Question 3.7: Find the first two natural frequencies of the following syst...
Find the first two natural frequencies of the following system model with the given accuracy ε of 0.001:
[M]{ü} + [K]{u} = 0 (3.105)
where
[M]=\begin{bmatrix} 20 & 5 & 0 \\ 5 & 10 & 5 \\ 0 & 5 & 10 \end{bmatrix} , [K]=\begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{bmatrix}
Equation (3.105) can be turned into an eigenvalue problem with three DoF (N = 3) as.
[K]\left\{u\right\}-\lambda _{i} [M]\left\{u\right\}=0 where \lambda _{i}=w^{2}_{i} (3.106)
Since the first two natural frequencies (M = 2) of the system model is interested, let assume that an initial vector as \left\{{u}\right\} _{2\times 3} , i.e.,
[u]_{23}=[u_{1} u_{2} ]=\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}
Let i = 1 and [X_{1}]= [u]_{23} to initialize the iterative process.
Iteration 1. Solving the equation set [K][Y_{i} ] = [M][X_{i} ] yields
[Y_{i}]=\begin{bmatrix} 25 & 20 \\ 30 & 35 \\ 30 & 40 \end{bmatrix}
The reduced eigenvalue problem at this iteration becomes
[K]^{*}[u]^{*}-\lambda [M]^{*}[u]^{*} =0
where
[K]^{*}=[Y_{i} ]^{T}[K][Y_{i} ] =\begin{bmatrix} 650 & 575 \\ 575 & 650 \end{bmatrix}
[M]^{*}=[Y_{i} ]^{T}[M][Y_{i} ] =1.0\times 10^{4} \begin{bmatrix} 4.7 & 5.1125 \\ 5.1125 & 5.725 \end{bmatrix}
The eigenvalues and eigenvectors of the above reduced problem can be found as
\lambda _{1}= 0.1052, \lambda _{2}= 0.0113, and [u]^{*}=\begin{bmatrix} 0.7455 & 0.0450 \\ -0.6665 & 0.9990 \end{bmatrix}Next, [X_{i} ] can be updated as
[X_{i}]=normc([Y_{i} ][u]^{*} )=normc\left\lgroup\begin{bmatrix} 5.3075 & 21.1040 \\ -0.9625 & 36.3137 \\ -4.295 & 41.3087 \end{bmatrix} \right\rgroup = \begin{bmatrix} 0.7697 & 0.3582 \\ -0.1396 & 0.6164 \\ -0.6229 & 0.7012 \end{bmatrix}Iteration 2. Solving the equation set [K][Y_{i} ]= [M][X_{i} ] with the updated [X_{i} ] yields
[Y_{i} ]= \begin{bmatrix} 7.1085 & 31.8025 \\-0.4800 & 53.3581 \\ -7.4069 & 63.4522 \end{bmatrix}
The reduced eigenvalue problem at this iteration becomes
[K]^{*} [u]^{*}-\lambda [M]^{*}[u]^{*} =0
where
[K]^{*}=[Y_{i} ]^{T}[K][Y_{i} ]= \begin{bmatrix}650 & 575 \\ 575 & 650 \end{bmatrix}
[M]^{*}=[Y_{i} ]^{T}[M][Y_{i} ] =1.0\times 10^{4} \begin{bmatrix} 4.7 & 5.1125 \\ 5.1125 & 5.725 \end{bmatrix}
The eigenvalues and eigenvectors of the above reduced problem can be found as
\lambda _{1}= 0.1052, \lambda _{2}= 0.0113, and [u]^{*}=\begin{bmatrix} 1.000 & -0.0069 \\ 0.0054 & 1.0000 \end{bmatrix}Next, [X_{i} ] can be updated as
[X_{i}]=normc([Y_{i} ][u]^{*} )=normc\left\lgroup\begin{bmatrix} 7.2799 & 31.7526 \\ -0.1923 & 53.3601 \\ -7.0647 & 63.5019 \end{bmatrix} \right\rgroup = \begin{bmatrix} 0.7175 &0.3575 \\ -0.0190 & 0.6008 \\ -0.6963 & 0.7150 \end{bmatrix}Check the convergence: max\left(\left|\frac{\lambda ^{i+ 1}_{1}-\lambda^{i+ 1}_{1} }{\lambda^{i+ 1}_{1}} \right|,\left|\frac{\lambda^{i+ 1}_{1}-\lambda^{i+ 1}_{1} }{\lambda^{i+ 1}_{1}} \right| \right) =0.05>ε ; therefore, the iteration has to be continued.
Iteration 3. Solving the equation set [K][Y_{i} ] = [M][X_{i} ] with the updated [X_{i} ] yields
[Y_{i} ]=\begin{bmatrix} 7.1139 & 31.6789 \\ -0.0274 & 53.2034 \\ -7.0852 & 63.3574 \end{bmatrix}
The reduced eigenvalue problem at this iteration becomes
[K]^{*} [u]^{*}-\lambda [M]^{*}[u]^{*} =0
where
[K]^{*}=[Y_{i} ]^{T}[K][Y_{i} ] =1.0\times 10^{3} \begin{bmatrix} 0.1514 & -0.0000 \\ -0.0000 & 1.5700 \end{bmatrix}
[M]^{*}=[Y_{i} ]^{T}[M][Y_{i} ] =1.0\times 10^{5} \begin{bmatrix} 0.0151 & -0.0000 \\ -0.0000 & 1.3908 \end{bmatrix}
The eigenvalues and eigenvectors of the above reduced problem can be found as
\lambda _{1}= 0.1001, \lambda _{2}= 0.0113, and [u]^{*}=\begin{bmatrix} 1.000 & -0.000 \\ 0.000 & 1.0000 \end{bmatrix}Next, [X_{i} ] can be updated as
[X_{i}]=normc([Y_{i} ][u]^{*} )=normc\left\lgroup\begin{bmatrix}7.1143 & 31.6788 \\ -0.0267 & 53.2034 \\ -7.0844 & 63.3575 \end{bmatrix} \right\rgroup = \begin{bmatrix} 0.7086 &0.3576 \\ -0.0027 & 0.6006 \\ -0.7056 & 0.7152 \end{bmatrix}Check the convergence: max\left(\left|\frac{\lambda ^{i+ 1}_{1}-\lambda ^{i}_{1} }{\lambda ^{i+ 1}_{1}} \right|,\left|\frac{\lambda ^{i+ 1}_{2}-\lambda ^{i}_{2} }{\lambda ^{i+ 1}_{2}} \right|\right) <\epsilon . The convergence condition is satisfied.
Finally, the first two natural frequencies are found as w_{1} = 0.3164 and w_{2} = 0.1063.