Multi-objective Design Optimization of Structural Geometric Nonlinearities for Response Attenuation of VLBI Antennae Subject to Aerodynamic Turbulence

Abstract

Background

Very-long-baseline interferometry (VLBI) is a technique employed in radio astronomy where a signal from a planetary radio source is collected on earth at multiple radio antennae. Working in a network, the time difference of the radio signal reception by the antennae is employed to achieve an effective aperture allowing for the development of a significantly high-resolution images of the space. However, the signal clarity of each individual antenna is dependent on its structural response under environmental conditions. This paper proposes a design optimization framework for VLBI antennae for their performance maximization under aerodynamic gust employing structural geometric nonlinearities.

Purpose

The dynamic aeroelastic response of the antenna is attenuated by optimizing actuation length changes in active structural elements to improve the pointing accuracy of the VLBI antenna under wind gust loads. No framework currently exists in the open literature that leverages the principles of tensegrity structures for vibration attenuation of VLBI antenna using control actuations, therefore, the need for such a framework has been established to design high-performance VLBI ground stations.

Methods

Dynamic aeroelastic gust analysis is performed considering the Power Spectral Density (PSD) with the Davenport spectrum (DS) statistical model and Tuned Discrete Gust analysis with a One-Minus Cosine gust profile. Analyses are performed for two different operating conditions using time-consistent loads (TCL) and time-consistent displacements (TCD).

Additionally, the effect of a varying number of active elements for control actuations is analyzed in the boomarm subsystem of the VLBI antenna while minimizing both the pointing error and total strain energy using an optimization framework employing a Multi-Objective Genetic Algorithm (MOGA).

Results

A case study was presented to showcase the proposed framework. A reduction of 82.6% with a total strain energy increase of 292.5% was obtained for the primary operating case under PSD gust excitation. On the other hand, at the increased mean wind speeds of the secondary operating case, the developed design algorithm was able to reduce the total pointing error by 80.9% but with a total strain energy increase of 825.3%. Similarly, for TDG analysis with the OMC excitation profile the optimization algorithm reduced the total pointing error by 51.6% with a TSE increase of 2098.1% and 80.5% with a TSE increase of 48.7% for the primary and secondary operating conditions, respectively, when compared to the uncontrolled response.

Conclusion

It is found that in all subcases analyzed the developed optimization framework successfully found the best response of the antenna using the utopian point method. These results confirm the effectiveness of the proposed method.

Appendices

Appendix A: Design Vectors for Best Individuals

The following tables give the design vectors of the best individuals for each subcase. Each vector represents an actuation length change in inches of the respective active element to induce prestress in the member and tune the geometric stiffness matrix for vibration attenuation. As the proposed framework is adaptive in nature a separate design vector is found for each specific subcase to ensure the best individual is tailored specifically to that subcase to achieve the best result.

Element ID	PSD Analysis with Davenport Spectrum
	Primary			Secondary
	16	32	44	16	32	44
85	0.034	− 0.139	− 0.269	− 0.059	− 0.075	− 0.079
55	0.015	0.002	− 0.135	− 0.010	0.001	− 0.130
65	0.270	− 0.134	− 0.135	− 0.604	− 0.235	− 0.137
75	− 0.021	0.003	− 0.135	− 0.042	0.013	− 0.042
2	0.290	0.084	0.049	0.130	− 0.020	0.090
5	0.349	0.085	0.049	0.174	0.050	0.048
57	0.827	0.222	0.280	0.832	1.219	0.297
56	1.786	0.575	0.719	1.607	0.623	0.722
94	0.301	− 0.096	− 0.477	− 0.668	− 0.141	− 0.479
10	0.730	0.041	− 0.086	0.263	0.030	0.041
12	0.317	0.034	0.049	0.301	0.035	0.050
63	0.944	0.211	0.280	0.090	0.099	0.279
64	1.556	0.605	0.585	1.365	0.606	0.715
95	0.786	− 0.220	− 0.477	− 0.450	− 0.446	− 0.480
9	− 0.357	0.036	− 0.086	0.288	− 0.059	0.057
13	0.424	0.036	− 0.454	0.296	0.113	0.040
67		0.229	0.280		0.200	0.310
66		0.620	0.585		0.623	0.729
96		− 0.444	− 0.244		− 0.432	− 0.259
18		0.119	− 0.086		0.117	0.046
21		0.097	− 0.086		0.044	0.082
87		0.236	0.280		0.265	0.295
86		0.481	0.953		0.506	0.789
92		− 0.117	− 0.477		− 0.121	− 0.770
28		0.018	0.416		− 0.036	0.051
30		0.092	0.282		1.042	0.012
77		0.207	0.415		0.230	− 0.330
76		0.491	1.087		0.585	0.715
98		− 0.100	− 0.109		− 0.106	− 0.478
19		0.109	0.049		0.074	− 0.129
22		0.160	− 0.086		− 0.018	0.119
83		− 0.086	0.648		− 0.190	0.283
84			1.087			0.747
91			− 0.244			− 0.485
29			0.282			0.219
31			0.049			0.057
73			0.280			0.280
74			0.719			0.452
97			− 0.244			0.142
3			− 0.086			0.854
4			0.282			0.058
53			0.514			0.350
54			0.585			0.784
93			− 0.244			− 0.467

Element ID	TDG Analysis with One-Minus Cosine
	Primary			Secondary
	16	32	44	16	32	44
85	− 0.006	− 0.129	− 0.140	− 0.225	− 0.132	− 1.228
55	0.003	0.018	− 0.135	0.017	0.000	− 0.135
65	− 0.582	− 0.134	− 0.135	− 0.121	− 0.133	− 0.135
75	0.013	0.001	− 0.135	0.175	0.021	− 0.135
2	− 0.146	2.008	0.048	0.352	0.084	0.049
5	0.172	0.097	− 0.242	0.620	0.084	0.049
57	1.162	0.227	1.242	1.062	0.232	0.280
56	1.333	0.460	0.717	1.805	0.936	0.719
94	− 0.822	0.930	− 0.477	− 0.181	− 0.100	− 0.477
10	0.039	0.024	0.044	0.304	0.036	0.049
12	0.333	0.131	0.054	0.558	0.036	− 1.044
63	0.924	0.218	0.280	1.135	− 0.272	0.280
64	1.213	0.661	0.719	1.883	0.606	0.719
95	− 0.522	− 0.441	− 0.478	0.019	− 0.442	− 1.570
9	0.311	− 0.316	0.049	0.293	− 0.536	0.049
13	0.299	0.040	0.048	0.541	0.036	0.049
67		0.631	0.281		0.228	− 0.812
66		0.609	0.724		0.606	0.719
96		− 0.447	− 0.522		− 0.417	− 0.477
18		0.584	0.034		0.117	0.049
21		0.100	0.049		0.220	0.049
87		0.231	0.280		0.238	0.280
86		0.470	0.720		− 0.146	− 0.374
92		0.421	− 0.478		− 0.098	− 0.477
28		0.083	0.048		0.084	0.049
30		0.741	0.066		0.084	0.049
77		0.238	0.285		0.232	0.280
76		0.496	0.719		0.750	0.719
98		− 0.099	− 0.477		− 0.296	− 0.477
19		0.162	0.049		0.252	1.141
22		0.153	0.084		0.153	0.049
83		− 0.163	0.277		− 0.174	0.280
84			0.719			0.719
91			− 0.478			− 0.477
29			0.047			0.049
31			0.048			0.049
73			0.279			0.280
74			0.719			0.719
97			− 0.477			− 0.477
3			0.048			0.049
4			0.050			0.049
53			0.280			0.280
54			0.719			− 0.374
93			− 0.477			− 0.477

Appendix B: Expanded Equations

The force influence matrix, ${{\varvec{S}}}_{{\varvec{F}}}$, defines the resulting effective prestress in each member due to a given active element length change and is defined as:

$${\textbf{S}}_{{\varvec{F}}}\Delta \textbf{L}={\textbf{P}}_{{{\varvec{k}}}_{\textbf{e}\textbf{f}\textbf{f}},}$$

$${\textbf{S}}_{{\varvec{F}}}=-{\textbf{W}}_{\textbf{s}}{\left({{\textbf{W}}_{\textbf{s}}}^{\textbf{T}}\textbf{G}{\textbf{W}}_{\textbf{s}}\right)}^{-1}{{\textbf{W}}_{\textbf{s}}}^{\textbf{T}},$$

where $\Delta \textbf{L}$ is a design vector representing the length change in the active members, this is kept zero for the non-active elements. ${{\varvec{W}}}_{\textbf{s}}$ is a matrix whose s columns pertain to the individual self-stress modes and is calculated with the null space of the equilibrium matrix, finally, G is the flexibility matrix. The effective prestress, ${{\varvec{P}}}_{{{\varvec{k}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}}$, is:

$${{\varvec{P}}}_{{{\varvec{k}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}}=\left({{\varvec{W}}}_{{\varvec{r}}}{{{\varvec{V}}}_{{\varvec{r}}}}^{-1}{{{\varvec{U}}}_{{\varvec{r}}}}^{\textbf{T}}\right)\Delta {\varvec{p}}-{{\varvec{W}}}_{{\varvec{s}}}\left[{\left({{{\varvec{W}}}_{{\varvec{s}}}}^{{\varvec{T}}}{\varvec{G}}{{\varvec{W}}}_{{\varvec{s}}}\right)}^{-1}{{{\varvec{W}}}_{{\varvec{s}}}}^{\textbf{T}}\left(\Delta {\varvec{L}}+{\varvec{G}}\left({{\varvec{W}}}_{{\varvec{r}}}{{{\varvec{V}}}_{{\varvec{r}}}}^{-1}{{{\varvec{U}}}_{{\varvec{r}}}}^{\textbf{T}}\right)\Delta {\varvec{p}}\right)\right],$$

where $\Delta {\varvec{p}}$ is a vector of element forces from the applied load. Next, the geometric stiffness matrix, ${{\varvec{K}}}_{{\varvec{G}}}$, using the effective prestress per element i is defined for an Euler–Bernoulli beam as follows:

$${{\varvec{K}}}_{{\varvec{G}}}\left({\varvec{i}}\right)=\frac{{{\varvec{P}}}_{{{\varvec{k}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}}({\varvec{i}})}{{{\varvec{l}}}_{{\varvec{k}}}({\varvec{i}})}\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& -1& 0\\ 0& 0& 1& 0& 0& -1\\ 0& 0& 0& 1& 0& 0\\ 0& -1& 0& 0& 1& 0\\ 0& 0& -1& 0& 0& 1\end{array}\right],$$

where ${{\varvec{l}}}_{{\varvec{k}}}$ is the element’s length. The following metric is used to determine a ranking of each element’s effectiveness to alter the geometric stiffness matrix and correspondingly the system response:

$${\boldsymbol{\Gamma }}_{{\varvec{j}}}=\sum_{{\varvec{i}}=1}^{{\varvec{m}}}{{\varvec{S}}}_{{\varvec{F}}}({\varvec{i}},{\varvec{j}}),$$

$$\boldsymbol{\Gamma }=\textbf{m}\textbf{a}\textbf{x}{\varvec{N}}\left({\boldsymbol{\Gamma }}_{{\varvec{j}}}\right),$$

where i and j represent the current row and column of an m x nel matrix, respectively, and N is chosen by the designer as the number of active elements desired.