Active Vibration Control Of Plate Structure Using Electromagnetic Transducer Based On H_∞ Optimized Positive Position Feedback

Table of Contents

Abstract

1. Introduction

2. Flexible Plate Structure Modeling
2.1 Experimental Model
2.2 Analytical Model
2.3 Modal Analysis
2.4 Numerical Model
2.5 Simulation Model of The Plate

3. Multi-modes MIMO HOPPF
3.1 PPF Controller
3.2. Multi-modes MIMO HOPPF Controller Parameter Selection

4. Simulation

5. Experiment

6. Summary

Acknowledgement

Conflict of interest statement

References

Active Vibration Control Of Plate Structure Using Electromagnetic Transducer Based On Optimized Positive Position Feedback

Zhonghui WU

Abstract

In this paper active vibration control (AVC) methodology is presented by the author using self-sensing magnetic transducers for a flexible plate structure. Optimized positive position feedback (HOPPF) controller is tested and verified for multi-modes multi-input-multi-output (MIMO) vibration suppression through simulation and experiment implement. Genetic algorithm (GA) searching is applied to obtain the optimal parameters of the controllers according to the minimization criterion solution to the norm of the whole closed-loop system.

Keywords

Optimized positive position feedback, Multi-Input-Multi-Output, Genetic Algorithm, Active Vibration Control

1. Introduction

Lightweight products and materials were employed by many designers to decrease the cross-sectional dimensions of the structures, improve dynamic performance and operating efficiency. The structures is becoming more flexible and susceptible. The harmful effects of unwanted vibration can be seen easily, especially at the moment the structure is operated at or near their natural frequencies or excited by disturbances which is coinciding with their natural frequencies [1]. Modal control is provided by vibration control engineers to suppress the vibration of flexible structures and has become the best choice for many years. Modal analysis and control refer to extract the interested mode signal from the structural response, decompose the dynamic equations of mechanical system into modal coordinates and design the a single degree-of-freedom oscillator in the modal domain [2,3].

Independent modal space control (IMSC) is proposed by Meirovitch, which can design the controller for each single mode and can be implemented independently to avoid the spillover to the residual modes [4-5]. But according to testing, it needs many sensors/ actuators as the number of modes which need to be controlled, it only can control a limited number of modes and not robust to uncertainties such as parameter fluctuation [6].

Moheimani raised resonant control method, which need to choose a high gain controller at the natural frequency of the flexible structure [7-12]. That controller has a decentralized characteristic from a modal control perspective and roll off quickly at the natural frequency of the structure to avoid spillover [13]. But it has limited method to increase damping to the structure and the performances is restricted [14].

Goh and Caughey provided Positive position feedback (PPF) after compared with other methods [15,16], and the stability was certified by Fanson J. L. [17,18]. PPF controller has several significant advantages and has been applied by many researchers that it is a reliable vibration control strategy to suppress the vibration of flexible systems [15-17, 19-33].

PPF had been modified [34-40] and had showed the robust ability [41-44]. In order to achieve better performance, Genetic Algorithms (GA) was chosen by the researcher to find the placement of sensor and actuator [45-46].

MIMO PPF controller has been verified by [47,14] on a cantilevered beam according to experimental implementation using pole placement and optimization method. [48] designed MIMO PPF controller on a flexible manipulator and through the GA method to find controller parameters to optimize result. [62] designed and compared HMPPF and HMVPF controller, which is followed [47,14] method to control multi-modes of the beam structure. MPVF controller was designed and provided by [64] for the vibration suppression of the beam.

[49-51] designed and verified GA method for tuning MIMO PPF controller. [52, 54] studied MIMO PPF controller based on the block-inverse technique to suppress the vibration of the grid and shell structure. Using PPF theory [55] gave a decentralized MIMO experimental compensation method for switched reluctance machine.

Through pseudo-inverse technique, [53] designed and proposed MIMO PPF controller. [56] presented the nonlinear MIMO PPF control method for both high and low amplitude vibration suppression of the flexible cantilever plate. MIMO PPF and PD combined controller is verified by [57] for the decoupled bending and torsional modes of the plate. In order to suppress vibration of sandwich plate, MIMO PPF and MIMO SISO were compared by [63].

In this paper, multi-modes multi-input-multi-output (MIMO) optimization PPF (HOPPF) controller will be studied by the author and applied the first time to suppress

Abbildung in dieser Leseprobe nicht enthaltenFigure 2.1 A thin plate in transverse vibrationAbbildung in dieser Leseprobe nicht enthaltenFigure 2.2 Transducer cross section [58] the vibration of a flexible plate structure.

2. Flexible Plate Structure Modeling

2.1 Experimental Model

In Fig. 2.1, we can see the schematic of the whole experimental plant t. An MDF board, which is screwed with four rubber legs, is placed on a table. On the MDF board, a uniform AL6061-T6 plate is screwed with three electromagnetic transducers (anticlockwise number 1, 2, 3). Another electromagnetic transducer is mounted as a disturbance noise shaker.

The mechanical the electric circuit model of the transducer are presented in Figure 2.2a, 2.2b separately. The measured mechanical parameters of the transducer are shown in Table 2.1. The transducer consists of a coil, producing a magnetic field when a current is fed through. The core is mounted inside the coil. The magnet of the core will be changed when the vibration is sensed from the plate [58]. The transducer is assumed to be mechanically identical. The natural frequencies of the whole plant can be obtained when the sweep signal is applied to the disturbance noise transducer.

The amplitude of the vibration is measured by accelerometers and shown on the oscilloscope.

Frequency domain results show that the first four modes, 27.1 Hz, 34.4 Hz, 40.5 Hz, 49.2 Hz, are the dominant modes for all the experimental models.

2.2 Analytical Model

This analytical model is including a thin plate and four supporting transducers.

A thin plate is assumed with dimensions: a b h. The Young's modulus of elasticity, density and Poisson's ratio of the plate are denoted by , 𝜌, and respectively. The deflection is in X and Y directions [59].

The partial differential equation (PDE) of a thin plate under transverse vibration can be written as [59]:

Abbildung in dieser Leseprobe nicht enthalten

2.3 Modal Analysis

Modal analysis methodology can be applied to find the solution to the PDE (2.1) for uniform thin plates with four supporting transducers under transverse vibration.

The typical PDE for flexible structures can be presented as [1]:

Abbildung in dieser Leseprobe nicht enthalten

and are linear homogeneous differential operators of order 2p and 2q respectively, and q p. Here, x, y is the spatial coordinate, which is defined over a domain . The general arbitrary input is denoted by f, which is distributed over . The boundary conditions corresponding to PDE (2.3) can be expressed as

Abbildung in dieser Leseprobe nicht enthalten

where i is a linear homogeneous differential operator of order less than or equal to 2p − 1.

The transfer function of the system can be derived as [1]:

Abbildung in dieser Leseprobe nicht enthalten

The Equation (2.5) is an infinite-dimensional transfer function because there is an infinite number of modes. That is a general solution of PDE (2.3). ( is the time-independent forcing term)

The solution, which is for a particular structure, can be solved through finding the eigenfunction ( ), the natural frequency ( ), and the structural damping ( ) of the structure [1].

The solution to the PDE (2.1) is assumed to be: (t, x, y)= , where and are the generalized coordinate and the eigenfunction, respectively [59].

Based on the orthogonality properties of the eigenfunctions , PDE of thin plate with four supporting transducers under transverse vibration can be solved independently for each mode [59].

According to the modal analysis solution, the transfer function from the actuator voltages (s) = [ ... , to the plate deflection (x, y, s) can be written as:

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