User:Eml4500.f08.gravy.bje/Notes3

Eigenvalues of the Global Stiffness Matrix K
For an unconstrained structure system, there are three possible rigid body motionsin 2-D (2 translation and 1 rotation).

Dynamic Evaluation Problem

 K v = λ Mv 

$$\ \underline{k} = \begin{bmatrix} K_{11}^{(1)} & K_{12}^{(1)} & K_{13}^{(1)} & K_{14}^{(1)} & 0 & 0  \\ K_{21}^{(1)}& K_{22}^{(1)} & K_{23}^{(1)} & K_{24}^{(1)} & 0 & 0 \\ K_{31}^{(1)} & K_{32}^{(1)} & (K_{33}^{(1)} + K_{11}^{(2)}) & (K_{34}^{(1)} + K_{12}^{(2)}) & K_{13}^{(2)} & K_{14}^{(2)} \\ K_{41}^{(1)}& K_{42}^{(1)} & (K_{43}^{(1)} + K_{21}^{(2)}) & (K_{44}^{(1)} + K_{22}^{(2)}) & K_{23}^{(2)} & K_{24}^{(2)} \\ 0 & 0 & K_{31}^{(2)} & K_{32}^{(2)} & K_{33}^{(2)} & K_{34}^{(2)} \\ 0 & 0 & K_{41}^{(2)} & K_{42}^{(2)} & K_{43}^{(2)} & K_{44}^{(2)}  \end{bmatrix}_{6x6}\,$$

$$\ \ = \begin{bmatrix} \frac{9}{16}& \frac{3\sqrt{3}}{16} & -\frac{9}{16} & -\frac{3\sqrt{3}}{16} & 0 & 0 \\  \frac{3\sqrt{3}}{16} & \frac{3}{16} & -\frac{3\sqrt{3}}{16} & -\frac{3}{16} & 0 & 0 \\ -\frac{9}{16} & -\frac{3\sqrt{3}}{16} & \frac{49}{16} & \frac{3\sqrt{3} - 40}{16} & -\frac{5}{2} & \frac{5}{2} \\ -\frac{3\sqrt{3}}{16} & -\frac{3}{16} & \frac{3\sqrt{3} - 40}{16} & \frac{43}{16} & \frac{5}{2} & -\frac{5}{2} \\  0 & 0 & -\frac{5}{2} & \frac{5}{2} & \frac{5}{2} & -\frac{5}{2} \\ 0 & 0 & \frac{5}{2} & -\frac{5}{2} & -\frac{5}{2} & \frac{5}{2}  \end{bmatrix}_{6X6}\,$$

MATLAB Code

Values of the stiffness matrix are assigned to matrix K >> K= [9/16 (3*sqrt(3))/16 -9/16 -(3*sqrt(3))/16 0 0; (3*sqrt(3))/16 3/16 -(3*sqrt(3))/16 -3/16 0 0; -9/16 -(3*sqrt(3))/16 49/16 ((3*sqrt(3)-40)/16) -5/2 5/2; -(3*sqrt(3))/16 -3/16 ((3*sqrt(3)-40)/16) 43/16 5/2 -5/2; 0 0 -5/2 5/2 5/2 -5/2; 0 0 5/2 -5/2 -5/2 5/2]

Values of K in decimal form K = 0.5625   0.3248   -0.5625   -0.3248         0         0    0.3248    0.1875   -0.3248   -0.1875         0         0   -0.5625   -0.3248    3.0625   -2.1752   -2.5000    2.5000   -0.3248   -0.1875   -2.1752    2.6875    2.5000   -2.5000         0         0   -2.5000    2.5000    2.5000   -2.5000         0         0    2.5000   -2.5000   -2.5000    2.5000

Matlab function to calculate eigenvalues >> d = eig(K)

List of the eigenvalues for global stiffness matrix K d = -0.0000  -0.0000    0.0000    0.0000    1.4705   10.0295

λ = 0, 0, 0, 0, 1.4705, 10.0295

It is important to note that four of the six eigenvalues are zero. These values represent zero stored elastic energy (rigid body at the modes).

FD Rel. on p. 10-1
$$(\underline{k}_{6x2}) (\underline{d}_{2x1}) = \underline{F}_{6x1} $$

$$\ \begin{bmatrix} K_{13}^{(1)} & K_{14}^{(1)} \\ K_{23}^{(1)} & K_{24}^{(1)} \\ (K_{33}^{(1)} + K_{11}^{(2)}) & (K_{34}^{(1)} + K_{12}^{(2)}) \\ (K_{43}^{(1)} + K_{21}^{(2)}) & (K_{44}^{(1)} + K_{22}^{(2)}) \\ K_{31}^{(2)} & K_{32}^{(2)} \\ K_{41}^{(2)} & K_{42}^{(2)} \end{bmatrix}_{6x2}\,$$ $$\ \begin{bmatrix} d_3 \\ d_4 \end{bmatrix}_{2x1}\,$$ $$\ \ = \begin{bmatrix} F_1 \\ F_2 \\ F_3 \\ F_4 \\ F_5 \\ F_6 \end{bmatrix}_{6x1}\,$$

$$\ \begin{bmatrix} -\frac{9}{16} & -\frac{3\sqrt{3}}{16} \\ -\frac{3\sqrt{3}}{16} & -\frac{3}{16} \\ \frac{49}{16} & \frac{3\sqrt{3} - 40}{16}\\ \frac{3\sqrt{3} - 40}{16} & \frac{43}{16} \\ -\frac{5}{2} & \frac{5}{2} \\ \frac{5}{2} & -\frac{5}{2} \end{bmatrix}_{6X2}\,$$ $$\ \begin{bmatrix} 4.3527 \\ 6.1271 \end{bmatrix}_{2x1}\,$$ $$\ \ = \begin{bmatrix} F_1 \\ F_2 \\ F_3 \\ F_4 \\ F_3 \\ F_4 \end{bmatrix}_{6x1}\,$$

$$F_1 = (-\frac{9}{16})(4.3527) + (-\frac{3\sqrt{3}}{16})(6.1271) = -4.4382$$

$$F_2 = (-\frac{3\sqrt{3}}{16})(4.3527) + (-\frac{3}{16})(6.1271) = -2.5624$$

$$F_5 = (-\frac{5}{2})(4.3527) + (\frac{5}{2})(6.1271) = 4.436$$

$$F_5 = (\frac{5}{2})(4.3527) + (-\frac{5}{2})(6.1271) = -4.436$$

Note: F3 and F4 were calculated in previous section to be 0 and 7 respectively