Published: 21 September 2023

Multiple periodic motions of a two degrees-of-freedom carbon fiber reinforced polymer laminated cylindrical shell

Ting Gao1
Jing Li2
Shaotao Zhu3
Ziyu Guo4
1, 2, 3, 4Interdisciplinary Research Institute, Faculty of Science, Beijing University of Technology, Beijing 100124, China
3Faculty of Information Technology, Beijing University of Technology, Beijing 100124, China
Corresponding Author:
Jing Li
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Carbon fiber reinforced polymer is a composite material, which is widely used in various engineering fields due to its excellent properties. We systematically discuss the influence of axial load amplitude parameters on the multiple periodic motions of carbon fiber reinforced polymer laminated cylindrical shell model. Based on the Melnikov vector function, the bifurcation regions of periodic orbits are obtained. It is found that the system has at most four periodic orbits under parameters conditions. Moreover, the phase portraits of periodic orbits are given by numerical simulation. The results offer an idea for parameter control of shell structure.

1. Introduction

Compared with conventional materials, carbon fiber reinforced polymer shows better heat resistance and corrosion resistance. It is a kind of lightweight and high strength engineering structural composite material, which has a positive impact on the development of aerospace, medical devices, rail transportation and other fields [1-4]. Therefore, in order to fully exploit the advantages of novel composite materials and maximize their role as structures in practical engineering applications, it is essential to study the dynamic characteristics of novel composite plate and shell structures.

A large number of scholars have studied carbon fiber composite materials and cylindrical shell structure. Under the internal resonance condition, Zhang et al. [5] discussed the response characteristic and complex dynamic behaviors of cylindrical shell model. Poul et al. [6] analyzed the enhancement effect of the plastic layer of carbon fiber reinforced thin steel sheet under shear load through experimental research. Lim [7] studied the natural frequencies of different plate and shell structures under free and simply supported boundary conditions with negative Poisson's ratio. Zhang et al. [8] devoted to study the influences of radial linear load and axial load at both ends of the carbon fiber reinforced composite laminated cylindrical shell on the nonlinear radial breathing vibration.

The literature review showed that there are many researches on mechanical property and nonlinear vibration behaviors of shell structures. However, little attention has been paid to the periodic or multiple periodic vibration behaviors of cylindrical shells structure. The development of periodic solutions theory [9-11] is conducive to a deep understanding of the vibration characteristics of the system in different parameter regions, and provides guidance for the vibration reduction design of nonlinear dynamics theory.

Motivated by this, we pay attention to the multiple periodic motions and its parameter control conditions of the carbon fiber reinforced polymer laminated cylindrical shell structure. In Section 2, original coupled system and averaged equation are given. In Section 3, the multiple periodic motions are studied by Poincaré map and the Melnikov function, and the bifurcation parameter is obtained. The number of periodic solutions in different parameter areas is 0, 2, 4 respectively. In Section 4, the phase configurations of periodic orbits are presented by MATLAB software. In Section 5, we give the conclusions.

2. Dynamic model and averaged equation

2.1. Dynamic model

We focus on the carbon fiber reinforced polymer laminated cylindrical shell structure subjected to different loads, which can be modeled by the two degrees-of-freedom governing equation [8]:

w¨i+ωi2wi+μiw˙i+r=02βi(r+1)w12-rw2r+r=03βi(r+4)w13-rw2r+βi8wip1cos(Ω2t)=Ficos(Ω1t) ,

wherei=1, 2. w1, w2 represent the vibration amplitude of the first and second order modesrespectively. μi represent damping coefficients. ωi are two linear natural frequencies. Fi are radial line load amplitude. p1 is axial load amplitude. Ωi are frequency under external excitation. βi=βijj=1,.,8, (i=1,2) represent the dimensionless coefficient.

2.2. Averaged equation

The system given in the above section is a four-dimensional non-autonomous system. We need to analyze the perturbation of Eq. (1). Considering the following 1:2internal resonance relationship:

ωi2=Ωi2/42-i+εσi , i=1,2 ,

where 0<ε<<1, σi are the detuning parameters.

We assume that Ωi=2. The following scale transformations are introduced as:

βijεβij, i=1,2; j=1,8, μiεμi, FiεFi .

For the convenience of calculation, we use the multiple scales method to obtain the averaged equation of the system with the help of MAPLE software:

x˙=N x+G(x) ,

where x=(x11,x12,x21,x22)TR4, G=(G11,G12,G21,G22)T is a vector-valued polynomial in variables of xij(i=1,2;j=1,2). N=i, i2, 2(Ni). Ni=j, 3-j2 , 2((-1)j+1ni). ni=μi2/23i+σi2/2i+2. The symbol g, lp, q(M) represents a p×q block matrix with (g,l)-th blockMand all other blocks are zero matrices [12].

Introducing the transformation GεG, system Eq. (4) can be rewritten as:

x˙i=JDHi (xi)+ε Gi(x) , i=1,2,

where xi=(xi1,xi2)TR2, Gi=(Gi1,Gi2)T, J=j , 3-j2 , 2(-1)j+1, (j=1,2). In addition, Hi(xi)=ni(xi12+xi22)/2, DHi(xi)=(Hi/xi1 , Hi/xi2)T. Therefore, the study of multiple periodic motions of original system Eq. (1) can be transformed into the study of bifurcation of multiple periodic solutions of the system Eq. (5).

2.3. Periodic motions

When ε=0, system Eq. (5) degenerates to two uncoupled Hamiltoniansystems on plane (x11,x12) and (x21,x22). Then each system has a family of periodic orbits: Γhi={xhi|Hi(xi)=hi}, (i=1,2) surrounding the origin of the system for hiK. Assuming that Γh ican be expressed as:

xi1=2hinicosnit+i-1t0i-1, xi2=2hinisinnit+i-1t0i-1, i=1,2 .

We introduce the curvilinear coordinates in the neighborhood of the invariant torus Γh1×Γh2. Define a global cross section Σ in the phase space, and construct thekth iteration of Poincaré map Pk: ΣΣ. We need to calculate the simple zero of the Melnikov function M=(M1,M2,M3)T, where:

Mi=ni02πj=1,22xijGijdt , i=1,2,

Let n2=2n1=2 , then we can obtain:

M1=a16cos (2t0)h1h2+a14sin (2t0)h1h2+2a111h12+a110h1h2 ,
M2=b1cos (2t0)h1h2+2a26sin (2t0)h1h2+a214h22+2a213h1h2 ,
M3=h2b2+h1b3+b4+2(a14-a16)h2cos (2t0)-b1h1/h2sin (2t0)+2a26h1/h2cos (2t0) ,

where b1=a28-a27, b2=2a19+a29, b3=2(2a112-a212), b4=2(a11+a21).

Our goal is to analyze the number of solutions for (t0,h1,h2) of Eqs. (6-8), so we let M=0. Assuming that a14=a16=0, a27=a28, from Eqs. (6-7), we can obtain:

h1=ηh2, h2=γsin2t0,

where η=-a1102a111, γ=-2a26η2a213c1+a214. Substituting Eq. (9) into Eq. (8), we have:


where x=sin(2t0), c1=(ηb3+b2)γ2, c2=2a26ηγ. When c22-4c1b4-4b42>0 and b40, the system has four periodic solutions.

3. Numerical simulation

We detect the existence and multiplicity of periodic vibrations by changing the value ofp1. The parameters conditions PC={μ1, μ2, σ1, σ2, β1, β2, f1,f2} are as follows:

β1=(1.5,0,-2,-1,1,2,3,1), β2=(-2,1,1,2,1,1,-4,1).

(1) The axial load amplitude p1=0<p10 (p10=41055547+34124401 is the critical parameter value). In this case, the system Eq. (1) is only subjected to the radial line load, and b4=0. There is a pair of periodic orbit: Γ0:{h1=0.074483543, h2=0.055862657}. Fig. 1 represents the phase configurations of the periodic orbits projected on various planes and spaces.

Fig. 1There are two periodic orbits when p1=0

There are two periodic orbits when p1=0

a)(x11, x12) plane

There are two periodic orbits when p1=0

b)(x21,x22) plane

There are two periodic orbits when p1=0

c)(x11,x12, x21) space

(2) The axial load amplitude 0<p1<p10. In this case, there are four periodic solutions. Figs. 2-3 represent the phase portraits of the periodic orbits projected on various planes and spaces when p1=1.5, p1=3.5. When p1=1.5, the four periodic orbits can be divided into two pairs:

Γ+:h1=0.155636048, h2=0.116727036,
Γ-:h1=0.0197166881, h2=0.0147875161.

When p1=3.5, the four periodic orbits can be divided into two pairs:


When p1=1.5 is increased to p1=3.5, the distance between two pairs of periodic orbits becomes closer and closer. The periodic orbit with larger amplitude and the periodic orbit with smaller amplitude are gradually increasing, and the former increases slower than the latter. It should be noted that we have tried multiple sets of different values of p1 to confirm whether the periodic motions continue in this manner. Only two sets of data are listed here.

Fig. 2There are four periodic orbits when p1=1.5

There are four periodic orbits when p1=1.5

a)(x11, x12) plane

There are four periodic orbits when p1=1.5

b)(x21,x22) plane

There are four periodic orbits when p1=1.5

c)(x11,x12, x21) space

Fig. 3There are four periodic orbits when p1=3.5

There are four periodic orbits when p1=3.5

a)(x11, x12) plane

There are four periodic orbits when p1=3.5

b)(x21,x22) plane

There are four periodic orbits when p1=3.5

c)(x11,x12, x21) space

(3) The axial load amplitude p1=p10. In this case, the system has two periodic solutions satisfying Γ1:{h1=0.415880440, h2=0.311910330}. The periodic vibration phase diagrams of the system are shown in Fig. 4. We can observe that the periodic orbits Γ+ and Γ- coincide into Γ1 when p1=p10.

Fig. 4There are two periodic orbits when p1=p10

There are two periodic orbits when p1=p10

a)(x11, x12) plane

There are two periodic orbits when p1=p10

b)(x21,x22) plane

There are two periodic orbits when p1=p10

c)(x11,x12, x21) space

When p10, we have analyzed the specific bifurcation phenomenon when p1passes through p10 from left to right. This change also occurs when p1 passes through -p10. As shown in Fig. 5, we use (x11,x12,p1) space to describe the whole process. We can clearly see that when p1 crosses 0 from left to right, the periodic trajectory Γ0 splits immediately, forming two pairs of periodic trajectories. With the increase of p1, the increase rate of the amplitude of Γ+ is less than that in Γ-, that is, gradually approaching. The periodic orbitsΓ+ and Γ- coincide at p1=p10 to form Γ1, and then disappear when p1>p10. Above, we only analyze the effect of the axial load amplitude on the number of periodic orbits and the form of spiral. When p10. The analysis of p1<0 is similar to it.

Fig. 5Bifurcation of periodic orbits in (x11,x12,p1) space

Bifurcation of periodic orbits in (x11,x12,p1) space

4. Conclusions

It is essential to study periodic motions of cylindrical shell structure which is closely intertwined with energy transfer and vibration analysis. In this paper, the Melnikov function is obtained, which is different from the traditional quantitative method. Under certain parameter conditions, it is clear from theoretical and numerical results that the system has multiple periodic motions. We take the axial load amplitudep1 as bifurcation parameter to study the evolution law of the periodic orbits, and obtain the detailed bifurcation diagram about the parameter of p1, as shown in Fig. 5. It is found that the periodic motion of cylindrical shells structure is more sensitive to axial load than radial line load, which is consistent with the results obtained in Ref. [8], and further illustrates the correctness of the theoretical analysis in this paper.


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About this article

24 March 2023
27 April 2023
21 September 2023
Mathematical models in engineering
carbon fiber reinforced polymer
multiple periodic motions
Melnikov function
phase portraits
axial excitation

This research was supported by National Natural Science Foundation of China (Grant No.12272011) and also supported by National Key R&D Program of China (Grant No. 2022YFB3806000).

Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflict of interest

The authors declare that they have no conflict of interest.