2.1 Concept of hinge equivalent rotational stiffness

Hinges are important components of a coilable mast that are used to connect longerons, battens and diagonal cables. A typical hinge of a coilable mast is shown in Fig. 2, which consists of two parts that can rotate relative to each other, Part A and Part B. Part A is rigidly connected with the battens and diagonal cables, and Part B is rigidly connected with the longeron. The central screw and central hole form the rotating pair between Part A and Part B.

Figure 2.

Typical hinge of a coilable mast.

Unlike the locked hinges of articulated masts, the main feature of coilable masts is that their hinges can rotate slightly after deployment. Therefore, modelling the hinge rotational effect is a key factor affecting the overall modelling accuracy of coilable masts. The simulation results show that the frequency response obtained by ignoring the hinge rotational effect in modelling is inconsistent with the experimental results in terms of natural frequencies and modal shapes. In this study, the concept of the hinge equivalent rotational stiffness K _hinge is proposed to describe the rotational stiffness of the hinge. The higher its value is, the more difficult it is for the hinge to rotate, and conversely, the easier it is to rotate. K _hinge is positive infinity for locked hinges and zero for the case of free rotation.

Figure 3.

Force analysis of hinge rotation.

The forces on the hinge during rotation are qualitatively analysed to facilitate understanding of the significance of the hinge equivalent rotational stiffness. In the case of small angle rotation, the diagonal cables play a major role in the hinge rotation process. As shown in Fig. 3, when the hinge and diagonal cables are in equilibrium, as shown by the dotted line, the tensions on the diagonal cables at positions 1–4 are equal. When the hinge rotates, the diagonal cables at positions 1 and 3 are pulled, and their tensions increase, whereas the tensions of the diagonal cables at positions 2 and 4 decrease or even slacken. Assuming that the tension at positions 1 and 3 at this time is ${F_{\rm{l}}}$ , the radial and tangential components of ${F_{\rm{l}}}$ are ${F_{{\rm{lr}}}}$ and ${F_{{\rm{lt}}}}$ , respectively. ${F_{{\rm{lt}}}}$ will produce a restoring moment to return the hinge to its equilibrium position, and this restoring moment will increase as the hinge rotational angle increases. In the case of a small angle of rotation, the elongations of the diagonal cables $\Delta l$ increase linearly with the torsional angle $\theta $ , as shown in Fig. 3, so the tensions of the diagonal cables and the restoring moment also increase linearly with the torsional angle; thus, at this time, K _hinge is a constant. In the case of a large angle of rotation, the longeron and battens connected to the hinge undergo nonnegligible deformation, making the force situation during the rotation of the hinge very complicated. At this time, the error of the linear assumption is large, and considering only K _hinge as a constant is not applicable to the modelling of this situation.

However, in practical engineering applications, the large angle of rotation of the hinge generally occurs only when the tip mass of the coilable mast is too large and the mast is highly flexible. In this case, the coilable mast may vibrate substantially or even buckle locally or completely under the effect of a large inertial force. Since this situation poses risks to the normal application of the coilable mast and the spacecraft platform with the mast, it is usually avoided at the design stage. Therefore, in most practical engineering applications, the hinge of the coilable mast does not experience a large angle of rotation, and the linear assumption made in this study, i.e., the case where K _hinge is constant, is applicable to most scenarios.

2.2 Model of the rotating hinge

On the basis of the concept of hinge equivalent rotational stiffness, a ‘two mass points – undetermined rotational stiffness’ (TMURS) hinge model is proposed, which is shown in Fig. 4. Unlike the traditional ‘one mass point’ hinge model, two mass points are used here to simulate the mass characteristics of the two parts of the hinge, Part A and Part B. A rotational connector is established between the two mass points, and the rotational connector is given a rotational stiffness K _hinge to describe the equivalent rotational stiffness of the hinge. The longerons, battens and diagonal cables are rigidly connected to the two mass points of the hinge.

Figure 4.

TMURS model of the rotating hinge.

K _hinge is an undetermined parameter that must be identified on the basis of experimental tests. The identification method for K _hinge is detailed in Section 3.0.

Remark: The nonlinearities of the hinge come mainly from the clearance, contact and collision of the hinge. They are ignored in the proposed hinge model in this paper. These nonlinearities are ignored because the clearance of the hinge is almost eliminated under the preloading of the diagonal cables, making the nonlinear characteristics of the hinge insignificant. Considering the nonlinearities introduced by hinge clearance, contact and collision on the basis of the proposed hinge model will further improve the accuracy of the overall model of the coilable mast. According to the literature, when considering the hinge clearance, contact and collision factors, the hinge is in different states under different excitations, resulting in different hinge stiffnesses. The change in the hinge stiffness leads to a change in the overall stiffness of the extension arm. This causes the natural frequency and dynamic response of the coilable mast to vary with the external excitations [Reference Zhang31, Reference Zhou and Zhou32]. Therefore, accurately predicting the model dynamic response while considering hinge nonlinearities is difficult.

2.3 Models of other components

In addition to rotating hinges, a coilable mast also contains other components: longerons, battens and diagonal cables. Longerons and battens can be modelled by beam units in the FEM. However, because they are more flexible than traditional truss structures are, longerons and battens are prone to large geometric deformations under external loads, resulting in geometrical nonlinearity. This problem can be solved by turning on the geometric large deformation switch in commonly used finite element software. Diagonal cables typically have a large tensile stiffness. However, unlike traditional truss structures, diagonal cables cannot withstand compressive loads. This characteristic of diagonal cables can be considered a material nonlinear problem. The diagonal cable material has a nonlinear constitutive relationship: the compressive stress is always 0, and the tensile stress varies linearly with the tensile strain.

Moreover, the modelling of diagonal cables also needs to consider preloads. The initial stress method is a method commonly used to apply preloads, which is realised by adding an initial stress field to diagonal cables. When the initial values of the preloads ${L_{{\rm{ini}}}}$ (the tensions of the diagonal cables during assembly) are known, their initial stress satisfies ${\sigma _{{\rm{ini}}}} = {L_{{\rm{ini}}}}/{S_{diag}}$ , where ${S_{diag}}$ is the cross-sectional area of the diagonal cable. Therefore, when a coilable mast model is established, an initial stress field should be given first, and then a static balance analysis should be carried out to obtain the rebalanced model. As shown in Fig. 5, Point 1 represents the stress of the diagonal cables under the initial preload, which is not equal to the actual stress of the diagonal cables after assembly of the coilable mast. Point 2 represents the stress of the diagonal cables after rebalancing analysis. The model after rebalancing analysis is the real model that considers the effect of diagonal cable preloads.

Figure 5.

Rebalancing process of coilable mast preloads.

3.1 Identification method

The hinge equivalent rotational stiffness K _hinge is related to the size of the hinge, the preloads of the diagonal cables and the configuration of the coilable mast, which differs for different coilable masts and is difficult to determine directly. Natural frequencies are inherent properties of structures, so K _hinge can be identified by targeting the experimental natural frequencies of the coilable mast.

Hammer test method (HTM) is a fast and convenient method for experimental modal analysis of structures and is widely used in engineering applications. Nevertheless, measuring the natural frequencies and modal shapes of a full-length coilable mast via the HTM is ineffective because they are usually too long to measure. Since coilable masts are typical array structures with ‘segments’ as the unit, single or multiple segments can reflect all their structural characteristics. Therefore, a single segment or multiple segments of the coilable mast can be selected as a scaled object instead of the full-length mast for hinge-equivalent rotational stiffness identification. In this study, two segments are selected to construct the scaled object instead of a single segment because the 2-segment scaled object can reflect the interactions between segments, as shown in Fig. 6. Except for the number of segments, the scaled object is completely consistent with the full-length object, including the configuration, diagonal cable preloads and boundary conditions.

Figure 6.

2-segment scaled object of a coilable mast.

In this way, the hinge equivalent rotational stiffness K _hinge is identified through the 2-segment scaled object first and then substituted into the full-length model of the coilable mast to verify its correctness. Specifically, the identification process can be divided into three phases, as shown in Fig. 7.

Figure 7.

Flow chart for the identification of the hinge equivalent rotational stiffness.

Phase 1 establishes the relationship between the rotational stiffness K in the dynamic model and the natural frequencies of the 2-segment coilable mast. Since it is difficult to establish an explicit relationship between K and the natural frequencies, the natural frequencies of the model with different values of K are analysed through finite element software, and the curve of the relationship between K and the natural frequencies is fit.

Phase 2 determines the hinge equivalent rotational stiffness K _hinge by comparing the analysis and test results of the natural frequencies of the scaled object. In this study, only the first– and second-order natural frequencies are selected as the target response to identify K _hinge. The first two modes are first-order torsion and bending modes, and their effective mass fractions are much greater than those of other higher-order modes. Therefore, coilable masts typically undergo first-order torsion and bending vibrations in the real world. Therefore, the model developed by using the first two orders of natural frequencies can reflect the main dynamic behaviour of the coilable mast. If the natural frequencies of the first two modes agree well with the test results, the accuracy of the model is acceptable. The following error function is defined to measure the error between the analysis results and the test results:

(1) \begin{equation}e = \alpha \left| {{f_1} - {f_{{\rm{t}}1}}} \right| + \beta \left| {{f_2} - {f_{{\rm{t}}2}}} \right|\end{equation}

where ${f_1}$ and ${f_{{\rm{t1}}}}$ are the analysis and test results of the first-order frequency, respectively, and where ${f_2}$ and ${f_{{\rm{t}}2}}$ are the analysis and test results of the second-order frequency, respectively. $\alpha $ and $\beta $ represent the proportions of the first- and second-order frequency errors in the error function, $\alpha + \beta = 1$ . When the error function is the minimum, the corresponding rotational stiffness K is the hinge equivalent rotational stiffness K _hinge.

Phase 3 involves verifying the correctness of K _hinge. The K _hinge obtained in Phase 2 to Phase 3 are used, and the full-length coilable mast model is established. Then, the first- and second-order natural frequencies are analysed and compared with the full-length test results to verify the correctness of K _hinge.

In summary, the process of identifying the hinge equivalent rotational stiffness K _hinge can be illustrated as the pseudocode in the following Algorithm 1.

Algorithm 1

Pseudocode for the process of identifying the hinge equivalent rotational stiffness K _hinge

3.2 Frequency response analysis based on the virtual hammer test

In the process of identifying K _hinge in Section 3.1, frequency response analysis is essential for obtaining the first two orders of the natural frequencies of the scaled model or the full-length model of the coilable mast. A virtual hammer test method (VHTM) is proposed in this subsection for frequency response analysis in the identification process. The VHTM is a method used to measure the frequency response of a structural model by simulating the HTM on a computer. In the VHTM, the hammer signal is simulated according to the hammer test for a real situation, and the response signal is calculated by the finite element software. The reason for using the VHTM instead of the modal analysis method for frequency response analysis is twofold. On the one hand, the use of the VHTM is intended to be consistent with the measurement principle of the HTM. On the other hand, the VHTM is more accurate than the modal analysis method is in determining the frequency response of nonlinear structures such as coilable masts because modal analysis cannot address the stiffness weakening effects of nonlinear components.

The flow chart of the frequency response analysis based on the VHTM is shown in Fig. 8. The analysis steps are as follows:

Figure 8.

Flow chart of the frequency response analysis based on the VHTM.

Step 1: The VHTM is performed via explicit dynamic analysis in finite element software, and the time-domain acceleration response is obtained at the response points. Unlike the multiexcitation input and single-response output (MISO) approach in the HTM, the VHTM adopts the single excitation input and multiresponse output (SIMO) approach. These two approaches are equivalent in frequency response analysis, but SIMO has higher computational efficiency because the response of all points can be obtained by one structural analysis.

Step 2: The frequency response function is solved by using the excitation and response signals obtained from the finite element software. Applying the fast Fourier transform (FFT) technique, the frequency response function is defined as the amplitude ratio of the acceleration response ${A_i}\left( \omega \right)$ to the excitation signal ${P_j}\left( \omega \right)$ :

(2) \begin{equation} {R_{ij}}\left( \omega \right) = {A_i}\left( \omega \right)/{P_j}\left( \omega \right)\end{equation}

where $\omega $ is the excitation frequency, i and j represent the numbers of response and excitation points, respectively, and ${A_i}\left( \omega \right)$ and ${P_j}\left( \omega \right)$ are the Fourier transforms of ${a_i}\left( t \right)$ and ${p_j}\left( t \right)$ , respectively. ${a_i}\left( t \right)$ and ${p_j}\left( t \right)$ are the time domain signals of the response and excitation, respectively. By averaging all the frequency response functions, the overall frequency response function of the structure can be obtained. For the SIMO method, the frequency response function of the structure can be simplified as follows:

(3) \begin{equation} {R_j}\left( \omega \right) = \frac{1}{n}\sum\limits_{i = 1}^n {{R_{ij}}\left( \omega \right)} \end{equation}

where $j$ is determined and where $n$ is the total number of response points.

Step 3: The natural frequencies are determined via a graphical method [Reference Ishimura, Minesugi and Kawano13]. The graphical method uses the characteristics of frequency response function curves to identify natural frequencies directly. When the frequency response function curve is obtained, the frequencies corresponding to its peaks are determined as the natural frequencies of the structure. The corresponding model shapes refer to the modal analysis results.

4.1 Hammer test conducted on the 2-segment coilable mast

A hammer test was carried out on the 2-segment coilable mast first. Except for the number of segments, the technical parameters of the 2-segment coilable mast in the hammer test were consistent with those of the full-length mast, as shown in Table 1. The schematic diagram and facilities of the hammer test are shown in Figs 9 and 10, respectively. The equipment used in the hammer test includes a force hammer, an accelerometer, two charge amplifiers and a computer. The force sensor was installed at the head of the force hammer to measure the excitation signal. The accelerometer was mounted on the top plate of the mast to measure the response signal under hammer excitation. The excitation signal and the response signal were input into the computer through two charge amplifiers to calculate the frequency response curves of the 2-segment coilable mast.

Figure 9.

Schematic diagram of the hammer test.

Figure 10.

Facilities of the hammer test (2-segment).

In the hammer test, each hinge except the bottom three hinges was treated as a node, and each node was divided into two excitation points, radial and tangential. The MISO method was used in the test because it is difficult to measure the acceleration response of all nodes under one excitation. To avoid the specificity of a single hammering excitation, three hammering excitations were performed at each excitation point, and the average of three responses was taken as the response of that point.

The frequency response curve of the 2-segment coilable mast was obtained, as shown in Fig. 11. As shown in the figure, the first two frequency peaks of the 2-segment coilable mast are obvious, and the corresponding frequencies are 58 Hz and 69 Hz. These two natural frequencies are used as target responses for the identification of the hinge equivalent rotational stiffness K _hinge.

Figure 11.

Frequency response of the 2-segment coilable mast (Test).

4.2 Modelling and parameter identification

According to the modelling methods in Section 2.0, the 2-segment scaled model of the coilable mast was established, as shown in Fig. 12. Figures (a) and (b) show the stress states before and after rebalancing analysis of the coilable mast, respectively. After the rebalancing analysis, the tensions of the diagonal cables change from 40 N to approximately 27.5 N, and there is good consistency in the tensions of the diagonal cables in different segments. The subsequent frequency response analyses based on the VHTM were performed on the rebalanced model.

Figure 12.

Dynamic model of the coilable mast (2-segment).

Then, the range of values for the hinge equivalent rotational stiffness K _hinge was given, and the sample space KK: {K₁, K₂…, K _n } was constructed. In this study, the range of K _hinge was 100−5000 N·mm/rad, which is a large range that can ensure that the actual value falls within it. For sample space construction, a linear space containing 50 sample points at intervals of 100 was constructed, which can evenly appraise the effect of hinge rotational stiffness on the response. Moreover, the dimensions of 50 sample points can balance the solving efficiency and the fitting accuracy of the response curve.

A self-developed platform based on the cosimulation technique of MATLAB, Python and ABAQUS FEA was used to automatically model and analyse the frequency response of the coilable mast with different hinge rotational stiffnesses K. Then, the first- and second-order natural frequencies corresponding to different K values were found. Afterwards, the analysis results were fitted using the least squares method to obtain the variation curves of the natural frequencies with respect to the hinge rotational stiffness, as shown in Fig. 13. As shown in the figure, the first-order and second-order frequencies both increase with increasing hinge rotational stiffness K. The difference is that the first-order frequency tends to stabilise gradually with increasing K value, whereas the second-order frequency shows a continuous increase. According to Equation (1), taking both $\alpha $ and $\beta $ as 0.5 and ${f_{{\rm{t}}1}}$ and ${f_{{\rm{t2}}}}$ as 58 Hz and 69 Hz, the error function can be calculated, and its curve is shown in Fig. 13.

Figure 13.

Curve of the variation in natural frequency with K.

As shown in Fig. 13, the error curve has a minimum point within the given range. The corresponding K of this point is approximated to 1420 N·mm/rad, which is the identified hinge equivalent rotational stiffness K _hinge. The 2-segment coilable mast was remodelled and analysed using K _hinge, and the results were compared with the test results to preliminarily verify the correctness of the parameter identification process. A comparison of the frequency response curves obtained from the analysis and test is shown in Fig. 14. As shown in the figure, the analysis results of the first- and second-order natural frequencies are very close to the test results. The relative errors are 1.09% and 1.26%. The analysis of the sources of error are detailed in the next subsection.

Figure 14.

Comparison of the frequency response curves (2-segment, K = 1,420 N·mm/rad).

In this paper, the damping of the system is ignored in the proposed dynamic model of the coilable mast. There are two main reasons for this: (1) The damping of the structure system is very complicated and difficult to determine directly in the dynamic model; and (2) most deployable structures, such as the coilable mast [Reference Zhou and Zhou32], have very little damping, and ignoring the damping of the structure system does not have much of an effect on the model accuracy. If the damping is considered, then the following relationship is approximately satisfied between the natural frequency and the natural frequency when the damping is ignored: ${\omega _d} = {\omega _n}\sqrt {1 - {\zeta ^2}} $ , where ${\omega _d}$ is the natural frequency when damping is considered, ${\omega _n}$ is the natural frequency when damping is ignored, and $\zeta $ is the structural damping ratio. For a structure with low damping, such as a coilable mast, the structural damping ratio satisfies $0 \lt \zeta \lt 1$ ; then, ${\omega _d} \lt {\omega _n}$ . At this time, the first- and second-order frequency variation curves in Fig. 13 are lower than the existing curves. Since the experimentally obtained target frequencies used for parameter identification are unchanged, the identified hinge equivalent rotational stiffness K _hinge increases. In terms of the change in the frequency response curve in Fig. 14, the amplitude of the analysis curve decreases, but the change in the natural frequency cannot be directly determined. The change in the natural frequency is affected by the joint influence of damping and K _hinge, and the natural frequency decreases when the effect of damping is greater and increases when the effect of the change in K _hinge is greater.

4.3 Parameter verification

The identification result of the hinge equivalent rotational stiffness (K _hinge = 1,420 N·mm/rad) was preliminarily verified by a 2-segment coilable mast. In this subsection, the correctness of the identification results is verified by the full-length coilable mast.

The model of the full-length coilable mast is shown in Fig. 15. Figures (a) and (b) reflect the stress states of the coilable mast before and after the rebalancing analysis, respectively. The tensions of the diagonal cables change from 40 N to approximately 24.5 N, the magnitudes of which are close to those of the scaled model and have good consistency in different segments.

Figure 15.

Dynamic model of the coilable mast (full length).

Figure 16.

Facilities of the hammer test (full length).

First, the first two natural frequencies of the full-length coilable mast were tested. The facilities for the hammer test are shown in Fig. 16. The boundary conditions of the full-length coilable mast and measuring equipment in the hammer test are the same as those in the 2-segment coilable mast test. The test results of the first- and second-order natural frequencies of the full-length coilable mast are 13 Hz and 18 Hz, respectively.

Then, the frequency response analysis based on the VHTM was performed for the coilable mast after rebalancing analysis. The analysis results were compared with the test results, as shown in Fig. 17. The analysis results of the first- and second-order natural frequencies of the full-length coilable mast are 13.75 Hz and 19.13 Hz, respectively. The relative errors with respect to the test results are 5.77% and 6.28%, respectively. Although the relative error is larger than that of the 2-segment model, it is still within a small range, and the frequency response curves of the analysis and test results are close, which verifies the correctness of the identified hinge equivalent rotational stiffness K _hinge and the effectiveness of the modelling and identification methods proposed in this paper.

Figure 17.

Comparison of frequency response curves (full length, K = 1,420 N·mm/rad).

The sources of error are systematically analysed below on the basis of the comparison results shown in Figs. 14 and 17. As shown in the figures, the errors are reflected in two main aspects: (1) the amplitude error of the frequency response curves, and (2) the location error of the natural frequencies. The amplitude error of the frequency response curves is due mainly to ignoring the system damping of the coilable mast model. The sources of the error in the natural frequencies are relatively complex, and we consider that they come from two aspects: (1) parameter identification error and (2) model error. The parameter identification error is related to the number of sample points and the curve fitting accuracy, as shown in Fig. 13. Theoretically, increasing the number of sample points and choosing a more accurate fitting method can improve the parameter identification accuracy, but this approach significantly increases the computational cost. The model error arises from ignoring the model damping as well as the potential nonlinear properties of the hinge. A comparison of the results shown in Figs 14 and 17 reveals that the relative error of the full-length model is larger than that of the 2-segment scaled model. On the one hand, the hinge equivalent rotational stiffness is identified on the basis of the 2-segment scale model, which is slightly different from the hinge stiffness in the full-length model, making the identified stiffness have a larger error in the full-length model. On the other hand, because the damping of the full-length model is greater than that of the 2-segment scaled model and the number of hinges is greater, the model error has a greater effect on the error of the full-length model.