# Differential Equations of Motion for Naturally Curved and Twisted Composite Space Beams

Authors: Ying Hao; Wei He; Yanke Shi
Journal: Shock and Vibration
 (2018)

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## Abstract

The differential equations of motion for naturally curved and twisted elastic space beams made of anisotropic materials with noncircular cross sections, being a coupled system consisting of 14 second-order partial differential equations with variable coefficients, are derived theoretically. The warping deformation of beam’s cross section, as a new design factor, is incorporated into the differential equations in addition to the anisotropy of material, the curvatures of the rod axis, the initial twist of the cross section, the rotary inertia, and the shear and axial deformations. Numerical examples show that the effect of warping deformation on the natural frequencies of the beam is significant under certain geometric and boundary conditions. This study focuses on improving and consummating the traditional theories to build a general curve beam theory, thereby providing new scientific research reference and design principle for curve beam designers.


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## Body

## 1. Introduction

Composite space beams with noncircular cross sections have extensive applications in civil, mechanical, aircraft, and aerospace constructions given their outstanding engineering properties, such as specific strength, specific stiffness, weight, and fatigue life over metallic counterparts. The static and dynamic characteristics of composite beams have attracted considerable attention in recent decades.Elastic couplings occur among all the forms of deformation given the anisotropy of the material. Moreover, in-plane and out-of-plane displacements may be coupled. Therefore, the differential equations of motion, obtained by the 3D elastic theory under isothermal conditions, become more complex by allowing initial curvature and torsion. Euler-Bernoulli and Timoshenko beam theories are unable to accurately describe the behavior of composite beams.Many researchers have conducted profound and careful research on nonclassical influences, such as transverse shear deformations, torsion-related warping, and several elastic couplings that can develop into composite beams. The natural frequencies and modes of composite beams under various boundary conditions are obtained by using the first-order shear deformation theory [1]. Based on the third-order shear theory, the forced vibration response of composite beams is investigated [2]. Li et al. [3] discussed the influence of different support conditions on the natural frequency of composite beams using many numerical examples. In this study, the effect of shear deformation is noted. Ramalingeswara and Ganesan [4] and Zapfe and Lesieutre [5] delivered the natural frequencies of laminated beams by using higher order shear theory and iterative method.Most of abovementioned studies [1–5] focus on straight beams. The research on the dynamic behavior is far less in composite curved beams than in straight beams considering the curvature and torsion. Yildirim [6] studied the free vibration of symmetrically orthogonal laminated curved beams, in which rotary inertia and axial and shear deformation effects are considered. Tseng et al. [7] studied the free vibration of laminated curved beams with variable curvature based on Timoshenko theory. However, a transverse shear strain is constant along the thickness direction, and the shear shape factor was used in this research to adjust the lateral shear stiffness. The accuracy of the results significantly depends on a selected shear shape factor. Matsunaga [8, 9] used 1D high-order theory to solve the natural frequencies and buckling loads of composite curved beams subjected to axial concentrated force considering the shear deformation and moment of inertia. Baba and Thoppul [10] explored the flexural strength and vibration characteristics of sandwich laminated curved beams determined the effects of laminated debonding and curvature on their natural frequencies. Yildirim [11] derived the differential equations of motion of spatial curved bars of anisotropic materials, but the examples in this research remain straight beams.The abovementioned studies [1–11] included one or several items of the anisotropy of the rod material, the curvature and torsion of the rod axis, the effects of rotary inertia, and the shear and axial deformations. The effect of the warping deformation of the cross section on natural frequencies is disregarded. In recent years, many researchers have attempted to establish generalized curved beam theory with warping effect. Kim et al. [12] developed a finite element-based beam analysis for anisotropic beams with arbitrary shaped cross sections through asymptotic expansion method. Displacements at each order are split into fundamental and warping solutions in their research. The influence of the warping effect is analyzed through examples, thereby demonstrating the importance of including the warping degrees of freedom in the dynamic analysis [13]. Yu et al. [14] formulated a generalized finite element-based cross-sectional analysis for inhomogeneous, initial curved and twisted, anisotropic beams, in which the transverse shear deformation is classified as part of the warping field. Ghafari and Rezaeepazhand [15] discussed the free vibration of rotating composite beams with an arbitrary cross section on the basis of dimensional reduction method. These authors decomposed a 3D elasticity problem of the composite beam into a 2D cross-sectional and a 1D beam analysis. Recently, Sapountzakis and Tsiptsis [16–19] used the Analog Equation Method and the Isogeometric Tool to solve the static and dynamic analysis of the curved beams with arbitrary cross section taking into account nonuniform warping and shear deformation effects. Nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions in these papers.As an extension of [11, 20–22], the essential features and novel aspects of the present formulation compared with previous ones are summarized as follows.(i)
The improved beam theory is capable of spatial curved beams with arbitrary cross section, variable curvature, and composite material considering warping effect.(ii)
The warping function of unidirectional composite beam with rectangular cross section can be obtained using the separation of variables method discussed in [23].(iii)
Numerical examples are presented for two 3D curved beams with constant curvature and torsion. The results show that the effect of the warping deformation on natural frequencies of the beam is significant under certain geometric and boundary conditions.(iv)
The influence of the warping deformation on the natural frequency is enhanced with the increase in the anisotropy of materials.


## 2. Geometrical Characteristics of a Space Anisotropic Beam

Considering a naturally curved and twisted anisotropic beam in space in Figure1, t,n,b indicate the tangential, normal, and binormal unit vectors of the beam axis, respectively. A second rectangular Cartesian frame (s,ξ,η) is introduced to consider the initial twist of the cross section. The angle between the ξ- and n-axes is θs. For a smooth curve, the Frenet–Serret formulae are(1)t′=k1n,n′=-k1t+k2b,b′=-k2n,where ′=d/ds and s,k1,k2 are the arc coordinate, the curvature, and torsion of the curve, respectively. In order to take into account the initial twist of the cross section, a new Cartesian reference frame (x1,ξ,η) is introduced such that the x1-axis is in the direction of t. If the unit vectors of Oξ and Oη are represented by iξ and iη, then (2)iξ=ncos⁡θ+bsin⁡θiη=-nsin⁡θ+bcos⁡θ.Figure 1
Geometry of a naturally curved and twisted anisotropic beam.With simultaneous equations (1) and (2), we have(3)t′iξ′iη′=0kη-kξ-kη0kskξ-ks0tiξiη,where kξ=k1sin⁡θ, kη=k1cos⁡θ, ks=k2+θ′.


## 3. Constitutive Equations

The following assumptions are used in the present study: the beam material is linearly elastic, homogeneous, and anisotropic. The pretwisting of the cross section is neglectedkξ=0. The generalized Hooke’s law for linearly elastic materials is(4)σi=Cijejor(5)ei=Sijσji,j=1,2,…,6,where Cij, Sij are the stiffness and flexibility matrices, correspondingly. Let(6)σ1=σs,σ2=σξ,σ3=ση,σ4=τξη,σ5=τsη,σ6=τsξ,e1=ess,e2=eξξ,e3=eηη,e4=2eξη,e5=2esη,e6=2esξ.The following assumptions on stresses are valid in the classical rod theory.(7)σ2=σ3=σ4=0.σβ(β=1,5,6) can be represented as eβ(β=1,5,6) by solving (7) and (5) simultaneously. Then, the results are substituted to the residual expressions of (5), and the strain components em(m=2,3,4) are related to eβ(β=1,5,6) by (8)em=Smnαnβeβm=2,3,4;n,β=1,5,6.In (8),(9)α11=S55S66-S562Δ,α55=S11S66-S162Δ,α66=S11S55-S152Δ,α15=α51=S16S56-S15S66Δ,α16=α61=S15S56-S16S55Δ,α56=α65=S15S16-S11S56Δ;Δ=detS11S15S16S15S55S56S16S56S66.So, σβ(β=1,5,6) is only related to eβ(β=1,5,6) but unrelated to em(m=2,3,4) when (8) is substituted to (4). Then, (4) is reduced to(10)σk=Ckβ+CkmSmnαnβeβ≜Qkβeβ;k,β,n=1,5,6;m=2,3,4.The displacement components of an arbitrary point of the beam are given as [20–22, 24](11)us,ξ,η,t=Ws,ξ,η,tt+Us,ξ,η,tiξ+Vs,ξ,η,tiη.u(s,ξ,η,t) is not only a function of coordinates s,ξ,η, but also a function of time under dynamic loads, in which (12)Us,ξ,η,t=uξs,t-ηφss,t,Vs,ξ,η,t=uηs,t+ξφss,tWs,ξ,η,t=uss,t+ηφξs,t-ξφηs,t+αs,tφξ,η,where uξ(s,t), uη(s,t), us(s,t) are the three translations, φξ(s,t), φη(s,t), φs(s,t) are the three rotations, α(s,t) is a generalized warping coordinate which is used to describe the decay of deformation along beam length, and φ(ξ,η) is the warping function of Saint-Venant’s torsion. Let ess, esξ, esη be the axial and shear strain components, respectively,(13)gess=εs+ηωξ-ξωη+gφα′+ks∂φ∂ξη-∂φ∂ηξα2gesξ=Gξεξ-φη-ηωs+g∂φ∂ξ+kηφα2gesη=Gηεη+φξ+ξωs+g∂φ∂η+kξφα,in which Gξ, Gη are shear correction factors, and g=1-ξkη+ηkξ. Based on the assumption that the initial curvature of the beam is gentle, then g≈1. Let(14)εs=us′-kηuξ+kξuη,εξ=uξ′-ksuη+kηus,εη=uη′-kξus+ksuξ,ωs=φs′-kηφξ+kξφη,ωξ=φξ′-ksφη+kηφs,ωη=φη′-kξφs+ksφξ.Let, for convenience, the resultant forces and moments on the cross section be expressed asQ1=Qs, Q5=Qη, Q6=Qξ, M1=Ms, M5=Mη, M6=Mξ and shown as(15)Qi=∬AσidA;Mi=εijk∬AxjσkdA;i,j,k=1,5,6,where εijk is the permutation tensor and x1=s; x5=η; x6=ξ.The generalized warping moment is(16)T=∬Aφσ1dA.The constitutive equations expressed by six displacement functionsui(s), φi(s)i=s,ξ,η and the generalized warping coordinate α(s) can be obtained by substituting (10), (13), and (14) into the above formulations (15)-(16).


## 4. Equations of Motion

The external force and moments per unit length of the beam axis are expressed asqex(s,t)=qsex(s,t),qξex(s,t),qηex(s,t), mex(s,t)=msex(s,t),mξex(s,t),mηex(s,t). Also let qin(s,t)=qsin(s,t),qξin(s,t),qηin(s,t), min(s,t)=msin(s,t),mξin(s,t),mηin(s,t) be the inertial force and the inertial moment per unit length of the rod axis. Owing to the displacements and the rotations that are assumed to be infinitesimal, the equations of motion can be written as [11, 24](17)∂Qs,t∂s+qexs,t=qins,t,∂Ms,t∂s+t×Qs,t+mexs,t=mins,t,in which (18)qsins,t=∬Aρ∂2uss,t∂t2dξdη;qξins,t=∬Aρ∂2uξs,t∂t2dξdη;qηins,t=∬Aρ∂2uηs,t∂t2dξdη;msins,t=∬Aρξ∂2uηs,t∂t2dξdη-∬Aρη∂2uξs,t∂t2dξdη;mξins,t=∬Aρη∂2uss,t∂t2dξdη;mηins,t=-∬Aρξ∂2uss,t∂t2dξdη.The equation of motion in a scalar form can be obtained by substituting (12)–(14) into (17). The equations of ∂us(s,t)/∂s, ∂uξ(s,t)/∂s, ∂uη(s,t)/∂s, ∂φs(s,t)/∂s, ∂φξ(s,t)/∂s, ∂φη(s,t)/∂s, ∂α(s,t)/∂s can be obtained by solving the abovementioned equations simultaneously. The obtained results are substituted into the derivatives of the generalized warping moment to determine the final differential equations of motion of naturally curved and twisted elastic space beams of anisotropic materials.(19)∂Ts,t∂s=∬σsφdξdη′.


## 5. Applications

### 5.1. Differential Equations of Motion for the Examples

A simple example is developed here for illustration. Two cylindrical helical springs with rectangular cross section can be regarded as 3D curved beams with constant curvature and torsion (Figure2). The warping function of unidirectional composite beam with rectangular cross section (Figure 3) can be obtained using the separation of variables method [23]:(20)φξ,η=ξη-∑n=1,3,5∞cneαγnbαγn1+e2αγnbeαγnη-cneαγnbαγn1+e2αγnbe-αγnηsin⁡γnξ,where α=Gxξ/Gxη, γn=nπ/2a, cn=16a2-1(n-1)/2/n2π2, n=1,3,5⋯.Figure 2
Cylindrical helical spring.Figure 3
Unidirectional composite beam.The differential equations of motion for cylindrical helical springs with rectangular cross section made of unidirectional composite materials are described as follows:(21)∂uss,t∂s=kηuξs,t+ksQ11D1D10D3I2+D2D9I3-D12I3I2A11D32I2+D22I3-D11I2I3-ksQ11D4A11αs,t+1A11-Q11D12I2I3A11A11D32I2+A11D22I3-A11D11I2I3+Q11D12I2I3Qss,t-D1D2I3A11D32I2+A11D22I3-A11D11I2I3+Q11D12I2I3Mξs,t+D1D3I2A11D32I2+A11D22I3-A11D11I2I3+Q11D12I2I3Mηs,t+D1I2I3A11D32I2+A11D22I3-A11D11I2I3+Q11D12I2I3Ts,t,∂uξs,t∂s=-kηuss,t+ksuηs,t+φηs,t-D5Q66+kηD1Q66A66Gξαs,t+1A66GξQξs,t,∂uηs,t∂s=-ksuξs,t-φξs,t-D6Q55A55Gηαs,t+1A55GηQηs,t,∂φss,t∂s=kηφξs,t+-D7Q55+D8Q66+kηD2Q66I3Q55+I2Q66αs,t+1I3Q55+I2Q66Mss,t,∂φξs,t∂s=-kηφs+ksφη-D1D2I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3Qss,t+ksA11D10D3D2-D32D9-D2D12I3+D11D9I3+Q11I3D1D2D4-D1D9A11D32I2+D22I3-D11I2I3+Q11D12I2I3αs,t+1I2Q11-A11D22I3Q11I2A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mξs,t+A11D2D3Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mηs,t+A11D2I3Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Ts,t,∂φηs,t∂s=-ksφξs,t+D1D3I2A11D32I2+D22I3-D11I2I3+Q11D12I2I3Qss,t+ksA11D10D22-D3D2D9+D3D12I2-D10D11I2+Q11I2D1D1D10-D3D4A11D32I2+D22I3-D11I2I3+Q11D12I2I3αs,t+A11D2D3Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mξs,t+1I3Q11-A11D32I2Q11I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mηs,t-A11D3I2Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Ts,t,∂αs,t∂s=-ksA11D10D3I2+I3D2D9-I3D12I2+Q11D1D4I2I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3αs,t+D1I3I2A11D32I2+D22I3-D11I2I3+Q11D12I2I3Qss,t+A11D2I3Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mξs,t-A11D3I2Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mηs,t-A11I3I2Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Ts,t,∂Qss,t∂s=kηQξs,t+ρA∂2uss,t∂t2+ρD1∂2αs,t∂t2,∂Qξs,t∂s=-kηQss,t+ksQηs,t+ρA∂2uξs,t∂t2,∂Qηs,t∂s=-ksQξs,t+ρA∂2uηs,t∂t2,∂Mss,t∂s=kηMξs,t+ρI1∂2φss,t∂t2,∂Mξs,t∂s=Qηs,t-kηMss,t+ksMηs,t+ρI2∂2φξs,t∂t2+ρD2∂2αs,t∂t2,∂Mηs,t∂s=-Qξ-ksMξ+ρI3∂2φηs,t∂t2-ρD3∂2αs,t∂t2,∂Ts,t∂s=ρD11α¨+ρD1u¨s+ρD2φ¨ξ-ρD3φ¨η+D16Q55+D14Q66+2D15+D11kηkηQ66+D13ks2Q11-D7Q55-D8+D2kηQ662I3Q55+I2Q66+D32D102I2ks2Q11A11I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3-2ks2Q11A11D12D10D3I2+D9D12D2I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3+2D9D10D3D4ks2Q11A11A11D32I2+D22I3-D11I2I3+Q11D12I2I3+D122I2I3ks2Q11A11A11D32I2+D22I3-D11I2I3+Q11D12I2I3+2ks2Q112D1D4D10D3I2+D2D9I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3+D22D92I3ks2Q11A11I2A11D32I2+D22I3-D11I2I3+Q11D12I2I3-ks2Q11D92I2-ks2Q11D102I3-ks2Q112D42A11+D12D42I2I3ks2Q113A11A11D32I2+D22I3-D11I2I3+Q11D12I2I3-D62Q552A55-D5+D1kη2Q662A66αs,t+ksQ11D4D32I2+D22I3-D11I2I3-D1D10D3I2+D2D9I3-D12I3I2A11D32I2+D22I3-D11I2I3+Q11D12I2I3Qss,t+D5+D1kηQ66A66Qξs,t+D6Q55A55Qηs,t+D7Q55-D8+D2kηQ66I3Q55+I2Q66Mss,t-ksA11D10D2D3-D32D9-D12D2I3+D11D9I3+ksQ11D1I3D2D4-D1D9A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mξs,t-ksA11D9D2D3+D22D10+D12D3I2-D11D10I2+ksQ11D1I2D1D10-D3D4A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mηs,t+ksA11D9D2I3+D10D3I2-D12I2I3+ksQ11D4I2I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3Ts,t,where (22)A11=∬AQ11dξdη,A55=∬AQ55dξdη,A66=∬AQ66dξdη,D1=∬Aφdξdη,D2=∬Aφηdξdη,D3=∬Aφξdξdη,D4=∬A∂φ∂ξη-∂φ∂ηξdξdη,D5=∬A∂φ∂ξdξdηD6=∬A∂φ∂ηdξdη,D7=∬Aξ∂φ∂ηdξdη,D8=∬Aη∂φ∂ξdξdη,D12=∬Aφ∂φ∂ξη-∂φ∂ηξdξdηD9=∬Aη∂φ∂ξη-∂φ∂ηξdξdη,D10=∬Aξ∂φ∂ξη-∂φ∂ηξdξdη,D11=∬Aφ2dξdηD13=∬A∂φ∂ξη-∂φ∂ηξ2dξdη,D14=∬A∂φ∂ξ2dξdη,D15=∬Aφ∂φ∂ξdξdηD16=∬A∂φ∂η2dξdη.


### 5.2. Effects of Material on the Natural Frequencies with Warping

The geometric parameters and finite element meshes of the two beams are exactly the same: the cylinder diameterR=5 mm, the helix pitch angle α-=5∘, the number of active turns n=4, and the rectangular cross section of 2a=1 mm in the ξ-direction and 2b=0.4 mm in the η-direction. The only difference between them is the material properties. Beam 1 is made of isotropic materials (the density is ρ=7900kg/m3, the modulus of elasticity is E=206GPa, and Poisson’s ratio is μ=0.3), while Beam 2 is made of unidirectional composite materials (the modulus of elasticity in the three main directions is E1=181Gpa, E2=E3=10.3Gpa, the modulus of shearing of the three planes is G23=3.433Gpa, G12=G13=7.17Gpa, the main Poisson’s ratio is μ23=0.5, μ12=μ13=0.28, and the density is ρ=1600kg/m3). We partitioned Beam 1 (Beam 2) into 720 Solid45 (Solid46) elements using ANSYS for analysis. The total number of nodal points is 1452. The first five natural frequencies of Beam 1 and Beam 2 are listed in Tables 1 and 2. The first five normal mode shapes of beams and the generalized warping coordinate are demonstrated in Figures 4–7, respectively.Table 1
The first five frequencies of Beam 1 (made of isotropic material).
Mode number (Hz) FEM Present study (PS) Warping ignored Errors (%) Warping included Errors (%) 1 521.34 745.84 43.06 523.25 0.37 2 899.54 1009.30 12.20 897.71 0.20 3 915.97 1044.17 13.99 914.36 0.17 4 970.21 1122.22 15.67 973.88 0.38 5 1129.5 1338.65 18.52 1118.41 0.98Table 2
The first five frequencies of Beam 2 (made of unidirectional composite material).
Mode number (Hz) FEM Present study (PS) Warping ignored Errors (%) Warping included Errors (%) 1 351.54 512.80 45.87 358.21 1.89 2 675.19 981.32 45.34 687.63 1.84 3 787.86 1117.87 41.89 805.74 2.27 4 880.95 1200.09 36.23 903.08 2.51 5 940.96 1357.22 44.24 958.56 1.87Figure 4
First five mode shapes of Beam 1: (1) first mode (523.25 Hz); (2) second mode (897.71 Hz); (3) third mode (914.36 Hz); (4) fourth mode (973.88 Hz); (5) fifth mode (1118.41 Hz).Figure 5
First five mode shapes of Beam 2 (made of unidirectional composite material): (a) first mode (358.21 Hz); (b) second mode (687.63 Hz); (c) third mode (805.74 Hz); (d) fourth mode (903.08 Hz); (e) fifth mode (958.56 Hz).
(a) (b) (c) (d) (e)Figure 6
First five normal mode shapes of the generalized warping coordinate of Beam 1.
(a)
First mode (523.25 Hz) (b)
Second mode (897.71 Hz) (c)
Third mode (914.36 Hz) (d)
Fourth mode (973.88 Hz) (e)
Fifth mode (1118.41 Hz)Figure 7
First five normal mode shapes of the generalized warping coordinate of Beam 2.
(a)
First mode (358.21 Hz) (b)
Second mode (687.63 Hz) (c)
Third mode (805.74 Hz) (d)
Fourth mode (903.08 Hz) (e)
Fifth mode (958.56 Hz)


## 5.1. Differential Equations of Motion for the Examples

A simple example is developed here for illustration. Two cylindrical helical springs with rectangular cross section can be regarded as 3D curved beams with constant curvature and torsion (Figure2). The warping function of unidirectional composite beam with rectangular cross section (Figure 3) can be obtained using the separation of variables method [23]:(20)φξ,η=ξη-∑n=1,3,5∞cneαγnbαγn1+e2αγnbeαγnη-cneαγnbαγn1+e2αγnbe-αγnηsin⁡γnξ,where α=Gxξ/Gxη, γn=nπ/2a, cn=16a2-1(n-1)/2/n2π2, n=1,3,5⋯.Figure 2
Cylindrical helical spring.Figure 3
Unidirectional composite beam.The differential equations of motion for cylindrical helical springs with rectangular cross section made of unidirectional composite materials are described as follows:(21)∂uss,t∂s=kηuξs,t+ksQ11D1D10D3I2+D2D9I3-D12I3I2A11D32I2+D22I3-D11I2I3-ksQ11D4A11αs,t+1A11-Q11D12I2I3A11A11D32I2+A11D22I3-A11D11I2I3+Q11D12I2I3Qss,t-D1D2I3A11D32I2+A11D22I3-A11D11I2I3+Q11D12I2I3Mξs,t+D1D3I2A11D32I2+A11D22I3-A11D11I2I3+Q11D12I2I3Mηs,t+D1I2I3A11D32I2+A11D22I3-A11D11I2I3+Q11D12I2I3Ts,t,∂uξs,t∂s=-kηuss,t+ksuηs,t+φηs,t-D5Q66+kηD1Q66A66Gξαs,t+1A66GξQξs,t,∂uηs,t∂s=-ksuξs,t-φξs,t-D6Q55A55Gηαs,t+1A55GηQηs,t,∂φss,t∂s=kηφξs,t+-D7Q55+D8Q66+kηD2Q66I3Q55+I2Q66αs,t+1I3Q55+I2Q66Mss,t,∂φξs,t∂s=-kηφs+ksφη-D1D2I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3Qss,t+ksA11D10D3D2-D32D9-D2D12I3+D11D9I3+Q11I3D1D2D4-D1D9A11D32I2+D22I3-D11I2I3+Q11D12I2I3αs,t+1I2Q11-A11D22I3Q11I2A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mξs,t+A11D2D3Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mηs,t+A11D2I3Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Ts,t,∂φηs,t∂s=-ksφξs,t+D1D3I2A11D32I2+D22I3-D11I2I3+Q11D12I2I3Qss,t+ksA11D10D22-D3D2D9+D3D12I2-D10D11I2+Q11I2D1D1D10-D3D4A11D32I2+D22I3-D11I2I3+Q11D12I2I3αs,t+A11D2D3Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mξs,t+1I3Q11-A11D32I2Q11I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mηs,t-A11D3I2Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Ts,t,∂αs,t∂s=-ksA11D10D3I2+I3D2D9-I3D12I2+Q11D1D4I2I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3αs,t+D1I3I2A11D32I2+D22I3-D11I2I3+Q11D12I2I3Qss,t+A11D2I3Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mξs,t-A11D3I2Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mηs,t-A11I3I2Q11A11D32I2+D22I3-D11I2I3+Q11D12I2I3Ts,t,∂Qss,t∂s=kηQξs,t+ρA∂2uss,t∂t2+ρD1∂2αs,t∂t2,∂Qξs,t∂s=-kηQss,t+ksQηs,t+ρA∂2uξs,t∂t2,∂Qηs,t∂s=-ksQξs,t+ρA∂2uηs,t∂t2,∂Mss,t∂s=kηMξs,t+ρI1∂2φss,t∂t2,∂Mξs,t∂s=Qηs,t-kηMss,t+ksMηs,t+ρI2∂2φξs,t∂t2+ρD2∂2αs,t∂t2,∂Mηs,t∂s=-Qξ-ksMξ+ρI3∂2φηs,t∂t2-ρD3∂2αs,t∂t2,∂Ts,t∂s=ρD11α¨+ρD1u¨s+ρD2φ¨ξ-ρD3φ¨η+D16Q55+D14Q66+2D15+D11kηkηQ66+D13ks2Q11-D7Q55-D8+D2kηQ662I3Q55+I2Q66+D32D102I2ks2Q11A11I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3-2ks2Q11A11D12D10D3I2+D9D12D2I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3+2D9D10D3D4ks2Q11A11A11D32I2+D22I3-D11I2I3+Q11D12I2I3+D122I2I3ks2Q11A11A11D32I2+D22I3-D11I2I3+Q11D12I2I3+2ks2Q112D1D4D10D3I2+D2D9I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3+D22D92I3ks2Q11A11I2A11D32I2+D22I3-D11I2I3+Q11D12I2I3-ks2Q11D92I2-ks2Q11D102I3-ks2Q112D42A11+D12D42I2I3ks2Q113A11A11D32I2+D22I3-D11I2I3+Q11D12I2I3-D62Q552A55-D5+D1kη2Q662A66αs,t+ksQ11D4D32I2+D22I3-D11I2I3-D1D10D3I2+D2D9I3-D12I3I2A11D32I2+D22I3-D11I2I3+Q11D12I2I3Qss,t+D5+D1kηQ66A66Qξs,t+D6Q55A55Qηs,t+D7Q55-D8+D2kηQ66I3Q55+I2Q66Mss,t-ksA11D10D2D3-D32D9-D12D2I3+D11D9I3+ksQ11D1I3D2D4-D1D9A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mξs,t-ksA11D9D2D3+D22D10+D12D3I2-D11D10I2+ksQ11D1I2D1D10-D3D4A11D32I2+D22I3-D11I2I3+Q11D12I2I3Mηs,t+ksA11D9D2I3+D10D3I2-D12I2I3+ksQ11D4I2I3A11D32I2+D22I3-D11I2I3+Q11D12I2I3Ts,t,where (22)A11=∬AQ11dξdη,A55=∬AQ55dξdη,A66=∬AQ66dξdη,D1=∬Aφdξdη,D2=∬Aφηdξdη,D3=∬Aφξdξdη,D4=∬A∂φ∂ξη-∂φ∂ηξdξdη,D5=∬A∂φ∂ξdξdηD6=∬A∂φ∂ηdξdη,D7=∬Aξ∂φ∂ηdξdη,D8=∬Aη∂φ∂ξdξdη,D12=∬Aφ∂φ∂ξη-∂φ∂ηξdξdηD9=∬Aη∂φ∂ξη-∂φ∂ηξdξdη,D10=∬Aξ∂φ∂ξη-∂φ∂ηξdξdη,D11=∬Aφ2dξdηD13=∬A∂φ∂ξη-∂φ∂ηξ2dξdη,D14=∬A∂φ∂ξ2dξdη,D15=∬Aφ∂φ∂ξdξdηD16=∬A∂φ∂η2dξdη.


## 5.2. Effects of Material on the Natural Frequencies with Warping

The geometric parameters and finite element meshes of the two beams are exactly the same: the cylinder diameterR=5 mm, the helix pitch angle α-=5∘, the number of active turns n=4, and the rectangular cross section of 2a=1 mm in the ξ-direction and 2b=0.4 mm in the η-direction. The only difference between them is the material properties. Beam 1 is made of isotropic materials (the density is ρ=7900kg/m3, the modulus of elasticity is E=206GPa, and Poisson’s ratio is μ=0.3), while Beam 2 is made of unidirectional composite materials (the modulus of elasticity in the three main directions is E1=181Gpa, E2=E3=10.3Gpa, the modulus of shearing of the three planes is G23=3.433Gpa, G12=G13=7.17Gpa, the main Poisson’s ratio is μ23=0.5, μ12=μ13=0.28, and the density is ρ=1600kg/m3). We partitioned Beam 1 (Beam 2) into 720 Solid45 (Solid46) elements using ANSYS for analysis. The total number of nodal points is 1452. The first five natural frequencies of Beam 1 and Beam 2 are listed in Tables 1 and 2. The first five normal mode shapes of beams and the generalized warping coordinate are demonstrated in Figures 4–7, respectively.Table 1
The first five frequencies of Beam 1 (made of isotropic material).
Mode number (Hz) FEM Present study (PS) Warping ignored Errors (%) Warping included Errors (%) 1 521.34 745.84 43.06 523.25 0.37 2 899.54 1009.30 12.20 897.71 0.20 3 915.97 1044.17 13.99 914.36 0.17 4 970.21 1122.22 15.67 973.88 0.38 5 1129.5 1338.65 18.52 1118.41 0.98Table 2
The first five frequencies of Beam 2 (made of unidirectional composite material).
Mode number (Hz) FEM Present study (PS) Warping ignored Errors (%) Warping included Errors (%) 1 351.54 512.80 45.87 358.21 1.89 2 675.19 981.32 45.34 687.63 1.84 3 787.86 1117.87 41.89 805.74 2.27 4 880.95 1200.09 36.23 903.08 2.51 5 940.96 1357.22 44.24 958.56 1.87Figure 4
First five mode shapes of Beam 1: (1) first mode (523.25 Hz); (2) second mode (897.71 Hz); (3) third mode (914.36 Hz); (4) fourth mode (973.88 Hz); (5) fifth mode (1118.41 Hz).Figure 5
First five mode shapes of Beam 2 (made of unidirectional composite material): (a) first mode (358.21 Hz); (b) second mode (687.63 Hz); (c) third mode (805.74 Hz); (d) fourth mode (903.08 Hz); (e) fifth mode (958.56 Hz).
(a) (b) (c) (d) (e)Figure 6
First five normal mode shapes of the generalized warping coordinate of Beam 1.
(a)
First mode (523.25 Hz) (b)
Second mode (897.71 Hz) (c)
Third mode (914.36 Hz) (d)
Fourth mode (973.88 Hz) (e)
Fifth mode (1118.41 Hz)Figure 7
First five normal mode shapes of the generalized warping coordinate of Beam 2.
(a)
First mode (358.21 Hz) (b)
Second mode (687.63 Hz) (c)
Third mode (805.74 Hz) (d)
Fourth mode (903.08 Hz) (e)
Fifth mode (958.56 Hz)


## 6. Conclusion

The differential equations of motion for elastic space beams with noncircular cross sections and anisotropic materials have been derived systematically based on naturally curved and twisted beam theory. In the equations, the warping deformation of the cross section, the initial twist of the cross section, the shear and axial deformations, and the rotary inertia are studied theoretically.The results show that the effect of warping deformations on the natural frequencies of the beam is very significant under certain geometric and boundary conditions. The errors can be excessively large when the warping effect is disregarded. The influence of warping on the natural frequency is enhanced with the increase in the anisotropy of materials.



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*Source: 5015807-2018-05-15.xml*