reliability analysis of structures incorporating shunted...

Reliability analysis of structures incorporating shunted piezoelectric transducers for the purpose of passive vibration mitigation

L. R. Cunha1, N. S. Saad1, D. A. Rade1 1Laboratory of Structural Mechanics (LMEst) School of Mechanical Engineering (FEMEC) Federal University of Uberlandia (UFU) Campus Santa Monica – Av. João Naves de Ávila 2121 Building 1M, CEP 38408-100, Uberlândia, MG, Brazil e-mail: [email protected]

Abstract Among the passive techniques of vibration control, the use of piezoelectric transducers connected to electric circuits has been intensively investigated lately. However, as is the case of any engineering system, uncertainties affecting the physical and geometrical characteristics of the control device are unavoidable and prone to jeopardize the control performance. In this context, this paper is devoted to the numerical procedures intended for the evaluation of the reliability of structures containing shunted piezoelectric transducers. Reliability here is meant as the probability of complying with pre-defined control goals, given the probability distributions ascribed to the uncertain variables, which are modeled as continuous random variables. The performance goals are accounted for by the proper choice of the so-called limit state functions (LSF), which are computed from the structural responses. The reliability indices are computed by using First Order Reliability Method (FORM) and the results are compared to Monte Carlo Simulation (MCS) associated to Latin Hypercube Sampling (LHS). Numerical simulations are presented for a truss structure modeled by finite elements, containing a piezoelectric stack transducer connected to a resistive-inductive (resonant) shunt circuit. The comparison between FORM and MCS results enables to evaluate the accuracy and computational effort involved in the use of both methods.

1 Introduction

Structural systems are subjected to parametric uncertainties, which means that many of the variables of a structural design cannot be considered as deterministic; they have a stochastic nature and are frequently modeled as random variables, characterized by probability distributions [1; 6; 7; 10]. These uncertainties directly influence the performance, durability and compliance with design requirements. Considering the unavoidable presence of uncertainty, reliability analysis can be performed in order to assess the probability that a structural system will meet requirements such as those mentioned above, which are represented in terms of the so-called limit state functions (LSF). Various methods can be used to evaluate the reliability of complex structural systems of practical interest, for which the LSF can rarely be stated explicitly in terms of the uncertain variables; instead, as those variables intervene in a rather complicated way in the structural behavior, the LSF must be evaluated with the aid of finite element models. Direct Monte Carlo Simulations (MCS) can be used to calculate reliability, but it is largely known that this method generally requires a large number of simulations, which implies high computational cost to achieve the necessary convergence [10]. To circumvent this limitation, approximate numerical methods have been developed, among which some of the most widely used is the First Order Reliability Method – FORM, and the Second Order Reliability Method - SORM [6]. Both methods are based in local

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approximations of the LSF: FORM adopts a linear approximation while SORM is based on a quadratic one. Despite the fact that reliability analysis has become a relatively mature discipline, not so many research works have addressed actively or passively controlled structural systems, considering specific LSF applicable to this category of systems. In reference [8], the author investigates the use of various reliability methods applied to structures under different vibration control strategies, including dynamic vibration absorbers, shunted piezoceramics and active vibration control based on piezoelectric stack actuators. In this paper, one addresses the reliability analysis of structures whose vibrations are controlled passively by using piezoelectric actuators associated with passive electrical circuits, called shunt circuits. The principle behind this strategy is that the vibrational energy is transformed into electric energy through the direct piezoelectric effect and is transferred to the circuit where it is partially dissipated. Among the types of electric circuit used, RL circuits, known as resonant circuits, are considered as some of the most efficient [5]. Such circuits comprise an inductor and a resistor which are connected to the piezoelectric transducer that is assimilated to a capacitor, thus forming an RLC circuit. When coupled to a dynamic system, this device operates similarly to a dynamic vibration absorber (DVA). Just like DVAs, shunt circuits must be tuned, which means that the values of their electric parameters must be precisely chosen for vibration attenuation in a narrow frequency band. However, the characteristic values of electronic components are prone to variability, due to manufacturing process and temperature, which can lead to mistuning and, as a consequence, decrease of the control performance. In this scenario, it becomes essential to evaluate the probability that the system will comply with the design requirements, given the probability density functions ascribed to the uncertain variables considered. In this remainder, FORM is used to calculate the reliability of a plane truss modeled by using the finite element method, for which one of the members contains a piezoelectric stack actuator connected to a resonant shunt circuit. Three different LSF are defined based on typical operation and performance requirements. Uncertainties in the values of resistance and inductance of the shunt circuit are modeled as Gaussian random variables. Furthermore, Monte Carlo simulations are performed for the purpose of comparison and verification of FORM results. A comparative analysis of accuracy and computational effort involved in both methods is presented.

2 Methods of reliability analysis

To illustrate the concept of reliability, consider the case where a variable R describes the allowable load of a structural component and a variable S represents the actual applied load; it is assumed that both variables have a random nature, and can be represented by their probability density functions, and (Figure 1). In this figure, and designate their respective nominal (mean) values.

The probability of failure quantifies the relative frequency of events in which the value of the applied load exceeds the value of the allowable load, that is:

, (1)

and the reliability of the member is given as

fPRe −=1 . (2)

)(rf R )(sf S

NR NS

)( SRPPf ≤=

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Figure 1: Illustration to support the concept of reliability.

According to reference [6], the probability of failure calculated from the probability density functions, assuming the R and S are independent from each other, is given by:

( ) ( ) ( ) ( )∫∫ ∫∞∞

=

=

00 0

dssfsFdssfdrrfP SRS

s

Rf . (3)

For a limit state function ( )Xg involving n random variables [ ]Tnxxx 21=X , one has:

, (4)

where is the joint probability density function of the random variables .

As per its definition, the limit state function ( )Xg can be geometrically interpreted as a hypersurface separating the safety region from the failure region in the space of the random variables. Equation (4) can be considered as the fundamental equation of reliability analysis. However, in general, the joint PDF is practically impossible to obtain, especially in explicit form. Moreover, even if the joint PDF is known, it becomes quite a difficult task to calculate the integral indicated in Equation (4). One possible alternative is to make use of analytical approximations of this integral, as detailed in the next section.

2.1 First Order Reliability Method (FORM)

Considering the case of two random variables introduced in the previous section, FORM operates by determining the shortest distance between the point ),( SR µµ and the curve that represents the limit state function. For this purpose, the LSF is linearized at point P, called the design point. The method requires that the original (or physical) variables be transformed to a system of reduced coordinates ( )SR, in such a way that these new variables have Gaussian distributions with mean and standard deviation equal to 0 and 1, respectively. Thus, the following transformations are made:

R

RRRσµ−

= , (5)

∫ ∫≤

=

0)(

2121 ),,,(Xg

nnXf dxdxdxxxxfP

),,,( 21 nX xxxf nXXX ,,, 21

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S

SσμSS −

= , (6)

SRSR SRSRg µµσσ −+−=),( . (7)

In Figure 2, it can be noticed that the reliability index, denoted by β , represents geometrically the shortest distance between the origin of the reduced coordinate system and the curve that represents the limit state function in the transformed space, being defined as:

∗∗= XX Tβ , (8)

where *X designates the design point.

Knowing the value of β , the failure probability is calculated as:

)( β−Φ=fP , (9)

where Φ is the cumulative normal distribution function. This method becomes exact when all variables are statistically independent with normal probability distributions, and the function limit state is linear. Figure 3 illustrates the geometric interpretation of FORM.

Figure 2: Representation of the reliability estimation problem: (a) in the original coordinate system; (b) in

the reduced coordinate system.

Figure 3: Representation of FORM with non-linear limit state functions.

(b) (a)


2.2 Monte Carlo Simulation (MCS)

Monte Carlo Simulation can be used to estimate reliability directly. This method consists in sampling the random variables according to their PDFs and computing the corresponding values of the limit state function. The safety and failure probabilities are calculated by counting the number of samples that are in the security and the failure domains, respectively. Hence, the probability of failure is expressed as follows:

[ ]n

gIn

nP i

nif

f0)(1 ≤

== = X (10)

where fn corresponds to the number of samples found in the failure domain fD , and [ ]I is a counting operator, which accumulates the number of occurrences of its argument. Figure 4 illustrates reliability estimation using MCS for the case of two random variables and a linear LSF. The main disadvantage of this method is that in order to achieve statistical convergence, a large number of samples is usually required, which generally involves high computational cost.

Figure 4: Illustration of reliability analysis using MCS for two uncertain variables and a linear limit state

function.

The number of required samples increases for small probabilities of failure. According to Melchers [10] an error estimation is given by:

f

f

nPP

k−

±=1

ε (11)

with k equals 2 for the 95% confidence interval. This value is associated with a bilateral confidence interval based on a normal distribution with zero mean and unit standard deviation. To reduce the computational time, improved sampling techniques can be used. In the case of this research, the technique called Latin Hypercube Sampling (LHS) was used [3].


3 Shunted piezoelectric transducers

Figure 5 illustrates a piezoelectric transducer connected to a resonant (RL) shunt circuit and bonded to a host vibrating structure.

Figure 5: Scheme of a piezoelectric patch connected to a resonant shunt circuit and bonded to a host structure.

Similarly to what happens with a dynamic vibration absorber, the resonant shunt circuits must be tuned, which means that the values of their electrical resistance and inductance parameters must be accurately determined for the attenuation of vibrations of the host structure in a given range of frequencies.

According to Hagood and von Flotow [5], the electromechanical coupling coefficient plays the same role as the mass ratio in tuning a DVA. For these authors, this coefficient can be approximated as follows:

, (12)

where and are the n-th natural frequencies in open and closed circuit, respectively.

Knowing the value of , one can calculate the optimum values of resistance and inductance according to:

, (13a)

, (13b)

, (13c)

. (13d)

However, in practical conditions, the values of the electric characteristics of the shunt circuit are inevitably affected by uncertainties resulting from material composition, manufacturing process and temperature variations. Such uncertainties can lead to mistuning of the shunt circuit and, as a result, deterioration of the performance of the damping device.

ijK

( ) ( )( )2

222

En

En

Dn

ijKω

ωω −=

Dnω E

nω

ijK optR optL

21

2

ij

ijopt K

Kr

+=

21 ijopt K+=δ

En

Spi

optopt C

rR

ω=

Spinopt

opt CL

2)(1

ωδ=


4 Numerical example

Figure 6 illustrates the situation of interest in which one of the members of a plane truss comprises a stack-type piezoelectric actuator, which is connected to a RL circuit, as shown in detail. The physical and geometrical properties of the main structure and the actuator are provided in Table 1, in which subscripts b and p indicate the properties related to the metallic and piezoelectric material, respectively.

Properties Unit Symbol Steel PZT-5H

Young modulus e 11101.2 ×

9100.60 × Density

, e 0.860,7

0.800,7 Cross section Area

, e 4100.25 −×

4105.27 −×

Piezoelectric strain coefficient or - 12100.650 −×

Coefficient of dielectric

permittivity - 9100.33 −×

Curie temperature

- 0.250

Table 1: Properties of the elements of the truss [8].

Figure 6: finite element model of a plane truss comprising a piezoelectric stack actuator connected to a RL

shunt circuit.

As detailed in reference [9] the finite element-based equations of motion of the electromechanical system is expressed as:

( ) ( ) ( ) ( )ttV~tt FKKUUM =−+ (14a)

( ) ( ) ( )tQtVt~ =+ ΓUK . (14b)

where M is the mass matrix, K is the stiffness matrix, K~ is the electromechanical coupling matrix, F is the vector of external loads, Γ is the matrix of dielectric permittivity, U is the vector of mechanical degrees of freedom, )(tQ is the electric charge and )(tV is the voltage across the electrodes of the piezoelectric patch. For the the R L circuit, the following relation holds:

][ 2mN bE EY33

][ 3mkg ρ bρ pρ][ 2m A bA pA

]/[ NC ][ Vm 33d

][ mF T33ε

][ºC cT


. (15)

Associating Eqs. (14) and (15), the electromechanical equations of motion are found under the form:

( ) ( ) ( ) ( )tttt FZKZCZM =++ , (16)

where:

( ) ( )( )

=tQt

tU

Z ,

−

=

−=

−=

IK0K

KΓ0K0C

Γ0KMM T~,

R

~R,L

~L (17)

The equations above can be integrated numerically to obtain the dynamic responses in the time domain. Moreover, the frequency response function matrix can be calculated as follows:

. ( ) ( ) 12 −++−= KCMH ωωω j .

(18)

In all the cases considered in this paper, the interest is to mitigate the contribution of the first vibration mode to the dynamic responses of the structure. To give a sense on the influence of the electrical parameters on the damping performance of the shunt circuit, Figure 7(a) shows the time responses in terms of the vertical displacement of point P in direction y, designated as , to a step force of magnitude of 10 N applied at the same point and direction, while Fig. 7(b) depicts the amplitudes of the corresponding driving point FRF. Both types of response are presented for different values of R and the value of L fixed to 310475.1 × H. Also indicated in these figures are the responses corresponding to the optimal parameters of the shunt circuit, obtained according to equations (13) for tuning to the first natural frequency. Similarly, Fig. 8 shows the same time responses and FRFs for different values of the inductance L, and the value of R fixed to 3100.50 × ohms.

Figure 7: Influence of the value of R: (a) on the time domain response ; (b) on the amplitudes of the corresponding driving point FRF.

( ) ( ) ( )tQLtQRtV +=

)(32 tU

)(32 tU

(a) (b)


Figure 8: Influence of the value of L: (a) on the time domain response ; (b) on the amplitudes of the corresponding driving point FRF.

4.1 Reliability analysis

For the evaluation of the reliability of the damping device, the following limit state functions are considered.

(19a)

(19b)

(19c)

The limit state function refers to a design requirement that constrains the maximum peak value of time response of the structure at point P to an excitation step of magnitude of 10 N applied at that point to be less than 61094.9

max−×=tY m. Thus, this LSF can be interpreted as a constraint to the response

overshoot.

On the other hand, refers to an operational limit of the electric shunt resistor. Assuming that the allowable power dissipation of the resistor is 850.0

max=tP W, establishes safe operation when

this value is not exceeded.

Finally, describes another performance requirement, by limiting to 10% the exceedance of the maximum amplitude of the harmonic response expected for the optimized shunt circuit, in the vicinity of the first natural frequency. Table 2 shows the statistical characteristics of resistance R and inductance L, considered as uncertain variables. Their mean values correspond to the optimal tuning values, calculated according to Eq. (13c) and (13d).

Variable Distribution Mean Standard deviation

Normal (Gaussian) 3100.50 × )10.0(100.5 3 µ× Normal (Gaussian) 310475.1 × )10.0(10475.1 2 µ×

Table 2: Statistical properties of the random variables considered.

Figure 9(a) allows comparing the responses to a unit step excitation of the structure without control and with a shunt circuit with optimal values of R and L. Figure 9(b) shows the electric charge flowing

)(32 tU

( ) ( )XX YtYtg −= max1

( ) ( )XX PtPtg −= max2

( ) ( )XX YfYfg −= max3

( )X1g

( )X2g( )X2g

( )X3g

nX )(µ )(σ

1X R ][Ω

2X L ][H

)(tQ

(a) (b)


through the shunt circuit as a function of time. The derivative of this response provides the electric current, from which the value of the dissipated power in the resistor is calculated according to .

Figure 9: Unit step responses considering the mean model: (a) vertical displacement at point P; (b) electric

charge flowing through the shunt circuit Figure 10 shows the frequency responses corresponding to the displacement of point P and the electric charge, showing the attenuation of the resonant amplitudes in the vicinity of the first natural frequency. It is also apparent the typical behavior induced by a dynamic vibration absorber.

Figure 10: Frequency responses for the mean model: (a) associated with the motion of point P and the electrical current in the band [0-100 Hz]; (b) detail in the vicinity of the first natural frequency.

Table 3 shows the reliability results obtained by using FORM and MCS-LHS for the three limit state functions previously defined. The computations have been performed on a personal computer with an Intel Centrino Core Duo P3550 processor with frequency of 2.0 GHz, and 3 GB of RAM. As for MCS-LHS, three different numbers of samples have been used for the sake of verification of convergence, which is demonstrated in Fig. 11. Also shown in Table 3 are the computation times required for each reliability evaluation. It should be pointed out that, in this table, the errors indicated have been computed by assuming the reliability values obtained from MCS-LHS with 20,000 samples as being exact.

( )tI 2RIP =

(a) (b)

(a) (b)


LSF Method [%] Reliability [%] Comput.time Error [%]

FORM - 1.6731 4.72 95.28 0h 00min 14s 1.85

MCS-LHS 1,000 - 5.70 94.30 0h 40min 41 0.80 6,000 - 6.52 93.48 2h 30min 30s 0.07 20,000 - 6.44 93.55 8h 20min 32s -

FORM - 0.4458 32.79 67.21 0h 00min 30s 22.06

MCS-LHS 1,000 - 14.00 86.00 1h 57min 21s 0.27 6,000 - 13.93 86.07 10h 42min 32s 0.19 20,000 - 13.77 86.23 30h 10min 20s -

FORM - 0.3956 34.62 65.38 0h 00min 02s 136.30

MCS-LHS 1,000 - 75.10 24.90 0h 08min 34s 9.88 6,000 - 72.55 27.45 0h 58min 26s 0.66 20,000 - 72.38 27.63 2h 50min 13s -

Table 3: Reliability results obtained using FORM and MCS-LHS.

Figure 11: Convergence of MCS-LHS: (a) for ; (b) for ; (c) for .

n β fP

( )X1g

( )X2g

( )X3g

( )X1g ( )X2g ( )X3g

(a)

(b)

(c)


The results presented in Table 3 reveal that FORM provides correct reliability estimation for , and fail in evaluating the reliabilities associated to and . The examination of Figures 12 to 14 enables to understand this occurrence. These figures depict the safety and failure domains defined in accordance with each of the three limit state functions, to which the joint probability density functions are superimposed. For each LSF, representations in both physical and reduced coordinate systems are shown. Complementary, Figures 15 to 17 show the reliability indices and the design points provided by FORM. It can be noticed that the limit state functions present dissimilar behaviors: indeed, for the first LSF, the failure and safety domains are separated by a linear curve; for the second LSF, the failure domain is confined to a closed region surrounded by the safety domain; for the third LSF, the safety domain is confined to a closed region surrounded by the failure domain. Thus, only for the original geometrical interpretations involved in FORM apply. As for the other LSFs, ad hoc numerical procedures would be necessary to adapt FORM to these conditions.

Figure 12: Limit state function and probability distributions of the variables: (a) physical

coordinate system; (b) reduced coordinate system.





( )X1g( )X2g ( )X3g

( )X1g

( )X1g

( )X2g

( )X3g

(a) (b)

(a) (b)

(a) (b)


Figure 15: Design point of : (a) in the physical coordinate system; (b) reduced coordinate system,

indicating the reliability index β.

Figure 16: Design point of : (a) in the physical coordinate system; (b) reduced coordinate system, indicating the reliability index .

Figure 17: Design point of : (a) in the physical coordinate system; (b) reduced coordinate system, indicating the reliability index .

5 Conclusions

In this paper, two different procedures, First Order Reliability Method and Monte Carlo Simulation, have been used for the estimation of the reliability of structures containing piezoelectric transducers connected to shunt circuits for the purpose of passive vibration control. The procedures have been applied to a two-dimensional truss, for three particular choices of limit state functions. Certainly, the procedure described herein can accommodate various other types of limit state functions. The numerical results obtained enabled to evaluate the accuracy and the computational effort of FORM as compared to MCS. It was found out that for two of the limit state functions considered, owing to their atypical geometrical characteristics, FORM failed in the evaluation of reliability. In these cases, only MCS provided correct results. A general conclusion is that reliability analysis is a very useful procedure to be incorporated in the analysis and design of vibration control devices.

( )X1g

( )X2gβ

( )X3gβ

(a) (b)

(a) (b)

(a) (b)


Acknowledgements

The authors are grateful to the Brazilian federal research agencies CNPq and CAPES, the Minas Gerais State research agency FAPEMIG, and the National Institute of Science and Technology of Smart Structures in Engineering – INCT-EIE, for the financial support to their research work.

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Verlag (2007), 306p. [4] Den Hartog, J.P., Mechanical Vibrations, McGraw-Hill Book Company, Inc. (1956), 366p. [5] Hagood, N.W.; Von Flotow, A., Damping of Structural Vibrations with Piezoelectric Materials and

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New York: John Wiley & Sons (2000). [7] Lemaire, M., Structural Reliability, ISTE (2009). [8] Cunha, L.R., Reliability Analysis of Structures Subjected to Active and Passive Vibration Control,

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[9] Leo, D.J., Engineering Analysis of Smart Material Systems, New Jersey: John Wiley & Sons (2007). [10] Melchers, R.E., Structural Reliability Analysis and Prediction, Second edition, New York: John

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