Semi-Active Pulse-Switching Vibration Suppression Using Sliding Time Window

Abstract

The performance of pulse-switching vibration control technique is investigated using a new method for switching sequence, in order to enhance the vibration damping. The control law in this method which was developed in the field of piezoelectric damping is based on triggering the inverting switch on each extremum of the produced voltage (or displacement); however, its efficiency in the case of random excitation is arguable because of the local extremum detection process. The new proposed method for switching sequence is only based on the fact that the triggering voltage level was determined using windowed statistical examination of the deflection signal. Results for a cantilever beam excited by different excitation forces, such as stationary and nonstationary random samples, and pulse forces are presented. A significant decrease in vibration energy and also the robustness of this method are demonstrated.

1. Introduction

The ability to reduce the vibration amplitude over a wide frequency band is essential in the vibration control. The dependency of dynamic stiffness and damping properties of special materials such as viscoelastic materials on the excitation frequency or temperature variations cause other methods or materials to be considered [1, 2]. Several methods have been investigated for semi-active vibration control, using piezoelectric elements [3–10]. These methods are interesting because they do not rely on any operating energy as in active control. They consist of driving by a few solid-state switches requiring very little power. The common strategy of these methods is the electric boundary conditions modification of the piezoelectric elements. Synchronized Switch Damping (SSD) or pulse-switching technique which is implemented in this study consists of leaving the piezoelectric elements in open circuit, except during a very brief period of time, where the electric charge is either suppressed in a short circuit (SSDS) or inverted with a resonant network (SSDI). Corr and Clark experimented with SSDI method in the case of multimodal vibrations [5]. Also, they showed that the original SSD control law is not optimal in the case of wide band excitations.

In the case of wide band excitations, the optimization of the piezoelectric elements (size and location) as well as the switching network is not sufficient. It is therefore necessary to establish methods to define the accurate switch triggering time that operates exclusively on certain selected extrema, which would maximize the electric power produced by the piezoelectric elements [3]. In a similar way, this paper utilizes statistical analysis of the structural deflection to predict the optimum instants for the switching sequence in order to maximize the extracted energy and vibration damping. This new developed strategy allows an easy implementation of this damping technique for any type of excitation forces. In order to analyze this method, a beam equipped with piezoelements wired on a pulse-switching cell was experimented. The response of a cantilever beam subjected to pulse, stationary, and nonstationary random excitations has been considered. The stationary random excitation is the white noise and the nonstationary random excitations are the pulses of white noise shaped in time as a rectangle [11–14]. The following section describes the multimodal model used for simulation, as well as the pulse-switching device model. It is followed by a discussion on the strategy of semi-active vibration control using sliding time window and the proposed statistical method.

2. Theoretical Model of the Semi-Active Pulse-Switching Device

The differential equation for a beam with lateral motion is given as [15] 𝜌𝜕²𝑢(𝑥,𝑡)𝜕𝑡²()+𝑐𝜕𝑢𝑥,𝑡+𝜕𝜕𝑡𝜕𝑥²²𝜕𝐸𝐽²𝑢(𝑥,𝑡)𝜕𝑥²=𝑓(𝑥,𝑡), (1) where 𝑢(𝑥,𝑡) is the lateral beam deflection according to Figure 1. 𝐸 and 𝐽 represent Young’s modulus and the inertia moment of the cross-section. 𝜌, 𝑐, and 𝑓(𝑥,𝑡) are the mass, damping, and excitation force per unit length, respectively. This equation corresponds to Euler-Bernoulli assumptions. It is interesting to find the particular solution of (1) for a general excitation force per unit length. First, assume that the excitation force is given by 𝑓(𝑥,𝑡)=𝑝(𝑥)𝑓(𝑡), (2) where 𝑝(𝑥) is the load’s spatial distribution and 𝑓(𝑡) is the input time history. Second, we assume that the particular solution is given by 𝑢(𝑥,𝑡)=^𝑁_𝑖=1𝜑_𝑖(𝑥)𝑞_𝑖(𝑡), (3) where 𝑞_𝑖(𝑡) and 𝜑_𝑖(𝑥) are the modal coordinates and eigenmode functions, respectively, and 𝑁 is the number of modes. The eigenmode functions are also orthogonal, thus satisfying the relations ^𝑙₀𝜌𝜑_𝑖(𝑥)𝜑_𝑘(𝑥)𝑀𝑑𝑥=0,for𝑖≠𝑘,_𝑖,for𝑖=𝑘,_𝑙0𝑑²𝑑𝑥²𝑑𝐸𝐽²𝜑_𝑖(𝑥)𝑑𝑥²𝜑_𝑘(𝑥)𝐾𝑑𝑥=0,for𝑖≠𝑘,_𝑖,for𝑖=𝑘, (4) where 𝑀_𝑖 and 𝐾_𝑖 are the generalized mass and generalized stiffness for the 𝑖th mode, respectively, and 𝑙 is the cantilever beam length. After insertion of (2) and (3) into (1) and multiplying it by 𝜑_𝑘(𝑥)𝑑𝑥, integrating on length 𝑙 and employing the orthogonality condition of (4), along with assuming that damping is proportional to the mass and stiffness distributions, the result will be 𝑀_𝑖𝑞̈_𝑖+𝑐_𝑖𝑞̇_𝑖+𝐾_𝑖𝑞_𝑖=𝑄_𝑖𝑓(𝑡), (5) where 𝑞_𝑖(𝑡) can be determined from these equations. 𝑐_𝑖 is the 𝑖th generalized damping coefficient and 𝑄_𝑖 is the generalized excitation force [12] given by 𝑄_𝑖=^𝑙₀𝑝(𝑥)𝜑_𝑖(𝑥)𝑑𝑥. (6)

Figure 1 

Schematic diagram of cantilever beam with pulse switching circuit. 𝑢(𝑥,𝑡) is the beam deflection along the transverse direction (𝑦) and 𝑓(𝑡) is the excitation force at the tip of the beam (𝑥=𝐿).

For the cantilever beam problem represented in Figure 1, the concentrated external force 𝑓(𝑡) is applied at the free end of the beam and the piezoelectric patches are bonded on the structure, and then (5) becomes 𝑀_𝑖𝑞̈_𝑖+𝑐_𝑖𝑞̇_𝑖+𝐾_𝑒𝑖𝑞_𝑖=𝜑_𝑖(𝑙)𝑓(𝑡)−𝛼_𝑖𝑣_𝑖(𝑡), (7) where 𝐾_𝑒𝑖 is the short circuit generalized stiffness. The term 𝛼_𝑖𝑣_𝑖(𝑡) corresponds to the force due to the 𝑖th component of the voltage according to the macroscopic piezoelectric coefficient 𝛼_𝑖. In the open circuit condition, the piezoelectric voltage 𝑣_𝑖 due to the 𝑖th mode of vibration and the global piezoelectric voltage 𝑣 resulting from the modes superposition are 𝑣_𝑖=𝛼_𝑖𝐶₀𝑞_𝑖(𝑡)1;𝑣=𝐶₀^𝑁_𝑖=1𝛼_𝑖𝑞_𝑖(𝑡),𝐼=^𝑁_𝑖=1𝛼_𝑖𝑞̇_𝑖−𝐶₀𝑣.̇ (8)(9)

In this equation, 𝐶₀ is the capacitance of the piezoelectric elements, 𝑣 is the total piezoelectric voltage, and 𝐼 is the total outgoing current from the piezoelectric patches. In general, the piezoelectric patches are wired together in parallel. The open circuit stiffness 𝐾_𝑑𝑖 of the structure is related to 𝐾_𝑒𝑖, 𝛼_𝑖, and 𝐶₀ by 𝐾_𝑑𝑖=𝐾_𝑒𝑖+𝛼²_𝑖𝐶₀. (10) The eigenmode angular frequencies 𝜔_𝑑𝑖 in open circuit and 𝜔_𝑒𝑖 in short circuit are defined as 𝜔_𝑑𝑖=𝐾_𝑑𝑖𝑀_𝑖;𝜔_𝑒𝑖=𝐾_𝑒𝑖𝑀_𝑖. (11) The coupling coefficients 𝑘_𝑖 and mechanical quality factor 𝑄_𝑚𝑖 are defined as 𝑘²_𝑖=𝛼²_𝑖𝐾_𝑑𝑖𝐶₀=𝜔²_𝑑𝑖−𝜔²_𝑒𝑖𝜔²_𝑑𝑖;𝑄_𝑚𝑖=𝐾_𝑑𝑖𝑐_𝑖𝜔_𝑑𝑖. (12)

The various parameters previously defined can be identified (Table 1). In the proposed control technique, an electric circuit is connected to the piezoelectric elements in order to perform vibration damping. The goal is to maximize transferred energy that corresponds to the part converted into electrical energy. Multiplying each term of (9) by voltage and integrating over time shows that the transferred energy is sum of the electrostatic energy stored on the piezoelectric elements and the energy absorbed or dissipated by the electrical device as 𝐸_𝑡=^𝑁_𝑖=1𝛼_𝑖^𝑡₀𝑞̇_𝑖𝑣_𝑖1𝑑𝑡=2𝐶₀𝑣²+^𝑡₀𝑣𝐼𝑑𝑡. (13) Also, this nonlinear damping technique consists of adding a switching device in parallel with the piezoelectric elements. The current in the switching device is always zero except during the voltage inversion that takes place at each switch trigger. At each inversion, the energy extracted from the piezoelectric elements is equal to the difference in the electrostatic energy on the piezoelectric elements before and after the voltage inversion jump. The energy dissipated in the switching device is then given by ^𝑡₀1𝑣𝐼𝑑𝑡=2𝐶_0𝑘𝑣²_𝑘1−𝛾², (14) where 𝑣_𝑘 is the piezoelectric voltage just before the 𝑘th inversion, and 𝛾 is the inversion coefficient. In this case, the voltage inversion (switch) is not perfect, because one part of the energy stored on the piezoelectric capacitance is lost in the switching network (electronic switch and inductance). These losses are modeled by an electrical quality factor 𝑄_𝑖. The relationship between 𝑄_𝑖 and the voltage of the piezoelectric element before and after the inversion process is given by 𝑣_after=−𝛾𝑣_before=−𝑣_before𝑒^{−𝜋/2𝑄𝑖}. (15)

Table 1 

Modal parameters of the electromechanical structure.

In the following numerical simulations, the switching sequence is simply modeled by (15). It is evident that optimization of this technique (maximizing the energy extracted by switching device) is obtained by maximizing the sum of the piezoelectric voltage squared before each switch (14).

3. Strategy Enhancement of Pulse-Switching Vibration Control Using Sliding Time Window

In this case, by using pulse-switching strategy of control, which consists of triggering the inverting switch on each voltage extremum or strain extremum, the switching sequence is implemented based on statistical analysis of the deflection signal on a sliding time window. The statistical strategy is based on the idea that it allows the piezoelectric deformation to reach a significant value before allowing voltage inversion in which the energy stored on the piezoelements is higher. Recent developments have been aimed at the optimal switching strategy to have maximum damping. This strategy amplifies the amplitude of piezoelectric voltage. Consequently, the switching sequence or switching strategy allows maximizing ∑𝑣²_𝑘. Therefore, it is not the number of switching sequences which is important, but much more the instant of the switch. This point has been illustrated in Figure 2. In this case, in each instant the deflection signal 𝑢(𝑡) is studied during a given time window 𝑇_𝑒𝑠 just before the present time (a moving sliding window on the signal), and statistically probable deflection threshold 𝑢_𝑚 is determined from both the temporal average μ_𝑢 and standard deviation 𝜎_𝑢 of the signal during the observation period 𝑇_𝑒𝑠. Then, it is compared to the observation signal, to define the instant of the next switch. μ_𝑢 and 𝜎_𝑢 of the signal 𝑢(𝑡) on 𝑇_𝑒𝑠 are defined, respectively, as 𝜇_𝑢=1𝑇_{𝑒𝑠𝑇𝑒𝑠}𝑢(𝑡)𝜎𝑑𝑡,_𝑢=1𝑇_{𝑒𝑠𝑇𝑒𝑠}𝑢(𝑡)−𝜇_𝑢²𝑑𝑡^1/2. (16) The positive deflection threshold 𝑢_𝑚 will be derived as 𝑢_𝑚=𝜇_𝑢+𝛽𝜎_𝑢, (17) where 𝛽 is an arbitrary coefficient. In fact, the calculations in each instant of time on the sliding time window are to make a decision for the next switch. According to the general principle aimed at triggering the switch, once the voltage has reached a significant and statistically probable value, switching will occur when the absolute value of the deflection reaches the threshold 𝑢_𝑚 (switching condition |𝑢(𝑡)|>𝑢_𝑚). This method in the case of random vibration is very efficient. However, in this case the most obvious characteristic is being nonperiodic. The knowledge of the past history of random motion is adequate to predict the probability of occurrence of various displacement magnitudes, but it is not sufficient to predict the precise magnitude at a specific instant [16]. In this method, the history of signal in the past time window 𝑇_𝑒𝑠 is studied two times by average and standard deviation of the signal. Therefore, extremums estimation is better especially in the case of random vibration. Also, it amplifies the piezoelectric voltage amplitude (Figure 2).

Figure 2 

Illustration of two different strategies together for pulse switching control. —: original strategy with switching on each extremum; – –: alternative strategy consisting of selecting specific maximum; –·–: free end deflection.

4. Results in the Case of Vibration Damping

4.1. Vibration Damping Performance Analysis

In order to compare this control approach with uncontrolled case, either for simulation or experimental data, it is necessary to define quantities that will be evaluated for this comparison. The quantity used is related to the deflection of the structure. This quantity 𝐼_𝑢, which is a summation on the time variable for the considered modes, is the mean squared response of 𝑢(𝑥,𝑡) [17] and is defined as 𝐼_𝑢=𝑢(𝑥,𝑡)²=lim_𝜏→∞1𝜏^𝜏₀𝑢²(𝑥,𝑡)𝑑𝑡=_𝑖𝑘𝜑_𝑖(𝑥)𝜑_𝑘(𝑥)lim_𝜏→∞1𝜏^𝜏₀𝑞_𝑖(𝑡)𝑞_𝑘(𝑡)𝑑𝑡. (18) This quantity will be computed in this form for the simulation results. The displacement damping 𝐴_𝑢 is then evaluated from these quantities, computed with the uncontrolled case and then with the piezoelement driven with the desired control process, and is defined as 𝐴_𝑢𝐼=10log^{𝑢controlled}𝐼_{𝑢uncontrolled}. (19)

4.2. Numerical Results

The numerical sample is a cantilever steel beam equipped with piezoelectric patches. A sample force is applied to the free end of the clamped beam. The model ((7) and (9) principally) is simulated by numerical integration using the fourth-order Runge-Kutta algorithm. The simulations are carried out using the statistical method described previously. The signal considered for the definition of the threshold is 𝑢(𝑡). The observation time window 𝑇_𝑒𝑠 has to be twice the period of the lowest resonance frequency in order to give satisfactory results. This time should be sufficiently long to obtain a realistic image of the deflection especially for the lowest frequency mode and sufficiently short to maintain a good frequency response of the control.

The controlled responses of the cantilever beam when the applied excitations are pulse, stationary, and nonstationary random forces have been studied. The stationary random excitation is Gaussian white noise, and the nonstationary random excitation is a Gaussian white noise shaped in time by a rectangular envelope function. This appears as a pulse of white noise shaped as a rectangle (random shock loadings). These vibrations with stop and start driving are clearly nonstationary; however, the system experiences a series of transient vibration events. Transient vibration is nonstationary [18]. A simple model for a nonstationary random process with a time varying mean value is given by a process where each sample record is of the form 𝑓(𝑡)=𝑎(𝑡)𝑥(𝑡). Here, 𝑎(𝑡) is a deterministic function (known physical process) and 𝑥(𝑡) is a sample record from a stationary random process. The considered structure and the corresponding numerical data calculated using the expressions detailed in Section 2 are gathered in Tables 2 and 3, respectively.

Table 2 

Characteristics of the numerical structure.

Table 3 

Modal parameters of the considered system.

Figure 3 shows the variations of displacement damping 𝐴_𝑢 versus the 𝛽 coefficient under the same random shocks. The applicable interval of 𝛽, for which the displacement damping 𝐴_𝑢 stays minimum, is very large. This indicates that the extremum estimation in the case of statistical study is better and more reliable. The curve shows a sharp decrease in the displacement damping for 𝛽 values larger than √2, which would correspond to the extremum for a pure sinusoidal signal. This is very understandable, because in this simulation, the mechanical quality factor is equal to 𝑄_𝑚=500, and consequently the excitation signal is very well filtered by the system and locally close to a sinusoid. The result of this filtering and pseudosinusoidal response, especially in the structures with high values of 𝑄_𝑚, or small damping ratio, results in better extremum detection by pulse switching technique and therefore higher damping. In this method, the sinusoidal signal is the ideal signal to have an optimum switch. The optimum displacement damping is about −13 dB. But, by damping ratio increasing, the filtering decreases; consequently the signal is more complicated and the statistical switching is not optimum; thus damping decreases. The weak sensitivity of this control technique to the 𝛽 coefficient according to the large range of 𝛽 (approximately 0.1<𝛽<1.4) allows nearly optimal displacement damping for this method. In these calculations, 𝜏=300 T, where 𝑇=0.0571 s is the period of first resonant frequency.

Figure 3 

Variations of the displacement damping (𝐴𝑢) in dB versus the 𝛽 coefficients. The observed signal is the beam tip deflection 𝑢(𝐿,𝑡).

Figure 4 shows the white noise excitation shaped in time by a rectangular envelope function (rectangular pulse of white noise). The numerical constant (the effective time interval Δ𝑇) of rectangular pulse of white noise is equal to the actual time duration of the pulse and is 15 times the longest natural period of the mechanical system.

Figure 4 

The rectangular pulse of white noise excitation.

Figure 5 illustrates the beam tip displacement with and without the statistical pulse-switching control, for rectangular pulse of white noise sample of Figure 4. The damping achieved for a wide band semi-active method is remarkable; it is equal to −12.96 dB for 𝑄_𝑚=500 and 𝛽=0.8.

(a)

(b)

Figure 5 

Free end displacement of the beam: (a) without control; (b) in the case of pulse-switching control for the rectangle pulse of white noise excitation (Δ𝑇=15𝑇).

Figure 6 shows the beam tip displacement with and without statistical SSDI control for a pulse force. The pulse duration is 0.006 s, and the displacement damping for 𝑄_𝑚=500 and 𝛽=0.8 is more than −10.4 dB. The vibration damping using piezoelectric is related to mechanical strain induced in piezoelectric elements. Therefore, in the case of free vibration, at the beginning, the damping is well, but by vibration amplitude decreasing the piezoelectric strain decreases and consequently the damping performance decreases as well.

(a)

(b)

Figure 6 

Free end displacement of the beam: (a) without control; (b) in the case of control for the pulse excitation force.

Figure 7 shows the values of displacement damping for some excitation samples with different behaviors (harmonic, stationary and nonstationary random, random shock, and pulse forces). The results show that this method of control is not very sensitive to the type of excitation force. Since the sliding window is always moving with the signal and with the variations of signal during the time, the statistical values (𝜇 and 𝜎) change proportionally, as well. Therefore, the decided moments for the switch triggering and extremums detection are approximately optimum. This is a good advantage, because for a high range of excitation signals, it is not necessary to adjust 𝛽 for each force type to have optimum damping.

Figure 7 

The values of displacement damping for different sample forces: (1) harmonic, (2) stationary random, (3) nonstationary random, (4) random shock, and (5) pulse excitation.

5. Conclusion

The pulse-switching semi-active nonlinear control technique is interesting for structural damping applications, because it provides simultaneously good damping performance, good robustness, and very low power requirements. It is important to consider that this technique is simple enough to be self-powered. The main limitations are related to the case of complex or random excitation where the synchronization on the strain extremum is not trivial. The proposed work demonstrates improved performance over nearly the entire anticipated operational range. The results show that this method of control is not too sensitive to all of the types of excitation behaviors. The reason is that the sliding time window is always moving with the signal, and with the variations of signal during the time, the statistical values change proportionally as well. Therefore, the moments of switch triggering are approximately optimum. It can be said that the sliding time window is independent of the type of excitation signal. Then, the statistical analysis of the strain could allow defining the criteria to identify more accurate switching instants. However, this method requires a little knowledge of the signal which results in the definition of a tuning parameter 𝛽. The results show that this adjustment can be made very coarse. Also, because of the excitation signal filtering by the system, the extremum detection (switching on extremums is a characteristic of this method) becomes easier especially for the systems with higher values of 𝑄_𝑚. In general, the main idea of this paper deals with complex vibrations and focuses on development of the new strategy based on a simple statistical analysis of the displacement signal, allowing an easy implementation of this damping technique for any excitation force.