摘要
飞行器上恶劣的动力学环境会导致精密设备的精度损失,甚至失效。为了保护飞行器上精密设备,需要改善动力学环境。采用隔振技术,即在精密设备与安装基础之间加装具有振动抑制功能的隔振器,可能有效地降低设备的振动强度。其中,基于压电智能结构的主动隔振系统,由于具有控制效果好、重量轻、响应快、能源需求小、结构简单可靠等特点,对于要求苛刻的飞行器而言最为理想。因此本文以圆锥壳为基本结构,采用压电材料作为传感器及作动器,对精密设备主动隔振问题进行研究。
锥壳的振动特性是进行主动控制的基础,但锥壳隔振器有其特殊性。边界条件为大端固支、小端自由;与横向振动相比,隔振器的轴向振动、扭转振动及侧倾振动对隔振而言更为重要。为描述固支-自由的边界条件,构造了基于多项式及三角函数的模态函数。然后基于薄壳基本假定,给出了锥壳结构的几何方程和物理方程,并进一步给出了自由振动时应变能、动能的表达式。应用瑞利-里茨法推导了固有频率、模态函数的求解公式。在分析过程中,考虑了前述四种振动形式,并针对每一种振动形式,给出了适用的公式。
对于扭转振动,由于普通的压电传感器对面内剪切应变不敏感,因此提出了剪切式压电传感器。推导出了剪切式压电传感器的一般信号方程;然后根据振动分析的结果,代入锥壳的本构方程,得出了圆锥壳作扭转振动时的传感信号方程;并最终给出了固支-自由边界条件下的扭振传感信号。为分析轴向、侧倾及横向振动,提出了对角式压电传感器,给出了其通用表达式。对角式传感器的传感信号可分成四个部分,分别对应于四个应变分量。结合数值方法对传感信号进行了仿真分析,给出了模态信号、传感器贴片的传感信号的分布规律。并考虑了小端装有附加质量时各信号的分布特性。传感器输出信号的幅值取决于锥壳的模态应变,且与传感器位置、几何形状、压电材料特性等参数有关。由于传感器贴片的平均效应,其输出信号的幅值小于模态信号,但差别微小。因此可以用贴片的输出信号代替模态信号以测试结构的振动特性。此外,对输出信号的分析结果为确定传感器的最优位置提供了依据。
基于逆压电效应,提出了对角式压电作动器以控制锥壳结构的轴向、侧倾及横向振动。根据模态叠加法,使用各阶模态的模态参与系数及振型函数合成锥壳上任意点的响应。将锥壳和压电作动器的振动方程转换到模态空间,得出模态振动方程。然后根据逆压电效应推导了作动器的数学模型,给出了对角式压电作动器的模态控制力表达式。对角式压电作动器的模态控制力包含四个分量,分别对应于径向及环向作动器单位长度上的力和力矩。采用开环控制,对作动器贴片的控制行为进行了分析。在单位控制电压的作用下,对角作动器产生的模态控制力及各分量的相对大小随贴片的位置及模态变化。对角式作动器对锥壳隔振器的轴向、侧倾及横向振动均有模态控制力输出。在有附加质量的情况下,对角式压电作动器各贴片均输出模态控制力,且幅值差别较小。分析结果为随后的主动控制提供了依据。
将模态振动方程转换到状态空间,以对角式压电传感器的模态输出信号为系统输出,以对角式作动器在单位控制电压下的模态控制力为系统控制输入向量,建立主动控制系统的系统状态空间方程。采用线性二次型(LQR)最优控制,通过使性能准则函数最小求得最优控制。使用数值方法对控制系统进行仿真,分析了不同作动器的控制力及控制电压。
设计并制作了智能锥壳缩比模型,并搭建了实验平台,对提出的位移函数、分布式压电传感器与作动器理论进行了原理性验证。
Severe dynamic loads in the flighting vechiles may cause accuracy loss or even damage to the precision equipment inside. Therefore better dynamic environment is requied to protect the precision equipments, and vibration isolation system (VIS) can be employeed to achieve this, which mianly consists of a vibration isolator installed between the equipment and the base. Active VIS based on piezoelectric smart structure are more promising VIS, because of its better performance, light weight, rapid response and less energy cost. Therefore, based on a conical shell model and the piezoelectric sensor/actuator, the active vibration isolation of precision equipment in flighting vechiles are investigated.
The vibration characteristics of the conical shell are foundations of vibration control. Generally, the conical isolator is fixed to the base at the major end, and is free to vibrate at the minor end. The bending, axial and torsional vibrations are much more important for VIS than the transverse vibration. To analyze these vibrations, a set of modal functions are presented based polynomial and sine function. Based on the thin shell assumptions, the geometric and physical equations are given, the energy equations for clamped-free conical shell are derived. Then the natural frequencies and modal functions are solved by using a Rayleigh-Ritz method. During the derivation, the axial, torsional, bending and transverse vibrations are investigated and detailed equations for each vibration type are specified.
Traditional piezoelectric sensor used for thin shells are assumed not sensitive to in-plane shear strain, is not applicable for torsional vibration of conical isolator. Therefore a shear-type piezoelectric sensor is presented, the mathematical model is developed based the direct piezoelectric effect. The shear type sensor is applied to the conical shell based on the modal analysis. Then the boundary conditions are specified and the sensing signal equations are derived. For the axial, bending and transverse modes, a diagonal sensor is presented. The signal equations of diagonal sensor are derived, the total signal consists of four components related to the four strain components respectively. By using a numerical method, the distribution of sensing signal and modal signal is evaluated. Furthermore, the distributed sensing of conical shell with payload at the minor end is investigated. The sensing signals depend on the modal deformation, sensor location, sensor geometry, the piezoelectric material and so on. The amplitudes of sensing signal are usually less than the modal one because of the average effect, but the difference is very small. Therefore, the sensing signal can be used to investigate the structural vibration characteristics. The optimal locations for sensor patches can be determined according to the results.
In order to actively control the axial, bending and transverse vibrations of conical shell, a diagonal piezoelectric actuator is proposed. By using the mode superposition method, the total dynamic response can be represented by the summation of all participating natural modes and their respective modal participation factors. Therefore, the vibration equations are transformed into the mode space, and a modal vibration equation is obtained. Based on the inverse effect, the mathematical model of the diagonal actuator is derived and the equation of modal control force is achieved. The modal control force consists of four component related to the active force and moment per unit length in longitudinal and circumferential direction respectively. By using an open-loop control method, the modal control characteristics of diagonal actuator segments at different locations are investigated and compared with unit control voltage. The results prove that the diagonal actuator is applicable of active control of axial, lateral and transverse vibrations of conical shell. When rigid mass is fixed to the minor end, every actuator patch generates a similar modal control force. The optimal locations of actuator segments for the control of different natural modes are also investigated.
The modal vibration equation is transformed into the state space, the sensing signal of diagonal sensor is chosen as the system output vector, the modal control force of diagonal actuator is chosen as the system input vector, and the state space equation of the active vibration control equation is established. The optimal controller is obtained by minimizing the performance function. Then the optimal control force and corresponding control voltage are investigated using a numerical method.
A scaled conical shell model is build, and the experimental platform is established. Principle experiments on the modal functions, the diagonal piezoelectric sensor and actuator are performed to verify proposed theory.
引文
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