Probabilistic free vibration analysis of composite structures using the Modal Stability Procedure

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Probabilistic free vibration analysis of composite structures using the Modal Stability Procedure

Q Yin, Frédéric Druesne, P Lardeur

To cite this version:

Q Yin, Frédéric Druesne, P Lardeur. Probabilistic free vibration analysis of composite structures

using the Modal Stability Procedure. International Conference on Noise and Vibration Engineering

and International Conference on Uncertainty in Structural Dynamics, 2016, Leuven, Belgium. �hal-

02958288�

Probabilistic free vibration analysis of composite structures using the Modal Stability Procedure

Q. Yin

, F. Druesne

, P. Lardeur

1

Université de Technologie de Compiègne, Sorbonne Universités, Laboratoire Roberval Centre de Recherches de Royallieu, 60200, Compiègne, France

e-mail: [email protected], [email protected], [email protected]

Abstract

As shown by experimental results, the manufacturing of composite structures induces a quite important variability of the mechanical behavior. In this paper, the Modal Stability Procedure (MSP) based on one nominal finite element analysis and a fast Monte Carlo Simulation (MCS) is developed for probabilistic free vibration analysis of laminated composite structures modeled by finite elements. Material properties (elastic properties, densities…) and physical properties (thicknesses and fiber orientations) are considered as uncertain parameters and are represented by random variables. The variability of natural frequencies is evaluated with the MSP. Two examples: an eight-layer composite square plate and a stiffened ten-layer composite rectangular plate, are studied. The results are compared with those obtained by the direct MCS, considered as a reference, and those presented in the literature. The comparison shows that the MSP provides quite accurate results with high computational efficiency. Moreover, an error indicator which is able to approximately evaluate the error level due to the modal stability assumption is proposed.

1 Introduction

Nowadays composite structures are largely used in various industrial sectors. However, the uncertainties in material and physical properties of composite laminated structures lead to a noticeable variability of responses. Figure 1 shows the frequency response functions (FRFs), observed experimentally with a laser scanning vibrometer, of 8 theoretically identical composite plates. The results highlight a high variability level of FRFs, up to 40dB. Uncertainties in composite structures result in significant variability of dynamic response. Therefore, it is necessary to model the input variability to study its effects on dynamic analysis of composite structures in order to understand and control the phenomenon of output variability.

Figure 1: FRFs variability observed experimentally of 8 theoretically identical composite plates

Probabilistic analysis, which allows to predict the variability of response, has been applied in dynamic analysis of composite structures for several years. The Monte Carlo Simulation (MCS) is a robust method and often used as a reference. With the direct MCS, a large number of random trials are performed, generally leading to several thousands of finite element analyses. This method is used to study the variability of dynamic response for a composite truss by Allegri et al. [1] and for thin-walled composite beams by Piovan et al. [2]. However, this method is quite expensive in computational cost, several fast methods have thus been developed. The perturbation method is a largely used fast probabilistic approach.

Using this method, Singh et al. investigate the influence of uncertain material properties on the variability of natural frequencies for composite plates [3], spherical panels [4] and cylindrical panels [5]. Tripathi et al. [6] develop the perturbation method for conical shells. The variability of natural frequencies and FRFs for composite plates [7,8,9] and shallow doubly curved shells [10] is studied by Dey et al. using methods based on design of experiments. Chakraborty et al. [11] apply polynomial correlated function expansion to study the variability of natural frequencies for composite plates with uncertain material properties, thicknesses and fiber orientations.

However, for laminated composite structures research is still in progress to develop methods which are compatible with any standard finite element software and are able to fast evaluate the variability of response with large numbers of random variables, high levels of input variability and large size models. In statics, Yin et al. [12] have proposed the Certain Generalized Stresses Method (CGSM). In this study, the Modal Stability Procedure (MSP), based on a mechanical assumption of modal stability, is therefore developed for probabilistic free vibration analysis of laminated composite structures. This method has been applied to study the variability of vibration response for homogeneous [13,14] and sandwich [15]

structures. In this paper, the variability of natural frequencies for laminated composite structures is evaluated with the MSP, taking into account uncertain material and physical properties. Two examples are treated: an eight-layer composite square plate and a stiffened ten-layer composite rectangular plate. The accuracy of results is verified by comparing with those obtained by the direct MCS. The computational cost and an error indicator are also investigated.

2 Modal Stability Procedure

2.1 Presentation of the MSP

The MSP, a numerical method which allows to study the variability of natural frequencies and FRFs, has been applied in different contexts [13,14,15]. The studies presented in the literature show that the MSP is able to provide very satisfactory results. In this paper it is developed to study the variability of the natural frequencies of laminated composite structures. This method is based on a mechanical assumption: the mode shapes of the structure are considered independent of input parameters uncertainty.

For modal probabilistic analysis, the MSP requires only one single finite element analysis in the nominal configuration which is compatible with any standard finite element software, Abaqus [16] is used in this study. The mode shapes and modal strains, obtained by the nominal finite element analysis, are used in metamodels allowing the calculation of natural frequencies or FRFs. Once the nominal information has been obtained, a vector of random variables is generated to represent input uncertain parameters. Then a fast MCS is performed using the MSP formulation, which allows to evaluate natural frequencies or FRFs.

The variability (mean value, standard deviation, coefficient of variation and distribution) of natural

frequencies or FRFs is subsequently obtained. This simulation is of course much less expensive than the

direct MCS. The flowchart of the MSP is given in figure 2.

Figure 2: Flowchart of the MSP 2.2 MSP formulation for natural frequency variability

In this paper the MSP is developed to study the variability of natural frequencies of laminated composite plates using a multilayered composite shell theory with transverse shear effects. The structures may be modelled with triangular or quadrilateral shell finite elements. Thanks to the modal stability assumption, a formulation of natural frequency can be written. First of all, the eigenvalue problem in dynamic analysis is given by:

i



K    M (1)

where K and M are respectively the stiffness and mass matrix, 

and 

are respectively the eigenvector and the eigenvalue for the ith mode. Then the expression of the Rayleigh quotient is used in modal analysis and the angular frequency ω can be written as:

2

  K

M

   

  ⁽²⁾

In our study the subscript “0” is used to define the nominal case and the subscript “p” is used to define the perturbed case. The perturbed eigenvector 

can be expressed as:

  

p p

   (3)

According to the MSP assumption,  

is considered 0. Taking into account Equations (2) and (3), the perturbed angular frequency 

is written as:

   

   

0 0 0 0

2

0 0

0 0

   

 

   

T T

p p p p

p T T

p p p p

K K

M M

     

       ⁽⁴⁾

where the input variability is introduced into the stiffness matrix K

and the mass matrix M

. Considering that the structure is modeled by finite elements, elementary matrices are used and Equation (4) can therefore be transformed into:

0 0

2 1

0 0

1





 



e

e

n T

, j p , j , j j

p n T

, j p , j , j j

k m

 

   ⁽⁵⁾

where k

and m

are respectively the perturbed stiffness and mass matrix for the jth element. 

is the nominal elementary eigenvector for the jth element. In Equation (5), the denominator is proportional to the kinetic energy and the numerator expresses the internal strain energy. Our objective is to prevent the calculation of k

and m

using a finite element software. Moreover, k

and m

may not be available with a standard finite element software. So the numerator is developed using the expression of internal energy in each element:

1 2

j elem

int lm, j lm, j

V

     dV ⁽⁶⁾

with V

the volume, 

the stresses and 

the strains for the jth element. For a laminated composite structure, 

can also be expressed as:





1

 2     

p , j

elem T T T T T

int , j p , j , j , j p , j , j , j p , j , j , j p , j , j , j CS ,p , j , j

S

e A e e B B e D A dS

       (7)

where S

is the area of the jth element, e

, 

and 

are respectively membrane strains, curvatures and transverse shear strains for the jth element. They are considered to be nominal due to the MSP assumption and obtained by one finite element analysis. A

, B

, D

and A

are respectively the membrane, membrane-bending coupling, bending stiffness matrix and corrected transverse shear stiffness matrix for the jth element. These matrices contain the input variability.

For the denominator of Equation (5), m

is considered as a concentrated mass matrix and is written as:

1

 

p , j k p , j ,k p , j ,k p , j ,k d

m  S h I / n (8)

where n

is the number of layers, 

and h

are respectively the density and thickness for kth layer of jth element, n

is the number of nodes in the element, I is an identity matrix whose size is 6 n

× 6 n

because there are six degrees of freedom per node. In matrix I the translational inertia terms are unitary while the rotational inertia terms are assumed to be zero.

According to Equations (7) and (8), Equation (5) can be transformed into:





2 1

0 0

1 1



 

   

  

 

e p , j

e l

n T T T T T

, j p , j , j , j p , j , j , j p , j , j , j p , j , j , j CS ,p , j , j

j S

p n n T

, j , j p , j ,k p , j ,k p , j ,k d

j k

e A e e B B e D A dS

I S h / n

     

    ⁽⁹⁾

In Equation (9) it is observed that an integration must be evaluated. A Gauss numerical integration is used.

In our study, the strains are assumed constant and are calculated at the center of each element, which leads to a single Gauss point. Equation (9) then becomes:





2 1

0 0

1 1

e

e l

n T T T T T

, j p , j , j , j p , j , j , j p , j , j , j p , j , j , j CS ,p , j , j

j

p n n T

, j , j p , j ,k p , j ,k p , j ,k d

j k

e A e e B B e D A wdetJ

I A h / n

     

   



 

   

 

  ⁽¹⁰⁾

where w is the weighting coefficient and detJ is the determinant of the Jacobian matrix. Equation (10) is

assumed to be valid for any random value of uncertain parameters. It is used by a fast MCS to evaluate

perturbed natural frequencies for each trial with the expression   2  f . The statistical quantities such as

mean value, standard deviation, coefficient of variation and distribution are then calculated.

3 Eight-layer composite square plate

3.1 Presentation of the example

The first example, an eight-layer E-glass/epoxy composite square plate (300mm  300mm), has been studied by Chakraborty et al. [11] with polynomial correlated function expansion. The stacking sequence is asymmetric [45/-45]

. This plate is relatively thin with a ratio between the thickness and the length

0 012



h / l . . All the edges of the plate are considered to be clamped. Based on the convergence study, a 16 16  mesh with 256 S4R elements (see figure 3 which shows the first mode shape) is used for the probabilistic analysis. The material properties (density  , elastic moduli E

and E

, Poisson’s ratio 

, shear moduli G

, G

and G

), thicknesses h

and ply orientations 

are considered as random. As a result, there are 23 independent uncertain parameters in total. The mean value, the standard deviation, the coefficient of variation and the type of distribution of these uncertain parameters are given in table 1. The variability of the first four natural frequencies is studied.

Figure 3: First mode shape of the square composite plate

Number Uncertain parameter Mean Standard deviation Coefficient of variation Distribution

1 E

(MPa) 4 2 10 . 

^{1512 3.6% Lognormal}

2 E

(MPa) 1 13 10 . 

^{621.5 5.5% Lognormal}

3 

0.3 0.0042 1.4% Lognormal

4 G

(MPa) 4 5 10 . 

^{189 4.2% Lognormal}

5 G

(MPa) 4 5 10 . 

^{189 4.2% Lognormal}

6 G (MPa)

4×10

168 4.2% Lognormal

7  ( t mm /

) 1 9 10 . 

1 9 10 . 

^{1% Lognormal}

8-15 h -

h

(mm) 0.45 0.0135 3% Rayleigh

16-23 

- 

(°) - 1.732 - Uniform

Table 1: Description of uncertain parameters 3.2 Variability of natural frequencies

In this example, the variability of the first four natural frequencies is evaluated by a fast MCS using MSP formulation and a direct MCS. The latter is used as the reference. Figure 4 gives the mean value ^   ^f

and the standard deviation ^   ^f obtained by both methods with 10000 trials, as well as those presented

by Chakraborty et al. [11]. All the uncertain parameters are taken into account. It can be observed that the results obtained with the three approaches are very close for all the four natural frequencies. The error between MSP and direct MCS is lower than 0.7% for the mean value and 1.1% for the standard deviation.

So the MSP provides quite satisfactory results.

Figure 4: Variability of natural frequencies (Hz)

The distribution of natural frequencies is also investigated. The probability density functions of the first four natural frequencies is shown in figure 5. It can be observed that the curves of MSP and direct MCS are very close, with a small shift. The difference between the results obtained by MSP and Chakraborty et al. [11] is more considerable for modes 2 to 4. The authors think that this is due to the fact that Chakraborty et al. [11] used a coarse mesh.

Mode 1 Mode 2

Mode 3 Mode 4 Figure 5: Probability density of natural frequencies (Hz)

Results shown in figures 4 and 5 prove the computational accuracy of the MSP. Then each type of

uncertain parameters (material properties, thicknesses and fiber orientations) is taken into account

individually in order to highlight their influence on the variability of natural frequencies. Figure 6 gives

the coefficient of variation of natural frequencies ^{c.o.v. f}   evaluated by the MSP for each type of uncertain parameters as well as all of them. It can be observed that the output variability depends very little on the mode number. Taking into account all uncertain parameters leads to the highest variability.

Uncertain material properties bring a higher variability than uncertain thickness. However, the output variability is nearly 0 when fiber orientations are uncertain. The results show that the variability of natural frequencies is more sensitive to uncertain material properties and thicknesses.

Figure 6: Influence of different types of uncertain parameters on the variability of natural frequencies 3.3 Error indicator

In our study, the results calculated by the direct MCS are used as the reference to validate the proposed method. But its cost is very high, especially for large size structures because a great number of finite element analyses are performed in this method. The error indicator is therefore a quite interesting challenge, which allows to quickly predict the error level. The objective of the indicator is to estimate the error level of the results obtained by the proposed method MSP compared with the reference, without performing a direct MCS with a large number of trials.

Mode 1 Mode 2

Mode 3 Mode 4

Figure 7: Error indicator on the variability of natural frequencies

In this example, the variability of natural frequencies is evaluated by MSP and direct MCS with 10 or 10000 trials. The errors between these two methods are calculated for 10 trials using the same input data and compared with exact errors, which are those obtained with 10000 trials. The results are given in figure 7. For the mean value, the errors for 10 and 10000 trials are almost the same, which means 10 trials can predict the error level of mean value. For the standard deviation, the difference between the errors obtained with 10 and 10000 trials is higher but remains low. Therefore 10 trials are sufficient to give a good estimation of the error level of standard deviation. According to the above results, an error indicator using 10 trials is able to approximately assess the error level.

4 Stiffened ten-layer composite rectangular plate

4.1 Presentation of the example

The second example concerns a stiffened rectangular plate (figure 8) which finds its interest in many sectors in particular aeronautic and naval industry. A primary plate (210mm  460mm  2mm) is made of 10 layers of taffeta glass tissue and polyester. The tissue is laid in only one direction, which leads to a stacking sequence [0]

. A stiffener in form of inverted “T” is positioned on this plate, with base 50mm, height 25mm and thickness 4mm. It consists of two “L” parts which are made of 10 layers of the same material as the primary plate. The geometric shape of this stiffened plate is shown in figure 8. The primary plate and the stiffener are manufactured by the infusion process and assembled by adhesive bonding.

Figure 8: Photograph and geometric shape of the stiffened plate

The mechanical properties of the material are: E

 24.5GPa, E

 25GPa, 

 0.164, G

 4.36GPa,

13





G G 4.1GPa and   1.79g/cm

. Three groups of uncertain parameters are introduced and lead to four cases: in case 1, the material properties are considered as uncertain with input coefficient of variation

 E ,E ,

,G

  15

c.o.v.  ,G ,G ,  % ; in case 2, the layer thicknesses h

are uncertain with

 

^{ 15}

c.o.v. h % ; in case 3, the fiber orientations 

are uncertain with ^{ }  

^{  3} ; in case 4, all the

uncertain parameters are taken into account. These levels of input variability are high. For one given layer,

the nine parameters are assumed to be independent. The parameters of different layers are either

dependent, which means 9 random variables in total, or independent, which means up to 270 random

variables considering that there are three parts and each of them has ten layers. The variability of the first

ten frequencies is studied by MSP with 10000 trials. A mesh with 5704 elements is used here following a

convergence study. Figure 9 shows the first two modes shapes.

Mode 1 Mode 2 Figure 9: First two mode shapes of the stiffened composite plate 4.2 Variability of natural frequencies

In this study, the MSP with 10000 trials is used to evaluate the variability of the first ten natural frequencies. Moreover, the MSP and direct MCS using 10 trials are also applied as an error indicator.

Figure 10 gives the coefficient of variation of natural frequencies in case 1 with dependent or independent uncertain material properties, as well as the errors provided by the error indicator. In can be observed that the errors are very low, which means that the MSP provides accurate results. The variability level for each mode is moderate. The output variability is comprised between 9% and 11% with dependent uncertain parameters, which is lower than the input variability. With independent uncertain parameters, the output variability is much smaller, with a value comprised between 2.5% and 3.5%. The output variability levels are lower than the input variability levels, due to a compensation phenomenon which is more pronounced when the parameters are independent.

Dependent (7 variables) Independent (210 variables) Figure 10: Variability of natural frequencies with uncertain material properties (case 1)

The results of case 2 are shown in figure 11. It can be observed that the results obtained by MSP and direct MCS are quite close. When the uncertain thicknesses are dependent, the variability of natural frequencies is nearly constant with a value comprised between 14% and 15%, which is close to the input variability.

This output variability is higher than in case 1. With independent uncertain thicknesses, the variability of

each frequency is comprised between 3% and 4%.

Dependent (1 variable) Independent (30 variables) Figure 11: Variability of natural frequencies with uncertain thicknesses (case 2)

The results of case 3 are given in figure 12. In this case it can be observed that the output variability levels are very small and the errors between MSP and direct MCS are relatively large. The output variability levels are less than 1% with dependent parameters and less than 0.3% with independent parameters. In this context, the influence of fiber orientation uncertainty can be neglected considering the low output variability, compared with uncertain material properties and thicknesses.

Dependent (1 variable) Independent (30 variables) Figure 12: Variability of natural frequencies with uncertain fiber orientations (case 3)

At last the variability of natural frequencies is studied with all the uncertain parameters. The results are given in figure 13. It can be observed that the errors between the MSP and direct MCS are very small. The output variability of the first ten modes is comprised between 4% and 6% with independent uncertain parameters. While all uncertain parameters are dependent, the input variability leads to a quite large output variability, reaching 18%.

Dependent (9 variables) Independent (270 variables)

Figure 13: Variability of natural frequencies with all types of uncertain parameters (case 4)

5 Computational cost

Previous work proves that the MSP is able to provide accurate results, but the computational time is an important issue as well. This method consists of two parts: a nominal finite element analysis, which allows to obtain nominal mode shapes and modal strains, and a post-processing where 10000 MCS trials are performed with Matlab. The cost of the MSP is compared with the direct MCS, which leads to 10000 finite element analyses in our study. In order to better compare these two methods, the cost is evaluated by counting floating operations (flops). While the number of layers has a weak effect on the cost, an example of an eight-layer composite plate is studied.

The numbers of flops of one MSP trial and one direct MCS trial are compared. For one MSP trial, the number of flops is obtained by counting flops of each algebraic operation [14]. For one direct MCS trial, namely one finite element free vibration analysis, the number of flops is in the order of n n b

with n

the number of modes, n

the number of degrees of freedom and b the half-band of global stiffness matrix.

Figure 14 shows the number of flops of one MSP trial and one direct MCS trial for evaluating the variability of one mode. It is observed that the cost of direct MCS is always over 100 times bigger than the cost of MSP. Moreover, the number of flops of the direct MCS increases faster than the MSP with the number of elements, which means that the efficiency of the MSP is more competitive for a large size model.

Figure 14: Number of flops of one MSP trial and one direct MCS trial

Then the cost of MSP is evaluated by considering the number of equivalent finite element analyses, as shown in figure 15. Compared with the direct MCS which costs 10000 finite element analyses, the MSP is considerably more efficient. The larger the model size is, the less equivalent finite element analyses the MSP costs. It costs about one equivalent analysis for a model of over 100000 elements.

Figure 15: Number of equivalent finite element analyses for the MSP

6 Conclusion

In this paper, the Modal Stability Procedure (MSP) has been developed to study the variability of natural frequencies of laminated composite structures. Material properties (densities, elastic and shear moduli…) and physical properties (thicknesses, fiber orientations) are considered as uncertain parameters. The method is based on the mechanical assumption that the mode shapes are weakly sensitive to uncertain parameters. Thanks to this assumption, modal element strains and mode shapes are extracted following one finite element calculation in the nominal configuration. Then a fast MCS is carried out using a metamodel allowing the calculation of natural frequencies.

A first example concerning a laminated square plate and a second example concerning a stiffened laminated rectangular plate are studied in this paper. The variability of natural frequencies is evaluated.

The results calculated by the MSP are compared with those obtained by the direct MCS, considered as a reference, and those presented in the literature. The comparison shows that the MSP provides quite accurate results. In these examples, the variability of material properties and thicknesses has a more important effect on the output variability, while uncertain fiber orientations lead to a very small output variability. An error indicator using 10 trials has been proposed, which is able to predict the error level between MSP and direct MCS without performing a large number of finite element analyses. Compared with the direct MCS, the MSP is quite competitive in computational cost, in particular for large size models.

References

[1] G. Allegri, S. Corradi, M. Marchetti, Stochastic analysis of the vibrations of an uncertain composite truss for space applications, Composites Science and Technology, Vol. 66, No. 2 (2006), pp. 273- 282.

[2] M.T. Piovan, J.M. Ramirez, R. Sampaio, Dynamics of thin-walled composite beams: Analysis of parametric uncertainties, Composite Structures, Vol. 105 (2013), pp. 14-28.

[3] B.N. Singh, D. Yadav, N.G.R. Iyengar, Natural frequencies of composite plates with random material properties using higher-order shear deformation theory, International Journal of Mechanical Sciences, Vol. 43, No. 10 (2001), pp. 2193-2214.

[4] B.N. Singh, D. Yadav, N.G.R. Iyengar, Free vibration of laminated spherical panels with random material properties, Journal of Sound and Vibration, Vol. 244, No. 2 (2001), pp. 321-338.

[5] B.N. Singh, D. Yadav, N.G.R. Iyengar, Free vibration of composite cylindrical panels with random material properties, Composite Structures, Vol. 58, No. 4 (2002), pp. 435-442.

[6] V. Tripathi, B.N. Singh, K.K. Shukla, Free vibration of laminated composite conical shells with random material properties, Composite Structures, Vol. 81, No. 1 (2007), pp. 96-104.

[7] S. Dey, T. Mukhopadhyay, S.K. Sahu, G. Li, H. Rabitz, S. Adhikari, Thermal uncertainty quantification in frequency responses of laminated composite plates, Composites Part B:

Engineering, Vol. 80 (2015), pp. 186-197.

[8] S. Dey, T. Mukhopadhyay, S. Adhikari, Stochastic free vibration analysis of angle-ply composite plates - A RS-HDMR approach, Composite Structures, Vol. 122 (2015), pp. 526-536.

[9] S. Dey, T. Mukhopadhyay, A. Spickenheuer, S. Adhikari, G. Heinrich, Bottom up surrogate based approach for stochastic frequency response analysis of laminated composite plates, Composite Structures, Vol. 140 (2016), pp. 712-727.

[10] S. Dey, T. Mukhopadhyay, S. Adhikari, Stochastic free vibration analyses of composite shallow

doubly curved shells - A Kriging model approach, Composites Part B: Engineering, Vol. 70 (2015),

pp. 99-112.

[11] S. Chakraborty, B. Mandal, R. Chowdhury, A. Chakrabarti, Stochastic free vibration analysis of laminated composite plates using polynomial correlated function expansion, Composite Structures, Vol. 135 (2016), pp. 236-249.

[12] Q. Yin, F. Druesne, P. Lardeur, The Certain Generalized Stresses Method for static analysis of multilayered composite plates with variability of material and physical properties, Composite Structures, Vol. 140 (2016), pp. 360-368.

[13] É. Arnoult, P. Lardeur, L. Martini, The modal stability procedure for dynamic and linear finite element analysis with variability, Finite Elements in Analysis and Design, Vol. 47, No. 1 (2011), pp.

30-45.

[14] F. Druesne, M.B. Boubaker, P. Lardeur, Fast methods based on modal stability procedure to evaluate natural frequency variability for industrial shell-type structures, Finite Elements in Analysis and Design, Vol. 89 (2014), pp. 93-106.

[15] F. Druesne, M. Hamdaoui, P. Lardeur, E.M. Daya, Variability of dynamic responses of frequency dependent visco-elastic sandwich beams with material and physical properties modeled by spatial random fields, Composite Structures, Vol. 152 (2016), pp. 316-323.

[16] Abaqus Analysis User’s Guide version 6.14, Hibbitt, Karlsson & Sorensen, Pawtucket, RI (2014).