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I'm trying to investigate the stability of tensegrity structures, and this question is related to the second order test.
Suppose there is a vector space $W=\operatorname{span}\{\vec{w_1},\vec{w_2},...,\vec{w_s}\}$, and $\vec{w_1},\vec{w_2},...\vec{w_s}$ is a basis. Similarly, we have a vector space $V=\operatorname{span}\{\vec{v_1},\vec{v_2},...\vec{v_m}\}$. In literature, $W$ is the nullspace of the equilibrium matrix, whereas $V$ is the left-nullspace of the equilibrium matrix. $\{\mathrm{w}\}\in W$ is called a self-stress, and $\{\mathrm{v}\}\in V$ is the initial nodal velocities of the structure, also called a mechanism.
A linear operator $L$ produces a generalized Laplacian matrix $[\mathbf{L}]$ from a vector in $W$, with a fixed graph. A generalized Laplacian matrix is a symmetric one whose row sums and column sums are all $0$. The diagonal terms are the additive inverse of the sum of other entries on the corresponding row, respectively. (Other entries depend on the specific topology of the graph.)
$$\begin{array}{rccl} 4 & --- & --- & 3\\ | & \diagdown & \diagup & | \\ | & \diagup & \diagdown & | \\ 1 & --- & --- &2 \\ \end{array}$$
For example, the structure above has 4 nodes and 6 members. Sorry for the horrible diagram. I can't upload images. Member 1-2, 2-3, 3-4 and 1-4 are cables. In the middle of the rectangular, there are 2 strut members (that's not 4 struts): Member 1-3 and Member 2-4. Each member is assigned a number, with cables only getting positive numbers and struts negative ones. These numbers, put into a vector is the self-stress. The equilibrium matrix of this example is
$$[\mathbf{A}]=\begin{bmatrix} x_{1}-x_2 & 0 & 0 & x_1-x_4 & x_1-x_3 & 0\\ y_{1}-y_2 & 0 & 0 & y_1-y_4 & y_1-y_3 & 0\\ x_2-x_1 & x_2-x_3 & 0 & 0 & 0 & x_2-x_4\\ y_2-y_1 & y_2-y_3 & 0 & 0 & 0 & y_2-y_4\\ 0 & x_3-x_2 & x_3-x_4 & 0 & x_3-x_1 & 0\\ 0 & y_3-y_2 & y_3-y_4 & 0 & y_3-y_1 & 0\\ 0 & 0 & x_4-x_3 & x_4-x_1 & 0 & x_4-x_2\\ 0 & 0 & y_4-y_3 & y_4-y_1 & 0 & y_4-y_2\\ \end{bmatrix}$$
where $x$ and $y$ are coordinates of nodes.
The nullspace of $[\mathbf{A}]$ is $W$, whereas the left-nullspace is $V$.
Suppose we have ourselves a self-stress vector $\{\mathrm{w}\}$.
$$\{\mathrm{w}\}=\begin{Bmatrix} w_{12}\\ w_{23}\\ w_{34}\\ w_{14}\\ w_{13}\\ w_{24}\\ \end{Bmatrix}$$
Then the Laplacian matrix is constructed in the following way:
- If there is a member connecting Node $i$ and Node $j$, then $L_{ij}$ equals the negative value of the member's self-stress.
- If there is no member between two nodes, that entry is $0$.
- The diagonal terms are the sums of the self-stresses of the members connected to the nodes.
For this example, its Laplacian matrix is
$$[\mathbf{L}]=\begin{bmatrix} w_{12}+w_{14}+w_{13} & -w_{12} & -w_{13} & -w_{14}\\ -w_{12} & w_{12}+w_{23}+w_{24} & -w_{23} & -w_{24}\\ -w_{13} & -w_{23} & w_{13}+w_{23}+w_{34} & -w_{34}\\ -w_{14} & -w_{24} & -w_{34} & w_{14}+w_{24}+w_{34}\\ \end{bmatrix}$$
And we can denote $L$ as a function maps $\{\mathrm{w}\}$ to a $[\mathbf{L}_w]$. As long as the structure is the same, the function $L$ is the same.
Now a sufficient condition for a tensegrity to be second-order stable is:
for every $\{\mathrm{v}\}$ in $V$, there is a $\{\mathrm{w}\}$ in $W$ so that $\{\mathrm{v}\}^\mathbf{T}([\mathbf{L}]\otimes[\mathbf{I}_d])\{\mathrm{v}\}\geq0$ (only when $\{\mathrm{v}\}$ represents a rigid body motion can $\{\mathrm{v}\}^\mathbf{T}([\mathbf{L}]\otimes[\mathbf{I}_d])\{\mathrm{v}\}=0$)
where $[\mathbf{I}_d]$ is a identity matrix of order $d$ which is the dimension of the structure ($d=3$ for 3D problems).
So my question is, given $W$, $V$ and $L$, and the dimension of $W$ is more than 1, how do we test the above condition? Are there any related studies?
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Suppose we have a space of symmetric, real matrices, M, with some particular structure. And there is a vector space V (or it could be a subspace of V). Matrices in M may be positive-definite, may be not. But for every vector {v} in V, there is always a matrix [M] in M, so that the quadratic form is positive. Could somebody tell me what this subject is called, please? And where can I find more information about it? I've searched the web but they are all about positive-definite matrices, instead of a positive-definite space whose matrices don't have to be positive-definite.
