Tensegrity Finite Element Method (TsgFEM)

Welcome to **TsgFEM software!**

General Information

Our group focuses on the research of integrating structure and control design. We design structures based on the tensegrity paradigm to meet the specified objectives. These objectives can vary from minimizing the structure's mass to controlling the structure to meet specific performance. This software is intended to study the statics and dynamics of tensegrity systems based on FEM. The authors would like to make this open-source software to help other researchers interested in this field.

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Cite this work as:
Ma, S., Chen, M., Skelton, R.E., 2022. Tensegrity Finite Element Method. Journal of Open Source Software, 7(75), 3390. DOI: https://doi.org/10.21105/joss.03390

Long term archived version:

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This software aims to facilitate the statics and dynamics of Tensegrity systems based on FEM. The software allows modeling, structural design, nonlinear static, and dynamic FEM simulation of any tensegrity and truss systems. The contribution of this software is mainly in these three aspects.

Modeling:

1. Model any tensegrity structures by nodal coordinates and the nodes' connectivity information.
2. Specify the constraints of nodal coordinates (restricted motions of the X-, X-, or/and Z-directions of some specific nodes).
3. Normally, structure members in symmetric positions have the same force densities. They are allowed to be grouped in the software. #### Statics:
4. Prestress modes and mechanism modes analysis by singular value decomposition of equilibrium matrix.
5. Prestress design and minimal mass calculation considering yielding and buckling constraints.
6. Stiffness and stability analysis.
7. Solve the equilibrium equations with any given external forces considering the nonlinearity of geometry and material.
8. Simulate the forced motion by giving sequences of some nodes or changing the rest length of some members. #### Dynamics:
9. Linearized dynamics.
10. Modal analysis, calculate natural frequencies and mode shapes.
11. Nonlinear dynamics simulation of any tensegrity structures or truss systems with linear elastic, multilinear elastic, or plastic materials.

The name of this software TsgFEM is suggested to be pronounced as Tenseg FEM. TsgFEM allows users to perform accurate dynamics and statics analysis for any tensegrity structures, including: 1. Rigid body dynamics with acceptable errors. This is achieved by setting relatively high stiffness bars in the dynamics simulation. 2. Actuate FEM dynamics simulation with elastic or plastic deformations in the presence of various kinds of boundary conditions, such as fixing any nodes in any direction, applying static or dynamic external forces (i.e., gravitational force, some specified forces, or random seismic vibrations). 3. Accurate modal analysis, including natural frequency and corresponding modes. 4. An interface towards structural control by a compact state-space form of an accurate linear model. 5. Statics analysis allows one to do form-finding, static load analysis, and critical buckling analysis.

The software is being used to develop various structures not limited to tensegrity structures, such as torus, cable dome, tower, lander, D-Bar Structures. The software is also applicable to analyze tensile membranes (pure string to string network) and truss structures (pure bar to bar network).

Undergraduate linear algebra, material mechanics/continuum mechanics, finite element method, and some basic knowledge of MATLAB are required to understand the codes well. This software is developed based on: - 64-bit Windows - MATLAB - MATLAB Optimization Toolbox

Note: Win7/Win10/Mac OS/Linux/Win XP/Win Vista, the software is compatible with a MATLAB version later than 2009a. However, we encourage the user to run the software with the latest MATLAB release if possible. (More information about MATLAB versions can be found here: https://en.wikipedia.org/wiki/MATLAB). Since 'linprog' and 'fmincon' functions from MATLAB Optimization Toolbox are used in statics calculation, this toolbox should also be installed (more information can be found here: https://www.mathworks.com/products/optimization.html). Since commercial software embedded a lot more material database, for research purposes, one may also use - ANSYS

TsgFEM also provides an interface to ANSYS. The automatically generated interface file allows users to perform the FEM simulation in ANSYS. It is compatible with an ANSYS version later than 16.0. However, we also encourage the user to run the software with the latest ANSYS release if possible.

LICENSE

/* This Source Code Form is subject to the terms of the Mozilla Public
 * License, v. 2.0. If a copy of the MPL was not distributed with this
 * file, You can obtain one at http://mozilla.org/MPL/2.0/.

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Tensegrity Finite Element Method (TsgFEM) folder contains the following parts:

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setup.m

If one wants to start using the MOTES software, run the 'setup.m' first. Open MATLAB and run the 'setup.m' file, it will:

• Add all library functions to MATLAB path,
• Add the Software Verification and Examples,
• Add User Guide,
• Add Videos folder,
• Add JOSS paper.

Note: setup.m must be run every time MATLAB opens before any other file.

JossPaper

This folder contains the journal paper corresponding to the software, including source file documents and references. The journal paper will provide the background introduction, a summary of our work, applications, references, etc.

Function Library

This folder contains:

All the library functions for tensegrity statics and dynamics analysis are organized in this folder. By following instructions of statics and dynamics analysis from "User_Guide.pdf," one can perform the analysis.

Software Verification and Examples

This folder contains:

1. Statics Examples

Here, we give examples to verify and demonstrate the statics of this software.

1. Dynamics Examples

Here, we give examples to verify and demonstrate the dynamics of this software.

User Guide

This folder contains "User_Guide.pdf." The file provides a detailed description of how to use the software, various applications, and become a developer.

Videos

Some interesting tensegrity animation examples are shown in this folder.

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Help Desk:

We are open and willing to answer any question. Please state your problem clearly and use the following emails to contact: Muhao Chen: muhaochen@tamu.edu, Shuo Ma: mashuo@zjut.edu.cn. Thank you!

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Acknowledgment:

The authors want to thank the Dr. Kevin Mattheus Moerman, Dr. Patrick Diehl, and Mr. Rohit Goswami for their great help improving the software. They are really nice, patient, and professional researchers. Thank you, indeed!

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Join TsgFEM Community and Contribute

How to contribute

Feedback and contributions are appreciated. Please use the same terminology so that everybody can be on the same page.

1. Fork it
2. Submit a pull request OR send emails to the help desk.

We will reply to you ASAP.

Coding standards

• MATLAB (>= 2009a)
• Function input and output comments
• Use the same Nomenclature as follows

Nomenclature

Geometry:

N: initial node positions
n: nodal coordinate vector
C_b: bar connectivity
C_s: string connectivity
C: connectivity matrix of the structure
ne: the number of members
nn: the number of nodes
a: a vector containing the index of free nodal coordinate
b: a vector containing the index of fixed nodal coordinate
Ia: a matrix locating the free nodal coordinate
Ib: a matrix locating the fixed nodal coordinate
Gp: group matrix
l: length of members
l0: rest length of members    

Statics

A_1a: equilibrium matrix considering boundary constraints
V2: null space of equilibrium matrix
w0: external force
q_gp: force density vector in group
q: force density vector
t_gp: force vector in group 
t: force vector
index_b: number of bars
index_s: number of strings
A_b: cross sectional area of bars
A_s: cross sectional area of strings
A_gp: cross-sectional area of all members in a group
A: cross-sectional area of all members
E: Young's modulus of members
rho: material density of members
mass: mass vector of members
substep: substeps in static analysis
w_t: external force in substeps
dnb_t: forced motion of fixed nodal coordinate in substeps
l0_t: rest length of members in substeps
t_t: the force of members in substeps
n_t: nodal coordinate vector in substeps

Dynamics

tf: simulation time duration
dt: simulation time step
tspan: time steps in solving dynamics equation
out_tspan: time steps for output
lumped: use lumped or consistent mass matrix
M: mass matrix
D: damping matrix
omega: natural frequency
V_mode: free vibration mode
dnb_d_t: velocity of forced motion 
dnb_dd_t: acceleration of forced motion 
l_t: members length in time steps
nd_t: velocity of nodal coordinate in time steps