Optimized Multi-modes MIMO Positive Position Feedback Active Vibration for Plate Structure

Optimized Multi-modes Multi-Input-Multi-Output Positive Position Feedback Active Vibration for Plate Structure

Zhonghui WU

School of Computer Science, Engineering and Mathematics, Faculty of Science and Engineering, Flinders University, Australia

ABSTRACT

Active vibration control (AVC) methodology is presented by the author in this paper using bonded three self-sensing magnetic transducers for a flexible plate structure. Multi-modes multi-input-multi-output (MIMO) positive position feedback (PPF) controller is tested and verified for vibration suppression through simulation and experiment implement. Based on genetic algorithm (GA) searching, optimal parameters of the controllers can be obtained according to the minimization criterion which is the solution to the norm of the whole closed-loop system.

Keywords

Multi-Input-Multi-Output, Positive Position Feedback, Genetic Algorithm, Vibration Control, Optimization

1. Introduction

In order to decrease the cross-sectional dimensions of the structures, improve dynamic performance and operating efficiency, lightweight products and materials were employed by many designers. However, when the structures become flexible, the harmful effects of unwanted vibration can be seen from structures, they are becoming more susceptible.

The problem is extremely worse when they operate at or near their natural frequencies or when they are excited by disturbances that coincide with their natural frequencies [1].

*Corresponding Author Tel.: +61 0405060831; E-mail address: wu0286@flinders.edu.au

Modal control has become the best choice for vibration control engineers to suppress the vibration of flexible structures for many years. In general, modal analysis and control refer to the procedure of decomposing the dynamic equations of mechanical system into modal coordinates and designing the control system in this modal coordinate system [2]. It extracts the interested mode signal from the structural response. The engineer can design the controller in the modal domain and control a single degree-of-freedom oscillator in the similar way [3]. There are three main modal control methods that can be found in the literature for controlling multi-modes vibration in flexible structures: independent modal space control (IMSC), resonant control and positive position feedback (PPF) control.

Meirovitch provided the independent modal space control (IMSC). IMSC can design the controller for each single mode. The controller can be implemented independently, in that way there will be little spillover to the residual modes [4-5]. But the problem is obvious: it needs lots of sensors/ actuators as the number of modes, which need to be controlled, and it can control a limited number of modes. Furthermore, the control system is not robust to uncertainties such as parameter fluctuation. In order to solve these problems, engineers had tried to apply the robust control techniques to this area in the past two decades [6].

According to the investigation of the resonant characteristic of the flexible structures, Moheimani raised resonant control method [7-12]. The designer chose a high gain controller at the natural frequency of the flexible structure. The controller would roll off quickly away from the natural frequency in order to avoid spillover. It is also described as having a decentralized characteristic from a modal control perspective [13], thereby making it possible to treat each mode of the system in isolation. One of limited performances for resonant controllers is restricted increase damping to the structure [14].

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

After that, lots of people did simulations and experiments using the PPF theory for active vibration [20-33]. Based on the performance has already achieved, some researchers modify the structure of PPF [34-40]. Expect the normal advantages, some people also showed the robust ability of the PPF controller [41-44].

During the PPF controller design, in order to achieve good performance, some designer used Genetic Algorithms (GA) to choose the placement of sensor and actuator. [45] applied GA to find efficient location of sensor and actuator of a cantilevered composite plate, vibration reduction for the first three modes has been achieved using the coupled PPF. The designer has proved that the whole system is robust to parameter variations. [46] presented a GA method of optimal placement for the cantilever plate. According to adding PD and PPF together, the proposed control method by can suppress the vibration decay process and the smaller amplitude vibration effectively, which has been approved and can be seen from simulations and experimental results.

In order to achieve better performance, based the complicated coupled relations between controllers, some designers provided MIMO PPF controller.

For beam structure: MIMO PPF controller has been verified by [47,14] on a cantilevered beam according to experimental implementation. The author also adopted 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.

For grid structure: [49-51] proposed the use of the GA method for tuning MIMO PPF controller for grid structure. Based on the block-inverse technique, [52] provided MIMO PPF controller to suppress the vibration of the grid structure.

For plate structure: according to pseudo-inverse technique, [53] designed and proposed MIMO PPF controller. A nonlinear MIMO PPF control method is presented by [56], both high and low amplitude vibration suppression of the flexible cantilever plate was achieved. [57] provided MIMO PPF and PD combined controller to control the decoupled bending and torsional modes of the plate. MIMO PPF and MIMO SISO were compared by [63] to suppress vibration of sandwich plate.

For shell structure, using block-inverse technique. [54] studied and verified MIMO PPF controller for the first two vibration modes.

For switched reluctance machine, based on the studying of PPF theory [55] gave a decentralized MIMO experimental compensation method.

In this paper, the author will use the MIMO PPF controller based optimization through GA searching for multi-modes vibration control of a flexible plate structure. According to the author's understanding, this effort is the first time to apply to plate structure, similar methods can be only found on beam structure.

2. Model of Flexible Plate Structure

2.1 Experimental Model

The schematic of the whole system, which is designed as an experimental plant to test the controller, is presented in Fig. 2.1. A uniform AL6061-T6 plate mounts with screw on the MDF board using three electromagnetic transducers (anticlockwise number 1, 2, 3). Another electromagnetic transducer is mounted on the MDF board with screw as a disturbance noise shaker. The MDF board is placed on the table with four rubber legs, which are screwed to the MDF board.

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Figure 2.1 A thin plate in transverse vibration

The mechanical model of the transducer, which is used in the experimental plant, is presented in Figure 2.2a. The structure of the transducer is similar to a loudspeaker, which also consists of a coil, produces a magnetic field when a current is fed through [58]. The magnet of the core, which is mounted inside the coil, will be affected by the vibration of the plate [58]. The electric circuit model, which is used for this transducer, is shown in Figure 2.2b.

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Figure 2.2 Cross section of transducer [58]

It is supposed that the transducers are mechanically identical. The measured mechanical parameters of the transducers can be seen in Table 2.1.

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Table 2.1: Mechanical parameters of transducer

Applying a sweep signal to experimental model, then the natural frequencies of the whole plant are obtained. According to observing the frequency of the signal, the amplitude of the vibration, which is needed to test, can be measured and shown on the oscilloscope by accelerometers.

Experiments show that the dominant modes for all the experimental models are the first four modes: 27.1 Hz, 34.4 Hz, 40.5 Hz, 49.2 Hz.

2.2 Analytical Model

This analytical model concludes a thin plate and four supporting transducers. These main assumptions are based on theory from [59]: a thin plate 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.

PDE of a thin plate under transverse vibration [59]:

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2.3 Modal Analysis

Consider the typical PDE for flexible structures [1]:

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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 (2.103) can be expressed as

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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 according to [1]:

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Because there is an infinite number of modes, so the Equation (2.5) is an infinite-dimensional transfer function. This is a general solution of PDE (2.3). The solution for a particular structure is then solved by finding the eigenfunction (), the natural frequency [illustration not visible in this excerpt], and the structural damping [illustration not visible in this excerpt].

In order to solve the partial differential equation (2.1) for uniform thin plates with four supporting transducers under transverse vibration, the modal analysis methodology will be applied to find the solution. The solution is assumed to be: [illustration not visible in this excerpt] where[illustration not visible in this excerpt] and are the generalized coordinate and the eigenfunction, respectively. 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].

Based on the modal analysis solution, the transfer function from the actuator voltages,

[illustration not visible in this excerpt] to the plate deflection (x, y, s) can be written as

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where [illustration not visible in this excerpt] and is the damping ratio associated with the mode (m, n). Furthermore, [illustration not visible in this excerpt]will be affected by the properties of the plate, actuator place and the eigenfunctions .

2.4 Numerical Model

Using software tool ANSYS, numerical model is designed and shown in Fig 2.3

2.5 Simulation Model of The Plate

Due to the MIMO properties, any one of the outputs on the top plate is induced by superposing all the inputs and their related dynamics. Hence, the dynamics of the whole system can be expressed by output-input

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Figure 2.3 ANSYS Model

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Figure 2.4 Block diagram of PPF control

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Figure 2.5 Amplitude Response

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Figure 2.6 Phase Response

open loop system superposition full description.

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where

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Ui input through the transducer i to the top plate, i = 1, 2, 3; ud input to shaker; yi output at the position on the top plate, i = 1, 2, 3; - dynamic between the shaker and transducer; tmn dynamic between transducer m and n, m = 1, 2, 3, n,m= 1, 2, 3.

3. Multi-modes MIMO PPF Controller

3.1 PPF Controller

Positive position feedback (PPF) controller was first proposed, applied and verified using experimental implement by Caughey and Goh in 1982. Its simplicity and robustness has attracted many researchers to adopt and apply in flexible structural vibration controlling [16]. According to observing, the researchers find that PPF is different from other control laws, the PPF controller is insensitive to uncertain natural damping ratios of the structure [16]. The measurement of position is positive, then it is fed into the compensator. And at the same time, the position signal from the compensator, which is fed back to the structure later, is also positive [18]. This character makes the PPF controller quite fit for collocated actuator/sensor pairs. Equations 3.1 and 3.2 show the structure and compensator equations in the scalar case [16]:

Structure: [illustration not visible in this excerpt]

Compensator: [illustration not visible in this excerpt]

where g is the scalar gain (positive), e is the modal coordinate (structural), η is the filter coordinate (electrical), [illustration not visible in this excerpt] are the structural frequencies, are the filter frequencies, [illustration not visible in this excerpt] are the structural damping ratios and are the filter damping ratios. This non-dynamic stability criterion is characteristic of the positive position feedback system.

PPF compensator can be seen in Equation (3.3) as the second-order transfer function. The transfer function form will be used in this text for deriving the properties of the control system [16].

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