CHAPTER 4 : ACTIVE CONTROL OF RADIATED SOUND POWER OF VIBRATING STRUCTURES
The active control of radiated sound power from a vibrating structure requires knowledge of sound generation and propagation. Reduction of the structural response via active vibration control or control of the near-field acoustic field does not guarantee that the far-field sound is attenuated accordingly. This chapter outlines the methods of attenuating radiated sound power from vibrating structures actively using feedforward control strategies. Numerical simulations of real-time control are presented for two cases in this chapter, a beam and a plate, with experimental results for a plate presented in Chapter 5. As mentioned in Chapter 2, at low frequencies the first few ARMs contribute the most to the sound power radiated from a vibrating structure. This also means, using superposition, that significant sound attenuation can be achieved by introducing one or more control actuators that cancels these first few ARMs.
In Chapter 3, the real time ARM estimates were developed. They are used to estimate the instantaneous ARM amplitudes, hence the radiated sound power of a vibrating structure. Thus, in this chapter, the controller is designed in a way that it minimises the real-time ARM amplitudes. Here, two control cases are considered. The first case (Section 4.3) is the reduction of radiated sound power from a vibrating beam using feedforward control. Assuming the disturbance signal is known, the controller model is obtained from the transfer function ratio of the primary path to the secondary path and reconstructed in discrete time using an FIR filter. The second control case, discussed in Section 4.4, is adaptive feedforward control of radiated sound power from a vibrating plate. The filtered-x least mean square (LMS) algorithm is used for the automatic adjustment of an FIR digital filter used as the controller. In each case, numerical simulations showing the control performances are presented.
Next, the optimal actuator locations for the second control case, i.e. when a baffled rectangular plate is the radiator, are determined through offline optimisation. Note that the second control case is verified experimentally and discussed in Chapter 5. Different control actuator locations give different control performance and changing the control actuator locations in the real environment is difficult. Therefore in the last part of this chapter, the offline optimisation of the actuator locations is performed using an ant colony optimisation (ACO) algorithm, although other optimisation algorithms can equally be used.
Cancellation of the first few acoustic radiation modes
As was seen in Chapter 2, the lower order ARMs are more efficient in radiating sound and generally contribute more to the radiated sound power. This also means that reducing the first few ARM amplitudes can give significant attenuation. In theory, at least j actuators are needed to control j ARM amplitudes . To obtain a significant sound attenuation, ideally the first j controlled ARM amplitudes should be zero, so that where Qc is the Ns x j matrix of controlled ARMs, v is the Ns x 1 vector of surface velocities, Ns is the number of structural sensors and yc is the j x 1 controlled ARM amplitudes. Furthermore, as a direct consequence of the linearity of the model, the velocity of the controlled plant can be obtained by the superposition principle by adding the part caused by the primary force, vp, to that caused by the j secondary forces, vs, i.e.
Now assume that the radiator is excited by a primary force Fp, Equation (4.2) can then be written as where Hp is the Ns x 1 vector of sensor transfer functions of the primary paths (i.e. from primary force to sensor locations), Hs is the Ns x j vector of sensor transfer functions of the secondary paths (i.e. the responses at the sensor locations due to the secondary forces), and Fs is the j x 1 vector of secondary forces. Substituting Equation (4.3) into Equation (4.1) and rearranging, gives as the vector of secondary forces required to completely cancel the j ARMs . Equation (4.4) can also be written as where
is the N x j vector of control frequency responses, relating Fs and Fp .
Equation (4.5) is a feedforward control law assuming Fp is known. If, on the other hand, some reference signal R is known, and Fp = H pr R , then the vector of feedforward control frequency responses becomes such that where
Feedforward active structural acoustic control applied to the vibrating beam
This section discusses feedforward control applied to the active structural acoustic control of a beam. There are two prerequisites to use the feedforward control strategy, which are: (i) the system under control is linear, and (ii) the secondary actuators are fully active . The simulations are performed under certain idealised circumstances, to illustrate the approach. In feedforward control there are at least two signal paths involved. The first one is the primary path. This path consists of everything from the reference signal to the error sensors. This includes the physical system, data converters, analogue anti-aliasing filters and reconstruction filters. The other signal path is called the secondary path, between the controller output and the error sensors.
Figure 4.1 shows the block diagram representation of the feedforward ASAC system used in this thesis. It can be seen from this figure that the same reference signal that drives the primary actuator is used as an input to the series of feedforward controllers. These controllers then produce signals which, when used to drive an appropriate actuation system, are able to reduce the targeted responses. In this case, the targeted responses to be reduced are the ARM amplitudes, hence reducing the overall radiated sound power.
Assume that j secondary forces are applied to control j ARMs. From Equation (4.6) the control transfer function to the kth secondary force is where k =1,2,3,…j. Equation (4.9) shows that the controller transfer function is highly dependent on the structural transfer function of the beam excited by the primary and secondary forces.
Real time implementation of feedforward controller
This section focuses on the design of controller filters to be implemented in real-time. The frequency response of the jth feedforward controller in Equation (4.9) can be written as The digital feedforward controller is designed in a manner similar to the causal-delayed FIR filter approach discussed in Section 3.2, which is found to be more accurate and stable than using IIR filters. Using the causal-delayed filter approach, a time-delay of the controller dc is introduced to the frequency response of the jth feedforward controller, which becomes H c, j (w ) exp (-iwd c / fs ) . By doing this, the controller filter will produce an approximation of the control force signal at time sample m+dc while the phase is linear with a time delay of dc samples. It is reported that optimal estimation of the jth controller frequency response Hc, j (w) requires 2dc+1 coefficients [67-68]. Hence Equation (4.11) can be written in discrete time as s=0
It is important to highlight that the secondary force signal Fs, j (m) , which is also the output of the controller filter, is only available after dc time samples. Moreover, since this control method is non-adaptive, a small error in the timing of the control signal will cause inaccurate control results. Therefore, the same amount of delay must be applied to the primary force reference signal to match the timing between signals from both primary and secondary paths. In numerical simulations, adding a time-delay to the input of the primary path of the system is not an issue.
Real-time simulation: Feedforward active structural acoustic control of beam radiator
This section demonstrates real time simulations of feedforward controller described in Section 4.3.1 in attenuating the sound power radiated from a vibrating beam, using Matlab and Simulink. The beam is simply supported at both ends with parameters given in Table 2.1. The simulation sampling frequency fs and duration are 1024 Hz and 10 seconds, respectively. For the frequency range between 0 Hz and the Nyquist frequency (i.e. fn =512 Hz), there are 3 resonance frequencies i.e. at 36.7 Hz, 147.0 Hz and 330.6 Hz, as shown in Table 2.2. The beam is excited with a random point force Fp (t) at point x0/l = 0.15. A total of 3 structural sensors, equally spaced a distance l/3 apart, are used to measure the surface velocities of the radiator. The distance between the radiator’s ends and the sensors closest from the edges are l/6, respectively. The real-time ARM amplitudes are then obtained by filtering the outputs of the structural sensors with the ARM filters described in Chapter 3.
In this simulation, three control cases are considered. These are cancellation of (i) the first, (ii) the first two, and (iii) the first three ARM amplitudes. Note that one control force is only able to control one ARM amplitude . Thus the number of control forces for cases (i), and (iii) are at least 1, 2, and 3, respectively. The locations of these actuators are determined by the ARM shapes and vibration modes shapes of the beam, as illustrated in Figure 2.2 and 2.7, respectively, as well as the radiation efficiencies of the ARMs in Figure 2.8. Generally, an actuator placed at a nodal point of a particular vibration mode will not be able to excite this mode of the structure. Due to the simply supported boundary conditions, exciting the structure at the end of the beam is also not possible. Moreover, to attenuate an individual ARM amplitude, the control actuator is best placed near the point where the ARM shape has the highest magnitude. In addition, the efficiency of the ARM to radiate sound power must also be considered. In Figure 2.8, the second ARM begins to radiate efficiently (i.e. efficiency > 0.1) around the second resonance frequency while the third ARM around the fifth resonance frequency (i.e. 918.5 Hz). Hence third ARM is not important here. Considering all these conditions, the chosen locations of the first, second and third secondary actuators are listed in Table 4.1. The configuration of the sensors and the actuators for this feedforward active structural acoustic control system is shown in Figure 4.2.
Frequency responses of the feedforward controllers
Each control transfer function to the secondary forces obtained from Equation (4.9) is reconstructed in the time domain using a 65th order FIR filter with 32-step delay. The same amount of delay is applied to the primary force reference signal to match the timing between signals from both primary and secondary paths. The calculation of the controller filter coefficients is performed by the Matlab function invfreqz and the frequency response of the implemented controller filter is obtained using the function freqz. The magnitudes of the FRFs of the theoretical and estimated feedforward controllers for the three cases considered are shown in Figure 4.3.
Overall, the implemented frequency responses of the controller are able to approximate the theoretical values reasonably well. Here, the phases of the controllers FRFs for all cases are linearly approximated with a dc time delays. Note that these FRFs represent the transfer function ratios of the primary path to the secondary paths. In Figure 4.3(a) the controller transfer function has a minimum value (i.e. almost zero) at 281 Hz, which means that almost no secondary force is needed to control the sound power at this frequency. On the other hand, having a peak in the plot means the sound power is difficult to control at that particular frequency. Generally the occurrences of the maximum or minimum values in the plots are due to the effects of the differences in the transfer function between the primary and secondary paths. If the control force is placed at the same or a symmetric location as the primary force, the controller frequency response is expected to be approximately constant across the whole frequency range.
Figure 4.4 shows the amplitudes of the first three ARM amplitudes when cancelling (a) the first, (b) the first two, and (c) the first three ARM amplitudes. The control is turned on at t = 5 seconds. Note that, here, only the first 2 ARMs are important i.e. have radiation efficiencies more than 0.1 (refer Figure 2.3). However, the third ARM amplitude is also included here to illustrate the control performance when more ARM amplitudes are controlled.
Overall, the targeted ARM amplitudes in each case are reduced significantly and almost immediately after the controllers are turned on. However, some of the non-targeted ARM amplitudes are affected as well. In Figure 4.4(a) for example, there is a small reduction in y3. There is also a slightly bigger reduction in y3 when the first two ARM amplitudes are cancelled. These happen because the shapes of the first and the third ARMs are both symmetric. Similarly, those ARMs with similar symmetry will typically experience a similar effect, and vice versa.
Figure 4.5 shows the spectra of the uncontrolled and controlled individual sound powers from the first ARM (W1), the second ARM (W2) and the third ARM (W3). It can be seen that cancelling the first ARM amplitude reduces W1 significantly and W3 slightly. This is due to the fact that the first and the third ARM shapes are symmetric. On the other hand, W2 is not affected as the location of the first control actuator is at the nodal point of the second ARM. It is also worth noting that the second resonant peak is not evident in W1 and the first and third resonant peaks are not evident in W2. As explained earlier, these phenomena are due to the fact that for both vibration and acoustic radiation modes, the odd modes are symmetric, while the even modes are antisymmetric. Around one resonance, one vibration mode dominates, and hence only ARMs with similar symmetry are strong radiators.
The spectra of the total radiated sound powers can be seen in Figure 4.6 for different numbers of control forces. The attenuations of the radiated sound power at the natural frequencies of the beam using up to three control forces are listed in Table 4.2. Generally, having more shakers and attempting to cancel more ARMs increases the attenuation as well as the control bandwidth. When the first ARM amplitude is cancelled, the sound power is attenuated up to the second resonance frequency, and the control bandwidth increases to 450 Hz when the first two ARM amplitudes are controlled. Moreover, cancellation of the first three ARM amplitudes attenuates the sound power for the whole frequency range of interest (i.e. 0 to 512 Hz). These happen due to the difference in radiation efficiencies of the ARMs, where the lower order ARMs are more efficient radiators (see Figure 2.3). For the frequency range of 0.05fn to 0.95fn (i.e. 26 Hz to 489 Hz), the frequency-averaged reduction achieved when cancelling the first, the first two and the first three ARM amplitudes are 3.1 dB, 14.5 dB and 63.8 dB, respectively.
Adaptive control of radiated sound power from a vibrating plate
This section discusses the application of adaptive feedforward control for attenuating the sound power radiated from a vibrating plate. This method is basically an advanced version of the non-adaptive feedforward control discussed in Section 4.3 and it is also more practical. This is because, in an actual physical system, a delay will always exist between the reference signal and the error signal as the reference signal is propagated through the plant transfer function. Since the non-adaptive feedforward controller relies heavily on the inverse of the transfer function, the system can give poor results due to filter approximation errors, causality issues and phase differences if this delay is not taken into account.
On the other hand, the adaptive controller can cope with effects caused by approximation errors or the delays by iteratively changing the controller filter coefficients so that the error signal is minimised. The optimal solution for the controller filter coefficients is obtained by minimising a cost function. To ensure this solution is optimal, the cost function is required to have a quadratic form. The hyper-parabolic surface characteristic of a quadratic function ensures the presence of a single, global minimum rather than having many local minima. Here, a decentralised control system is employed. In other words, there will be a number j of controllers, each meant to control a single error signal. In this research, the error signal is defined as the individual ARM amplitude.
Filtered-x LMS controller
The adaptive controller is designed by using the filtered-x least mean square (FxLMS) method. Here, two digital filters are required. The first filter is the estimator s(z) of the secondary path Gs(z), which is between the controller output and the error sensors e(m). The second filter is the controller filter. This filter is used to minimise the error signals from the primary path, Gp(z), which is between the input signal u(m) and the error signal e(m). Figure 4.7 illustrates the block diagram of the filtered-x LMS algorithm. From this figure, the desired signal from the primary path, dp(m), is given by
where K1 is the number of samples considered for the convolution of the primary path response, and gp,k are the coefficients of the primary path filter.
TABLE OF CONTENTS
TABLE OF CONTENTS
LIST OF FIGURES
CHAPTER 1 : INTRODUCTION
1.2 Literature review
1.6 Thesis outline
CHAPTER 2 : RADIATED SOUND POWER FROM VIBRATING STRUCTURES AND ACOUSTIC RADIATION MODES
2.2 Radiated sound power from a plane radiating surface
2.3 Acoustic radiation modes
2.4 Equation of motion of the radiators
CHAPTER 3 : TIME-DOMAIN ESTIMATION OF ACOUSTIC RADIATION MODE AMPLITUDES
3.2 Acoustic radiation modes estimated using FIR filters
3.3 Practical implementation of ARM FIR filters
3.4 Filter quality
3.5 Structural response estimation using IIR filters
3.6 Estimation of radiated sound power
3.7 Numerical simulations
CHAPTER 4 : ACTIVE CONTROL OF RADIATED SOUND POWER OF VIBRATING STRUCTURES
4.2 Cancellation of the first few acoustic radiation modes
4.3 Feedforward active structural acoustic control applied to the vibrating beam
4.4 Adaptive control of radiated sound power from a vibrating plate .
4.5 Optimisation of location of control actuators
CHAPTER 5 : EXPERIMENT
5.2 Test structure
5.3 Structural sensors and actuators
5.5 Compact RIO Controller programming
5.6 Calculation of the sound power radiated from the vibrating plate
5.7 Experimental results
CHAPTER 6 : CONCLUSIONS AND RECOMMENDATIONS
6.2 Recommendations for future work
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Active and adaptive control of noise transmission in structures