3. Active noise insulation control laws

#### 3.1 Sensor and actuator placement

From the control engineering's perspective, sensor/actuator selection and placement correlate to the system's observability and controllability. For instance, for an interesting mode, if the sensor is placed at the structure's nodal point, no informative response can be obtained. The measured signal is buried in noise. Similarly, if the actuator is placed along the interesting mode's nodal line, this mode will be uncontrollable.

To improve the control law's robustness and facilitate control law design, the collocated sensor/actuator configuration is favorable. Under such configuration, the transmission path between the collocated actuator and sensor is the nearest, generates a minimum phase system. In other words, the magnitude and phase responses between the sensor and actuator is unique and the pole/zero appears in an interlacing way. Correspondingly, as shown in Figure 4, the resonant peaks and anti-resonant through will appear one by one. In the phase plot, 180° phase lag is generated across the resonant frequency and 180° phase lead is generated across the anti-resonant frequency. If the two adjacent modes are near, the modes' coupling makes the phase lag being smaller than 180°. In summary, the interlacing property makes the phase lag's variation always within 180°, which facilitates the control law design.

In practical implementation, the sensor and actuator's physical properties make the collocation in a limited frequency range. This means the phase lag between the sensor/actuator pair will beyond 180° above certain frequency limit. Therefore, to meet the Nyquist criterion [13], the controller's gain must be limited within a certain range.

According to these mode shapes, it is shown that the 2-2 mode (even-even mode) shape radiates like a quadrupole acoustic source. In the low frequency range, for such vibration mode, the neighboring push-pull phenomenon makes acoustic cancellation occurs strongly. Similarly, acoustic cancellation also occurs strongly at the 1-2 mode (odd-even mode) and 2-1 mode (even-odd mode), which reduce these modes'sound radiation effectiveness. The 1-1 mode (odd-odd mode) radiates like a

Furthermore, the mode's radiation efficiencies can be quantized with different boundary conditions. For the simply supported boundary, the detailed equations can be found in Refs. [10, 11]. For the clamped boundary, the detailed equations can

These above analysis indicates that in the active sound insulation application, the

As shown in Figure 3, the sound pressure from the incident side is p<sup>1</sup> and the sound pressure from the transmitted side is p2. The structural displacement is η, damping coefficient is r, stiffness coeffcient is k, the dynamic system can be

The volume velocity is assumed continuous at the plate's two sides, and the

mη€ þ rη\_ þ kη ¼ p<sup>1</sup> � p<sup>2</sup> (1)

main interested modes will be odd-odd modes. At the resonant frequency, and assuming the displacement is uniform, the structure can be simplified into a

monopole, which has the highest sound radiation ability.

dynamic system with one degree of freedom.

corresponding acoustic impedance is:

be found in Ref. [12].

Simplified model for noise insulation study.

Noise and Vibration Control - From Theory to Practice

Figure 3.

expressed as:

84

#### 3.2 Negative acceleration feedback (NAF) control

The NAF control law utilize the structural acceleration signal as the input signal. Then the control output is generated according to the NAF control law, which is written as:

$$H\_{NAF}(j\alpha) = \lg \frac{\alpha\_c^2}{-\alpha^2 + 2\alpha\_c \zeta\_{NAF} j\alpha + \alpha\_c^2} \tag{5}$$

increased for a certain frequency. Because the controller can generate larger output at this frequency. However, the control performance will be reduced substantially when the target frequency is slightly biased from the realistic resonant frequency. If the damping value is too large, the control authority will be weak, and the added

Effective Low Frequency Noise Insulation Adopting Active Damping Approaches

The merit of NAF controller is: the control authority is exerted to one specific mode. The controller rolls-off at 40 dB/dec above the target control frequency. This roll-off characteristic can suppress high frequency noise effectively, which is favor-

Concerning multiple modes suppression, the same number of NAF controllers will be needed and the NAF controllers should be connected in parallel. However, the authors do not recommend using NAF control law to suppress more than three modes. One important reason is, when the interesting modes are adjacent, because the controller's output does not rolling off sufficiently, the control interferences are

Direct velocity feedback (DVF) control utilizes the structural velocity as the feedback signal directly. The controller is a constant gain, which can generate active

Because the DVF control law does not provide roll-off property, the control authority is limited for practical experimental test. However, in the sound insulation application, to realize global reduction, wideband active damping is required

To fulfill this task, the DVF control law must be modified and Filtered Velocity Feedback (FVF) control is proposed to improve the control performance. The FVF control law behaves like an electrical dynamic absorber, which can be deduced

�ω<sup>2</sup> þ 2ωnζFVFjω þ ω<sup>2</sup>

n

(6)

In the frequency domain, the FVF controller can be expressed as:

Bode plot of the FVF controller with different control damping ratio parameters.

HFVFð Þ¼ <sup>j</sup><sup>ω</sup> <sup>g</sup> <sup>j</sup><sup>ω</sup> <sup>þ</sup> <sup>2</sup>ωnζFVF

generated, making the control law's tuning process being complicated.

active damping effectiveness will be small.

DOI: http://dx.doi.org/10.5772/intechopen.85427

3.3 Filtered velocity feedback (FVF) control

able to guarantee the control gain.

damping to the structure [15].

essentially.

Figure 6.

87

according to Ref. [16].

where g is the control gain, ω<sup>c</sup> is the targeting control angular frequency,ζNAF is the controller's damping ratio parameter.

According to the analysis given in [14], at the targeting control angular frequency, the NAF control law generates active damping to the structure.

For a given control frequency, with different control damping parameters, the NAF controller's bode plot values is shown in Figure 5.

The damping ratio value should be paid attention for practical implementation. Clearly, if a lower damping ratio value is adopted, the control authority will be

Figure 4. Interlacing property of the collocated sensor/actuator configuration.

Figure 5. Bode plot of the NAF controller with different control damping ratio parameters.

#### Effective Low Frequency Noise Insulation Adopting Active Damping Approaches DOI: http://dx.doi.org/10.5772/intechopen.85427

increased for a certain frequency. Because the controller can generate larger output at this frequency. However, the control performance will be reduced substantially when the target frequency is slightly biased from the realistic resonant frequency. If the damping value is too large, the control authority will be weak, and the added active damping effectiveness will be small.

The merit of NAF controller is: the control authority is exerted to one specific mode. The controller rolls-off at 40 dB/dec above the target control frequency. This roll-off characteristic can suppress high frequency noise effectively, which is favorable to guarantee the control gain.

Concerning multiple modes suppression, the same number of NAF controllers will be needed and the NAF controllers should be connected in parallel. However, the authors do not recommend using NAF control law to suppress more than three modes. One important reason is, when the interesting modes are adjacent, because the controller's output does not rolling off sufficiently, the control interferences are generated, making the control law's tuning process being complicated.

#### 3.3 Filtered velocity feedback (FVF) control

Direct velocity feedback (DVF) control utilizes the structural velocity as the feedback signal directly. The controller is a constant gain, which can generate active damping to the structure [15].

Because the DVF control law does not provide roll-off property, the control authority is limited for practical experimental test. However, in the sound insulation application, to realize global reduction, wideband active damping is required essentially.

To fulfill this task, the DVF control law must be modified and Filtered Velocity Feedback (FVF) control is proposed to improve the control performance. The FVF control law behaves like an electrical dynamic absorber, which can be deduced according to Ref. [16].

In the frequency domain, the FVF controller can be expressed as:

$$H\_{FVF}(j\alpha) = \lg \frac{j\alpha + 2\alpha\_n \zeta\_{FVF}}{-\alpha^2 + 2\alpha\_n \zeta\_{FVF} j\alpha + \alpha\_n^2} \tag{6}$$

Figure 6. Bode plot of the FVF controller with different control damping ratio parameters.

3.2 Negative acceleration feedback (NAF) control

Noise and Vibration Control - From Theory to Practice

NAF controller's bode plot values is shown in Figure 5.

Interlacing property of the collocated sensor/actuator configuration.

Bode plot of the NAF controller with different control damping ratio parameters.

the controller's damping ratio parameter.

written as:

Figure 4.

Figure 5.

86

The NAF control law utilize the structural acceleration signal as the input signal. Then the control output is generated according to the NAF control law, which is

where g is the control gain, ω<sup>c</sup> is the targeting control angular frequency,ζNAF is

For a given control frequency, with different control damping parameters, the

The damping ratio value should be paid attention for practical implementation. Clearly, if a lower damping ratio value is adopted, the control authority will be

c �ω<sup>2</sup> þ 2ωcζNAFjω þ ω<sup>2</sup>

c

(5)

HNAFð Þ¼ <sup>j</sup><sup>ω</sup> <sup>g</sup> <sup>ω</sup><sup>2</sup>

According to the analysis given in [14], at the targeting control angular frequency, the NAF control law generates active damping to the structure.

where g is the control gain, ω<sup>n</sup> is the controller's upper frequency,ζFVF is the controller's damping ratio.

For a given control frequency, the FVF controller's Bode plot with different control damping values is shown in Figure 6.

As shown in Figure 6, in the low frequency, the controllers behaves like a DVF controller but the control output rolls off at 20 dB/dec after the controller's upper frequency. Therefore, active damping is generated in the low frequency range.

Unlike the NAF controller, the damping ratio of FVF controller should be high enough to generate sufficient gain below the upper frequency. For instance, ζFVF ¼ 0:7 is a good choice.

If acceleration signal is used as the control input instead of velocity signal, the modified FVF control law becomes into:

$$H\_{\rm FVF}'(j\omega) = \lg \frac{j\alpha + 2\alpha\_n \zeta\_{\rm FVF}}{j\alpha \left(-\alpha^2 + 2\alpha\_n \zeta\_{\rm FFF} j\alpha + \alpha\_n^2\right)}\tag{7}$$

interesting bandwidth's upper frequency. However, the achieved sampling rate is also restricted by the controller. When the control law requires intensive computation effort and the controller is not fast enough, the control law will not be finished in time. In such circumstance, to meet the deterministic mechanism, the sampling

Effective Low Frequency Noise Insulation Adopting Active Damping Approaches

In summary, to determine the control law's updating rate, the control bandwidth, computation burden and real-time target's performance should be taken into

Filters are widely used in active control systems, which can suppress excessive noise signal. Crucially, if the control law itself does not have the roll-off characteristic in the high frequency, low pass filter will be always favorable. The high pass

The filter's cut-off frequency and order should be determined with respect to the

As shown in Figure 8, the sound insulation test was performed in an anechoic room. The control target is a stiffened CFRP plate and the dimension parameter is 840 mm 840 mm 1 mm. The plate is mounted on an aluminum cavity, with a thickness of 16 mm. A loudspeaker is placed inside the cavity, which generates acoustic excitation the stiffened plate. The CFRP plate is clamped to the aluminum cavity using screws. The edges between the CFRP plate and aluminum box are

According to the sound insulation mass law, because the box's thickness is much larger than the CFRP plate, the sound transmission path from the internal cavity to

control plant. For the realization form, analog filter is recommended. Because

filter can also be utilized to suppress the ultra-low frequency noise.

analog filter has much less propagation delay than the digital filter.

rate should be reduced.

4.3 Filter

Figure 7.

5. Case studies

89

sealed to prohibit air leakage.

consideration simultaneously.

Real-time control platforms for rapid prototyping.

DOI: http://dx.doi.org/10.5772/intechopen.85427

### 4. Implementation considerations

#### 4.1 Real-time control platform

To implement the control law, a real-time operating system (RTOS) is required to maintain the control law operating in a deterministic way. Normally, the sensing signal is sampled from the analog input channel, and the data acquisition device is switched to hardware-timed single point mode. The real-time control system guarantees the control law being updated within the period of two consecutive sampling points. After the control algorithm is updated, the output signal is sent to the analog output channel.

In general, the control law's thread should have the highest priority during the execution. This mechanism makes the control law can be implemented in a deterministic way. The other functions of the RTOS, such as signal monitoring, data logging and network communications are usually assigned to low execution priorities.

For rapid control system prototyping, some examples of the real-time target are: PXI target [17], CompactRIO target [18], Speedgoat target [19] and dSPACE target [20]. These real-time platforms' corresponding photo is shown in Figure 7.

#### 4.2 Delay minimization

With respect to the feedback control system, time delay is highly unwanted. However, inside the feedback control system, each component can introduce some kind of delay, which generates adverse influence to system's stability and performance.

For a pure delay system, the phase lag (in degree) can be expressed as:

$$
\rho = -\mathsf{S7.3} \times \left. \alpha \times T \right| \tag{8}
$$

where ω is the angular frequency and T is the time caused by system delay.

If the sensor and actuator are placed in the collocated configuration, the physical signal's propagation delay from the actuator to the sensor is minimized. Therefore, the digital control system becomes the primary time delay source, which has close relationship with the sampling rate.

To minimize the delay caused by data sampling, the sampling rate should be high enough. In general, the sample rate should be at least 10 times higher than the Effective Low Frequency Noise Insulation Adopting Active Damping Approaches DOI: http://dx.doi.org/10.5772/intechopen.85427

Figure 7. Real-time control platforms for rapid prototyping.

interesting bandwidth's upper frequency. However, the achieved sampling rate is also restricted by the controller. When the control law requires intensive computation effort and the controller is not fast enough, the control law will not be finished in time. In such circumstance, to meet the deterministic mechanism, the sampling rate should be reduced.

In summary, to determine the control law's updating rate, the control bandwidth, computation burden and real-time target's performance should be taken into consideration simultaneously.

## 4.3 Filter

where g is the control gain, ω<sup>n</sup> is the controller's upper frequency,ζFVF is the

For a given control frequency, the FVF controller's Bode plot with different

enough to generate sufficient gain below the upper frequency. For instance,

FVFð Þ¼ <sup>j</sup><sup>ω</sup> <sup>g</sup> <sup>j</sup><sup>ω</sup> <sup>þ</sup> <sup>2</sup>ωnζFVF

As shown in Figure 6, in the low frequency, the controllers behaves like a DVF controller but the control output rolls off at 20 dB/dec after the controller's upper frequency. Therefore, active damping is generated in the low frequency range. Unlike the NAF controller, the damping ratio of FVF controller should be high

If acceleration signal is used as the control input instead of velocity signal, the

To implement the control law, a real-time operating system (RTOS) is required to maintain the control law operating in a deterministic way. Normally, the sensing signal is sampled from the analog input channel, and the data acquisition device is switched to hardware-timed single point mode. The real-time control system guarantees the control law being updated within the period of two consecutive sampling points. After the control algorithm is updated, the output signal is sent to the analog

In general, the control law's thread should have the highest priority during the execution. This mechanism makes the control law can be implemented in a deterministic way. The other functions of the RTOS, such as signal monitoring,

For rapid control system prototyping, some examples of the real-time target are: PXI target [17], CompactRIO target [18], Speedgoat target [19] and dSPACE target [20]. These real-time platforms' corresponding photo is shown in Figure 7.

With respect to the feedback control system, time delay is highly unwanted. However, inside the feedback control system, each component can introduce some kind of delay, which generates adverse influence to system's stability and performance. For a pure delay system, the phase lag (in degree) can be expressed as:

where ω is the angular frequency and T is the time caused by system delay. If the sensor and actuator are placed in the collocated configuration, the physical signal's propagation delay from the actuator to the sensor is minimized. Therefore, the digital control system becomes the primary time delay source, which has close

To minimize the delay caused by data sampling, the sampling rate should be high enough. In general, the sample rate should be at least 10 times higher than the

φ ¼ �57:3 � ω � T (8)

data logging and network communications are usually assigned to low

jω �ω<sup>2</sup> þ 2ωnζFVFjω þ ω<sup>2</sup>

n (7)

controller's damping ratio.

ζFVF ¼ 0:7 is a good choice.

control damping values is shown in Figure 6.

Noise and Vibration Control - From Theory to Practice

modified FVF control law becomes into:

4. Implementation considerations

4.1 Real-time control platform

output channel.

execution priorities.

4.2 Delay minimization

relationship with the sampling rate.

88

H0

Filters are widely used in active control systems, which can suppress excessive noise signal. Crucially, if the control law itself does not have the roll-off characteristic in the high frequency, low pass filter will be always favorable. The high pass filter can also be utilized to suppress the ultra-low frequency noise.

The filter's cut-off frequency and order should be determined with respect to the control plant. For the realization form, analog filter is recommended. Because analog filter has much less propagation delay than the digital filter.
