Research on touchdown performance of soft-landing system with flexible body

2.1. Model of the soft landing system

In order to meet the functional requirement of soft- landing gear system, and make the buffer work stably and reliably, the four-leg suspension landing buffer with aluminum honeycomb absorbing energy would the best model according to the features of different soft-landing gear system and the requirement of the lunar exploration program of China Change’s Project.

As Fig. 1 shows, every landing leg consists of three parts. They are four groups of main buffer leg (primary strut), auxiliary buffer struts (secondary strut) and footpad. It is the universal joint who makes the primary strut and secondary strut connect to the body of the landing gear system. The secondary strut and the primary strut are connected by ball joint, and it is the same to the primary strut and footpad. The deployable and lockable mechanism is settled at the inner of one of secondary strut of every group, which should be called multi-functional secondary strut. Another secondary strut is made up by tensile buffer aluminum honeycomb, compression buffer aluminum honeycomb band piston rod, and the inner structure of the strut is similar with multi-functional secondary strut. Without deployment and locking mechanism, it is just used for buffering, so it is called single functional secondary strut [18].

Fig. 1Schema of soft-landing gear mechanism

2.2. Multi-mass model analysis

Equation for the analysis were developed corresponding to the two-dimensional touchdown dynamics model shown in Fig. 2. This model is adequate for the analysis of either three or four legged vehicle. Motion takes place in a plane such that two legs contact the surface simultaneously for four-legged vehicle. It is assumed that the vehicle has feet of sufficient area to prevent penetration of the lunar surface [19]. In this model, the lander body is simplified to a simple mass - spring system based on the rigid mode of structure vibration motion and first n dynamics equations of vibration modes of a lunar Lander, which module quality and the number of spring depends on the order of flexible modes and accuracy requirements of system [20]. The simplified model of lander body model analysis and the body equivalent principle are as follows:

1) The natural vibration frequency of the simplified model and the actual modal frequencies of the lander body are the same.

2) The vertical response of the intersection of simplified model / buffer system and the landing / buffer system are the same under the same load.

Fig. 2Simplified model of the lander body modal analysis

Where in Fig. 2:

$m_n$ – simulation of $n$th order modal mass of lander,

$K_n$ – simulation of $n$th order modal spring stiffness of elastic effect,

$Z_n$ – the vertical displacement of the mass $m_n$,

$C_n$ – simulation of $n$th order modal flexible damping coefficient.

Based on the rigid mode of the lander structure vibration and the $n$ order dynamic equation of the vibration mode the dynamic equation is:

where $\left\{z\right\}={(z_0,z_1,\cdots ,z_n)}^T$, $\left\{p\right\}={(p_0,p_1,\cdots ,p_n)}^T$, $p_1=\cdots =p_n=0$.

The dynamic equation of lander body in modal coordinates (include rigid mode) is:

where $\left\{q\right\}={(q_0,q_1,\cdots ,q_n)}^T,q_0$.

Corresponding lander sinking mode with unit vertical displacement:

$\left[\tilde{M}\right]$ – generalized mass diagonal matrix,

$\left[\tilde{K}\right]$ – generalized stiffness diagonal matrix,

$\left[\xi \right]$ – modal damping coefficient diagonal matrix.

According to the former equivalence principle, there is:

Let:

Thus:

where:

Put equation (6) into (1), collated as:

Compare formula (2) with formula (7), the right options are found identical. The equivalent condition of the simplified model and the original lander body attribute to the coefficient matrix of the formula (2) corresponding to the formula (7) in the left:

Undetermined coefficients ${\alpha }_{ij}$, parameters $m_i$, $k_i$ and $c_i$ in the simplified model can be determined in equation (8). Since ${\stackrel{-}{K}}_{ii}$ and ${\stackrel{-}{M}}_{ii}$ satisfying the conditions that ${\stackrel{-}{K}}_{ii}={\omega }_i^2{\stackrel{-}{M}}_{ii}$, ensuring the free vibration frequency of the simplified model with the original landing body are consistent.

3.1. The finite element model of the wholly landing simulation

The finite element of the soft-landing system is created by simplifying the related apparatus, as shown in Fig. 3, this model is designed by Nanjing Astronautics and Aeronautics University. The soft-landing system includes the main strut, the secondary strut, the buffering, the footpad and the compression release apparatus [20-22]. The main strut is equipped with the buffering which can relieve the landing impact. The secondary strut is equipped with the stretching and locking apparatus which is used to stretch and lock before impact. The compression release apparatus achieves to compress and release the footpad which can avoid excessive subsidence and relieve the sliding resistance. Because the landing leg is composed of honeycomb whose wall thickness is far less than its length, the cylindrical shells can be simplified as the shells unit and the collector as the mass unit and the footpad as the shells unit [23-24]. The shells unit is divided as grids with QUAD4. The piston rod and the outer cylinder of the landing leg is made up of aluminum alloy 7A09T6 whose performance parameters (density, elastic modulus, Poisson’s ratio) are given by: $\rho =2800\mathrm{}\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$, $E=71\mathrm{}\mathrm{G}\mathrm{P}\mathrm{a}$, $\mu =0.33$. The static friction coefficient, the dynamic friction coefficient and the exponential decay coefficient of the aluminum alloy are given by: ${\mu }_s=0.06$, ${\mu }_k=0.03$, $\beta =0.1$. After connecting every member and defining the touching if the aluminum honeycomb is equivalent to the Lagrange unit Hex8, the finite element model of the prototype is divided as: $86130$ units, $77832$ connectors, as shown in Fig. 4.

Fig. 3Schema of landing gear system

a)

b)

The buffering is composed of the strong aluminum honeycomb and the weak aluminum honeycomb. As shown in Fig. 5, the resilient section and the dense segment of the honeycomb are small after simplification. Assuming the carrying capacity $F_1=$ 16.4 kN, $F_2=$ 35.4 kN.

Then we have the corresponding buffering stroke $S_1=$ 150 mm, $S_2=$ 300 mm.

Fig. 4Finite element model for soft-landing impacting

Fig. 5Undergone sketch of the two-stage honeycomb buffering

3.2. Simulation analysis of landing performance

In this paper, the landing conditions are set as follows: the value of the gravitational field acceleration is $9.8\mathrm{m}/{\mathrm{s}}^2$, the lander is released vertically from the origin at the speed of $4.43\mathrm{m}/\mathrm{s}$ and its four legs land on the ground at the same time, the module on the landing performance is rigid, the total mass is $1800\mathrm{k}\mathrm{g}$.

Acceleration response and buffering stroke are the basic evaluation for the honeycomb during the landing. In order to reduce the influence of high-frequency response, fast Fourier transform is used to do frequency spectrum analysis of the original data. At the same time, we adopt a series of low-pass filters of frequency of $90\mathrm{H}\mathrm{z}$ which is processed for it. Fig. 6 represents simulation curves of the lander’s acceleration response which has been processed by a series of low-pass filters.

Fig. 6Acceleration time histories of lunar landing

Two-stage aluminum honeycomb buffering apparatus is adopted by primary buffering, as shown in Fig. 7. The primary aluminum honeycomb is compressed in $0\sim 45\mathrm{m}\mathrm{s}$, when the value of acceleration remains about $5\mathrm{g}$. In $45\sim 55\mathrm{m}\mathrm{s}$, the acceleration increases dramatically and then to its peak value of $9.61\mathrm{g}$, in $55\sim 85\mathrm{m}\mathrm{s}$ the value of acceleration oscillates around $7.5\mathrm{g}$, and then the secondary aluminum honeycomb comes into play. In $85\sim 110\mathrm{m}\mathrm{s}$, the lander reduces its acceleration rapidly to finish the process of landing until its value oscillates around $-1\mathrm{g}$ and attitude tends to stabilize. Throughout the process of landing impact, the fluctuations of acceleration response are mainly caused by buffer and the dynamic response of structures, where the response higher than $90\mathrm{H}\mathrm{z}$ has been filtered.

The cushioning effect of the main and secondary buffering is mainly reflected on absorption of aluminum honeycomb to impact energy during the landing. As shown in Fig. 7, the weak aluminum honeycomb on main and secondary buffering begins to work from $0\mathrm{m}\mathrm{s}$; at $55\mathrm{m}\mathrm{s}$, energy absorption capacity of weak aluminum honeycomb is depleted. At the same time, the strong aluminum honeycomb begins to work. As a result, the intensity of lander’s acceleration response increases. It is not until the aluminum honeycomb on main and secondary buffering finish stopping the service that the value of lander’s acceleration response starts to decrease rapidly.

Fig. 7The energy absorption of main and secondary honeycomb