Wave control of a flexible space tether based on elastic metamaterials

This study proposes a spider‐web elastic metamaterial to suppress vibrations in space slender structures, such as flexible space tethers. The metamaterial consists of unit cells that are periodically distributed on the space tether to obtain band gaps. The finite element model of the unit cell is established by employing the absolute nodal coordinate formulation (ANCF) due to the large deformation of the structure. The eigenfrequencies and corresponding vibration modes of the unit cell are obtained by ANCF. Moreover, the band gap of the unit cell is calculated based on the phonon crystal theory. The relationship between the vibration modes and the band gaps is analyzed. Finally, an experiment is conducted to verify the vibration transmission characteristics of finite period cells. The results show the effectiveness of the spider‐web elastic metamaterial for vibration suppression of a flexible tether. This study provides insights into the use of elastic metamaterials for vibration isolation in space tether systems.


| INTRODUCTION
The space system has a wide application potential, but its large swept area increases the risk of collision between the system and space debris, Active control usually consumes propellant or energy, such as propellant for thruster control and electrical energy for tension control. In addition, passive dampers are still required for tether motions that are beyond the range of active control. Compared with active vibration control, passive vibration control does not consume additional energy. It is simpler and more reliable by absorbing or dissipating energy through additional components.
However, the lack of passive vibration absorbers for space tether systems highlights the significance of the research in this area.
The vibration absorber consisting of periodic structures proposed by Al Ba'ba'a et al. 6 provides a promising approach for passive vibration control of space tether systems.
Elastic metamaterials, also known as phononic crystals, are constructed by periodic artificial structures, which are usually described by crystal cells and lattices in crystallography. When the elastic wave propagates in a phononic crystal, a special dispersion relation is formed due to the periodicity of the structure. The frequency range without the dispersion relation curve is called the band gap, in which the elastic wave propagation is suppressed.
Nevertheless, the elastic waves in other frequency ranges will propagate without loss under the effect of dispersion relation.
Using these characteristics of the phononic crystal, specific structures were designed to block or regulate the propagation of elastic waves. Zhu et al. 7 proposed a chiral-lattice-based elastic metamaterial beam, which has the capacities of broadband vibration suppression and load bearing by using multiple embedded local resonators. Qian et al. 8 proposed the idea that the transverse waves can be controlled by the longitudinal vibrations.
With the help of a nonuniform distribution of shunting circuits, Liu et al. 9 improved the vibration suppression effect of a finite hybrid piezoelectric phononic crystal beam. Yoon et al. 10,11 presented a pendulum-type elastic metamaterial for absorbing the vibration of a tether system, but it has a high-frequency range of the band gap.
In conclusion, elastic metamaterials are widely used for vibration suppression. Nevertheless, elastic metamaterials are rarely used in vibration suppression of space tethers.
In summary, there is a demand to study the passive vibration damping of tethered satellites. The elastic metamaterial has a reliable performance in low-frequency vibration suppression and leads to fast decay of the elastic wave in the band gap. Therefore, one can take advantage of this property to design an elastic metamaterial for passive vibration control of slender structures (such as tethers and inflatable tubes) in tethered satellite systems. In this paper, an elastic metamaterial structure is designed and the energy band for the wave control problem of slender structures in tethered satellite systems is calculated.
The rest of the paper is organized as follows. In Section 2, a finite element modeling and simulation method is proposed for an elastic metamaterial with large deformation. A spider-web metamaterial is presented to suppress the vibration of flexible space tethers in Section 3. The band gap of the metamaterial is calculated and the mechanism of the band gap is analyzed and discussed in Section 4. An experimental setup to verify the vibration transmission characteristics of finite period cells is described in Section 5.
Finally, conclusions are given in Section 6.

| THEORETICAL FOUNDATION
The theoretical foundation is introduced in this section to develop the model of the tether and harmonic oscillator. Elastic metamaterials are usually periodic structures. The elastodynamics characteristics of elastic metamaterials can be described by the graph of a dispersion curve, which is similar to the band structure in solid-state physics. Therefore, the elastic dynamics and solid-state physics are the theoretical basis for the study of the elastic metamaterials. In this section, the band theory used to describe the band gap and the research methods of the band gap are introduced. There are many methods to calculate the band gap, such as the transfer matrix method, plane wave expansion method, finite difference time domain method, multiple scattering method, finite element method, and so forth. The structure of the spiderweb elastic metamaterial is complex, and the propagation of elastic waves in the structure cannot be expressed analytically. Therefore, the finite element method is adopted in this paper. Due to the elongated characteristics of the tether, common finite element software cannot add Bloch boundary conditions to the beam element. The finite element method based on the absolute nodal F I G U R E 1 Special gradient-deficient beam element.
F I G U R E 2 Midsurface and nodes of a thin shell element.
HUANG ET AL.
| 59 coordinate formulation (ANCF) is therefore used to model the tether and harmonic oscillator.

| Band theory
Floquet 12 gave the general solution of the differential equation with periodic coefficients, which has a special form. When Bloch 13 explained the conductivity of metal based on quantum mechanics, he gave a similar rule, which is called the Bloch theorem. According to this theorem, the displacement field in the periodic structure can be expressed as where k is the wave vector, ω is the angular frequency, i is the imaginary unit, and t is the time. r x ( ) k has the same period as the periodic medium, which is defined as follows: where R is the crystal translation vector.
From Equation (1), it is straightforward to obtain the wave function by applying the separation of the variable method, that is, On the basis of the property of the translation operator of translation, one has According to Equation (4), the wave function of the whole infinite periodic structure can be expressed as the wave function of a unit cell F I G U R E 3 Geometry of the unit cell of the spider-web metamaterial.
T A B L E 1 Design parameters of the spider-web metamaterial.   F I G U R E 6 Real and imaginary parts of finite element model. F I G U R E 7 Energy band structure with the spider-web resonator.

| Spatial gradient-deficient beam element
The spatial gradient-deficient beam element of ANCF 14 is used to describe the deformation of the flexible tethers, as shown in Figure 1.
Since tethers are slender, the torsional deformation of the element is neglected, and it has only half of the degrees of freedom as the fully parametrized beam element.
The global position of a point on the centerline of the beam where S is the shape function matrix, namely, where ξ is the dimensionless local coordinates and q is the vector of nodal coordinates, that is, where r i and r j are the global position vectors of points A and B, r i,x and r j,x are the gradients of the position vectors.
The mass matrix of the beam element is where ρ is the density, L is the length of the element, and A is the area of the cross section.
The strain energy of the beam element can be divided into two parts: where E is Young's modulus, I is the bending stiffness, ε is the strain, and κ is the curvature.
By calculating the first derivative of strain energy, the elastic force can be obtained as follows: is the curvature at the current configuration. r x and r xx can be written as The Jacobian matrix of the elastic force is

| Thin shell element
The thin shell element of ANCF 15,16 is used to describe the deformation of the spider-web resonator, as shown in Figure 2.
F I G U R E 8 Energy band structure without the spider-web resonator.
F I G U R E 9 Vibration modes at key points: From Figure 2, the vector of nodal coordinates of a shell element can be written as T  ,  T  ,  T  T  ,  T  ,  T  T  ,  T  ,  T  T  ,  T  , T T The total strain energy of the thin shell element is given by where ε mid denotes the Green-Lagrange strain vector of the midsurface of the shell element, ε k denotes the bending strain vector of the shell element, E is the fourth-order elastic tensor of the plane stress problem, and V 0 is the initial volume of the element. By calculating the first and the second derivatives of the strain energy, the elastic forces and its Jacobi matrix can be obtained, and the detailed derivation is shown in Bloch. 13

| DEVELOPMENT OF THE ELEMENT MODEL
The design of the elastic metamaterial structure for tether vibration isolation is different from that of the general elastic metamaterial. Since the scale of tethers in one direction is much larger than the other two scales, the vibration isolation structure can only be designed outside the tether. In other words, it is only possible to design the resonators that are periodically attached to the tether outside. The band gap of the locally resonan elastic metamaterial is usually generated by the vibration coupling of the resonator and the substrate. Therefore, the resonator used for the tether vibration isolation must be attached to the tether, and its vibration should be coupled with the vibration of the tether.
Considering the low inherent frequency of the tether itself, the structure of the resonator should not only have a very soft part as a spring, but also have a larger mass or a denser part as a mass block. On the basis of all these factors and inspired by the spider web, a spider-web-like resonator, which is periodically attached to the tether, is designed in this section.

| Design of the geometry and parameters of the spider-web metamaterial
Inspired by the spider web, this paper presents a similar vibrationabsorbing structure, as shown in Figure 3.

| Calculation of the eigenfrequencies
For the unit cell of the spider-web metamaterial, the eigenfrequencies are extracted. Table 2 shows the eigenfrequencies of one unit cell with a hinged boundary.
When the boundary condition is the hinged boundary, the mode shapes are obtained with the ANCF method, as shown in Figure 5.

| Method of calculation of the band gap
In this study, the periodic boundary condition 17,18 is adopted on the top and bottom of the unit cell to simulate infinite periodic structures. The displacement field can be divided into the real part and imaginary part from Equation (4) and Euler formula. r(x) represents the coordinates of the point at the bottom of the unit cell, denoted as q b . r(x + R) represents the coordinates of the point on the top of the unit cell, denoted as q t . Therefore, Equation (4) can be rewritten as By equating the real and imaginary parts on the left and right sides of the equation, the periodic boundary conditions can be written as two constraint equations as Then, two identical finite element models are constructed by using the same grids, so as to simulate the real and imaginary parts of The above approach is implemented in the finite element method by creating two identical finite element models with the same mesh, the same material parameters, and the same initial configuration. The only difference between two identical finite element models is the number of the degrees of freedom. These two identical models are used to simulate the real and imaginary solution domains, as shown in Figure 6.
It has been pointed out in Section 2 that the nodal coordinates of the spatial gradient-deficient ANCF beam element include the global position vector r and the global slope vector r x . Then, the constraint equation can be further refined into the relationship between the individual nodal coordinates as follows:

| Numerical result
According to the method introduced in Section 4.1, the energy band structure of the elastic metamaterial is calculated, as shown in Figure 7. For comparison, the energy band structure of the tether without the spider-web resonator is also calculated, as shown in Figure 8. The energy band structure in Figure 7 illustrates that three relatively wide band gaps are found. The first band gap can be observed between 11.72 and 19.78 Hz, the second band gap can be observed between 33.6 and 83.05 Hz, and the third band gap can be observed between 110.37 and 186.77 Hz. The band structure of the case without the spider-web harmonic oscillator was also calculated, as shown in Figure 8. And there will be no band gap without the spider-web resonator.

| Mechanism analysis of the band gap
The appearance of the band gap is directly related to the vibration mode of frequencies at the edge of the band gap. To study the mechanism of the band gap, four key points are selected, as shown in  considered, the band gap should be wider. Figure 9D shows that internal resonance occurs in the spider-web resonator at the cut-off frequency of the second band gap.
Moreover, compared with Figure 5, the vibration mode on the edge of the band gap in Figure 9 is

| Influence of thickness of the ring
The energy band structure of the metamaterial is shown in Figure 11 when the other geometric parameters are kept constant and the thickness of the ring is taken as 0.001, 0.0009, 0.0008, 0.0007, 0.0006, and 0.0005 m.
The first, second, and fourth dispersion curves in Figure 11A are

| Influence of the spoke width
The energy band structure of the metamaterial is shown in Figure 12 when the other geometric parameters are kept constant and the widths of the spokes are taken as 0.005, 0.006, 0.007, 0.008, 0.009, and 0.01 m.
In Figure 12,

| Experimental design and instrumentation
The spider-web resonator is made by three-dimensional (3D) printing, and the resonator is strung on a Kevlar tether. The material of the 3D printing is ABS, whose Young's modulus is 2600 MPa and density is 1160 kg/m 3 .
The resonators prepared by the 3D printing are shown in Figure 13. The diameter of the ring is 100 mm, the thickness is 1 mm, and the width is 15 mm. There are three spokes in the middle of the resonator. The angle between two adjacent spokes is 120°. The length of the spokes is 50 mm.
The thickness is 1 mm. And the width is 5 mm. There is a hole at the intersection of the three spokes; the inner diameter of the hole is 1 mm, the outer is 3 mm, the thickness of the hole is 1 mm, and the height is 5 mm. The Kevlar rope with a diameter of 1 mm is passed through the hole and fixed on the resonator with 502 glues.
The instruments used in this experiment are listed in Table 3.
The experiment design is shown in Figure 14.

| Result analysis
The parameters such as the sensitivity of the sensor and the input mode are set on the software Enterprise Data Management (EDM) accompanying the data acquisition system, and a sweep task plan is added with an F I G U R E 15 Experimental setup: (A) metamaterial structures and exciter, (B) workstation, and (C) data acquisition system and power amplifier. excitation displacement amplitude of 1 mm, a frequency range from 2 to 100 Hz, and a sweep speed of 0.5 oct/min. The results of the data acquired by the sweep task are shown below.
The variation of the acceleration with frequency at the excitation point is shown in Figure 16A, where the acceleration is quadratic with respect to frequency, which is also consistent with the theory. The logarithmic spectrum of the acceleration at the measurement point is shown in Figure 16B, and the curves reach the peak at 5.45, 11.47, and 17.14 Hz. The frequency of these third-order intrinsic vibrations is so low, and the energy is so high that the elastic metamaterial structure cannot isolate vibration. It can also be seen from the vibration transmission characteristic curves in Figure 16C that the acceleration generated by the excitation is not decayed at the measurement point.
From the vibration transmission characteristic curve, it can be found that the vibration isolation effect of the metamaterial becomes gradually pronounced from 18 Hz, and the vibration isolation effect is obvious in the range of 30-40 Hz. In the sweeping process, it is found that the vibration isolation effect is particularly noticeable when the excitation frequency is 32 Hz. Therefore, a 32-Hz standing frequency task plan on the EDM software is added to observe the vibration transmission in the metamaterial structure, as shown in Figure 17.
From Figure 17A, two or three cells at the bottom vibrate more vigorously, while the cells at the top hardly vibrate, and most of the vibrational energy gathers in the cells at the bottom and dissipates.
As shown in Figure 17B, the vibration of the spider-web resonator and the vibration of the tether are coupled, and the vibrations of the adjacent cells are in an opposite phase, and therefore, a good vibration isolation effect is achieved.
The above experimental results show that the mechanism of band gap generation is quite consistent with the simulation results, but there are some deviations in the frequency range. The main reason is that there must be some errors in the processing of the spider-web resonator.
Although it is machine-processed, the 3D printer is heated to melt the consumables, and the ABS is easy to produce warpage after cooling, resulting in deviations between the specimen and the simulated model.

| CONCLUSION
In this study, a spider-web metamaterial is presented to attenuate the the band gap generation mechanism is in good agreement with that given by the simulation results, but the frequency range has some deviations.