Piezoelectric line moment actuator for active radiation control from light-weight structures

Mechanical Systems and Signal Processing 96 (2017) 260–272

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Piezoelectric line moment actuator for active radiation control from light-weight structures Vojtech Jandak a, Petr Svec b, Ondrej Jiricek a,⇑, Marek Brothanek a a b

Czech Technical University in Prague, Faculty of Electrical Engineering, Technicka 2, 166 27 Prague, Czech Republic Skoda Auto, a.s., Czech Republic

a r t i c l e

i n f o

Article history: Received 25 November 2015 Received in revised form 24 March 2017 Accepted 3 April 2017

2008 MSC: 23-557 Keywords: Acoustics Piezoelectricity Line moment actuator ASAC

a b s t r a c t This article outlines the design of a piezoelectric line moment actuator used for active structural acoustic control. Actuators produce a dynamic bending moment that appears in the controlled structure resulting from the inertial forces when the attached piezoelectric stripe actuators start to oscillate. The article provides a detailed theoretical analysis necessary for the practical realization of these actuators, including considerations concerning their placement, a crucial factor in the overall system performance. Approximate formulas describing the dependency of the moment amplitude on the frequency and the required electric voltage are derived. Recommendations applicable for the system’s design based on both theoretical and empirical results are provided. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Active noise control techniques can provide a reasonable alternative to passive ones for attenuation of low-frequency noise. Besides other applications, they seem to be feasible for controlling sound radiated from a light-weight structure by application of vibrational forces or moments directly to the radiating structure. This technique is commonly called active structural acoustic control (ASAC) [8] and can also be used for improvement of transmission loss of such structure without significant increase of mass, with applications in aircraft or car industry. Recently, a number of studies have been carried out in order to find a suitable actuator for practical applications. Some of them are based on classical electrodynamic construction [5]. However, for theoretical studies as well as experimental verification of ASAC systems, various piezoelectric transdusers based on lead-zircanium-titanium (PZT) piezoceramics or polymer-polyvinylidene-fluoride (PVDF) foil are most commonly used [4,17,9,14]. Both types of piezoelectric actuators can act as sensors or actuators, however, PVDF transucers are usually used only for sensing. As this paper is focused on the construction and application of new actuator, the word transducer hereinafter means actuator. Vibrating plate or shell structures are induced by actions of mechanical forces and moments, both of which can be distributed over certain area or concentrated into several points or along lines or curves (e.g. along the edges). Transducers configured to act as (point) force actuators can be represented by a small piezoelectric patch. This type of actuator can be suitable for ASAC using a decentralized control strategy e.g. [16,2,20]. The second possibility of excitation of a structure is ⇑ Corresponding author. E-mail address: [email protected] (O. Jiricek). http://dx.doi.org/10.1016/j.ymssp.2017.04.003 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.

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by the moment of force, which can be advantageous in specific applications. This can be realized as pair of force actuators operating in opposite phases [11] or by another configuration using the bending moments of single-layer (unimorph) [12] or double-layer (bimorph) piezoelectric stripes [21]. This paper presents a new type of line moment actuator useful for excitation of thin plates and shells in ASAC experiments. The proposed actuator consists of piezoelectric bimorphs clamped on one side to a metallic handle. When external electric voltage is applied, the bimorphs start to oscillate and the bending moment arises along the handle as the result of inertial force. Using linearized piezoelectric state equations, simplified formulas are derived for amplitude spectrum of generated bending moment. It is shown that the moment is linearly dependent upon the applied voltage. The actuator uses the mechanical resonance of the stripes to maximize the generated moment. For analysis and description, we will assume a thin rectangular plate with an attached actuator. For the reader’s comfort, the thin plate theory will be briefly summarized and discussed. 2. Bending waves in thin plates At low frequencies (when plate thickness is small compared to structural wavelength), the most important type of mechanical waves that couple with the surrounding fluid and radiate sound are bending waves. The longitudinal waves are hard to excite and the transverse deflection is caused only by the Poisson effect which is small compared with bending wave deflection, and shear waves become important only at high frequencies. As the bending waves are described in detail by many authors (e.g. [7,19]) we will only briefly summarize the Kirchoff theory and show the corrections for rotary inertia and shear deformation which is assumed in the Midlin theory. Let’s assume a homogeneous and isotropic thin plate with negligible damping and thickness 2h induced mechanically so that the bending wave is travelling in it. Under the Kirchhoff assumption, the bending wave equation can be derived

€3 ¼ f ; DD22 u3 þ 2hqu

ð1Þ

where D is bending stiffness u3 is the normal component of plate displacement and f is the external forcing function, which will be discussed later in this article, and D2 is the two-dimensional Laplace operator. Using the Midlin theory, one may easily obtain equations of motion of a thin plate for bending (flexural) waves 3

Mab;b  Q a ¼ q €3 Q a;a ¼ 2hqu

2h € a; u 3

ð2Þ ð3Þ

where the internal moments and shear forces are given by equations

Z M ab ¼ Z Qa ¼

h

h h

h

x3 sab dx3 ;

ð4Þ

sa3 dx3 :

ð5Þ

The transverse forces do not take into account the parabolic distribution of shear stress along the thickness of the plate, which leads to warping of the plate’s cross-section. The discrepancy in total energy forces us to introduce a correction factor j [3]

Z Q a ¼ j2

h

h

sa3 dx3 ¼ j2 l2hðua  x3;a Þ

For the case presented above the factor

ð6Þ

j is (see e.g. [10, pp. 68])

j  0:76 þ 0:3m: 2

ð7Þ

It is useful to recall that the factor j is related to Rayleigh’s surface waves and also describes the ratio of Rayleigh’s waves phase speed (cR ) to the speed of shear waves (cs ). The approximate equation above is the estimation of a real root of equation

 2 3  2 2     cR cR 8ð2  mÞ c2R 8  ¼ 0:  8 þ 1m 1m c2s c2s c2s From the physical standpoint, only the root smaller than one will be of our interest. 2.1. Forcing functions The bending waves are excited in the thin plate by actions of external forces, point forces acting perpendicularly on the surface or moments within the surface. Assuming only the forces acting normally to the surface and in-plane oriented moments, the virtual work of these external sources is

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Z

Z T i dui dS ¼ S

S

ðMe1 du2  M e2 du1 þ T e3 du3 ÞdS

ð8Þ

where M e1 and M e2 are external moments acting along the x and y axes respectively and T e3 is the external surface force acting perpendicularly to the plate surface. The resulting forcing function is then

f ¼ T e3 þ

D

j2 l2h

D2 T e3

ð9Þ

The special case of point force located at x0 ; y0 can be described using the Dirac distribution

f ¼ T e3 dðx  x0 Þdðy  y0 Þ:

ð10Þ

When the distributed moment load is considered, the forcing function assumes the following form

f ¼

@Me1 @Me2  : @y @x

ð11Þ

In practice, most important are especially the point and line moments. The point moment with an amplitude M B can be written as

Me ¼ MB dðx  x0 Þdðy  y0 Þ:

ð12Þ

When the latter case (line moment) is acting along the x axis at y ¼ y0 between the points x0 a x1 with amplitude M L , the excitation function becomes

Me1 ¼

ML dðy  y0 Þ½Hðx  x0 Þ  Hðx  x1 Þ x1  x0

ð13Þ

where HðxÞ stands for the Heaviside (unit step) function. 3. Piezoelectric bimorph The piezoelectric effect represents the introduction of an electric field in elastic solids as a result of deformation caused by external forces, while the converse piezoelectric effect means deformation of the material due to an applied electric field. This effect can be observed in many crystals or ceramics like silicone dioxide or PZT – lead-zirconium-titanium based ceramics. The linearized theory describes the relation among stress s, strain e, electric field E and electric displacement D by the piezoelectric constitutive equations [1]. Assuming adiabatic processes, the respective constitutive equations can be written in the following form

Dm ¼ dml sl þ mk Ek ;

ð14Þ

em ¼ slm sl þ dmk Ek ;

ð15Þ

where  is permitivity, d is the piezoelectric strain constant and s is elastic compliance. Using compressed matrix notation, the Latin indexes assume the values from 1 to 3 and the Greek indexes assume the values from 1 to 6. The piezoelectric bimorph is a sandwich structure consisting of at least two piezoelectric layers and a central electrode. Typically, a thin varnish layer is applied to isolate and protect the bimorph [6,21,18]. The piezoelectric layers are identically polarized so when connected in parallel to the external voltage as depicted in Fig. 1, the first layer extends while the second contracts, causing the bimorph to bend. Let us assume linear strain distribution along the thickness of the bimorph during its bending

Fig. 1. Realization of the piezoelectric bimorph.

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e11 ðzÞ ¼ Cz þ eð0Þ

263

ð16Þ

where C and eð0Þ are constants. If both piezoelectric layers are of the same thickness, the neutral plane will be in the center of the bimorph, so eð0Þ ¼ 0. The stress in the central electrode with strain given by (16) can be calculated using Hooke’s law

Ys Cz 1  m2s

ss11 ðzÞ ¼

ð17Þ

were Y s is the Young modulus and ms is the Poisson ratio of the central electrode. From the constitutive Eq. (14), one easily obtains the stress distribution in the piezoelectric layers under the linear strain distribution assumption (16)

sp11 ¼

  Yp d31 U ; Cz  hp 1  m2p

ð18Þ

where U is the applied voltage. The constant C is determined from the moment equilibrium requirement, which results in 2

C¼

2

3d31 U Y p ½ðhs þ hp Þ  hs  2hp Y ½ðh þ h Þ3  h3  þ Y 1m2p h3 p s p s 1m2 s s

ð19Þ

s

where hp and hs are seen in Fig. 1. When the central electrode is assumed to have negligible thickness, Eq. (19) can be simplified to the form

C¼

3d31 U 2

2hp

:

ð20Þ

At low frequencies, the motion of the bimorph can be appoximated by the model of a cantilever beam with the moment load at its free end. The previous analysis showed that the bimorph can be appoximated by a thin prismatic beam excited by moment M 0 at its end. Assuming harmonic electric voltage is applied to the bimorph, the excitation moment will be also harmonic with amplitude

M0 ¼

Yp Yp IC ¼ hp d31 U: 1  m2p 1  m2p

ð21Þ

where I is the moment of inertia of the bimorph with a negligible central electrode. The solution for the displacement of the cantilever beam with harmonic moment load can be expressed as

uðx; tÞ ¼

1 M0 X An Un ðxÞ sinðxtÞ qbhl n¼0 x2n ð1 þ igÞ  x2

ð22Þ

where b is width, l is the length and h ¼ 2hp þ 2hs is the thickness of the bimorph. Un ðxÞ are eigenfunctions

Un ðxÞ ¼ ½coshðbn xÞ þ cosðbn xÞ 

coshðbn lÞ þ cosðbn lÞ ½sinhðbn xÞ  sinðbn xÞ sinhðbn lÞ þ sinðbn lÞ

ð23Þ

and An are their amplitudes and g represents the structural damping factor of the material. The amplitudes of the contributing modes are calculated using the boundary conditions

An ¼ bn ð½sinhðbn lÞ þ sinðbn lÞ 

coshðbn lÞ þ cosðbn lÞ ½coshðbn lÞ  cosðbn lÞÞ sinhðbn lÞ þ sinðbn lÞ

ð24Þ

and the bn coefficients ensue from the solution of the equation

cosðbn lÞ coshðbn lÞ ¼ 1:

ð25Þ

Considering that in harmonic motion, an element of the piezoelectric stripe at distance x from the clamped end oscillates with amplitude u, the resulting amplitude of the dynamic bending moment produced by one bimorph clamped to the holder can be estimated

Z

l

Mdyn ¼

qbhx2 xuðx; xÞdx ¼ 0

1 X M0 n¼0

2 ð1 n

l x

An x2 þ igÞ  x2

Z

l

xUn ðxÞdx:

ð26Þ

0

4. Line moment actuator To create an moment actuator using a bimorph, there are several possibilities, however in all cases the bimorph(s) must be fixed in some light-weight holder. To fix the bimorph perpendicularly to the structure creates symmetric structure, but

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Fig. 2. Dependence of the moment of force (re 1 N m) on frequency for a single bimorph with length of 40 mm (blue) and 60 mm (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

seems to be unpractical. Another possibility is to fix the actuator parallel to the excited structure. Using the pair of the bimorphs operating in opposite phases as a see-saw results in a symmetric actuator (see below). For realization of actuators, two types of bimorphs with different free lengths were used. This dimension measured from the clamped end to the free end is therefore always smaller than the total length. The total dimensions of the two bimorphs were 60  20  0:6 mm and 40  20  0:6 mm with free lengths 51 and 31 mm respectively. Using (26), the amplitude spectrum for both types of bimorphs that formed the actuator were calculated and both lengths are depicted in Fig. 2. It can be seen that the amplitude reaches its maximum at the frequencies of structural resonance of the bimorph stripe. The actuators are most efficient at frequencies close to the first resonance frequency of the bimorphs they consist of. However, the useful frequency range starts at 60% of the resonance frequency and the high frequency limit is approximately triple the resonant frequency. At these low frequencies, the sound radiated by the bimorphs may be omitted as the hydrodynamic shortcut and small area of oscillation make their contribution in real systems negligible. The sound is radiated only by the surrounding baffle and the mechanical power injected by the actuator is used to modify its vibration to minimize either the sound pressure level at discrete point(s) or the total radiated acoustic power.

Fig. 3. Displacement of the free end of the bimorph versus excitation voltage.

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Fig. 4. Scheme of the moment actuator realized from eight bimorphs.

The displacement of the free end of the bimorph was masured to verify the simplified linear model represented by Eqs. (21) and (22). The single 60 mm bimorph was excited by the usual low-frequency amplifier connected in series with a resistor of 1 kX to protect the amplifier. The measured results at the first resonance frequency of the bimorph (95 Hz) are in Fig. 3. It is obvious that in the range of applied voltage used in our experiments, the linear model is an acceptable approximation. Note that the voltage used for excitation of the bimorphs is relatively small compared to other piezoelectric applications (see e.g. [20]). The actuator used in the experiments presented in the practical part of this article consists of one or several pairs of bimorphs placed opposite each other, clamped at on side to a metallic holder using screw connections, as depicted in Fig. 4, where d is the length of the metallic holder (equal to 96 mm), a is the width of the metallic holder (equal to 4 mm), b is the width of the single bimorph (20 mm), and l is the free length of the bimorph (51 mm). Obviously, the correct phase matching is crucial for formation of moment behavior (the opposite bimorphs must oscillate with the opposite phases). The holder of the actuator is then glued to the structure to be excited forming the line moment excitation. The mechanical power flow in the structure may be studied by calculating the structural intensity. It is defined

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Fig. 5. Square normal velocity distribution on the plate excited by line moment.

Fig. 6. Total flux of structural intensity on the plate excited by line moment.

Fig. 7. Experimental set-up containing chipboard cabinet with loudspeaker and the actuator on the steel plate.

I ¼ sv

ð27Þ

where s is the stress tensor and v is the velocity vector. In thin plates, the formulas for calculating the components of the inplane structural intensity from the known normal velocity may be derived using the Kirchhoff assumptions [15]. For harmonic oscillations at frequency x, the structural intensity vector is

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Fig. 8. Eight bimorphs line moment actuator fixed at the mock-up steel plate.

I¼

  EI 1m I rðr2 vÞv  r2 vr2 v  r  r  ðvrv Þ ; 2x 2

ð28Þ

The intensity vector consists of the bending moment contribution, shear force contribution and twisting moment contribution, which can be easily calculated using the Fourier transform of (28) as proposed by Pascal [15]. By calculating the divergence of the structural intensity, the injected power can be calculated, to estimate the system performance and efficiency. For studying the theoretical parameters, the large plate with one line moment actuator attached in the centre was modeled. Reflected waves at the plate edges were reduced by an additional damping factor. this allows us to study the power transmission between actuator and structure. The model was solved in the frequency domain using the modal superposition method with simply supported boundary conditions, as this leads to simple eigenfunctions without loss of generality. The normal velocity of the plate and total structural intensity, calculated according to (1) and (28) respectively, indicate the dipole character of the field generated by the line moment, as can be seen in Figs. 5 and 6. It implies that this kind of excitation is the most effective when placed along the nodal lines of the structure.

Fig. 9. Calculated structural intensity for mode of 75 Hz.

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Fig. 10. Modeled attenuation as a function of position of actuator.

Fig. 11. Modeled attenuation for the area corresponding to the experiment.

5. Simple ASAC with line moment actuator The behavior of a moment actuator on thin plates was studied in the previous paragraph. Now, it is worth examining the performance of the system with a single secondary actuator in a real situation. The experimental mock-up for actuator tests consists of a steel plate with dimensions of 58 cm  58 cm and thickness of 0.6 mm fixed on a rigid chipboard cabinet containing a baffle with a loudspeaker. The steel plate was actuated by an incident sound wave radiated by a loudspeaker. The ASAC system was modeled to achieve a maximum attenuation at the position of the error microphone located 0.5 m in front of the center of the plate, as shown in Fig. 7. The eight-bimorph actuator realized according to the description in the previous paragraph is shown in Fig. 8. The electronics enable the setting of the phase shift between excitation signals as well as setting of the optimal magnitudes. For the purpose of this study, the mode of 75 Hz was selected; therefore the bimorphs with a length of 60 mm were used for construction of the actuator providing sufficient performance at the selected frequency. 5.1. Numerical simulation The numerical model of a simply supported steel plate in an infinite baffle with a plane sound wave as a primary excitation was created for the excitation frequency of 75 Hz. The presence of the actuator is neglected in this model, as its weight is small compared to the weight of the steel plate. Primary vibrations of the plate including structural intensity calculated according to Eq. (28) are depicted in Fig. 9.

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Fig. 12. Measured normal surface velocity on steel plate.

Fig. 13. Sound pressure level generated by actuator under constant actuator power input.

The maximal attenuation of ASAC for various positions of the actuator located on the modeled plate was calculated using the superposition principle. The actuator was successively positioned on the plate with its base parallel with axis x in a uniform grid (6 columns  56 rows). The excitation level was the same for all positions and was set to the level for which the maximum attenuation was reached in the optimal position. Attenuation as a function of the position of the actuator is shown in Fig. 10, and it can be seen that the maximum attenuation can be reached when the actuator is fixed just at the edge of the plate where the nodal line is assumed. However, the modeled attenuation of 60 dB is not reachable in practical realization of this experiment, as the actuator due to its mechanical parameters can be fixed in the area limited by the white dotted rectangle in Fig. 10. Model results for this area are zoomed in Fig. 11 and this area was measured in the following measurement. 5.2. Experimental verification The operational deflection shape measured by the laser scanning vibrometer (289 measurement positions) shows good agreement between the model and our experiment. Measured results can be found in Fig. 12. Under laboratory conditions it is not sufficiently illuminating to use a real feed-forward ASAC system as, except for the worst positions of the actuator, the system suppresses the sound pressure level at the error microphone position to the back-

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Fig. 14. Used positions of actuator in experimental verification.

Fig. 15. Attenuation as a function of secondary excitation.

ground noise level. To be able to compare modeled and measured results, the sound pressure level in the error microphone position generated by the moment actuator has to be measured. This level as a function of the position for constant actuator power input (5.25 V in this experiment) is in Fig. 13. Measurements were realized in a grid of 5 columns and 11 rows, as illustrated in Fig. 14 in the form of black lines where the actuator was successively fixed. After the actuator efficiency test, the whole ASAC system described in Fig. 7 with the loadspeaker as a primary source was measured. The actuator was placed in the optimal position which corresponds to the maximal sound pressure level in Fig. 13. Attenuation as a function of secondary excitation under the condition of the optimal phase shift between the primary source and the actuator is shown in Fig. 15 for model and measured results at each optimal position. The secondary level of the actuator excitation, where the maximal attenuation was reached, was set to 0 dB. The same procedure was used for calculation of model results. The red curve shows the model results for the optimal position, which is at the edge of the steel plate

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Fig. 16. Recalculated attenuation as a function of position of the actuator.

(see Fig. 10, position (0.3; 0.0) m); the blue curve was obtained by measurement for the actuator in the position where it maximally excited the plate (position (0.3; 0.15) m). Positions of the actuator are not exactly the same for the model and the experiment, as it is not possible to fix the actuator precisely at the edge of the plate. If we would like to compare the data measured with a constant actuator power input with the calculated model results, it is necessary to recalculate the measured sound pressure levels in Fig. 13 using attenuation as a function of the actuator excitation shown in Fig. 15. The resulting recalculated attenuation is presented in Fig. 16. The attenuation is strongly sensitive to the actuator excitation around its maximum; therefore this process resulted in three maxima corresponding to the positions with maximum measured sound pressure levels. Measured levels were rounded to whole numbers taking into account various components of uncertainty of measurement (caused by additional mass of the actuator, quality of sticking, number of positions and precision of location, e.g.). This measured result is strongly comparable with the modeled result shown in Fig. 11, where the only applicable area of the plate is zoomed. 6. Conclusions This paper is concerned with a piezoelectric line moment actuator designed for broadband active structural acoustics control of light-weight structures. After a brief summary of the bending waves in thin plates and piezoelectric bimorph behavior, the construction of the bimorph moment actuator is presented. Its actuation capabilities were modeled and measured when fixed on a thin metal plate. Possible application of the developed actuator was then demonstrated on a simple active structural acoustics system based in suppression radiation by a selected structural mode of a rectangular plate fixed on a chipboard cabinet. Relatively good agreement can be seen comparing the measured and model results, taking into account the extremely significant influence of boundary conditions at the edges of the plate caused by deformation in the course of cutting the metal sheet and realization of its fixing. This fact also results in asymmetry of the measured results in Figs. 13 and 16. The influence of the actuator fixing is not negligible as well, as it is not possible to glue it in the same way twice. According to the presented theory, the actuator should be placed at the nodal line, and according to our previous studies the best results are achieved when active structural intensity flows through the nodal line when primary excitation is applied. Presented actuator has a potential to be used for broad-band active control of sound radiation from light-weight structures or for active control of sound transmission through thin structures [13]. The main advantages of the presented actuator consist in its light weight, reasonable dimensions and low voltage. As a result, it seems likely that this actuator will find success in practical applications. Acknowledgements This work was supported by the CTU student project No. SGS16/221/OHK3/3T/13 Measurement, modeling and evaluation methods in acoustics. References [1] ANSI/IEEE 176-1987, IEEE Standard on Piezoelectricity, The Institute of Electrical and Electronics Engineers, Inc, 1988.

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