Modal analysis of a Variable Stiffness Composite Laminated plate with diverse boundary conditions: Experiments and modelling

Journal Pre-proofs Modal analysis of a Variable Stiffness Composite Laminated plate with diverse boundary conditions: experiments and modelling Ana Margarida Antunes, Pedro Ribeiro, José Dias Rodrigues, Hamed Akhavan PII: DOI: Reference:

S0263-8223(19)34141-8 https://doi.org/10.1016/j.compstruct.2020.111974 COST 111974

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Composite Structures

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Please cite this article as: Antunes, A.M., Ribeiro, P., Rodrigues, J.D., Akhavan, H., Modal analysis of a Variable Stiffness Composite Laminated plate with diverse boundary conditions: experiments and modelling, Composite Structures (2020), doi: https://doi.org/10.1016/j.compstruct.2020.111974

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Modal analysis of a Variable Stiffness Composite Laminated plate with diverse boundary conditions: experiments and modelling Ana Margarida Antunesa; Pedro Ribeiroa,*, José Dias Rodriguesa, Hamed Akhavana aDEMec/INEGI,

Faculdade de Engenharia, Universidade do Porto, R. Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal * Corresponding author. Tel.: +351 22 508 1721. E-mail address: [email protected] Abstract. The modes of vibration of a Variable Stiffness Composite Laminate were obtained by

experimental

modal

analysis

and

compared

with

the

ones

resulting

from

theoretical/mathematical models. Three types of boundary condition were considered: CFFF, CFCF and FFFF, where C stands for clamped and F for free edges. Frequency response functions were experimentally obtained and employed to identify natural frequencies, modal damping ratios and mode shapes of vibration, using methods known as CMIF - Peak picking and circle-fit. The identified natural frequencies and mode shapes of vibration were compared with the ones resulting from models based on Classical Plate Theory and on First-order Shear Deformation Theory. Although two massive, stiff, steel blocks were bolted with the plate inbetween in order to approach a clamped boundary, the modal properties are still significantly influenced by the flexibility of the resulting fixture. After introducing springs along the boundaries in the mathematical model, to better represent a “real clamped” boundary, quite good agreement between theoretical and experimental results was obtained. The experimental results here presented can be used to validate theoretical models of Variable Stiffness Composite Laminated plates.

Keywords: Variable Stiffness Composite Laminated Plate; vibrations; experimental modal analysis; boundary conditions.

1. Introduction In recent years, composite laminates reinforced with curvilinear fibres stimulated a large interest [1-3]. This is justified by the enlarged design space provided by this type of laminates, which allows designers to fulfil specific requirements in a more efficient way, potentially leading to better performance and weight savings. Because the stiffness changes spatially, these laminates are a type of Variable Stiffness Composite Laminates and often the acronym VSCL is used to designate them. VSCL plates can be used in applications where vibrations are a major concern, notably in aeronautics [4] and wind turbines [5]. Therefore, a number of papers specifically addressing vibrations of VSCL have been published. Some of these works are mentioned in this paragraph; in-depth literature reviews on diverse aspects related to mechanics of VSCL, including vibrations, can be found in [1-3]. Honda et al. [6] proposed a multi-objective optimization method to maximize fundamental frequencies or in-plane strengths, while minimizing the average curvatures of fibres. Vescovini and Dozio [7] computed linear modes of vibration of “monolithic” and sandwich plates with external face-sheets reinforced by curvilinear fibres, and critical buckling loads of sandwich plates, resorting to the Ritz method. Tornabene et al. [8] investigated the modes of vibration of singly and doubly-curved panels reinforced by curvilinear fibres. In a follow-up work [9], modes of vibration of composite sandwich plates and doubly curved shells with variable stiffness were investigated; a considerable variation in the natural frequencies of vibration was achieved by varying the fibre orientation. Linear modes of vibration and buckling modes of prestressed stiffened VS laminated plates subjected to a uniform in-plane end shortening were studied in [10]. Parametric studies showed that, depending on the in-plane load and boundary condition, the linearly varying fibre paths can increase the fundamental frequency of those plates. Viglietti et al. [11] analysed a thin-walled composite box and a NACA 2415 wing. It was found that curvilinear fibres can be used to modify only the desired frequencies, e.g., bending or torsion, without undesired effects on the others. The dynamic stability of a curved VSCL panel subjected to periodic axial compression load was studied in [12]. The linear modes of vibration of variable stiffness laminated composite rectangular plates were studied by Houmat in [13], following a three-dimensional elasticity theory combined with the p-version of the finite element method (FEM) . Also resorting to the p-version FEM, the research group related to this work has carried out a number of analysis on vibrations of VSCLs, addressing linear and non-linear modes of vibration, and dynamic response to external forces, including aeroelastic behaviour [14-19]. 2

In spite of all the interest triggered by VSCL, there is a lack of experimental data on vibration analysis of tow placed laminated panels. To the best of the authors knowledge, there are only two publications: reference [20], a precursor to the present paper, where a shorter work on a VSCL plate with free boundaries is presented, and reference [21], a recent work where vibration tests were performed on a variable-stiffness cylinder with free-free boundary conditions, experimentally identifying natural frequencies and mode shapes. In this paper, experimental modal analysis is carried out on a rectangular Variable Stiffness Composite Plate with three boundary conditions: all edges free (FFFF), one edge clamped and the remaining edges free (CFFF), two opposite edges clamped and the other two free (CFCF). Natural frequencies of vibration, mode shapes of vibration and modal damping ratios are identified. The data presented can be used by other researchers to validate their own theoretical/mathematical models. The experimental results are compared with data computed using two p-version type finite element models, one based on the Classical Plate Theory (CPT) and another based on the First-order Shear Deformation Theory (FSDT) [22]. Although thick steel blocks were used to fix the plate, attempting to approach a clamped boundary, it is found that the stiffness of the grounding structure is far from that of a theoretical clamp, a common occurrence [23]. In order to model the “nearly clamped” or “real clamped” boundary conditions, springs are introduced along the boundaries in the mathematical models, and the values of their stiffness are adjusted so that the models approach the experimental setup. 2. Theoretical models The experimental data will be compared with results from two mathematical/theoretical models for vibrations of VSCL plates, one based on Classical - or Kirchhoff’s - Plate Theory (CPT) and the other on First-order Shear Deformation Theory (FSDT, also known as ReissnerMindlin [24, 25]). The remainder of this section is divided in two parts: the first is a synopsis of the derivation of the ordinary differential equations of motion, given for the sake of completeness; the second part addresses the imposition of the boundary conditions. With the exception of details regarding boundary conditions, the FSDT model is a simplified version of the one presented in [18] for cylindrical shallow shells, and the CPT one is the linear part of the model presented in [15], where geometric non-linearity was considered.

3

2.1. Ordinary differential equations of motion Only bending and small displacements are considered, so the First-order Shear Deformation theory displacement field is simply given by

u  x, y, z, t   zy0  x, y, t  ,

v  x, y, z, t    zx0  x, y, t  ,

w  x, y, z, t   w0  x, y, t  ,

(1)

with u (x,y,z,t), v(x,y,z,t) and w(x,y,z,t) representing displacement components along coordinate axis x, y and z (Figure 1). Functions x0  x, y, t  and y0  x, y, t  represent rotations of a line perpendicular to the middle plane about axis x and y, respectively. These rotations follow the right-hand

rule.

In

the

CPT

displacement

y0  x, y, t   w,0x  x, y, z, t  and

field,

x0  x, y, t   w,0y  x, y, z, t  , where “,x” indicates differentiation with respect to x.

Figure 1. Coordinate system and displacement components.

In the FSDT model, w0  , , t  , x0  x, y, t  and y0  x, y, t  are written as T w  w0  , , t   ff  ,   0   0 x  , , t      0   , , t   0  y  

0

 q t  w        q x  t   0  T   q  t  ff y  ,      y  0

ff x  , 

T

0

(2)

where ff i  ,  are vectors of shape functions and qi  t  are vectors of generalised coordinates. Symbols  and  represent non-dimensional coordinates, which are here related to x and y by x

a , 2

y

b , 2

(3)

with a representing the plate length and b the plate width (Figure 1). It transpires from equation (3) that one p-version finite element suffices in the problem at hand, given that we are analysing 4

a rectangular plate. This feature discharges us from discussing continuity questions between ptype elements, which are addressed in [26, 27]. The number of shape functions and generalised coordinates on Equation (2) is increased until satisfactory convergence is achieved. In the CPT p-element, it is assumed that the transverse displacement obeys a relation equal to the one given for w0  , , t  in Equation (2). Although the shape functions here employed have been frequently used (for example, in [14-16, 18, 28-33]), their expressions are now repeated, since knowing these functions is important to understand how the boundary conditions were implemented in the model. The bi-dimensional shape functions are given by products of one-dimensional shape functions in  and . Taking, for example, the shape functions for the transverse displacements, each term of vector ff w  ,  is given by ffi  ,   fi   fi  

i  1, 2, po2

 i 1  i  Int    1, i  i  po  i  1  po 

(4)

with “Int()” indicating the integer part of the number between brackets. A total of po shape functions is employed in each direction, resulting in po2 bi-dimensional shape functions. The computational code is written so that the same type and number of shape functions are used in

 and ; this observation also applies to the rotational degrees of freedom in the FSDT. Concerning the transverse displacements, the following four cubic functions 1 3 1 1 1 1 1 f1        3 , f2        2   3 , 2 4 4 4 4 4 4 1 3 1 1 1 1 1 f3         3 , f 4          2   3 2 4 4 4 4 4 4

(5)

plus higher order polynomials given by

f r   

 1  2r  2n  7 !! r 2 n1 , r>4,  2n n ! r  2n  1 ! n 0

INT  r 2 

n

(6)

are used. The hierarchical set of polynomials given by (6) has advantageous properties and has been time and again successfully employed (see, e.g., [32] and references therein); according to the literature, these polynomials were derived from Rodrigue’s form of Legendre

5

polynomials [31]. At =-1 and =1, f r   is equal to zero and so is its derivative with respect to . In the FSDT model, a set of polynomials is required for the rotations as well. We use two linear functions, which are equal to 1 at one boundary and to zero at the other t1()=

1 1 – , 2 2

t2()=

1 1 +  2 2

(7)

and, as higher order polynomials,

tr   

 1  2r  2n  5!! r 2 n1 , r > 2.  2n n ! r  2n  1 ! n 0

INT  r 2 

n

(8)

Functions ti() have been successfully employed to approach rotations in FSDT pversion type elements in [15, 27, 34, 35] and in Third-order Shear Deformation Theory (TSDT) p-elements in, amongst other references, [14, 33]. Both for x  t  and for y  t  , p shape functions are necessary in each direction, resulting in p2 bi-dimensional shape functions for each rotation. Considering free and undamped vibrations, employing the principle of virtual work, with strains and stresses related by expressions that can be found in [15, 18], the ordinary differential equations of motion are derived. In the FSDT case, they have the following form

Μ 33 0  w 44  0 Μ x  0 0 

 ..  q t  0   w   K 33 K 34   ..   43 0  qx  t    K  K 44  K b44    K 53 K 54  K 54 Μ55y   ..  b  qy  t      

  q w  t   0       K 45  K b45  qx  t    0  ,  0   K 55  K 55 b  q  t    y    K 35

(9)

ii ij where Μk and Kk are, respectively, consistent mass and stiffness matrices. Subscript k

indicates the physical origin of the matrix, with symbols  and b standing for shear and bending, respectively. Superscripts i and j range from 3 to 5 in order to facilitate the identification of the matrices in reference [15], where a similar model was applied to analyse geometrically nonlinear vibrations of VSCLs. The detailed expressions of the mass and stiffness matrices are given in Appendix 1. The double dot over the generalised coordinates indicates that they are differentiated twice with respect to time.

6

In the classical, or Kirchhoff, plate theory, the free vibration equations of motion are even simpler ..

33 M 33 w qw t   K b qw t   0 .

(10)

33 The mass matrix Mw is equal to the mass matrix that appears in Kirchhoff type,

laminated composites reinforced by straight fibres; it is given by equation (8a) of reference [31]. The stiffness matrix K 33 b is defined by equation (A6) of reference [18]. For the sake of completeness, these two matrices are written in Appendix 1. Irrespectively of the boundary conditions, the number of degrees of freedom of the FSDT model is nFSDT  po2  2 p2 and the number of degrees of freedom of the CPT model is

nCPT  po2 . The user can select independently the number of shape functions for each displacement component. However, to limit the number of case studies, we decided to use the same number, i.e. po=p. 2.2. Implementation of boundary conditions As stated in the introduction, three boundary conditions are of interest here: all edges free (FFFF), one edge clamped and the three other edges free (CFFF), two opposite edges clamped and two opposite edges free (CFCF). The way boundary conditions were considered in the model is now explained in some detail, because modelling clamped boundary conditions so that they approach the experimental ones can constitute an issue. Free boundaries To approach the FFFF case, one must employ shape functions given by equation (5) both in the CPT and in the FSDT models; in the FSDT model, shape functions given by equation (7) must also be used. Naturally, the p-elements are enriched with shape functions from equations (6) and (8) until convergence is achieved. Clamped boundaries One way to simulate a clamped boundary would be removing, from the stiffness and mass matrices, the rows and the columns related to the generalised coordinates that correspond to the displacement components to prescribe; this option was not followed here. Instead, a “penalty method”, often used in structural mechanics [36, 37], was chosen to approach a

7

(theoretical) clamped boundary. In the FSDT p-element, the penalty method was implemented as described in the ensuing lines, which address the CFFF example. If boundary =-1 is clamped, then certainly

y0  1,, t   x0  1,, t   0, w0  1,, t   0 ,

(11)

and we also impose – though imposing this condition is not an universal option amongst researchers - that the middle surface of the plate is perpendicular to the clamp by enforcing w0  1, , t   0. 

(12)

To obtain a displacement field that approximately respects equations (11) and (12), the diagonal elements of the stiffness matrix that multiply by generalised coordinates related to shape functions t1(), both for  y0  , , t  and for x0  , , t  , and the diagonal elements of the stiffness matrix that multiply by coordinates related to shape functions f1() and f2() are increased by adding a large number, , known as “penalty parameter”. After a number of test cases, it was found that by using  =10p max(K), with p = 3∼5, as penalty parameter (with max(K) representing the element of matrix K with

maximum absolute value), good approximations to ideal clamps were obtained. Instead of one, two penalty parameters can be employed, on a sub-matrix basis (one parameter for the transverse displacements, the other one for the rotations). The approach in the CPT p-element is very similar, but one has to constrain the appropriate derivatives of w0, instead of  y0 and x0 . To verify the theoretical models for CFFF plates, comparisons were carried out with [38], where a laminated composite plate with straight fibres is analysed; the natural frequencies and mode shapes of vibration obtained are shown in [39]. Tests were also performed on CFFF and CFCF plates in isotropic materials, and the natural frequencies and mode shapes of vibration were compared with the ones given in [40]. In all cases, the agreement was very good. We should nonetheless inform that numerical issues occurred when carrying out numerical tests with models that employed penalty parameters. First, it was verified that the penalty parameter should not be arbitrarily large, because extremely large values lead to poor conditioning of the stiffness matrix and, consequently, to errors in the solution of the eigenvalue problem required to obtain the natural frequencies and natural mode shapes of vibration. This kind of problem is mentioned on page 146 of [37], in a static analysis setting. With the computational code here employed, ill-conditioning did not have noticeable 8

consequences on the static problems addressed (simple tests, not included in this paper), where the displacement field due to an external force was determined. Without intending to delve into theoretical numerical analysis, it is not surprising that ill-conditioning of the stiffness matrix affected more the eigenvalue than the static equilibrium problem, because the former requires more mathematical operations, also involves the mass matrix and, in some methods, the inversion of matrices. An extremely large, artificial, stiffness term will necessarily be related to a relatively small term in the mass matrix, leading to numerical difficulties, as stated in [41]. The issue mentioned in the former paragraph is more manifest in the FSDT than in the CPT model and becomes noticeable when many shape functions are employed. Our tests indicate that if about (the number depends on the eigenvalue algorithm and on the number of digits used in the computations) 14 shape functions per dimension are not enough to achieve an accurate approximation with a model employing a penalty parameter, one should consider refining the mesh, instead of further increasing the number of shape functions. Boundaries supported on distributed springs As will be seen later, even using very stiff blocks to hold the plates in the experimental setup, the real boundaries can significantly differ from theoretical clamps. To better approach the actual boundary conditions, a mathematical model where two opposite sides of the plate are connected to translational (a “translational spring” is also known as “linear spring”, with “linear” indicating that the springs elongate/contract along one direction or line) and torsional springs distributed along the boundaries, as represented in Figure 2. Kw1 and Kw2 represent the stiffness per unit length of translational springs; K y1 and K y 2 the torsional stiffness per unit length of torsional springs that rotate about the y axis; Kx1 and Kx 2 the torsional stiffness per unit length of torsional springs that rotate about the x axis. In all distributed spring stiffnesses, indexes 1 and 2 respectively indicate the boundaries =-1 and =1. Edges connected to springs along the boundaries can be designated as “elastically restrained edges” [42], [43]. In a FSDT setting, the virtual works done by the distributed forces and moments due to the presence of the springs are

9

b 2

1

b 2

1

 WKFSDT    K w w  -1, , t   w  -1,  d    K w w 1, , t   w 1,  d 1

w

1

b 2

1

1

2

1

b 2

(14)

b 2

(15)

 WKFSDT    K  y  -1, , t   y  -1,  d    K  y 1, , t   y 1,  d  1

y

y1

1

1

1

b 2

1

y2

2

2

1

   K x  -1, , t  x  -1,  d    K x 1, , t  x 1,  d  WKFSDT  x

1

x1

1

1

1

x2

2

2

(13)

Figure 2. Schematic representation of a plate on distributed springs.

 K 33 w1 ,

K 44x

and

K 55y

 

 ..  q w  t   K 33  K 33 Μ 33   0 0 w1   w   ..   44 43 0  qx  t     K   0 Μ x     0 Μ55y   .. K 53  0  qy  t    

K 34 K 44  K b44  K 44x K  K 54

10

  q  t   0    w    45 45 K  Kb   q x  t     0   0  55   K 55  K 55 q t   b  K y     y     K 35

54 b

(16)

If the Classical Plate Theory is applied, the virtual work done by the distributed force exerted by the distributed translational springs is still given by Equation (13). The virtual works associated with the rotational springs become b 2

 WK    K   w,x  -1, , t      w,x  -1,   d    K 1

1

y

y1

1

1

b 2

 WK    K  w,y  -1, , t     w,y  -1,   d    K 1

y

1

x1

y2

1

1

x2

  w 1, , t      w 1,   b2 d ,x

,x

 w 1, , t     w 1,   b2 d ,y

,y

(18)

   K 44x

and

K 55y



3. Plate properties and experimental setup 3.1 Characteristics of the plate The plate analysed was manufactured using a Fibre Placement Machine, at the National Aerospace Laboratory, NLR, the Netherlands. It is a rectangular plate with 10 layers and the following geometric properties (Figure 3): a = 0.4 m, b = 0.3 m, h = 0.0018 m. The thickness, represented by h, is not actually constant; the value adopted is an average value.

Figure 3. Representation of a reference curvilinear fibre path  k(x). As represented in Figure 3, the angle the fibre makes with respect to the x axis changes linearly with x [44], so that the angle in lamina k is defined as 11

(17)



k

 x 

2 T1k  T0k  a

x  T0k

(19)

T0k and T1k represent the fibre’s angle at the lamina’s centre and edges, respectively. This fibre k

k

path is represented by < T0 | T1 . If we consider the whole plate to be analysed here - that is, before



k 0

clamping

it

-

its

layup

| 1k , 0k | 1k , 0k | 1k , 0k | 1k , 90º | 90º

is

the

following:

 , with  =30º and  =10º. s

k 0

k 1

The material of the plate is Hexply AS4/8552, a high performance material for aerospace structures, whose properties can be consulted in Table 1. The values of E1, E2, G12, and v12 were provided by the manufacturer, with the indication that they were obtained following ASTM D3039 [45], ASTMD695 [46] and ASTMD3518 [47]. These values are average ones; for example, the value used for E1 is the mean of the modulus obtained in tension and the modulus obtained in compression, with the latter two moduli also resulting from averages of values from diverse tension/compression tests. The mass density was obtained by weighting the plate and calculating the mass per unit volume; hence, even here there are uncertainties, because the thickness of the plate is not constant. Values for the transverse shear moduli were not provided, hence it was assumed that G13 is equal to G12 and that G23 is smaller, as usually occurs in fibre reinforced composites. Table 1. Material properties of the VSCL plate tested Property Value Units Longitudinal Young modulus, E1

126.3

GPa

Transverse Young modulus, E2

8.765

GPa

In-plane shear modulus, G12

4.92

GPa

Transverse shear modulus, G13

4.92

GPa

Transverse shear modulus, G23

3.35

GPa

Poisson’s ratio, v12

0.334

-

Mass density, 

1556.7

kgm-3

The experimental tests in a condition that approaches a free plate were performed before the present work, by hanging the plate so that it stands vertically [20]; details on the setup implemented for those tests are given in the same reference. The following two sub-sections solely regard the CFFF and CFCF cases and are based on [39], which contains additional details. 12

3.2 Experimental setup To approach a clamped boundary condition in one edge of the plate (CFFF), two steel blocks were bolted together, fixing the plate between them. The CFCF condition is achieved in a similar way. The bolts were tightened using a dynamometric wrench. The edges clamped were the ones with 300 mm length. Photographs of the plate when clamped on one edge and on two opposite edges are shown in Figure 4.

(a)

(b)

Figure 4. (a) CFFF plate and (b) CFCF plate. When the two edges are clamped, expression (19) is valid as it stands, but with the k k k k following parameters: a=0.324 m, T0 =30º, T1 =13.8º ( T1k in < T0 | T1  is the orientation of the

fibre at the clamp). When only one edge is clamped, the following expression should be used instead of (19)

2 T1k  T0k   a x0       T0k  2 2 k

(20)

with  representing the length of the whole plate (=0.4 m), a the length of the plate between the clamp and the opposite free edge (here a =0.36 m), and x0 the length of the part of the plate clamped between the steel blocks (x0=-a). Non-dimensional coordinate  still goes from -1 to

13

k k k k 1, and T0 =30º, T1 =10º ( T1k in < T0 | T1  is the orientation of the fibre at the end of the whole

plate, including the part between the steel blocks). 3.3 Measurement system and procedure The excitation is produced by an electromagnetic shaker (LDS V201) suspended from a frame and connected to a lightweight force transducer (Brüel & Kjær Type 8203; mass 3.2 g) attached to the plate through a flexible rod (Figure 4). The shaker is driven by a power amplifier, which is connected to multi-channel analyser PULSETM Type 3560C, controlled by PULSETM software from Brüel & Kjær. The force transducer and the accelerometers are also connected to the analyser. Accelerations were measured on the accessible nodes of a 1010 grid, which was drawn on the whole plate. The number of accessible nodes is 109 in the CFFF plate and 108 in the CFCF case (Figure 5). Two sets of measurements were performed, with excitation points marked as 1 and 2 in Figure 5. Random excitations were applied in frequency ranges wide enough to excite the first seven or eight modes of vibration. An estimator usually known as H1(), which is the ratio between the cross spectrum of excitation and response and the autospectrum of the excitation [23], was used to obtain the FRFs here shown; a second estimator, H2(), was employed to compute the coherence γ2() of each frequency response function (FRF) [39]. Forty averages were performed in each measurement and the Hanning window [23] was applied.

(a)

(b)

Figure 5. Measurement grid and excitation points 1 and 2 (dimensions in mm): (a) CFFF plate and (b) CFCF plate.

14

Two different accelerometers were used in the measurements. Because it provided data with less noise in clamped nodes, Brüel & Kjær Type 4507, mass 4.8 g, was used for measurements in those nodes. ENDEVCO Type 27A11, mass 0.8 g, was used in the remaining nodes, to avoid excessively increasing the mass in specific locations of the plate. The identification of modal parameters from the experimentally obtained frequency response functions was performed in stages. First of all, the FRFs from the measurements made with each excitation point were imported to ARTeMIS Modal [48], which then processes them and presents the corresponding CMIF (Complex Mode Indicator Function). The CMIF is defined by the square of the singular values of the FRF, obtained from the singular value decomposition of the FRF submatrix. For the extraction of natural frequencies and mode shapes, a method named “CMIF - Peak picking" available in the software is used [39]; it does not identify damping ratios. The obtained mode shapes are usually complex, so they were first transformed to the Real domain and, afterwards, exported to MATLAB® in order to compute the contour plots shown in Section 4. The peaks of the CMIF “occur approximately at the damped natural frequencies” [49] whilst our mathematical models provide undamped natural frequencies. As we will see, the modal damping ratios are quite small, so the differences between damped and undamped natural

frequencies

are

minor.

 [23]code

4. Results and analyses The chief goals of this section are to present the modal properties identified experimentally and verify the adequacy of the models introduced in Section 2. A number of details regarding verifications on the quality of the measurements and on the accuracy of the experimental modal identification are not shown here, interested readers can consult [39]. 4.1 CFFF plate Frequency response functions were experimentally obtained on all points indicated in Figure 5 (a), for both excitation points; the frequency range was 10 Hz to 210 Hz and the frequency resolution 0.125 Hz (1600 spectral lines). Figure 6 shows two CMIFs, each obtained using all FRFs per excitation point. These CMIFs allow one to have a good understanding of the values

15

of natural frequencies of vibration. The 4th and 7th modes are the only ones that can solely be

CMIF (dB)

CMIF (dB)

identified from a single set of FRF data.

Frequency (Hz) (a) 1st excitation point

Frequency (Hz) (b) 2nd excitation point

Figure 6. CMIFs of all frequency response functions, CFFF plate, as provided by ARTeMIS modal software.   i Table exp

2   i

CPT

 i

FSDT

  are

given

between

brackets



iTheor  iexp  iexp

(21)

with Theor either replaced by CPT or by FSDT. Deliberately, the expression for i does not take the absolute value of the weighted difference, so that the sign of i conveys whether the estimation is larger or smaller than the experimental value. In the CPT model, po=12; in the FSDT model, po= p =12.

16

Table 2. Experimental and theoretical (ideal clamp) natural frequencies, in Hz, CFFF plate, po= p12. M od e

1

iexp 

 CPT  i   FSDT17.16 i (18.3%)

2

     38.57 (2.87%)

3

    85.78 (0.384%)

4

     107.3 (18.8%)

5

6

     131.9 (11.7%)

     186.7 (1.55%)

7

     214.4 (12.2 %)

All theoretical natural frequencies of vibration on Table 2 are above the experimental ones; this is expectable, since the steel blocks certainly permit some level of displacement and cross-section rotation. The 2nd, 3rd and 6th theoretical natural frequencies are quite close to the experimental ones, with relative errors below 3.5%. However, the other frequencies present unsatisfactory errors, achieving more than 18% in the first and fourth modes. FSDT performs only marginally better than CPT, because the plate is quite thin. The theoretical mode shapes of vibration are generally similar to the experimental ones [39], with the maximum dissimilarity occurring on the 4th natural mode shape of vibration (shown in Figure 7).

Mode 4 - experimental

Mode 4 - theoretical

Figure 7. Experimental (ARTeMIS modal, CMIF – Peak Picking) and theoretical fourth mode shape of vibration, CFFF plate. Although the steel clamps are fairly thick and rigid, with measured accelerations that have rather small magnitudes (as transpires from the mode shapes shown in Figure 8), it results from the previous comparison that the “experimental clamp” is still somewhat far from imposing a theoretical clamped boundary. Therefore, the CPT and FSDT models with distributed springs at the boundaries were used in an effort to approach the real boundary condition. After a few attempts, the following values were selected for the stiffness per unit 17

length of the distributed springs: Kw1 =106 Nm-2; K y1 =1.5103 N and Kx1 = 109 N. Because this boundary is not truly a clamped boundary, when mentioning the theoretical model we will designate it as RFFF, where ‘R’ stands for rigid (with large, but finite, stiffness). When referring to experiments, these boundary conditions will be designated as CFFF, like in Table 2, or as RFFF, like in Table 3, depending on the context. Table 3 provides the values obtained with the distributed springs replacing the ideal clamped boundary in the models, along with the experimental ones. On average, the predicted natural frequencies of vibration became closer to the experimental results by replacing the ideal clamp by distributed springs, with the root-mean-square error (RMSE) of the CPT predictions falling from 12.4% to 4.52%, and the RMSE of the FSDT predictions falling from 11.6% to 4.37%. The maximum absolute value of the difference decreases from about 19% to about 8%. Table 3. Experimental and theoretical natural frequencies, in Hz, RFFF case, po= p13 Mode

1 14.500

2 37.500

3 85.750

4 90.375

5 118.125

6 183.875

iCPT

14.77 (1.88%)

35.50 (-5.34%)

79.36 (-7.45%)

94.22 (4.24 %)

120.0 (1.58 %)

183.4 202.3 (-0.251%) (5.92%)

iFSDT

14.71 (1.47%)

35.22 (-6.06%)

78.99 (-7.89%)

93.94 (3.94 %)

118.2 182.6 200.2 (0.0847%) (-0.682%) (4.83%)



exp i

7 191.000

As shown in Figure 8, the mode shapes of the CPT theoretical model with distributed springs are similar to the ones identified from experiments. The improvement in the prediction of the fourth mode shape can be verified by comparing Figure 4 with Figure 7. The 1st mode shape is the one where the largest differences between theory and experiments occur; the coherence decreased at lower frequencies and the first mode was the hardest mode to define by experimental modal analysis [39]. One possible reason for the difficulties is the lower sensitivity of the lightweight accelerometer at lower frequencies. We should note here that large excitation amplitudes were avoided, in order to minimize non-linear effects in the dynamic response. In these and in the following cases, the transverse component of the mode shapes computed using the FSDT models are always extremely similar to the ones of the respective CPT models, so only the latter are represented.

18

Mode 1 - experimental

Mode 1 – theoretical

Mode 2 - experimental

Mode 2 - theoretical

Mode 3 - experimental

Mode 3 - theoretical

Mode 4 - experimental

Mode 4 - theoretical

19

Mode 5 - experimental

Mode 5 - theoretical

Mode 6 - experimental

Mode 6 - theoretical

Mode 7 - experimental

Mode 7 - theoretical

Figure 8. Experimental (ARTeMIS modal, CMIF – Peak Picking) and theoretical (CPT) mode shapes, RFFF plate. The estimates for the modal damping ratios (i) obtained using the circle-fit method [23] are shown in Table 4, together with the respective standard deviations (i). The modal damping ratios shown are mean values of estimates from eight selected FRFs (four per excitation point). The VSCL tested is lightly damped, with the damping ratios varying from about 0.4% to almost 2%. The in-house code that applies the circle-fit method allows one to easily control and assess the quality of the results. Nonetheless, there is some uncertainty on the estimations, with, in particular, a rather high standard deviation for the first mode. 20

Table 4. Experimental modal damping ratios (ξi %) and standard deviations (i %),  Mode 1 2 3 4 5 6 7 1.965 0.500 0.647 0.496 0.517 0.580 0.479 i

i

0.679

0.122

0.093

0.032

0.060

0.070

0.047

We would like to end this section with a word about convergence and the number of significant digits of the values shown in the tables. The experimental results were written with the number of digits provided by default by the modal identification software; clearly, they are not all significant digits. In the FSDT model, CFFF case, po = p =12 appears to lead to values of the natural frequencies that are precise – as far as convergence is concerned, under the hypothesis that support this model and with the penalty parameter chosen - until, depending on the mode, the 3rd or the 4th digit. The number of one-dimensional shape functions employed in the CPT model was also po = 12. Regarding the RFFF case, apparently po = 13 is enough to accurately compute the natural frequencies up to their third or fourth significant digit in the CPT model. Although the FSDT values were also given with four digits, the same level of convergence was not confirmed with the FSDT model using po= p =13.

4.2 CFCF plate During the test on the CFCF plate, temperature was monitored in order to safeguard the plate from noteworthy contractions/expansions, which could significantly modify the modes of vibration.  Figure 9 The frequency range of the FRF measurements was 0-400 Hz, with a frequency resolution of f=0.125 Hz (3200 spectral lines). The two excitation points chosen can be seen in Figure 5 (b).

21

CMIF (dB)

CMIF (dB)

Frequency (Hz) (a) 1st excitation point

Frequency (Hz) (b) 2nd excitation point

Figure 9. CMIF of all frequency response functions, CFCF plate, as provided by ARTeMIS modal software.   i Table exp

5  i

CPT

 i

FSDT

  are

given

between

brackets  [39]        

22

Table

5.

 po= p11. M od e†

1

2

3

4

5

iexp

95.125

117.625

165.000

258.000

275 .75 0

153.1

211.7

315.7

 

 

 

iCPT 136.7  

iFSDT 136.5

(43.5%)

152.7

211.1

314.6

(29.9%)

(27.9%)

(21.9%)

6

307 .25 0 379. 403 9 .0 (37. (31. 7%) 2%) 378 401 .4 .5

7

8

361.625

388.375

486.1

473.7

 

 

483.7 (33.8%)

471.9 (21.5%)

(37. (30. 2%) 7%) † The mode number follows the values of the experimental natural frequencies; the theoretical values were chosen by matching the mode shapes, an exercise that was not always possible (further explanations on the main text).

The introduction of the second clamp clearly led to larger differences between the theoretical ideal CFCF plate and the corresponding experimental results, not only in the values of natural frequencies, but also in the mode shapes. Therefore, the CPT and FSDT models with distributed springs at the boundaries will be again employed, in order to approach the real boundary conditions. After a few numerical tests, it was decided to use the following values for the stiffnesses per unit length: Kw1 = Kw2 = 4106 Nm-2; K y1 = K y 2 =1.2103 N and Kx1 = Kx 2 =109 N. Following the reasoning explained before, we will designate these boundary conditions as RFRF. Two reasons justify the fact that stiffness values differ from the CFFF case: (1) due to mounting restrictions, the dimensions of the part of the plate that stands in-between the steel blocks is not the same in the CFCF and CFFF cases; (2) the torque applied in the two setups was slightly different (10 Nm in the CFFF case and 8 Nm in the CFCF case). The comparison between the first eight experimental and theoretical natural frequencies is exhibited in Table 6. In the CPT model, po= 13 apparently leads to values with 4 significant digits (actually, in most modes po= 11 is enough). More shape functions are required in the FSDT model; the values given were computed with po=p=15.

23

Table 6. Experimental and theoretical natural frequencies, in Hz, RFRF case; po=13 in CPT, po= p15 in FSDT. Mode



exp i

iCPT iFSDT

1 95.125

2 117.625

3 165.000

4 258.000

5 275.750

6 307.250

7 361.625

8 388.375

97.35

116.1

175.7

277.5

286.3

321.6

405.4

441.4

(2.34%)

(-1.28%)

(6.48%)

(7.57%)

(3.81%)

(4.67%)

(12.1%)

(13.7%)

97.19 (2.17%)

114.0 (-3.05%)

172.8 (4.75%)

275.2 (6.68%)

284.1 (3.02%)

308.0 (0.235%)

392.7 (8.59 %)

437.0 (12.5%)

The improvement achieved by considering springs distributed along the boundaries is very large, even though some relative differences are still somewhat high, particularly in the higher modes. The FSDT model provided lower natural frequencies, since it is more flexible and takes inertia effects more into account than the CPT model. In most cases, closer values to the experimental ones were computed using the FSDT model. The experimental and theoretical mode shapes of the RFRF plate are presented in Figure 10. The theoretical mode shapes shown were obtained using the CPT model; the transverse components of the mode shapes using FSDT are very similar. Contrasting with the CFCF theoretical models, now all the predicted mode shapes correspond to the experimental ones, ordered according to the magnitude of the natural frequencies, and the correlation between the mode shapes is very good. There is a significant improvement by considering distributed springs, particularly in the 4th and 5th mode shapes, which were very badly approached by the CFCF models [39].

Mode 1 - experimental

Mode 1 – theoretical

24

Mode 2 - experimental

Mode 2 - theoretical

Mode 3 - experimental

Mode 3 - theoretical

Mode 4 - experimental

Mode 4 - theoretical

Mode 5 - experimental

Mode 5 - theoretical

25

Mode 6 - experimental

Mode 6 - theoretical

Mode 7 - experimental

Mode 7 - theoretical

Mode 8 - experimental

Mode 8 - theoretical

Figure 10. Experimental (ARTeMIS modal, CMIF – Peak Picking) and theoretical (CPT) mode shapes, RFRF plate. The experimental modal damping ratios were as well determined for the RFRF (CFCF) case. The estimates obtained and the respective standard deviation  can be consulted in Table 7. Most modal damping ratios are higher than the ones identified in the CFFF case, possibly because the introduction of a second fixing feature increased dissipation at the boundaries. In most modes, a good fitting of a circle was achieved when using circle-fit method; nevertheless, a significant variability between the values provided by different FRFs occurred in some cases [39], as transpires from the values of the standard deviations.

26

Table 7. Experimental modal damping ratiosξi %) and standard deviations (i %), R Mod 1 2 3 4 5 6 7 8 e

i

i

































































4.3 FFFF plate The experimentally identified natural frequencies and mode shapes of vibration of the FFFF plate were published in [20]. Table 8 contains the experimental and theoretical natural frequencies of vibration. In the CPT model po=11, in the FSDT po =p=21 (the values with po =p=11 differed, at most, 0.175 % from the latter). Overall, good agreement between experiments and theory is observed, with the FSDT model performing only slightly better than the CPT model. Table 8. Experimental [20] and theoretical natural frequencies, in Hz, FFFF case, po=11 in CPT, po= p1 in FSDT. Mode



exp i

iCPT iFSDT

1 50.897

2 57.522

3 85.777

4 109.28

5 120.14

6 162.36

7 195.64

49.19

51.51

87.93

110.8

120.7

165.7

208.0

(-3.35%)

(-10.4%)

(2.50%)

(1.42%)

(0.47%)

(2.04%)

(6.31%)

51.48

87.83 (2.39%)

110.5 (1.13%)

120.3 (0.135%)

165.5 (1.93%)

207.2 (5.93 %)

49.09 (-3.35%)

(-10.5%)

Mode 1 - experimental

Mode 1 – theoretical

27

Mode 2 - experimental

Mode 2 - theoretical

Mode 3 - experimental

Mode 3 - theoretical

Mode 4 - theoretical

Mode 4 - experimental

Mode 5 - experimental

Mode 5 - theoretical

Mode 6 - experimental

Mode 6 - theoretical

28

Mode 7 - experimental

Mode 7 - theoretical

Figure 11. Experimental and theoretical (CPT) mode shapes, FFFF plate. The mode shapes obtained by experimental modal analysis are reproduced from [20] with permission from the editors. The modal damping ratios were also identified by the authors of [20], using the even though they were not published. The values obtained are given in Table 9. Table 9. Experimental modal damping ratios, FFFF case. Mode 1 2 3 4 0.57 1.16 0.33 0.43 i

5 0.69

6 1.11

7 0.57

4.4 About the differences between theory and experiments Much was said in the previous sections about the requirement of correctly modelling the boundary conditions. Nevertheless, there are other reasons for the differences between the predictions provided by the theoretical models and the experimentally obtained values. Obviously, the measuring equipment interferes with the system under analysis, as even lightweight accelerometers and force transducers have mass and, in a smaller fraction, alter the stiffness of the system. The material constants also play a role. The in-plane elastic and shear moduli were provided by the supplier of the plates and should be approximately correct. However, we used average values in the mathematical model, since the moduli had different values in tension and compression. The transverse shear moduli – which only have a small importance in this thin plate - were not given by the supplier and we estimated the values after an analysis of the available literature.

29

The mathematical models here employed do not take into account the gaps and overlaps that appear in laminates with curvilinear fibres, manufactured using Automatic Tow Placement Technology [3, 50-52]. The formulation is based on the assumption that fibres are perfectly shifted, all obeying the same analytical formula, but in truth this does not occur, as only the central fibre of each tow is shifted and the other fibres are parallel to it. Although the curvilinear fibre paths of the plate analysed were chosen so that these imperfections are not significant (as well as to avoid fibre kinking and wrinkles), the plate is not as ideally outlined. This is visible to a naked eye inspection and, as written on section 2, was verified when the thickness was measured at various points. Furthermore, one of the surfaces of the real panels is smooth (the surface on the side that is located against the mould surface when the panels are manufactured), whilst the other surface is uneven. This unevenness is increased by the aforementioned gaps and overlaps, which lead to build-down and build-up of fibres in certain areas.

5. Conclusion

Experimental modal analyses were performed on a rectangular composite laminated plate with curvilinear carbon fibres (VSCL), in order to identify its modal properties. The data provided contributes to fill a void in the available literature on VSCL plates, which is extensive in numerical data, but where experimental modal analysis was almost absent. Three boundary conditions were considered: one edge clamped and the other three free (CFFF), two opposite edges clamped and the other two free (CFCF), four edges free (FFFF). The mode shapes and natural frequencies identified via experimental modal analysis can be used by other researchers, to verify their models. Damping ratios were also provided. The experimental results were here compared with the predictions computed using mathematical models based on Classical Plate Theory (CPT) and on First-order Shear Deformation Theory (FSDT). In the FFFF case, where the real boundary conditions are closely approached by the theoretical models, both mathematical models provided natural frequencies and mode shapes reasonably close to the experimental ones. The differences found between theory and experiments are justified by diverse reasons, from the non-exact conformity between the assumed and the real orientation of the fibres to the elastic properties assumed, that are average values and do not consider differences in tension and compression.

30

If it is well-known that theoretical clamped boundary conditions are difficult to achieve in practice, we did not expect the differences between the theoretical and real clamps to be so large, because rather stiff steel blocks were used to hold the plates. The consideration of translational and torsional springs distributed along the boundaries led to theoretical models that predicted the experimentally obtained natural frequencies and natural mode shapes of vibration with much better accuracy. The plate analysed is quite thin, therefore it is not surprising that the CPT and FSDT models provide similar results. Naturally, the natural frequencies of the FSDT model are lower than the ones of the CPT model, so the FSDT model performs marginally better. The difference between CPT and FSDT slightly increased with the introduction of distributed springs in the models, possibly because the torsional springs only lead to independent sub-matrices in the latter. The small advantage of the FSDT model augments with the mode order, due to the increasing importance of transverse shear and rotatory inertia.

Acknowledgements The authors would like to acknowledge Dr. João Amorim for his kind help in the laboratory. This research was carried out in the framework of project Nº 030348, POCI-01-0145-FEDER030348, "Laminated composite panels reinforced with carbon nanotubes and curvilinear carbon fibres for enhanced vibration and flutter characteristics”, funded by FEDER, through Programa Operacional Competitividade e Internacionalização – COMPETE 2020, and by National Funds (PIDDAC), through FCT/MCTES.

Appendix 1 – Stiffness and mass matrices due to bending of the plate The detailed expressions of the bending and shear stiffness matrices of the FSDT p-type element are similar to equations (38) and (39) of reference [34], D12 ( x) D13 ( x)  -ff, x  y  D11 ( x )   -ff, y   0 D ( x ) D ( x ) D ( x )  12 22 23  ff,xx    D13 ( x) D23 ( x) D66 ( x)  -ff  y T  ,y y T

K b44  54 K b

 K 33  43 K   K 53 



-ff, xy K    K    0 45 b 55 b

K 34 K 44 K 54

0 ff,yx

 ff,wx K 35     K 45    ff y   0 K 55  

ff,wy  T   C11 ( x) C12 ( x)  ff,wx 0    wT x  C12 ( x ) C22 ( x )   ff, y -ff   31

ff

y T

0

0   T ff,yx  d , T  ff,xx  

0  d , T -ff x 

(A1.1)

(A1.2)

but with the matrices D(x) and C(x) of VSCL [15]:

 U1  k  x   U 4  k  x    0 3 3 1 0 0  z z    k k 1    k k k 0 1 0  cos  2 k  x    U  x  D x   0      U 4   x   U1   x    2 3 0 0 0  k 1   U 5  k  x    0 0  n

1 U 3  k  x     1  0

1 1 0

0 0 0  cos  4 k  x    U 2  k  x    0  1  1

0 U 3  k  x    0  1

0 0 1

1    1 sin  4 k  x    ,  0  

0 0 1

1 1  sin  2 k  x    0 

(A1.3)

n 1 0 1   U 7  k  x    C  x     hk  U 6  k  x       0 1 0     k 1 0  1  sin  2 k  x    U 7  k  x      1 0  

0 cos  2 k  x     1

(A1.4)

.

The expressions of Ui, i=1-7, can be found in [15]. Letter  in equation (A1.4) represents a shear correction factor; the value used was 5/6 [24]. The expressions of the mass matrices due to the transverse displacement and to the cross section rotation are equal to the expressions written in equation (42) of reference [34]: Μ 33  w  0   0

0 Μ44x 0

 w wT  ff ff 0    0    h  0    Μ55y    0 

0 h2 y y T ff ff 12 0

    d 0   2 h x x T  ff ff  12 0

(A1.5) 1

33 The matrix M 33 w of the CPT p-element is equal to the matrix M w of equation (A1.5).

Stiffness matrix K 33 b of the CPT p-element is, as written in equation (A6) of reference [18], the following: w wT w wT w wT w wT K33 b   ff,xx D11  x ff,xx  ff,xx D12  x ff,yy  2ff,xx D13  x ff,xy  ff,yy D12  x ff,xx + 

w wT D23  x ff,xyw T  2ff,xyw D13  x ff,xxw T  2ff,xyw D23  x ff,yy  +ff D22  x ff,xyw T +2ff,yy w ,yy

wT 4ff,xyw D33  x ff,yy d

32

(A1.6)

Appendix 2 – Matrices due to boundary springs Matrix K 33 w1 , due to the translational springs, is 1 1 b b w wT K 33  -1,  d  1 K w2 ff w 1,  ff wT 1,  d w1  1 K w1 ff  -1,  ff 2 2

(22)

 rotational springs are 1 1 2 2 K 44x   Kx ff,w  -1,  ff,wT  -1,  d   Kx ff,w 1,  ff,wT 1,  d 2 1 1  1 b b

K 55y 



1

1

K  y ff ,w  -1,  ff ,wT 1

1 2b 2b d    K  y f ,w 1,  ff ,wT 1,  2 d  2 2  1 a a

(23)

(24)

  are 1 b b K  x ff  x  -1,  ff  x  -1,  d    K  x ff  x 1,  ff  x 1,  d  1 2 1 1 2 2

K 44x 



1

K 55y 



1

1

K  y ff 1

y

 -1,  ff   -1,  y

1 b b   d    K  y ff y 1,  ff y 1,  d  2 1 2 2

(25)

(26)

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Highlights:  The modes of vibration of laminated composite plates reinforced by curvilinear fibres were determined experimentally and theoretically/numerically  The theoretical models were adjusted by considering flexible boundaries  The experimental data presented can be useful for model validation CRediT author statement Ana Margarida Antunes: Investigation, Validation, Data Curation, Visualization, Writing Review & Editing, Software. Pedro Ribeiro: Methodology, Software, Writing - Original Draft, Funding acquisition, Supervision, Project administration. José Dias Rodrigues: Investigation, Methodology, Writing - Review & Editing, Supervision. Hamed Akhavan: Writing - Review & Editing, Resources. Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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