CN101562409B

CN101562409B - Piezoelectric structure damping control object compensation method

Info

Publication number: CN101562409B
Application number: CN2009100840041A
Authority: CN
Inventors: 姚军; 王晓慧; 叶建华; 金有刚; 李德峰; 谢汉兴
Original assignee: Beihang University
Current assignee: Beihang University
Priority date: 2009-05-12
Filing date: 2009-05-12
Publication date: 2012-06-20
Anticipated expiration: 2029-05-12
Also published as: CN101562409A

Abstract

A piezoelectric structure damping control object compensation method, the method has four specific steps. Step 1: set the substrate; Step 2: set the actuator piece; Step 3: set the sensor piece; Step 4: adjust the gain g _c . The damping control method provided by the invention overcomes the deficiencies in the prior art and eliminates the feedback of the local excitation strain of the piezoelectric actuator, so that the mathematical model of the piezoelectric structure returns to the theoretical model. It is a damping control method with ingenious design and simple operation, which can be widely used in the technical field of active control of structural vibration.

Description

Physical Compensation Method for Damping Control of Piezoelectric Structures

(一)技术领域(1) Technical field

本发明涉及一种压电结构阻尼控制方法，尤其涉及一种压电结构阻尼控制实物补偿法，属结构振动主动控制技术领域。The invention relates to a piezoelectric structure damping control method, in particular to a piezoelectric structure damping control object compensation method, which belongs to the technical field of structural vibration active control.

(二)背景技术(2) Background technology

目前，众多文献中记载的压电结构阻尼控制采用的是典型控制模式——速度负反馈构成的闭环系统。本发明中用压电梁作为典型结构来解释典型控制模式原理以及本发明的控制方法，但该方法也适用于各种其他结构。At present, the damping control of piezoelectric structures recorded in many literatures adopts a typical control mode - a closed-loop system composed of speed negative feedback. In the present invention, the piezoelectric beam is used as a typical structure to explain the principle of a typical control mode and the control method of the present invention, but the method is also applicable to various other structures.

图1是众多文献采用的典型控制模式示意图，从图可见：Figure 1 is a schematic diagram of a typical control mode adopted by many literatures. It can be seen from the figure:

1、在梁1的下表面贴有一压电作动片2，上表面同一位置贴有同一尺寸的压电测量片3并后接运算放大器5和电阻4构成应变率传感器；这种作动器和传感器“同位”配置能保证接成1. A piezoelectric actuator 2 is attached to the lower surface of the beam 1, and a piezoelectric measuring sheet 3 of the same size is attached to the same position on the upper surface, followed by an operational amplifier 5 and a resistor 4 to form a strain rate sensor; this actuator The "same position" configuration with the sensor can ensure the connection

2、速度负反馈后的系统的稳定性。2. The stability of the system after speed negative feedback.

2、压电应变率传感器输出u_s经增益g_s后与外激励电压u_e相减，经功放增益g_a后激励压电作动片构成闭环。2. The output u _s of the piezoelectric strain rate sensor is subtracted from the external excitation voltage u _e after the gain g _s , and the piezoelectric actuator is excited to form a closed loop after the power amplifier gain g _a .

典型控制方法所带来的缺点和不足有：The disadvantages and deficiencies brought about by typical control methods are:

理论上阻尼比会随着回路增益g_sg_a而增大。但是，实验结果并非如此令人满意。随着g_sg_a的增大，系统却趋向于不稳定，产生高频或很低频的自激振动，而此前模态阻尼比增量还未达到期望的要求如0.1。Theoretically the damping ratio will increase with the loop gain g _s g _a . However, the experimental results are not so satisfactory. With the increase of g _s g _a , the system tends to be unstable, producing high-frequency or very low-frequency self-excited vibrations, and the increment of the modal damping ratio has not yet reached the desired requirement such as 0.1.

观察实测到的应变开环频响，发现与理论模型有显著不同：Observing the measured strain open-loop frequency response, it is found that it is significantly different from the theoretical model:

1.相频特性：在共振区下落π/2后不久却恢复到接近于零。1. Phase-frequency characteristic: it returns to close to zero shortly after falling π/2 in the resonance region.

2.幅频特性：在过共振区后并非以40db/oct斜率一直衰减下去，而是恢复到一个近似常量。2. Amplitude-frequency characteristics: After passing through the resonance region, it does not attenuate with a slope of 40db/oct, but returns to an approximate constant.

(三)发明内容(3) Contents of the invention

本发明的目的是提供一种压电结构阻尼控制实物补偿法，它克服了现有技术中的不足，是一种设计巧妙，操作简单的阻尼控制方法。The purpose of the present invention is to provide a piezoelectric structure damping control material compensation method, which overcomes the deficiencies in the prior art, and is a damping control method with ingenious design and simple operation.

现将发明原理作以下介绍：Invention principle is introduced as follows now:

1.引言1 Introduction

改变结构阻尼，历来是振动控制的最基本课题之一，更是振动抑制最主要的手段——增大阻尼。从传统的摩擦器和集中式阻尼器到近二十年来的分布式阻尼材料等被动控制手段，各个手段均是利用附属结构的与基体结构变形速度有关的耗能特性达到增大结构阻尼的目标。经典的主动控制阻尼技术的典型代表是用速度负反馈构成的闭环系统，在单自由度系统，在理论和实践中非常成功，对多自由度系统，会出现观测溢出和控制溢出等问题的困扰，可能导致控制效果变差，甚至失去稳定性。Changing structural damping has always been one of the most basic issues of vibration control, and it is the most important means of vibration suppression - increasing damping. From the traditional friction device and centralized damper to the passive control methods such as distributed damping materials in the past two decades, all methods use the energy dissipation characteristics of the auxiliary structure related to the deformation speed of the base structure to achieve the goal of increasing structural damping. . A typical representative of classic active control damping technology is a closed-loop system composed of negative velocity feedback. In single-degree-of-freedom systems, it is very successful in theory and practice. For multi-degree-of-freedom systems, problems such as observation overflow and control overflow will appear. , may lead to poor control effect and even loss of stability.

速度负反馈阻尼控制的另一中心问题是有什么合适的传感器和作动器以及相配套的设备。对于大型柔性结构，更需要有大批分布式的传感器和作动器。压电材料的发展，正好适应了人们长期对这种传感器和作动器企盼的要求，从而近20年来，关于用压电片作动器和传感器的速度负反馈结构主动阻尼控制——所谓的“电子阻尼”的研究蓬勃开展起来。Another central problem of speed negative feedback damping control is what suitable sensors and actuators and matching equipment are available. For large flexible structures, a large number of distributed sensors and actuators are required. The development of piezoelectric materials has just adapted to the long-term expectations of people for this kind of sensors and actuators. Therefore, in the past 20 years, the active damping control of the speed negative feedback structure using piezoelectric actuators and sensors—the so-called Research on "electronic damping" is flourishing.

首先，在2节压电结构建模的基础上，以压电悬臂梁本地激励——传感速度负反馈阻尼控制为例展开论述，发现结果远不如预期的那么好。对实验现象特别是压电作动器——传感器的频响分析和计算机仿真试验探究其原因，在于压电结构这种特殊的压电“应变激励——应变响应”带来的“局部激励应变”干扰。这导致了我们对2节压电结构建模的反思，在3.2节提出了定性的但已足够实用的修正。接着，在3.3节，提出了局部激励应变补偿方案——让压电结构数学模型回归到2节的理论模型中去，结果在实验中控制的阻尼得到了大幅提高。First of all, based on the modeling of the piezoelectric structure in Section 2, the local excitation of the piezoelectric cantilever beam-sensing speed negative feedback damping control was discussed as an example, and the results were found to be far less than expected. The reason for the experimental phenomenon, especially the piezoelectric actuator-sensor frequency response analysis and computer simulation test, lies in the "local excitation strain" brought by the special piezoelectric "strain excitation-strain response" of the piezoelectric structure. "interference. This leads to a rethinking of our modeling of piezoelectric structures in Section 2, where qualitative but practical enough corrections are proposed in Section 3.2. Then, in Section 3.3, a local excitation strain compensation scheme is proposed—let the mathematical model of the piezoelectric structure return to the theoretical model in Section 2. As a result, the damping controlled in the experiment has been greatly improved.

2.压电结构的传递函数模型2. Transfer function model of piezoelectric structure

在一个结构上贴一些压电片，利用它们的正压电效应测量结构的振动状态，把这些压电片及其附属装置称为“压电传感器”；另外再贴一些压电片并用压电激励，其负压电效应将会激起结构振动，把这些压电片及其附属装置称为“压电作动器”。基体结构、压电作动器和压电传感器合在一起的机电耦合结构称为“压电结构”。Paste some piezoelectric sheets on a structure, and use their positive piezoelectric effect to measure the vibration state of the structure. These piezoelectric sheets and their accessories are called "piezoelectric sensors"; Excitation, its negative piezoelectric effect will excite structural vibrations, and these piezoelectric sheets and their accessories are called "piezoelectric actuators". The electromechanical coupling structure of the base structure, piezoelectric actuator and piezoelectric sensor is called "piezoelectric structure".

2.1压电陶瓷片的四类压电方程2.1 Four types of piezoelectric equations for piezoelectric ceramic sheets

压电体的机——电本构关系也即压电方程描述其机械量(应力T和应变S)及电学量(电场强度E和电位移D)间的耦合关系。本节将导出理论和实验研究中用到的压电陶瓷片(属6mm点群晶体)的四类压电方程。The mechanical-electrical constitutive relation of the piezoelectric body, that is, the piezoelectric equation describes the coupling relationship between its mechanical quantity (stress T and strain S) and electrical quantity (electric field strength E and electric displacement D). This section will derive four types of piezoelectric equations for the piezoelectric ceramic sheet (belonging to 6mm point group crystal) used in theoretical and experimental research.

设压电薄片平面内坐标为x、y，法线方向也即极化方向为z。本文中将以力学分析中惯用的符号为主，但在本节及以后有关章节分析压电片为主的论述中，为保持压电体分析的原貌，将沿用压电体分析文献中的一般符号，二者对照表如下：Let the in-plane coordinates of the piezoelectric sheet be x, y, and the normal direction, that is, the polarization direction, be z. In this paper, the usual symbols in mechanical analysis will be used as the main ones, but in this section and the subsequent chapters in the analysis of piezoelectric sheets, in order to maintain the original appearance of piezoelectric body analysis, the general symbols in piezoelectric body analysis literature will be used. Symbols, the comparison table between the two is as follows:

轴 axis 应力 stress 应变 Strain 从力学角度 From a mechanical point of view x y z x y z σ_x σ_y σ_z τ_yz τ_zx τ_xy σ _x σ _y σ _z τ _yz τ _zx τ _xy ε_x ε_y ε_z γ_yz γ_zx γ_xy ε _x ε _y ε _z γ _yz γ _zx γ _xy 从压电角度 From a piezoelectric point of view 1 2 3 1 2 3 T₁ T₂ T₃ T₄ T₅ T₆ T ₁ T ₂ T ₃ T ₄ T ₅ T ₆ S₁ S₂ S₃ S₄ S₅ S₆ S ₁ S ₂ S ₃ S ₄ S ₅ S ₆

压电陶瓷作为6mm点群晶体，有2种最基本的压电效应方程；其一，电学短路(E₁＝E₂＝E₃＝0)条件下的正压电效应As a 6mm point group crystal, piezoelectric ceramics have two basic piezoelectric effect equations; one is the positive piezoelectric effect under the condition of electrical short circuit (E ₁ =E ₂ =E ₃ =0)

$[\begin{matrix} {D D.}_{11} \\ {D D.}_{22} \\ {D D.}_{33} \end{matrix}] = = [\begin{matrix} 00 & 00 & 00 & 00 & {d d}_{1515} & 00 \\ 00 & 00 & 00 & {d d}_{2525} & 00 & 00 \\ {d d}_{3131} & {d d}_{3232} & {d d}_{3333} & 00 & 00 & 00 \end{matrix}] [\begin{matrix} {T T}_{11} \\ {T T}_{22} \\ {T T}_{33} \\ {T T}_{44} \\ {T T}_{55} \\ {T T}_{66} \end{matrix}],, (({E E.}_{11} = = {E E.}_{22} = = {E E.}_{33} = = 00)) - - - - - - ((22 - - 11))$

其二，力学自由(T_i＝0，i＝1-6)条件下的逆压电效应Second, the inverse piezoelectric effect under the condition of mechanical freedom (T _i =0, i=1-6)

$[\begin{matrix} {S S}_{11} \\ {S S}_{22} \\ {S S}_{33} \\ {S S}_{44} \\ {S S}_{55} \\ {S S}_{66} \end{matrix}] = = [\begin{matrix} 00 & 00 & {d d}_{3131} \\ 00 & 00 & {d d}_{3232} \\ 00 & 00 & {d d}_{3333} \\ 00 & {d d}_{2525} & 00 \\ {d d}_{1515} & 00 & 00 \\ 00 & 00 & 00 \end{matrix}] [\begin{matrix} {E E.}_{11} \\ {E E.}_{22} \\ {E E.}_{33} \end{matrix}],, (({T T}_{i i} = = 00,, i i = = 11 - - 66)) - - - - - - ((22 - - 22))$

其中E_i和D_i分别为压电体沿i轴的内部场强和极面电位移，d_ij为有关i和j轴的压电常数，有Among them, E _i and D _i are the internal field strength and polar surface electric displacement of the piezoelectric body along the i axis, respectively, and d _ij is the piezoelectric constant about the i and j axes.

d_p＝d₃₁＝d₃₂ (2-3)d _p =d ₃₁ =d ₃₂ (2-3)

其中下标p标志“压电片”，本文以后将一直沿用这一下标。Among them, the subscript p marks "piezoelectric film", and this subscript will be used in this article.

对于我们关心的2维压电片，有For the 2D piezoelectric sheet we care about, we have

T₃＝0和E₁＝E₂＝0 (2-4)T ₃ =0 and E ₁ =E ₂ =0 (2-4)

并且我们也只关心D₃、S₁和S₂，方程(2-1)(2-2)成为And we only care about D ₃ , S ₁ and S ₂ , equation (2-1)(2-2) becomes

D₃＝d₃₁T₁+d₃₂T₂＝d_p(T₁+T₂)(E₃＝0) (2-5)D ₃ =d ₃₁ T ₁ +d ₃₂ T ₂ =d _p (T ₁ +T ₂ )(E ₃ =0) (2-5)

$[\begin{matrix} {S S}_{11} \\ {S S}_{22} \end{matrix}] = = [\begin{matrix} {d d}_{3131} \\ {d d}_{3232} \end{matrix}] {E E.}_{33} = = [\begin{matrix} {d d}_{p p} \\ {d d}_{p p} \end{matrix}] {E E.}_{33},, (({T T}_{11} = = {T T}_{22} = = 00)) - - - - - - ((22 - - 66))$

在力学自由(T₁＝T₂＝0)条件下，压电薄片作为一个普通电容，还有介电方程Under the condition of mechanical freedom (T ₁ ＝T ₂ ＝0), the piezoelectric sheet acts as an ordinary capacitor, and the dielectric equation

${D D.}_{33} = = {ϵ ϵ}_{p p}^{T T} {E E.}_{33},, (({T T}_{11} = = {T T}_{22} = = 00)) - - - - - - ((22 - - 77))$

其中ε_p ^T是力学自由(T₁＝T₂＝0，由上标T标志)条件下的介电系数where ε _p ^T is the dielectric coefficient under the condition of mechanical freedom (T ₁ =T ₂ =0, marked by superscript T)

作为弹性体的压电薄片，有电学短路(E₃＝0)条件下的弹性本构关系As an elastic piezoelectric thin film, there is an elastic constitutive relationship under the condition of electrical short circuit (E ₃ =0)

$[\begin{matrix} {S S}_{11} \\ {S S}_{22} \end{matrix}] = = \frac{11}{{E E.}_{p p}} [\begin{matrix} 11 & - - {μ μ}_{p p} \\ - - {μ μ}_{p p} & 11 \end{matrix}] [\begin{matrix} {T T}_{11} \\ {T T}_{22} \end{matrix}],, (({E E.}_{33} = = 00)) - - - - - - ((22 - - 88))$

其中E_p和μ_p分别为压电片的弹性模量和泊桑比。Where E _p and μ _p are the elastic modulus and Poisson's ratio of the piezoelectric film, respectively.

结合方程(2-5)——(2-8)，得到压电片的机——电耦合本构方程Combining equations (2-5)-(2-8), the mechanical-electrical coupling constitutive equation of the piezoelectric sheet is obtained

$[\begin{matrix} {S S}_{11} \\ {S S}_{22} \\ {D D.}_{33} \end{matrix}] = = [\begin{matrix} \frac{11}{{E E.}_{p p}} & \frac{- - {μ μ}_{p p}}{{E E.}_{p p}} & {d d}_{p p} \\ \frac{- - {μ μ}_{p p}}{{E E.}_{}} & \frac{11}{{E E.}_{p p}} & {d d}_{p p} \\ {d d}_{p p} & {d d}_{p p} & {ϵ ϵ}_{p p}^{T T} \end{matrix}] [\begin{matrix} {T T}_{11} \\ {T T}_{22} \\ {E E.}_{33} \end{matrix}] - - - - - - ((22 - - 99))$

它全面地反映了压电片6个量(4个机械量T₁、T₂、S₁、S₂和2个电学量D₃和E₃)间的耦合关系，既包含了正压电效应，又包含了逆压电效应。It comprehensively reflects the coupling relationship between the 6 quantities of the piezoelectric film (4 mechanical quantities T ₁ , T ₂ , S ₁ , S ₂ and 2 electrical quantities D ₃ and E ₃ ), including the positive piezoelectric effect , including the inverse piezoelectric effect.

在不同的场合，用不同的自变量和因变量将更为方便。为此，把方程(2-9)称为第一类压电方程；从中，还可以导得与其等价的其它三类压电方程。In different situations, it is more convenient to use different independent and dependent variables. For this reason, the equation (2-9) is called the first kind of piezoelectric equation; from it, other three kinds of piezoelectric equations equivalent to it can also be derived.

第二类压电方程Piezoelectric equation of the second kind

$[\begin{matrix} {T T}_{11} \\ {T T}_{22} \\ {D D.}_{33} \end{matrix}] = = [\begin{matrix} \frac{{E E.}_{p p}}{11 - - {μ μ}_{p p}^{22}} & \frac{{μ μ}_{p p} {E E.}_{p p}}{11 - - {μ μ}_{p p}^{22}} & - - {e e}_{p p} \\ \frac{{μ μ}_{p p} {E E.}_{p p}}{11 - - {μ μ}_{p p}^{22}} & \frac{{E E.}_{p p}}{11 - - {μ μ}_{p p}^{22}} & - - {e e}_{p p} \\ {e e}_{p p} & {e e}_{p p} & {ϵ ϵ}_{p p}^{S S} \end{matrix}] [\begin{matrix} {S S}_{11} \\ {S S}_{22} \\ {E E.}_{33} \end{matrix}] - - - - - - ((22 - - 1010))$

其中压电系数where the piezoelectric coefficient

${e e}_{p p} = = \frac{{E E.}_{p p} {d d}_{p p}}{11 - - {μ μ}_{p p}} - - - - - - ((22 - - 1111))$

而and

${ϵ ϵ}_{p p}^{S S} = = {ϵ ϵ}_{p p}^{T T} - - 22 {e e}_{p p} {d d}_{p p} = = {ϵ ϵ}_{p p}^{T T} - - \frac{22 {E E.}_{p p} {d d}_{p p}^{22}}{11 - - {μ μ}_{p p}} - - - - - - ((22 - - 1212))$

为压电片在夹持(S₁＝S₂＝0，以上标S标志)条件下的介电常数。is the dielectric constant of the piezoelectric sheet under clamping conditions (S ₁ =S ₂ =0, marked with S above).

第三类压电方程Piezoelectric equation of the third kind

$[\begin{matrix} {S S}_{11} \\ {S S}_{22} \\ {E E.}_{33} \end{matrix}] = = [\begin{matrix} \frac{11}{{E E.}_{p p}} - - \frac{{d d}_{p p}^{22}}{{ϵ ϵ}_{p p}^{T T}} & \frac{- - {μ μ}_{p p}}{{E E.}_{p p}} - - \frac{{d d}_{p p}^{22}}{{ϵ ϵ}_{p p}^{T T}} & {g g}_{p p} \\ \frac{- - {μ μ}_{p p}}{{E E.}_{p p}} - - \frac{{d d}_{p p}^{22}}{{ϵ ϵ}_{p p}^{}} & \frac{11}{{E E.}_{p p}} - - \frac{{d d}_{p p}^{22}}{{ϵ ϵ}_{p p}^{T T}} & {g g}_{p p} \\ - - {g g}_{p p} & - - {g g}_{p p} & \frac{11}{{ϵ ϵ}_{p p}^{T T}} \end{matrix}] [\begin{matrix} {T T}_{11} \\ {T T}_{22} \\ {D D.}_{33} \end{matrix}] - - - - - - ((22 - - 1313))$

其中压电系数where the piezoelectric coefficient

${g g}_{p p} = = \frac{{d d}_{p p}}{{ϵ ϵ}_{p p}^{T T}} - - - - - - ((22 - - 1414))$

第四类压电方程The fourth kind of piezoelectric equation

$[\begin{matrix} {T T}_{11} \\ {T T}_{22} \\ {E E.}_{33} \end{matrix}] = = [\begin{matrix} \frac{{E E.}_{p p}}{11 - - {μ μ}_{p p}^{22}} + + \frac{{e e}_{p p}^{22}}{{ϵ ϵ}_{p p}^{S S}} & \frac{{μ μ}_{p p} {E E.}_{p p}}{11 - - {μ μ}_{p p}^{22}} + + \frac{{e e}_{p p}^{22}}{{ϵ ϵ}_{p p}^{S S}} & - - {h h}_{p p} \\ \frac{{μ μ}_{p p} {E E.}_{p p}}{11 - - {μ μ}_{p p}^{22}} + + \frac{{e e}_{p p}^{22}}{{ϵ ϵ}_{p p}^{S S}} & \frac{{E E.}_{p p}}{11 - - {μ μ}_{p p}^{22}} + + \frac{{e e}_{p p}^{22}}{{ϵ ϵ}_{p p}^{S S}} & - - {h h}_{p p} \\ - - {h h}_{p p} & - - {h h}_{p p} & \frac{11}{{ϵ ϵ}_{p p}^{S S}} \end{matrix}] [\begin{matrix} {S S}_{11} \\ {S S}_{22} \\ {D D.}_{33} \end{matrix}] - - - - - - ((22 - - 1515))$

其中压电系数where the piezoelectric coefficient

${h h}_{p p} = = \frac{{e e}_{p p}}{{ϵ ϵ}_{p p}^{S S}} = = \frac{{E E.}_{p p} {d d}_{p p}}{{ϵ ϵ}_{p p}^{S S} ((11 - - {μ μ}_{p p}))} - - - - - - ((22 - - 1616))$

压电片常用于如梁之类的一维应力结构中，T₂≡0，相应的四类压电方程成为Piezoelectric sheets are often used in one-dimensional stress structures such as beams, T ₂ ≡0, and the corresponding four types of piezoelectric equations become

$[\begin{matrix} S_{1} \\ D_{3} \end{matrix}] = [\begin{matrix} \frac{1}{E_{p}} & d_{p} \\ d_{p} & ϵ_{p}^{T} \end{matrix}] [\begin{matrix} T_{1} \\ E_{3} \end{matrix}]$ (第一类T₂≡0) (2-17) $[\begin{matrix} S_{1} \\ {D.}_{3} \end{matrix}] = [\begin{matrix} \frac{1}{{E.}_{p}} & d_{p} \\ d_{p} & ϵ_{p}^{T} \end{matrix}] [\begin{matrix} T \end{matrix}] [\begin{matrix} _{1} \\ {E.}_{3} \end{matrix}]$ (first type T ₂ ≡0) (2-17)

$[\begin{matrix} T_{1} \\ D_{3} \end{matrix}] = [\begin{matrix} E_{p} & - e_{p} \\ e_{p} & ϵ_{p}^{S} \end{matrix}] [\begin{matrix} S_{1} \\ E_{3} \end{matrix}]$ (第二类T₂≡0) (2-18) $[\begin{matrix} T_{1} \\ {D.}_{3} \end{matrix}] = [\begin{matrix} {E.}_{p} & - e_{p} \\ e_{p} & ϵ_{p}^{S} \end{matrix}] [\begin{matrix} S \end{matrix}] [\begin{matrix} _{1} \\ {E.}_{3} \end{matrix}]$ (Second type T ₂ ≡0) (2-18)

$[\begin{matrix} S_{1} \\ E_{3} \end{matrix}] = [\begin{matrix} \frac{1}{E_{p}} \frac{ϵ_{p}^{S}}{ϵ_{p}^{T}} & g_{p} \\ - g_{p} & \frac{1}{ϵ_{p}^{T}} \end{matrix}] [\begin{matrix} T_{1} \\ D_{3} \end{matrix}]$ (第三类T₂≡0) (2-19) $[\begin{matrix} S_{1} \\ {E.}_{3} \end{matrix}] = [\begin{matrix} \frac{1}{{E.}_{p}} \frac{ϵ_{p}^{S}}{ϵ_{p}^{T}} & g_{p} \\ - g_{p} & \frac{1}{ϵ_{p}^{T}} \end{matrix}] [\begin{matrix} T \end{matrix}] [\begin{matrix} _{1} \\ {D.}_{3} \end{matrix}]$ (The third type T ₂ ≡0) (2-19)

$[\begin{matrix} T_{1} \\ E_{3} \end{matrix}] [\begin{matrix} \frac{1}{E_{p}} \frac{ϵ_{p}^{T}}{ϵ_{p}^{S}} & - h_{p} \\ - h_{p} & \frac{1}{ϵ_{p}^{S}} \end{matrix}] [\begin{matrix} S_{1} \\ D_{3} \end{matrix}]$ (第四类T₂≡0) (2-20) $[\begin{matrix} T_{1} \\ {E.}_{3} \end{matrix}] [\begin{matrix} \frac{1}{{E.}_{p}} \frac{ϵ_{p}^{T}}{ϵ_{p}^{S}} & - h_{p} \\ - h_{p} & \frac{1}{ϵ_{p}^{S}} \end{matrix}] [\begin{matrix} S \end{matrix}] [\begin{matrix} _{1} \\ {D.}_{3} \end{matrix}]$ (The fourth type T ₂ ≡0) (2-20)

其中in

e_p＝E_pd_p (2-21)e _p = E _p d _p (2-21)

${g g}_{p p} = = {d d}_{p p} / / {ϵ ϵ}_{p p}^{T T} - - - - - - ((22 - - 22 twenty two))$

${h h}_{p p} = = {e e}_{p p} / / {ϵ ϵ}_{p p}^{S S} = = \frac{{E E.}_{p p} {d d}_{p p}}{{ϵ ϵ}_{p p}^{S S}} - - - - - - ((22 - - 23 twenty three))$

${ϵ ϵ}_{p p}^{S S} = = {ϵ ϵ}_{p p}^{T T} - - {e e}_{p p} {d d}_{p p} = = {ϵ ϵ}_{p p}^{T T} - - {E E.}_{p p} {d d}_{p p}^{22} - - - - - - ((22 - - 24 twenty four))$

2.2二种压电陶瓷片应变传感器2.2 Two kinds of piezoelectric ceramic strain sensors

见图3c，设压电片二极间通过电阻R构成回路。由第四类压电方程(2-15)，压电片内场强为As shown in Figure 3c, it is assumed that the two electrodes of the piezoelectric film form a circuit through a resistor R. According to the fourth kind of piezoelectric equation (2-15), the field strength inside the piezoelectric sheet is

${E E.}_{33} ((t t)) = = - - {h h}_{p p} (({S S}_{11} ((t t)) + + {S S}_{22} ((t t)))) + + {D D.}_{33} ((t t)) / / {ϵ ϵ}_{p p}^{S S} - - - - - - ((22 - - 2525))$

输出电压为The output voltage is

${u u}_{p p} ((t t)) = = {δ δ}_{p p} {E E.}_{33} ((t t)) = = - - {δ δ}_{p p} {h h}_{p p} (({S S}_{11} ((t t)) + + {S S}_{22} ((t t)))) + + (({δ δ}_{p p} / / {ϵ ϵ}_{p p}^{S S})) {D D.}_{33} ((t t)) - - - - - - ((22 - - 2626))$

其中δ_p为压电片厚度。单位面积压电片提供的电流为Where δ _p is the thickness of the piezoelectric sheet. The current provided by the piezoelectric sheet per unit area is

$i i ((t t)) = = - - {\overset{\cdot &Center Dot;}{D D.}}_{33} ((t t)) = = \frac{{u u}_{p p} ((t t))}{R R} = = \frac{- - {δ δ}_{p p} {h h}_{p p}}{R R} (({S S}_{11} ((t t)) + + {S S}_{22} ((t t)))) + + \frac{{δ δ}_{p p}}{{ϵ ϵ}_{p p}^{S S} R R} {D D.}_{33} ((t t)) - - - - - - ((22 - - 2727))$

从中可得关于电位移D₃(t)的一阶微分方程From it we can get the first order differential equation about the electric displacement D ₃ (t)

$\frac{R R {ϵ ϵ}_{p p}^{S S}}{{δ δ}_{p p}} {\overset{\cdot &Center Dot;}{D D.}}_{33} ((t t)) + + {D D.}_{33} ((t t)) = = {ϵ ϵ}_{p p}^{S S} {h h}_{p p} (({S S}_{11} ((t t)) + + {S S}_{22} ((t t)))) - - - - - - ((22 - - 2828))$

在频域内的解为(在本文中，一个变量在时域t，频域ω或拉氏域s内将采用同一符号)The solution in the frequency domain is (in this paper, a variable will use the same sign in the time domain t, frequency domain ω or Laplace domain s)

${D D.}_{33} ((jω jω)) = = \frac{{h h}_{p p} {ϵ ϵ}_{p p}^{}}{11 + + jω jω \frac{R R {ϵ ϵ}_{p p}^{S S}}{{δ δ}_{p p}}} (({S S}_{11} ((jω jω)) + + {S S}_{22} ((jω jω)))) - - - - - - ((22 - - 2929))$

代回(2-27)，得到在频域内Substitute back to (2-27) to get in the frequency domain

$i i ((jω jω)) = = - - jω jω {D D.}_{33} ((jω jω)) = = \frac{- - jω jω {h h}_{p p} {ϵ ϵ}_{p p}^{}}{11 + + jω jω \frac{R R {ϵ ϵ}_{p p}^{S S}}{{δ δ}_{p p}}} (({S S}_{11} ((jω jω)) + + {S S}_{22} ((jω jω)))) - - - - - - ((22 - - 3030))$

考虑二种特例。其一，R＝0，有(注意方程2-16)Consider two special cases. One, R=0, has (note equation 2-16)

i(jω)＝-jωe_p(S₁(jω)+S₂(jω)) (2-31)i(jω)＝-jωe _p (S ₁ (jω)+S ₂ (jω)) (2-31)

$i i ((t t)) = = - - {e e}_{p p} (({\overset{\cdot &Center Dot;}{S S}}_{11} ((t t)) + + {\overset{\cdot &Center Dot;}{S S}}_{22} ((t t))))$

压电片提供的电流与其沿1和2轴的应变率之和成正比；其二，R很大，以致在频域内有The current provided by the piezoelectric sheet is proportional to the sum of the strain rates along the 1 and 2 axes; secondly, R is so large that there are

$ω ω \frac{R R {ϵ ϵ}_{p p}^{S S}}{{δ δ}_{p p}} > > > > 11 - - - - - - ((22 - - 3232))$

方程(2-30)，将近似为Equation (2-30), will be approximated as

$i i ((jω jω)) \approx \approx \frac{- - {δ δ}_{p p} {h h}_{p p}}{R R} (({S S}_{11} ((jω jω)) + + {S S}_{22} ((jω jω)))) - - - - - - ((22 - - 3333))$

$i i ((t t)) \approx \approx \frac{- - {δ δ}_{p p} {h h}_{p p}}{R R} (({S S}_{11} ((t t)) + + {S S}_{22} ((t t))))$

压电片提供的电流与其沿1和2轴的正应变之和成正比The current provided by the piezoelectric sheet is proportional to the sum of its positive strains along the 1 and 2 axes

方程(2-30)或(2-31)和(2-33)构成了压电片用作结构应变率或应变传感器的基础。Equations (2-30) or (2-31) and (2-33) form the basis for piezoelectric sheets used as structural strain rate or strain sensors.

设把压电片“理想”地粘贴到一个结构上——所谓“理想粘贴”是指压电片粘贴面处的应变与结构的当地应变相同；又设压电片很薄，可以忽略其应变沿厚度的变化，这样，压电片平面域内的应变S₁和S₂分别与结构的当地应变ε_x和ε_y一致Assume that the piezoelectric sheet is "ideally" pasted on a structure—the so-called "ideal paste" means that the strain at the pasted surface of the piezoelectric sheet is the same as the local strain of the structure; and the piezoelectric sheet is so thin that its strain can be ignored variation along the thickness such that the strains _S1 and _S2 in the planar domain of the piezoelectric sheet coincide with the local strains _εx and _εy of the structure, respectively

S₁(x，y，t)＝ε_x(x，y，t) S₂(x，y，t)＝ε_y(x，y，t) (2-34)S ₁ (x, y, t) = ε _x (x, y, t) S ₂ (x, y, t) = ε _y (x, y, t) (2-34)

2.2.1压电应变率传感器2.2.1 Piezoelectric strain rate sensor

把压电片直接接到一个线性运算放大器的负输入端(见图3a)。因为负输入端为“虚地”，电位近似为零，因此方程(2-31)近似成立，运放输出电压为Connect the piezo directly to the negative input of a linear op amp (see Figure 3a). Because the negative input terminal is a "virtual ground", the potential is approximately zero, so the equation (2-31) is approximately true, and the output voltage of the op amp is

${u u}_{s the s} ((t t)) = = - - {R R}_{f f} {&Integral; &Integral; &Integral; &Integral;}_{{Ω Ω}_{S S}} i i ((x x,, y the y,, t t)) dxdy dxdy = = {R R}_{f f} {e e}_{p p} {&Integral; &Integral; &Integral; &Integral;}_{{Ω Ω}_{S S}} (({\overset{\cdot &Center Dot;}{ϵ ϵ}}_{x x} ((x x,, y the y,, t t)) + + {\overset{\cdot &Center Dot;}{ϵ ϵ}}_{y the y} ((x x,, y the y,, t t)))) dxdy dxdy - - - - - - ((22 - - 3535))$

其中R_f为运放反馈电阻，Ω_s为压电片遍及的结构区域，它表明：压电片——运放组合作为结构的“应变率传感器”，输出电压正比于结构的当地正应变率之和的积分，当压电片面积足够小时，将趋于直接正比于结构的当地点正应变率之和。Among them, R _f is the feedback resistance of the operational amplifier, and Ω _s is the structure area covered by the piezoelectric film, which shows that the combination of the piezoelectric film and the operational amplifier acts as a "strain rate sensor" of the structure, and the output voltage is proportional to the local positive strain rate of the structure The integral of the sum, when the piezoelectric sheet area is small enough, will tend to be directly proportional to the sum of the local normal strain rates of the structure.

2.2.2压电应变传感器2.2.2 Piezoelectric strain sensor

把压电片经一大电阻R接入运放的负输入端(见图3b)，压电片处境将如同图3c，在条件(2-32)下，运放输出电压为Connect the piezoelectric piece to the negative input terminal of the operational amplifier through a large resistance R (see Figure 3b), the situation of the piezoelectric piece will be as shown in Figure 3c, under the condition (2-32), the output voltage of the operational amplifier is

${u u}_{s the s} ((t t)) = = - - {R R}_{f f} {&Integral; &Integral; &Integral; &Integral;}_{{Ω Ω}_{s the s}} i i ((x x,, y the y,, t t)) dxdy dxdy$

$= = \frac{{δ δ}_{p p} {h h}_{p p} {R R}_{f f}}{R R} {&Integral; &Integral; &Integral; &Integral;}_{{Ω Ω}_{s the s}} (({ϵ ϵ}_{x x} ((x x,, y the y,, t t)) + + {ϵ ϵ}_{y the y} ((x x,, y the y,, t t)))) dxdy dxdy,, ((\frac{ωR ωR {ϵ ϵ}_{p p}^{S S}}{{δ δ}_{p p}} > > > > 11)) - - - - - - ((22 - - 3636))$

它表明：压电片——运放组合作为结构的“应变传感器”，输出电压正比于结构的当地正应变之和的积分，当压电片面积足够小时，将趋于直接正比于当地点正应变之和。It shows that: the combination of piezoelectric film and operational amplifier is used as a "strain sensor" of the structure, and the output voltage is proportional to the integral of the sum of the local positive strains of the structure. When the area of the piezoelectric film is small enough, it will tend to be directly proportional to the local positive strain. sum of strains.

因为运放正输入端的输入阻抗非常大，因此把压电片直接接到运放的正输入端(见图3b)，也将构成应变传感器。Because the input impedance of the positive input of the op amp is very large, connecting the piezoelectric film directly to the positive input of the op amp (see Figure 3b) will also constitute a strain sensor.

对于一维应力结构(T₂(x，t)＝σ_y(x，t)＝0)，相应于方程(2-35)的应变率和(2-36)应变传感器输出分别为For a one-dimensional stress structure (T ₂ (x, t) = σ _y (x, t) = 0), the strain rate and (2-36) strain sensor output corresponding to equation (2-35) are respectively

${u u}_{s the s} ((t t)) \approx \approx {b b}_{p p} {R R}_{f f} {e e}_{p p} {&Integral; &Integral;}_{{Ω Ω}_{s the s}} {\overset{\cdot &Center Dot;}{ϵ ϵ}}_{x x} ((x x,, t t)) dx dx - - - - - - ((22 - - 3737))$

${u u}_{s the s} ((t t)) \approx \approx \frac{{b b}_{p p} {δ δ}_{p p} {h h}_{p p} {R R}_{f f}}{R R} {&Integral; &Integral;}_{{Ω Ω}_{s the s}} {ϵ ϵ}_{x x} ((x x,, t t)) dx dx - - - - - - ((22 - - 3838))$

其中b_p为压电片宽度。注意其中的压电常数e_p和h_p由方程(21-24)确定。Where b _p is the width of the piezoelectric sheet. Note that the piezoelectric constants e _p and h _p are determined by equations (21-24).

2.3压电梁的传递函数模型2.3 Transfer function model of piezoelectric beam

我们将在待控制的结构上粘贴若干压电片，利用其正压电效应作结构应变或应变率传感器(“压电传感器”)；同时，又另外贴若干压电片并受电压激励，利用其逆压电效应作为作动器(“压电作动器”)。在本文中，把这种同时带有压电传感器和作动器的结构称为“压电结构”，如“压电梁”、“压电板”等。We will paste some piezoelectric sheets on the structure to be controlled, and use its positive piezoelectric effect as a structural strain or strain rate sensor (“piezoelectric sensor”); Its inverse piezoelectric effect acts as an actuator ("piezoelectric actuator"). In this paper, the structure with both piezoelectric sensors and actuators is called "piezoelectric structure", such as "piezoelectric beam", "piezoelectric plate" and so on.

梁作为一种最基本的连续结构，因其力学模型简单，有简洁的物理意义明确的解析解，常常被选用作新型振动控制技术发展的基础研究对象；此外，实际工程中有大量的柔性结构确实也可归为梁的范畴，因此，对压电梁的建模及振动控制成为人们最为关注的研究课题之一，发展也较为成熟。但是要说明的是，本发明是以梁作为示例来讨论，该发明的方法适用于其他各种结构，只是结构的理论建模会有不同。As the most basic continuous structure, the beam is often selected as the basic research object for the development of new vibration control technology because of its simple mechanical model and clear analytical solution with clear physical meaning; in addition, there are a large number of flexible structures in actual engineering Indeed, it can also be classified into the category of beams. Therefore, the modeling and vibration control of piezoelectric beams has become one of the most concerned research topics, and the development is relatively mature. However, it should be noted that the present invention is discussed with a beam as an example, and the method of the present invention is applicable to various other structures, but the theoretical modeling of the structure will be different.

2.3.1梁上的压电作动器分析2.3.1 Analysis of the piezoelectric actuator on the beam

参阅图4，设梁的长、宽和高分别为L，b和h，分别对应于x，y和z轴。设在梁的展度(x₁，x₂)内上下表面各贴有一宽度为b_p，厚度为δ_p的压电陶瓷片，极化方向都沿正z轴，同时外加激励电压u_a(t)。Referring to Fig. 4, let the length, width and height of the beam be L, b and h respectively, which correspond to the x, y and z axes respectively. Assuming that the upper and lower surfaces of the beam's spread (x ₁ , x ₂ ) are each pasted with a piezoelectric ceramic sheet with a width of b _p and a thickness of δ _p , the polarization direction is along the positive z axis, and an excitation voltage u _a ( t).

设梁为Bernoulli-Euller梁。因为梁是一维应力结构(σ_y(x，z，t)≡0)，因此压电片也处一维应力状态(T₂(x，t)≡0)，据第二类压电方程(2-18)，上表面压电片截面应力为Let the beam be a Bernoulli-Euller beam. Because the beam is a one-dimensional stress structure (σ _y (x, z, t)≡0), the piezoelectric sheet is also in a one-dimensional stress state (T ₂ (x, t)≡0), according to the second piezoelectric equation (2-18), the cross-sectional stress of the piezoelectric sheet on the upper surface is

T₁＝E_pS₁-e_pE₃(e_p＝E_pd_p) (2-39)T ₁ ＝E _p S ₁ -e _p E ₃ (e _p ＝E _p d _p ) (2-39)

其中压电片内场强The field strength inside the piezoelectric film

E₃＝u_a/δ_p (2-40)E ₃ = u _a /δ _p (2-40)

设压电片在梁上是“理想”粘贴的(参阅条件2-34)，梁的上表面应变为Assuming that the piezoelectric film is "ideally" pasted on the beam (see condition 2-34), the upper surface of the beam should be

ε_x0＝ε_x|_z＝h/2＝S₁ (2-41)ε _x0 = ε _x | _{z = h/2} = S ₁ (2-41)

把它和(2-40)一起代回(2-39)，有Substituting it and (2-40) back into (2-39), we have

${T T}_{11} = = {E E.}_{p p} {ϵ ϵ}_{x x 00} - - \frac{{E E.}_{p p} {d d}_{p p}}{{δ δ}_{p p}} {u u}_{a a} - - - - - - ((22 - - 4242))$

由对称性可知下表面压电片截面应力为-T₁，从而压电片对梁的激励弯矩为From the symmetry, it can be seen that the cross-sectional stress of the piezoelectric sheet on the lower surface is -T ₁ , so the excitation bending moment of the piezoelectric sheet on the beam is

$M m = = - - (({T T}_{11} {b b}_{p p} {δ δ}_{p p})) h h = = - - {E E.}_{p p} {b b}_{p p} {δ δ}_{p p} h h {ϵ ϵ}_{x x 00} + + \frac{h h {b b}_{p p} {E E.}_{p p} {d d}_{p p}}{{δ δ}_{p p}} {u u}_{a a} - - - - - - ((22 - - 4343))$

将上述激励力矩折算到梁的截面弯矩Convert the above excitation moment to the section bending moment of the beam

$M m = = \frac{22 EI EI {ϵ ϵ}_{x x 00}}{h h},, ((I I = = \frac{11}{1212} b b {h h}^{33})) - - - - - - ((22 - - 4444))$

其中E和I分别为梁的弹性模量和截面惯矩。联解方程(2-43)和(2-44)有where E and I are the elastic modulus and section moment of inertia of the beam, respectively. The joint solutions of equations (2-43) and (2-44) have

$M m ((t t)) = = \frac{h h {b b}_{p p} {E E.}_{p p} {d d}_{} p p}{11 + + 66 \frac{{E E.}_{p p} {δ δ}_{p p} {b b}_{p p}}{Ehb Ehb}} {u u}_{a a} ((t t)) - - - - - - ((22 - - 4545))$

方程(2-45)表明Equation (2-45) shows that

(1)因为u_a(t)不随坐标x而变，因此压电作动器的截面激励力矩M(t)也不随x而变，即为在压电片遍及的梁展度(x₁，x₂)内的均匀力矩；或者，也可以看作是在压电片两端x₁和x₂处施加了一对反向的力矩M(t)(见图4)。(1) Since u _a (t) does not change with the coordinate x, the cross-sectional excitation moment M(t) of the piezoelectric actuator does not change with x, that is, the beam spread (x ₁ , x ₂ ) within the uniform moment; or, it can also be seen as a pair of opposite moments M(t) applied at the two ends of the piezoelectric sheet x ₁ and x ₂ (see Figure 4).

(2)压电作动器的激励能力大小除直接正比其压电常数d_p外，还取决于压电片与梁间的截面刚度比(刚度匹配)。弹性模量E_p较大的压电陶瓷片比弹性模量小得多的压电薄膜(PVDF)有较大的驱动能力，这正是我们选用压电陶瓷片为作动器的基本原因之一。(2) In addition to being directly proportional to the piezoelectric constant _dp , the excitation capacity of the piezoelectric actuator also depends on the section stiffness ratio (stiffness matching) between the piezoelectric sheet and the beam. The piezoelectric ceramic sheet with a larger elastic modulus E _p has a greater driving ability than the piezoelectric film (PVDF) with a much smaller elastic modulus. This is one of the basic reasons why we choose piezoelectric ceramic sheets as actuators. one.

2.3.2微元压电作动器到传感器的传递函数2.3.2 Transfer function from microelement piezoelectric actuator to sensor

按照梁的模态理论，梁的挠度按其固有振型展开为According to the modal theory of the beam, the deflection of the beam is expanded according to its natural mode shape as

$w w ((x x,, t t)) = = {Σ Σ}_{r r = = 11}^{\infty \infty} {Φ Φ}_{r r} ((x x)) {q q}_{r r} ((t t)) - - - - - - ((22 - - 4646))$

其中Φ_r(x)和q_r(t)分别为第r阶固有振型和相应的模态坐标，其在拉氏域(s)内的运动方程为where Φ _r (x) and q _r (t) are the r-th order natural mode shape and the corresponding mode coordinates respectively, and its motion equation in the Lagrangian domain (s) is

${q q}_{r r} ((s the s)) = = \frac{{f f}_{r r} ((s the s))}{{m m}_{r r} {s the s}^{22} + + {k k}_{r r}} - - - - - - ((22 - - 4747))$

其中m_r，k_r，f_r(s)分别为第r阶模态质量、刚度和模态广义力。Among them m _r , k _r , f _r (s) are the rth order modal mass, stiffness and modal generalized force respectively.

施于x＝x_a和x＝x_a+dx_a处的一对反向力矩M(t)产生的模态广义力Modal generalized force generated by a pair of opposite moments M(t) applied at x=x _a and x=x _a +dx _a

${f f}_{r r} (({x x}_{a a},, t t)) = = M m ((t t)) (({Φ Φ}_{r r}^{x x} (({x x}_{a a} + + d d {x x}_{a a})) - - {Φ Φ}_{r r}^{x x} (({x x}_{a a})))) = = M m ((t t)) {Φ Φ}_{r r}^{xx xx} (({x x}_{a a})) d d {x x}_{a a} - - - - - - ((22 - - 4848))$

其中上标x代表对x求导；代回(2-47)和(2-46)，梁在拉氏域内的响应为Where the superscript x represents the derivative of x; substituting (2-47) and (2-46), the response of the beam in the Laplace domain is

$w w ((x x,, {x x}_{a a},, s the s)) = = {Σ Σ}_{r r = = 11}^{\infty \infty} \frac{{Φ Φ}_{r r} ((x x)) {f f}_{r r} (({x x}_{a a},, t t))}{{m m}_{r r} {s the s}^{22} + + {k k}_{r r}} = = M m ((s the s)) d d {x x}_{a a} {Σ Σ}_{r r = = 11}^{\infty \infty} \frac{{Φ Φ}_{r r} ((x x)) {Φ Φ}_{r r}^{xx xx} (({x x}_{a a}))}{{m m}_{r r} {s the s}^{22} + + {k k}_{r r}} - - - - - - ((22 - - 4949))$

现设在x＝x_a处贴有一对宽为b_p，长为dx_a的压电作动器，根据2.3.1节的分析，只要把方程(2-45)代入(2-49)，就可以得到梁对压电作动器的响应Suppose now that a pair of piezoelectric actuators with width b _p and length dx _a are pasted at x=x _a , according to the analysis in section 2.3.1, as long as equation (2-45) is substituted into (2-49), The response of the beam to the piezoelectric actuator can be obtained

$w w ((x x,, {x x}_{a a},, s the s)) = = \frac{h h {E E.}_{p p} {d d}_{} p p}{11 + + 66 \frac{{E E.}_{p p} {δ δ}_{p p} {b b}_{p p}}{Ehb Ehb}} (({b b}_{p p} d d {x x}_{a a})) {u u}_{a a} ((s the s)) {Σ Σ}_{r r = = 11}^{\infty \infty} \frac{{Φ Φ}_{r r} ((x x)) {Φ Φ}_{r r}^{xx xx} (({x x}_{a a}))}{{m m}_{r r} {s the s}^{22} + + {k k}_{r r}} - - - - - - ((22 - - 5050))$

相应的，梁的表面应变为Correspondingly, the surface strain of the beam is

${ϵ ϵ}_{x x 00} ((x x,, {x x}_{a a},, s the s)) = = \frac{h h}{22} {w w}^{xx xx} ((x x,, {x x}_{a a},, s the s)) - - - - - - ((22 - - 5151))$

现设在x＝x_s处贴有一宽为b_ps，厚为δ_ps，长为dx_s(其中δ_ps，h_ps等变量中的下标s标志“压电传感器”，以与压电作动器的相应量区别)的微元压电片并后接运放构成压电应变传感器(图3b)，根据方程(2-38)，(2-51)和(2-50)有传感器输出电压Assume now that x=x _s is pasted with a width of b _ps , a thickness of δ _ps , and a length of dx _s (the subscript s in variables such as δ _ps and h _ps marks "piezoelectric sensor", to work with piezoelectric The corresponding amount of the actuator is different) the microelement piezoelectric sheet is connected with the op amp to form a piezoelectric strain sensor (Figure 3b), according to the equation (2-38), (2-51) and (2-50) have sensor output Voltage

${u u}_{s the s} ((s the s)) = = \frac{{δ δ}_{ps ps} {h h}_{ps ps} {R R}_{f f}}{R R} {ϵ ϵ}_{x x 00} (({x x}_{s the s},, {x x}_{a a},, s the s)) {b b}_{ps ps} d d {x x}_{s the s}$

$= = α α {u u}_{a a} ((s the s)) {Σ Σ}_{r r = = 11}^{\infty \infty} \frac{{Φ Φ}_{r r}^{xx xx} (({x x}_{s the s})) {Φ Φ}_{r r}^{xx xx} (({x x}_{a a}))}{{m m}_{r r} {s the s}^{22} + + {k k}_{r r}} (({b b}_{ps ps} d d {x x}_{s the s})) (({b b}_{p p} d d {x x}_{a a})) - - - - - - ((22 - - 5252))$

其中in

$α α = = \frac{h h {δ δ}_{ps ps} {h h}_{ps ps} {R R}_{f f}}{22 R R} \frac{h h {E E.}_{p p} {d d}_{} p p}{11 + + 66 \frac{{E E.}_{p p} {b b}_{p p} {h h}_{p p}}{Ehb Ehb}} - - - - - - ((22 - - 5353))$

由此可知，从单位面积(b_pdx_a＝1)压电作动器的激励电压u_a到单位面积(b_psdx_s＝1)压电传感器的输出电压u_s的传递函数为It can be seen that the transfer function from the excitation voltage u _a of the piezoelectric actuator per unit area (b _p dx _a =1) to the output voltage u _s of the piezoelectric sensor per unit area (b _ps dx _s =1) is

$H h (({x x}_{s the s},, {x x}_{a a},, s the s)) = = \frac{{u u}_{s the s} ((s the s))}{{u u}_{a a} ((s the s))} = = α α {Σ Σ}_{r r = = 11}^{\infty \infty} \frac{{Φ Φ}_{r r}^{xx xx} (({x x}_{s the s})) {Φ Φ}_{r r}^{xx xx} (({x x}_{a a}))}{{m m}_{r r} {s the s}^{22} + + {k k}_{r r}} - - - - - - ((22 - - 5454))$

同理，如果用的是压电应变率传感器(图3a)将有Similarly, if a piezoelectric strain rate sensor (Fig. 3a) is used, there will be

$H h (({x x}_{s the s},, {x x}_{a a},, s the s)) = = \frac{{u u}_{s the s} ((s the s))}{{u u}_{a a} ((s the s))} = = {α α}_{v v} {Σ Σ}_{r r = = 11}^{\infty \infty} \frac{{Φ Φ}_{r r}^{xx xx} (({x x}_{s the s})) {Φ Φ}_{r r}^{xx xx} (({x x}_{a a}))}{{m m}_{r r} {s the s}^{22} + + {k k}_{r r}} - - - - - - ((22 - - 5555))$

其中in

${α α}_{v v} = = \frac{- - h h {R R}_{f f} {e e}_{ps ps}}{22} \frac{h h {E E.}_{p p} {d d}_{} p p}{11 + + 66 \frac{{E E.}_{p p} {b b}_{p p} {δ δ}_{p p}}{Ehb Ehb}} - - - - - - ((22 - - 5656))$

2.3.3有限尺寸压电作动器到传感器的传递函数2.3.3 Transfer function from finite size piezoelectric actuator to sensor

现设压电作动片和传感片都是有限尺寸的，参数d_p、b_p、d_ps、b_ps等不随梁坐标x而变。根据方程(2-54)，从有限尺寸压电片作动器到有限尺寸压电片应变传感器的传递函数为Assume that both the piezoelectric actuator and sensor are of finite size, and the parameters d _p , b _p , d _ps , b _ps etc. do not change with the beam coordinate x. According to equation (2-54), the transfer function from the finite size piezoelectric actuator to the finite size piezoelectric strain sensor is

$H h (({Ω Ω}_{s the s},, {Ω Ω}_{a a},, s the s)) = = \frac{{u u}_{s the s} ((s the s))}{{u u}_{a a} ((s the s))} = = α α {b b}_{ps ps} {b b}_{p p} {&Integral; &Integral;}_{{Ω Ω}_{s the s}} d d {x x}_{s the s} {&Integral; &Integral;}_{{Ω Ω}_{a a}} H h (({x x}_{s the s},, {x x}_{a a},, s the s)) d d {x x}_{a a}$

其中in

Ω_s，Ω_a分别代表压电传感片和作动片遍及的梁的区域。如果用的是应变率传感器，相应的有Ω _s and Ω _a respectively represent the area of the beam covered by the piezoelectric sensor piece and the actuator piece. If a strain rate sensor is used, the corresponding

3压电结构阻尼控制和局部激励应变补偿3 Piezoelectric structure damping control and local excitation strain compensation

3.1压电梁本地速度负反馈阻尼控制实验探索3.1 Experimental Exploration of Local Velocity Negative Feedback Damping Control of Piezoelectric Beam

图5是众多文献采用的典型控制模式，在背景技术部分也给出了详细的图示，这里将作定量的说明，以便更好理解背景技术部分中的论述。Fig. 5 is a typical control mode adopted by many documents, and a detailed diagram is also given in the background technology part, and a quantitative description will be made here to better understand the discussions in the background technology part.

依据2节梁的建模，有开环传递函数方程(方程2-59)Based on the modeling of a 2-section beam, there is an open-loop transfer function equation (Equation 2-59)

其中

见方程(2-58)。A是压电作动片和传感片遍及的梁区域。注意2节建模时是梁的上下表面各贴一作动片，这里是只在下表面贴一片，但从梁的总体弯曲响应来说，相当于方程(3-1)的系数α_v减半。in

See equation (2-58). A is the area of the beam over which the piezoelectric actuator and sensing plates are spread. Note that when modeling in Section 2, a moving piece is pasted on the upper and lower surfaces of the beam, but here only one piece is pasted on the lower surface, but from the perspective of the overall bending response of the beam, it is equivalent to halving the coefficient _αv of equation (3-1).

按图5右边的控制框图，可以求到闭环传递函数为According to the control block diagram on the right side of Figure 5, the closed-loop transfer function can be obtained as

${\overset{&OverBar; &OverBar;}{H h}}_{v v} ((s the s)) = = \frac{{u u}_{s the s} ((s the s))}{{u u}_{e e} ((s the s))} = = \frac{{g g}_{a a} {H h}_{v v} ((s the s))}{11 + + {H h}_{v v} ((s the s)) {g g}_{a a} {g g}_{s the s}} - - - - - - ((33 - - 22))$

设梁的各阶模态固有频率是离散的，在第r阶固有频率Ω_r邻近，其余各阶模态的响应可以忽略不计，从而有Assuming that the natural frequencies of each order mode of the beam are discrete, and the rth order natural frequency Ω _r is close to it, the responses of the other order modes can be ignored, so that

代回(3-2)有Replacing (3-2) with

由此可见，第r阶模态增大了阻尼系数It can be seen that the rth mode increases the damping coefficient

或阻尼比or damping ratio

随着回路增益g_sg_a而增大，看来是很理想的。It seems ideal to increase with the loop gain g _s g _a .

但是，实验结果并非如此令人满意。随着g_sg_a的增大，系统却趋向于不稳定，产生高频或很低频的自激振动，而此前模态阻尼比增量还未达到期望的要求如0.1。However, the experimental results are not so satisfactory. With the increase of g _s g _a , the system tends to be unstable, producing high-frequency or very low-frequency self-excited vibrations, and the increment of the modal damping ratio has not yet reached the desired requirement such as 0.1.

观察实测到的应变开环频响(图6b)，发现它与理论模型的Observing the measured strain open-loop frequency response (Figure 6b), it is found that it is consistent with the theoretical model

H(jω)＝H_v(jω)/jω (3-7)H(jω)＝ _Hv (jω)/jω (3-7)

(见图6a)有显著不同：(See Figure 6a) There are significant differences:

(1)相频特性：在共振区下落π/2后不久却恢复到接近于零。(1) Phase-frequency characteristic: it returns to close to zero shortly after falling π/2 in the resonance region.

(2)幅频特性：在过共振区后并非以40db/oct斜率一直衰减下去，而是恢复到一个近似常量。(2) Amplitude-frequency characteristics: After passing through the resonance region, it does not attenuate with a slope of 40db/oct, but returns to an approximate constant.

对压电作动器特别是传感器的建模再分析和计算机仿真试验，表明这一差别来源于作动器局部激励应变对压电测量片应变的干扰，而这正是约束闭环系统反馈增益不能提高的一个极为重要的原因。The modeling reanalysis and computer simulation tests of the piezoelectric actuator, especially the sensor, show that this difference comes from the interference of the local excitation strain of the actuator on the strain of the piezoelectric measuring piece, and this is why the feedback gain of the constrained closed-loop system cannot An extremely important reason for improvement.

3.2对压电结构建模的反思和修正3.2 Reflection and revision of piezoelectric structure modeling

在2节压电结构建模时，我们实质上沿用了传统的激励——传感模式的一种默认：作动器激励只产生结构的总体变形而忽略施力处的局部变形；传感器只感受结构的总体变形而忽略同位作动器激励产生的局部变形的影响。对于传统的力或基础激励——加速度计响应之类，这种假设确实是足够合理的，而今在压电作动片——压电传感片这种“应变激励——应变传感”的新情况下，就可能成为问题了。When modeling the piezoelectric structure in section 2, we essentially followed the traditional excitation—a default of the sensing mode: the actuator excitation only produces the overall deformation of the structure and ignores the local deformation at the force application point; the sensor only senses The overall deformation of the structure is considered and the influence of local deformation generated by the excitation of co-located actuators is neglected. For the traditional force or basic excitation-accelerometer response, this assumption is indeed reasonable enough, but now in the "strain excitation-strain sensing" of the piezoelectric actuator-piezoelectric sensor Under new circumstances, this may become a problem.

以压电梁为例，压电作动片应变ε_a中包括两部分Taking the piezoelectric beam as an example, the piezoelectric actuator strain ε _a includes two parts

ε_a＝-ε_m+ε_ta (3-8)ε _a = -ε _m +ε _ta (3-8)

其中ε_m是梁的总体弯曲应变响应，而ε_ta是因其作为激励源而产生的局部激励应变分量。压电传感片的应变ε_s也包括相应的两部分where _εm is the overall bending strain response of the beam, and _εta is the local excited strain component due to it being the excitation source. The strain ε _s of the piezoelectric sensor sheet also includes two corresponding parts

ε_s＝ε_m+ε_ts (3-9)ε _s = ε _m + ε _ts (3-9)

其中ε_ts是它对压电作动片局部激励应变ε_ta经结构传播而来的响应。这样，从压电驱动电压u_a到传感器应变ε_s的传递函数为Where ε _ts is its response to the local excitation strain ε _ta of the piezoelectric actuator propagated through the structure. In this way, the transfer function from the piezoelectric driving voltage u _a to the sensor strain ε _s is

${H h}_{ϵ ϵ} ((s the s)) = = \frac{{ϵ ϵ}_{s the s} ((s the s))}{{u u}_{a a} ((s the s))} = = \frac{{ϵ ϵ}_{m m} ((s the s))}{{u u}_{a a} ((s the s))} + + \frac{{ϵ ϵ}_{ts ts} ((s the s))}{{u u}_{a a} ((s the s))} = = \frac{{ϵ ϵ}_{m m} ((s the s))}{{u u}_{a a} ((s the s))} + + \frac{{ϵ ϵ}_{ts ts} ((s the s))}{{ϵ ϵ}_{ta ta} ((s the s))} \frac{{ϵ ϵ}_{ta ta} ((s the s))}{{u u}_{a a} ((s the s))} - - - - - - ((33 - - 1010))$

其中的第一部分相当于在2节建模中得到的传递函数；第二部分中的因子ε_ta(s)/u_a(s)取决于压电作动片的激励特性，可视为常量；而另一因子ε_ts(s)/ε_ta(s)则取决于应力波在结构中的传播特性，与传感片与作动片的相对位置有关，在二片同位配置时，因为二者相距最近，将达到最大。The first part is equivalent to the transfer function obtained in Section 2 modeling; the factor ε _ta (s)/u _a (s) in the second part depends on the excitation characteristics of the piezoelectric actuator and can be regarded as a constant; The other factor ε _ts (s)/ε _ta (s) depends on the propagation characteristics of the stress wave in the structure, and is related to the relative position of the sensor plate and the action plate. When the two plates are arranged in the same position, because the two The closest distance will be the maximum.

这样，对2节的压电建模，至少在传感片离作动片很近时，必须引入一代表作动片局部激励应变在结构中传递特性的修正项。例如方程(3-1)，应改为In this way, for the piezoelectric modeling of Section 2, at least when the sensing piece is very close to the moving piece, a correction term must be introduced to represent the transfer characteristics of the local excitation strain of the moving piece in the structure. For example, equation (3-1), should be changed to

对这种修正项H_t(s)的定量分析将是一个值得研究的课题，在有多个压电片激励时，会显得更复杂。令人欣慰的是，它是不难通过实验参数识别来估计并通过一些简单的策略去消除或“补偿”的，从而使压电结构数学模型又回归到“常规”中去，这将是3.3节要研究的问题。The quantitative analysis of this correction term H _t (s) will be a topic worth studying, and it will appear more complicated when there are multiple piezoelectric slices excited. The good news is that it is not difficult to estimate through experimental parameter identification and eliminate or "compensate" through some simple strategies, so that the mathematical model of the piezoelectric structure returns to "normal", which will be 3.3 The question to be researched.

在计算机仿真试验中我们引入了常修正项H_t(s)＝Const，当它从0开始增大时，可以看到频响曲线将从常规的模式(图6a)逐渐转向“压电模式”(图6b)。在实验中，我们加测离作动片不远处(“异地”)的压电传感片的频响曲线u_st(jω)/u_a(jω)/(jω)，表示在图6c中，它已很接近于常规频响模式(图6a)了；这是因为作动片局部激励应变的传播随距离迅速衰减的缘故(圣维南原理)。In the computer simulation experiment, we introduced a constant correction term H _t (s) = Const, when it increases from 0, it can be seen that the frequency response curve will gradually turn from the conventional mode (Figure 6a) to the "piezoelectric mode" (Fig. 6b). In the experiment, we additionally measured the frequency response curve u _st (jω)/u _a (jω)/(jω) of the piezoelectric sensor not far from the actuator ("off-site"), which is shown in Figure 6c , which is very close to the conventional frequency response mode (Fig. 6a); this is because the propagation of the local excitation strain of the actuator decays rapidly with distance (Saint-Venant's principle).

在2节，我们都是以上下两面贴有压电片一起构成压电作动器的，似乎与现在分析的只有一面压电片的有所不同，其实不然。事实上，这时压电传感片直接感受作动片激励应变分量，ε_ts(s)/ε_ta(s)＝1，从而使局部激励应变的响应变得更大。In section 2, we all have piezoelectric actuators with piezoelectric sheets on the upper and lower sides. It seems to be different from the current analysis of only one piezoelectric sheet, but it is not. In fact, at this time, the piezoelectric sensing piece directly feels the excitation strain component of the actuator, ε _ts (s)/ε _ta (s)=1, so that the response to the local excitation strain becomes larger.

3.3压电作动器局部激励应变的补偿3.3 Compensation of local excitation strain of piezoelectric actuator

记remember

开环频响(3-11)成为The open loop frequency response (3-11) becomes

${H h}_{v v} ((s the s)) = = \frac{{u u}_{s the s} ((s the s))}{{u u}_{a a} ((s the s))} = = {b b}_{ps ps} {b b}_{p p} {α α}_{v v} s the s (({H h}_{m m} ((s the s)) + + {H h}_{t t} ((s the s)))) - - - - - - ((33 - - 1313))$

图5的控制框图成为图7(暂且没有ΔH(s)支路)。闭环频响为The control block diagram of Fig. 5 becomes Fig. 7 (without the ΔH(s) branch for now). The closed loop frequency response is

${\overset{&OverBar; &OverBar;}{H h}}_{v v} ((s the s)) = = \frac{{u u}_{s the s} ((s the s))}{{u u}_{e e} ((s the s))} \approx \approx \frac{{g g}_{a a} {b b}_{ps ps} {b b}_{p p} {α α}_{v v} s the s (({H h}_{m m} ((s the s)) + + {H h}_{t t} ((s the s))))}{11 + + {g g}_{s the s} {g g}_{a a} {b b}_{ps ps} {b b}_{p p} {α α}_{v v} s the s (({H h}_{m m} ((s the s)) + + {H h}_{t t} ((s the s))))} - - - - - - ((33 - - 1414))$

相应于窄带内的近似方程(3-3)和(3-4)Approximate equations (3-3) and (3-4) corresponding to the narrow band

变得相当复杂，不再有简洁的阻尼增量解(3-5)或(3-6)。becomes quite complicated, and there is no longer a neat damped incremental solution to (3-5) or (3-6).

事实上，正是由于H_t(s)的客观存在，在构成闭环系统后，损坏了弯曲振动的控制品质。设法消除H_t(s)的影响成为提高压电结构控制质量的一件大事。In fact, it is precisely because of the objective existence of H _t (s) that the control quality of bending vibration is damaged after the closed-loop system is formed. Trying to eliminate the influence of H _t (s) becomes a major issue in improving the control quality of piezoelectric structures.

见图7，从理论上说，没有太大的难处：增加一支旁路校正环节See Figure 7, theoretically, there is not much difficulty: add a bypass correction link

ΔH(s)＝-H_t(s) (3-17)ΔH(s)=- _Ht (s) (3-17)

就可以了。这就是说，这一环节补偿将把压电结构的传递函数模型回归到2节导出的结果中去。That's it. That is to say, this link compensation will regress the transfer function model of the piezoelectric structure to the results derived in Section 2.

综合以上所述，本发明通过以下技术方案实现的：把与压电梁上的作动片和传感片相同的二压电片贴在一个材料和厚度与梁基体相同的小基片上构成了“压电补偿片”，它的作动片与梁上的作动片受同样的激励，而传感片后接运放，其安排也与梁上的传感器一样。Based on the above, the present invention is realized through the following technical solutions: two piezoelectric sheets identical to the actuating sheet and sensing sheet on the piezoelectric beam are pasted on a small substrate whose material and thickness are the same as the beam matrix to form a "Piezoelectric compensation sheet", its actuating sheet and the actuating sheet on the beam receive the same excitation, and the sensor sheet is connected to the operational amplifier, and its arrangement is the same as that of the sensor on the beam.

由于采用上述技术方案，本发明提供的阻尼控制方法消除了压电作动片局部激励应变的反馈，从而使压电结构的数学模型又回归到理论模型中去。Due to the adoption of the above technical solution, the damping control method provided by the present invention eliminates the feedback of the local excitation strain of the piezoelectric actuator, so that the mathematical model of the piezoelectric structure returns to the theoretical model.

见图2所示，一种压电结构阻尼控制实物补偿法，以梁为示例来说明，该方法具体步骤如下：As shown in Figure 2, a piezoelectric structure damping control physical compensation method is illustrated by taking a beam as an example. The specific steps of this method are as follows:

步骤一：设置基体。取一个小基片8，该基片的材料和厚度与梁1都相同。Step 1: Set up the base. Take a small substrate 8, the same material and thickness as the beam 1.

步骤二：设置作动片。把与梁1上的作动片2相同的作动片9贴在小基片8上，作动片9与梁1上的作动片2受同样的激励。Step 2: Set the animation. Paste the same actuating piece 9 as the actuating piece 2 on the beam 1 on the small substrate 8, and the actuating piece 9 and the actuating piece 2 on the beam 1 are excited in the same way.

步骤三：设置传感片。把与梁1上的测量片3相同的测量片10贴在小基片8的另一面，与原先的作动片9正对。测量片10后接运算放大器12和电阻11，与梁1上的测量片3后面的电阻4和运算放大器5一样，并且测量片10，运算放大器12和电阻11的连接关系也与测量片3，电阻4和运算放大器5的连接关系一样。Step 3: Set up the sensor sheet. Paste the same measuring piece 10 as the measuring piece 3 on the beam 1 on the other side of the small substrate 8, facing the original actuating piece 9. Measuring sheet 10 is followed by operational amplifier 12 and resistor 11, the same as resistor 4 and operational amplifier 5 at the back of measuring sheet 3 on the beam 1, and measuring sheet 10, the connection relationship between operational amplifier 12 and resistor 11 is also the same as measuring sheet 3, The connection relationship between the resistor 4 and the operational amplifier 5 is the same.

步骤四：调节增益g_c。因为基片面积很小，在梁的非高频模态频带内，基片的弯曲应变可以忽略不计，只有拉压应变，从而它的从作动片应变到传感片的传播将可以直接模拟梁上作动片局部激励拉压应变到传感片的传播，或者说，二者的拉压应变传递函数ε_ts(s)/ε_ta(s)相同或成比例。这样，只要调节增益g_c就可以达到“补偿”条件(3-17)Step 4: Adjust the gain g _c . Because the substrate area is very small, in the non-high frequency modal frequency band of the beam, the bending strain of the substrate is negligible, only the tension and compression strain, so its propagation from the actuator strain to the sensor sheet can be directly simulated on the beam The propagation of tensile and compressive strains from the actuating piece to the sensing piece is locally excited, or in other words, the transfer functions of the tension and compression strains ε _ts (s)/ε _ta (s) of the two are the same or proportional. In this way, the "compensation" condition can be achieved only by adjusting the gain g _c (3-17)

$ΔH ΔH ((s the s,, {g g}_{c c})) = = \frac{Δ Δ {u u}_{s the s} ((s the s,, {g g}_{c c}))}{{u u}_{a a} ((s the s))} = = - - {H h}_{t t} ((s the s)) - - - - - - ((33 - - 1818))$

g_c应接近于-1。g _c should be close to -1.

该方法的优点及功效是：它消除了压电作动片局部激励应变的反馈，从而使压电结构的数学模型又回归到理论模型中去。所以，它是一种设计巧妙，操作简单的阻尼控制方法。The advantages and effects of this method are: it eliminates the feedback of the local excitation strain of the piezoelectric actuator, so that the mathematical model of the piezoelectric structure returns to the theoretical model. Therefore, it is a damping control method with ingenious design and simple operation.

(四)附图说明(4) Description of drawings

图1压电梁本地速度负反馈阻尼控制示意图Fig.1 Schematic diagram of piezoelectric beam local velocity negative feedback damping control

图2局部激励应变补偿下的阻尼控制示意图Fig.2 Schematic diagram of damping control under local excitation strain compensation

图3压电片传感器示意图(a应变率传感器 b应变传感器 c基本电路)Figure 3 Schematic diagram of piezoelectric sensor (a strain rate sensor b strain sensor c basic circuit)

图4梁上压电作动器分析示意图Fig.4 Analysis schematic diagram of piezoelectric actuator on the beam

图5压电梁本地速度负反馈阻尼控制示意图Fig.5 Schematic diagram of piezoelectric beam local velocity negative feedback damping control

图6应变开环频响示意图(a理论模型 b实测——本地 c实测——异地)Figure 6 Schematic diagram of strain open-loop frequency response (a theoretical model b actual measurement - local c actual measurement - remote)

图7局部激励应变的影响及补偿示意图Fig.7 Schematic diagram of influence and compensation of local excitation strain

其中图中符号说明如下：The symbols in the figure are explained as follows:

1梁，2作动片，3测量片，4电阻Rf，5运算放大器，6增益g_s，7增益g_a，1 Beam, 2 Action piece, 3 Measuring piece, 4 Resistance Rf, 5 Operational amplifier, 6 Gain g _s , 7 Gain g _a ,

8小基片，9作动片，10测量片，11电阻，12运算放大器，13增益g_c。8 small substrates, 9 action pieces, 10 measurement pieces, 11 resistors, 12 operational amplifiers, 13 gain g _c .

(五)具体实施方式(5) Specific implementation methods

见图2所示，本发明是一种压电结构阻尼控制实物补偿法，以梁为示例来说明，该方法具体步骤如下：As shown in Figure 2, the present invention is a kind of piezoelectric structure damping control physical compensation method, which is illustrated by taking a beam as an example. The specific steps of the method are as follows:

步骤二：设置作动片。把与梁1上的作动片2相同的作动片9贴在小基片8上，作动片9与梁上的作动片2受同样的激励。Step 2: Set the animation. Stick the same actuating piece 9 as the actuating piece 2 on the beam 1 on the small substrate 8, and the actuating piece 9 and the actuating piece 2 on the beam receive the same excitation.

步骤三：设置传感片。把与梁1上的测量片3相同的测量片10贴在小基片8的另一面，与原先的作动片9正对。测量片10后接运算放大器12和电阻11，与梁1上的测量片3后面的电阻4和运算放大器5一样，并且测量片10，运算放大器12和电阻11的连接关系也与测量片3，电阻4和运算放大器5的连接关系一样。Step 3: Set up the sensor sheet. Paste the same measuring piece 10 as the measuring piece 3 on the beam 1 on the other side of the small substrate 8, facing the original actuating piece 9. Measuring sheet 10 is followed by operational amplifier 12 and resistor 11, the same as resistor 4 and operational amplifier 5 at the back of measuring sheet 3 on beam 1, and measuring sheet 10, the connection relationship between operational amplifier 12 and resistor 11 is also the same as measuring sheet 3, The connection relationship between the resistor 4 and the operational amplifier 5 is the same.

g_c应接近于-1。g _c should be close to -1.

Claims

1. A piezoelectric structure damping control object compensation method, is characterized in that: the method concrete steps are as follows:

Step 1: set the substrate; take a small substrate (8), the material and thickness of the substrate are the same as that of the beam (1);

Step 2: actuating sheet is set; the second actuating sheet (9) identical with the first actuating sheet (2) on the beam (1) is pasted on the small substrate (8), and the second actuating sheet ( 9) The same excitation as the first actuating piece (2) on the beam;

Step 3: Set the sensing sheet; paste the second measuring sheet (10) identical to the first measuring sheet (3) on the beam (1) on the other side of the small substrate (8), and make the same as the original second measuring sheet (10). The moving piece (9) is opposite; the second measuring piece (10) is connected with the operational amplifier (12) and the resistor (11), and the resistor (4) and the operational amplifier (11) at the back of the first measuring piece (3) on the beam (1) Amplifier (5) is the same, and the second measuring sheet (10), the connection relation of operational amplifier (12) and resistance (11) is also with the first measuring sheet (3), the connection of resistance (4) and operational amplifier (5) the same relationship;

Step 4: Adjust the gain g _c ; because the substrate area is very small, in the non-high frequency mode frequency band of the beam, the bending strain of the substrate can be ignored, only the tension and compression strain, so its strain from the actuator to the sensor can directly simulate the propagation of tension and compression strain locally excited on the actuator on the beam to the sensing piece, that is, the tension and compression strain transfer function ε _ts (s)/ε _ta (s) of the two is the same, that is, proportional, so , as long as the gain g _c is adjusted, the "compensation" condition can be achieved,

ΔH ΔH ((s the s,, {g g}_{c c})) = = \frac{{Δu Δu}_{s the s} ((s the s,, {g g}_{c c}))}{{u u}_{a a} ((s the s))} = = - - {H h}_{t t} ((s the s))

g _c should be close to -1.