Finite element linear static structural analysis and modal analysis for Lunar Lander

3.1. Constraints and loads

The upper surface of the finite element model is constrained in all six degrees of the freedom. The loads corresponding to the case $a=$4 g are applied on the bottom surfaces of the four footpads as uniform pressure accounting for the forces applied by the lunar surface. Linear static structural analysis is submitted to MSC.Nastran to study the response of the landing system in terms of deflection and stress.

3.2. Presentation of the displacement

Fig. 3 shows the displacement contour of the results from the linear static structural analysis. It can be inferred from the Fig. 3 that the loads applied on the footpads bend the landing-gear assemblies from the joint point outwards. With the increase in the height, displacements appearing on the finite elements of the main struts are reduced from 0.0614 m to zero. With the increase in the distance to the center of the model, the displacements appearing on the finite elements of the secondary struts are on the increase.

Fig. 3Displacement contour

The displacements are maximum at the outermost points of the footpads, and the maximum strain ${\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is equal to 0.0614 m. The landing-gear radius ($R$) of the model is equal to 2.2 m, then the ${\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}/R$ is 2.8 %, which is a smaller rate. So, the stiffness of the structure is enough. However, the displacements on the inner cylinders of the main struts reduce quickly with the increase of the height, so the inner cylinders of the main struts bend to the out side sharply. So, the inner cylinders of the main struts are the weakness parts of the structure in the term of stiffness.

3.3. Presentation of the displacement

Fig. 4 displays the stress contour of the results from the linear static structural analysis. It can be inferred from the Fig. 4 that the distribution of the stress on the landing-gear struts is not uniform. Several stress-larger areas appear on the outside of the primary struts (Fig. 4(a)), what is attributed to the sharp bend of the primary struts (as the displacement contour shown in Fig. 3). At the same time, the phenomenon of stress concentration occurs on the junction between the primary struts and the secondary struts, as shown in Fig. 4(b).

It is found that the maximum stress generates inner side of the junction between the primary strut and the secondary strut. The maximum stress ${\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is equal to 210 Mpa which is less than allowed stress of material 484 Mpa. However, the junction between the primary struts and the secondary struts, and the inner cylinders of the primary struts are also dangerous parts in the term of strength under this operating condition.

Fig. 4Stress contour

a) Looking from the outside

b) Looking from the inside

4.1. The basic theory for the modal analyses

Multiple-degree-of-freedom systems (MDOF) are represented by the following Eq. (9):

where $M\text{,}$$C$and $K$are respectively $n\times n$ mass, damping, and stiffness matrices, the $n$ dimensional vector variable $x\left(t\right)$ indicates a position of a lumped mass $M_i$ on the structure, and the $n$ dimensional vector variable $f\left(t\right)$ describes the externally applied force.

Transforming Eq. (1) to the Laplace domain (assuming zero initial conditions) yields:

With $Z\left(s\right)$ the dynamic stiffness matrix:

The transfer function matrix $H\left(s\right)$ between displacement and force vectors, $X\left(s\right)=H\left(s\right)F\left(s\right)$, equals the inverse of the dynamic stiffness matrix:

With the numerator polynomial matrix $N\left(s\right)$ given by:

And the common-denominator polynomial $d\left(s\right)$, also known as the characteristic polynomial:

When the damping is small, the roots of the characteristic polynomial $d\left(s\right)$ are complex conjugate pole pairs, ${\lambda }_m$ and ${\lambda }_m^{*}$, $m=1,...,N_m$, with the number of modes system. The transfer function can be rewritten in a pole-residue form:

The residue matrices $R_m^{*}$, $m=\mathrm{1,2},...,N_m$ are defined by:

It can be shown that the rank of the matrix $R_m$ is $m$ that $R_m$ can be decomposed as:

with ${\psi }_m$ a vector representing the “modal shape” of mode $m$. From Eq. (7), it is obvious that the full transfer function matrix is completely characterized by the modal parameters, the poles ${\lambda }_m=-{\sigma }_m+i{\omega }_{d,m}$, and the mode shape vectors ${\psi }_m$, $m=\mathrm{1,2},...,N_m$.

4.2. Constraints for the modal analyses

The bottom of the rocket nozzle frame is fixed in all six degrees of freedom, while the other parts of the system are free in all six degrees of freedom. Modal analysis with the clamped boundary conditions is submitted to MSC.Nastran to extract the first six modes, and the natural frequencies and their corresponding vibration mode shapes are obtained.

4.3. Presentation of results

The pre-processing and post-processing of the finite element data are accomplished using MSC/Patran. The natural frequencies and their corresponding mode shapes are recovered, plotted, and animated in three dimensions to provide a visual understanding of the dynamic response.

Only the lower frequencies and vibration mode shapes are of interest because they adequately describe the dynamic behavior of the model [17-19]. Table 1 lists the frequencies and maximum displacements of the first six flexible modes.

As shown in Table 1, the frequency range in the analysis is from 9 Hz to 36 Hz, while the range of their corresponding maximum displacements is from 0.07 m to 0.5 m. The corresponding vibration mode shapes for the frequencies of the lunar lander are shown in Fig. 5.

The first mode shape is the rotation of the entire model around the $z$ axis (Fig. 5(a)), while the second mode shape is the rotation of the entire model around the $x$ axis (Fig. 5(b)). As shown in the Fig. 5(c), the third mode shape is the stretching along the $y$ axis direction. It is noteworthy that the bottom of the rocket nozzle is fixed in all six degrees of freedom, so flexible deformation occurs on the bottom surface of the model.

Fig. 5Sketch of the mode shapes of the first six modes

a) The 1st mode

b) The 2rd mode

c) The 3rd mode

d) The 4th mode

e) The 5th mode

f) The 6th mode

As shown in Fig. 5(d), the vibration shape of the 4th mode is the inward bend of two primary struts around the junction points between the primary struts and the module, while the others bend outwards. All the four primary struts bend inwards for the 5th mode, whereas all of them bend outwards for the 6th mode. To sum up, the vibration shapes for the mode 4, mode 5 and mode 6 are the bends of the primary struts around the junction points between the primary struts and the module.

As shown in Table 1, the value of each frequency is close to the contiguous one. It indicates that the lunar lander is likely to resonate, when the frequency of the external excitation is in the range from 9 Hz to 36 Hz. Then, small external exciting forces can result in important deformation, and possibly, damage can be induced in the structure of the lunar lander.

It is referred from the Fig. 5 that the maximum displacement of each mode shape is at the outside of the footpads. In particular, the maximum displacement for the 6th mode is up to 0.498 m. In other word, the primary struts of the landing-gear assemblies and the bottom surface of the module are the weakness parts of the lunar lander in the term of modal.