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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In general, mechanical equipment such as cars, airplanes, and machine tools all operate with constant frequency characteristics. These constant working characteristics should be controlled if the dynamic performance of the equipment demands improvement or the dynamic characteristics is intended to change with different working conditions. Active control is a stable and beneficial method for this, but current active control methods mainly focus on vibration control for reducing the vibration amplitudes in the time domain or frequency domain. In this paper, a new method of dynamic frequency characteristics active control (DFCAC) is presented for a flat plate, which can not only accomplish vibration control but also arbitrarily change the dynamic characteristics of the equipment. The proposed DFCAC algorithm is based on a neural network including two parts of the identification implement and the controller. The effectiveness of the DFCAC method is verified by several simulation and experiments, which provide desirable results.

Over the past several decades, structural vibration control has attracted the attention of many scholars and has led to many achievements both in theory and application. Active control has attracted much more attention in recent years because it has many attractive properties, such as flexibility and adaptability.

Balas [

Frequency-domain active control methods which are based on linear quasi-static assumptions can avoid the effects of transient response. They are suitable for active structural control with a periodic response. Noise and data reduction can be implemented relatively easily, and it can significantly reduce the computational complexity [

Therefore, a new method of dynamic frequency characteristics active control (DFCAC) is proposed first to deal with this problem. In general, mechanical equipment such as cars, airplanes, and machine tools all operate with constant frequency characteristics. These constant working characteristics should be controlled if the dynamic performance of the equipment must be improved or if the dynamic characteristics are changed to satisfy different working conditions. For example, in order to improve the stability, robustness of the equipment and the comfort to the operators, the frequency components of resonance should be avoided, interference, and some new frequency components should be added to the frequency response of the equipment to improve the stability.

Based on the frequency-domain active control methods, the error criterion is derived aiming at changing the frequency response in the frequency domain, and then the new DFCAC method is established. Compared with the aforementioned active control methods, there are some difficulties and advantages in constructing the DFCAC method. First, the objective function of the DFCAC method can be any desired frequency signal. Besides, the objective signal is a frequency signal that can reflect the frequency properties expected to the equipment. Second, the signals, the parameters and the control process are in the frequency-domain in the DFCAC method, which can save processing time associated with Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT) computation and which is convenient for control processes. A frequency error is chosen for iteration in the DFCAC method. The global error and frequency node error are combined together as the error criterion, which can improve the flexibility, adaptability and anti-jamming capability of the DFCAC method.

In Section 2, the DFCAC method is constructed using neural networks. The error criterion consisting of global and frequency node error is introduced in Section 3. In Section 4, simulations and experiments under different working conditions are implemented and the efficiency of the DFCAC method is verified.

A neural network is constituted by many nonlinear cells and is widely used in mechanisms, aviation, banking, finance, amusement, and so forth. Neural network is very good at modeling complex and nonlinear relationships between input and output variables and is suitable for adaptive control. Hence, a neural network is chosen as the main algorithm in the DFCAC method.

The network structures of the NNI and NNC are shown in _{1}…_{n}, _{1}…_{n} and _{1}… _{n} are the actuating parameters of the actuators. _{ji}_{j}_{lj}_{l}

In the NNI, a linear transfer function is used for the output layer. Following the Widrow-hoff principle [^{i}^{i}

The network NNC has three layers, an input layer U, hidden layer O and output layer X. A tangent sigmoid transfer function is used for the hidden layer, and a linear transfer function is used for the output layer. The global frequency error

Here, _{j}_{l}

To increase the learning rate and reliability, the correction formulas for the NNC based on the momentum gradient descent procedure can be obtained as follows.

For the weights and biases connecting the input and hidden layers:

For the weights and biases connecting the hidden and output layers,

Here,

As mentioned in Section 2, the global frequency error _{k}_{k}

There are some reasons and advantages for constructing this error criterion. First, constructing the frequency error

Second, the stopping criterion for each iteration in

If the global frequency error alone is taken as the stopping criterion, it may make the control process require more time or even be non-convergent. As shown in

If the frequency node error alone is taken as the stopping criterion, it may lead to some unacceptable control results. As shown in

Therefore, taking the global and frequency node errors together as the error criterion can avoid the two aforementioned problems. Furthermore, it can improve the flexibility, adaptability and anti-interference ability of the DFCAC method. Background noise in engineering cannot be avoided or eliminated generally, but the anti-interference ability of the DFCAC method can be improved by appropriately adjusting err_goal1 in

The DFCAC method is implemented computationally according to the framework of the DFCAC method in

As shown in

_{x}_{y}^{10} N/m^{2}; Poisson ratio ^{3}. The dynamics of the controlled flat plate is calculated by multivariable wavelet finite element method (MWFEM) [

As shown in

As shown in

To verify the effectiveness of the DFCAC method in practical applications, some experimental results are discussed in this section. The experimental setup is shown in

Control platform (NI PXI-8108)

2.53 GHz Intel Core 2 Duo T9400 dual-core processor

1 GB (1 × 1 GB DIMM) 800 MHz DDR2 RAM standard, 4 GB maximum

10/100/1000BASE-TX (gigabit) Ethernet, ExpressCard/34 slot, and 4 Hi-Speed USB ports

Integrated hard drive, GPIB, serial, and other peripheral I/O

D/A (NI PXI-6733)

8 high-speed digital I/O lines; two 24-bit counters; digital triggering

Onboard or external update clock

PXI trigger bus for synchronization with DAQ, motion, and vision products

Power amplifier (GF-10 Far East Vibration, Beijing, China)

The maximum power: 10 W

The maximum current: 1 A

The peak voltage: 10 V

Frequency Range: 5 Hz–20,000 Hz

Size: 280 mm × 200 mm × 120 mm

Actuator (JZ-1 Far East Vibration, Beijing, China)

The maximum driving force: 2N

Working frequency range: 5 Hz–1,000 Hz

The maximum displacement: ±1.5 mm

The maximum no-load acceleration: 67 g

The maximum current: 0.5 A

Size: Φ50 mm × 160 mm

Sensor (PCB 352C34)

Sensitivity: 100 mV/g

Working frequency range: 0.3 Hz–15 kHz

Measurement range: ±50 g pk

Resolution: 0.00015 g

Temperature range: −54 °C–+93 °C

Size: Φ50 mm × 160 mm

Data acquisition card (NI PXI-4472B)

Ability to synchronize up to 5000 channels in a PXI system

24-bit resolution ADCs with 110 dB dynamic range

±10 V input range or ±31 V with SMB-120 cable

8 simultaneously sampled vibration-optimized analog inputs at up to 102.4 kS/s

Software-configurable AC/DC coupling and IEPE conditioning

The controlled flat plate

As shown in

According to the aforementioned experimental setup and corresponding algorithm in the simulation, the program was implemented by using LABVIEW. Several experiments under different working conditions were conducted to verify the efficiency.

Working condition 1_Flat plate with one controlled actuator and motor interference

As shown in

Working condition 2_Flat plate with two controlled actuators

As shown in

Working condition 3_Flat plate with one constant actuator and one controlled actuator

As shown in

On the basis of neural networks, a dynamic frequency characteristics active control method was constructed and implemented. First, neural network identification and controller systems were constructed. Then the modified formulas to the corresponding weights and biases were derived by a gradient descent procedure, and a momentum factor was brought in to improve the convergence rate. Then, an error criterion was constructed to improve the flexibility, adaptability and anti-interference ability of the method. Following that process, several simulations and experiments under different working conditions were conducted to validate the constructed algorithm. Through comparison of the control results, it can be seen that the proposed method achieves the control objective. Therefore, the DFCAC method constructed in this paper can not only achieved active vibration control, but also arbitrarily change the dynamic characteristics of mechanical devices.

This work was supported by the key project of the National Natural Science Foundation of China (No. 51035007), Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 2007B33) and FOK YING TUNG Education Foundation (No.121052) and Shaanxi Province Project (No. 2011kjxx06).

_{∞}control for vehicle active suspension systems

The control frame of the DFCAC system.

The network structures of the (

The comparison between different time- and frequency-domain signals (

The global frequency error illustration (

The frequency node error illustration (

The flow chart of the DFCAC method.

Flat plate with one actuator.

The controlled flat plate in simulation.

The first five mode shapes of the flat plate in simulation.

Controlled results of the simulation in working condition 1 (

Flat plate with two actuators.

Controlled results of the simulation in working condition 2 (

The experimental setup for the DFCAC system.

The data flow diagram and equipment model.

The controlled flat plate in experiment.

Experimental schematic diagram for working condition 1.

Experimental control results for working condition 1 (

Experimental schematic diagram for working condition 2.

Experimental control results for working condition 2 (

Experimental schematic diagram for working condition 3.

Experimental control results for working condition 3 (

The first five natural frequencies of the controlled plate in simulation.

Method(DOFs) | _{1} |
_{2} |
_{3} |
_{4} |
_{5} |
---|---|---|---|---|---|

MWFEM (484) | 2,069.4 | 2,464.6 | 4,048.2 | 5,654.0 | 6,017.8 |

Ansys Shell63 (38400) | 2,067.4 | 2,462.8 | 4,065.7 | 5,471.7 | 5,705.4 |

The first three natural frequencies of the controlled plate in experiment.

Method | _{1} (Hz) |
_{2} (Hz) |
_{3} (Hz) |
---|---|---|---|

MWFEM | 118.86 | 290.68 | 354.81 |

Test | 111.21 | 261.73 | 350.98 |