# Theoretical Analysis on the Nonlinear Free Vibration of a Tri-Cross String

Authors: Bo Pan; Jingda Tang; Ryuichi Tarumi; Fulin Shang; Yanbo Wang; Weijun Zhang; Xueyan Zhang
Journal: Shock and Vibration
 (2017)

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## Abstract

Here we present a theoretical analysis on the nonlinear free vibration of a tri-cross string system, which is an element of space net-antennas. We derived the governing equations from Hamilton’s principle and obtained a linearized solution by the standard perturbation method. The semi-analytical solutions of the governing equations have not been provided referring to the solution of plate vibrating problem. This analysis revealed that natural frequencies of the tri-cross string depend on the vibration amplitude due to the geometrical nonlinearity in the constitutive equation. The geometric parameters, such as the diameters and the lengths of the constituent strings, also affect the frequency through the nonlinearity of the tri-cross string. The nonlinear natural frequency shows coupled characteristic; that is, the natural frequency of the tri-cross string varies with that of the constituent strings, but the contribution of each constituent string to the natural frequency is in different proportions.


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## Body

## 1. Introduction

Recently, we have analyzed the nonlinear dynamic response of the cross string, the simplest net structure of strings such as the space antennas, using Hamilton’s principle and the perturbation method [1]. Tri-cross string is another simplest net structure with odd constituent strings among all the multi-cross strings, except for the cross string. It is difficult to discuss their oscillation characteristics (such as natural frequency) by a theoretical analysis because of the complex nonlinearity. However, it is necessary to provide an analytical result to compare with experiments to obtain insight on the motion law of the structure and guide the structural design.Natural vibration of an initially stressed and linearly elastic string with small amplitudes has been extensively investigated by theoretical analysis, simulations, and experimental researches. However, the relevant researches on the initially stretched string with finite amplitudes are still rare, despite they are quite important for the engineering problems. The Kirchhoff string equation is accepted as a good first approximation of the nonlinear behavior in the transverse direction of a string [2]. Molteno and Tufillaro studied qualitatively the agreement between the analytical results obtained via the truncated Kirchhoff string equation and the experimental results [3]. Various numerical schemes, such as the Galerkin method, were developed to simulate the large-amplitude vibrating problem of the elastic strings based on the nonlinear Kirchhoff string model [4]. Xiong and Hutton used Hamilton’s principle to obtain the governing equation and the boundary conditions of a multi-guided rotating string [5], and they proved that Hamilton’s principle could be used for the derivation of nonlinear vibration problems as we presented in a recent paper [1]. Later on, more complicated conditions of single string vibrating problem were analyzed, such as moving boundary conditions [6], unsteady vibration varying length [7], and the influence of friction force [8]. The transverse vibration of nonlinear strings is governed by partial differential equations. A finite difference method has been developed to obtain the numerical solutions of these equations using the perturbation method [9], different from the treatment of Carrier [10, 11]. The mentioned studies mainly focus on the vibrating characteristics of a single elastic string. The experimental and computational researches of networks and space antennas most focus on the cases of linear vibration [12, 13] and linear design [14, 15], while the complex nonlinear behavior of the multistring networks is still an open issue. However, few theoretical works on the multistring networks were found in the literature. In this work, we further consider the problem of a tri-cross string, which is another simplest net structure with odd constituent strings and the theoretical studies on tri-cross strings are necessary, especially for the space antennas.In this study, we studied the free nonlinear vibration response of a tri-cross string, composed of three single strings as shown in Figure1, and attempted to provide a theoretical method to analyze the nonlinear features of the tri-cross string. The governing equations of the nonlinear vibration of the tri-cross string are derived from Hamilton’s principle and solved by the perturbation method. As the traditional potential-function method used for the nonlinear vibration of a single string is quite difficult to adopt for a tri-cross string, novel semi-analytical solutions of the governing equations were provided. Based on the obtained solutions, we discuss the influence of the nonlinear feature and coupling properties on the total frequency of the tri-cross string. The results show that the contribution of each constituent string to the tri-cross natural frequency is in different proportions. Moreover, the frequencies of the tri-cross string are dependent on the material properties of its constituent strings. The work may provide theoretical guide for the free vibration of net structures such as space antennas with flexible strings.Figure 1
(a) The initial state of a tri-cross string. (b) The oscillating state of a tri-cross string.
(a)
(b)


## 2. Derivation of the Motion Equations

### 2.1. Model

As shown in Figure1, the three single strings are connected to each other by two joints (later referred to as node) in the tri-cross string, and these two nodes will keep connected during vibration. This assembled structure known as tri-cross string can be treated as another kind of the simplest net of strings except for the cross string in [1]. In this study, the condition of geometrical nonlinearity and material linearity is considered. It means the motion of the tri-cross string is transverse and one-dimensional with large amplitude, but the elongation of the string during vibration is not negligible. Furthermore, the strains in these strings remain elastic in the linear range. In nonlinear analysis, the tension in the tri-cross should vary and can usually be expressed as the function of position and time, that is, T(x,t). For simplicity, the tension force can be further treated as the only function of time T(t) [2, 11]. In Figure 1, the axis is applied to connect to each string. The strings connected toy1-axis andy2-axis are jointed to the string ofx-axis, which are calledy1-axis andy2-axis for short, respectively. During the vibration, the vibrated modal ofy1-axis andy2-axis can be different orders and would differ in the characteristics of vibrating plane. However, they are both connected to thex-string and affected by each other by the constraint of nodes. Considering the real structure of net strings such as space antennas, the nonrigid boundary conditions and the finite span of this tri-cross string are adopted in this paper. For the sake of simplicity, the three single strings are chosen to be made of the same material as well.In the following section, we will try to simplify the structure of the tri-cross string and transfer it into a dynamic model. At first, the whole jointed tri-cross string structure can be divided into three single strings with two nodal forces applied onto each of them. Considering jogged two strings of the whole tri-cross string, the separated nodal force on each of the jogged two strings has the same amplitude but in the opposite direction. In this study, the jogged tri-cross string will be disassembled into the three strings separately, and one of them will be chosen as an object substituted into Hamilton’s principle. In this way, the governing equation of one single string belonging to the jogged tri-cross string structure can be obtained, and it can demonstrate the common properties of each single string.Under the above assumptions, the governing equations of the vibration of the tri-cross string are derived as follows.


### 2.2. Hamilton’s Principle

The Lagrange integral functionJ of a single string can be expressed as(1)J=∫t1t2Ltdt,whereJ is obtained by integrating Lagrange functionL with respect to the timet andL is expressed as(2)Lt=Kt-Ut,whereK is the kinetic energy andU is the potential energy of the system.As the derivation method and process of the mechanics models for the three constituent single strings of the tri-cross string are similar, it is convenient to present the derivation of the mathematical expressions ofK(t) andU(t) of one string.In the present problem, the vibrating direction is vertical tox-y plane. Therefore, the expression of the kinetic energyK(t) is shown as(3)Kt=12∫0l0ρ∂w∂t2dx,where l0 is the length of the string in the initial static equilibrium position, ρ is the mass per unit length and is the function ofx,w is the displacement of the string, andx is the longitudinal coordinate.The total potential energyU(t) of one string consists of one part due to tension workUT(t) and the other part coming from the external excitationUF(t), i.e.,U(t) =UT(t) +UF(t). As shown in Figure 2, a string in the equilibrium position presents an initial tension T0 and its initial lengthl0. We take an infinitesimal element of the displaced string as an object, and its length in the deformed configuration is ds. In the equilibrium position under initial tension, the length of this element is dx, under the definition of initial length being dx0. It is assumed that the string remains linearly elastic under the varying tension during the vibration, and the tension T(x,t) can be expressed as (4)Tx,t=EAε,in which the total strain of the element ε is(5)ε=ds-dx0dx0=dsdxdxdx0,dxdx0=1+ε0,where ε0 is the initial strain. According to the geometrical relations, the vibrating string away from the initial equilibrium position can be expressed as (6)dsdx=1+∂w∂x2.By inserting (5) and (6) into (4), (4) can be expressed as(7)Tx,t=EA1+ε01+∂ω∂x2-1,whereE is Young’s modulus andA is the cross-sectional area. For simplicity and emphasis of geometrically nonlinear behavior, the tension in the one-dimensional and flexible string can be regarded as the function of time only [2, 11]. With the solution of the nonlinear vibration problem of two-dimensional thin plates, averaging the stress in one axis is one effective method. In our previous study, the tension is presented under the same assumption [1] (8)Tt=T0+EAl0∫0l01+∂wx,t∂x2dx-l0or(9)Tt=EA+T0l0∫0l01+∂wx,t∂x2dx-l01+ε0.Using (9), the tension work stored in the potential energy of the one single string with a length of dx can be expressed as(10)dUTt=EA+T0l0∫0l01+∂w∂x2dx-l01+ε01+∂w∂x2dx-dx.Integrating (10) along the string, we can obtain(11)UTt=EA+T04l0∫0l0∂w∂x2dx·∫0l02ε01+ε0+∂w∂x2dx.On the other hand, one external excitation work stored in the potential energy is provided as(12)UFt=∫0l0Fjoint1t·δx-xjoint·wstx,yjoint1,yjoint2,tdx,where Fjoint1(t), Fjoint2(t) denote the nodal forces at the joints, whose coordinates are (x1,joint, yjoint1,yjoint2), (x2,joint, yjoint1,yjoint2), respectively; wst represents the displacement of the whole tri-cross string; δ(x) is Dirac delta function. Figure 3 shows the relation between the (inner) force and the excitation. In fact, the nodal force should be regarded as an inner constraint force by considering the whole tri-cross string. By dividing the tri-cross string into three strings, the nodal forces in the same position but belonging to two different jogged strings have the same amplitude and opposite direction.Figure 2
Displacement of thex-string component of the cross string in Figure 1.Figure 3
Modeling of the tri-cross string by separating the three strings of the structure and adding a joint force to the joint as the joint condition.For one single string with more than one joint with other constituent strings under multiple excitations, say, alongx-direction, (12) may be extended as(13)UFt=∫0l0∑i=1nFjointit·δx-xi,joint·wstx,yjoint1,yjoint2,tdx.According to Hamilton’s principle, (1) can be expressed by substituting (11), (12), (13) into (1) as(14)δJ=δ∫t1t2Kt-UTt-UFtdt=0.


### 2.3. Governing Equations

According to (14) and the differential variational principle, the governing equation of the x-string can be obtained as(15a)ρ∂2w∂t2-EA+T0l0∫0l0∂w∂x2dx+ε0l01+ε0·∂2w∂x2=Fjoint1t·δx-x1,joint+Fjoint2t·δx-x2,joint,y1=yjoint1,y2=yjoint2.Similarly, by applying the kinetic energyK(t) and the potential energyU(t) ony1-string andy2-string, we can obtain the governing equations ofy1-string andy2-string as(15b)ρ∂2w′∂t2-EA+T1l1∫0l1∂w′′∂y12dy+ε1l11+ε1·∂∂y1∂w′∂y1=-Fjoint1t·δy1-yjoint1,x=x1,joint,y2=yjoint2,(15c)ρ∂2w′′∂t2-EA+T2l2∫0l2∂w′′∂y22dz+ε2l21+ε2·∂∂y2∂w′′∂y2=-Fjoint2t·δy2-yjoint2,x=x2,joint,y1=yjoint1in which l1,l2 are the lengths ofy1-string andy2-string in the static equilibrium position, respectively;w, w′,w′′ are the displacements ofx-string,y1-string, andy2-string, respectively; and wst is the displacement of the whole tri-cross string structure,(16)w=wstx,yjoint1,yjoint2,t,w′=wstx1,joint,y1,yjoint2,t,w′′=wstx2,joint,yjoint1,y2,t.The boundary conditions corresponding tox-string,y1-string, andy2-string are shown as follows:(17a)w=0,atx=0,x=l0,(17b)w′=0,aty1=0,y1=l1,(17c)w′′=0,aty2=0,y2=l2,and the initial conditions corresponding tox-string,y1-string, andy2-string are (18)wstx,y1,y2t=0x=xi,y1=yj1,y2=yk2,=wstxi,yj1,yk2,0,where wst(xi,yj,zk,0) is the initial displacement.It is well known that the natural frequency with nonlinearity of a geometrically nonlinear string is composed of the linear natural frequency of the string and the nonlinear fluctuation frequency [16, 17]. Based on the above derivations, it is obvious that the vibration of the tri-cross string is coupled with each other due to the two joint nodes. In the following, we will discuss the nonlinearity and the coupling behavior among the x-string,y1-string, andy2-string with the two jogged mechanical connections.


## 2.1. Model

As shown in Figure1, the three single strings are connected to each other by two joints (later referred to as node) in the tri-cross string, and these two nodes will keep connected during vibration. This assembled structure known as tri-cross string can be treated as another kind of the simplest net of strings except for the cross string in [1]. In this study, the condition of geometrical nonlinearity and material linearity is considered. It means the motion of the tri-cross string is transverse and one-dimensional with large amplitude, but the elongation of the string during vibration is not negligible. Furthermore, the strains in these strings remain elastic in the linear range. In nonlinear analysis, the tension in the tri-cross should vary and can usually be expressed as the function of position and time, that is, T(x,t). For simplicity, the tension force can be further treated as the only function of time T(t) [2, 11]. In Figure 1, the axis is applied to connect to each string. The strings connected toy1-axis andy2-axis are jointed to the string ofx-axis, which are calledy1-axis andy2-axis for short, respectively. During the vibration, the vibrated modal ofy1-axis andy2-axis can be different orders and would differ in the characteristics of vibrating plane. However, they are both connected to thex-string and affected by each other by the constraint of nodes. Considering the real structure of net strings such as space antennas, the nonrigid boundary conditions and the finite span of this tri-cross string are adopted in this paper. For the sake of simplicity, the three single strings are chosen to be made of the same material as well.In the following section, we will try to simplify the structure of the tri-cross string and transfer it into a dynamic model. At first, the whole jointed tri-cross string structure can be divided into three single strings with two nodal forces applied onto each of them. Considering jogged two strings of the whole tri-cross string, the separated nodal force on each of the jogged two strings has the same amplitude but in the opposite direction. In this study, the jogged tri-cross string will be disassembled into the three strings separately, and one of them will be chosen as an object substituted into Hamilton’s principle. In this way, the governing equation of one single string belonging to the jogged tri-cross string structure can be obtained, and it can demonstrate the common properties of each single string.Under the above assumptions, the governing equations of the vibration of the tri-cross string are derived as follows.


## 2.2. Hamilton’s Principle

The Lagrange integral functionJ of a single string can be expressed as(1)J=∫t1t2Ltdt,whereJ is obtained by integrating Lagrange functionL with respect to the timet andL is expressed as(2)Lt=Kt-Ut,whereK is the kinetic energy andU is the potential energy of the system.As the derivation method and process of the mechanics models for the three constituent single strings of the tri-cross string are similar, it is convenient to present the derivation of the mathematical expressions ofK(t) andU(t) of one string.In the present problem, the vibrating direction is vertical tox-y plane. Therefore, the expression of the kinetic energyK(t) is shown as(3)Kt=12∫0l0ρ∂w∂t2dx,where l0 is the length of the string in the initial static equilibrium position, ρ is the mass per unit length and is the function ofx,w is the displacement of the string, andx is the longitudinal coordinate.The total potential energyU(t) of one string consists of one part due to tension workUT(t) and the other part coming from the external excitationUF(t), i.e.,U(t) =UT(t) +UF(t). As shown in Figure 2, a string in the equilibrium position presents an initial tension T0 and its initial lengthl0. We take an infinitesimal element of the displaced string as an object, and its length in the deformed configuration is ds. In the equilibrium position under initial tension, the length of this element is dx, under the definition of initial length being dx0. It is assumed that the string remains linearly elastic under the varying tension during the vibration, and the tension T(x,t) can be expressed as (4)Tx,t=EAε,in which the total strain of the element ε is(5)ε=ds-dx0dx0=dsdxdxdx0,dxdx0=1+ε0,where ε0 is the initial strain. According to the geometrical relations, the vibrating string away from the initial equilibrium position can be expressed as (6)dsdx=1+∂w∂x2.By inserting (5) and (6) into (4), (4) can be expressed as(7)Tx,t=EA1+ε01+∂ω∂x2-1,whereE is Young’s modulus andA is the cross-sectional area. For simplicity and emphasis of geometrically nonlinear behavior, the tension in the one-dimensional and flexible string can be regarded as the function of time only [2, 11]. With the solution of the nonlinear vibration problem of two-dimensional thin plates, averaging the stress in one axis is one effective method. In our previous study, the tension is presented under the same assumption [1] (8)Tt=T0+EAl0∫0l01+∂wx,t∂x2dx-l0or(9)Tt=EA+T0l0∫0l01+∂wx,t∂x2dx-l01+ε0.Using (9), the tension work stored in the potential energy of the one single string with a length of dx can be expressed as(10)dUTt=EA+T0l0∫0l01+∂w∂x2dx-l01+ε01+∂w∂x2dx-dx.Integrating (10) along the string, we can obtain(11)UTt=EA+T04l0∫0l0∂w∂x2dx·∫0l02ε01+ε0+∂w∂x2dx.On the other hand, one external excitation work stored in the potential energy is provided as(12)UFt=∫0l0Fjoint1t·δx-xjoint·wstx,yjoint1,yjoint2,tdx,where Fjoint1(t), Fjoint2(t) denote the nodal forces at the joints, whose coordinates are (x1,joint, yjoint1,yjoint2), (x2,joint, yjoint1,yjoint2), respectively; wst represents the displacement of the whole tri-cross string; δ(x) is Dirac delta function. Figure 3 shows the relation between the (inner) force and the excitation. In fact, the nodal force should be regarded as an inner constraint force by considering the whole tri-cross string. By dividing the tri-cross string into three strings, the nodal forces in the same position but belonging to two different jogged strings have the same amplitude and opposite direction.Figure 2
Displacement of thex-string component of the cross string in Figure 1.Figure 3
Modeling of the tri-cross string by separating the three strings of the structure and adding a joint force to the joint as the joint condition.For one single string with more than one joint with other constituent strings under multiple excitations, say, alongx-direction, (12) may be extended as(13)UFt=∫0l0∑i=1nFjointit·δx-xi,joint·wstx,yjoint1,yjoint2,tdx.According to Hamilton’s principle, (1) can be expressed by substituting (11), (12), (13) into (1) as(14)δJ=δ∫t1t2Kt-UTt-UFtdt=0.


## 2.3. Governing Equations

According to (14) and the differential variational principle, the governing equation of the x-string can be obtained as(15a)ρ∂2w∂t2-EA+T0l0∫0l0∂w∂x2dx+ε0l01+ε0·∂2w∂x2=Fjoint1t·δx-x1,joint+Fjoint2t·δx-x2,joint,y1=yjoint1,y2=yjoint2.Similarly, by applying the kinetic energyK(t) and the potential energyU(t) ony1-string andy2-string, we can obtain the governing equations ofy1-string andy2-string as(15b)ρ∂2w′∂t2-EA+T1l1∫0l1∂w′′∂y12dy+ε1l11+ε1·∂∂y1∂w′∂y1=-Fjoint1t·δy1-yjoint1,x=x1,joint,y2=yjoint2,(15c)ρ∂2w′′∂t2-EA+T2l2∫0l2∂w′′∂y22dz+ε2l21+ε2·∂∂y2∂w′′∂y2=-Fjoint2t·δy2-yjoint2,x=x2,joint,y1=yjoint1in which l1,l2 are the lengths ofy1-string andy2-string in the static equilibrium position, respectively;w, w′,w′′ are the displacements ofx-string,y1-string, andy2-string, respectively; and wst is the displacement of the whole tri-cross string structure,(16)w=wstx,yjoint1,yjoint2,t,w′=wstx1,joint,y1,yjoint2,t,w′′=wstx2,joint,yjoint1,y2,t.The boundary conditions corresponding tox-string,y1-string, andy2-string are shown as follows:(17a)w=0,atx=0,x=l0,(17b)w′=0,aty1=0,y1=l1,(17c)w′′=0,aty2=0,y2=l2,and the initial conditions corresponding tox-string,y1-string, andy2-string are (18)wstx,y1,y2t=0x=xi,y1=yj1,y2=yk2,=wstxi,yj1,yk2,0,where wst(xi,yj,zk,0) is the initial displacement.It is well known that the natural frequency with nonlinearity of a geometrically nonlinear string is composed of the linear natural frequency of the string and the nonlinear fluctuation frequency [16, 17]. Based on the above derivations, it is obvious that the vibration of the tri-cross string is coupled with each other due to the two joint nodes. In the following, we will discuss the nonlinearity and the coupling behavior among the x-string,y1-string, andy2-string with the two jogged mechanical connections.


## 3. Solution of the Governing Equations

In the integrodifferential nonlinear governing equations (15a), (15b), and (15c), the second terms all present the term ∂w/∂x. It is difficult to obtain the analytical solutions of the above integrodifferential nonlinear governing equations. In this study, modal superposition method is adopted here to solve these equations with semi-analytical formula first, and the solutions were used to analyze their responses with coupled behavior.The boundary and initial conditions of the tri-cross string are one kind of a net or special plate; thus similarly the tri-trigonometric series can be employed to expand the displacement of the tri-cross string as(19)wstx,y,z,t=∑m=1M∑n1=1N1∑n2=1N2amn1n2fmn1n2tsinmπxl0sinn1πy1l1sinn2πy2l2,in which amn1n2 is the mn1n2th initial amplitude of the vibration and fmn1n2(t) presents the dimensionless time-domain function. In the tri-cross string, the time dependence ofx-string,y1-string, andy2-string is exactly the same. Moreover, in order to make (9) satisfy the initial condition, assumefmn1n2(t=0)=1. Considering the geometrical structure in Figure 1, the magnitude of wst is zero when x≠x1,joint or y1≠yjoint1, y1≠yjoint1 or y2≠yjoint2, and x≠x2,joint or y2≠yjoint2. Lining to (8) and (19), all amn1n2s should be equal to zero based on the initial and boundary conditions. Therefore, it is obvious that wst is nonzero only at (x,yjoint1,yjoint2) or (x1,joint,y1,yjoint2,t) or (x2,joint,yjoint1,y2,t) according to the geometrical structure shown in Figure 1.Considering the corresponding external excitation terms in (15a), (15b), and (15c), we can describe the nodal forces in modal form as trigonometric series(20a)Fjoint1t·δx-x1,joint=∑m=1MFm1∗∗sinmπxl0,(20b)Fjoint2t·δx-x2,joint=∑m=1MFm2∗∗sinmπxl0,(20c)-Fjoint1t·δy1-yjoint1=∑n1=1N1Fn11∗sinn1πy1l0,(20d)-Fjoint2t·δy2-yjoint2=∑n2=1N2Fn22∗sinn2πy2l2,where Fm1∗∗,Fm2∗∗,Fn11∗, and Fn22∗ are the trigonometric modal parameters.In this paper, we aim to consider the effect of geometrically nonlinear displacement. The length of the string varies rather than being constant. Meanwhile, we suppose the weak nonlinearity of this problem such that the strains inside the tri-cross string are not quite large during vibration. Thus, the amplitudes of displacement (amn1n2) are small in magnitude; as a result, the nodal force is also quite small. By substituting (19) and (20a) and (20b) into (15a), we can rewrite the governing equation (15a), for simplicity,(21a)ρf¨mn1n2t+EA+T0l0∫0l0∑i=1M∑j=1N1∑k=1N1∑l=1N2∑h=1N2iπl02aijlaikhfijltfikhtcos2iπxl0·sinjπyjoint1l1sin⁡lπyjoint2l2·sin⁡kπyjoint1l1sin⁡hπyjoint2l2+ε0l01+ε0·mπl02fmn1n2t=Fm1∗∗+Fm2∗∗Syz1∗∗amn1n2or (21b)ρf¨mn1n2t+EA+T02∑i=1M∑j=1N1∑k=1N1∑l=1N2∑h=1N2iπl02aijlaikhfijltfikht·sin⁡jπyjoint1l1sinlπyjoint2l2·sin⁡kπyjoint1l1sin⁡hπyjoint2l2·fmn1n2tmπl02+EA+T0l0ε0l01+ε0·mπl02fmn1n2t=Fm1∗∗+Fm2∗∗Syz1∗∗amn1n2,where(22a)Syz1∗∗=∑n1=1N1∑n2=1N2sinn1πyjoint1l1sinn2πyjoint2l2,Sxz1∗=∑m=1M∑n1=1N2sinmπx1,jointl0sinn2πyjoint2l2,(22b)Syz2∗∗=∑n1=1N1∑n2=1N2sinn1πyjoint1l1sinn2πyjoint2l2,Sxy2∗=∑m=1M∑n1=1N1sinmπx2,jointl0sinn1πyjoint1l1.The orthogonality of trigonometric functions is applied in the derivation of (21a) and (21b). By inserting (19) and (20b) into (15b), we can get similar governing equation fory1-string as (23a)ρf¨mn1n2t+EA1+T12∑i=1M∑k=1M∑j=1N1∑l=1N2∑h=1N2jπl12aijlakjhfijltfkjht·siniπx1,jointl0sinlπyjoint2l2·kπx1,jointl0sinhπyjoint2l2·fmn1n2tn1πl12+EA1+T1l1ε1l11+ε1·n1πl12fmn1n2t=Fn11∗Sxz1∗amn1n2.Similarly, substituting (19) and (20c) into (15c), we have similar form of the governing equation fory2-string as(23b)ρf¨mn1n2t+EA2+T22∑i=1M∑k=1M∑j=1N1∑l=1N1∑h=1N2hπl22aijhaklhfijhtfklht·siniπx2,jointl0sinjπyjoint1l1·kπx2,jointl0sinlπyjoint1l1·fmn1n2tn2πl22+EA2+T2l2ε2l21+ε2·n2πl22fmn1n2t=Fn22∗Sxy2∗amn1n2.In order to connect (21a), (23a), and (23b), (20a), (20b), (20c), and (20d) should be rewritten as follows:(24a)∫0l0Fjoint1t·δx-x1,jointsinmπxl0dx=∫0l0∑m=1MFm1∗∗sinmπxl0sinmπxl0dx,(24b)∫0l0Fjoint2t·δx-x2,jointsinmπxl0dx=∫0l0∑m=1MFm2∗∗sinmπxl0sinmπxl0dx,(24c)∫0l1-Fjoint1t·δy-yjoint1sinn1πy1l1dy1=∫0l1∑n1=1N1Fn11∗sinn1πy1l1sinn1πy1l1dy1,(24d)∫0l2-Fjoint2t·δy2-yjoint2sinn2πy2l2dy2=∫0l2∑n2=1N2Fn22∗sinn2πy2l2sinn2πy2l2dy2.Based on (24a)–(24d), one has(25a)Fm1∗∗=-Bmn11Fn11∗,(25b)Fm2∗∗=-Bmn22Fn22∗,where(26)Bmn11=l1sinmπx1,joint/l0l0sinn1πyjoint1/l1=l1l0x1my1n1,Bmn22=l2sinmπx2,joint/l0l0sinn2πyjoint2/l2=l2l1x2my2n2,x1m=sinmπx1,jointl0,x2m=sinmπx2,jointl0,y1n1=sinn1πyjoint1l1,y2n2=sinn2πyjoint2l2.In the perturbation method, a properly chosen small perturbation parameter should be provided during the solution. For achieving this purpose, we defined(27)gmn1n2t=EA+T0EA1+T1EA2+T2mπl0n1πl1n2πl2l0T0l1T1l2T2fmn1n2t,where(28)Dmn1n2=EA+T0EA1+T1EA2+T2mπl0n1πl1n2πl2l0T0l1T1l2T2.From (27), fmn1n2(t) can be written as(29)fmn1n2t=mπl0-1n1πl1-1n2πl2-1T0EA+T0l0T1EA1+T1l1T2EA2+T2l2gmn1n2t.By substituting (29) into (22a) and (22b), the governing equation ofx-string can be transformed from (21a) and (21b) to (30)g¨mn1n2t+T02T12T22l0-4amn1n22m24ρπ2EA+T0EA1+T12EA2+T22∑i=1M∑j=1N1∑k=1N1∑l=1N2∑h=1N2aijlaikhjklhamn1n22y1jy1ky2ly2hgijltgikht·gmn1n2t+ω0m2gmn1n2t=Dmn1n2Fm1∗∗+Fm2∗∗ρSyz1∗∗amn1n2.Similarly, the governing equation of y1-string becomes (31)g¨mn1n2t+T02T12T22l1-4amn1n22n124ρπ2EA+T02EA1+T1EA2+T22∑i=1M∑k=1M∑j=1N1∑l=1N2∑h=1N2aijlakjhiklhamn1n22x1ix1ky2ly2hgijltgkjht·gmn1n2t+ω0n1′2gmn1n2t=Dmn1n2Fn11∗ρSxz1∗amn1n2and the governing equation of y2-string becomes(32)g¨mn1n2t+T02T12T22l2-4amn1n22n224ρπ2EA+T02EA1+T12EA2+T2∑i=1M∑k=1M∑j=1N1∑l=1N1∑h=1N2aijhaklhijklamn1n22x2ix2ky1ly1hgijhtgklht·gmn1n2t+ω0n2′′2gmn1n2t=Dmn1n2Fn22∗ρSxy2∗amn1n2.As the three single strings have the identical displacements at the joint, the joint condition can be defined as follows:(33a)wx1,joint,t=w′yjoint1,t=wstx1,joint,yjoint1,yjoint2,t,(33b)wx2,joint,t=w′′yjoint2,t=wstx2,joint,yjoint1,yjoint2,t.Define (34)ω0m2=T0ρmπl02,where ω0m is the linear natural frequency of x-string, proportional to the initial tension T0 and the square of the string length l0. Similarly, the linear natural frequency of y1-string and y2-string can be defined as(35)ω0n1′2=T1ρ·n1πl12,(36)ω0n2′′2=T2ρ·n2πl22.Three small perturbation parameters can be defined as (37)εmn1n2x=T02T12T22l0-4amn1n22m24ρπ2EA+T0EA1+T12EA2+T22,(38)εmn1n2y1=T02T12T22l1-4amn1n22n124ρπ2EA+T02EA1+T1EA2+T22,(39)εmn1n2y2=T02T12T22l2-4amn1n22n224ρπ2EA+T02EA1+T12EA2+T2,(40)εmn1n2x=cmn1x-y1εmn1n2y1,cmn1x-y1=l0-4l1-4m2n12EA+T0EA1+T1,(41)εmn1n2x=cmn2x-y2εmn1n2y2,cmn2x-y2=l0-4l2-4m2n22EA+T0EA2+T2.By substituting (24a), (24b), (24c), (24d), (27), (36), (37), (38), and (39) into (30), (31), and (32), we obtain(42)g¨mn1n2t+εmn1n2x1+bmn1y1+bmn2y2∑i=1M∑j=1N1∑k=1N1∑l=1N2∑h=1N2pjklhgijltgikhtgmn1n2t+bmn1y1cmn1x-y1∑i=1M∑k=1M∑j=1N1∑l=1N2∑h=1N2qiklhgijltgkjht·gmn1n2t+bmn2y2cmn2x-y2∑i=1M∑k=1M∑j=1N1∑l=1N1∑h=1N2Vijklgijhtgklht·gmn1n2t+ω0m2+bmn1y1ω0n1′2+bmn2y2ω0n2′′21+bmn1y1+bmn2y2gmn1n2t=0,in which(43)pjklh=aijlaikhjklhamn1n22y1jy1ky2ly2h,qiklh=aijlakjhiklhamn1n22x1ix1ky2ly2h,Vijkl=aijhaklhijklamn1n22x2ix2ky1ly1h,(44)bmn1y1=Sxz1∗Bmn11Syz1∗∗,bmn2y2=Sxy2∗Bmn22Syz2∗∗.By adopting the perturbation method, gmn1n2(t) can be decomposed by means of the perturbation expansion(45)gmn1n2t=g0mn1n2t+εmn1n2x1+bmn1y1+bmn2y2g1mn1n2t+⋯.The frequency of the tri-cross string is also expressed as the following expressions with a small perturbation parameter(46)ωmn1n22=ω0m2+bmn1y1ω0n1′2+bmn2y2ω0n2′′21+bmn1y1+bmn2y2+εmn1n2x1+bmn1y1+bmn2y2α1+εmn1n2x1+bmn1y1+bmn2y22α2+⋯.By inserting (45) and (46) into (42), we obtain the following relation accompanied with the perturbation parameter (εmn1n2x/1+bn1my+bn1n2z)0(47)g¨0mn1n2t+ωmn1n22g0mn1n2t=0.By satisfying fmn1n2(0)=1, g0mn1n2(t) can be taken as(48)g0mn1n2t=Dmn1n2cosωmn1n2t.The terms accompanied with the perturbation parameter (εmn1n2x/1+bmn1y1+bmn2y2)1 are(49)g¨1mn1n2t+ωmn1n22g1mn1n2t=α1g0mn1n2t-∑i=1M∑j=1N1∑k=1N1∑l=1N2∑h=1N2pjklhg0ijltg0ikhtg0mn1n2t+bmn1y1cmn1x-y1∑i=1M∑k=1M∑j=1N1∑l=1N2∑h=1N2qiklhg0ijltg0kjht·g0mn1n2t+bmn2y2cmn2x-y2∑i=1M∑k=1M∑j=1N1∑l=1N1∑h=1N2Vijklg0ijhtg0klht·g0mn1n2t.From (49), g1mn1n2(t) can be obtained as(50)g1mn1n2t=Dmn1n24∑i=1M∑j=1N1∑k=1N1∑l=1N2∑h=1N2pjklhDijlDikhcosωijlt+ωikht+ωmn1n2t-cosωmn1n2tωijl+ωikh+ωmn1n22-ωmn1n22+cosωijlt+ωikht-ωmn1n2t-cosωmn1n2tωijl+ωikh-ωmn1n22-ωmn1n22+cosωijlt-ωikht+ωmn1n2t-cosωmn1n2tωijl-ωikh+ωmn1n22-ωmn1n22+cosωijlt-ωikht-ωmn1n2t-cosωmn1n2tωijl-ωikh+ωmn1n22-ωmn1n22+Dmn1n24bmn1y1cmn1x-y1∑i=1M∑k=1M∑j=1N1∑l=1N2∑h=1N2qiklhDijlDkjh·cosωijlt+ωkjht+ωmn1n2t-cosωmn1n2tωijl+ωkjh+ωmn1n22-ωmn1n22+cosωijlt+ωkjht-ωmn1n2t-cosωmn1n2tωijl+ωkjh-ωmn1n22-ωmn1n22+cosωijlt-ωkjht+ωmn1n2t-cosωmn1n2tωijl+ωkjh-ωmn1n22-ωmn1n22+cosωijlt-ωkjht-ωmn1n2t-cosωmn1n2tωijl+ωkjh-ωmn1n22-ωmn1n22+Dmn1n24bmn2y2cmn2x-y2∑i=1M∑k=1M∑j=1N1∑l=1N1∑h=1N2VijklDijhDklhcosωijht+ωklht+ωmn1n2t-cosωmn1n2tωijh+ωklh+ωmn1n22-ωmn1n22+cosωijht+ωkjht-ωmn1n2t-cosωmn1n2tωijl+ωkjh+ωmn1n22-ωmn1n22+cosωijht-ωkjht+ωmn1n2t-cosωmn1n2tωijl+ωkjh+ωmn1n22-ωmn1n22+cosωijht-ωklht-ωmn1n2t-cosωmn1n2tωijh+ωklh+ωmn1n22-ωmn1n22(51)α1=14∑i=1M∑j=1N1∑k=1N1∑l=1N2∑h=1N2pjklhDijlDikh+14bmn1y1cmn1x-y1∑i=1M∑k=1M∑j=1N1∑l=1N2∑h=1N2qiklhDijlDkjh+14bmn2y2cmn2x-y2∑i=1M∑k=1M∑j=1N1∑l=1N1∑h=1N2VijklDijhDklh.gmn1n2(t) include the linear part g0mn1n2(t) and 1st-order perturbation parameter (εmn1n2x/1+bmn1y1+bmn2y2)1 term g1mn1n2(t)(52)gmn1n2t=g0mn1n2t+εmn1n2x1+bmn1y1+bmn2y2g1mn1n2tand lastly, fmn1n2(t) in (29) can be determined as(53)fmn1n2t=cosωmn1n2t+εmn1n2x1+bn1my+bn1n2z1Dmn1n2g1mn1n2t.Thus, the circular frequency of the tri-cross string can be obtained as(54)ωmn1n22=ω0m2+bmn1y1ω0n1′2+bmn2y2ω0n2′′21+bmn1y1+bmn2y2+εmn1n2x1+bmn1y1+bmn2y2α1=ω0m2+bmn1y1ω0n1′2+bmn2y2ω0n2′′21+bmn1y1+bmn2y2+εmn1n2x1+bmn1y1+bmn2y214∑i=1M∑j=1N1∑k=1N1∑l=1N2∑h=1N2pjklhDijlDikh+14bmn1y1cmn1x-y1∑i=1M∑k=1M∑j=1N1∑l=1N2∑h=1N2qiklhDijlDkjh+14bmn2y2cmn2x-y2∑i=1M∑k=1M∑j=1N1∑l=1N1∑h=1N2VijklDijhDklh.Therefore, the frequency of the tri-cross string is in the following form:(55)fmn1n2st=12πω0m2+bmn1y1ω0n1′2+bmn2y2ω0n2′′21+bmn1y1+bmn2y2+εmn1n2x1+bmn1y1+bmn2y2α1.According to the initial conditions, the amplitude amn1n2 can be obtained by the following relation:(56)∑m=1M∑n1=1N1∑n2=1N2amn1n2fmn1n2tsinmπxil0sinn1πyj1l1sin⁡n2πyk2l2=wstxi,yj1,yk2,0,where m=1,2,…,M;n1=1,2,…,N1;n2=1,2,…,N2.In this study, approximate semi-analytic formulas on the natural analysis of the structural displacements and frequencies are presented. To approach the exact solution as close as possible, large value ofM,N1,N2 should be chosen, where the number of the chosen sampling points on the three strings should be identical as well. To improve the accuracy of the solution derived from the above perturbation method, the order of magnitude of the parameter (εmn1n2x/1+bmn1y1+bmn2y2) also needs to be increased. Next we will demonstrate it with various numerical examples.


## 4. Results and Discussion

We choose the geometrical and mechanical properties of the tri-cross string as follows:l0=1 m, l1=0.95 m, l2=1.05 m, x1,joint=0.32l0,x2,joint=0.64l0, yjoint1=0.33l1, yjoint2=0.63l2,E= 210 × 109 Pa, ρ = 0.1543 kg/m, ε = 0.2×10−3. The three strings all have the same diameter, that is, 0.005 m. The magnitude of the maximum displacement is taken to be 10−3 m. The perturbation parameters (εmn1n2x/1+bmn1y1+bmn2y2)   (m,n1,n2=1,2,…,N) are chosen to be much smaller than unity with a maximum value of 10−7.In order to discuss the accuracy of the solution equation (55), Table 1 presents the first three-order natural frequencies with different M,N1, and N2. It is shown that (55) with 4 or 5 perturbation terms already provides sufficient accuracy for the current tri-cross string structure. This fact confirms that the perturbation procedure with the above parameters is sufficient for the problem considered here. In the following discussions, we will choose M=N1=N2=5.Table 1
The first three frequencies with different amplitudes (Hz).
w
s
t
(
x
=
0.5
,
y
j
o
i
n
t
1
=
0.33
,
y
j
o
i
n
t
2
=
0.63,0
)
/
m
3 terms (N = 3)
4 terms (N = 4)
5 terms (N = 5)
f
111
s
t
f
222
s
t
f
333
s
t
f
111
s
t
f
222
s
t
f
333
s
t
f
111
s
t
f
222
s
t
f
333
s
t
1.2 × 10−3
37.74
75.60
114.53
37.03
74.03
111.96
36.95
73.82
111.63
2.4 × 10−3
37.89
75.92
115.05
37.22
74.38
112.52
37.46
74.77
112.92
3.0 × 10−3
38.02
76.16
115.43
37.56
74.66
113.95
37.82
75.47
114.24
4.0 × 10−3
38.27
76.67
116.26
38.66
75.25
114.86
38.59
76.97
115.92Until now, the nonlinear free vibration analysis of a tri-cross string with geometrical linearity cannot be accurately simulated with a commercial finite element code to provide a numerical solution because the nonlinearity from varying arc length and the coupled characteristics caused by jogged joints are quite complicated. In this work, we focus on the discussion of the nonlinearity and couple characteristic of the prestressed tri-cross string.With similar process, we can obtain the nonlinear frequency of one single string. The frequencies of the three strings corresponding to the tri-cross string are given below in an explicit form:(57)fmΙ=12πω0m2+am2EA+T0l0-22ρT02mπl02·12am2y1mmπl02+∑i=1N12ai2y1iiπl02,fmΙΙ=12πω0m′2+am2EA1+T1l0′-22ρT12mπl12·12am2x1mmπl12+∑i=1N12ai2x1iiπl12,fmΙΙI=12πω0m′′2+am2EA2+T2l0′-22ρT22mπl22·12am2y2mmπl22+∑i=1N12ai2y2iiπl22.In the following discussions, the natural frequencies of the tri-cross string will be compared with those of the corresponding constituent single strings. Comparing (55) with (57), it can be seen that (55) includes not only linear frequency [(ω0m)2+bmn1y1(ω0n1′)2+bmn2y2(ω0n2′′)2]/(1+bmn1y1+bmn2y2), combined with the linear frequencies from the three single strings, but also nonlinear fluctuation (εmn1n2x/1+bmn1y1+bmn2y2α1), where the terms such as pjklhDijlDikh reflect the coupling characteristics from the jogged strings.In the following sections, we will discuss the nonlinear frequencies of the tri-cross string by varying structural parameters, such as the vibration amplitude, the diameter of one string, the diameters of two strings, the length of one string, and the lengths of two strings.Figure4 shows the first four natural frequencies of the whole tri-cross string(f111st,f222st,f333st,f444st) with different initial amplitudes wst(x=0.5l0,yjoint1,yjoint2,0). First, the natural frequencies with the same mode in the three constituent strings, that is, f111st,f222st,f333st,f444st, are chosen in the discussion. It is seen that the natural frequencies of the tri-cross string and the three single strings increase with increasing amplitudes. Moreover, the frequencies of the tri-cross string are among those of the three single strings, reflecting the averaging effect of the three strings. It is one kind of the coupling effect of the tri-cross string. The total frequencies are not only the average of the nonlinear frequencies of the three constituent single strings in any way. To some extent, the total frequency fmn1n2st can be regarded as one kind of nonlinear combination of frequencies of the three constituent single strings. Therefore, the variation tendency of the frequency of the tri-cross string presents a nonlinear response by varying the amplitudes of the tri-cross string shown in Figure 4. If geometries of the constituent single strings are changed, the differences among the tri-cross string and the three single strings become rather complicated according to the results from our calculations.Figure 4
The influence of the vibration amplitudewst(x=0.5l0,yjoint1,yjoint2,0) on the first four frequencies of the strings: (a) the first mode f111st, (b) the second mode f222st, (c) the third mode f333st, and (d) the fourth mode f444st.
(a)
(b)
(c)
(d)Figure5 shows the influence of increasing the diameter of one constituent string on the first four natural frequencies of the strings, with the initial displacement of 1×10-3 m at the point with the maximum magnitude. By setting l0=1 m, l1=0.95 m, l2=1.05 m, the initial natural frequencies of the single strings are different from each other, which has been shown in Figure 5. In the case of increasing the diameter of one single string, its own frequencies increase and the frequencies of the whole tri-cross string also increase. It is shown that the change of the frequencies of the whole tri-cross string accompanies the varying of diameter of y1-string. In contrast, the x-string and y2-string have not been changed while changing the diameter of y1-string. The results demonstrate that changing the diameter of one single string does not change the frequencies of the whole structure and the frequencies of the other two constituent strings too much.Figure 5
The influence of the diameter of one constituent string on the first four frequencies of the strings: (a) the first modef111st, (b) the second mode f222st, (c) the third mode f333st, and (d) the fourth mode f444st.
(a)
(b)
(c)
(d)In the following, we study the effect of changing the diameters of two strings. Figure6 depicts the profiles of the first four natural frequencies of the strings, by increasing the diameters of two constituent strings (with d2=d3(A1=1/4πd22,A2=1/4πd32)), with the initial displacement of 1×10-3 m at the point in maximum magnitude. By setting l0=1 m, l1=0.95 m, l2=1.05 m, the initial natural frequencies of the single strings are different from each other, which has been shown in Figure 6. When the diameters of these two strings increase, their own frequencies increase and the frequencies of the whole tri-cross string also increase. Similar to the case of changing the diameter of only one string, the tri-cross string changes its frequencies less than that of the y1-string. The frequency of the y2-string does not change too much as well. This confirms the important fact again that the frequencies of tri-cross string cannot be treated as a mere combination of the nonlinear frequencies of the three constituent strings in any way.Figure 6
The influence of diameters of two constituent strings on the first four frequencies of the strings: (a) the first modef111st, (b) the second mode f222st, (c) the third mode f333st, and (d) the fourth mode f444st.
(a)
(b)
(c)
(d)In addition to the diameters, the lengths of the strings also have profound influence on the vibration of the tri-cross string; see Figure7 for the effect of the length ratio of the two strings l1/l0 (here l0 and l2 keep constant) on the first four natural frequencies of the cross string (f111st,f222st,f333st,f444st), with the initial displacement being 1×10-3 m at the point with the maximum magnitude. With the consideration of increasing the string length ratio, it is indicated that the natural frequencies of the tri-cross string at different modes will decrease. The length ratio factor by varying the length of one string provides pronounced effect on the tri-cross string, however, not as that on the varying single string. That is to say, the frequency of the tri-cross string is determined by the properties of its constituent strings, but the response sensitivity of the tri-cross string on local structural variation decreases due to its coupling characteristics and nonlinear response.Figure 7
The influence of the length of one constituent string on the first four frequencies of the strings: (a) the first modef111st, (b) the second mode f222st, (c) the third mode f333st, and (d) the fourth mode f444st.
(a)
(b)
(c)
(d)Furthermore, we increase the lengths ofx-string and y2-string proportionally. The first four natural frequencies of the tri-cross string (f111st,f222st,f333st,f444st) resulting from the influence of the length ratio of the two strings l1/l0,l2/l0 (here l0 and l2/l1 keep constant) are shown in Figure 8, where the initial displacement is 1×10-3 m at the point with the maximum magnitude. With the consideration of increasing the string length ratio, it is indicated that the natural frequencies of the tri-cross string at different modes will decrease. Compared with the results shown in Figure 7, the frequencies of tri-cross string are much closer to the frequencies of the single strings which increase its own lengths. The reason could be that the variation of the two constituent strings is more effective on the whole structure than that of single constituent string.Figure 8
The influence of lengths of two constituent strings on the first four frequencies of the strings: (a) the first modef111st, (b) the second mode f222st, (c) the third mode f333st, and (d) the fourth mode f444st.
(a)
(b)
(c)
(d)Figure9 shows the first four natural frequencies of the tri-cross string (f212st,f222st,f232st,f242st) with different initial amplitudes wst(x=0.5l0,yjoint1,yjoint2,0). It is also shown that the natural frequencies of the tri-cross string and the three single strings increase with increasing amplitudes. More importantly, the frequencies (f212st,f222st,f232st,f242st) show that the order of the modes is not a simple integer. It agrees with (55), where the subscript and symbol mn1n2 in the expression provide a multiple order of the natural frequencies of the tri-cross string. In another way, the subscript and symbol mn1n2 in the expression mean the different orders of modes of the constituent strings. Therefore, the results in Figure 9 and (55) reflect the coupling characteristics that are the contribution from the different orders of modes of the constituent strings.Figure 9
The influence of the vibration amplitudewst(x=0.5l0,yjoint1,yjoint2,0) on the first four frequencies of the strings: (a) the first mode f212st, (b) the second mode f222st, (c) the third mode f232st, and (d) the fourth mode f242st.
(a)
(b)
(c)
(d)


## 5. Conclusion

In this paper, we derived the governing equations of the tri-cross string from Hamilton’s principle. The analytical solution of the coupled structure is obtained by using the perturbation method. Furthermore, the accuracy and convergence analysis of the analytical solution has been carefully conducted. As for the properties of natural frequencies of the vibrating tri-cross string, the results show that the overall feature of the tri-cross string is combined with the nonlinear characteristics and coupling features; that is, the natural frequencies of the tri-cross string are composed of the linear natural frequencies of the three constituent strings and the nonlinear parts. For the nonlinear part, we use the analytical solution to study the effect of the structural parameters on the nonlinear frequencies of the tri-cross string, such as vibration amplitudes, diameters and lengths, and the different orders of natural modes of the constituent strings. The results demonstrated that the contribution of each constituent string to the tri-cross natural frequency is in different proportions. In addition, the frequencies of the tri-cross string are dependent on the material properties of its constituent strings as well. More importantly, the configuration of a string net or space antennas (here the simplest net, i.e., a tri-cross string) tends to decrease the sensitivity to local structural variation. The solution obtained in this work may provide a theoretical reference for similar free vibration problem of net structures such as space antennas with flexible strings.



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*Source: 3935686-2017-11-20.xml*