Vibration prediction of thin-walled composite I-beams using scaled models 1
2
3
Mohamad Eydani Asl
*
, Christopher Niezrecki, James Sherwood, Peter Avitabile 4
5
Department of Mechanical Engineering, University of Massachusetts Lowell, One University Avenue, Lowell, Massachusetts 01854 6
7
8
Abstract 9
10
Scaled models of large and expensive structures facilitate in understanding the physical behavior of the large structure during 11
operation but on a smaller scale in both size and cost. These reduced-sized models also expedite in tuning designs and material 12
properties, but also could be used for certification of the full-scale structure (referred to as the prototype). Within this study, the 13
applicability of structural similitude theory in design of partially similar composite structures is demonstrated. Particular emphasis 14
is placed on the design of scaled-down composite I-beams that can predict the fundamental frequency of their corresponding 15
prototype. Composite I-beams are frequently used in the aerospace industry and are referred to as the back bone of large wind 16
turbine blades. In this study, the governing equations of motion for free vibration of a shear deformable composite I-beam are 17
analyzed using similarity transformation to derive scaling laws. Derived scaling laws are used as design criteria to develop scaled-18
down models. Both complete and partial similarity is discussed. A systematic approach is proposed to design partially similar 19
scaled-down models with totally different layup from those of the full-scale I-beam. Based on the results, the designed scaled-20
down I-beams using the proposed technique show very good accuracy in predicting the fundamental frequency of their prototype. 21
22
Keywords: Wind turbine blade, Similitude, Sub-component, Composite I-beam, Distorted layup scaling 23
24
25
1. Introduction 26
27
28
The certification process for a wind turbine blade starts with coupon testing of the materials that are used in the manufacture of the 29
blade and is finalized with full-scale testing of the blade. Coupon testing is relatively quick and cost-effective, but it is not fully 30
representative of the structural integrity of the blade. In contrast, full-scale testing is time consuming and very expensive (e.g. 31
hundreds thousands of dollars). As blade lengths continue to increase, the logistics of full-scale testing become more challenging 32
and the test time increases as the resonant frequency of the blade decreases. Therefore, the total time for doing a specific number 33
of fatigue cycles increases. Subcomponent testing can bridge the gap between coupon and full-scale testing of the blade. If 34
meaningful scaled-down models can be designed that are representative of their parent components in the full-scale blade, then 35
blade certification can be expedited and the confidence of blade manufacturers for introducing new materials (e.g. bio-based 36
resins) into the industry can grow. 37
38
Interest and activity in the testing of wind-turbine-blade subcomponents has gained momentum in recent years and this interested 39
has led to a number of case studies of a variety of parts of utility-scale wind turbine. Mandell et al. [1] tested composite I-beams 40
with flanges and shear webs out of components which are used in the cross section of wind turbine blades. Stiffness and strain 41
measurements that were observed in a four-point bending test of the beams were in agreement with predictions from simple beam 42
theory and finite element analysis. Cairns et al. [2] studied the root section of the blade where the root specimens represented a 43
single insert of a blade root into the hub joint. The primary focus of the study was manufacturing, but a significant amount of static 44
and fatigue strength data were generated by performing pull-out tests. Mandell et al. [3] conducted a study with the focus on skin-45
stiffener intersections and sandwich panel closeout. Their goal was to predict skin-stiffener fracture loads and evaluate 46
performance at locations where the sandwich panel transitions into the normal laminate. 47
48
A few studies investigated the performance of adhesive joints and bond lines of a wind turbine blade using subcomponent testing. 49
The idea was to test the static and fatigue properties of the shear web to spar cap bond under stress states that are representative of 50
those seen during field service of a wind turbine blade. Sayer et al. [4] proposed an asymmetric three-point bending test where 51
they used a custom beam configuration which has come to be called a Henkel beam. It was meant to give a comparable 52
*
Corresponding author. Tel.: +1 978 934 2584; fax: +1 978 934 3048.
E-mail addresses: Mohamad_Eydaniasl@student.uml.edu (M. E. Asl), Christopher_Niezrecki@uml.edu (C. Niezrecki), james_sherwood@uml.edu (J. Sherwood),
Peter_Avitabile@uml.edu (P. Avitabile).
© 2016. This manuscript version is made available under the Elsevier user license
http://www.elsevier.com/open-access/userlicense/1.0/
combination of bending moment and shear forces as a three-point bending test while reducing the stress concentrations at the 53
clamped end. The specimen was used for a parametric study, investigating the influence of the design and manufacturing variables 54
on shear web to spar cap adhesive joints [5]. Zarouchas et al. [6] performed a static four-point bending test on two symmetric I-55
beams. The tested I-beams represented the spar cap and shear web structure inside a wind turbine blade. 56
57
Although subcomponent testing is usually categorized as laboratory-scale test rigs, there are a few mid-level blade test approaches 58
that fall into the subcomponent category. Jensen [7] conducted a study of the structural static strength of a box girder of a 34-m 59
wind turbine blade, loaded in flap-wise direction. A combination of experimental and numerical work was used to address the 60
critical failure mechanisms of the box girder of wind turbine blade. White et al. [8] developed a dual-axis test setup on a truncated 61
37-m wind turbine blade which combined resonance excitation with forced hydraulic loading to reduce the total test time required 62
for evaluation. Subcomponents in this category are known as “Designed Subcomponents”. Such subcomponents provide an 63
accurate evaluation of the structural performance of the blade and can be used to characterize the dominant failure modes. 64
However, such tests are involved, expensive and hard to implement on a laboratory scale. 65
66
The designed subcomponent (referred to as the “model”) regardless of the size and complexity needs to be correlated with the full-67
scale component (referred to as the “prototype”, see Fig.1). The connection between the scaled model and the prototype must be 68
based on the structural parameters that predict the behavior of the system under consideration. Similitude theory can extract 69
scaling laws from the governing equations of the system to connect the response of the scaled-down model to the prototype. To 70
take advantage of the scaling laws, a properly scaled model should be designed to work well with the derived scaling law. In other 71
words, the designed scaled model should be able to predict the response the prototype accurately by using the derived scaling 72
laws. Otherwise, the experimental data of the scaled model cannot be correlated with the prototype, and therefore, the designed 73
model cannot be representative of its corresponding prototype. As the model is not always an exact scaled-down replica of the 74
prototype, a logical methodology should be implemented to design scaled models that can be used with scaling laws to predict the 75
response of the prototype. 76
77
Similitude theory is an analytical tool for determining the necessary and sufficient conditions of similarity between two systems. 78
These similarity conditions may be derived directly from the governing equations of the system which lead to more specific 79
similarity conditions than dimensional analysis. Simitses and Rezaeepazhand [9] established a technique that could be applied 80
directly to the governing equation of a system to derive the scaling laws. The derived scaling laws are then used to predict or 81
estimate the response of a prototype by the response of its associated model. The buckling response of an orthotropic and 82
symmetric cross-ply laminated plate was investigated as a benchmark in that study. In later studies, they analyzed the vibration of 83
scaled laminated rectangular plates [10]. Additionally, they studied the effect of axial and shear load on stability of scaled 84
laminated rectangular plates [11, 12]. According to their results, the scaling laws that are obtained directly from the governing 85
equations can be used with perfect accuracy for cross-ply laminates, while for the angle-ply laminates the scaling laws did not 86
show good accuracy. Later, this method was extensively used in their works regarding the vibration response of laminated shells 87
[13, 14]. 88
89
The design of scaled models for a structure made of composite materials is more challenging than the same structural shape made 90
of isotropic materials. Laminated structures cannot be scaled down to any arbitrary size because of the practical difficulties in 91
scaling the thickness of the plies of a laminate. Design of scaled-down composite models with the same layup as the prototype 92
will be limited by manufacturing constraints because only fabrics with specific thicknesses are available in industry. Therefore, 93
making scaled-down composite models with a completely similar lamination scheme as the prototype is hard to implement. 94
Therefore, use of partially similar scaled models can be considered as an alternative. Although ply-level scaling [9] within the 95
scope of complete similarity has been implemented successfully, the design of partially similar models is still lacking a systematic 96
methodology. 97
98
Within this study, which is an extension of Authors’ previous work on design of scaled composite models [15-17], similitude 99
theory is applied to the governing equations of motion for vibration of a thin-walled composite I-beam [18] to design scaled-down 100
composite I-beams which are representations of the spar caps and shear web of a utility scale wind turbine blade [19]. This paper 101
presents the first work to design partially similar laminated models with totally different layups than their prototype using a 102
systematic approach to predict the vibration response of the prototype. The main strength of the proposed approach compared to 103
the numerical assessments is that a representative scaled model can be used for validation of the numerical estimation of the 104
structural parameters as numerical assessments require experimental validation. Use of the scaled-down models to assess the 105
natural frequencies of the full-scale structure provides an inexpensive approach to validate the numerical estimation of the 106
frequencies using experimental data prior to full-scale testing of the structure. However, the proposed approach has its own 107
limitations in terms of the design and manufacturing of the scaled model. The design of the scaled model should follow a logical 108
methodology to ensure the similarity between the model and the prototype. Moreover, the feasibility of manufacturing of the 109
designed model needs to be considered in the design procedure. Both of the mentioned limitations are analyzed carefully in this 110
study to design the models that are similar to their prototype and practical to be manufactured. The models designed with the 111
proposed approach are shown to have a very good accuracy in predicting the fundamental frequency of their corresponding 112
prototype. 113
114
When combined into an I-beam, the spar caps and shear web constitute the backbone of wind turbine blades. Thus, they are a 115
major structural component of the blade and make for an interesting case study for subcomponent test design. In this study, 116
similitude theory is applied to the subcomponent test concept, to develop scaled-down models that are representative of the 117
dynamic characteristics of the I-beam structure inside a utility-scale blade. Governing equations for vibration of a simply-118
supported thin-walled shear deformable composite I-beam are considered to derive the scaling laws. Derived scaling laws are used 119
as design criteria to develop scaled models that can accurately predict the fundamental frequency of the prototype. Within this 120
analysis, the geometry and ply scheme of the prototype were based on a portion of Sandia National Laboratories 100-m long wind 121
turbine blade near the maximum chord. Both complete and partial similarity cases are studied based on the considered layup for 122
the prototype. Models with different sizes and ply schemes are designed that can accurately predict the fundamental frequency of 123
their corresponding prototype. 124
125
126
127
Fig. 1. Prototype (top) and Models with different scales and layups (bottom) showing 128
the associated displacement boundary conditions. 129
130
131
132
2. Description of the mathematical model 133
134
This section presents the governing equations of motion for flexural vibration of a shear-deformable thin-walled composite I-beam 135
shown in Fig. 2. The closed-form solution is derived for the natural frequencies of the I-beam with a simply-supported ends. 136
Similarity transformation is subsequently applied to derive the scaling laws. The objective is to design scaled-down beams that 137
can be used to predict the fundamental flexural frequency of the prototype using scaling laws. For this study, free vibration in the 138
y-direction in the absence of thermal effects for a symmetric I-beam is considered. 139
140
141
142
Fig. 2. Geometry of the I-beam and the associated reference frame [18]. 143
144
A shear-deformable model was considered in this study as shear flexibility has a remarkable effect on natural frequencies of the 145
laminated beams [20]. Neglecting all coupling effects due to the symmetric geometry, the equations governing free flexural 146
vibration in y-direction are given by [18] and are the well-known Timoshenko beam equations: 147
148
(1a)
(1b)
where
denotes the rotation of the cross section with respect to x axis shown in Fig. 2, V the displacement in the y direction, the 149
prime
is used to indicate differentiation with respect to z and the dot
is used to denote differentiation with respect time. The 150
density and area of cross section are expressed by and A, respectively, and
is the area moment of inertia with respect to the x-151
axes. The terms
and
are shear and flexural rigidity of thin walled composite (
com
) with respect to the x-axis, 152
respectively which could be expressed as: 153
154
(1c)
(1d)
155
where
and
are elements of extensional, coupling and bending stiffness matrices for a composite layup [21]. 156
The repeated index denotes summation where index varies from 1 to 3 where the indices 1 and 2 represent the top and bottom 157
flanges, respectively, and 3 is for the web, as shown in Fig. 2, and
denotes width of the flanges and web. Although 158
employment of the in-thickness shear deformability may result in over-prediction of the frequencies specially for shorter beams 159
and higher modes [22], considered model in this study was shown to predict the fundamental frequency of the I-beam geometry 160
accurately and in good agreement with analytical solution and finite element results [18]. The closed-form solution for flexural 161
natural frequencies in the y-direction may be directly calculated for the simple-support boundary condition as [18]: 162
163
(2)
164
where n indicates the mode number. 165
166
3. Scaling laws for vibration of composite I-beams 167
168
Natural frequencies for vibration of a simply supported shear-deformable composite I-beam are described by Eq. (2). To derive 169
the scaling laws, it is assumed that all the variables of the governing equations for the prototype (
can be connected to their 170
corresponding variables in a scaled model (
by a one-to-one mapping. Then, the scale factor for each variable can be defined 171
as
which is the ratio of each variable of the prototype to that of the scaled model. Rewriting Eq. (2) for the model 172
and prototype and applying similarity transformation, the scaling laws can be extracted as follows based on the standard similitude 173
procedure [9]: 174
175
176
(3)
(4a)
(4b)
Eq. (3) is referred to as the design scaling law and is a prerequisite for deriving the constitutive scaling laws Eq. (4a-b). Design 177
scaling law Eq. (3) denotes that the ratio of the flexural rigidity to the shear rigidity must be equal to the square of the length of the 178
two scales for complete similarity between two shear-deformable beams to exist. With Eq. (3) satisfied, the ratio of the natural 179
frequencies between the two scales can be obtained using constitutive scaling laws Eq. (4a-b). Eq. (4a) predicts the natural 180
frequencies of the prototype using the flexural stiffness ratio of the model and the prototype (i.e. λ
EI
) while Eq. (4b) 181
predicts the natural frequencies of the prototype using the shear stiffness ratio of the model and the prototype (i.e. 182
λ
GA
). However, if models are designed such that they satisfy Eq. (3) where λ
l
2
= λ
EI
/ λ
GA
then Eqs. (4a) and (4b) yield 183
the same results in predicting the natural frequencies of the prototype. Upon expanding Eq. (3) by using the definition of 184
the flexural and shear rigidities as provided in Eq. (1c-1b), specific design scaling laws can be written. For the specific case of 185
assuming that the flanges and shear web of the I-beam are identical, Eq. (1b) and Eq. (1c) can be simplified as follows: 186
187
(5)
(6)
188
The second term of Eq. (5) vanishes for a symmetric layup for the flanges and the web, i.e. B
11
0. Also the third term in Eq. (5) 189
is negligible for a thin-walled beam, i.e. D
11
0. This study focuses on the fundamental frequency (n=1), but the same 190
methodology can be utilized for the higher modes. Upon applying the similarity transformation to Eq. (5) and Eq. (6), the 191
following scaling laws can be found: 192
193
(7)
(8)
By substitution of Eq. (7) and Eq. (8) into Eq. (3), the following scaling laws are derived for a symmetric thin-walled laminated I-194
beam: 195
196
(9)
(10)
197
Eq. (9-10) are specific versions of Eq. (3) which are valid for a thin-walled composite I-beam with a symmetric layup. To use the 198
constitutive response scaling laws Eq. (4a-b) to predict the vibration frequencies of the prototype using a scaled model, Eqs. (9-10) 199
must be valid between the prototype and the scaled model. Eq. (9) can be satisfied by assuming that that model and the prototype 200
have the same geometrical aspect ratio. However, validation of Eq. (10) depends on the respective layup schemes of the prototype 201
and of the model. 202
203
In the next sections of this study, validation of Eq. (10) is investigated by considering different layups for both prototype and 204
scaled models. To have Eq. (9) satisfied for all the case studies, all models are assumed to have the same geometrical aspect ratio 205