Dynamics evaluation of structural buckling (April 18, 2016)
Dr. Arita who graduated from our university in March 2016 considered the deployment repeatability of gossamer structure, and proposed a novel formulation for the criteria of the robustness of the deployment against the disturbance and/or deviation of the design paraneters. She validated her formulation by numerical examples, and got Dr. Engineering. The part of her research was published in Mechanical Engineering Journal.
Please refer the doctral thesis of Dr. Arita and the reference[1], and [2] for the detail of the researh. Only the essential point of her research is introduced in this web page.
There are four essential points in her research.
 The rigid mode for the structure under modtion (with deformation) was defined exactly, the method to decompose the increment of the displacement at each time step into elastic deformation and rigid body displacement is shown.

There was shown the method to decompose the eigen mode for zero or negative eigen value of the tangent stiffness matrix during the motion into the rigid body mode and the elastic deformation mode by usin the resulta obtained in No.1, and the "dynamics" buckling mode during the motion was defined.
 There was shown the analytical solution of the disturbance force with minimum norm at arbitrary time that causes the buckling mode deformation and the corresponding displacement vector along with the buckling mode vector.
 "The possibility of the buckling bifurcation " and "the amount of the deformation at the buckling" during the motion are defined theoretically by the above formulation.
The stiffness matrix in the deformed state is generally different from that in the undeformed state. Let us denote the position vector in the undeformed state as \({\mathbf{X}}\), that in the deformed state as \({\mathbf{x}}\), and the stiffness matrix as \({\mathbf{K}}({\mathbf{x}},{\mathbf{X}})\). Then the rigid body mode vector during the motion, \({{{\mathbf{\tilde q}}}_i}\), is defined as \({\mathbf{K}}({\mathbf{x}},{\mathbf{x}}){{{\mathbf{\tilde q}}}_i} = {\mathbf{0}}\). Using this definition, we can obtain all of the translational rigid motion, rotational rigid motion, and the local deformation of the membrane in the slack region. If we define \({{{\mathbf{\tilde q}}}_i}\) as \({\mathbf{K}}({\mathbf{x}},{\mathbf{X}}){{{\mathbf{\tilde q}}}_i} = {\mathbf{0}}\), several problems occur such that the rotational rigid motion is not involved in the rigid body mode because of the effect of the geometrical stiffness.
The subspace \({\mathbf{\tilde Q}} = \left\langle {{{{\mathbf{\tilde q}}}_1},\,{{{\mathbf{\tilde q}}}_2}, \cdots ,\,{{{\mathbf{\tilde q}}}_n}} \right\rangle \) that includes the rigid mode vector \({{{\mathbf{\tilde q}}}_i}\) as the base vector represents the rigid body space at each time step.
You may think "the buckling model vector" at arbitrary time can be defined as the eigen vector of the stiffness matrix\({\mathbf{K}}({\mathbf{x}},{\mathbf{X}})\) that has nonpositive eigen value. But this buckling mode vector contains the vector component that is in the above rigid body space. If there exists the eigen vector \(\eta \) that satisfies \({\mathbf{K}}({\mathbf{x}},{\mathbf{X}}){\boldsymbol{\eta}} = \lambda {\boldsymbol{\eta}} \) and \(\lambda \leqslant 0\), we can obtain the omponent \({{\boldsymbol{\eta}} ^*}\) normal to \({{\mathbf{\tilde Q}}}\), and define the unit vector \({{\mathbf{e}}^*} \equiv {{\boldsymbol{\eta}} ^*}/\left {{{\boldsymbol{\eta}} ^*}} \right\). Then, we can define \({{\mathbf{e}}^*}\) as the buckling mode vector ("dynamics buckling mode vector"). If \({{\boldsymbol{\eta}} ^*} = {\mathbf{0}}\), we can conclude that \({{\boldsymbol{\eta}} ^*}\) is a rigid body mode vector and there does not occur the buckling bifurcation (prelase refer the right figure).
By the way, the unrepeatability of the deployment can be caused by the variation of the initial condition, the deviation (change) of any pysical parameter, the buckling bifurcation by the distarbance force. This research focuses on the buckling bifircation, which is difficult to predict, and assumes that the structure that does not cause the buckling bifurcation at any time for all the possible distarbances has the heighest deployment repeatability. But it may be quite difficult to design such a structure. Thus this research evaluates "how large distarbance causes the motion along the buckling mode", and assumes that " the motion has lower repeatability if the norm of the distrbance necessary to cause the motion along the bucking mode direction is smaller and the required duration of the distarbance is shorter".
Then, Dr. Arita derived such a necessary distarbance and its norm, the required duration, and the amount of the displacement at the bifurcation theoretically (\(\Delta {\mathbf{x}}\) in the above left figure. She defined the distarbance norm as the DF value (Disturbance Force value), and the norm of the displacement at each node ( \(\alpha \) multipy the norm of the component of \({{\mathbf{e}}^*}\) at the node) as the BD value (Buckling Displacement value). She evaluated the robustness against the distarbance by the DF value and the BD value. For example, if the DF value is smaller, the structure can have the buckling bifurcation more easily, but even if the DF value is small, there occurs little problem if the BD value is small because the motion does not deviate largely from the predicted motion (nominal motion).
The easiness of the buckling bifurcation depends on the velocity and the acceleration. In fact, the DF value and the BD value depend on the velocity and the acceleration, so that we can evaluate the deployment motion including both of the structural design and the motion. In addition, we can investigate whichi nodes (which reagion in the structure) can deviate larger at the buckling bifurcation by using the BD value that is defined at every node, and we can change the design of such region so that the BD value is reduced. Thus the BD value gives a suggestion for the modification of the design.
THe below figure illustrates the numerical example of the spin deployment of the membrane which is similar to that of IKAROS (You can see the enlarged figure if you click each figure). The yellow contour represents the BD value at each node, which shows that the center tether has large buckling displacement as expected. The red line represents the slack member, which shows that there are many slack members around the center of the edges during the later stage of the deployment. This means that the tensile stress is hardly applied to the center of the edge, which is also observed in the deployment of rectangular membrane.
Dr. Arita analyzed the deployment of the membrane structure that is similar to SPROUT, which showed that the compressive stiffness of the membrane drives the deformation and the edge of the triangular membrane can be slacked easiliy as expected in the experiments.
We hope this research will be applied to the design and development of spacecrafts and will help the developers/users to launch any deployable "wonderfull" structure in the future that is said to be impossible to launch into space,
This research was inspired by the research "Establishment of prediction method of dynamics of large gossamer multibody space structures and understanding of their dynamics
" supporetd by JSPS KAKENHI.
References
[1]  Shoko Arita, Yasuyuki Miyazaki, A study of dynamic evaluation of structural buckling, Mechanical Engineering Journal, Vol.2, Paper No.1500677, pp.18, March 31, 2016, DOI: 10.1299/mel.1500677. 
[2] 
Shoko Arita, Yasuyuki Miyazaki, A Proposal of Dynamics Evaluation Method of Structural Buckling and Instability, Proceedings of 31th Symposium on Aerospace Structure and Materials, pp.16, December 8, 2015, ISAS/JAXA, Sagamiahra, Kanagawa Japan, https://repository.exst.jaxa.jp/dspace/handle/ais/560948. 
P.S. (March 31, 2017)
Dr. Arita is working as an Assistant professor in Shizuoka University.
