# Free Vibration of Composite Thin-Walled Beams with Chordwise Asymmetric Closed Cross-Sections

Authors: Keun-Taek Kim
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

In this paper, some analytical results via extended Galerkin method on free vibration characteristics of an anisotropic composite beam, which is modeled as a nonuniform thin-walled structure with a chordwise asymmetric closed cross-section and corrected the warping functions, are newly presented. For this study, nonclassical parameters such as warping restraint, transverse shear flexibility, and structural couplings induced by two special configurations, such as circumferentially uniform stiffness (CUS) and circumferentially asymmetric stiffness (CAS), are incorporated. And also, design parameters of the beam associated with preset angles, pretwist angles, taper ratios, and section ratios are additionally investigated. The results of this study could play an important role in more efficient designs of composite thin-walled beams.


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## Body

## 1. Introduction

In the theoretical explicit analyses of thin-walled beams, the closed cross-sections of most aircraft wings and rotor blades have been using almost symmetrical contours in the related literatures based on the author’s knowledge [1–9]. A thin-walled beam theory originally developed by Song in his doctoral dissertation [2] has been extensively used in the literature for the study of dynamic response [3, 5, 6], structural control [4, 7], and static aeroelasticity [8, 9].However, in general, aircraft wings have an asymmetrical cross-section in the chordwise direction. Therefore, in this study, we investigated cross-sectional characteristics through the mathematical modeling of asymmetric cross-section and the correction of the warping function. In order to compensate the warping function of the asymmetric cross-section in the chordwise direction, the shape function of the cross-section contour is derived on the basis of an arbitrary pole of the cross-section and the shape function based on the pole is determined so as to satisfy the equilibrium conditions of the warping constraint. And then, the relation of the geometric cross-section is modified based on the pole of the new position and the warping function is corrected and applied to the governing equation.In this paper, the beam modeled as a chordwise asymmetric cross-section is analyzed for free vibration characteristics numerically, and in order to investigate the effect of the asymmetry of the cross-section, a comparative study was conducted on a symmetrical cross-section model with the same chordwise length and cross-section thickness as a special case of the considered model. In the mathematical modeling, warping restraint, transverse shear flexibility, and structural couplings induced by circumferentially uniform stiffness (CUS) and circumferentially asymmetric stiffness (CAS) configurations [1, 4] are considered.In addition, the present paper examined the effects of the following parameters, such as preset angle, pretwist angle, taper ratio, and section ratio on the eigenfrequencies of the beam. Also, we compared the case where the warping function was corrected and the case where the warping function was not corrected. As a result, in this study, numerical examples were carefully analyzed and compared to investigate the effect of various design parameters on the natural frequencies that provide the basis for dynamic characteristics of beams.The results presented as families of curves or tables, which may be used for the direct estimation of the eigenfrequencies for any given beam.


## 2. Mathematical Modeling

### 2.1. Basic Assumptions

The following basic assumptions have been used in this study [1, 2, 4, 9]:
(i)
The cross-sections do not deform within their own planes, i.e., no in-plane deformations are allowed
(ii)
Transverse shear effects are incorporated. Moreover, the transverse shear strainsγxy and γyz are uniform over the entire beam cross-section
(iii)
Although the rate of twist is assumed as constant in Saint-Venant torsion, it cannot be assumed to be constant along the beam axis while a warping restraint exists but it has been assumed to be a function of the axis coordinates in this paper
(iv)
Primary and secondary warping effects are sufficiently important to be included. The first one is related to the warping displacement of points on the midline cross-section and the second one is related to the points off the midline contour
(v)
In the absence of an internal pressure field, the resultant hoop stress is negligibly small as compared to the remaining stresses
(vi)
The beam has a nonuniform and asymmetric laminar flow airfoil profile, and in this sense, the beam is considered to be linearly tapered in both the horizontal and vertical planes
(vii)
Deformations are small and linear elasticity theory has been applied


### 2.2. Geometric Configuration of the Model

In this study, Figure1 shows a chordwise asymmetric closed cross-section of a composite thin-walled cantilevered beam composed of 2 contours, that is, biconvex and ellipse. The pole (Pm) of the section is located at connection points of the 2 different contours on the chord line.Figure 1
A chordwise asymmetric closed cross-section.And also, the considered beam in Figure2 is modeled as an aircraft wing with preset angle (α0, so-called initial angle of attack), pretwist angle (γ0), taper ratio (σ), and section ratio (ξ) for the analysis.Figure 2
The mathematical model.


### 2.3. Governing Equations

In principle of the theory for beams, shells and plates, and massive bodies by using the equations of three-dimensional continuum theory should be able to derive with taking advantage of the factors which serve to distinguish each type of structure. In this sense, the theory of plates and shells constitutes a two-dimensional approximation of the three-dimensional elasticity theory, while solid cross-section and thin/thick-walled beams are both one-dimensional approximations of the three-dimensional continuum theory [1, 2].Therefore, using the equations of the 3-D continuum theory, the considered thin-walled beam model was induced to 1-D equations as follows but omitted the process of the transformation in this paper; see reference [1].
(1)ux,y,z,t=u0z,t−ys,z−ndxdsϕz,t,vx,y,z,t=v0z,t−xs,z−ndydsϕz,t,wx,y,z,t=w0z,t+xs,zθyz,t+ys,zθxz,t−Fws,zϕ′z,t+ndydsθyz,t−dxdsθxz,t−as,zϕ′z,t,where s denotes the arc length measured along the circumferential coordinate (whose origin is arbitrarily but conveniently chosen) and n denotes the coordinate in the thickness direction. In addition,
(2)θxz,t=γyzz,t−v0′z,t,θyz,t=γxzz,t−u0′z,t,Fws,z=∫0srns,z−ψs,zds=F1s,as,z=−xs,zdxds−ys,zdyds=F2n,s,rns,z=xs,zdyds−ys,zdxds,ψs,z=∮rns,zds∮ds:torsionalfunction,where ∮⋅ds is the integral along the closed midline contour and rns=xsl+ysm is a geometric quantity (see Figure 3), where l=cosn,x, m=sinn,x denote direction cosines. Also, F1s is the primary warping function and F2n,s is the secondary warping function.Figure 3
The coordinate system.The boundary value problem, which consists of differential equations and boundary conditions, can be conveniently derived using the extended Hamilton’s principle, which is as follows:(3)∫t1t2δT−δV+δWdt=0,δu0=δv0=δw0=δθy=δθx=δϕ=0att=t1,t2,where δT and δV denote the kinetic energy and strain energy, respectively, and δW is the virtual work due to nonconservative forces.As a result, the governing equations and boundary conditions are as follows but also omitted the process of the transformation in this paper; see reference [1].


#### 2.3.1. Governing Equations

(4)δu0:a41w0′+a42θy′+a43θx′+a44θy+u0′+a45θx+v0′−a46ϕ″+a47ϕ′′−b1u¨0−b2ϕ¨=0,δv0:a51w0′+a52θy′+a53θx′+a54θy+u0′+a55θx+v0′−a56ϕ″+a57ϕ′′−b1v¨0+b3ϕ¨=0,δw0:a11w0′+a12θy′+a13θx′+a14θy+u0′+a15θx+v0′−a16ϕ″+a17ϕ′′−b1w¨0+b3θ¨y+b2θ¨x−b7ϕ¨=0,δθy:a21w0′+a22θy′+a23θx′+a24θy+u0′+a25θx+v0′−a26ϕ″+a27ϕ′′−a41w0′+a42θy′+a43θx′+a44θy+u0′+a45θx+v0′−a46ϕ″+a47ϕ′−b3w¨0+b5θ¨y+b6θ¨x−b9ϕ¨′+b15θ¨y−b13θ¨x−b17ϕ¨′=0,δθx:a31w0′+a32θy′+a33θx′+a34θy+u0′+a35θx+v0′−a36ϕ″+a37ϕ′′−a51w0′+a52θy′+a53θx′+a54θy+u0′+a55θx+v0′−a56ϕ″+a57ϕ′−b2w¨0+b6θ¨y+b4θ¨x−b8ϕ¨′−b13θ¨y+b14θ¨x+b16ϕ¨′=0,δϕ:a71w0′+a72θy′+a73θx′+a74θy+u0′+a75θx+v0′−a76ϕ″+a77ϕ′′−a61w0′+a62θy′+a63θx′+a64θy+u0′+a65θx+v0′−a66ϕ″+a67ϕ′−b4+b5+b14+b15ϕ¨−b3v¨0+b2u¨0−−b7w¨0−b9θ¨y−b8θ¨x+b10ϕ¨′−b17θ¨y+b16θ¨x+b18ϕ¨′=0.


#### 2.3.2. Boundary Conditions

(5)δu0:a41w0′+a42θy′+a43θx′+a44θy+u0′+a45θx+v0′−a46ϕ″+a47ϕ′=0,δv0:a51w0′+a52θy′+a53θx′+a54θy+u0′+a55θx+v0′−a56ϕ″+a57ϕ′=0,δw0:a11w0′+a12θy′+a13θx′+a14θy+u0′+a15θx+v0′−a16ϕ″+a17ϕ′=0,δθy:a21w0′+a22θy′+a23θx′+a24θy+u0′+a25θx+v0′−a26ϕ″+a27ϕ′=0,δθx:a31w0′+a32θy′+a33θx′+a34θy+u0′+a35θx+v0′−a36ϕ″+a37ϕ′=0,δϕ:a71w0′+a72θy′+a73θx′+a74θy+u0′+a75θx+v0′−a76ϕ″+a77ϕ′−a61w0′+a62θy′+a63θx′+a64θy+u0′+a65θx+v0′−a66ϕ″+a67ϕ′′−−b7w¨0−b9θ¨y−b8θ¨x+b10ϕ¨′−b17θ¨y+b16θ¨x+b18ϕ¨′=0,δϕ′:a61w0′+a62θy′+a63θx′+a64θy+u0′+a65θx+v0′−a66ϕ″+a67ϕ′=0,where aiji,j=1,2,…,6 are stiffness coefficients and bjj=1,2,…,18 are mass coefficients.


## 2.1. Basic Assumptions

The following basic assumptions have been used in this study [1, 2, 4, 9]:
(i)
The cross-sections do not deform within their own planes, i.e., no in-plane deformations are allowed
(ii)
Transverse shear effects are incorporated. Moreover, the transverse shear strainsγxy and γyz are uniform over the entire beam cross-section
(iii)
Although the rate of twist is assumed as constant in Saint-Venant torsion, it cannot be assumed to be constant along the beam axis while a warping restraint exists but it has been assumed to be a function of the axis coordinates in this paper
(iv)
Primary and secondary warping effects are sufficiently important to be included. The first one is related to the warping displacement of points on the midline cross-section and the second one is related to the points off the midline contour
(v)
In the absence of an internal pressure field, the resultant hoop stress is negligibly small as compared to the remaining stresses
(vi)
The beam has a nonuniform and asymmetric laminar flow airfoil profile, and in this sense, the beam is considered to be linearly tapered in both the horizontal and vertical planes
(vii)
Deformations are small and linear elasticity theory has been applied


## 2.2. Geometric Configuration of the Model

In this study, Figure1 shows a chordwise asymmetric closed cross-section of a composite thin-walled cantilevered beam composed of 2 contours, that is, biconvex and ellipse. The pole (Pm) of the section is located at connection points of the 2 different contours on the chord line.Figure 1
A chordwise asymmetric closed cross-section.And also, the considered beam in Figure2 is modeled as an aircraft wing with preset angle (α0, so-called initial angle of attack), pretwist angle (γ0), taper ratio (σ), and section ratio (ξ) for the analysis.Figure 2
The mathematical model.


## 2.3. Governing Equations

In principle of the theory for beams, shells and plates, and massive bodies by using the equations of three-dimensional continuum theory should be able to derive with taking advantage of the factors which serve to distinguish each type of structure. In this sense, the theory of plates and shells constitutes a two-dimensional approximation of the three-dimensional elasticity theory, while solid cross-section and thin/thick-walled beams are both one-dimensional approximations of the three-dimensional continuum theory [1, 2].Therefore, using the equations of the 3-D continuum theory, the considered thin-walled beam model was induced to 1-D equations as follows but omitted the process of the transformation in this paper; see reference [1].
(1)ux,y,z,t=u0z,t−ys,z−ndxdsϕz,t,vx,y,z,t=v0z,t−xs,z−ndydsϕz,t,wx,y,z,t=w0z,t+xs,zθyz,t+ys,zθxz,t−Fws,zϕ′z,t+ndydsθyz,t−dxdsθxz,t−as,zϕ′z,t,where s denotes the arc length measured along the circumferential coordinate (whose origin is arbitrarily but conveniently chosen) and n denotes the coordinate in the thickness direction. In addition,
(2)θxz,t=γyzz,t−v0′z,t,θyz,t=γxzz,t−u0′z,t,Fws,z=∫0srns,z−ψs,zds=F1s,as,z=−xs,zdxds−ys,zdyds=F2n,s,rns,z=xs,zdyds−ys,zdxds,ψs,z=∮rns,zds∮ds:torsionalfunction,where ∮⋅ds is the integral along the closed midline contour and rns=xsl+ysm is a geometric quantity (see Figure 3), where l=cosn,x, m=sinn,x denote direction cosines. Also, F1s is the primary warping function and F2n,s is the secondary warping function.Figure 3
The coordinate system.The boundary value problem, which consists of differential equations and boundary conditions, can be conveniently derived using the extended Hamilton’s principle, which is as follows:(3)∫t1t2δT−δV+δWdt=0,δu0=δv0=δw0=δθy=δθx=δϕ=0att=t1,t2,where δT and δV denote the kinetic energy and strain energy, respectively, and δW is the virtual work due to nonconservative forces.As a result, the governing equations and boundary conditions are as follows but also omitted the process of the transformation in this paper; see reference [1].


### 2.3.1. Governing Equations

(4)δu0:a41w0′+a42θy′+a43θx′+a44θy+u0′+a45θx+v0′−a46ϕ″+a47ϕ′′−b1u¨0−b2ϕ¨=0,δv0:a51w0′+a52θy′+a53θx′+a54θy+u0′+a55θx+v0′−a56ϕ″+a57ϕ′′−b1v¨0+b3ϕ¨=0,δw0:a11w0′+a12θy′+a13θx′+a14θy+u0′+a15θx+v0′−a16ϕ″+a17ϕ′′−b1w¨0+b3θ¨y+b2θ¨x−b7ϕ¨=0,δθy:a21w0′+a22θy′+a23θx′+a24θy+u0′+a25θx+v0′−a26ϕ″+a27ϕ′′−a41w0′+a42θy′+a43θx′+a44θy+u0′+a45θx+v0′−a46ϕ″+a47ϕ′−b3w¨0+b5θ¨y+b6θ¨x−b9ϕ¨′+b15θ¨y−b13θ¨x−b17ϕ¨′=0,δθx:a31w0′+a32θy′+a33θx′+a34θy+u0′+a35θx+v0′−a36ϕ″+a37ϕ′′−a51w0′+a52θy′+a53θx′+a54θy+u0′+a55θx+v0′−a56ϕ″+a57ϕ′−b2w¨0+b6θ¨y+b4θ¨x−b8ϕ¨′−b13θ¨y+b14θ¨x+b16ϕ¨′=0,δϕ:a71w0′+a72θy′+a73θx′+a74θy+u0′+a75θx+v0′−a76ϕ″+a77ϕ′′−a61w0′+a62θy′+a63θx′+a64θy+u0′+a65θx+v0′−a66ϕ″+a67ϕ′−b4+b5+b14+b15ϕ¨−b3v¨0+b2u¨0−−b7w¨0−b9θ¨y−b8θ¨x+b10ϕ¨′−b17θ¨y+b16θ¨x+b18ϕ¨′=0.


### 2.3.2. Boundary Conditions

(5)δu0:a41w0′+a42θy′+a43θx′+a44θy+u0′+a45θx+v0′−a46ϕ″+a47ϕ′=0,δv0:a51w0′+a52θy′+a53θx′+a54θy+u0′+a55θx+v0′−a56ϕ″+a57ϕ′=0,δw0:a11w0′+a12θy′+a13θx′+a14θy+u0′+a15θx+v0′−a16ϕ″+a17ϕ′=0,δθy:a21w0′+a22θy′+a23θx′+a24θy+u0′+a25θx+v0′−a26ϕ″+a27ϕ′=0,δθx:a31w0′+a32θy′+a33θx′+a34θy+u0′+a35θx+v0′−a36ϕ″+a37ϕ′=0,δϕ:a71w0′+a72θy′+a73θx′+a74θy+u0′+a75θx+v0′−a76ϕ″+a77ϕ′−a61w0′+a62θy′+a63θx′+a64θy+u0′+a65θx+v0′−a66ϕ″+a67ϕ′′−−b7w¨0−b9θ¨y−b8θ¨x+b10ϕ¨′−b17θ¨y+b16θ¨x+b18ϕ¨′=0,δϕ′:a61w0′+a62θy′+a63θx′+a64θy+u0′+a65θx+v0′−a66ϕ″+a67ϕ′=0,where aiji,j=1,2,…,6 are stiffness coefficients and bjj=1,2,…,18 are mass coefficients.


## 2.3.1. Governing Equations

(4)δu0:a41w0′+a42θy′+a43θx′+a44θy+u0′+a45θx+v0′−a46ϕ″+a47ϕ′′−b1u¨0−b2ϕ¨=0,δv0:a51w0′+a52θy′+a53θx′+a54θy+u0′+a55θx+v0′−a56ϕ″+a57ϕ′′−b1v¨0+b3ϕ¨=0,δw0:a11w0′+a12θy′+a13θx′+a14θy+u0′+a15θx+v0′−a16ϕ″+a17ϕ′′−b1w¨0+b3θ¨y+b2θ¨x−b7ϕ¨=0,δθy:a21w0′+a22θy′+a23θx′+a24θy+u0′+a25θx+v0′−a26ϕ″+a27ϕ′′−a41w0′+a42θy′+a43θx′+a44θy+u0′+a45θx+v0′−a46ϕ″+a47ϕ′−b3w¨0+b5θ¨y+b6θ¨x−b9ϕ¨′+b15θ¨y−b13θ¨x−b17ϕ¨′=0,δθx:a31w0′+a32θy′+a33θx′+a34θy+u0′+a35θx+v0′−a36ϕ″+a37ϕ′′−a51w0′+a52θy′+a53θx′+a54θy+u0′+a55θx+v0′−a56ϕ″+a57ϕ′−b2w¨0+b6θ¨y+b4θ¨x−b8ϕ¨′−b13θ¨y+b14θ¨x+b16ϕ¨′=0,δϕ:a71w0′+a72θy′+a73θx′+a74θy+u0′+a75θx+v0′−a76ϕ″+a77ϕ′′−a61w0′+a62θy′+a63θx′+a64θy+u0′+a65θx+v0′−a66ϕ″+a67ϕ′−b4+b5+b14+b15ϕ¨−b3v¨0+b2u¨0−−b7w¨0−b9θ¨y−b8θ¨x+b10ϕ¨′−b17θ¨y+b16θ¨x+b18ϕ¨′=0.


## 2.3.2. Boundary Conditions

(5)δu0:a41w0′+a42θy′+a43θx′+a44θy+u0′+a45θx+v0′−a46ϕ″+a47ϕ′=0,δv0:a51w0′+a52θy′+a53θx′+a54θy+u0′+a55θx+v0′−a56ϕ″+a57ϕ′=0,δw0:a11w0′+a12θy′+a13θx′+a14θy+u0′+a15θx+v0′−a16ϕ″+a17ϕ′=0,δθy:a21w0′+a22θy′+a23θx′+a24θy+u0′+a25θx+v0′−a26ϕ″+a27ϕ′=0,δθx:a31w0′+a32θy′+a33θx′+a34θy+u0′+a35θx+v0′−a36ϕ″+a37ϕ′=0,δϕ:a71w0′+a72θy′+a73θx′+a74θy+u0′+a75θx+v0′−a76ϕ″+a77ϕ′−a61w0′+a62θy′+a63θx′+a64θy+u0′+a65θx+v0′−a66ϕ″+a67ϕ′′−−b7w¨0−b9θ¨y−b8θ¨x+b10ϕ¨′−b17θ¨y+b16θ¨x+b18ϕ¨′=0,δϕ′:a61w0′+a62θy′+a63θx′+a64θy+u0′+a65θx+v0′−a66ϕ″+a67ϕ′=0,where aiji,j=1,2,…,6 are stiffness coefficients and bjj=1,2,…,18 are mass coefficients.


## 3. Numerical Analysis and Discussion

### 3.1. Material Properties and Geometrical Dimensions

Wing structures modeled as a cantilevered thin-walled beam of a chordwise asymmetric closed cross-section made of T300/5208 carbon epoxy material are considered for numerical simulations. Their geometrical dimensions and material properties are shown in Table1.Table 1
Material properties and geometrical dimensions.
E1=206.75 GPa, E2=E3=5.17 GPa, G1=3.10 GPa, G2=G3=2.55 GPa
ν21=ν31=0.00625, ν32=0.25, ρ=1528.15 kg/m3
Chord length:c+a=0.2 m, section thickness: b=ξ∗c
Semispan:L=2.0 m, section contour thickness: h=0.004 m


### 3.2. Correction of the Primary Warping Functions

The warping displacement in pure twist can be obtained as the product of the warping function and the rate of twist [1, 2].
(6)ws,z,t=F1sϕ′z,t,where F1s=∫0sλ−rnds+C, the constant C being determined by satisfying the following warping equilibrium condition:
(7)∮F1sds=0,thus yielding
(8)C=∮F1ds∮ds.In equation (7), ∮⋅ds denotes the integration around the whole midline contour of the cross-section.Note that fulfillment of∮F1ds=0 implies
(9)∮K∂w∂sds=0,K=1,2,…,N,and vice-versa [1].From the above relations, equations (6)–(9), the warping functions could be corrected for the model (see Figure 4) as the following procedures.Step 1.
In advance, the geometrical shape functions of the contour are derived as follows:
For an ellipse contour, fromx2/a2+y2/b2=1,
(10)x=±acosφ,y=±bsinφ.
For a biconvex contour, fromx2+y+Rcosθ02=R2 at the upper,
(11)x=±Rsinφ,y=±Rcosφ−Rcosθ0.
And fromx2+y−Rcosθ02=R2 at the lower,
(12)x=±Rsinφ,y=±Rcosφ+Rcosθ0.
And also, the length and the area of the contours are(13)Ω=Ωe+Ωb,β=βe+βb,Ωb=R22θ0−sin2θ02,Ωe=πab2,βe=2cEm,βb=2Rθ0,where Em is the complete elliptical integral (m=1−b2/a2, where a≥b), Ω and β are the all area and all length of the contours, and ⋅e and ⋅b mean the physical quantities of the ellipse and biconvex contours, respectively.
From the above relations, we have the geometrical shape functions of the contour as follows:(14)s0≤s≤s1:x1P=−RsinsR,y1P=b−R+RcossR,s1≤s≤s2:x2P=−Rsinβb−sR,y2P=b−R+Rcosβb−sR,s2≤s≤s3:x3P=−Rsinπβb−sβe,y3P=−bcosπβb−sβe,s3≤s≤s4:x4P=−Rsinπβb−sβe,y4P=−bcosπβb−sβe.Step 2.
And then, for correction of the warping functions, so as to satisfy equilibrium condition (7), the pole (Pm) should be translated to the new one (Pn) and also the shape functions (equation (14)) modified as follows:
(15)s0≤s≤s1:x1P=xc−RsinsR,y1P=b−R+RcossR,s1≤s≤s2:x2P=xc−Rsinβb−sR,y2P=b−R+Rcosβb−sR,s2≤s≤s3:x3P=xc−Rsinπβb−sβe,y3P=−bcosπβb−sβe,s3≤s≤s4:x4P=xc−Rsinπβb−sβe,y4P=−bcosπβb−sβe,where xc is a translation quantity along the chordwise (see Figure 4), that is,
(16)xc=accscθ0/22−2Em+πθ0−πθ0cosθ0+Emsin2θ04cθ0+aEmsinθ0.
To verify the above results through an example, in case of the noncorrected primary warping functions (equation (14)), the function’s graph can be depicted in Figure 5. In Figure 5, the sum of the area based on the x-axis is not zero. That means that the warping function should be corrected [1]. Therefore, from the corrected warping functions (equations (15) and (16)), Figure 6 can be drawn and shows that the all area is zero. As a result, the above relations, equations (15) and (16), will be applied to governing equations and boundary conditions of the beam for analysis.Figure 4
The translated geometrical parameters of the section (Model A).Figure 5
Noncorrected primary warping functions of Model A.Figure 6
Corrected primary warping functions of Model A.


### 3.3. Characteristics of the Cross-Section

For a survey of the characteristics of the chordwise asymmetric closed cross-section (Model A), using the corrected warping functions, mass coefficientsbj and stiffness coefficients aij of equations (4) and (5) were found. The details of the mass and stiffness coefficients could be found in reference [1], which are equations (4.1-4) for mass coefficients and Tables 4.3-1 for stiffness coefficients of chapter 4. The results of the nonzero coefficients were provided in Tables 2 and 3 for comparing between Model A and a symmetric one (Model C, see Figure 7) according to a preset angle (α0) and a pretwist angle (γ0) of CUS and CAS configurations for each. In the tables, “S” entries are valued elements of the symmetric section (Model C) and “A” entries are those of the asymmetric section (Model A) in addition to the results of Model C.Table 2
Mass coefficients of Model A and Model C for CUS and CAS.
b1
b2
b3
b4
b5
b6
b7
b8
b9
b10
b11
b12
b13
b14
b15
b16
b17
b18
α0=0 and γ0=0
S
A
S
S
A
S
S
S
S
A
S
α0≠0 or γ0≠0
S
A
A
S
S
S
A
A
S
S
S
S
A
A
STable 3
Stiffness coefficients of Model A and Model C.
(a)
CUS:α0=0 and γ0=0
aij
1
2
3
4
5
6
7
1
S
A
S
2
S
S
A
3
S
S
A
4
S
A
5
S
6
Sym.
S
7
S
(b)
CAS:α0=0 and γ0=0
aij
1
2
3
4
5
6
7
1
S
A
S
2
S
A
3
S
A
S
4
S
5
S
S
6
Sym.
S
A
7
S
(c)
CUS:α0≠0 or γ0≠0
aij
1
2
3
4
5
6
7
1
S
S
A
S
2
S
A
S
A
S
3
S
S
S
A
4
S
A
S
5
S
A
6
Sym.
S
7
S
(d)
CAS:α0≠0 or γ0≠0
aij
1
2
3
4
5
6
7
1
S
S
A
S
A
2
S
A
S
A
A
A
3
S
A
A
S
S
4
S
A
A
5
S
S
6
Sym.
S
A
7
SFigure 7
Model B and Model C for comparing.And then, stiffness coefficientsaij=aji in cases of (α0=0° and γ0=0°) and (α0=4° and γ0=10°) were depicted in Figures 8 and 9 for CUS and CAS at η=1.0, σ=0.6, and ξ=0.4. These figures show that a14, a15, a24, a25, a27, a34, a35, a37, a45, a46, a56, and a67 have asymmetric values and the others have symmetric values at a ply angle of 90°.Figure 8
Stiffness coefficientsaij=aji vs. ply angle of Model A for CUS.Figure 9
Stiffness coefficientsaij=aji vs. ply angle of Model A for CAS.Furthermore, Table4 shows that some stiffness coefficients of the model have different values between α0=0° and γ0=0° and α0≠0° or γ0≠0° and some are the same or the others only appeared in case of α0≠0° or γ0≠0°. And especially, when i=j, aii have all the same values for CUS and CAS configurations.Table 4
Nonzero stiffness coefficients of Model A.
CUS
aij=aji
Different values
a11, a17, a25, a34, a66, a77
Same values
a12, a22, a27, a33, a36, a44, a46, a55
Onlyα0≠0 or γ0≠0
a13, a23, a26, a37, a45, a56
CAS
aij=aji
Different values
a11, a66, a67, a77
Same values
a12, a14, a22, a24, a33, a36, a37, a44, a55, a56
Onlyα0≠0 or γ0≠0
a13, a15, a23, a25, a26, a27, a34, a35, a45, a46


### 3.4. Eigenfrequency of the Model

#### 3.4.1. Stiffness and Mass Matrices

The eigenvalue problem is discretized by representing in the form via extended Galerkin method (EGM):(17)u0z,t=∑j=1Najψj1z,v0z,t=∑j=1Nbjψj2z,w0z,t=∑j=1Ncjψj3z,θyz,t=∑j=1Ndjψj4z,θxz,t=∑j=1Nejψj5z,ϕz,t=∑j=1Nfjψj6z,where ψjr are trial functions and have to satisfy the geometrical boundary conditions at z=0 and also, aj, bj, cj, dj, ej, and fj are arbitrary coefficients. Now, the discretization is accomplished directly in the Hamilton’s principle functional and then written in a matrix form as follows:
(18)⋯⋮Kpq⋮⋯aj⋮fj−ω2⋯⋮Mpq⋮⋯aj⋮fj=Q1⋮Q6,where Kpq and Mpq are stiffness and mass matrices, respectively, and Qrr=1,2,…,6 is a force vector.Similarly, for the survey of the characteristics of the chordwise asymmetric closed cross-section (Model A), using the corrected warping functions, the mass and stiffness matrices of equation (18) were found. The results of the nonzero coefficients were provided in Tables 5 and 6 for comparing between Model A and a symmetric one (Model C, see Figure 7) according to a preset angle (α0) and a pretwist angle (γ0) of CUS and CAS configurations for each. In the tables, “S” entries are valued elements of the symmetric section (Model C) and “A” entries are those of the asymmetric section (Model A) in addition to Model C.Table 5
Stiffness matrixKpq of Model A and Model C.
(a)
CUS:α0=0 and γ0=0
p
q
1
2
3
4
5
6
1
S
S
S
A
2
S
S
S
3
S
A
S
4
S
S
A
5
S
A
6
Sym.
S
(b)
CAS:α0=0 and γ0=0
p
q
1
2
3
4
5
6
1
S
S
S
2
S
S
S
3
S
S
4
S
S
5
S
6
Sym.
S
(c)
CUS:α0≠0 or γ0≠0
p
q
1
2
3
4
5
6
1
S
S
S
S
A
2
S
S
S
A
3
S
A
A
S
4
S
S
A
5
S
A
6
Sym.
S
(d)
CAS:α0≠0 or γ0≠0
p
q
1
2
3
4
5
6
1
S
S
S
S
S
S
2
S
S
S
S
S
3
S
S
S
4
S
S
S
5
S
S
6
Sym.
STable 6
Mass matrixMpq of Model A and Model C.
(a)
CUS & CAS:α0=0 and γ0=0
p
q
1
2
3
4
5
6
1
S
2
S
S
3
S
S
4
S
S
5
S
6
Sym.
S
(b)
CUS & CAS:α0≠0 or γ0≠0
p
q
1
2
3
4
5
6
1
S
S
2
S
S
3
S
S
S
4
S
S
S
5
S
S
6
Sym.
SAt this point, Table5 shows that the stiffness matrix for CAS configuration is composed of the same elements between Model A and Model C, in cases of α0=0 and γ0=0 and α0≠0 or γ0≠0. And also, mass matrices have all the same values for CUS and CAS configurations at any case.


#### 3.4.2. Effects of Preset Angle and Pretwist Angle on Eigenfrequency

In order to investigate the effects of preset and pretwist angles on eigenfrequencies calculated for Model A, both CUS and CAS configurations. The results are shown in Figure10 and Table 7.Figure 10
Eigenfrequencies vs. ply angle of Model A viaα0=0 and γ0=0 for CUS and CAS.Table 7
Eigenfrequencies of Model A viaα0=0 and γ0=0 for CUS and CAS.
CUS (°)
ω1 (rad/s)
ω2 (rad/s)
ω3 (rad/s)
θ
α0=0, γ0=0
α0=4, γ0=10
α0=0, γ0=0
α0=4, γ0=10
α0=0, γ0=0
α0=4, γ0=10
0
51.1874
51.2014
110.305
110.096
260.03
260.587
45
63.8732
63.9046
150.383
149.814
324.784
326.814
90
301.001
301.106
595.863
594.734
1068.34
1068.17
135
63.8728
63.8881
150.383
150.215
324.785
324.6
180
51.1874
51.2014
110.305
110.096
260.03
260.587
0
51.1874
51.2008
110.305
110.096
260.03
260.57
45
64.6077
64.8452
159.239
158.803
328.334
329.249
90
301.001
301.106
595.863
594.734
1068.34
1068.17
135
63.9356
63.6511
161.385
160.896
324.406
325.731
180
51.1874
51.2008
110.305
110.096
260.03
260.57In Figure10 and Table 7, preset and pretwist angles have no near effects on eigenfrequencies; nevertheless, the differences exist but very negligibly small.


#### 3.4.3. Effects of Correction of the Primary Warping Functions

Next, based on ply angles, the effects of corrected (CW) and noncorrected (NCW) warping functions on eigenfrequencies were estimated for Model A, both CUS and CAS configurations. The results are shown in Figure11 and Table 8.Figure 11
Eigenfrequencies vs. ply angle of Model A via CW and NCW for CUS and CAS.Table 8
Eigenfrequencies of Model A via CW and NCW for CUS and CAS.
CUS (°)
ω1 (rad/s)
ω2 (rad/s)
ω3 (rad/s)
θ
CW
NCW
CW
NCW
CW
NCW
0
51.1874
51.1852
110.305
110.303
260.03
260.026
45
63.8732
63.8757
150.383
150.38
324.784
324.81
90
301.001
301.359
595.863
595.858
1068.34
1065.79
135
63.8728
63.8744
150.383
150.381
324.785
324.812
180
51.1874
51.1852
110.305
110.303
260.03
260.026
0
51.1874
51.1852
110.305
110.303
260.03
260.026
45
64.6077
64.0606
159.239
159.237
328.334
328.122
90
301.001
301.362
595.863
595.858
1068.34
1065.83
135
63.9356
63.7094
161.385
161.381
324.406
325.027
180
51.1874
51.1844
110.305
110.303
260.03
260.028In Figure11 and Table 8, the effects of corrected (CW) and noncorrected (NCW) warping functions on eigenfrequencies were observed to have very small differences for ply angles. However, for precise calculation, the primary warping function should be properly corrected according to the warping equilibrium condition from equations (6)–(9).


#### 3.4.4. Effects of Geometrical Parameters

From now on, the effects of geometrical parameters (α0, γ0, σ, and ξ) on eigenfrequencies were investigated for Model A, both CUS and CAS configurations.(1) Preset Angle,α0. After calculating the eigenfrequencies via preset angles α0, the results are shown in Figure 12. Generally, from the results, the preset angle effects on the eigenfrequencies are very insignificant.Figure 12
Eigenfrequencies vs. ply angle of Model A viaα0=0 for CUS and CAS.(2) Pretwist Angle,γ0. Also, after calculating the eigenfrequencies via pretwist angles γ0, the results are shown in Figure 13. As a result, lower-order eigenfrequencies were not almost affected by the pretwist angle but the higher modes increase the effects which is associated with the preset angles being higher.Figure 13
Eigenfrequencies vs. ply angle of Model A viaγ0=0 for CUS and CAS.(3) Taper Ratio,σ. Next, the eigenfrequencies via taper ratios σ of the beam were found. The results are shown in Figure 14. In this case, the taper ratio has more effect on the eigenfrequencies related to the higher modes. The first and second eigenfrequencies decrease which is associated with the taper ratio being higher and the third or more eigenfrequencies increase along the taper ratio being higher.Figure 14
Eigenfrequencies vs. ply angle of Model A viaσ for CUS and CAS.(4) Section Ratio,ξ. Finally, the eigenfrequencies via section ratios ξ of the beam were computed. The results are shown in Figure 15.Figure 15
Eigenfrequencies vs. ply angle of Model A viaξ for CUS and CAS.Figure15 shows that the eigenfrequencies via section ratios have irregular trends for each. This is because of the characteristic that circles, squares, etc. have no values for primary warping functions [1], that is, see Figure 16, the primary function of the ellipse contour of Model A has almost zero values at ξ=0.7716. As a result, the effects of section ratio on the eigenfrequencies exist but the substantial factor of the effects is not section ratios but contour shapes.Figure 16
Primary warping function of Model A atξ=0.7716.


#### 3.4.5. Comparison of Model B and Model C

For the comparison with a symmetric section, whena=c in Model A, Model A is fixed to Model B (Figure 7). And then, the eigenfrequencies of Model B and Model C were compared to each other. The results are shown in Figure 17.Figure 17
Eigenfrequencies of Model B and Model C for CUS and CAS.In Figure17, Model C has symmetric eigenfrequencies at a ply angle of 90°, but on the contrary, Model B has not. Also, it is notable that CUS and CAS have the same values at ply angles 0°, 90°, and 180° for each, and especially, ω2 of CAS of Model B has an observable asymmetric curve of eigenfrequency which is one of vertical transverse shear vibration modes.


## 3.1. Material Properties and Geometrical Dimensions

Wing structures modeled as a cantilevered thin-walled beam of a chordwise asymmetric closed cross-section made of T300/5208 carbon epoxy material are considered for numerical simulations. Their geometrical dimensions and material properties are shown in Table1.Table 1
Material properties and geometrical dimensions.
E1=206.75 GPa, E2=E3=5.17 GPa, G1=3.10 GPa, G2=G3=2.55 GPa
ν21=ν31=0.00625, ν32=0.25, ρ=1528.15 kg/m3
Chord length:c+a=0.2 m, section thickness: b=ξ∗c
Semispan:L=2.0 m, section contour thickness: h=0.004 m


## 3.2. Correction of the Primary Warping Functions

The warping displacement in pure twist can be obtained as the product of the warping function and the rate of twist [1, 2].
(6)ws,z,t=F1sϕ′z,t,where F1s=∫0sλ−rnds+C, the constant C being determined by satisfying the following warping equilibrium condition:
(7)∮F1sds=0,thus yielding
(8)C=∮F1ds∮ds.In equation (7), ∮⋅ds denotes the integration around the whole midline contour of the cross-section.Note that fulfillment of∮F1ds=0 implies
(9)∮K∂w∂sds=0,K=1,2,…,N,and vice-versa [1].From the above relations, equations (6)–(9), the warping functions could be corrected for the model (see Figure 4) as the following procedures.Step 1.
In advance, the geometrical shape functions of the contour are derived as follows:
For an ellipse contour, fromx2/a2+y2/b2=1,
(10)x=±acosφ,y=±bsinφ.
For a biconvex contour, fromx2+y+Rcosθ02=R2 at the upper,
(11)x=±Rsinφ,y=±Rcosφ−Rcosθ0.
And fromx2+y−Rcosθ02=R2 at the lower,
(12)x=±Rsinφ,y=±Rcosφ+Rcosθ0.
And also, the length and the area of the contours are(13)Ω=Ωe+Ωb,β=βe+βb,Ωb=R22θ0−sin2θ02,Ωe=πab2,βe=2cEm,βb=2Rθ0,where Em is the complete elliptical integral (m=1−b2/a2, where a≥b), Ω and β are the all area and all length of the contours, and ⋅e and ⋅b mean the physical quantities of the ellipse and biconvex contours, respectively.
From the above relations, we have the geometrical shape functions of the contour as follows:(14)s0≤s≤s1:x1P=−RsinsR,y1P=b−R+RcossR,s1≤s≤s2:x2P=−Rsinβb−sR,y2P=b−R+Rcosβb−sR,s2≤s≤s3:x3P=−Rsinπβb−sβe,y3P=−bcosπβb−sβe,s3≤s≤s4:x4P=−Rsinπβb−sβe,y4P=−bcosπβb−sβe.Step 2.
And then, for correction of the warping functions, so as to satisfy equilibrium condition (7), the pole (Pm) should be translated to the new one (Pn) and also the shape functions (equation (14)) modified as follows:
(15)s0≤s≤s1:x1P=xc−RsinsR,y1P=b−R+RcossR,s1≤s≤s2:x2P=xc−Rsinβb−sR,y2P=b−R+Rcosβb−sR,s2≤s≤s3:x3P=xc−Rsinπβb−sβe,y3P=−bcosπβb−sβe,s3≤s≤s4:x4P=xc−Rsinπβb−sβe,y4P=−bcosπβb−sβe,where xc is a translation quantity along the chordwise (see Figure 4), that is,
(16)xc=accscθ0/22−2Em+πθ0−πθ0cosθ0+Emsin2θ04cθ0+aEmsinθ0.
To verify the above results through an example, in case of the noncorrected primary warping functions (equation (14)), the function’s graph can be depicted in Figure 5. In Figure 5, the sum of the area based on the x-axis is not zero. That means that the warping function should be corrected [1]. Therefore, from the corrected warping functions (equations (15) and (16)), Figure 6 can be drawn and shows that the all area is zero. As a result, the above relations, equations (15) and (16), will be applied to governing equations and boundary conditions of the beam for analysis.Figure 4
The translated geometrical parameters of the section (Model A).Figure 5
Noncorrected primary warping functions of Model A.Figure 6
Corrected primary warping functions of Model A.


## 3.3. Characteristics of the Cross-Section

For a survey of the characteristics of the chordwise asymmetric closed cross-section (Model A), using the corrected warping functions, mass coefficientsbj and stiffness coefficients aij of equations (4) and (5) were found. The details of the mass and stiffness coefficients could be found in reference [1], which are equations (4.1-4) for mass coefficients and Tables 4.3-1 for stiffness coefficients of chapter 4. The results of the nonzero coefficients were provided in Tables 2 and 3 for comparing between Model A and a symmetric one (Model C, see Figure 7) according to a preset angle (α0) and a pretwist angle (γ0) of CUS and CAS configurations for each. In the tables, “S” entries are valued elements of the symmetric section (Model C) and “A” entries are those of the asymmetric section (Model A) in addition to the results of Model C.Table 2
Mass coefficients of Model A and Model C for CUS and CAS.
b1
b2
b3
b4
b5
b6
b7
b8
b9
b10
b11
b12
b13
b14
b15
b16
b17
b18
α0=0 and γ0=0
S
A
S
S
A
S
S
S
S
A
S
α0≠0 or γ0≠0
S
A
A
S
S
S
A
A
S
S
S
S
A
A
STable 3
Stiffness coefficients of Model A and Model C.
(a)
CUS:α0=0 and γ0=0
aij
1
2
3
4
5
6
7
1
S
A
S
2
S
S
A
3
S
S
A
4
S
A
5
S
6
Sym.
S
7
S
(b)
CAS:α0=0 and γ0=0
aij
1
2
3
4
5
6
7
1
S
A
S
2
S
A
3
S
A
S
4
S
5
S
S
6
Sym.
S
A
7
S
(c)
CUS:α0≠0 or γ0≠0
aij
1
2
3
4
5
6
7
1
S
S
A
S
2
S
A
S
A
S
3
S
S
S
A
4
S
A
S
5
S
A
6
Sym.
S
7
S
(d)
CAS:α0≠0 or γ0≠0
aij
1
2
3
4
5
6
7
1
S
S
A
S
A
2
S
A
S
A
A
A
3
S
A
A
S
S
4
S
A
A
5
S
S
6
Sym.
S
A
7
SFigure 7
Model B and Model C for comparing.And then, stiffness coefficientsaij=aji in cases of (α0=0° and γ0=0°) and (α0=4° and γ0=10°) were depicted in Figures 8 and 9 for CUS and CAS at η=1.0, σ=0.6, and ξ=0.4. These figures show that a14, a15, a24, a25, a27, a34, a35, a37, a45, a46, a56, and a67 have asymmetric values and the others have symmetric values at a ply angle of 90°.Figure 8
Stiffness coefficientsaij=aji vs. ply angle of Model A for CUS.Figure 9
Stiffness coefficientsaij=aji vs. ply angle of Model A for CAS.Furthermore, Table4 shows that some stiffness coefficients of the model have different values between α0=0° and γ0=0° and α0≠0° or γ0≠0° and some are the same or the others only appeared in case of α0≠0° or γ0≠0°. And especially, when i=j, aii have all the same values for CUS and CAS configurations.Table 4
Nonzero stiffness coefficients of Model A.
CUS
aij=aji
Different values
a11, a17, a25, a34, a66, a77
Same values
a12, a22, a27, a33, a36, a44, a46, a55
Onlyα0≠0 or γ0≠0
a13, a23, a26, a37, a45, a56
CAS
aij=aji
Different values
a11, a66, a67, a77
Same values
a12, a14, a22, a24, a33, a36, a37, a44, a55, a56
Onlyα0≠0 or γ0≠0
a13, a15, a23, a25, a26, a27, a34, a35, a45, a46


## 3.4. Eigenfrequency of the Model

### 3.4.1. Stiffness and Mass Matrices

The eigenvalue problem is discretized by representing in the form via extended Galerkin method (EGM):(17)u0z,t=∑j=1Najψj1z,v0z,t=∑j=1Nbjψj2z,w0z,t=∑j=1Ncjψj3z,θyz,t=∑j=1Ndjψj4z,θxz,t=∑j=1Nejψj5z,ϕz,t=∑j=1Nfjψj6z,where ψjr are trial functions and have to satisfy the geometrical boundary conditions at z=0 and also, aj, bj, cj, dj, ej, and fj are arbitrary coefficients. Now, the discretization is accomplished directly in the Hamilton’s principle functional and then written in a matrix form as follows:
(18)⋯⋮Kpq⋮⋯aj⋮fj−ω2⋯⋮Mpq⋮⋯aj⋮fj=Q1⋮Q6,where Kpq and Mpq are stiffness and mass matrices, respectively, and Qrr=1,2,…,6 is a force vector.Similarly, for the survey of the characteristics of the chordwise asymmetric closed cross-section (Model A), using the corrected warping functions, the mass and stiffness matrices of equation (18) were found. The results of the nonzero coefficients were provided in Tables 5 and 6 for comparing between Model A and a symmetric one (Model C, see Figure 7) according to a preset angle (α0) and a pretwist angle (γ0) of CUS and CAS configurations for each. In the tables, “S” entries are valued elements of the symmetric section (Model C) and “A” entries are those of the asymmetric section (Model A) in addition to Model C.Table 5
Stiffness matrixKpq of Model A and Model C.
(a)
CUS:α0=0 and γ0=0
p
q
1
2
3
4
5
6
1
S
S
S
A
2
S
S
S
3
S
A
S
4
S
S
A
5
S
A
6
Sym.
S
(b)
CAS:α0=0 and γ0=0
p
q
1
2
3
4
5
6
1
S
S
S
2
S
S
S
3
S
S
4
S
S
5
S
6
Sym.
S
(c)
CUS:α0≠0 or γ0≠0
p
q
1
2
3
4
5
6
1
S
S
S
S
A
2
S
S
S
A
3
S
A
A
S
4
S
S
A
5
S
A
6
Sym.
S
(d)
CAS:α0≠0 or γ0≠0
p
q
1
2
3
4
5
6
1
S
S
S
S
S
S
2
S
S
S
S
S
3
S
S
S
4
S
S
S
5
S
S
6
Sym.
STable 6
Mass matrixMpq of Model A and Model C.
(a)
CUS & CAS:α0=0 and γ0=0
p
q
1
2
3
4
5
6
1
S
2
S
S
3
S
S
4
S
S
5
S
6
Sym.
S
(b)
CUS & CAS:α0≠0 or γ0≠0
p
q
1
2
3
4
5
6
1
S
S
2
S
S
3
S
S
S
4
S
S
S
5
S
S
6
Sym.
SAt this point, Table5 shows that the stiffness matrix for CAS configuration is composed of the same elements between Model A and Model C, in cases of α0=0 and γ0=0 and α0≠0 or γ0≠0. And also, mass matrices have all the same values for CUS and CAS configurations at any case.


### 3.4.2. Effects of Preset Angle and Pretwist Angle on Eigenfrequency

In order to investigate the effects of preset and pretwist angles on eigenfrequencies calculated for Model A, both CUS and CAS configurations. The results are shown in Figure10 and Table 7.Figure 10
Eigenfrequencies vs. ply angle of Model A viaα0=0 and γ0=0 for CUS and CAS.Table 7
Eigenfrequencies of Model A viaα0=0 and γ0=0 for CUS and CAS.
CUS (°)
ω1 (rad/s)
ω2 (rad/s)
ω3 (rad/s)
θ
α0=0, γ0=0
α0=4, γ0=10
α0=0, γ0=0
α0=4, γ0=10
α0=0, γ0=0
α0=4, γ0=10
0
51.1874
51.2014
110.305
110.096
260.03
260.587
45
63.8732
63.9046
150.383
149.814
324.784
326.814
90
301.001
301.106
595.863
594.734
1068.34
1068.17
135
63.8728
63.8881
150.383
150.215
324.785
324.6
180
51.1874
51.2014
110.305
110.096
260.03
260.587
0
51.1874
51.2008
110.305
110.096
260.03
260.57
45
64.6077
64.8452
159.239
158.803
328.334
329.249
90
301.001
301.106
595.863
594.734
1068.34
1068.17
135
63.9356
63.6511
161.385
160.896
324.406
325.731
180
51.1874
51.2008
110.305
110.096
260.03
260.57In Figure10 and Table 7, preset and pretwist angles have no near effects on eigenfrequencies; nevertheless, the differences exist but very negligibly small.


### 3.4.3. Effects of Correction of the Primary Warping Functions

Next, based on ply angles, the effects of corrected (CW) and noncorrected (NCW) warping functions on eigenfrequencies were estimated for Model A, both CUS and CAS configurations. The results are shown in Figure11 and Table 8.Figure 11
Eigenfrequencies vs. ply angle of Model A via CW and NCW for CUS and CAS.Table 8
Eigenfrequencies of Model A via CW and NCW for CUS and CAS.
CUS (°)
ω1 (rad/s)
ω2 (rad/s)
ω3 (rad/s)
θ
CW
NCW
CW
NCW
CW
NCW
0
51.1874
51.1852
110.305
110.303
260.03
260.026
45
63.8732
63.8757
150.383
150.38
324.784
324.81
90
301.001
301.359
595.863
595.858
1068.34
1065.79
135
63.8728
63.8744
150.383
150.381
324.785
324.812
180
51.1874
51.1852
110.305
110.303
260.03
260.026
0
51.1874
51.1852
110.305
110.303
260.03
260.026
45
64.6077
64.0606
159.239
159.237
328.334
328.122
90
301.001
301.362
595.863
595.858
1068.34
1065.83
135
63.9356
63.7094
161.385
161.381
324.406
325.027
180
51.1874
51.1844
110.305
110.303
260.03
260.028In Figure11 and Table 8, the effects of corrected (CW) and noncorrected (NCW) warping functions on eigenfrequencies were observed to have very small differences for ply angles. However, for precise calculation, the primary warping function should be properly corrected according to the warping equilibrium condition from equations (6)–(9).


### 3.4.4. Effects of Geometrical Parameters

From now on, the effects of geometrical parameters (α0, γ0, σ, and ξ) on eigenfrequencies were investigated for Model A, both CUS and CAS configurations.(1) Preset Angle,α0. After calculating the eigenfrequencies via preset angles α0, the results are shown in Figure 12. Generally, from the results, the preset angle effects on the eigenfrequencies are very insignificant.Figure 12
Eigenfrequencies vs. ply angle of Model A viaα0=0 for CUS and CAS.(2) Pretwist Angle,γ0. Also, after calculating the eigenfrequencies via pretwist angles γ0, the results are shown in Figure 13. As a result, lower-order eigenfrequencies were not almost affected by the pretwist angle but the higher modes increase the effects which is associated with the preset angles being higher.Figure 13
Eigenfrequencies vs. ply angle of Model A viaγ0=0 for CUS and CAS.(3) Taper Ratio,σ. Next, the eigenfrequencies via taper ratios σ of the beam were found. The results are shown in Figure 14. In this case, the taper ratio has more effect on the eigenfrequencies related to the higher modes. The first and second eigenfrequencies decrease which is associated with the taper ratio being higher and the third or more eigenfrequencies increase along the taper ratio being higher.Figure 14
Eigenfrequencies vs. ply angle of Model A viaσ for CUS and CAS.(4) Section Ratio,ξ. Finally, the eigenfrequencies via section ratios ξ of the beam were computed. The results are shown in Figure 15.Figure 15
Eigenfrequencies vs. ply angle of Model A viaξ for CUS and CAS.Figure15 shows that the eigenfrequencies via section ratios have irregular trends for each. This is because of the characteristic that circles, squares, etc. have no values for primary warping functions [1], that is, see Figure 16, the primary function of the ellipse contour of Model A has almost zero values at ξ=0.7716. As a result, the effects of section ratio on the eigenfrequencies exist but the substantial factor of the effects is not section ratios but contour shapes.Figure 16
Primary warping function of Model A atξ=0.7716.


### 3.4.5. Comparison of Model B and Model C

For the comparison with a symmetric section, whena=c in Model A, Model A is fixed to Model B (Figure 7). And then, the eigenfrequencies of Model B and Model C were compared to each other. The results are shown in Figure 17.Figure 17
Eigenfrequencies of Model B and Model C for CUS and CAS.In Figure17, Model C has symmetric eigenfrequencies at a ply angle of 90°, but on the contrary, Model B has not. Also, it is notable that CUS and CAS have the same values at ply angles 0°, 90°, and 180° for each, and especially, ω2 of CAS of Model B has an observable asymmetric curve of eigenfrequency which is one of vertical transverse shear vibration modes.


## 3.4.1. Stiffness and Mass Matrices

The eigenvalue problem is discretized by representing in the form via extended Galerkin method (EGM):(17)u0z,t=∑j=1Najψj1z,v0z,t=∑j=1Nbjψj2z,w0z,t=∑j=1Ncjψj3z,θyz,t=∑j=1Ndjψj4z,θxz,t=∑j=1Nejψj5z,ϕz,t=∑j=1Nfjψj6z,where ψjr are trial functions and have to satisfy the geometrical boundary conditions at z=0 and also, aj, bj, cj, dj, ej, and fj are arbitrary coefficients. Now, the discretization is accomplished directly in the Hamilton’s principle functional and then written in a matrix form as follows:
(18)⋯⋮Kpq⋮⋯aj⋮fj−ω2⋯⋮Mpq⋮⋯aj⋮fj=Q1⋮Q6,where Kpq and Mpq are stiffness and mass matrices, respectively, and Qrr=1,2,…,6 is a force vector.Similarly, for the survey of the characteristics of the chordwise asymmetric closed cross-section (Model A), using the corrected warping functions, the mass and stiffness matrices of equation (18) were found. The results of the nonzero coefficients were provided in Tables 5 and 6 for comparing between Model A and a symmetric one (Model C, see Figure 7) according to a preset angle (α0) and a pretwist angle (γ0) of CUS and CAS configurations for each. In the tables, “S” entries are valued elements of the symmetric section (Model C) and “A” entries are those of the asymmetric section (Model A) in addition to Model C.Table 5
Stiffness matrixKpq of Model A and Model C.
(a)
CUS:α0=0 and γ0=0
p
q
1
2
3
4
5
6
1
S
S
S
A
2
S
S
S
3
S
A
S
4
S
S
A
5
S
A
6
Sym.
S
(b)
CAS:α0=0 and γ0=0
p
q
1
2
3
4
5
6
1
S
S
S
2
S
S
S
3
S
S
4
S
S
5
S
6
Sym.
S
(c)
CUS:α0≠0 or γ0≠0
p
q
1
2
3
4
5
6
1
S
S
S
S
A
2
S
S
S
A
3
S
A
A
S
4
S
S
A
5
S
A
6
Sym.
S
(d)
CAS:α0≠0 or γ0≠0
p
q
1
2
3
4
5
6
1
S
S
S
S
S
S
2
S
S
S
S
S
3
S
S
S
4
S
S
S
5
S
S
6
Sym.
STable 6
Mass matrixMpq of Model A and Model C.
(a)
CUS & CAS:α0=0 and γ0=0
p
q
1
2
3
4
5
6
1
S
2
S
S
3
S
S
4
S
S
5
S
6
Sym.
S
(b)
CUS & CAS:α0≠0 or γ0≠0
p
q
1
2
3
4
5
6
1
S
S
2
S
S
3
S
S
S
4
S
S
S
5
S
S
6
Sym.
SAt this point, Table5 shows that the stiffness matrix for CAS configuration is composed of the same elements between Model A and Model C, in cases of α0=0 and γ0=0 and α0≠0 or γ0≠0. And also, mass matrices have all the same values for CUS and CAS configurations at any case.


## 3.4.2. Effects of Preset Angle and Pretwist Angle on Eigenfrequency

In order to investigate the effects of preset and pretwist angles on eigenfrequencies calculated for Model A, both CUS and CAS configurations. The results are shown in Figure10 and Table 7.Figure 10
Eigenfrequencies vs. ply angle of Model A viaα0=0 and γ0=0 for CUS and CAS.Table 7
Eigenfrequencies of Model A viaα0=0 and γ0=0 for CUS and CAS.
CUS (°)
ω1 (rad/s)
ω2 (rad/s)
ω3 (rad/s)
θ
α0=0, γ0=0
α0=4, γ0=10
α0=0, γ0=0
α0=4, γ0=10
α0=0, γ0=0
α0=4, γ0=10
0
51.1874
51.2014
110.305
110.096
260.03
260.587
45
63.8732
63.9046
150.383
149.814
324.784
326.814
90
301.001
301.106
595.863
594.734
1068.34
1068.17
135
63.8728
63.8881
150.383
150.215
324.785
324.6
180
51.1874
51.2014
110.305
110.096
260.03
260.587
0
51.1874
51.2008
110.305
110.096
260.03
260.57
45
64.6077
64.8452
159.239
158.803
328.334
329.249
90
301.001
301.106
595.863
594.734
1068.34
1068.17
135
63.9356
63.6511
161.385
160.896
324.406
325.731
180
51.1874
51.2008
110.305
110.096
260.03
260.57In Figure10 and Table 7, preset and pretwist angles have no near effects on eigenfrequencies; nevertheless, the differences exist but very negligibly small.


## 3.4.3. Effects of Correction of the Primary Warping Functions

Next, based on ply angles, the effects of corrected (CW) and noncorrected (NCW) warping functions on eigenfrequencies were estimated for Model A, both CUS and CAS configurations. The results are shown in Figure11 and Table 8.Figure 11
Eigenfrequencies vs. ply angle of Model A via CW and NCW for CUS and CAS.Table 8
Eigenfrequencies of Model A via CW and NCW for CUS and CAS.
CUS (°)
ω1 (rad/s)
ω2 (rad/s)
ω3 (rad/s)
θ
CW
NCW
CW
NCW
CW
NCW
0
51.1874
51.1852
110.305
110.303
260.03
260.026
45
63.8732
63.8757
150.383
150.38
324.784
324.81
90
301.001
301.359
595.863
595.858
1068.34
1065.79
135
63.8728
63.8744
150.383
150.381
324.785
324.812
180
51.1874
51.1852
110.305
110.303
260.03
260.026
0
51.1874
51.1852
110.305
110.303
260.03
260.026
45
64.6077
64.0606
159.239
159.237
328.334
328.122
90
301.001
301.362
595.863
595.858
1068.34
1065.83
135
63.9356
63.7094
161.385
161.381
324.406
325.027
180
51.1874
51.1844
110.305
110.303
260.03
260.028In Figure11 and Table 8, the effects of corrected (CW) and noncorrected (NCW) warping functions on eigenfrequencies were observed to have very small differences for ply angles. However, for precise calculation, the primary warping function should be properly corrected according to the warping equilibrium condition from equations (6)–(9).


## 3.4.4. Effects of Geometrical Parameters

From now on, the effects of geometrical parameters (α0, γ0, σ, and ξ) on eigenfrequencies were investigated for Model A, both CUS and CAS configurations.(1) Preset Angle,α0. After calculating the eigenfrequencies via preset angles α0, the results are shown in Figure 12. Generally, from the results, the preset angle effects on the eigenfrequencies are very insignificant.Figure 12
Eigenfrequencies vs. ply angle of Model A viaα0=0 for CUS and CAS.(2) Pretwist Angle,γ0. Also, after calculating the eigenfrequencies via pretwist angles γ0, the results are shown in Figure 13. As a result, lower-order eigenfrequencies were not almost affected by the pretwist angle but the higher modes increase the effects which is associated with the preset angles being higher.Figure 13
Eigenfrequencies vs. ply angle of Model A viaγ0=0 for CUS and CAS.(3) Taper Ratio,σ. Next, the eigenfrequencies via taper ratios σ of the beam were found. The results are shown in Figure 14. In this case, the taper ratio has more effect on the eigenfrequencies related to the higher modes. The first and second eigenfrequencies decrease which is associated with the taper ratio being higher and the third or more eigenfrequencies increase along the taper ratio being higher.Figure 14
Eigenfrequencies vs. ply angle of Model A viaσ for CUS and CAS.(4) Section Ratio,ξ. Finally, the eigenfrequencies via section ratios ξ of the beam were computed. The results are shown in Figure 15.Figure 15
Eigenfrequencies vs. ply angle of Model A viaξ for CUS and CAS.Figure15 shows that the eigenfrequencies via section ratios have irregular trends for each. This is because of the characteristic that circles, squares, etc. have no values for primary warping functions [1], that is, see Figure 16, the primary function of the ellipse contour of Model A has almost zero values at ξ=0.7716. As a result, the effects of section ratio on the eigenfrequencies exist but the substantial factor of the effects is not section ratios but contour shapes.Figure 16
Primary warping function of Model A atξ=0.7716.


## 3.4.5. Comparison of Model B and Model C

For the comparison with a symmetric section, whena=c in Model A, Model A is fixed to Model B (Figure 7). And then, the eigenfrequencies of Model B and Model C were compared to each other. The results are shown in Figure 17.Figure 17
Eigenfrequencies of Model B and Model C for CUS and CAS.In Figure17, Model C has symmetric eigenfrequencies at a ply angle of 90°, but on the contrary, Model B has not. Also, it is notable that CUS and CAS have the same values at ply angles 0°, 90°, and 180° for each, and especially, ω2 of CAS of Model B has an observable asymmetric curve of eigenfrequency which is one of vertical transverse shear vibration modes.


## 4. Conclusion

A lot of effort and attention are needed in deriving and correcting the warping functions from the geometric relationship of the chordwise asymmetric closed cross-sections. However, this is an indispensable procedure for initial sizing through analytical methods in the early stages of design development, such as aircraft wings and rotor blades.For this purpose, in this paper, a model with a chordwise asymmetric closed cross-section (Model A) of a composite thin-walled beam was selected considering the contour shape for generalization of chordwise asymmetric closed cross-sections. And then, mathematical modeling was carried out by modifying the warping functions of the asymmetric cross-section and considering the transverse shear effect, warping restraint effect, and geometrical parameters, such as preset angle (α0), pretwist angle (γ0), taper ratio (σ), and section ratio (ξ) of the beam.Based on the mathematical modeling with the corrected warping function, mass coefficients, stiffness coefficients, and eigenfrequencies of the beam with a chordwise asymmetric closed cross-section were investigated for the CUS and CAS configurations of Model A. In addition, the present paper examined the effects of the following parameters: preset angle, pretwist angle, taper ratio, and section ratio of the beam. The results are presented as families of curves and tables, which may be used for the direct estimation of the eigenfrequencies for any given beam. Also, a comparative study was conducted on the symmetrical cross-section model (Model C) and the symmetrical cross-section model (Model B) with same chord length and section thickness and the results were also provided.From the above results, it is shown that the theoretical explicit analysis of the thin-walled beam with chordwise asymmetric closed cross-sections through the correction technique of the warping functions proposed in this study is possible and can be applied to the field of dynamic characteristics and aeroelasticity and so on. As a future research work, other lay-up schemes which may yield different elastic couplings might be considered. Moreover, in order to obtain more practical results, the multicell theory as well as stiffeners (spars and ribs) of the wing should be considered.



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*Source: 7503097-2019-02-25.xml*