A fuzzy α-cut optimization analysis for vibration control of laminated composite smart structures under uncertainties

Accepted Manuscript

A fuzzy α-cut optimization analysis for vibration control of laminated composite smart structures under uncertainties Marcos D.F. Awruch , Herbert M. Gomes PII: DOI: Reference:

S0307-904X(17)30618-2 10.1016/j.apm.2017.10.002 APM 12002

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

16 March 2017 5 September 2017 1 October 2017

Please cite this article as: Marcos D.F. Awruch , Herbert M. Gomes , A fuzzy α-cut optimization analysis for vibration control of laminated composite smart structures under uncertainties, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.10.002

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Highlights  It is presented a new methodology to account uncertainty propagation in composite materials.  The LQG control performance over a smart composite is evaluated under Fuzzy uncertainty.  A heuristic approach for envelope evaluation of composite behaviour is proposed.  The approach allows a large number of uncertain parameters without increased numerical cost.  Small uncertainties from multiple sources resulted in important output variability.

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A fuzzy α-cut optimization analysis for vibration control of laminated composite smart structures under uncertainties Marcos D. F. Awruch*, Herbert M. Gomes† *

Mechanical Engineering, Federal University of Santa Maria, Av. Roraima 1000, †

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97105-900, Santa Maria, RS, Brazil, [email protected],

Mechanical Engineering, Federal University of Rio Grande do Sul Rua Sarmento Leite, 425, 90050-170, Porto Alegre, RS, Brazil [email protected]

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Summary: As a high performance and specific mechanically tailored material, composite structures present high structural behavior sensitivity to small variations in geometry or material properties. Some of these uncertainties are intrinsically related to the manufacturing process but others are usual uncertainty in material properties. A Fuzzy interval analysis methodology is proposed in this paper in order to evaluate the

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structural behavior of laminated composite smart structures under vibration control and uncertain parameters. A Particle Swarm Optimization algorithm is used to search

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for input parameters in order to minimize and maximize system outputs by an antioptimization and thus build their envelopes. Each input parameter has interval values

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defined by -cut levels and interpolated by the Fuzzy reasoning. This methodology is applied to a smart structure of laminated composite material with attached piezoelectric

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patches and controlled by an optimal control regulator. System’s interval outputs like natural frequency, mechanical vibration, and electric control input energy are

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investigated taking into account uncertainties in the composite and piezoelectric material properties, ply angles and layer thickness. Finally, it is concluded that these uncertainties may affect the control performance since it was found significant modifications in the dynamic behavior and therefore this should be accounted in the design stages.

Key words: Vibration Control, Smart Structures, Possibilistic Theory, Fuzzy Interval

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Analysis, Uncertainties Quantification, Laminated Composite.

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INTRODUCTION In the past few years, substantial attention has been paid to active vibration

control of smart and lightly damped flexible structures in several fields of civil, mechanical and aerospace engineering. The applications extend from vibration mitigation on tower structures and motion control of robotic systems to satellite solar

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panels and energy harvesting devices. In order to satisfy precision control and lightweight requirements, smart materials such as piezoelectric and shape memory alloys are frequently integrated into laminated composite structures as sensors or actuators. The use of piezoceramic material is a field with ongoing investigation and application and its advantages include low-power consumption, fast response time, a

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wide variety of shapes and sizes and easy implementation.

An important step related to any structural modelling is the uncertainties identification and quantification present in the actual system, taking it into account in the analysis stage. The two major types of uncertainties are classified as aleatory and epistemic uncertainties. The aleatory uncertainty is generally modelled as probabilistic

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variables with statistical information assumed based on experimental data (loads, boundary conditions, material property variability). On the other hand, epistemic

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uncertainty comes from the lack of data information (statistical parameter uncertainty), imperfect modelling, simplified assumptions or lack of awareness of the phenomena

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being modelled.

Laminated composite structures are known for their challenges during the

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fabrication processes that might generate uncertainties in some properties or even defects such as interlaminar voids, fiber misalignment, residual stresses, variation in ply thickness, just to name a few [1, 2]. To deal with the uncertainty in a project, it is

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possible to use a probabilistic approach, but in that case, it is desirable to have enough and reliable information about the random variables, such as mean values, moments and distribution types [3]. Usually, in many engineering applications, there are not enough measurements or knowledge of some parameters, or even they are measured with not sufficient accuracy. When statistical data cannot be obtained or the information is imprecise, a possibilistic approach is preferable. Possibilistic methods deal with the extreme scenarios, or the problem’s output envelopes, giving neither information about

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their probabilistic distribution nor their correlations. The mathematical framework using this methodology includes Dempster-Shafer evidence theory [4]. Moreover, it is well proven the efficacy in founding extreme bounds by interval analysis in comparison with simple Monte Carlo simulations [4, 5, and 6]. According to Potter (2009) [7], even for statistical data given by suppliers (e.g.

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mass/unit area of a prepreg composite), the measured probability density function does not meet usual specifications (showing even bimodal shape). According to Roy et al. (2017) [8], the development of the Design Allowable Database requires an evaluation of multiple batches of composites with the associated construction of very large mechanical and other physical property databases. In addition, to quantify uncertainty in

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the structural performance, (1) numerous subcomponents may have to be fabricated requiring expensive tooling, and (2) one must then perform expensive tests on these elements to determine their long-term performance. Furthermore, interval values can be easily treated numerically. For convergence of statistics using MC simulations framework, several numerical simulations are necessary that is opposite to the most

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accurate and less numerically processing and time-consuming results obtained using a Fuzzy or Possibilistic framework. This is the main motivation behind this research. BRIEF BIBLIOGRAPHICAL REVIEW

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The problem that arises in designing composites components with associated

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uncertainty is well posed by Antonio [9]. A comprehensive survey of the techniques used for sensitivity evaluation and the uncertainty propagation is presented in this paper

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but limited to sensitivity-based methods. It is suggested that the effects of uncertainty in mean values of mechanical properties of composites are very sensitive for reliability

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index evaluations.

There are some new studies where the focus in the analysis of uncertainties in

composites, such as [10], makes use of the first order reliability method (FORM) with a high fidelity shear deformable laminated model, considering uncertainties associated with fiber orientation and ply thickness. Lopez et al. [11] present a comparison between the FORM and the polynomial chaos representation for the reliability analysis, where loads, strength properties and fiber orientation angles for each layer are considered as

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random variables. In Goyal and Kapania [12] paper, angles and properties of the laminate are considered as having an uncertainty degree, so the reliability problem is analyzed based on these uncertainties. There are also experimental studies trying to take into account sources of uncertainties, as Lekou et al. [13] study where they try to obtain the estimation of measurement uncertainties in the properties of composite materials. A

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good survey on methodologies used for uncertainty evaluation in composite structures can be found in Sing and Grover [14].

Zhang et al. [15] propose an interval Monte Carlo method for structural reliability evaluation and report that the Interval Probability of Failure obtained using the interval MC tends to be wider than that from a Bayesian approach, which

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incorporates the probability information into the confidence bounds. Chowdhury and Adhikari [16] proposed a High-Dimensional Model Representation (HDMR) metamodel for a dynamic analysis with fuzzy uncertain variables and validated the results that are compared to a direct Monte Carlo Simulation. This approach presents an accurate estimation of the system response and also a lower computational effort than a

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direct complete simulation. Chakraborty and Sam [17] made a relevant study on methods transforming possibilistic variables to equivalent probabilistic variables,

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enabling a hybrid uncertainty approach to different problems. The vibration control performance of structures under the presence of

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uncertainty is an important field of study, for example in earthquake isolation systems in civil engineering [18,19] or in dynamic control in aerospace applications [20], among others. There are few studies related to the performance of piezoelectric controllers for

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composite materials under uncertainty properties. Most of them are related to the use of Robust Control (H norm) and LMI (Linear matrix inequalities). Koroishi et al. [21]

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present a robust control applied to a piezoelectric actuator bonded to a composite structure based on a Linear Matrix Inequalities applied to a Linear Quadratic Regulator. The inclusion of uncertainty in the analysis is performed by introducing a variation of 10% in the dynamic matrix of the structural model (but unfortunately, not assigning the sources of such uncertainty). The statistics of 100 samples are generated using Monte Carlo Simulation associated to Latin Hypercube sampling. Envelope curves are presented in order to evaluate the performance of the LMI/LQR controller. In the case

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of structural uncertainty, it is reported that the conventional robust control outperformed the proposed LMI/LQR control. Trevilato [22] uses a similar approach based on LMI, but in his work, the way the uncertainty is introduced in the model is different: the ply angle orientation and the stiffness of the finite elements attached to the clamped end of a cantilever beam are defined as the only sources of uncertainty. The tested control

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techniques are the traditional LQR and the H robust control. The expected better performance of the H control compared to the LQR in the presence of uncertainties was reported.

Choi [23] investigates the effects of actuator uncertainty in the vibration control of a smart beam structure using a Robust Control incorporating the hysteretic behavior

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of the actuator and parameter variations for the system uncertainty. It was reported that the formulation was experimentally verified and presented good performance. Marinova et al. [24] simulate damage in a smart composite beam using the Linear Fractional Transformation (LFT) implemented in H control theory. The LFT assumes a feedforward using a Gain matrix of uncertainty, bounded to the unitary norm that relates

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the system outputs to a new control force that comes from a different control channel of the system. The uncertainty is treated as differential mass, damping and stiffness

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matrices in the LFT framework. Based on the simulated numerical results a suboptimal H control law is implemented and they conclude the vibration suppressions are

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effective in different loading situations. In a different way, Moutsopolou et al. [25] modelled uncertainties in the whole mass, stiffness and damping matrices as variations

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around mean values and an H criterion, using  analysis to take into account the worst scenario for uncertain disturbances and noise in the system. Unfortunately, there is no

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assignment for the source of the uncertainty present in the variations of the structural matrices.

In this work it is applied a fuzzy α-cut optimization methodology, which can be

described as a multiple interval analysis process, associated with a heuristic algorithm working as anti-optimization in order to search the output boundaries for an example case, studying the uncertainty propagation. No papers were found that deal with control performance of piezo-laminated composites under uncertainties with possibilistic parameters and a heuristic approach. The PSO (Particle Swarm Optimization) algorithm

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is used as the tool for the minimum and maximum boundary search. Herein the finite element model, based on the First-Order Shear Deformation Theory (FSDT), is a laminated composite plate with embedded piezoelectric actuators and sensors controlled by Linear Quadratic Regulator/Gaussian (LQR/LQG) controllers. Multiple sources of uncertainties are considered such as thickness, ply orientation and material proprieties

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for the composite and piezoceramic components. The analyzed output performance parameter is the integral over time of the kinetic and potential energy of the controlled beam, as well as the electric control input energy. The spatial displacement along time and the first natural frequency envelopes are shown as well.

3.1

MATHEMATICAL MODELLING

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Finite Element Model for composite and piezoelectric material The composite laminated plate is modeled using the classical FSDT (First Order

Shear Deformation Theory) [26] where the Kirchhoff assumptions are relaxed and

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constant shear deformation is assumed along the ply thickness, so moderately thick plates and shells can be modelled. Other hypotheses are also considered such as the

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perfect bonding between layers; the assumption the resin between plies is infinitesimally thin; and that each layer has a uniform thickness. Furthermore, the

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modifications in the angle-ply configurations could induce bending–stretching and bending-twisting coupling effects depending on the generated asymmetry of the

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laminate [27]. The displacements field has the following form:

where

(

)

(

)

(

)

(

)

(

)

(

)

)

(

(

are the displacements field and

functions. In this case, and

(1)

) are the unknowns

are the displacements of a point on the plane

represents a rotation of a transverse normal about y and x axis, respectively.

The linear strains associated to the displacements field are thus obtained as:

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(2) )

(

)

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(

A triangular three node finite element with 6 DoF per node (including drilling degree of freedom) is implemented for FSDT that is also used as a framework for the

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structural piezoelectric lamina (TCGC-T9 element, see [28]). Mindlin plate theory with Lagrange interpolation functions enhanced with drilling effect are used to approximate

]

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[

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the displacements field. So, these main assumptions lead to:

[

(3)

[

]

]

are the curvatures (derivatives of the rotation angles),

strains,

is the ply height referred to the mid-plane.

are the mid-plane

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where

Piezoelectric elements might behave nonlinearly at high voltages, so it is

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necessary to remain at low voltage levels and use the linear constitutive relations defined by IEEE standards [29] which are usually considered as a good representation of those materials. The constitutive relations are described as: (4) where

is the stress vector,

strain vector,

represents the electric displacement vector,

is the electrical field vector,

is the

is the laminate elastic tensor and finally,

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is the piezoelectric constants matrix and represents the dielectric constant matrix. As previously mentioned, it is used a finite element applicable for plates and shells that takes into account the coupling effects of membrane and bending. This finite element is extended by adding an extra DoF for each piezoelectric layer present in the element [30].

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Most piezoelectric materials used as actuators in structural mechanics have a plate aspect and orthotropic mechanical properties. Furthermore, the nonlinear behavior is only prominent at higher voltages. Therefore, here in this paper, it is assumed linearity for the piezoelectric material structural behavior. It is also assumed that the electric field (E) and the electric displacement field (D) are uniform and normal to the

[ where

]

[

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plate mid-plane, being the polarization orientation perpendicular to it. ] and

is the applied voltage on the k-th lamina and

lamina thickness.

(5)

is the piezoelectric k-th

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Detailed steps for assumed interpolation functions, the complete integration scheme for the triangular element and assembling of the stiffness and mass matrices can

Coupled electro-mechanical system of equations

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be found in [28] and [31].

The problem consists of a laminated composite plate with embedded piezoelectric

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patches placed as actuators and sensors used for the device’s vibration control. The globally coupled equation of motion and electrical charge of this type of system can be

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cast as follows [32]:

where

[

and

mass matrix,

̈ ][ ̈ ]

̇ ][ ̇ ]

[

][

]

[

]

(6)

are displacement and potential electrical fields, respectively,

is the

is the damping matrix,

is the

electrical charge vector, and, finally,

[

is the mechanical force vector and

is the mechanical stiffness,

is the electric stiffness

are the electro-mechanical coupling stiffness matrix.

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Considering the case where the electrical charge

, and dividing the

electrical DoF into separate actuating and sensing capabilities, Eq. (6) can be rewritten as: ̈

(7) ̇

where the

and

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(8) notation is indicated for sensors and actuators matrices and vectors,

respectively, and

.

The potential, kinetic and electrical energies for a given time instant is defined

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as [33]:

4.1

(11)

OPTIMAL MODAL CONTROL

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(10)

̇

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̇

(9)

Modal state space formulation

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To reduce the problem order of multiple degrees of freedom, especially for complex structures in finite elements, it is usual to work with a truncated modal model

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[34], where only the most important modes are considered in the simulation (usually lower modes of vibration are the most easily excitable). The transformation to the

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modal form is performed starting from the spatiotemporal separation hypothesis: ()

()

where (t) is the time-dependent modal coordinates vector and

(12) the time-independent

modal matrix that can be found solving the eigenvalue problem: ( in addition column of

*

+,

)

(13)

being the natural frequencies of the structure and each

the corresponding eigenvector.

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Taking into account Eq. (7) and Eq. (12), the following equation of motion is obtained in modal coordinates: ̈

is the identity matrix, considering orthonormal eigenvectors ( *

)

and

being the damping ratio of the ith mode (it is assumed a

+ with

proportional damping).

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where

(14) ̇

In the next step, it is convenient to work in the state space model to reduce the second order differential equations, Eq. (7), to a set of first order ones. Defining the state space vector

as: ]

[ ̇]

(15)

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[

It is possible to arrive at the following system of equations: ̇

where

is the control input force and

some external mechanical disturbance. The

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state space matrices are defined as:

]

[

] [

(17)

] ]

Modern control formulation

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[

[

(16)

Modern control theory is usually applied in the case of systems with multiple

inputs and outputs, using time domain instead of frequency domain as the classical control theory. The optimal control objective [35] is to determine the control signal that will make a process be controlled and at the same time optimize a performance index. For the LQR control, the main objective it is to minimize the performance index :

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∫( where

)

is semi positive definite matrix and

(18)

is strictly positive definite matrix, both

defined by the system designer using some criterion. Those weighting matrices are defined beforehand in this paper by an optimization looking for a balance between

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mechanical energy and control force (applied voltage) that should result in less mechanical potential and kinetic energy in the cantilever plate, but with control voltages not exceeding the piezoceramic electric threshold potential for depolarization. Since there is only one channel of control for the presented examples, and

, by a diagonal matrix: [

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once set

, and

is defined

(19)

Assuming control forces that are proportional to the space state vector, Eq. (20), the objective is to find the gain vector

(20)

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minimize in full state feedback [36].

applied to the state space variables in order to

and

matrices:

(22)

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for

(21)

is the Riccati matrix, defined by the solution of the algebraic Riccati Equation

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where

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It can be shown that the solution to obtain this gain reads:

In this work, it is used an LQR (Linear Quadratic Regulator) or a Linear

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Quadratic Gaussian (LQG) control, which is composed partially of a Linear Quadratic Regulator (LQR) and a state vector observer. It is used the Kalman filter as an observer which allows the state vector estimations based upon the output y and inferring the plant and measurement noises, assumed as uncorrelated white noises. The feedback information for the LQG control is given by the sensor measurements and the estimation of the state is built by the Kalman filter. Since not all states are available to the LQG, the error between the true and

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̂. Adding the plant and measurement noise

estimated states can be defined as to the system, gives: ̇

(23) Those white noises are uncorrelated, with zero mean and covariance intensity and . The estimator state is given by: ̂̇ where

̂

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matrices

(

( ̂))

(24)

is the observer gain, found in a similar fashion as the previous

gain, by the Riccati equation, using

and , instead of the weighting matrices

̇ [ ] ̇

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. The closed-loop system is then defined by: [

][ ]

[

]

[

][ ]

and

(25)

The control forces are defined in this case based on the estimated states as: ̂

Fuzzy α-cut optimization

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5.1

FUZZY ANALYSIS

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(26)

Fuzzy Set Theory was first presented by Zadeh [37] in his seminal paper. A fuzzy set defines a class of objects that have some sort of belongings (pertinence) to a

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group of objects. Differently, from traditional Set Theory that has crisp borders, a fuzzy set is composed of a support function (known as membership function) with values that

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can lie between 0 and 1, thus being possible to deal with vague or incomplete information. In 1978, Zadeh [38] released the Possibilistic Theory using fuzzy sets that

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played a similar role to the probability theory for definite stochastic distributions. The α-cut (or α-levels) methodology divides the fuzzy set into well-defined crisp real intervals with α-levels of belongingness or relevance ( ) and can be defined generically for a fuzzy variable A as: { where ̃ is a fuzzy set and

̃| ( )

is an element of ̃ ,

+

(27)

is the α-level cut k that represents

some sort of uncertainty (α-cuts ranges from 0, complete uncertainty, to 1, complete

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[

certainty). This interval can also be defined as

] , where

represent the minimum and maximum values for the

and

set, respectively. Figure 1

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shows the graphical representation of a fuzzy set ̃ .

Figure 1. Fuzzy set ̃ and its sub sets

and

for i to k α-cuts.

In order to study and analyze input or model parameter uncertainties modeled as

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fuzzy sets, it is applied the fuzzy α-cut optimization methodology (Möller et al. [39]). This methodology consists in multiple interval anti-optimization analyses along

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membership functions for the output that is itself a fuzzy set composed of fuzzy numbers, as expected. So,

̃

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where ̃ means the fuzzy inputs,

(̃ ̃

̃ )

(28)

is the number of inputs,

some linear or non-

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linear deterministic function and ̃ a fuzzy output set. This can also be understood as uncertainty propagation in terms of intervals, since the uncertainty of the inputs and parameters will be propagated to the outputs.

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The fuzzy α-cut optimization consists in finding for each pertinence level, the

minimum and maximum value for the desired output. Therefore, the problem consists of multiple optimization direction searches, two (minimization and maximization) for each α-cut that the fuzzy set was discretized. To solve Eq. (28), the problem can be stated as: ( [

]

) (29)

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where

is a specific desired output and for a different output, another combination of

optimal

might arise.

Once defined the desired discretization for the α-levels, the uncertainty propagation is performed for each α-level cut. Fig. 2 represents the proposed problem

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for a two input and one output problem ̃ , ̃ and ̃ :

Figure 2. Uncertainty propagation for two input and one output problem at defined α-cut

5.2

level.

Particle Swarm Optimization The particle swarm optimization (PSO) is a heuristic optimization algorithm

inspired by the observation of the social behavior of beings such as fish schooling,

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insects swarming and birds flocking. Kennedy and Eberhart [40] introduced this algorithm in 1995. The method is based on the social influence and social learning, so the exchange of information between individuals may lead them to solve complex problems. This algorithm was chosen due to simplicity in implementation and reported robustness in finding the global optimum. The PSO is used as the main engine for the

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fuzzy α-cut optimization analysis. As stated by Li et al. [41], it involves a number of particles, which have a defined position and velocity, and they are initialized randomly in a multidimensional search space of a cost function (a modification of the objective function to handle with constraints in the problem). Each particle represents a potential solution for the problem and the measure of suitability is the cost function value. The

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set of particles is generally referred as “swarm”. These particles “fly” through the multidimensional space. They have three essential reasoning capabilities: inertia, the memory of their own best position and knowledge of the global or neighborhoods best position.

The basic particle parameters, position, and velocity, are updated throughout

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each iteration by the following equations:

e

)

(

)-

(30)

are the updated velocity and actual one, respectively, of particle i with

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where

(

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,

respect to design variable j. In the same way

and

are the particle position.

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is the best position found so far from the self-historical path by particle i while is the best position found so far in the swarm until that iteration k.

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random numbers between 0 and 1, while

e

e

are

are cognitive parameters that represent

the confidence of the particle on its own result or on the swarm best result (both set as 2

in this paper). The

parameter is used to avoid the divergence behavior in the

algorithm, and it was proposed by Bergh and Engelbrecht [42]. Finally,

is the inertia

factor, introduced in the original PSO by Shi e Eberhart [43], and means the importance of the past particle velocity on the searching procedure generating such a momentum in the direction of search.

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As a metaheuristic algorithm, PSO makes few or no assumptions smoothness or differentiability about the problem being optimized and thus it is not prone to get stuck in local minima. Furthermore, the algorithm present stochastic operators (random numbers) and this reinforces this virtue. The good behavior and robustness of the algorithm in searching very large spaces of candidate solutions are referred in the

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literature and benchmarked with several optimization problems (Taherkhani and Sababaksh, [44]). The use of metaheuristic PSO as optimization engine in the α-cut level analysis (anti-optimization) is one of the interesting innovative propositions of the paper since it provides extra security against local minima. 6

NUMERICAL EXAMPLES

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Being a multidisciplinary field, smart materials presents a variety of uncertainty sources, from materials properties to circuitry and controlling features. This analysis deals with the composite material and piezoceramic properties uncertainties and the resulting variability, or uncertainty propagation in the structural behavior. In this

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example, it is studied a cantilever composite plate. The cantilever beam is the most used structure for experimental and numerical benchmark/validation of control strategies.

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According to Kim et al. [45], this setup is the widely used setup in energy harvesting. Raaja et al. [46] used a piezoelectric cantilever beam as a MEMS accelerometer to investigate SHM applications. Since the idea of this paper is to deal with many sources

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of uncertainties in the parameters of the controlled composite structure, it was chosen a classical and well-behaved test case to take better account of uncertainties effects.

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In the example, it is studied a cantilever composite plate consisted of 4 layers

with 30 cm in length and 4.5 cm in width, embedded with piezoceramic patches (lead

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zirconate titanate, PZT) to control free vibration due to a suddenly applied force of 0.5 N for a short period (5 ms) at all free nodes at the tip of the beam. The piezoelectric patch is positioned at the fixed end and next to the lateral edge as shown in Fig. (3). This position choice is based on the bending and twisting vibration modes (present in the first 4 modes) that should be controlled and that was previously defined by an optimization procedure.

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Figure 3. Finite element mesh of the cantilever composite plate with the location of the piezoelectric patches (dark elements) and the applied forces.

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The nominal properties of the composite material plate and piezoelectric layer properties are given in Table 1. The piezoceramic electric threshold potential for depolarization is ±200 V. In this configuration, the first 4 vibration modes have been selected to be controlled and for simulation purposes, the modal FE model including the

modes with value = 2%.

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first 10 vibration modes is used. It was used a weakly modal damping ratio for all

PZT [29]

Graphite epoxy

[45º/-45º/-45º/45º]

Density

ρ = 1600 kg/m3

ρ = 7600 kg/m3

hc = 0.5 mm

hp = 0.25 mm

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Stacking sequence

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Layer thickness

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Young’s moduli

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Shear moduli

Poisson’s ratio

Piezoelectric constant

Electrical permissivity Electrical potential limit

E1 = 172.5 GPa E2 = 6.9 GPa

E1 = E2 = 63.0 GPa

G12 = G13 = 3.45 GPa

G12 = G13 =G23 =24.6

G23 = 1.38 GPa

GPa

= 0.25

= 0.28 =

= 10.62 C/m2 = 15.50 nF/m ±200 V

Table 1. Composite laminate and PZT piezoceramic properties.

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6.1

Nodal displacement envelope In this example, it is considered an uncertainty in the fiber orientation for each layer,

independently; therefore, there are four uncertain parameters. It is established that those inputs are modeled as fuzzy sets in a symmetrical triangular shape and it is discretized in five α-levels, plus the nominal one (

). For each level, there is an interval

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varying around the nominal value of ±1.25º, ±2.50º, ±3.75º, ±5.00º and ±6.25º (Fig. 4). For this example, it is used the LQR controller and the PZT patches are placed only on the upper face of the cantilever, located on the darker elements of Fig. (3). The weighting matrix

was optimized (PSO) for the minimum mechanical energy in the

nominal case, respecting the PZT limitation of ±200 V. The values used in this example =2.810×106 and

=1.184.

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are

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Figure 4. Fuzzy sets for layer orientation of 45º (left) and -45º (right)

The intention is to build a displacement envelope along time (minimum and

maximum possible values) for the free tip corner node displacement (1 uncertain

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output). This type of analysis might be computationally expensive since two optimizations are performed for each time step and for each α-level in order to obtain the Fuzzy Intervals. Fig. (5) to Fig. (9) show the envelopes for different α-levels and built with a discretization of 125 time intervals for 1 second simulation. In this example, with 24 individuals, each upper or lower limit for each -cut, comprising 125 single independent optimizations resulted around 40,000 simulations.

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Figure 5. Displacement envelop for α=0 (±6.25º)

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Figure 6. Displacement envelop for α =0.2 (±5.00º)

Figure 7. Displacement envelop for α =0.4 (±3.75º)

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Figure 8. Displacement envelop for α =0.6 (±2.50º)

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Figure 9. Displacement envelop for α =0.8 (±1.25º)

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Analyzing envelopes in Figure 5 to 9, it is possible to note that at the beginning of the transient behavior, most configurations have a similar path and the interval is

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narrow. However, as time goes by that interval becomes smooth indicating that there is not an increase in the displacement interval (it closely accommodate the periodic

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motion) but mostly in the uncertainty phase (several periodic displacement histories may occur). That effect is due to different vibration phases for different ply configurations that fulfill the wave gaps of the envelope presented at the beginning. It is interesting to note that the stacking sequence for each instant of time is different, so there is not a single case that would follow exactly one of the boundaries. Overlapping the envelopes for different α-levels and adding, in red, the nominal curve (α=1, the tip of the triangle) it results in Fig. (10).

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Figure 10. Overlapped envelopes for different α-levels (red is the nominal response,

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darker areas representing high α-levels, light lines representing lower α-levels ).

Analyzing each time instant, getting vertical cuts through the curves of Fig. (10),

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it is possible to see the fuzzy sets for those outputs, depicted in Fig. (11).

Figure 11. Fuzzy sets of displacement for some time instants.

Another study that is possible to be performed is the displacement variability

interval for each time instant (distance between boundaries), in order to define where is the critical time instant with the highest degree of displacement uncertainty. Fig. (12) shows these results for ply angle intervals ±1.25º and ±6.25º.

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Figure 12. Displacement variability interval ( the difference between lower and upper

6.2

Frequency domain envelope

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boundaries along time).

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Following the previous example, considering the same structure and uncertainty, the objective here is to build the frequency response for the referred node displacement.

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The Power Spectral Density (PSD) envelope was built around the first natural frequency (13.28 Hz for the nominal structure), and the algorithm performed the minimization and maximization of the PSD amplitude in a range of 5 Hz and 20 Hz, discretized in 0.5 Hz

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width spectral lines. The overlapped plot for different α-cuts and the nominal value (red) are shown in Fig. (13). Frequencies are shown as fuzzy sets are presented in Fig.

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(14), for some frequency steps.

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Figure 13. Overlapped envelopes for different α-levels.

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Figure 14. PSD tip displacement variability as fuzzy set intervals for some discrete frequencies.

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In a different way, if the interested output is some natural frequency itself, the

optimization is performed to find the extreme cases for each α-level on that specific vibration mode. In the case of the first natural frequency, the resulted fuzzy set is shown in Fig. (15).

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6.3

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Figure 15. Fuzzy set for the first natural frequency. Multiple uncertainty sources and energy analysis

Emulating a more realistic scenario where there might be different small sources of uncertainties in the project, this study analyzes the uncertainty propagation influence

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in system’s output (considered as mechanical and electrical energies related to the vibration energy and used controlling forces). In this numerical example, it will be used

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the LQG. The PZT sensors are attached to the lower face of the plate collocated to the actuators as shown in Fig. (3). The used LQG parameters are

=6.353×106 and

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=4.478. Since a Kalman filter is necessary to predict the system states, plant and sensors noises,

and

need to be defined. Their spectral densities are hard to obtain

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in real cases, so those matrices are treated as parameters of the controller like matrices. Here it is used

and

and

.

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The uncertain fuzzy input parameters are modeled as symmetrical triangles

where the lowest α-level (α=0) have a ±3% variability from the nominal value and, the fibers orientation for each layer, a variability of ±3º, defined in Table 2. Therefore, there are 22 uncertain parameters in total, considering PZT actuator, sensor, and the composite Graphite-Epoxy properties. The fuzzy sets are discretized in four α-levels plus the nominal one.

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Maximum

Fibers Ɵ1 and Ɵ3

42º

48º

Fibers Ɵ2 and Ɵ4

-48º

-42º

G-E layer thickness

0.485 mm

0.515 mm

G-E E1

167.33 GPa

177.68 GPa

G-E E2

6.69 GPa

7.11 GPa

PZT thickness

0.2425 mm

PZT E1 and E2

61.11 GPa

PZT e31 and e32

10.30 C/m2

PZT ξ33

15.03 nF/m

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Minimum

0,2575 mm 64.69 GPa

10.94 C/m2

15.97 nF/m

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Table 2. Uncertain parameters for α =0

For the nominal structure, the controlled and free vibration plots are presented in Fig. (16) and the force control is shown in Fig. (17). The integration over time of the mechanical energy (sum of kinetic and potential energies) in this case resulted in

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Emec=6.35×10-5J·s and the electrical input energy resulted in Eele=-1.18×10-4 J·s.

Figure 16. Time response for the LQG (red) and not controlled (blue).

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Figure 17. Control voltages applied to the actuators. Depolarization limit of 200V is not

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exceeded.

The analyzed outputs are presented in Fig. (18) and Fig. (19) as fuzzy sets and shown as a percentage of variation with respect to the nominal value. Those results were

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obtained using the PSO algorithm, as proposed previously in the paper and by Monte Carlo simulation with different population sizes. For the PSO algorithm, with 80

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individuals, each -cut limit anti-optimization took 8 to 10 iterations to converge (change in the objective function) leading to a total number of structural analysis around 6000 simulations. First, it is easy to notice that even with the higher attempt and

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computational effort, with 100,000 simulations, the Monte Carlo simulations could not obtain the extreme values in an accurate way like in the representation of the fuzzy set

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(PSO). It is interesting to observe that the fuzzy set results are not symmetrical, like the Fuzzy sets for the inputs, and have a high variability for the critical case, which are the

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maximum energies values. Although the inputs were established with relatively small variability, even for low α-levels, the propagation is considerably high, reaching up to 21% of mechanical energy increase (related to the value in the nominal structure) and up to 46% more electrical control energy.

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Figure 18. Fuzzy number for mechanical energy variation using PSO and Monte Carlo

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(MC) simulations.

Figure 19. Fuzzy number for electrical input energy variation using PSO and Monte

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Carlo (MC) simulations.

CONCLUSIONS Smart materials with piezoelectric control are applied in different fields of

engineering. The uncertainty propagation can be used to estimate the final envelope for

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structural behavior. In the best of the author’s knowledge, there are no papers that deal with control performance of piezo-laminated composites under uncertainties with possibilistic parameters and a heuristic approach. The proposed methodology allows accounting several uncertainty parameters that are classically treated in a Monte Carlo framework.

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In this paper, this high-cost approach (MC) is faced with a heuristic (PSO) antioptimization approach that shows advantages in finding extreme values for the uncertain variables. The obtained results using this technique encompass those obtained by traditional MC simulations.

In this paper, the extreme values for displacements, spent energy, natural

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frequencies, etc., could be found using the interval analysis together with a heuristic algorithm approach resulting in accurate solutions and computational time-savings when compared to simple Monte Carlo simulations. In the last example, even with 100,000 simulations for the uncertain variables, Monte Carlo technique was unable to be as accurate as the results from the Fuzzy Interval, especially for lower -cuts levels.

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The process to obtain that type of envelope can be computationally expensive in the case of Fuzzy interval analysis, since a high number of optimizations are necessary. On

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the other hand, when a specific uncertain output variable is sought, the simulation might be affordable and easily performed, depending only on the size of the problem being simulated. Considering a low variability in the uncertain parameters, it is shown that the

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system’s output values presented substantial uncertainty as their fuzzy sets

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representation were widely spread for low α-levels. It is important to note that this methodology is capable to deal with a large number of uncertain parameters problems, and being the main bottleneck, only the performance of the optimization algorithm.

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The importance of the displacement envelope presented in this paper is of

concern for robotic applications in obstacle avoidance. The natural frequency modification analyzed is important since it may affect the interaction behavior like in flutter problems or resonance like in harvesting problems, so their variability should be taken into account in the design stage. Regarding the variability of the electrical input energy, a substantial increase in this parameter might affect applications where low consumption control systems are necessary like in aerospace applications. Finally, as

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indicated by the last example, taking into account the vibration energy uncertainty by the mechanical energy presented a useful metric tool to check the effectiveness of the controller. In view of practical applications, this fuzzy α-cut optimization method is not intrusive and may well adapt to several existing codes and software, allowing the

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technique to deal with a wide range of problems. Material property uncertainties and fabrication defects are a reality and may cause significant modifications in the dynamic behavior of structures and controllers. Therefore, the authors conclude that, in these cases, it is always advisable to investigate the design performance under the presence of

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uncertain parameters and especially in extreme case scenarios.

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