**6.1 Brake squeal**

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0

1

**Figure 7.** Amplification of dissipated energy with enhanced SSDI technique versus amplitude ratio *r*<sup>A</sup>

in the high frequency oscillations during the open switch phases. One can realize that in the standard switching law the switching always occurs exactly during the first mode extrema. At these moments, the voltage at the piezoceramics might be increased or decreased by the influence of the higher frequency, so that in mean this effect cancels out. With the enhanced switching law, the switch is always triggered when the second mode is maximum and augments therefore the voltage buildup. However, the switching is no longer occuring in phase with the first mode velocity, which reduces slightly the energy dissipation. For more

Obviously the increase in energy dissipation grows with the second mode amplitude. But also the frequency ratio *r*<sup>f</sup> = Ω2/Ω<sup>1</sup> between the first and second mode has an influence. The higher the second frequency, the smaller is the period time *T*<sup>2</sup> and therefore the timeframe. This means that the second mode maximum is in average closer to the first mode maximum, which is ideal for the energy dissipation. Figure 7 shows the amplification of energy dissipation versus the frequency ratio *r*<sup>f</sup> and the amplitude ratio *r*<sup>A</sup> = *q*ˆ2/*q*ˆ1. It can be concluded that for a given frequency ratio *r*<sup>f</sup> (this ratio is approximately 2*π* = 6.26 for the clamped beam), the energy dissipation grows linearly with the second mode amplitude. Additionally, the energy dissipation grows with a higher second frequency. Theoretically, for very low second mode amplitude, this enhanced switching law actually might give less damping than the standard law (the borderline is marked by a red line). This is due to the non-optimal phase shift of the switching signal, which is not in exact antiphase with the first

mode velocity anymore. But these regions are practically not very relevant.

6.26

*r*f

10 0 0.5 1 1.5 2 2.5 3 3.5

¯*E*diss,enh./*E*diss,std.

and frequency ratio *r*f.

details the reader is referred to [12].

1

2

*r*A

Brake noise that is dominated by frequencies above 1 kHz is usually called 'brake squeal'. It is widely accepted that brake squeal is caused by friction induced vibrations. A friction characteristic that is decreasing with relative velocity results in an energy input and can excite vibrations. Other works explain the instability with nonconservative restoring forces [6, 18]. This mechanism does not need the assumption of a decreasing friction characteristic, and it is not depending on certain damping properties. Although the brake function itself is not affected by these vibrations, the generated noise marks a significant comfort problem. Brake squeal remains unpredictable, even state-of-the-art FE analyses cannot cope with the complexity of the problem. Therefore, brake manufacturers typically reduce the tendency to squeal in a time consuming process of designing, building and testing of prototypes in a mostly empirical way.

Recently, the use of piezoceramics has been investigated for the suppression of brake squeal [22] in an active feedback control. The authors succeeded in controlling the squaling, however this method requires sensing electronics, complex amplifiers and a power supply. Therefore, this technology is expensive and unsuitable for many applications like automotive brakes. Piezoelectric shunt damping for brake squeal control might be a cheaper alternative.

### *6.1.1 Brake prototype and stability analysis*

Before designing the shunt damping network, the stability of the brake is studied using a multibody system, as shown in Figure 8. This model has been introduced in [13] to simulate

**Figure 8.** Brake model and disc eigenform.

the efficiency of linear *LR* and *LRC* shunts as well as a feedback control for brake squeal suppression. The two brake pads are modelled as rigid bodies and the contact area is represented as a layer with distributed stiffness and damping properties. Both pads have two translational degrees of freedom (out-of-plane and in-plane direction) and stay in contact with the brake disc. The coefficient of friction *μ* between disc and pads is assumed to be constant. The brake disc is described as an annular disc according to the Kirchhoff plate theory. Only the mode with four nodal diameter and one nodal circle is considered, this mode is depicted in Figure 8, as the corresponding frequency agrees best with the squealing frequency. The rotation of the disc introduces gyroscopic terms. Further more, the brake model contains nonconservative restoring forces as a result of the friction forces in the contact area between the pads and the disc. These forces can be identified in the unsymmetric stiffness matrix. Because of these forces, the mechanical model is possibly unstable. This can be shown by a complex eigenvalue analysis, as reported in Figure 9. The stability of the brake system

**Figure 9.** Imaginary part versus real part of the eigenvalues of the uncontrolled brake.

is determined by the largest real part of the eigenvalues, termed *λ*max. A variation of the coefficient of friction *μ* shows the influence of the nonconservative restoring forces. Without friction forces, *μ* = 0, the brake is asymptotically stable, as *λ*max is negative. With increasing friction, two pairs of eigenvalues move in opposite direction. The system becomes unstable above a critical friction force *μ*crit with *λ*max(*μ* = *μ*crit) = 0. The imaginary part corresponds to the squealing frequency, and is termed Ωsq.

Figure 10 shows the prototype disc brake at the Institute of Dynamics and Vibration Research with three piezoelectric stack actuators. Their forces act in the same direction as the brake pressure so that the out-of-plane vibrations of the brake disc can be influenced. The piezoceramics are placed between the inboard brake pad and the brake piston and protected by a cap construction against shear forces and debris. Other publications propose a similar placement of the actuators, for example the 'smart pads' [23] which include the piezoceramics directly into the back side of the brake pads. Another possibility is to place the actuators within the brake piston [1]. Three piezoelectric stack actuators with circular cross section 708 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges Shunted Piezoceramics for Vibration Damping - Modeling, Applications and New Trends <sup>15</sup> Shunted Piezoceramics for Vibration Damping – Modeling, Applications and New Trends 709

**Figure 10.** Prototype disc brake with embedded piezoceramics.

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represented as a layer with distributed stiffness and damping properties. Both pads have two translational degrees of freedom (out-of-plane and in-plane direction) and stay in contact with the brake disc. The coefficient of friction *μ* between disc and pads is assumed to be constant. The brake disc is described as an annular disc according to the Kirchhoff plate theory. Only the mode with four nodal diameter and one nodal circle is considered, this mode is depicted in Figure 8, as the corresponding frequency agrees best with the squealing frequency. The rotation of the disc introduces gyroscopic terms. Further more, the brake model contains nonconservative restoring forces as a result of the friction forces in the contact area between the pads and the disc. These forces can be identified in the unsymmetric stiffness matrix. Because of these forces, the mechanical model is possibly unstable. This can be shown by a complex eigenvalue analysis, as reported in Figure 9. The stability of the brake system

−2 −1.5 −1 −0.5 0 0.5 1

is determined by the largest real part of the eigenvalues, termed *λ*max. A variation of the coefficient of friction *μ* shows the influence of the nonconservative restoring forces. Without friction forces, *μ* = 0, the brake is asymptotically stable, as *λ*max is negative. With increasing friction, two pairs of eigenvalues move in opposite direction. The system becomes unstable above a critical friction force *μ*crit with *λ*max(*μ* = *μ*crit) = 0. The imaginary part corresponds

Figure 10 shows the prototype disc brake at the Institute of Dynamics and Vibration Research with three piezoelectric stack actuators. Their forces act in the same direction as the brake pressure so that the out-of-plane vibrations of the brake disc can be influenced. The piezoceramics are placed between the inboard brake pad and the brake piston and protected by a cap construction against shear forces and debris. Other publications propose a similar placement of the actuators, for example the 'smart pads' [23] which include the piezoceramics directly into the back side of the brake pads. Another possibility is to place the actuators within the brake piston [1]. Three piezoelectric stack actuators with circular cross section

**Figure 9.** Imaginary part versus real part of the eigenvalues of the uncontrolled brake.

�(*λ*)

*μ* ↑

*μ* = *μ*crit

−4000

to the squealing frequency, and is termed Ωsq.

−Ωsq/2*π*

−2000

�(*λ*)

2*π*

0

2000

4000

Ωsq/2*π*

and material FPM 231 from company MARCO are used. They are designed to withstand brake pressures exceeding 30 bar and temperatures up to 200◦C. This is certainly not enough for typical temperatures during strong brakings, but enough for principal feasibility studies in the lab. It is possible to connect all piezoceramics with *LR*- or *LRC*-shunts. When the SSDI-technique is used, one of the ceramics (typically the middle one) is used as a sensor and the remaining two are shunted.

### *6.1.2 Modeling of the combined system and control of brake squeal*

The tuning of the resonant *LR*- and *LRC*-shunts is done like it is described in [16] for an assumed squealing frequency of *f*sq ≈ 3400Hz. The results for a passive *LR* and two negative capacitance shunts with different capacitance ratios *δ* are shown in Figure 11. The maximum real part *λ*max is given versus the squealing frequency *f*sq. The squealing frequency of the brake model is artificially changed by multiplicating the stiffness matrix by a constant term, which results in a change of all eigenfrequencies of the system.

All three networks are capable to stabilize the brake when tuned precisely, as *λ*max is negative. However, the frequency bandwidth in which the brake is stable is very narrow for the passive *LR*-shunt. Practically this frequency range is not enough for a robust suppression of the brake squealing, as it might occur in a broad range due to the many possible eigenfrequencies of the brake. As expected, the negative capacitance networks perform better. The maximum reduction of *λ*max is equal to that achievable with *LR*-networks, but this occurs in a broader frequency range. The closer the capacitance value is tuned to −1, the better the performance results.

#### *6.1.3 Measurements on the brake test rig*

Measurements are conducted on the brake test rig with the modified brake using the following procedure to experimentally determine the frequency bandwidth of the damping effect: The passive *LR* or active *LRC* shunt is disconnected from the piezoceramics, and the brake

**Figure 11.** Stability of the brake model for *LR*-shunt (*δ* = 0) and *LRC*-shunts with *δ*<sup>1</sup> = −0.6 and *δ*<sup>2</sup> = −0.88.

pressure and disc speed is varied until a proper and steady squealing arises. During the tests, this usually happens for pressures between 8 and 15 bar and velocities of 23 rpm of the beake disc. The squealing frequency could be located at approx. 3400Hz. After this, the shunt is connected to the electrodes, and the inductance and resistance are set to the calculated optimum values. Afterwards, the inductance value is reduced until the damping effect is vanishes, as the network is too strongly mistuned. This is the initial value of the inductance at the beginning of each measurement.

During the measurements, the shunt is periodically connected and disconnected for 10 seconds. After each cycle, the inductance is increased so that in the following 10 seconds of connection the shunt is tuned to a constant, new frequency. In the first half of each measurement, the electrical resonance frequency is successively tuned closer to the squealing frequency and the damping effect grows. In the middle of the measurement, the shunt is tuned nearly perfectly, and the effect is maximized. In the second half, the mistuning grows again as the inductance value is further increased, and the damping effect is diminished. The measurement is stopped when no squealing reduction is noticeable anymore. This procedure is repeated for different *LR* and *LRC*-shunts.

During the measurements, the sound pressure is recorded with a microphone, which is located in a distance of 50 cm from the brake. In the upper plot of Figure 12 the sound pressure is given versus the time for one exemplary measurement. In the lower plot, the corresponding sound pressure level (SPL) and the inductance values are shown. As shown, during the measurement time of more than 3 minutes, the SPL of the squealing brake remained nearly constant within 95-100 dB. In the very first and last switchings between connection and disconnection of the shunt, nearly no reduction in the SPL is noticed, as the mistuning is too strong. In the middle of the measurement the squealing stops immediately after connecting the shunt and starts again after disconnecting. The remaining sound without the squealing is environmental noise,

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2600 2800 3000 3200 3400 3600 3800 4000 4200

Squealing frequency *f*sq [Hz]

**Figure 11.** Stability of the brake model for *LR*-shunt (*δ* = 0) and *LRC*-shunts with *δ*<sup>1</sup> = −0.6 and

pressure and disc speed is varied until a proper and steady squealing arises. During the tests, this usually happens for pressures between 8 and 15 bar and velocities of 23 rpm of the beake disc. The squealing frequency could be located at approx. 3400Hz. After this, the shunt is connected to the electrodes, and the inductance and resistance are set to the calculated optimum values. Afterwards, the inductance value is reduced until the damping effect is vanishes, as the network is too strongly mistuned. This is the initial value of the inductance

During the measurements, the shunt is periodically connected and disconnected for 10 seconds. After each cycle, the inductance is increased so that in the following 10 seconds of connection the shunt is tuned to a constant, new frequency. In the first half of each measurement, the electrical resonance frequency is successively tuned closer to the squealing frequency and the damping effect grows. In the middle of the measurement, the shunt is tuned nearly perfectly, and the effect is maximized. In the second half, the mistuning grows again as the inductance value is further increased, and the damping effect is diminished. The measurement is stopped when no squealing reduction is noticeable anymore. This procedure

During the measurements, the sound pressure is recorded with a microphone, which is located in a distance of 50 cm from the brake. In the upper plot of Figure 12 the sound pressure is given versus the time for one exemplary measurement. In the lower plot, the corresponding sound pressure level (SPL) and the inductance values are shown. As shown, during the measurement time of more than 3 minutes, the SPL of the squealing brake remained nearly constant within 95-100 dB. In the very first and last switchings between connection and disconnection of the shunt, nearly no reduction in the SPL is noticed, as the mistuning is too strong. In the middle of the measurement the squealing stops immediately after connecting the shunt and starts again after disconnecting. The remaining sound without the squealing is environmental noise,

−1

at the beginning of each measurement.

is repeated for different *LR* and *LRC*-shunts.

*δ* = 0 (LR) *δ* = −0.6 *δ* = −0.88

−0.5

*λ*max [-]

*δ*<sup>2</sup> = −0.88.

0

0.5

**Figure 12.** Sound pressure and SPL during one measurement with stepwise varied inductance.

which has been measured as high as 75 dB, and is dominated by the sound of the electric motor that drives the brake disc.

The performance of the shunted piezoceramics is evaluated by the reduction of the mean SPL during each 10 seconds of connection and disconnection for every inductance value. In Figure 13 this reduction is given versus the indunctance (normalized to the optimal value). The figure shows the results for the passive *LR* shunt (*δ* = 0) and two different *LRC* shunts

**Figure 13.** Reduction in SPL versus inductance tuning for *LR*- and *LRC*-shunts.

with the same capacitance ratios as in the simulations reported in Figure 11. It can be seen that the maximum reduction for each shunt is achieved for the perfectly tuned shunts (*L* ≈ *L*opt respectively *η* ≈ 1). In these cases, all shunts - including the passive *LR* shunt - are capable to suppress the squealing totally, as predicted by the simulations. The differences in the maximum reduction can be explained by different strength of the squealing. Naturally, a weak squealing delimits the maximum possible reduction compared to a strong squealing.

From the inductance ratio *L*/*L*opt, the frequency ratio between the electrical eigenfrequency and the squealing frequency can be re-calculated. Defining the state 'silent' and 'squealing' by an arbitrary threshold of 12 dB SPL-reduction, the brake is stabilized in a range of Δ*f* = 40Hz for the passive *LR* shunt. With actice *LRC*-shunts, the stabilized range covers Δ*f* = 212Hz with *δ* = −0.66 and Δ*f* = 950Hz with *δ* = −0.88. These results show a good accordance with the simulation results in Figure 11. However, some influences like the heating up of the piezoceramics lead to a reduction of the piezoelectric effect so that the performance at the end of each measurement is slightly lower than in the beginning.
