This paper extends the resonance frequency detuning (RFD) vibration reduction approach to cases of turbomachinery blade mistuning. Using a lumped parameter mistuned blade model with included piezoelectric elements, this paper presents an analytical solution of the blade vibration in response to frequency sweep excitation; direct numerical integration confirms the accuracy of this solution. A Monte Carlo statistical analysis provides insight regarding vibration reduction performance over a range of parameters of interest such as the degree of blade mistuning, linear excitation sweep rate, inherent damping ratio, and the difference between the open-circuit (OC) and short-circuit (SC) stiffness states. RFD reduces vibration across all degrees of blade mistuning as well as the entire range of sweep rates tested. Detuning also maximizes vibration reduction performance when applied to systems with low inherent damping and large electromechanical coupling.

## Introduction

Resonance frequency detuning is a transient approach that detunes the response from the excitation to decrease vibration. It relies on a stiffness perturbation induced by altering the electrical boundary conditions of piezoelectric elements at some optimal time as the excitation frequency of a certain engine order sweeps through a resonance crossing, where optimal is defined as the switch time that minimizes the blades' maximum vibration amplitude. By locating these piezoelectric elements in regions of high modal strain within a system, switching between the open-circuit (OC) and the short-circuit (SC) configuration induces a nearly instantaneous global change in system stiffness from a high-stiffness state to a low-stiffness state and vice versa. References [8] and [9] provide a more thorough explanation of the RFD concept. The previous analysis toward a single-degree-of-freedom (SDOF) system shows that RFD performance is a function of the excitation sweep rate, damping, piezoelectric electromechanical coupling, and, most importantly, the time of the stiffness perturbation, otherwise known as the switch trigger. The authors' previous study of a SDOF system identified the piezoelectric coupling coefficient as the most prominent parameter influencing the optimal switch trigger [9]. A follow-up study extended the analysis to a system where the SDOF assumption no longer holds; such a system included two oscillators with included piezoelectric elements and the masses lightly coupled resulting in two closely spaced modes [10]. Results indicate that RFD performance suffers when an open-circuit resonance peak and an adjacent short-circuit resonance peak overlap. This paper builds on these previous studies by extending RFD analysis toward the particular issue of blade mistuning.

The paper is organized as follows: Sec. 2 presents the necessary background associated with piezoelectric materials and their utilization in vibration reduction. A subsequent discussion outlines some potential drawbacks of the previous piezoelectric-based vibration reduction methods in a turbomachinery application as well as the alleviation of these drawbacks offered by RFD. Section 3 presents a lumped parameter blade mistuning model and the derivation of the associated equations of motion in nondimensional form. It provides an analytical approach to solve these equations of motion subjected to frequency sweep excitation and direct numerical integration provides validation. Section 4 presents the results of this analysis, including the methodology for determining the optimal switch trigger. It also presents Monte Carlo simulations to investigate to what extent the degree of blade mistuning, sweep rate, damping, and the difference between stiffness states affects RFD performance. Section 5 offers concluding remarks as well as future research directions.

## Background

Researchers have exploited the electromechanical coupling of piezoelectric materials for use in sensors, actuators, vibration energy harvesters, and vibration reduction applications. The vibration reduction approaches can be categorized broadly into one of three realms: passive, semi-active, and active. In these approaches, a surface-mounted or embedded piezoelectric element transforms the mechanical stress induced from free or forced vibrations into the electrical domain. Attaching a shunt circuit across the piezoelectric electrodes allows manipulation of this transformed electrical energy to provide the vibration reduction.

For the passive approaches, passive circuit elements in the attached shunt dissipate the energy in the electrical domain by means of a resistor (R-shunt) or resistor and inductor (RL-shunt) that, in conjunction with the capacitance of the piezoelectric element, forms a resonant circuit [11,12]. These passive methods require tuning of the circuit parameters to target a specific vibration frequency; the resonant shunt has greater vibration damping performance over the resistive shunt, but at the expense of operating bandwidth.

Recently, researchers developed resonant shunted piezoelectric systems with application toward mistuned blisk assemblies that show promising vibration reduction performance [1316]. Due to the limited operating bandwidth of these resonant shunts, performance of these approaches can be sensitive to frequency drift arising from environmental effects and foreign object damage over the lifespan of the blisk assembly. Variations in the mistuning pattern may also arise from the “blending” repair process [17]. Such variations in the mistuning pattern over time require retuning the circuit parameters to the altered target frequency; developments of adaptive control strategies show promise in addressing this issue [16]. Furthermore, targeting multiple resonance crossings can quickly increase the complexity of the shunt circuit, which raises implementation concerns due to the stringent size constraints associated with a turbomachinery environment. For example, each additional resonance crossing would require a separate current blocking or current flowing circuit branch containing circuit elements tuned to the specific targeted frequency [18,19]. Additional drawbacks include the large inductance requirements for targeting lower vibration frequencies; synthetic inductors that utilize digital signal processors or operational amplifiers to increase the effective inductance of a circuit are available, but require power to operate [20,21].

Semi-active approaches address some of these aforementioned drawbacks associated with passive approaches by extending the operating bandwidth and reducing the inductance requirements for piezoelectric-based vibration reduction. These approaches are semi-active in the sense that power is needed only to open and close a switch in an otherwise passive shunt circuit. In a common approach, this switch is open as the system moves away from equilibrium and closed at an extremum point. The duration the switch remains closed, as well as the circuit elements included in the shunt, depend on the specific approach employed [2224]. These methods require four switches per vibration cycle and power to control the switch. Conceivably, targeting transient passages of resonance in the kHz range would require thousands of switches. Although these methods do not require tuning passive circuit elements, a tuning in the time domain is necessary and requires additional knowledge of the local phase of vibration such that the switches occur at the optimal point in the vibration cycle to achieve maximum vibration reduction.

The RFD approach seeks to simplify piezoelectric-based vibration reduction approaches for application toward a turbomachinery environment. RFD requires only two switches per resonance crossing: one switch to detune the response from the excitation and reduce vibration, and one switch to revert to the initial stiffness state following the resonance crossing. The switch trigger can also occur at any point in the vibration cycle, although switching at the point of peak kinetic energy (i.e., zero displacement) can alleviate concerns regarding exciting transients of other vibration modes upon switching from a high- to low-stiffness state at the point of peak strain energy (i.e., displacement extremum) [25]. Finally, RFD utilizes a switch in the shunt circuit rather than passive circuit elements, thus enabling the approach to target multiple resonance crossings; however, the timing of the optimal switch trigger does depend on the electromechanical coupling present for the targeted mode of vibration, so a time domain tuning of this switch trigger for each mode may be necessary. To address this tuning issue, the authors proposed a control strategy utilizing a measurement of the time-varying piezoelectric voltage to apply the switch trigger [26].

The use of piezoelectric materials in a turbomachinery application poses several implementation challenges. These materials lose their piezoelectric properties at temperatures greater than their Curie temperature. In general, higher-temperature materials have lower coupling, creating a tradeoff between higher temperature limits (and cooling requirements) and electromechanical coupling [27]. Therefore, the current technology restricts these systems to target vibration located in the fan blades and cold-side of the compressor where, coincidentally, high-cycle fatigue is most problematic [28]. These systems also require an implementation that does not disturb the airflow around the blade or compromise the blades' structural integrity. Locating the piezoelectric element on the disk rather than the blades has shown some promise [15,16]. Another possible implementation involves coating the fibers of a composite laminate in regions of high modal strain with piezoelectric material to create a network of internal “patches” to reduce vibration in a targeted mode [29]. Such an approach seems increasingly promising with the introduction of composite fan and even compressor blades [30].

A lumped parameter model serves as a simplified representation of a blisk assembly capable of qualitatively capturing the mistuning phenomenon, thus providing an analytical tool to investigate RFD performance. This model consists of N blades subjected to engine order C excitation, where a SDOF oscillator represents the rth blade with mass m, damping coefficient c, stiffness k, and excitation f. Placing the piezoelectric elements p in parallel with the stiffness and damping elements in each oscillator represents a blade-mounted system and creates the desired electromechanical coupling. Finally, a stiffness component kc provides the interblade coupling between adjacent blades. Figure 1 illustrates this lumped parameter model. Assuming a constant mass, damping, coupling stiffness, and identical piezoelectric elements, varying the blade stiffness generates the mistuned system.

### Equations of Motion.

By balancing the forces on the rth blade, the coupled electromechanical equations of motion for an N-blade system are
$mx¨r+cx˙r−kcxr−1+(kr+kp+2kc)xn−kcxr+1+θvr=fr$
(1)

$−θxr+Cpvr=qr, ∀r=1,2,…,N$
(2)
where kp is the stiffness of the piezoelectric element in the short-circuit state, θ is the electromechanical coupling term, and Cp is the piezoelectric capacitance. The assumption that the piezoelectric elements located on each blade are identical leads to constant values for these parameters. Also, vr is the piezoelectric voltage and qr is the generated charge for the piezoelectric element located on the rth blade. For a tuned system, the nominal blade and the piezoelectric stiffness in the short-circuit state (i.e., vr = 0) are k0 and kp; the total blade short-circuit stiffness is then ksc,0 = k0 + kp. The stiffness variation of the rth blade arises through inclusion of the term δr, which is a random sample of a normal distribution with zero mean and standard deviation σδ. Thus, the mistuned system stiffness in the short-circuit state is
$ksc,r=ksc,0(1+δr)$
(3)
In the open-circuit state (i.e., qr = 0), Eq. (1) becomes
$mx¨r+cx˙r−kcxr−1+(ksc,r+θ2Cp+2kc)xr−kcxr+1=fr$
(4)
which means the blade stiffness in the open-circuit state is
$koc,r=ksc,r+θ2Cp$
(5)
The difference between the open- and short-circuit stiffness is defined as
$Δk=koc,r−ksc,r=θ2Cp$
(6)
and is constant for each blade. The equations of motion written in purely mechanical form are
$mx¨r+cx˙r−kcxr−1+[[koc,r−(1−s0)Δk]+2kc]xr−kcxr+1=fr$
(7)
where s0 is a binary number based on the stiffness state of the piezoelectric element such that
$s0={0,short-circuit state 1,open-circuit state$
(8)

To simplify this analysis, the switches for each blade occur simultaneously. A more optimal approach may be to switch each blade independently, thus causing more complexity in the associated dynamics. For instance, independent switching will lead to additional variations in blade stiffness, and consequently, will alter the dynamic response. Also note that this analysis assumes that only one switch is made from the open-circuit to the short-circuit state. If a switch was conversely made from the short- to the open-circuit state, a generated charge would appear across the piezoelectric element with a magnitude that depends on the mechanical displacement at the time of the switch. A constant forcing term would then appear on the right-hand side of Eq. (7) but is beyond the scope of this analysis.

Furthermore, RFD reduces vibration associated with engine order excitation and resonance passages; it is inherently a transient approach. As such, a swept sinusoidal provides the excitation with constant amplitude F0 and a phase at the rth blade
$φr(t)=ωrate2t2+ω0t+ψr$
(9)
where ωrate is the linear frequency sweep rate, ω0 is the initial excitation frequency, and ψr is the constant portion of the phase at the rth blade given as
$ψr=2πC(r−1)N$
(10)
which is an integer multiple (r − 1) of the interblade phase angle.

### Nondimensional Equations of Motion.

Casting the equations of motion in nondimensional form generalizes the analysis in preparation of a parametric study. Using the parameters in the open-circuit state, the nondimensionalized time and displacement are
$t¯=koc,0mt=ωoct and x¯r=koc,0F0xr$
(11)
The damping ratio is also defined as
$ζ0=c2koc,0m$
(12)
The nondimensional equations of motion are then
$x¯′′r+2ζ0x¯′r−k¯cx¯r−1+[[1−(1−s0)Δk¯]+(1−Δk¯)δr+2k¯c]x¯r−k¯cx¯r+1=f¯r(t¯)$
(13)
where $( )′$ indicates a derivative with respect to the nondimensional time and an overbar indicates a variable in nondimensional form. Also, the phase of excitation in terms of the nondimensional variables is
$φ¯r=12ω¯ratet¯2+ω¯0t¯+ψr$
(14)
and the nondimensional sweep rate $ω¯rate$ and the initial operating frequency $ω¯0$ are
$ω¯rate=ωrate(ωoc,0)2 and ω¯0=ω0ωoc,0$
(15)
Written in matrix form, the equations are
$Ix¯″+2ζ0Ix¯′+[K¯oc−(1−s0)Δk¯I]x¯=F¯(t¯)$
(16)

where the first term in the stiffness matrix corresponds to the mistuned open-circuit stiffness matrix $K¯oc$ and includes the stiffness perturbations due to mistuning. The second term incorporates the stiffness perturbations resulting from a switch between the open- and short-circuit states, where I is the N × N identity matrix.

### Modal Equations of Motion.

A solution for Eq. (16) is now desired. Direct numerical integration schemes will lead to large computation times as the number of blades increases or frequency sweep rate decreases, thus warranting an alternate approach. Markert and Seidler previously derived an analytical solution for a SDOF system subjected to frequency sweep excitation [31]. Application of this solution requires a transformation of Eq. (16) into the modal coordinate system using the matrix of mode shapes $Φ$ calculated for the system in the open-circuit state (i.e., s0 = 1)
$x¯=Φp$
(17)
where p is the vector of modal coordinates. The modal equations of motion are then
$Ip″+2ζ0Ip′+[Ωoc−(1−s0)Δk¯I]p=G$
(18)
where $Ωoc$ is a diagonal matrix whose nonzero elements correspond to the nondimensional modal frequencies in the open-circuit state $(ωoc,r)2$ and G is the modal forcing vector. The equation of motion for the rth mode is then
$pr″+2ζ0pr′+[(ωoc,r)2−(1−s0)Δk¯]pr=gr$
(19)
In the short-circuit state (i.e., s0 = 0), the modal equation is
$pr″+2ζ0pr′+[(ωoc,r)2−Δk¯]︸(ωsc,r)2pr=gr$
(20)
The effective piezoelectric coupling factor $keff,r2$ for the rth mode can also be found
$keff,r2=(ωoc,r)2−(ωsc,r)2(ωoc,r)2=Δk¯(ωoc,r)2$
(21)
and the short-circuit natural frequency is
$ωsc,r=ωoc,r1−keff,r2$
(22)
Compactly, the modal equation of motion is
$pr″+2ζ0pr′+ωr2pr=gr$
(23)

where ωr is the rth modal frequency, depending on the operating state (open-circuit or short-circuit) of the piezoelectric elements.

Furthermore, the rth component of the modal forcing vector is
$gr=∑s=1Nϕs,rf¯s$
(24)
From examination of the above equation, gr is a linear combination of N swept sinusoidal waves with various amplitudes and phases. An expression for this combination of sinusoidal waves as a single wave with amplitude G0,r and phase λr is
$gr=G0,r sin(12ω¯ratet¯2+ω¯0t¯+λr)$
(25)
where
$G0,r=[∑s=1Nϕs,r cos(ψs)]2+[∑s=1Nϕs,r sin(ψs)]2$
(26)

$λr=tan−1[∑s=1Nϕs,r sin(ψs)∑s=1Nϕs,r cos(ψs)]$
(27)

### Analytical Response for Swept Excitation.

To stay consistent with the derivation in Ref. [31], set Eq. (19) into a similar form by scaling the already nondimensional time such that
$t̂r=ωr(t¯−t¯0)$
(28)
where $t¯0$ is the initial time of operation in a stiffness state. For example, $t¯0=0$ at start-up with operation in the open-circuit state, and $t¯0=t¯sw$ after the switch to the short-circuit state. The phase of excitation in scaled form is then
$φ̂r=12ω̂rate,rt̂r2+ω̂0,rt̂r+λ̂r$
(29)
where
$ω̂rate,r=ω¯rateωr2$
(30)

$ω̂0,r=ω¯ratet¯0+ω¯0ωr$
(31)

$λ̂r=(12ω¯ratet¯02+ω¯0t¯0+λr)$
(32)
Finally, the scaled modal displacement and damping ratio are
$p̂r=ωr2Grpr and ζ̂r=1ωrζ0$
(33)
The resulting scaled modal equation of motion is
$p̂°°r+2ζ̂rp̂°r+p̂r=sin φ̂(t̂r)$
(34)
Note here that $(°)$ is a derivative with respect to the scaled time $t̂r$. The scaled modal response is then
$p̂r=|Qr(t̂r)| sin [φ̂r(t̂r)+βr(t̂r)]$
(35)

where $Qr(t̂r)$ is the time-dependent complex amplitude and $βr(t̂r)$ is the time-dependent phase difference between the response and the excitation. Markert and Seidler provide a full derivation and explanation of the terms comprising this complex amplitude, which is omitted here for brevity [31].

### Analytical Solution Validation.

The analytical solution was validated using the fourth-order Runge–Kutta (here, Matlab's ode45) numerical integration to solve Eq. (16). This approach requires recasting Eq. (16) in first-order form. To this end, choosing the displacement and the velocity of each mass as the states produces the state vector
$y={xx′}$
(36)
The first-order equations of motion are then
$y′=Ay+B$
(37)
where
$A=[0I−[K¯oc−(1−s0)Δk¯I]−2ζ0I],B={0F¯}$
(38)

The example system consists of N = 13 blades subjected to engine order C = 3 excitation with sweep rate $ω¯rate=1×10−4$, damping ζ0 = 0.001, and coupling stiffness $k¯c=0.02$. The mistuning for this specific 13-blade system arises from the stiffness variation with a standard deviation σδ = 0.03. An arbitrary switch trigger ωsw = 0.993 initiates the stiffness state switch. Additionally, the difference between stiffness states is $Δk¯=0.05$, which would correspond to $keff2=5%$ for a single blade decoupled from the system. To the authors' knowledge, a full blade with embedded piezoelectric elements has yet to be built so this piezoelectric coupling factor represents an estimate of what is to be expected. However, a previous study idealized a blade as a flat trapezoidal plate with a surface bonded piezoelectric patch; that plate exhibited coupling factors of $keff2=8.86%$ for the two-stripe mode and $keff2=6.91%$ for the third bending mode, so a value of $keff2=5%$ is likely attainable for many modes [8].

Figure 2 displays the responses (normalized by the open-circuit maximum peak) for the first four blades as a function of the nondimensional excitation frequency; the analytical oscillatory response and response magnitude obtained from ode45 are both shown. The ode45 response magnitude envelopes the analytical response very well, lending confidence to the accuracy of the analytical approach as adapted here.

## Performance Analysis of Resonance Frequency Detuning

Since turbomachinery blade mistuning can originate during the manufacturing process or due to wear of the blades over the service life of the engine, the variation between blades is an inherently random process. A Monte Carlo study facilitates estimating the performance statistics associated with the RFD vibration reduction approach. This analysis also employs the system model parameters presented in the example from Sec. 3.5. Furthermore, RFD relies on an appropriate selection of the frequency-based stiffness switch trigger to achieve the maximum vibration reduction performance for any given system.

### Resonance Frequency Detuning Optimal Switch Trigger.

A previous study showed the optimal switch trigger for a SDOF system was primarily a function of the electromechanical coupling [9]. For systems with multiple modes influencing the response, this switch trigger may no longer be optimal; however, the methodology for obtaining the switch still applies. First, a brute force approach tests a range of various switches and the maximum peak response magnitudes for each blade are calculated. The switch is optimal if it minimizes the maximum response peak of the entire set of blades (i.e., the lowest “worst-case” blade vibration for a particular system), although an alternate performance metric may be to minimize the peak RMS magnitude of the entire set. For example, consider the N = 13 blade system previously described with σδ = 0.03 and subject to engine order C = 3 excitation. (Note here that this is one specific mistuned system included to visualize the optimal switch trigger determination process.) Figure 3(a) shows the maximum peak blade response normalized by the maximum peak response in the mistuned open-circuit configuration. For an uncoupled SDOF system, the optimal switch for a coupling coefficient of $keff2=5%$ occurs at ωSDOF = 0.993, but that switch trigger does not provide the optimal peak vibration reduction in the mistuned case. The optimal switch trigger for this mistuned system is actually ωopt = 1.012 (21% vibration reduction compared to 10% at ωSDOF = 0.993). Note that for a delayed switch application such that ωsw > 1.06, the system operates in the open-circuit state for the entirety of the resonance passage, resulting in the open-circuit peak magnitude; for early switch application such that ωsw < 0.9, the system operates in the short-circuit state for the entirety of the resonance passage, resulting in the short-circuit peak magnitude. Also note that for some nonoptimal switch triggers, the maximum peak response magnitudes may even be greater than the untreated open-circuit case.

Furthermore, Fig. 3(b) shows the maximum peak magnitudes for each blade in the system normalized by the peak blade magnitude in the baseline, open-circuit configuration. For the baseline case, the first blade has the largest peak magnitude. For a switch trigger occurring at ωSDOF = 0.993, the peak response decreases for nearly all the blades, including the first blade. For the optimal switch trigger occurring at ωopt = 1.012, there is ample vibration reduction for the first blade, as well as a further reduction in the overall curve. To further present the effectiveness of the current approach, Fig. 3(c) shows the normalized RMS response magnitudes for the entire set of N = 13 blades taken at each time instant. Clearly, the SDOF switch trigger provides some vibration reduction, but the optimal switch trigger produces the greatest overall performance.

### Monte Carlo Simulations.

In this section, Monte Carlo simulations provide insight toward the effects that various parameters have on RFD performance. Such parameters include the degree of mistuning σδ, the sweep rate $ω¯rate$, the damping ratio ζ0, and the difference in stiffness states $Δk¯$. For each set of parameters, obtaining a representative distribution of data required testing 500 separate systems. For each run, the optimal switch ωopt determination follows the process outlined in Sec. 4.1. The 99th percentile amplification factor provides a measure of RFD performance. Calculation of this metric involves identifying the maximum blade response for the entire set of blades and normalizing this magnitude with respect to the maximum peak for the tuned system in the open-circuit state. The 99th percentile amplification factor is then the normalized magnitude above which only 1% of the data set lies. Analyzing 500 systems may not provide enough samples to obtain an accurate value of the 99th percentile amplification factor; however, the maximum blade response will tend toward a Weibull distribution [6]. Consequently, fitting the amplification factor data from the Monte Carlo simulations with a Weibull distribution produces a more accurate value for the 99th percentile amplification factor than solely using the 500 data points. As such, the results presented in this section utilize the 99th percentile amplification factor obtained from the Weibull fit.

The first system parameter investigated is the degree of blade mistuning using 50 values of σδ ∈ [0, 0.08] with system parameters: N = 13, C = 3, $ω¯rate=1×10−4, ζ0=0.001$, and $Δk¯=0.05$. Figure 4 shows the 99th percentile amplification factor as a function of the degree of mistuning for the open-circuit case (i.e., no switch application) that acts as the mistuned baseline, the SDOF switch trigger ωSDOF, and the optimal switch trigger ωopt. Also shown is the peak amplitude for the baseline tuned system in the open-circuit case as well as the optimal RFD response for that tuned system. For the mistuned system with no switch application, the amplification factor increases sharply as σδ increases initially; however, the amplification factor increases more slowly and even decreases as σδ increases further. A similar trend can be seen in the literature for harmonic excitation. For the optimal mistuned switch trigger ωopt, RFD reduces vibration across all σδ values relative to the mistuned open-circuit case; however, RFD relative performance degrades as σδ increases. For small σδ values, the set of modal frequencies of the system is confined to a small range and, in the short-circuit state, the entirety of these resonance peaks are shifted away from the open-circuit peaks. Therefore, the optimal switch trigger application from the open- to the short-circuit state occurs at an excitation frequency that is larger than the short-circuit resonance frequencies. Increasing σδ, however, means the resonance peaks span a much larger range of frequencies. As such, not all of the short-circuit resonance peaks have been passed at the time of the switch to the short-circuit state, thus resulting in passages through some short-circuit peaks and a degradation in the vibration reduction performance. For these larger σδ values, multiple switch triggers (i.e., switching blades independently) may improve performance. The performance of RFD using the SDOF optimal switch trigger exhibits similar trends; again, reducing vibration relative to the mistuned baseline, but accounting for mistuning in finding the optimal switch trigger significantly improves performance.

Figure 5 shows the effects on the 99th percentile amplification factor (calculated from the Weibull fit previously described) of several design parameters using a constant value σδ = 0.03 for all simulations. These parameters include the sweep rate $ω¯rate$, damping ratio ζ0, and the difference in stiffness states $Δk¯$. Figure 5(a) shows the effect of varying $ω¯rate$ while holding the other parameters constant at ζ0 = 0.001 and $Δk¯=0.05$. As $ω¯rate$ increases, the peak response for the open-circuit mistuned baseline first increases until a maximum at $ω¯rate=2×10−4$, then decreases. For these quicker sweeps, the excitation passes through multiple resonance peaks, with each successive resonance peak compounding onto the vibrational response causing the amplification increase. Further increases in the sweep rate over a value of $ω¯rate>2×10−4$ cause the excitation to pass through the resonance frequencies rapidly and generate little vibration, causing the reduction in the amplification factor shown. (Note that this critical sweep rate of 2 × 10−4 is for this set of parameter values only.) Conversely, as the sweep rate slows, the response begins to approach that of harmonic excitation and the compounding effect of these multiple peaks diminishes, also resulting in a decreased amplification factor. Also note that RFD vibration reduction performance for the mistuned system remains relatively constant across all values of $ω¯rate$ relative to the open-circuit mistuned system, in stark contrast to the decreased RFD performance in the tuned system as $ω¯rate$ increases. It is also interesting to see that the performance associated with the SDOF switch approaches the open-circuit mistuned baseline for slow sweeps, while approaching the optimal switch trigger for quicker sweeps. Because the SDOF switch trigger remains constant for each trial, this effect indicates that the optimal switch trigger may vary as a function of the sweep rate, originally delayed past this SDOF switch trigger for slower sweeps, and approaching this point as the sweep rate increases.

Figure 5(b) shows the effect of varying ζ0 while holding the other parameters constant at $ω¯rate=1×10−4$ and $Δk¯=0.05$. Increasing the inherent system damping broadens the resonance peaks and reduces their overall magnitudes. As such, increasing ζ0 diminishes the compounding effect of sweeping through multiple resonance peaks, leading to decreased vibration for the open-circuit mistuned case. Additionally, optimal RFD provides greater relative vibration reduction performance for lower damping values for both the tuned and mistuned cases. Again, the performance of the SDOF switch trigger resides between optimal switching and the mistuned baseline.

Finally, Fig. 5(c) shows the effect of varying the difference in stiffness states $Δk¯$, a direct consequence of increasing the amount of electromechanical coupling on each blade, while holding the other parameters constant at $ω¯rate=1×10−4$ and ζ0 = 0.001. Unsurprisingly, varying $Δk¯$ has no effect on the open-circuit mistuned response as there is no switch to the short-circuit state. The slight deviations below $Δk¯=0.10$ from the otherwise constant line are artifacts of the statistical nature of the Monte Carlo analysis; a greater number of runs should lead to zero deviation. As $Δk¯$ increases, the short-circuit resonance peaks increasingly shift away from the open-circuit peaks, thus maximizing vibration reduction performance for RFD in both the tuned and mistuned cases.

## Conclusions

The vibration reduction performance and optimal switching for RFD was previously established for a SDOF system. For the case of turbomachinery blade mistuning, variations in blade properties lead to a frequency splitting and multiple peaks to be scattered around the resonance frequency of a single blade and the SDOF assumption no longer holds. The optimal switch trigger that corresponds to maximum vibration reduction for an uncoupled SDOF system is no longer optimal when applied to systems with regions of high modal density where multiple modes influence the vibration response associated with resonance passages. Instead, this paper shows the architecture used to achieve the optimal switch trigger for each mistuning pattern that reduces the magnitude of the peak blade response. The optimal trigger for a SDOF system is mainly a function of the electromechanical coupling coefficient; however, several other factors may influence this optimal trigger as applied to mistuned systems. One such factor may include the sweep rate as multiple peaks at each resonance crossing can compound and influence the overall response. A second factor may also include the engine order excitation that can excite separate sets of modes for a particular system; as such, the maximum blade response may occur at a frequency that depends on the particular engine order, thus influencing the optimal switch trigger. This dependence of the optimal switch trigger on an increased number of design parameters warrants further development of a response-based switch [26]. The switches connected to the piezoelectric elements on each blade were also assumed to switch simultaneously; future work should examine the performance of independently switching the piezoelectric elements.

Furthermore, the random nature of mistuning necessitated using a Monte Carlo analysis to understand the influence of various parameters on RFD performance. An analytical solution was presented and validated to enable the computational efficiency necessary to perform the large number of simulations required for such an analysis. The SDOF switch trigger reduces vibration across most test cases; however, updating the switch trigger maximizes performance for mistuned systems. This optimal vibration reduction performance is a function of the degree of blade mistuning, the sweep rate, the inherent modal damping, and the electromechanical coupling coefficient. Increasing the degree of mistuning and the sweep rate degrade performance, while increasing the electromechanical coupling and decreasing the system damping maximize performance.

## Acknowledgment

The authors gratefully acknowledge support from the Office of Naval Research monitored by Dr. Joseph Doychak, Dr. Knox Millsaps, and Dr. Steven Martens. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Office of Naval Research, the U.S. Navy, or the U.S. Government.

## Funding Data

• Office of Naval Research (Grant Nos. N00014-13-1-0538 and N00014-17-1-2527).

## Nomenclature

• A, B =

state matrix, input vector

•
• c, ζ0 =

damping coefficient, damping ratio

•
• C =

engine order

•
• Cp, θ =

piezoelectric capacitance, electromechanical coupling factor

•
• F, fr =

mechanical force vector, rth element

•
• G, gr =

modal force vector, rth element

•
• G0,r, λr =

rth modal force magnitude, constant portion of phase

•
• $keff2$ =

effective modal coupling factor

•
• K, kr =

system stiffness matrix, rth element

•
• m =

lumped mass component

•
• p, pr =

vector of modal displacements, rth element

•
• Q, β =

time-dependent amplitude, swept response phase

•
• qr, vr =

piezoelectric charge, voltage on the rth blade

•
• s0 =

stiffness state parameter

•
• t =

time

•
• x, xr =

mechanical displacement vector, rth element

•
• y =

state vector

•
• δr =

stiffness variation of the rth blade

•
• σδ =

•
• $Φ$ =

matrix of mode shapes stored columnwise

•
• φr =

phase of excitation for the rth blade

•
• ψr =

phase component of the rth blade due to engine order excitation

•
• ωr =

the rth modal frequency

•
• ω0 =

initial operating frequency

•
• ωrate =

linear frequency sweep rate

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