Yu FAN, Yu HU, Ygung WU, Lin LI

a School of Energy and Power Engineering, Beihang University, Beijing 100191, China

b Beijing Key Laboratory of Aero-engine Structure and Strength, Beijing 100191, China

c Sino-French Engineering School, Beihang University, Beijing 100191, China

KEYWORDS Blade;Dry friction damping;Experiments;Finite element method;Piezoelectric;Synchronized switch damping

Abstract The Synchronized Switch Damping(SSD)is regarded as a promising alternative to mitigate the vibration of thin-walled structures in aero-engines, especially for blades or bladed disks.The common manner is to shunt the switch circuit independently to a single piezoelectric structure.This paper is aimed at exploring a novel way of using the SSD, i.e., the SSD is interconnected between two piezoelectric structures or substructures.The damping mechanism, performance,and effective range of the interconnected SSD are studied numerically and experimentally.First,based on a dual cantilever beam finite element model,the time domain and frequency domain modeling and solving methods of the interconnected SSD are deduced and validated.Then,the influence of the amplitude and phase relationship on the damping effect of the interconnected SSD is numerically studied and compared with the shunted SSD.A self-sensing SSD control board is developed,and experimental studies are carried out.The results show that the interconnected SSD establishes an additional energy channel between the corresponding piezoelectric structures.When the amplitudes of the two cantilever beams are different,the interconnected SSD balances the vibration level of each beam.When the amplitudes of the two cantilever beams are the same, if the appropriate interconnection manner is selected according to the phase, the resonance peak can be reduced by more than 30%.When the vibration is in-phase/out-of-phase, the damping generated by the interconnected SSD in a cross/parallel manner is even more significant than the shunted SSD.Furthermore, this novel connection scheme reduces the number of SSD circuits in half.Finally, for engineering applications, we implement the proposed damping technology to the finite element model of a typical dummy bladed disk.A piezoelectric damping ratio of 13.7% is achieved when the amount of piezo material is only 10% of blade mass.Compared with traditional friction dampers,the major advancements of the interconnected SSD are:(A)it can reduce the vibration level of blades without friction interface; (B) the space constraint is overcome, i.e., the vibration energy is not necessarily dissipated independently in one sector or through physically adjacent blades, and instead, the dissipation and transfer of vibrational energy can be realized between any blade pair.If a specific gating circuit is adopted to adjust the interconnection manner of the SSD, vibration mitigation under variable working conditions with different engine orders will be expected; (C)designers do not need to worry about the annoying nonlinearities related to working conditions anymore.

1.Introduction

Dry friction is the important damping source in blades or bladed disks of in-service aero-engines.Typical frictional damping structures include blade shrouds, under-platform dampers,and friction rings.Owing to complex stuck,slip,separation behaviors and the states’transition on the contact interface,1dry friction includes strong local nonlinearities to the host structure, and the friction damping effect is related to the vibration state and the normal preload2–3.

For the next generation of advanced aero-engines, replacing bladed disks with integrally bladed disks is the trend to improve reliability, resulting in a decrease in the number of friction interfaces.On the other hand, maneuvering flights become more and more frequent, and deviation from the target working condition makes dry friction damping design annoying due to the nonlinear nature of friction.Therefore,to develop effective alternative damping strategies is in demand.

Piezoelectric materials are smart materials with the advantages of lightweight,high energy density,wide bandwidth,and ease of integration.The piezoelectric effect is the cornerstone of piezoelectric materials,which guarantees the mutual conversion of mechanical energy and electrical energy.Based on the electromechanical coupling property, piezoelectric materials can be used as sensors, actuators, energy harvesters, and even dampers.

From the mathematical viewpoint, based on the corresponding relationship between mechanical variables and electrical variables in the dynamics equation of piezoelectric structures, i.e., electromechanical analogy,4one can realize specific mechanical functions through circuit design.Specifically,for linear electrical components,inductance can be compared to mass in the mechanical field, resistance can be considered as damping, and capacitance can be regarded as stiffness.Electrical parameters are much easier to adjust than mechanical parameters.Therefore,piezoelectric materials provide a potential way to achieve vibration suppression under variable working conditions.

The resistive and resonant shunting circuits are initially proposed by Hagood and Flotow5to damp the vibration of mechanical structures.From the perspective of electromechanical analogy,these passive damping strategies are equivalent to constructing viscous damping and dynamic vibration absorbers by electrical elements.Schwarzendahl et al.6carried out experimental studies on the passive shunted damping to blades.The damping effect of the resistive shunt is weak(i.e.,resonance amplitude drops by less than 10%7).Resonant shunted damping can achieve significant single-mode vibration control, but the damping effect is very sensitive to the inductance.8It is necessary to well tune the circuit before use, and a slight change of the vibration frequency will cause great degradation of the damping.This narrow band nature restricts the resonant shunted damping to the fixed working condition but not the variable one.Another drawback of the resonant shunted damping is that an excessive inductance value is needed for low-frequency vibration, (e.g., around 10–1000H for frequency less than 100 Hz), and the corresponding physical inductor will be very heavy.

To overcome these drawbacks and realize broadband vibration control while keeping the simplicity of implementation,semi-passive piezoelectric shunted damping including negative capacitance shunt,9adaptive resonant shunt,10–11and Synchronized Switch Damping (SSD)12attracts more and more attention since the 2000 s.The synchronized switch damping,firstly proposed by Richard et al.,13seems to be the most promising one in engineering.Owing to the self-adaptive broadband nature, it is insensitive to the excitation frequency and the variation of structural characteristics or electric parameters, and it can achieve multi-modal vibration reduction.14–15In addition, only low power is needed to operate the switch, which can even be realized with self-powered technologies in the domain of energy harvesting16–17.

The classic working mechanism of the SSD for single-mode control is first to sense the deformation of the structure, and then close the shunted switch instantly when the strain reaches the maximum, so that the voltage on the piezoelectric electrodes shifts 90° compared with the open-circuit state, and always generates a force that hinders the motion of the structure, thereby producing the damping effect.For multi-mode control, Corr and Clark18proposed a control law from the modal energy rate aspect for the SSD: by controlling the switch at the right instants, the modal energy change rate is kept less than zero,so that the energy is always dissipated during vibration.In general,co-located piezo transducers must be used to monitor the vibration.Owing to the nonlinear nature in the discontinuous switch closing procedures, the research carried out by Liu et al.19demonstrated that the hysteresis behavior of the SSD and dry friction are similar.From the perspective of electromechanical analogy,the synchronized switch can be compared to the friction damper.

There are several variants of SSD developed by researchers,including the Synchronized Switch Damping on Short-circuit(SSDS),13the Synchronized Switch Damping on Inductance(SSDI),20–21the Synchronized Switch Damping on Voltage source (SSDV),22and the Synchronized Switch Damping on Negative Capacitance (SSDNC).23Compared with SSDS, the other three types amplify the voltage on the electrodes when the switch works although the amplification mechanisms are different, therefore better damping is expected.Considering the implementation in engineering, the SSDV and SSDNC need the self-adaptive external voltage source and the negative capacitance, respectively, increasing drastically the complexity of the electric circuit.In contrast, the SSDI only connects a small inductor(i.e.,less than 100 mH)to the switch.This also bypasses the unrealistic large inductance required by the passive resonant shunt.

Due to the nonlinear processing of voltage, the numerical simulation of the piezoelectric structures shunted with SSD is more complicated than linear piezoelectric structures.Classic methods such as Runge-Kutta or Newmark can be executed to catch the time history.When one is more interested in the steady-state response,the time integral will take a long time to reach the steady state.Therefore,the well-known Harmonic Balance Method (HBM) can be used to extract the Frequency Response Function(FRF)instead.Although the HBM is more efficient than the time integral for the steady-state response simulation, it will also be very time-consuming when large-scale finite element models are under investigation, and even sometimes the convergence problem might occur.Fortunately, Liu et al.19found the quasi-linear behavior of SSD through nonlinear modal analyses and steady-state response analyses.Inspired by this,Yan24and Wu25et al.proposed equivalent linearization methods to accelerate the simulation.Combined with the Reduced-Order Model(ROM)of the linear piezoelectric structure proposed by Thomas et al.,26the time cost by the equivalent linearization will be reduced to only 1%of the HBM.Thanks to these methods,a lot of numerical studies were carried out27–29to reveal the damping mechanism of different types of SSD and to preliminarily demonstrate their feasibility.

Most of these aforementioned studies on the mechanism level have been experimentally verified.Developing a small-sized,stable,and effective SSD control board is an essential and inevitable step for experimental research.The switch and the control circuit can be realized either through two complementary Metal Oxide Semiconductor Field Effect Transistors(MOSFETs)triggered by an Microcontroller Unit (MCU)20,29or the real-time simulation system containing acquisition and output modules such as dSpace.30For the former,although smaller in size and easier to integrate into engineering structures, many improvements are still to be made, such as to avoid false triggering of the switch when the same piezoelectric patch is used for both vibration monitoring and suppression,and to reduce the phase lag caused by filtering.The latter is mostly for laboratory purposes rather than engineering applications because real-time simulation equipment brings significant additional weight.

There are only a few studies on the application of synchronous switch damping to blades or bladed disks vibration suppression.Bachmann et al.31integrated the Lead Zirconium Titanate (PZT) ceramic material into a full composite fan blade and tested the resonant shunted damping and synchronized switch damping.The total damping increased by 500%-600% at the cost of additional weight of 23%–32%.In the studies of Liu32and Wu25et al., the feasibility of the SSD on bladed disks is numerically investigated, and one SSD is connected independently to one piezoelectric patch on each blade.From the viewpoint of electromechanical analogy, the shunted SSD can be compared to the dry friction between the blade root and groove.In this way, the vibration energy is only consumed within the corresponding sector, and the same number of SSD control boards as the number of blades is required.

Other studies24,33are aimed at tailoring the bandgap through periodic electromechanically coupled structures with SSD.One of them made by Bao et al.33is very interesting.In their work, one SSD is interconnected between two piezoelectric patches to form a new cell, and this new manner permits enhancing the attenuation in particular frequency bands compared with the traditional manner of SSD (i.e., each SSD is independently connected to one cell).

In bladed disks, all blades vibrate at the same frequency with a certain inter-blade phase depending on the Engine Order(EO).Blades with the same phase can be set as a group.If every two blades (or two groups of blades) are interconnected with one SSD circuit, they will interact with each other through the electrical channel.The hysteresis characteristics of the interconnected SSD will be similar to those of the blade shrouds.It is obvious that the number of control boards can be reduced by at least half compared with the shunted SSD,which is beneficial to cost savings.More importantly, owing to the self-adaptation property of the SSD,it has the potential for vibration suppression of blades under variable working conditions theoretically.Moreover, unlike shrouded blades where the energy is only dissipated or transferred between two adjacent blades, the interconnected SSD is free of contact interface and can break the space constraints of blades (i.e.,any two blades with an appropriate phase difference can interact through the electrical energy channel),and thus it can overcome the drawback that the blade shroud is not effective in terms of damping towards certain EO.

Intuitively, we can expect that the above-mentioned interconnected SSD has a bright future.The premise is that we need to answer the following three questions:

(1)What is the performance of the interconnected SSD,is it better or close to that of the shunted SSD?

(2)How does the amplitudes and phase difference affect the damping effect of the interconnected SSD?

(3) How much damping can be generated for a typical bladed disk?

These three questions are exactly what we want to clarify in this paper.

Specifically, first, the dynamic equations of a general discrete model damped by the interconnected SSD are established, and the corresponding numerical methods in the time domain and the frequency domain are derived (Section 2).Then a dual cantilever beam model which is the simplification of a pair of blades is used as an example to reveal the vibration suppression mechanism of the interconnected SSD numerically(Section 3).Section 4 is dedicated to experimental verification.For completeness,the SSD control circuit newly developed by our research group is also presented in this section.Finally,to demonstrate the feasibility of engineering, a typical dummy bladed disk finite element model is taken as an example to quantitatively illustrate the output damping of the proposed technology in Section 5.

2.Problem formulation

2.1.Two manners of interconnected SSD

In this study,only the SSDS and SSDI are under investigation because these two types are relatively simple concerning the control circuit.On the contrary, the SSDV needs additional voltage sources, and stable negative capacitance is not easy to realize for the SSDNC.

If the SSD is connected between the electrodes of two piezoelectric patches, it is termed the interconnected SSD.There are two interconnection manners,the first is the parallel manner(Fig.1 top),that is,the SSD is connected between the electrodes at the same position (e.g., ①+ to SSD to ②+) of the two piezoelectric patches, and the other electrodes of the two piezoelectric patches are short-circuit (e.g., ①- to ②-);the second is the cross manner (Fig.1 bottom), that is, the SSD connects the electrodes crossly of the two piezoelectric patches(e.g.,①+to SSD to ②-),and the not-used electrodes are short-circuit (e.g., ①- to ②+)0.2.2.Modeling in time domain.

A general mechanical system can be divided into multiple substructures.Each substructure is mechanically independent or coupled (Fig.2).The SSD can be interconnected between every two piezoelectric patches bonded to the underlying substructures.Without loss of generality, taking a structure containing only two substructures as an example, we explain the mechanism of the interconnected SSD and derive the dynamic equations of the system in the time domain.

Different from the shunted SSD, the interconnected SSD closes the switch when the voltage between the two connected electrodes belonging to different piezoelectric patches reaches the extremums and opens it again when the voltage is zero(SSDS) or reverse (SSDI).The electric boundary condition of the two piezoelectric structures changes twice in one vibrational period, and the piezoelectric patches exert force on the respective substructures.The dynamic of the system with the interconnected SSD is piecewise linear, that is, it is at the open-circuit state most of the time, and it is at the interconnected state only at the moment the switch is closed.Based on this, the time domain governing equations can be directly given, and we will only show the case where the SSD is interconnected in parallel:

Fig.1 Schematic of parallel manner (top) and cross manner(bottom) of interconnected SSD.

Fig.2 Schematic of an arbitrary structure with multiple piezoelectric patches interconnected with SSD.

In Eq.(1),Dmirefers to the structural matrix for the ith substructure,where D can be M,K or C for the mass matrix,stiffness matrix, and damping matrix, respectively; Dmijstands for the mechanically coupled matrix between the substructures i and j,and if these two substructures are mechanically independent, these coupling terms will be zero; Kmpiis the electromechanically coupled matrix of the ith substructure; Cpidenotes the capacitance of the piezoelectric patch bonded to the ith substructure; x and Virefer to the displacement vector and voltage between the electrodes of the piezoelectric patch i respectively,which are the function of the time;fiis the excitation force vector, and the excitation frequency at each substructure is assumed to be identical; Qidenotes the electric charge stored in the capacitance.

The first equation of Eq.(1) describes the dynamic of the system when the switch is open, and the second equation works when the switch is closed.The switch will be triggered only at the moment the voltage difference between the two piezo patches reaches the local extremums, i.e.,

where ‘－’and ‘+’represent the instants closely before and after the present moment respectively.The closing time of the switch is very short, which is equal to half of the electric resonance period for SSDI and to nearly zero for SSDS.The closing duration of the switch for the interconnected SSDI is given by

Since the switch closing is instantaneous, the velocity and displacement of the structures do not change at this moment.There is only discontinuity of the voltage on each piezoelectric patch.Therefore, the entire process of the system in the time domain can be regarded as the sequential connection of multiple half vibrational periods at the open-circuit state, and the initial condition of each half period can be regarded as a sudden change in the voltage due to the switch closing.

Based on this, we can rewrite the dynamic equations into the state space in each consecutive half period during which the switch is not triggered:

By solving the above equation,the voltage initial condition can be deduced.

Combining Eq.(5), Eq.(6), and Eq.(7), we can use the Runge-Kutta algorithm(e.g.,ode45 in MATLAB)to simulate the dynamic performance of the system in the time domain.

2.2.Modeling in frequency domain

Here we use the idea of equivalent linearization to derive the impedance of the interconnected SSD.This idea originates from Ref.,24which is only applicable to the case where the parameters of the two piezoelectric substructures are the same.This paper expands this method,and the two substructures can be different.

First, according to the Norton theorem, we derive the equivalent current source and the equivalent capacitance of the interconnected piezoelectric substructures:

where Inortonis the Norton current source.The equivalent capacitance has been already expressed by Eq.(4).The schematic of the equivalent circuit is shown in Fig.3.

At the open-circuit state,the voltage drop between the two piezoelectric patches is calculated by

If the system is under single harmonic excitation, the first harmonic of the displacement will play a decisive role.Therefore, it is reasonable to truncate the voltage to the first harmonic:

where B is the voltage amplitude, ω refers to the excitation angular frequency, and φ is the phase condition.Without loss of generality,the initial phase of the excitation can be adjusted so that φ is 0 for ease of expression.

When the system reaches the steady state, the voltage can be regarded as the superposition of the sinusoidal function and the crenel function:

Fig.3 Schematic of equivalent circuit of interconnected SSD according to Norton theorem.

Then,we derive the governing equations of the system with the interconnected SSD in the frequency domain by simplifying Eq.(17):

Fig.4 Schematic of linearized impedance of interconnected SSD.

It is worth noting that the aforementioned governing equations either in the time domain or in the frequency domain can be easily extended to the cases where the SSDs are crossly interconnected and where structures contain more piezoelectric patches.

3.Numerical studies

Fig.5 Finite element model of a dual piezoelectric beam system with interconnected SSD.

In this section,we use the finite element model of a dual piezoelectric cantilever beam system shown in Fig.5 to study the performance of the interconnected SSD numerically both in the time domain and frequency domain.The two piezoelectric beams are identical.Two cantilever beams can be regarded as two simplified blades without considering disk coupling.The objective of this study is:

-to mutually verify the correctness of the time domain and frequency domain algorithms proposed in the previous section;

- to compare the optimal damping effect of the interconnected SSD and that of the shunted SSD;

- to reveal the influence of the amplitude and phase difference between the two blades on the damping.

The finite element model is established in Ansys.The beams are made of aluminum and modeled by SOLID 45 elements.The piezoelectric patches are made of PZT-5H and modeled by SOLID 5 elements.The parameters of the model are listed in Table 1.

The vibration of the cantilever beams is captured near the free ends(yellow dot in Fig.5).The single point harmonic excitation is applied perpendicular to the beam near the clamped end of each beam (red dot in Fig.5):

where ωerefers to the excitation frequency.

The first bending mode and the second bending mode are selected as the target modes.At the open-circuit state, the modal frequencies are 8.8 Hz and 52.4 Hz for mode one and mode two, respectively.The corresponding modal shapes are displayed in Fig.6.

3.1.Time domain results

For different combinations of the excitation amplitude and the phase difference (F1,F2,φ) the time history of the response amplitudes and the voltage of the dual-beam interconnected with the SSD in a parallel manner are given.Note that the excitation frequency is ωe= 8.8 Hz in the vicinity to the first resonance, and the structural damping ratio is set to 1%.

Fig.6 The first two bending modes of piezoelectric cantilever beam at open-circuit state.

For Case 1, two beams vibrate out of the phase with the same amplitude.When the vibration reaches the steady state,the time domain response of the system interconnected by SSDI(γ=0.6)is shown in Fig.7.The result is reasonable since the voltage on SSD (orange dashed line in Fig.7(a)) respects the theoretical waveform, i.e., the voltage magnification and inversion effect exist.The switch closes when the displacement difference of the two beams(blue solid line in Fig.7(a))reaches the extremums, otherwise the switch is open.The voltage on each piezoelectric patch is also out of phase (Fig.7(b)), and the voltage on SSD is the sum of the voltage on each piezoelectric patch.

If we set γ to zero,the SSDI degenerates to SSDS(Case 2).We notice in Fig.8(a)that when the switch is open,the voltage is proportional to the deformation difference of the two beams, and when the switch closes, the voltage on SSD drops to zero.Owing to the periodic operation of the switch,there is voltage inversion.Since there is no inductance in the circuit,the voltage magnification does not exist.That is why the damping effect of the interconnected SSDS is lower than that of the SSDI, and we can see that the displacement amplitudes of the two beams interconnected by SSDS(blue lines in Fig.8(b)) at the resonance are larger than those by SSDI.

To clarify the influence of the amplitude, starting from the limit case,we remove the excitation on beam 2.For Case 3,the interconnected SSDI is used with the excitation parameters(F1,F2) = (10, 0), indicating that only the first beam is under excitation.The system takes more time to obtain the steady state, and the vibration amplitude gradually increases for the second beam.Finally, both two beams vibrate in the steady state(Fig.9(a)).The voltage on each piezoelectric patch is different in amplitude and has a phase difference of 180° (Fig.9(b)).In Fig.9(c), the area of the green hysteresis loop marked with ‘+’is proportional to the energy converted from the mechanical field to the electrical field in one period of beam 1 at the steady state.The area of the red hysteresis loop marked with‘-’is proportional to the input mechanical energy of beam 2.We can conclude that the interconnected SSDI creates a new energy channel coupling the two mechanically independent substructures, and the energy flows from the substructure at the higher energy level to the substructure at the lower energy level.

Table 1 Parameters of finite element model.

Fig.7 Case 1: Interconnected SSDI in a parallel manner, (F1 F2 φ) = (10, 10, π).

Fig.8 Case 2: Interconnected SSDS in a parallel manner, (F1 F2 φ) = (10, 10, π).

Theoretically, the crossly interconnected SSD for in-phase vibration is equivalent to the interconnected SSD in parallel for out-of-phase vibration, and it is validated also by numerical simulation in this work.For the sake of being concise, the results of the crossly interconnected SSD are not displayed in this paper.

When the two beams do not vibrate out of phase (or in phase), how does the interconnected SSDI work? In Case 4,the corresponding parameters are (F1,F2,φ) = (10, 10, 5π/6).Compared with Case 1, the waveform of the voltage on each piezoelectric patch changes because the instant of the switch closing does not coincide anymore with the deformation extremums of each beam (Fig.10(b)).The converted energy in one period for the beam i is given as

We can derive from Eq.(20) that the less the phase difference between the voltage and the deformation rate, the larger the converted energy.Therefore, the phase difference between the deformation rate and the basic component of the voltage on the underlying piezoelectric patch is not zero, leading to a negative effect on the damping.The hysteresis loop in Fig.10(c) also explains the degraded damping effect due to the phase difference.On the corners of the hysteresis loop,there are negative energy dissipation zones marked with ‘-’,which reduce the total damped energy.This can explain why the response amplitude of Case 4(Fig.10(a))is larger than that of Case 1.

3.2.Frequency domain results

Based on the linearization formula (Eq.(18)), the steady-state response of the dual-beam system interconnected with the SSD can be efficiently calculated in the resonant zone.The consistency of the frequency domain method and the time domain method is validated as shown in Fig.11(a), where the relative error is less than 3%.

We first compare the damping effect generated by the interconnected SSD and the shunted SSD when two beams vibrate out of the phase with the same vibration amplitude.When(F1,F2,φ) = (10, 10, π), the damping effect of the interconnected SSDS in parallel (marked with green circle) equals the case where SSDS are shunted independently to each beam(red dashed line),whereas the vibration reduction by the interconnected SSDI (blue circle marker) is even better than the shunted SSDI (purple dashed line) both for the 1B (Fig.11(a))and 2B(Fig.11(b))mode.This is because the parallel connection of the SSDI changes the equivalent capacitance and inversion factor jointly.The interconnected SSDI (over 70%vibration reduction) generates larger damping than the interconnected SSDS (40% vibration reduction).

Fig.9 Case 3:Interconnected SSDI in a parallel manner,(F1 F2)= (10, 0).

If only one beam is excited,i.e.,(F1,F2)=(10,0),compared with the open-circuit state(blue solid line for beam 1,red solid line for beam 2), for the interconnected SSDI, the response amplitude of the excited beam decreases(blue line marked with circles),and meanwhile,the unexcited beam is activated by the force generated from the attached piezoelectric patch (red line marked with circles) for both mode 1B (Fig.12(a)) and 2B(Fig.12(b)).The resonance amplitude for beam 2 is still lower than beam 1.From the energy perspective, this phenomenon reveals that an electric channel coupling these two beams is established, where the vibration energy flows from the excited beam to the unexcited one.

Fig.10 Case 4:Interconnected SSDI in a parallel manner,(F1 F2 φ) = (10, 10, 5π/6).

Another typical case is that the excitation amplitudes for different substructures are the same,but there is a phase difference between substructures, usually encountered in the nodaldiameter vibration of bladed disks.The performance of the interconnected SSDI depends on the phase difference.Taking the first mode as an example(Fig.13(a)),the optimal damping effect is achieved when φ=π for the interconnected SSDI in a parallel manner (around 75% vibration reduction), and when φ=0 or 2π, the damping is zero.Concerning the interconnected SSD in a cross manner (Fig.13(b)), the trend is opposite, i.e., the optimal damping is realized when the phase difference is 0, and there is no damping effect when the phase difference is π.

We can also find that the resonant amplitudes of beam 1 and beam 2 are different when the phase difference is not 0 or π (Fig.13).In more details, the frequency response curves under different phase conditions for the interconnected SSDI in a parallel manner are shown in Fig.14.In addition,two resonant peaks even appear for φ=5π/6 (Fig.14(c)) and φ=7π/6 (Fig.14(e)).

Fig.11 FRFs of dual cantilever beam vibrating out of phase with the same amplitude under different electric boundary conditions,(F1 F2 φ) = (10, 10, π).

Fig.12 Comparison of FRF at open-circuit state and interconnected SSDI: only one beam is under excitation, (F1 F2) = (10, 0).

Fig.13 Relationship between resonant amplitude and phase difference (φ) for the first mode: comparison of interconnected SSDI and open-circuit state.

This phenomenon can be explained from the modal aspect.We can solve directly the eigenvalue problem of Eq.(18) to obtain the modal information of the system with the interconnected SSDI.For the first mode, two beams vibrate In Phase(IP) following 1B modal shape (Fig.15(a)).The modal frequency is 8.8 Hz, and no electric charge is generated.For the second mode, two beams vibrate Out of Phase (OOP) following 1B modal shape (Fig.15(b)).The modal frequency is 9.7 Hz, and an electric charge exists.The natural frequencies of these two modes are close to each other.Therefore, in the frequency band that we are interested in, both of the two modes are possible to be excited.When the two beams are excited with a phase difference of π, only the second mode(1B-OOP) of the system will be excited in the resonance zone,so that charges are produced and thus the damping effect is generated.On the contrary, if the phase difference is 0, only the first mode (1B-IP) will be excited.In this case, no charge is generated, that is why the piezoelectric damping is zero.When the phase difference is neither 0 nor π, both of the two modes participate in the motion.Taking φ=5π/6 as an example, we can see that the second peak on the red curve appears owing to the participation of the second mode.The amplitude of beam 2 is lower than that of beam 1 due to the superposition of the IP and OOP modes.

Fig.14 FRFs under different phase conditions for interconnected SSDI in a parallel manner.

Fig.15 Modal shapes of dual piezoelectric cantilever beam interconnected with SSD.

This double resonant peaks phenomenon can also occur in the cases where the double beams vibrate at different levels.The frequency response curves are shown in Fig.16.When(F1F2φ)=(10,10,π),there is one resonance peak for the second beam because only the corresponding mode is excited(Fig.16(e)).But in some other cases ((F1F2φ) = (10 5 π) in Fig.16(c)and(F1F2φ)=(10,7.5,π)in Fig.16(d)),the second peak appears.Similarly, the above phenomenon can also be explained by modal superposition.

As far as the influence of the dual-beam vibration level on the vibration reduction effect of the interconnected SSD is concerned, the relationship between the resonant amplitude of each beam and the ratio of the excitation level is displayed in Fig.17.When the ratio of the vibration levels is low, e.g.,F2/F1=0.1,the amplitude of beam 1 whose vibration is larger decreases compared with the open-circuit state, while the amplitude of beam 2 increases.As shown in cases ((F1F2φ)= (10, 0, π) in Fig.16(a) and (F1F2φ) = (10, 2.5, π) in Fig.16(b)), the interconnected SSD balances the vibration between the two beams.However, when F2/F1>0.3, the vibration of both the two beams interconnected with SSDI(green lines) is reduced compared with the open-circuit state(blue lines).We can also find that the damping effect of the interconnected SSDI is close to or even better than that of the shunted SSD (red lines) if F2/F1is larger than 0.6.

Sometimes due to the pasting process or the material properties, the electromechanical coupling capacity of the piezoelectric patch on each substructure will be diverse, which can be reflected in the electromechanically coupled matrix Kmpi.Supposing that there is 10% deviation of the electromechanically coupled matrix when two piezoelectric cantilevers are excited at the same amplitude and with the phase difference of 180°, the FRFs are shown in Fig.18.The impact of the electromechanical coupling deviation is: (A) both the resonant frequency and the amplitudes of the two beams interconnected with SSDI are different; (B) the damping effect of the interconnected SSDI is still slightly better than the shunted SSDI.

From the above numerical studies, the following are summarized:

(1) The correctness of the frequency domain method and the time domain method are crossly verified.

Fig.17 Relationship between resonant amplitude and excitation amplitude ratio (F2/F1) for the first mode: comparison of interconnected SSDI in a parallel manner, open-circuit state,and shunted SSDI.

Fig.16 FRFs with different excitation amplitude ratios: comparison of different electric boundary conditions.

(2) The damping generated by the interconnected SSDI is significant in a quite large range in terms of the phase difference in some cases, and the vibration reduction effect of the interconnected SSDI is even better than the shunted SSDI.

(3) There is a threshold for the ratio of vibration amplitudes.When the ratio is smaller than this threshold, the interconnected SSD plays a role in balancing the vibration levels of the two substructures; when the ratio is greater than the threshold, both substructures will be damped.

(4)The performance of the interconnected SSDI is not very sensitive to the small deviation of the electromechanically coupled matrix.

4.Experimental verification

Fig.18 FRFs of cantilever beam system with 10% deviation of electromechanically coupled matrix, (F1,F2,φ) = (10, 10, π):comparison of open-circuit state, shunted SSDI, and interconnected SSDI.

In this section,we first introduce the test bench.The hardware framework and the software logic of the newly developed selfsensing SSD control circuit are given in detail.Technological progress is also highlighted and summarized.Then test results are discussed, and the findings from the numerical simulation are validated.

4.1.Test device

4.1.1.SSD control board

The hardware framework of the newly developed SSD circuit is shown in Fig.19(a).The workflow of the SSD board is first to input the voltage signal directly from the electrodes of the piezoelectric patches.Then after the Analogto-Digital Conversion (ADC) process, the extremum of the scaled input signal is detected in a Micro Controller Unit(MCU).Next, the MCU commands the MOSFET (MOS)switch to close for a specific period, inducing the voltage inversion.

Fig.19 Schematic of self-sensing SSD control circuit.

The technological progress of the control board is presented as follows:

(1) The SSD is self-sensing, i.e., no additional input (e.g.,the displacement signal of the beam) is required to determine the extremum.The signal used for extremum detection is exactly the voltage on the piezoelectric patches for vibration reduction.The difficulty is that, first, the voltage on the SSD is beyond the limit of the MCU; second, when SSD works,the voltage is not sinusoidal anymore.The voltage oscillates in the L-C circuit when the switch is closed, resulting in an inevitable local extremum due to the unavoidable slight delay.To tackle the first problem,we use several Operational Amplifiers(OP-AMP)to scale the voltage signal proportionally,thus the waveform is kept and meanwhile the amplitude is acceptable.Concerning the second obstacle,we prevent false triggering by setting the silent time through the MCU, i.e., after the switch is triggered, the extreme value detection is no longer performed within the time of T/4.

(2)The voltage on the SSD can be observed or recorded by an oscilloscope or a data acquisition system without any interference.If we directly connect a voltmeter in parallel to the SSD to measure the voltage between the electrodes of the piezoelectric patches, the voltage on the piezoelectric patches will be smaller than the real value,and the observation voltage is not accurate.The reason is that the impedance of the measuring instrument is not infinite, the capacitance of the piezoelectric patch is relatively small, and there will be current flowing into the voltmeter.In order to ensure that the SSD voltage is observed without affecting the damping effect, the OP-AMP circuit is adopted to isolate the input signal.Finally,the voltage on the SSD can be measured on the corresponding pins.

(3)The reliability is improved.The board has low heat generation and has withstood the test of 150 h of continuous work.This is owing to the addition of the sampling process in the MCU data analysis, thus some redundant data are filtered especially for low-frequency vibration,and the computational cost is reduced.

Fig.20 Control board of self-sensing SSD.

(4) The control board is small in size (Fig.20), and only a 36 V DC power supply is needed.The software and the hardware communicate in the MCU.The logic of the software is displayed in Fig.19(b).Filtering,extreme value detection, switch closing time estimation, and false trigger prevention are implemented through digital signal processing in the MCU.The low-pass filtering is realized by the median filtering algorithm.Concerning the extremum detection,if the absolute value of the data point increases first and then decreases for 5 consecutive filtered data points, the local extremum will be identified.The switch closing time is the half period of the oscillation circuit consisting of the equivalent intrinsic capacitance of piezoelectric patches and the inductance for SSDI, and as short as possible for SSDS theoretically.However, due to measurement errors of the capacitance and the inductance, the calculated switch closing time will be inaccurate.Since the damping effect of the SSDI is sensitive to the switch closing time (around 100 μs), an interface for artificially inputting the closing time is added.

4.1.2.Experimental system

The experimental model consists of two aluminum cantilever beams bonded with piezoelectric patches near the clamped ends,which is consistent with the simulation model.The newly developed SSD board is interconnected between the two piezo patches(i.e.,+input of the SSD control board to+electrode of the piezo patch on beam 2, -input of the control board to + electrode of the piezo patch on beam 1).

A 36 V DC power supply is used to feed the SSD board.Each beam is excited by an electromagnetic actuator (JZ-10,Fmax=100 N) near the clamped end.Two sinusoidal signals with amplitude and phase relationships are generated by a signal generator with two output channels for the two actuators.The actuators input the amplified voltage from the corresponding amplifiers and generate the excitation force.Two laser displacement sensors (Panasonic HL-G108-S-J) are used to capture the vibration of the beams near the free ends.The output signals from laser displacement sensors are connected to the data acquisition device Oros-34.The schematic and photo of the experimental system are shown in Fig.21.

4.2.Test results

Fig.22 Time domain displacement of two beams interconnected with SSDI in parallel at resonances (experimental results): mode 1(8.95 Hz) vs.mode 2 (53.9 Hz), two beams vibrate out of phase with nearly the same amplitude.

The interconnected SSDI is effective for both the first and the second mode when two beams vibrate out of the phase with the same amplitude.As shown in Fig.22, the response amplitudes at the corresponding resonant frequencies for both the two beams drop by over 30% when the SSDI works.For the second mode, the damping effect is even better (38% vibration reduction).The vibration reduction ratio is smaller than the numerical estimation.This is normal for the following reasons: first, there is an inevitable delay for the switch triggering in practice, leading to the negative effect on damping; second, the electromechanical coupling capacities of piezoelectric structure deviate from the nominal value due to material properties, processing, and pasting.In Section 4.3, we will calibrate the numerical model according to the experimental results.

Fig.23 Comparison of resonant amplitude reduction ratio(experimental results): interconnected SSDI vs.interconnected SSDS vs.shunted SSDI, two beams vibrate out of phase with nearly the same amplitude for mode 1.

The performance of the interconnected SSDI is much more effective than the interconnected SSDS, and slightly better than the shunted SSDI when two beams vibrate out of the phase with the same amplitude.As displayed in Fig.23, the resonant peak of the first mode drops by around 31%, 8%,and 23% for the interconnected SSDI, interconnected SSDS,and shunted SSDI respectively.We can use the voltage signal to explain this result.The voltage on the interconnected SSDI(green dotted line in Fig.24(a))is the difference of the voltage on each piezo patch (red solid line for the voltage on piezo patch 1 and blue dashed line for the voltage on piezo patch 2 in Fig.24(a)).Therefore, the maximal absolute value of the voltage on the SSDI control board for the interconnected is larger than the shunted one(Fig.24(c)),and thus larger damping is generated.The voltage inversion appears when the switch closes for the interconnected SSDI owing to the inductance, and on the contrary, there is no voltage lifting for the interconnected SSDS(Fig.24(b)).It can be seen that the maximum voltage of the interconnected SSDS is also lower than the interconnected SSDI.The above phenomenon indicates that the dissipated energy for the interconnected SSDS is lower.

The damping effect of the interconnected SSDI in a parallel manner reaches the best when two beams vibrate out of phase and decreases with the deviation of the phase difference from 180°.This trend is consistent with the numerical results.As shown in Fig.25 and Fig.26, setting the open-circuit state as reference, the resonance drop for the interconnected SSDI at φ=90ois only 12.5%.When the phase difference changes within the range of [150°, 210°], the attenuation of the vibration reduction effect is not obvious.

From the perspective of voltage (Fig.27), the shift of the phase difference causes the voltage amplitude on the SSD board(green dotted line)to drop,which is positively correlated with the deviation of the phase difference from 180°.Concerning the voltage on the electrodes of each piezoelectric patch(red line for piezo patch 1 and blue dashed line for piezo patch 2), the deviation of the phase difference induces that the voltage on each piezo patch is not inverse at the time when the voltage on the SSD board reaches the extremum, hence the applied voltage and the deformation rate of the underlying beam are not synchronous.This phase advance (piezo 1) or lag (piezo 2) is the reason for the decrease in damping effect.

Fig.24 Time domain voltage at the 1st resonant frequency(experimental results) for interconnected SSDI, interconnected SSDS, and shunted SSDI: two beams vibrate out of phase with nearly the same amplitude.

For the case where the vibration amplitudes of the two beams are different, first, a limit case is chosen in the experimental study.Only beam 1 is under excitation, whereas beam 2 is free.As demonstrated in Figs.28(a) and (b), when the interconnected SSDI works, the amplitude of beam 1 drops and beam 2 begins to vibrate.The coupling provided by the interconnected SSD balances the response of the two beams.In contrast, when the amplitude ratio is large (F2/F1=0.5),the vibration levels of both beam1 and beam 2 decrease(Figs.28(c)and(d)).Therefore,there must be a specific amplitude ratio dividing the function of the interconnected SSD into’higher one decreases, lower one rises’, and ’both beams decline’.The reason for the above phenomenon is the coexistence of the energy balance mechanism and the damping mechanism of the interconnected SSD.The dominant mechanism determines the behavior of the interconnected SSD.Concerning the mistuning of bladed disks,the amplitude of each blade will be different,and the vibration will be concentrated on certain blades.Regardless of the above mechanisms,it is expected to suppress the response amplification phenomenon of bladed disks by interconnected SSD.

The tests on the interconnected SSD in a cross manner were also carried out, the results are similar to those in the parallel manner except for the phase lag.For the sake of conciseness,they are not displayed in this paper.

4.3.Model calibration

The experimental results verify qualitatively the numerical results in Section 3, but quantitatively, the differences are not negligible.The reason is that the model parameters used in Section 3 are nominal values and have not been corrected.For example, the material properties of piezo patches, the structural damping ratio and the voltage inverse factor γ are potential error sources.The purpose of Section 3 is to reveal the mechanism of the interconnected SSD and to clarify how the amplitude and phase difference affect the damping.In this section, we focus on whether the numerical method can precisely predict the dynamic of SSD damped structures after model calibration.Therefore, we first calibrate the finite element model according to experiment data, then calculate the steady-state response, and quantitatively compare the results with the experiments.

In the case where the two beams vibrate out of phase with the same amplitude, by setting the experimental results in the open-circuit state as a reference, the structural damping ratio is corrected to ξ=1.54%.According to the time domain voltage in experiments, we calibrate the inverse factor γ = 0.56 and 0.67 at the first and the second resonance, respectively.Based on the multimeter measurement, the capacitance of the piezo patch is corrected to 76 nF, which is almost double the nominal value.The modified material properties of the piezo patches are listed in Appendix for more details.

After model calibration, simulation and experimental results show good agreement when the SSD is connected(Fig.29(a) and Fig.29(b)).The relative error is less than 5.4%.The resonant peak of the first mode drops by around 29.8% and 23.5% for the interconnected SSDI and shunted SSDI, respectively.The resonant peak of the second mode drops by around 38.1% for the interconnected SSDI.All damping performance above is in accordance with the test results in Section 4.2.

In summary, the experimental and numerical results verify each other quantitatively after model calibration.If we use the impedance with a higher quality factor and reduce the resistance in the control board, better performance will be expected.

Fig.25 FRFs of interconnected SSDI for mode 1 (experimental results): two beams vibrate with different phases and nearly the same amplitude.

Fig.26 Relationship between resonant amplitude and phase difference for interconnected SSDI (experimental results): two beams vibrate with nearly the same amplitude for mode 1.

5.Application

In this section, a typical dummy bladed disk finite element model is used as an example to estimate the damping performance of the proposed technology and to illustrate the feasibility in engineering quantitatively.

The bladed disk is modeled with NASA-ROTOR 37 profile(Fig.30(a)) and there are overall 36 sectors.The engine order excitation is applied at the middle of the blade tips on the blade basin(red dot in Fig.30(b)).The displacement at the observation point (yellow dot in Fig.30(b)) is taken to estimate the vibration level of the bladed disk.The inner diameter is fully constrained.

The bladed disk is modeled by SOLID 45 elements,and the material is steel with Young’s modulus 280 GPa, Poisson’s ratio 0.3, mass density 7600 kg/m3, and structural damping ratio 0.87%.The PZT-5H piezoelectric patches are modeled by SOLID 5 elements.The inverse factor is set to 0.6.

The full-circle FE model has around 107DOFs.A Reduced-Order Modeling (ROM) is indispensable to accelerate the simulation.The cyclic reduction technique cannot be used in this case because the electrical circuit breaks the cyclic periodicity.In this study,the Craig-Bampton(CB)modal synthesis is adopted in each sector, and therefore the inner DOFs are replaced by truncated modal coordinates.To reduce the dimension of the inter-sector boundaries,we project the implicated motion to the basis composed of the interface modes proposed in Ref.34and Ref.35Thanks to these ROM techniques, more than 99.9% of the DOFs are reduced.The number of DOFs of the FE model is reduced to 796,which consists of two parts: (A) 1 voltage DOF and 20 modal DOFs in each sector; (B) 40 interface modal DOFs.

The size, shape, and position of piezoelectric patches are determined by the topological optimization method,36aimed at maximizing the Modal Electromechanical Coupling Factor(MEMCF) under the constraint of 10% added mass of each blade.This method is based on the following: (A) the optimized piezo damping performance solely depends on the MEMCF7; (B) the piezoelectric material should be placed where the local weighted strain (i.e., the absolute value of the overall equivalent electric field) reaches the maximum.During the topological optimization, the elements are sorted by the weighted strain in descending order and replaced by the piezoelectric material until the piezo material mass ratio exceeds the predetermined threshold.

The modal frequency versus Nodal Diameter Index (NDI)diagram of the bladed disk at the open-circuit state is displayed in Fig.31(a), where only the first four modal groups are plotted.We set the first-order mode with NDI 6 as the target(Fig.31(b)).Following the topological optimization procedure, the piezoelectric material is distributed to the blade root(red area in Fig.30(b)), and the corresponding MEMCF is 0.078.

The topology of the interconnected SSD is shown in Fig.32, which is determined by the target modal shape (i.e.,the phase of each blade).For this mode, the inter-blade phase is π/3.We divide the blades into three groups.In Group I,sectors 1,7,...,31 vibrate in one phase, while sectors 4,10,...,34 vibrate in another phase, with a difference of π.In Group I,first, the sectors that vibrate in the same phase are connected in parallel separately to form two subnetworks.Then, one SSD is interconnected between the two subnetworks.The connection is similar for Groups II and III.Finally, only 3 SSD boards are used.

Fig.27 Time domain voltage at the first resonant frequency for interconnected SSDI(experimental results),two beams vibrate with the same amplitude: comparison of phase difference.

Fig.28 Time domain displacement of two beams interconnected with SSDI at the first resonance(experimental results):comparison of different amplitude ratios.

Fig.29 FRFs of two beams vibrating out of phase with the same amplitude after model calibration by numerical simulation:comparison of different electric boundary conditions.

Fig.30 FE model.

Fig.31 Modal information of blade disk embedded with piezoelectric patches at open-circuit state.

Setting the open-circuit state as a reference, it can be observed in Fig.33 that the resonant peak drops by around 94.2% for the interconnected SSDI according to the steadystate response analysis, and the corresponding piezoelectric damping ratio estimated by the half-power method is 13.68%.The vibration suppression is impressive.

Fig.32 Scheme of interconnected SSD on bladed disk.

Fig.33 FRFs (peak-to-peak) of bladed disk vibrating in the targeting mode (NDI 6, mode group 1) frequency (with logarithmic ordinate): comparison of open-circuit state, shunted SSDI,and interconnected SSDI.

6.Conclusions

In this paper,we propose an interconnected SSD for the vibration suppression of blades.Through numerical simulation and experiments, the damping performance of this new strategy under different working conditions is analyzed in depth.It is preliminarily proved that the interconnected SSD can be used to reduce the vibration level of blades under variable working conditions with different engine orders.This technology is friction interface free, and the space constraint of the interacting blade pairs can be overcome.

A novel SSD control board with a self-sensing function is developed.The voltage on the SSD interconnected between the electrodes of the piezoelectric patches can be directly used to realize the extremum detection, avoiding the use of additional channels to record the structural displacement.Another essential technical progress is that the damping effect will not be interfered with when the voltage signal is under observation.

The main conclusions are drawn as follows:

(1) The damping effect of the interconnected SSD depends on the phase difference between the two blades.If they vibrate out of phase/in phase with the same amplitude, the damping effect of the interconnected SSDI in a parallel manner/cross manner is even better than that of the shunted SSDI.It is effective for multiple modes, and the resonant amplitude can be reduced by over 30%.As the phase difference moves away from the optimal value, the damping effect gradually decreases.

(2) The interconnected SSD can establish an additional energy channel between spatially non-adjacent blades, which can balance their vibration levels.The damping mechanism and energy balance mechanism coexist.When the vibration levels of the two blades differ greatly,the energy balance mechanism is dominating; when the difference in vibration level is smaller, the damping mechanism plays a major role.

(3) The interconnected SSD is not sensitive to the slight deviation of substructures.The electromechanical coupling capacities of the two cantilever beams in the experiment are not completely the same due to the pasting process of piezoelectric materials, but the influence on the damping effect is not significant.

(4)A piezoelectric damping ratio of 13.7%is achieved for a typical FE model of the bladed disk when the amount of piezo material is only 10% of blade mass, indicating a promising prospect in engineering.

Future work is dedicated to the numerical and experimental studies of the interconnected SSD on mistuned bladed disks.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was funded by the Major Projects of Aero-Engines and Gas Turbines,China (Nos.J2019-IV-0005-0073 and J2019-IV-0023-0091), the Aeronautical Science Foundation of China (No.2019ZB051002), China Postdoctoral Science Foundation (No.2021M700326), and the Advanced Jet Propulsion Creativity Center,China (Nos.HKCX2020-02-013, HKCX2020-02-016 and HKCX2022-01-009).

Appendix.The modified material properties of piezo patches are:

The direction of the above matrix is determined according to IEEE standard on piezoelectricity, 1988: 1-x, 2-y, 3-z, 4-yz, 5-xz, 6-xy.