CN113707119B - Active regulation and control method for piezoelectric acoustic metamaterial - Google Patents
Active regulation and control method for piezoelectric acoustic metamaterial Download PDFInfo
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Abstract
The invention discloses an active regulation and control method of piezoelectric acoustic metamaterial, which specifically comprises the following steps: for one-dimensional piezoelectric acoustic metamaterials with limited length and limited number of local resonance units, firstly, an electromechanical coupling model is established, then, on the premise that the piezoelectric acoustic metamaterials generate negative dynamic stiffness, the band gap range and the formation conditions of the piezoelectric acoustic metamaterials are calculated, then, a transfer function is deduced and frequency compensation is carried out, and finally, the transfer function is digitalized, so that the digital programmable piezoelectric acoustic metamaterials are realized; the active regulation and control method provided by the invention can be used for conveniently carrying out parameter configuration according to the target frequency in real time, and realizes obvious band gap and super-strong vibration attenuation exceeding 40dB in a low frequency range; in addition, the piezoelectric acoustic metamaterial is a multiple-input multiple-output (MIMO) system, the design complexity of the MIMO system is greatly reduced by the active regulation method, and a bridge is built for large-scale application of an active control algorithm in the piezoelectric acoustic metamaterial.
Description
Technical Field
The invention belongs to the technical field of vibration control, and particularly relates to an active regulation and control method of a piezoelectric acoustic metamaterial.
Background
The piezoelectric acoustic metamaterial is formed by attaching a piezoelectric material in a controlled structure in an embedding or sticking mode, connecting the piezoelectric material with a shunt circuit to form a local resonance unit, and forming a periodic structure by the local resonance unit; the piezoelectric acoustic metamaterial has wide application prospect in the field of vibration reduction and noise reduction due to the advantages of light weight, flexibility, high design freedom and the like.
The local resonance unit of the periodic structure has the characteristic of an elastic band gap, and under the action of the band gap, the propagation of elastic waves in the structure is blocked; the traditional shunt circuit is realized based on analog electronic devices, for example, a piezoelectric sheet is connected with an LR shunt circuit to generate local resonance, so as to realize a tunable vibration absorber; in order to enhance the local resonance band gap caused by the passive resonance shunt circuit, an enhancement shunt circuit (a-R circuit) is proposed, but the piezoelectric acoustic metamaterial formed by the analog shunt circuit is difficult to tune in real time and has poor control effect at low frequency.
In order to solve the limitation of the analog shunt circuit, a digital impedance technology is provided, the technology realizes a synthetic impedance circuit by a microcontroller, a required impedance can be established between the voltage on the piezoelectric element and the current flowing out, the existing digital controller realized based on the AR circuit adopts a pole-zero method to adjust poles and zeros in a Young modulus transfer function, negative dynamic stiffness is not directly considered, parameter configuration is inaccurate, the formation of a piezoelectric acoustic metamaterial band gap depends on whether the negative dynamic stiffness can be generated, the dynamic stiffness of the system can be influenced by changing the parameters of the shunt circuit, and flexible adjustment of the local resonance band gap is realized.
Disclosure of Invention
According to the active regulation and control method for the piezoelectric acoustic metamaterial, the negative dynamic stiffness is used as an important parameter to realize final regulation and control, so that the regulation of the local resonance band gap is more flexible and accurate.
The technical scheme adopted by the invention is as follows:
an active regulation and control method of piezoelectric acoustic metamaterial is implemented according to the following steps:
s1, establishing an electromechanical coupling model of a piezoelectric acoustic metamaterial;
s2, calculating the band gap range and the formation condition of the piezoelectric acoustic metamaterial on the premise that the piezoelectric acoustic metamaterial generates negative dynamic stiffness according to the electromechanical coupling model of S1;
s3, deducing a transfer function of the control unit according to the electromechanical coupling model of the S1, compensating the resonant frequency, and forming a band gap with adjustable range and depth on the premise of meeting the band gap forming condition of the S2;
s4, digitizing the transfer function of the S3 to realize active regulation and control of the piezoelectric acoustic metamaterial.
Preferably, in the step S1, an electromechanical coupling model of the piezoelectric acoustic metamaterial is built, specifically:
according to the control unit, an electromechanical coupling model of the piezoelectric acoustic metamaterial is established by utilizing a dynamic equation of the composite beam structure, wherein the electromechanical coupling equation of the metamaterial structure is subjected to decoupling treatment, and the r-th order modal response can be expressed as:
wherein H is r (s) is Laplacian transform of the mode coordinates corresponding to the r-th order mode, Q r (s) is Laplacian transformation of modal excitation force of composite beam structure, s is Laplacian operator, ζ r Is the mechanical damping ratio omega corresponding to the r-order mode of the composite beam structure r Is the natural frequency of the order r of the composite beam structure, alpha is a dimensionless parameter related to electromechanical coupling effect, C p Is the internal equivalent capacitance of the piezoelectric sheet, and beta is the voltage amplification factor of the voltage amplification circuit;
Z 0 (s) =1/(Cs), Z(s) =ls+r, where C is the feedback capacitance of the charge amplifier and L, R is the inductance and resistance, respectively, in series with the piezoelectric plate in the enhanced shunt circuit.
Preferably, in the step S2, the band gap range and the formation condition of the piezoelectric acoustic metamaterial are calculated according to the condition that the electromechanical coupling model generates negative dynamic stiffness in the piezoelectric acoustic metamaterial, specifically:
the band gap range is:
band gap formation conditions:
in the above, ω t Is the resonant frequency of an inductance-resistance series shunt circuit, meets omega t 2 ＝1/(LC p )。
Preferably, in S3, the transfer function is derived and the resonant frequency is compensated according to the electromechanical coupling model, specifically:
s31, obtaining a general form of a transfer function according to the control unit is as follows:
in the above equation, γ is the additional gain of the control output.
S32, according to the band gap range and the formation condition, for the resonance frequency omega in S31 t Frequency compensation is carried out:
ω t ＝ω c +Δω c
in the above, Δω c Is the compensation frequency omega c Is the center frequency of the band gap.
Preferably, in the step S4, the transfer function is digitized, so as to realize active regulation and control of the piezoelectric acoustic metamaterial, specifically:
the transmission function is discretized by using a zero-order retainer according to the sampling time, and then the discretized second-order transmission function is converted into a differential equation form and is input into a digital controller, so that the active regulation and control of the piezoelectric acoustic metamaterial are realized.
Preferably, the control unit includes a pair of piezoelectric elements and an enhanced shunt circuit, and the pair of piezoelectric elements and the enhanced shunt circuit are connected.
Preferably, the enhanced shunt circuit includes a charge amplifying circuit, a voltage amplifying circuit, a resistor and an inductor which are arranged in series, one piezoelectric element of the pair of piezoelectric elements is connected with the charge amplifying circuit, and the other piezoelectric element forms the piezoelectric actuator by using the inverse piezoelectric effect.
Compared with the prior art, the active regulation and control method of the piezoelectric acoustic metamaterial provided by the invention aims at the one-dimensional piezoelectric acoustic metamaterial with limited length and limited number of local resonance units, firstly establishes an electromechanical coupling model, then calculates the band gap range and the formation condition of the piezoelectric acoustic metamaterial on the premise that the piezoelectric acoustic metamaterial generates negative dynamic stiffness, deduces a transfer function and performs frequency compensation, and finally digitizes the transfer function, thereby realizing the digital programmable piezoelectric acoustic metamaterial.
The active regulation and control method provided by the invention can be used for conveniently carrying out parameter configuration according to the target frequency in real time, and realizes obvious band gap and super-strong vibration attenuation exceeding 40dB in a low frequency range; in addition, the piezoelectric acoustic metamaterial is a multiple-input multiple-output (MIMO) system, and the active regulation and control method provided by the invention greatly reduces the design complexity of the MIMO system.
Drawings
FIG. 1 is a flow chart of an active control method of a piezoelectric acoustic metamaterial provided by an embodiment of the invention;
FIG. 2 is a schematic circuit diagram of a local resonance unit of an enhanced shunt circuit in an active control method of a piezoelectric acoustic metamaterial according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of a control unit of an active control method for piezoelectric acoustic metamaterial according to an embodiment of the present invention;
fig. 4 is a frequency response diagram of an active regulation method of a piezoelectric acoustic metamaterial provided by an embodiment of the present invention.
The drawings are marked with the following description: 1-enhancement type shunt circuit, 2-charge amplifying circuit and 3-voltage amplifying circuit.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
In the description of the present invention, it should be clearly understood that terms such as "vertical", "horizontal", "longitudinal", "front", "rear", "left", "right", "upper", "lower", "horizontal", and the like indicate an orientation or a positional relationship based on that shown in the drawings, and are merely for convenience of describing the present invention, and do not mean that the apparatus or element referred to must have a specific orientation or position, and thus should not be construed as limiting the present invention.
The following description provides various embodiments of the invention that may be substituted or combined between different embodiments, and thus the invention is also to be considered as embracing all possible combinations of the same and/or different embodiments described. Thus, if one embodiment includes feature A, B, C and another embodiment includes feature B, D, then the present application should also be considered an embodiment that includes one or more of all other possible combinations of A, B, C, D, although such an embodiment may not be explicitly recited in the following.
The embodiment of the invention provides an active regulation and control method of a piezoelectric acoustic metamaterial, which is shown in fig. 1 and is implemented specifically according to the following steps:
s1, establishing an electromechanical coupling model of a piezoelectric acoustic metamaterial;
s2, calculating the band gap range and the formation condition of the piezoelectric acoustic metamaterial on the premise that the piezoelectric acoustic metamaterial generates negative dynamic stiffness according to the electromechanical coupling model of S1;
s3, deducing a transfer function according to the electromechanical coupling model of the S1, compensating the resonant frequency, and forming a band gap with adjustable range and depth on the premise of meeting the band gap forming condition of the S2;
s4, digitizing the transfer function of the S3 to realize active regulation and control of the piezoelectric acoustic metamaterial.
In this way, by adopting the method, the embodiment firstly establishes an electromechanical coupling model for the one-dimensional piezoelectric acoustic metamaterial with limited length and limited number of local resonance units, then calculates the band gap range and the formation condition of the piezoelectric acoustic metamaterial on the premise that the piezoelectric acoustic metamaterial generates negative dynamic stiffness, deduces a transfer function and compensates the resonance frequency, and finally digitizes the transfer function to realize the digital programmable piezoelectric acoustic metamaterial.
The active regulation and control method provided by the invention can be used for conveniently carrying out parameter configuration according to the target frequency in real time, and realizes obvious band gap and non-traditional super-strong vibration attenuation exceeding 40dB in a low frequency range; in addition, the piezoelectric acoustic metamaterial is a multiple-input multiple-output (MIMO) system, the design complexity of the MIMO system is greatly reduced by the active regulation method provided by the embodiment, and a bridge is built for large-scale application of an active control algorithm in the piezoelectric acoustic metamaterial.
In particular embodiments:
in the step S1, an electromechanical coupling model of the piezoelectric acoustic metamaterial is established, and specifically comprises the following steps:
according to the control unit, a dynamic equation of the composite beam structure under the action of external input voltage is obtained:
wherein EI is flexural rigidity of the composite beam structure under the condition of short circuit of the piezoelectric element, w (x, t) is transverse deflection of the beam at the position x and time t, m is mass of the composite beam structure per unit length, θ is electromechanical coupling term under physical coordinates, v j (t) is the potential difference between the electrodes of the j-th pair of piezoelectric elements, the number of the piezoelectric elements being j=1, 2,..s, each pair of piezoelectric elements being from x=x j L Beginning to x j R Ending, total length Deltax j ＝x j R -x j L H(s) is a Heaviside function and f (x, t) is the lateral force per unit length distributed on the composite beam.
The transverse deflection of the composite beam structure is unfolded in a mode space, and the front N-order vibration mode is intercepted, and the method comprises the following steps:
wherein the number of modalities is r=1, 2 r (t) is the mode corresponding to the r-th order modeState coordinate expression, phi r (x) Is the mass normalization eigenfunction of the mode shape corresponding to the mode of the r-th order.
The unit length transverse force f (x, t) which is distributed and acted on the composite beam is equivalently replaced by concentrated force, so that the vibration problem is simplified, and the composite beam structure is obtained when x=x f Modal excitation force q at r (t)：
An electromechanical coupling model of the piezoelectric acoustic metamaterial is established, wherein an electromechanical coupling equation of the metamaterial structure is subjected to decoupling treatment, and the r-th order modal response can be expressed as follows:
wherein H is r (s) is Laplacian transform of the mode coordinates corresponding to the r-th order mode, Q r (s) is Laplacian transformation of modal excitation force of composite beam structure, s is Laplacian operator, ζ r Is the mechanical damping ratio omega corresponding to the r-order mode of the composite beam structure r Is the natural frequency of the order r of the composite beam structure, alpha is a dimensionless parameter related to electromechanical coupling effect, C p Is the internal equivalent capacitance of the piezoelectric sheet, and beta is the voltage amplification factor of the voltage amplification circuit;
Z 0 (s) =1/(Cs), Z(s) =ls+r, where C is the feedback capacitance of the charge amplifier and L, R is the inductance and resistance, respectively, in series with the piezoelectric plate in the enhanced shunt circuit.
More specifically:
the control unit comprises a pair of piezoelectric elements and an enhanced shunt circuit, and the piezoelectric elements are connected with the enhanced shunt circuit.
The enhanced shunt circuit 1 comprises a charge amplifying circuit 2, a voltage amplifying circuit 3, a resistor and an inductor which are arranged in series, wherein one piezoelectric element of a pair of piezoelectric elements is connected with the charge amplifying circuit, and the other piezoelectric element forms a piezoelectric actuator by utilizing the inverse piezoelectric effect.
The pair of piezoelectric elements and the enhanced shunt circuit together form a local resonance unit, as shown in fig. 2, which is a schematic circuit diagram of the local resonance unit of the enhanced shunt circuit, wherein the local resonance unit comprises a pair of piezoelectric elements and the enhanced shunt circuit, one piezoelectric element of the pair of piezoelectric elements is connected with a charge amplifying circuit, the function is equivalent to that of a piezoelectric sensor, the other piezoelectric element forms a piezoelectric actuator by utilizing the inverse piezoelectric effect, and each piezoelectric element can be represented as a current source connected with an internal capacitor of the piezoelectric actuator in parallel.
The enhanced shunt circuit 1 is composed of a charge amplifying circuit 2, a voltage amplifying circuit 3 and an inductance-resistance series circuit, the whole framework composed of the enhanced shunt circuit and the piezoelectric element can be regarded as a typical active control system, and the resonance principle is a main control strategy.
In particular embodiments:
the piezoelectric element was selected according to the parameters in table 1 below:
TABLE 1
In particular embodiments:
the composite beam structure is made of stainless steel, eight pairs of piezoelectric patches are closely attached to the upper surface and the lower surface of the base beam, the distances among the piezoelectric patches are the same, and the piezoelectric patches are symmetrically distributed on the symmetry axis in the length direction of the composite beam structure. PZT-5H is selected as the piezoelectric patch, the polarization directions of the piezoelectric elements attached in pairs are opposite, and the dimensions and the structural parameters of the matrix beam are shown in Table 2:
TABLE 2
In the step S2, negative dynamic stiffness is generated according to an electromechanical coupling model and the piezoelectric acoustic metamaterial, and the band gap range and the formation condition of the piezoelectric acoustic metamaterial are calculated, specifically:
the band gap range is:
band gap formation conditions:
in the above, ω t Is the resonant frequency of an inductance-resistance series shunt circuit, meets omega t 2 ＝1/(LC p )。
As shown by the band gap range, the formation conditions and the parameter comparison experiments, the larger the values of alpha and beta are, the width and depth of the band gap formed by the piezoelectric acoustic metamaterial are increased; the larger R is, the smaller the width and depth of the band gap are, and when the R value is too large, the piezoelectric metamaterial does not have the band gap.
Specifically, let the
Bringing s=jω into the above, with in the frequency domain
ω′ r 2 ＝a r (ω)+b r (ω)j
Wherein the method comprises the steps of
If a is r (ω)<0, ω' r 2 The real part of (2) is smaller than zero, which can be regarded as that the piezoelectric acoustic metamaterial realizes equivalent negative dynamic stiffness under the (r) th mode, and the frequency range where omega is locatedThe circumference is the band gap range. At the same time b r (ω) is always greater than zero, the effect of which can be understood to be an increase in equivalent damping for piezoelectric acoustic metamaterials. Thus, the enhanced shunt circuit achieves vibration suppression through the combined action of equivalent negative dynamic stiffness and equivalent damping.
In one embodiment:
in the step S3, a transfer function is deduced according to the band gap range and the formation condition, and the resonance frequency is compensated, specifically:
s31, obtaining a general form of a transfer function according to the control unit is as follows:
in the above equation, γ is the additional gain of the control output.
Fig. 3 is a schematic diagram of a digital programmable piezoelectric acoustic metamaterial control unit, where the digital programmable piezoelectric acoustic metamaterial is composed of a plurality of periodically distributed piezoelectric metamaterial control units, and each control unit is identical. The digital equivalent is carried out on the circuit principle of the enhanced shunt circuit, so that the digital programmable piezoelectric acoustic metamaterial is obtained, wherein the general form of the transfer function of the control unit is shown as above;
s32, according to the band gap range and the formation condition, for the resonance frequency omega in S31 t Frequency compensation is carried out:
ω t ＝ω c +Δω c
in the above, Δω c Is the compensation frequency omega c Is the center frequency of the band gap.
In active regulation using transfer functions, given a target frequency, the center frequency of the desired band gap coincides with the target frequency, but the center frequency of the local resonant band gap formed by the digital circuit will be lower than the target frequencyTherefore, the resonance frequency omega in the transfer function must be adjusted t And compensating. C according to transfer function p And C it is known that adjusting the additional gain γ of the control output is substantially equivalent to adjusting the electromechanical coupling strength α, adjusting γ, β and R, achieving regulation of the width and depth of the bandgap. It should be noted that after the control system reaches dynamic balance, the control output voltage of the digital controller cannot be increased any more by continuously increasing γ and β, so that the performance is not enhanced any more, and the system divergence is caused.
In one embodiment:
in the step S4, the transfer function is digitalized, so that the active regulation and control of the piezoelectric acoustic metamaterial are realized, specifically:
the transmission function is discretized by using a zero-order retainer according to the sampling time, and then the discretized second-order transmission function is converted into a differential equation form and is input into a digital controller, so that the active regulation and control of the piezoelectric acoustic metamaterial are realized.
Specifically:
the charge signal can be converted into a voltage signal suitable for the acquisition range of the ADC through a proper charge amplifying circuit, and the logic of the controller is completely realized by using an embedded program. The program logic calculation process is completed in real time inside the microcontroller. The transfer function is digitized by a microcontroller, the continuous time transfer function is discretized by using a zero-order retainer according to sampling time, then the discretized second-order transfer function is converted into a form of a differential equation, and finally the digitization is realized by a corresponding filter structure:
y(k)＝a 0 x(k)+a 1 x(k-1)+a 2 x(k-2)-b 1 y(k-1)-b 2 y(k-2)
where x (k) is an input signal at time k, y (k) is an output signal at time k, a 0 ，a 1 ，a 2 Is the feedforward filter coefficient, b 1 ，b 2 Is the feedback filter coefficient.
In this embodiment, the sampling frequency is chosen to be 10kHz.
Under the condition that all piezoelectric elements are grounded to realize short circuit, sweep frequency experiments are carried out on piezoelectric acoustic metamaterials, a signal generator provides sweep frequency signals, the frequency is linearly changed from 5Hz to 200Hz, a data acquisition device acquires exciting voltage output by the signal generator as system input, acquires voltage signals output by an accelerometer as system output, and the amplitude-frequency characteristics of the system can be obtained by utilizing a signal processing method according to input and output data as shown in figure 4. The natural frequency 105.6Hz is selected as the target frequency of the digital programmable piezoelectric acoustic metamaterial, and a vibration control experiment is carried out nearby the target frequency, so that the metamaterial can be seen to generate an obvious band gap, and obvious vibration attenuation exceeding 40dB is achieved.
The digital controller can adjust parameters wirelessly and in real time, thereby easily controlling the frequency, width and depth of the band gap. The active regulation and control method provided by the invention greatly reduces the design complexity of the MIMO system, and builds a bridge for large-scale application of the active control algorithm in the piezoelectric acoustic metamaterial.
The present invention is not limited to the above-mentioned embodiments, and any changes or substitutions that can be easily understood by those skilled in the art within the technical scope of the present invention are intended to be included in the scope of the present invention. Therefore, the protection scope of the present invention should be subject to the protection scope of the claims.
Claims (5)
1. The active regulation and control method of the piezoelectric acoustic metamaterial is characterized by comprising the following steps of:
s1, establishing an electromechanical coupling model of a piezoelectric acoustic metamaterial;
s2, calculating the band gap range and the formation condition of the piezoelectric acoustic metamaterial on the premise that the piezoelectric acoustic metamaterial generates negative dynamic stiffness according to the electromechanical coupling model of S1;
s3, deducing a transfer function according to the electromechanical coupling model of the S1, compensating the resonant frequency, and forming a band gap with adjustable range and depth on the premise of meeting the band gap forming condition of the S2;
s4, digitizing the transfer function of the S3 to realize active regulation and control of the piezoelectric acoustic metamaterial;
the electromechanical coupling model of the piezoelectric acoustic metamaterial is established in the step S1, and specifically comprises the following steps:
according to the control unit, an electromechanical coupling model of the piezoelectric acoustic metamaterial is established by utilizing a dynamic equation of the composite beam structure, wherein the electromechanical coupling equation of the metamaterial structure is subjected to decoupling treatment, and the r-th order modal response can be expressed as:
wherein H is r (s) is Laplacian transform of the mode coordinates corresponding to the r-th order mode, Q r (s) is Laplacian transformation of modal excitation force of composite beam structure, s is Laplacian operator, ζ r Is the mechanical damping ratio omega corresponding to the r-order mode of the composite beam structure r Is the natural frequency of the order r of the composite beam structure, alpha is a dimensionless parameter related to electromechanical coupling effect, C p Is the internal equivalent capacitance of the piezoelectric sheet, and beta is the voltage amplification factor of the voltage amplification circuit;
Z 0 (s) =1/(Cs), Z(s) =ls+r, where C is the feedback capacitance of the charge amplifier, and L, R is the inductance and resistance, respectively, in series with the piezoelectric plate in the enhanced shunt circuit;
in the step S2, according to an electromechanical coupling model, on the premise that the piezoelectric acoustic metamaterial generates negative dynamic stiffness, the band gap range and the formation condition of the piezoelectric acoustic metamaterial are calculated, and specifically, the band gap range and the formation condition are as follows:
band gap range Δω bg The method comprises the following steps:
band gap formation conditions:
in the above, ω t Is the resonant frequency of an inductance-resistance series shunt circuit, meets omega t 2 ＝1/(LC p )。
2. The method for actively controlling a piezoelectric acoustic metamaterial according to claim 1, wherein the step S3 derives a transfer function according to an electromechanical coupling model, specifically:
s31, obtaining a transfer function K according to the control unit m The general form of(s) is:
in the above formula, γ is the additional gain of the control output;
s32, according to the band gap range and the formation condition, for the resonance frequency omega in S31 t Frequency compensation is carried out:
ω t ＝ω c +Δω c
in the above, Δω c Is the compensation frequency omega c Is the center frequency of the band gap.
3. The method for actively controlling the piezoelectric acoustic metamaterial according to claim 2, wherein the step S4 is to digitize a transfer function to realize the active control of the piezoelectric acoustic metamaterial, specifically:
the transmission function is discretized by using a zero-order retainer according to the sampling time, and then the discretized second-order transmission function is converted into a differential equation form and is input into a digital controller, so that the active regulation and control of the piezoelectric acoustic metamaterial are realized.
4. A method of active conditioning of a piezoelectric acoustic metamaterial according to any of claims 2 to 3, wherein the control unit comprises a pair of piezoelectric elements and an enhanced shunt circuit, the pair of piezoelectric elements being connected to the enhanced shunt circuit.
5. The method according to claim 4, wherein the enhanced shunt circuit comprises a charge amplifying circuit, a voltage amplifying circuit, a resistor and an inductor which are arranged in series, one piezoelectric element of the pair of piezoelectric elements is connected with the charge amplifying circuit, and the other piezoelectric element forms the piezoelectric actuator by using the inverse piezoelectric effect.
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