wide band range. Moreover, being the resistor negligible, the switched shunt performsmore thermal stability. The absence of an external power supply injecting energy into the system guarantees an unconditional stability of the control. The paper at hand is concernedwith the formalisation of a semi-analytical solution describing the displacement and pressure field within an air column Fourier expansion solution for a switched shunt... 501 subjected to switch control, implemented to a piezo actuator. This approach ensures quick estimation of the efficiency of the control actingwithin an enclo- sure, exploitable as thepreliminarydesign reference for further,more complex, numerical modelling. Theacoustic field is describedby the telegraph equationwhose solutionhas been expressed as aFourier expansion.Thedisturbance (sinusoidal excitation) and control action (given by a long plate actuated by a piezo) have been supposed localised on two boundaries of a 1D domain. At first, a preliminary validation of the solution has been addressed, trac- king series coefficients vs. harmonic order behaviour. Secondly, modal shapes have been plotted for the first two resonances without control. Then the con- trol authority has been investigated, computing the displacement vs. time as well as vs. amplification gain at fixed locations along the column axis. The related information has been expressed in terms of the punctual and averaged sound pressure level. Larger authority has been observed for the 2nd normal frequency: 24.6dBof attenuation, with a 10-times amplification. Evenwithout any amplification, coherently with the semi-passive nature of this control, a reduction of 3.8dB has been estimated. In order to assess the model validity and to point out possible limits, a tailored experimental prototype; more in detail – a duct instrumented with microphones along the generatrix with a loudspeaker and a pzt controlled panel mounted on the bases, will bemanufactured. The abovementionedmodel can be further enriched by introducing impe- dance for damping layer applications. Moreover, the solution validity can be extended in the frequency band, even though paying in terms of complexity, introducing the effects of structural inertial and damping actions. Finally, different and more complex geometries, of major interest for real applications (axial-symmetric, tapered, etc), could be investigated by forma- lising the problem through tailored boundary conditions. References 1. Ameduri S., Ciminello M., Concilio A., 2007, Semi-analytical solution of a structural system controlled by a switched shunt architecture,Proceeding of the 36th International Congress and Exhibition onNoise Control Engineering – Session ANVC”Active Noise andVibration Control”, Istanbul, Turchia, Paper IN07-032 502 S. Ameduri, M. Ciminello 2. Boyce W.E., DiPrima R.C., 2008, Elementary Differential Equations and Boundary Value Problems, JhonWiley & Sons Inc, seventh edition, 621-655 3. 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Preumont A., 1997,Vibration Control of Active Structures: An Introduction, Kluwer Academic Publishers 504 S. Ameduri, M. Ciminello 28. Richard C., Guyomar D., Audigier D., Bassaler H., 2000, Enhanced semi-passive damping using continuous switching and Isolation, SPIE, 3989, 288-299 29. Richard C., Guyomar D., Audigier D., Ching G., 1999, Semi-passive damping using continuous switching of a piezoelectric device, Society of Photo- Optical Instrumentation Engineering (SPIE) Conference Series, 3672, 104-111 30. Timoshenko S.P., Woinowsky-Krieger S., 1959, Theory of Plates and Shells, McGraw-Hill International Editions, second edition, 4-6 Rozwiązanie Fouriera dla przewodu akustycznego z bocznikującym układem sterowania Streszczenie W pracy zaprezentowano pół-analityczne rozwiązanie zagadnienia redukcji po- ziomu ciśnienia akustycznego w przewodzie za pomocą zsynchronizowanej metody bocznikowania. Opis pola przemieszczeńwewnątrz jednowymiarowej kolumny powie- trza sformalizowano rozwinięciemw szereg Fouriera przy uwzględnieniu sinusoidalnie zmiennych zaburzeń na prawym i lewym brzegu przewodu. Dla przypadku z wy- łączonym układem sterowania rozwiązanie obliczono w całej dziedzinie czasu przy utrzymaniu stałych parametrów obwodu elektrycznego, natomiast dla trybu stero- wanego dokonano analizy, dzieląc przedział czasu na fragmenty. W każdym z nich charakterystyczne cechy układu sterowania (np. napięcie przykładane do piezoelek- tryków, natężenie pola elektrycznego...) dobranowodpowiedni sposób.Na podstawie obserwacji przemieszczeń porównano poziom ciśnienia akustycznego dla warunków z wyłączonym i włączonym sterowaniem oraz z i bez wzmocnienia sygnału. Manuscript received July 6, 2009; accepted for print November 16, 2009