User:Eml4507.s13.team2/Report5

Honor Pledge
On our honor, we used Dr. Vu-Quoc notes Fead.s13.sec53b as reference.

Given: A spring-mass-damper system
A spring-mass-damper system with the following properties is given.



Solve by hand the general eigenvalue problem
Solve the generalized eigenvalue problem for the spring-mass-damper system using the following data given.

$$ m_1=3, m_2=2 $$

$$ k_1=10, k_2=20, k_3=15 $$

Solution
$$ d=\begin{Bmatrix} d_1\\ d_2 \end{Bmatrix} $$

$$ F=\begin{Bmatrix} F_1\\ F_2 \end{Bmatrix} $$

$$ M=\begin{bmatrix} m_1&0\\ 0&m_2 \end{bmatrix} $$

$$ K=\begin{bmatrix} k_1+k_2&-k_2 \\ -k_2&k_2+k_3 \end{bmatrix} $$

$$ [\mathbf{K}-\gamma \mathbf{I}]\mathbf{x}=\left ( \begin{bmatrix} k_1+k_2&-k_2 \\ -k_2&k_2+k_3 \end{bmatrix} -\gamma \begin{bmatrix} 1 &0 \\ 1 & 0 \end{bmatrix}\right ) \begin{Bmatrix} x_1\\ x_2 \end{Bmatrix}=\begin{Bmatrix} 0\\ 0 \end{Bmatrix} $$

$$ [\mathbf{K}-\gamma \mathbf{I}]\mathbf{x}=\left ( \begin{bmatrix} 30&-20 \\ -20&35 \end{bmatrix} -\gamma \begin{bmatrix} 1 &0 \\ 1 & 0 \end{bmatrix}\right ) \begin{Bmatrix} x_1\\ x_2 \end{Bmatrix}=\begin{Bmatrix} 0\\ 0 \end{Bmatrix} $$

$$ det\begin{bmatrix} 30-\gamma & -20\\ -20& 35-\gamma \end{bmatrix}=\gamma ^2-65\gamma -650=0 $$

Eigenvalues:

$$ \gamma_1=12.34 $$

$$ \gamma_2=52.66 $$

Eigenvectors:

$$ \gamma _1=12.34 $$

$$ \mathbf{x_1}=\begin{bmatrix} -0.7497\\ -0.6618 \end{bmatrix} $$

$$ \gamma _2=-52.66 $$

$$ \mathbf{x_2}=\begin{bmatrix} 0.6618\\ -0.7497 \end{bmatrix} $$

Honor Pledge
On our honor, this problem was solved on our own with reference to Dr. Vu-Quoc's Fead.s13.sec53b lecture notes located here and our team's solution to 4.4.

Given: A 10-bar truss system


Given: $$ A= 0.5 $$ $$ E= 5 $$ $$ \rho = 2 $$ $$ L_{12}=L_{24}=L_{46}=1 $$ $$ L_{23}=L_{45}=1$$

Solve the generalized eigenvalue problem
Solve the generalized eigenvalue problem for the above truss and display the lowest 3 eigenpairs.

Solution


%%Problem 5.2 %%Given A=.5; E=5; rho=2;

m1=(rho*A*1)/2; %%mass of vertical and horizontal m2=(rho*A*sqrt(2))/2; %%mass of diagonals

%Stiffness k=[0.353553391	0.353553391	1	0	-1.353553391	-0.353553391	0	0	0	0	0	0; 0.353553391	1.353553391	0	-1	-0.353553391	-0.353553391	0	0	0	0	0	0;   0	0	-0.646446609	0.353553391	1	0	0	0	-0.353553391	-0.353553391	0	0; 0	-1	0.353553391	2.353553391	0	0	0	-1	-0.353553391	-0.353553391	0	0; -0.353553391	-0.353553391	0	0	0.707106781	0.707106781	-0.353553391	-0.353553391	0	0	0	0; -0.353553391	-0.353553391	0	0	0.707106781	1.707106781	-0.353553391	-0.353553391	0	-1	0	0;    0	0	0	0	-0.353553391	-0.353553391	1.353553391	0.353553391	-1	0	0	0; 0	0	0	-1	-0.353553391	-0.353553391	0.353553391	2.353553391	0	0	0	-1; 0	0	-0.353553391	-0.353553391	0	0	-1	0	1.707106781	0.707106781	-0.353553391	-0.353553391; 0	0	-0.353553391	-0.353553391	0	-1	0	0	0.707106781	1.707106781	-0.353553391	-0.353553391; 0	0	0	0	0	0	0	0	-0.353553391	-0.353553391	0.353553391	0.353553391; 0	0	0	0	0	0	0	-1	-0.353553391	-0.353553391	0.353553391	1.353553391];

%lumped mass M=[(m1+m2) 0 0 0 0 0 0 0 0 0 0 0; 0 (m1+m2) 0 0 0 0 0 0 0 0 0 0; 0 0 (3*m1+m2) 0 0 0 0 0 0 0 0 0; 0 0 0 (3*m1+m2) 0 0 0 0 0 0 0 0; 0 0 0 0 (2*m1+2*m2) 0 0 0 0 0 0 0; 0 0 0 0 0 (2*m1+2*m2) 0 0 0 0 0 0; 0 0 0 0 0 0 (3*m1+m2) 0 0 0 0 0; 0 0 0 0 0 0 0 (3*m1+m2) 0 0 0 0; 0 0 0 0 0 0 0 0 (2*m1+2*m2) 0 0 0; 0 0 0 0 0 0 0 0 0 (2*m1+2*m2) 0 0; 0 0 0 0 0 0 0 0 0 0 (m1+m2) 0; 0 0 0 0 0 0 0 0 0 0 0 (m1+m2)];

%eigenvalues(D) and eigenvectors(V) [V D]=eig(k,M)

%position p=[0 1 1 2 2 3; 0 0 1 0 1 0];

%nodes n=[1 1 2 3 3 2 2 4 5 4; 2 3 3 5 4 5 4 5 6 6];

%1st deformation for i=1:2:11 x((i+1)/2)=p(1,((i+1)/2))+D(1,1)*V(i,1); end for i=2:2:12 y((i/2))=p(2,(i/2))+D(1,1)*V(i,1); end %plot the truss system for i=1:10 p1=n(1,i); p2=n(2,i); xc=[x(p1),x(p2)]; yc=[y(p1),y(p2)]; axis([0 2 -1 1]) plot(xc,yc,'b') hold on end

MATLab Output

>> Report5_2

V =

Columns 1 through 9

0.2973   1.0000    0.3448   -0.7095   -0.3655    1.0000   -1.0000    0.8201   -1.0000    1.0000   -0.1361    0.9860    0.2081    0.3629    0.8935    0.7835   -0.4606    0.3840   -0.1510   -0.6098   -0.0356    0.1009   -0.1029    0.1064    0.2450   -0.8041   -0.2087   -0.9179    0.0212   -0.3869   -0.7796   -0.3931    0.6772   -0.1592   -0.0889    0.2193   -0.2409    0.1539   -0.1437    0.4407    0.2957   -0.2120    0.5372   -0.8793   -0.1879   -0.3957    0.0597   -0.1436    1.0000    0.2129    0.2962    0.3782    0.6422   -0.1079    0.0721   -0.0027    0.0826   -0.4463   -0.9254   -0.3554    0.8254    1.0000    0.2381    0.9021    0.0305   -0.3848    0.2221   -0.7068   -0.2207   -0.8824   -0.0983   -0.0711    0.2986   -0.0710   -0.2206    0.1456    1.0000    0.0337   -0.5634    0.7647    0.3433    0.4099   -0.0408   -0.1438   -0.6182    0.4756   -0.6776    0.5985   -0.4195   -0.1293   -0.2686   -0.0461    0.2980    0.1683   -0.3457   -0.1880   -0.9047   -0.9492    0.6353   -0.9996    0.0038    1.0000    0.0477    0.2713   -0.9654   -0.8322    0.3576   -0.1804

Columns 10 through 12

1.0000   1.0000   -1.0000   -0.3675    1.0000    1.0000   -0.0991    1.0000   -1.0000   -0.1877    1.0000    1.0000   -0.0337    1.0000   -1.0000    0.2448    1.0000    1.0000   -0.1345    1.0000   -1.0000   -0.0598    1.0000    1.0000   -0.2573    1.0000   -1.0000    0.1983    1.0000    1.0000    0.0092    1.0000   -1.0000   -0.0661    1.0000    1.0000

D =

Columns 1 through 9

2.1553        0         0         0         0         0         0         0         0         0   -0.4421         0         0         0         0         0         0         0         0         0    1.6342         0         0         0         0         0         0         0         0         0    1.1985         0         0         0         0         0         0         0         0         0    1.3133         0         0         0         0         0         0         0         0         0    0.7937         0         0         0         0         0         0         0         0         0    0.5736         0         0         0         0         0         0         0         0         0    0.2890         0         0         0         0         0         0         0         0         0    0.1110         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0

Columns 10 through 12

0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    0.0693         0         0         0   -0.0000         0         0         0    0.0000

Honor Pledge
On our honor, this problem was solved on our own with reference to Dr. Vu-Quoc's Fead.s13.sec53b lecture notes.

Problem Statement
Let there be a multiple degrees-of-freedom system as seen in Figure 1



Find the eigenvectors
1. Find the eigenvectors such that
 * {| style="width:100%" border="0"

$$  \displaystyle x_{11}=x_{12}=1 $$
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 * }
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in the matrix below
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$$  \displaystyle \begin{bmatrix} \mathbf x_{1} & \mathbf x_{2} \end{bmatrix} = \begin{bmatrix} x_{ij} \end{bmatrix}=\begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix} $$
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Plot the eigenvectors
2. Plot the eigenvectors and compare the mode shapes where
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$$  \displaystyle x_{21}=x_{22}=1 $$ And comment on the eigenvectors in both cases
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Create an animation of the mode shapes
3. Create an animation for each mode shape using the gif format

Part 1: Find eigenvector with given assumption
In the given two-dof system:
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$$  \displaystyle K=\begin{bmatrix} 3 & -2\\ -2 & 5 \end{bmatrix} $$      (3.1)
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 * 
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$$  \displaystyle \gamma_{1,2}=4 \mp \sqrt{5} $$      (3.2)
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 * 
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The eigenvectors can be obtained using
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$$  \displaystyle [\mathbf K - \gamma \mathbf I]\mathbf x = \mathbf 0 $$      (3.3)
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 * 
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Assuming
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$$  \displaystyle x_{11}=x_{12}=1 $$      (3.4) The matrix becomes
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 * 
 * }
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$$  \displaystyle \left [ K-\gamma _{1}I \right ]X=\begin{bmatrix} -1+\sqrt{5} &-2 \\ -2 & 1+\sqrt{5} \end{bmatrix}\begin{Bmatrix} 1\\ x_{2} \end{Bmatrix}=\begin{Bmatrix} 0\\ 0 \end{Bmatrix} $$      (3.5) and
 * style="width:95%" |
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 * 
 * }
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$$  \displaystyle \left [ K-\gamma _{2}I \right ]X=\begin{bmatrix} -1-\sqrt{5} &-2 \\ -2 & 1-\sqrt{5} \end{bmatrix}\begin{Bmatrix} 1\\ x_{2} \end{Bmatrix}=\begin{Bmatrix} 0\\ 0 \end{Bmatrix} $$      (3.6)
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Therefore the equations become
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$$  \displaystyle (-1+\sqrt{5})-2x_{2}=0 $$      (3.7)
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$$  \displaystyle (-1-\sqrt{5})-2x_{2}=0 $$      (3.8)
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 * 
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$$  \displaystyle x_{21}=(-1+\sqrt{5})/2 $$      (3.9)
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 * 
 * }
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$$  \displaystyle x_{22}=(-1-\sqrt{5})/2 $$      (3.10) The eigenvector matrix becomes
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 * 
 * }
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$$  \displaystyle X = \begin{Bmatrix} 1 & 1\\ (-1+\sqrt{5})/2 & (-1-\sqrt{5})/2 \end{Bmatrix} $$      (3.11)
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Part 2: Plot eigenvectors and compare

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$$  \displaystyle x_{21}=x_{22}=1 $$      (3.12) The matrix becomes
 * style="width:95%" |
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 * 
 * }
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$$  \displaystyle \left [ K-\gamma _{1}I \right ]X=\begin{bmatrix} -1+\sqrt{5} &-2 \\ -2 & 1+\sqrt{5} \end{bmatrix}\begin{Bmatrix} x_{1}\\ 1 \end{Bmatrix}=\begin{Bmatrix} 0\\ 0 \end{Bmatrix} $$      (3.13) and
 * style="width:95%" |
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 * 
 * }
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$$  \displaystyle \left [ K-\gamma _{2}I \right ]X=\begin{bmatrix} -1-\sqrt{5} &-2 \\ -2 & 1-\sqrt{5} \end{bmatrix}\begin{Bmatrix} x_{1}\\ 1 \end{Bmatrix}=\begin{Bmatrix} 0\\ 0 \end{Bmatrix} $$      (3.14)
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 * 
 * }

Therefore the equations become
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$$  \displaystyle (-1+\sqrt{5})x_{1}-2=0 $$      (3.15)
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 * }
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$$  \displaystyle (-1-\sqrt{5})x_{1}-2=0 $$      (3.16)
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 * 
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$$  \displaystyle x_{11}=2/(-1+\sqrt{5}) $$      (3.17)
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 * }
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$$  \displaystyle x_{12}=2/(-1-\sqrt{5}) $$     (3.18) The eigenvector matrix becomes
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 * }
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$$  \displaystyle X = \begin{Bmatrix} 2/(-1+\sqrt{5}) & 2/(-1-\sqrt{5})\\ 1 & 1 \end{Bmatrix} $$     (3.19)
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 * <p style="text-align:right">
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The plot of the first mode

The plot of the second mode

When both modes are compared on the same graph

The eigenvectors are exactly the same when plotted against each other. Mode11 and Mode12 are both apart of Mode1, and Mode21 and Mode22 are both apart of Mode2.

Part 3: Animation
Using the following animation equation from Fead.s13.sec53b notes:


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$$  \displaystyle y = \mathbf X_i sin(\omega_it) $$     (3.20)
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 * }


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$$  \displaystyle \omega_i = \sqrt{\gamma_i} $$     (3.21)
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$$  \displaystyle T_i=1/\omega_i $$     (3.22)
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We arrive at the following:
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$$  \displaystyle \omega_1 = 1.328 rad/s $$     (3.23)
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$$  \displaystyle \omega_2 = 2.497 rad/s $$     (3.24)
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$$  \displaystyle T_1 = 0.7529 s^-1 $$     (3.25)
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$$  \displaystyle T_2 = 0.4004 s^-1 $$     (3.26)
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 * }

Graphs and Animations
The shape of Mode 1 looks like

The shape of Mode 2 looks like

Mode 1 gif animation located here http://makeagif.com/i/_nsr0o

Mode 2 gif animation located here http://makeagif.com/i/OdD8T9

Honor Pledge
On our honor, we referenced problem R3.6 from EML4507.s13.team2/Repor4

Problem Statement
Given: Material information for a plane truss Continuation of R3.6, involving the plane truss shown below.



The mass density of the material is given to be $$ \rho = 5,000 \ kg/m^3 $$

Find the lumped mass matrix, the 3 lowest eigenpairs, and plot the 3 lowest mode shapes.

MATLAB Code
Readout of eigenvalues and eigenvectors

D =

Columns 1 through 7

-3.4909e-008           0            0            0            0            0            0 0 3.1099e-008            0            0            0            0            0 0           0  8.9492e-008            0            0            0            0 0           0            0  4.0921e+006            0            0            0 0           0            0            0  1.3605e+007            0            0 0           0            0            0            0  2.4654e+007            0 0           0            0            0            0            0  2.8135e+007 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0

Columns 8 through 14

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0   4.118e+007            0            0            0            0            0            0 0 5.3747e+007            0            0            0            0            0 0           0  6.0392e+007            0            0            0            0 0           0            0  7.5968e+007            0            0            0 0           0            0            0  1.0563e+008            0            0 0           0            0            0            0  1.3332e+008            0 0           0            0            0            0            0  1.9204e+008 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0

Columns 15 through 21

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0  1.9856e+008            0            0            0            0            0            0 0 2.0526e+008            0            0            0            0            0 0           0  2.2958e+008            0            0            0            0 0           0            0  2.5299e+008            0            0            0 0           0            0            0  2.8646e+008            0            0 0           0            0            0            0  2.9447e+008            0 0           0            0            0            0            0  3.2176e+008 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0

Columns 22 through 28

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0  3.3174e+008            0            0            0            0            0            0 0 3.6902e+008            0            0            0            0            0 0           0  3.7378e+008            0            0            0            0 0           0            0  3.8857e+008            0            0            0 0           0            0            0  4.1168e+008            0            0 0           0            0            0            0  4.5178e+008            0 0           0            0            0            0            0  4.7778e+008

V =

Columns 1 through 6

0.092063     0.42067      0.27171     -0.24191      0.20606      0.61593      0.73897     -0.51572      0.17906      0.74673     -0.51818     -0.14572      0.19592      0.22885       0.4105      0.20048     -0.36981      0.72724      0.73897     -0.51572      0.17906      0.76074     -0.55198     -0.16391     0.092063      0.42067      0.27171     -0.19682     0.096574       0.4535      0.63512      -0.3239     0.040281      0.15633      0.33341     0.010399      0.19592      0.22885       0.4105      0.19678     -0.34716      0.64656      0.63512      -0.3239     0.040281      0.20202      0.28381    -0.046857     0.092063      0.42067      0.27171    -0.091692    -0.030513      0.12531      0.53127     -0.13209    -0.098503     -0.37374      0.55159      0.17953      0.19592      0.22885       0.4105      0.14761     -0.18966      0.45334      0.53127     -0.13209    -0.098503     -0.33532      0.63251       0.1653     0.092063      0.42067      0.27171     0.040395    0.0019793     -0.20383      0.42741     0.059727     -0.23729     -0.57153     -0.08336    -0.016936      0.19592      0.22885       0.4105     0.040395   -0.0019793      0.20383      0.42741     0.059727     -0.23729     -0.57153      0.08336     0.016936     0.092063      0.42067      0.27171      0.14761      0.18966     -0.45334      0.32356      0.25154     -0.37607     -0.33532     -0.63251      -0.1653      0.19592      0.22885       0.4105    -0.091692     0.030513     -0.12531      0.32356      0.25154     -0.37607     -0.37374     -0.55159     -0.17953     0.092063      0.42067      0.27171      0.19678      0.34716     -0.64656      0.21971      0.44336     -0.51485      0.20202     -0.28381     0.046857      0.19592      0.22885       0.4105     -0.19682    -0.096574      -0.4535      0.21971      0.44336     -0.51485      0.15633     -0.33341    -0.010399     0.092063      0.42067      0.27171      0.20048      0.36981     -0.72724      0.11585      0.63517     -0.65364      0.76074      0.55198      0.16391      0.19592      0.22885       0.4105     -0.24191     -0.20606     -0.61593      0.11585      0.63517     -0.65364      0.74673      0.51818      0.14572

Columns 7 through 12

0.27807     0.42384      0.15054     -0.34098    -0.059514      0.83481     -0.41608      -0.1569     -0.07994     -0.41932     -0.32303       0.1441     -0.29109      -0.1631     -0.25248      0.22739      0.87401      0.22501     -0.47639     -0.19259     -0.10544      -0.5758     -0.49083      0.27464     0.067727      0.20444     0.029876      -0.5338     -0.38109      0.40489       0.5402      0.46425      0.25781       0.3778      0.36859     -0.25536     -0.25424     -0.13287     -0.19142      0.16559      0.57522      0.11806      0.54119      0.64115       0.4275     0.066927     0.050024      0.16092   -0.0096049       0.2682      0.20058     -0.49071     -0.45559     -0.30141    -0.026362     -0.60658     -0.59257    -0.025052     0.016698     -0.15281    0.0039061     0.037019     0.030338      0.35549      0.19874     -0.36021      0.17428     -0.40575     -0.61188     -0.27316    -0.096797     0.090029      0.10902      0.17501     -0.07142     -0.41642     -0.28724     -0.60897     -0.69164     0.029703      0.62875      0.18793      0.13372    -0.078668      0.10902     -0.17501     -0.07142      0.41642     -0.28724     -0.60897     -0.69164    -0.029703      0.62875     -0.18793      0.13372    -0.078668    0.0039061    -0.037019     0.030338     -0.35549      0.19874     -0.36021      0.17428      0.40575     -0.61188      0.27316    -0.096797     0.090029   -0.0096049      -0.2682      0.20058      0.49071     -0.45559     -0.30141    -0.026362      0.60658     -0.59257     0.025052     0.016698     -0.15281     -0.25424      0.13287     -0.19142     -0.16559      0.57522      0.11806      0.54119     -0.64115       0.4275    -0.066927     0.050024      0.16092     0.067727     -0.20444     0.029876       0.5338     -0.38109      0.40489       0.5402     -0.46425      0.25781      -0.3778      0.36859     -0.25536     -0.29109       0.1631     -0.25248     -0.22739      0.87401      0.22501     -0.47639      0.19259     -0.10544       0.5758     -0.49083      0.27464      0.27807     -0.42384      0.15054      0.34098    -0.059514      0.83481     -0.41608       0.1569     -0.07994      0.41932     -0.32303       0.1441

Columns 13 through 18

0.39472     0.58017     -0.36804    -0.029885       0.2574     -0.63744      0.11336    0.0068501   -0.0082754    0.0070421    -0.010763     0.065714     -0.86377      0.52241      0.49337    -0.011226     -0.76821      0.35892      0.28335     0.050438    -0.077702     0.092251      0.32517     -0.47469      0.27656     -0.22204      0.11128      0.11355       0.1204      0.18868      -0.3005     -0.38416      0.27334      0.20946       0.5475     -0.56333     -0.34557      0.07095     0.052545  -0.00085696     0.025426    -0.049686     0.044478       0.3642      -0.1968      -0.2944     -0.67527      0.64524     -0.26072     -0.58511      0.44006    -0.051352     -0.26552      0.33385       0.1237     0.057759     -0.65173     -0.57274     -0.21933    0.0083882      0.34411     -0.56953     -0.40985    -0.085054      0.44413     0.079253     -0.21839     -0.08089      0.59958      0.62543      0.15132      0.13121     -0.63107      0.13935     -0.13291    -0.080994     -0.17547     -0.21593      0.26379      0.30744     0.017204      0.79156     0.056774      0.28071      0.63107     -0.13935     -0.13291     0.080994     -0.17547      0.21593     -0.26379     -0.30744     0.017204     -0.79156     0.056774     -0.28071     -0.34411      0.56953     -0.40985     0.085054      0.44413    -0.079253      0.21839      0.08089      0.59958     -0.62543      0.15132     -0.13121      0.26072      0.58511      0.44006     0.051352     -0.26552     -0.33385      -0.1237    -0.057759     -0.65173      0.57274     -0.21933   -0.0083882      0.34557     -0.07095     0.052545   0.00085696     0.025426     0.049686    -0.044478      -0.3642      -0.1968       0.2944     -0.67527     -0.64524     -0.27656      0.22204      0.11128     -0.11355       0.1204     -0.18868       0.3005      0.38416      0.27334     -0.20946       0.5475      0.56333      0.86377     -0.52241      0.49337     0.011226     -0.76821     -0.35892     -0.28335    -0.050438    -0.077702    -0.092251      0.32517      0.47469     -0.39472     -0.58017     -0.36804     0.029885       0.2574      0.63744     -0.11336   -0.0068501   -0.0082754   -0.0070421    -0.010763    -0.065714

Columns 19 through 24

0.34468   -0.083395     -0.59515     0.056167     -0.24865     0.055097     -0.11916    -0.021461      0.16666      0.22032     -0.70663     -0.76713       0.4231      0.13759     0.067272     -0.89652      0.63921     0.036916      0.41221     0.066013     -0.37208     -0.44706       1.0697       1.1248     -0.14467      0.14418      0.74912     -0.19288      0.22938     -0.41394      0.49371       0.2887      0.21402     -0.20211    -0.062593     -0.21076     -0.12231    -0.044732    -0.030132      0.44183     -0.42226    -0.025178     -0.41345     -0.17022    -0.039726      0.13422       0.1086       0.1923      0.27508       0.3355      0.12963       -0.136      -0.1915      0.27435      0.36828      0.61944      0.33657     0.084389    -0.050219     0.023178     -0.58889     -0.13515     0.095554      0.43035      0.29075      0.31703     -0.46923      -0.5221     -0.34922      0.10684     0.040685      0.13132      0.12248      0.21572     -0.51433      0.66948   -0.0036594      0.13832    -0.073913      0.72472     -0.12197      0.23208     0.025416       0.1134      0.12248     -0.21572     -0.51433     -0.66948   -0.0036594     -0.13832    -0.073913     -0.72472     -0.12197     -0.23208     0.025416      -0.1134     -0.58889      0.13515     0.095554     -0.43035      0.29075     -0.31703     -0.46923       0.5221     -0.34922     -0.10684     0.040685     -0.13132      0.27508      -0.3355      0.12963        0.136      -0.1915     -0.27435      0.36828     -0.61944      0.33657    -0.084389    -0.050219    -0.023178     -0.12231     0.044732    -0.030132     -0.44183     -0.42226     0.025178     -0.41345      0.17022    -0.039726     -0.13422       0.1086      -0.1923     -0.14467     -0.14418      0.74912      0.19288      0.22938      0.41394      0.49371      -0.2887      0.21402      0.20211    -0.062593      0.21076       0.4231     -0.13759     0.067272      0.89652      0.63921    -0.036916      0.41221    -0.066013     -0.37208      0.44706       1.0697      -1.1248      0.34468     0.083395     -0.59515    -0.056167     -0.24865    -0.055097     -0.11916     0.021461      0.16666     -0.22032     -0.70663      0.76713

Columns 25 through 28

0.41508     0.13022     -0.36482       0.3124      0.24871      0.51075     -0.37982      0.21327     -0.33188      0.69784     -0.70038      0.41825     -0.33224     -0.59907      0.36768     -0.18545     -0.66249      0.22382      0.33064     -0.45015    -0.043417     0.092746      0.06165    -0.085966      0.24844     -0.59495      0.72349     -0.48099    -0.041278     -0.19366      0.14598    -0.079588      0.65126     -0.50493    -0.096539      0.52184     0.012598     -0.13784    -0.011063     0.096382     -0.29148     0.039471      -0.5283      0.51818     0.053448     0.014578     -0.10034     0.081482     -0.45892      0.41969     -0.22946     -0.54001      0.04812      0.10057    -0.055857    -0.093242      0.45892      0.41969      0.22946     -0.54001     -0.04812      0.10057     0.055857    -0.093242      0.29148     0.039471       0.5283      0.51818    -0.053448     0.014578      0.10034     0.081482     -0.65126     -0.50493     0.096539      0.52184    -0.012598     -0.13784     0.011063     0.096382     -0.24844     -0.59495     -0.72349     -0.48099     0.041278     -0.19366     -0.14598    -0.079588      0.66249      0.22382     -0.33064     -0.45015     0.043417     0.092746     -0.06165    -0.085966      0.33188      0.69784      0.70038      0.41825      0.33224     -0.59907     -0.36768     -0.18545     -0.41508      0.13022      0.36482       0.3124     -0.24871      0.51075      0.37982      0.21327

Honor Pledge
On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions. MATLAB code was based on that from our own Problem 3.1 located here.

Given: A 2-Bar Truss System
The following 2-bar truss system is given in FEAD.F08 notes on p.5-4 and p.11-3.



The properties of each element are given in the table below.

Plot the eigenvectors
Plot the eigenvectors corresponding to the zero eigenvalues of the 2-bar truss system.

Given: A 4-Bar Truss System
The following 4-bar truss system including support is given in FEAD.F08 notes on p.21-3.



The properties of each element are given in the table below.

Plot the eigenvectors
Plot the eigenvectors corresponding to the zero eigenvalues for the truss system.

Finding the eigenvectors
The following MATLAB code is used to find the global stiffness matrix and solve for the eigenvalues and eigenvectors of the 2-bar truss.

The results of the code are given below, where K is the stiffness matrix, V is a matrix of column eigenvectors, and D is a diagonal matrix of the corresponding eigenvalues.

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 V = -0.1322   0.3884   -0.6488    0.1712    0.6174   -0.0139   -0.0614   -0.9125   -0.1730    0.0809    0.3565   -0.0080   -0.4668   -0.0888   -0.4571    0.1009   -0.5409    0.5123    0.5182   -0.0860   -0.5050    0.2026   -0.4329   -0.4904   -0.4328   -0.0263   -0.1808   -0.7246   -0.0765   -0.4984    0.5522   -0.0235   -0.2287   -0.6228    0.0765    0.4984 D = -0.0000        0         0         0         0         0         0   -0.0000         0         0         0         0         0         0    0.0000         0         0         0         0         0         0    0.0000         0         0         0         0         0         0    1.4705         0         0         0         0         0         0   10.0295

Plotting the eigenvectors
Looking at the matrix D, the first 4 columns contain zero eigenvalues. Hence, the first 4 columns of the matrix V are the eigenvectors which correspond to zero eigenvalues. These eigenvectors are plotted with the following MATLAB code, repeated for each eigenvector.

The plotted eigenvectors are shown below.









Finding the eigenvectors
The following MATLAB code is used to find the global stiffness matrix and solve for the eigenvalues and eigenvectors of the 4-bar truss with support.

The results of the code are given below, where K is the stiffness matrix, V is a matrix of column eigenvectors, and D is a diagonal matrix of the corresponding eigenvalues. K = Columns 1 through 6 2.1213   2.1213         0         0   -2.1213   -2.1213    2.1213    8.1213         0   -6.0000   -2.1213   -2.1213         0         0    6.0000         0   -6.0000         0         0   -6.0000         0    6.0000         0         0   -2.1213   -2.1213   -6.0000         0    8.1213    2.1213   -2.1213   -2.1213         0         0    2.1213    8.1213         0         0         0         0         0         0         0         0         0         0         0   -6.0000  Columns 7 through 8 0        0         0         0         0         0         0         0         0         0         0   -6.0000         0         0         0    6.0000 V = Columns 1 through 6 0.2910  -0.6675    0.0645    0.2562   -0.5948    0.0000    0.1575    0.4183    0.4471    0.0177   -0.1586   -0.5000   -0.1462   -0.2866    0.5408    0.0639    0.4362    0.5000    0.1575    0.4183    0.4471    0.0177   -0.4362    0.5000   -0.1462   -0.2866    0.5408    0.0639    0.1586   -0.5000    0.5947    0.0374   -0.0292    0.2101    0.1586   -0.0000    0.3402   -0.1935    0.0975   -0.9150    0.0000   -0.0000    0.5947    0.0374   -0.0292    0.2101    0.4362   -0.0000  Columns 7 through 8 -0.0000   0.2150   -0.2887    0.4914   -0.2887    0.2764    0.2887   -0.2764    0.2887   -0.4914   -0.5774   -0.4914    0.0000   -0.0000    0.5774    0.2764 D = Columns 1 through 6 -0.0000        0         0         0         0         0         0   -0.0000         0         0         0         0         0         0   -0.0000         0         0         0         0         0         0    0.0000         0         0         0         0         0         0    3.8183         0         0         0         0         0         0   12.0000         0         0         0         0         0         0         0         0         0         0         0         0  Columns 7 through 8 0        0         0         0         0         0         0         0         0         0         0         0   12.0000         0         0   16.6670

Plotting the eigenvectors
Looking at the matrix D, the first 4 columns contain zero eigenvalues. Hence, the first 4 columns of the matrix V are the eigenvectors which correspond to zero eigenvalues. These eigenvectors are plotted with the following MATLAB code, repeated for each eigenvector.

The plotted eigenvectors are shown below.









Honor Pledge
On our honor, this problem was solved on our own with reference to Dr. Vu-Quoc's Fead.s13.sec53b lecture notes and the following websites: ibk and engineering

Problem Statement
Description: We have a two degrees of freedom spring-mass-damper system as shown in the following figure. Note: This is the same given problem setup for problem 3.3 from our team's Report 3([]) The displacement (d), force (F), mass (M), damping (C), and stiffness (K) matrices are given to be:


 * {| style="width:100%" border="0"

$$  \displaystyle d = \left[\begin{array}{cc} d_{1} \\ d_{2} \\ \end{array} \right] $$     (6.0)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$  \displaystyle F = \left[\begin{array}{cc} F_{1} \\ F_{2} \\ \end{array} \right] $$     (6.1)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$  \displaystyle M = \left[\begin{array}{cc} m_{1} & 0 \\ 0 & m_{2} \\ \end{array} \right] $$     (6.2)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$  \displaystyle C = \left[\begin{array}{cc} (c_{1}+c_{2}) & -c_{2} \\ -c_{2} & (c_{2}+c_{3}) \\ \end{array} \right] $$     (6.3)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$  \displaystyle K = \left[\begin{array}{cc} (k_{1}+k_{2}) & -k_{2} \\ -k_{2} & (k_{2}+k_{3}) \\ \end{array} \right] $$     (6.4)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

We are given the following numerical values:
 * {| style="width:100%" border="0"

$$  \displaystyle m_{1}=3, m_{2}=2 $$     (6.5)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$  \displaystyle c_{1}=\frac{1}{2}, c_{2}=\frac{1}{4}, c_{3}=\frac{1}{3} $$     (6.6)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$  \displaystyle k_{1}=10, k_{2}=20, k_{3}=15 $$     (6.7)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$  \displaystyle F_{1}(t)=0, F_{2}(t)=0 $$     (6.8)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$  \displaystyle d_{1}(0)=-1, d_{2}(0)=2, d'_{1}(0)=0, d'_{2}(0)=0 $$     (6.9)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

Objective: Solve this problem using the modal superposition method.

Solution
Method: The modal superposition (MS) method is a method of solving the general equation of motion:


 * {| style="width:100%" border="0"

$$  \displaystyle M\ddot{U}+C\dot{U}+KU=F(t) $$     (6.10)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

The basic concept is to use free vibrations mode shapes to uncouple equation 6.10, the general equation of motion, and to put these uncoupled equations in terms of "modal coordinates." We can then solve each of these modal coordinate equations independently and then use the superposition of each of these equations to give the solution to our original equation of motion. Hence the name, modal superposition.

Starting with the general equation of motion shown in equation 6.10, we make a transformation using matrices P and X, where P is a square matrix of order n (n being the number of degrees of freedom in the system) and X is also of order n and is time dependent.
 * {| style="width:100%" border="0"

$$  \displaystyle U=PX $$     (6.11)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

We then substitute this into equation 6.10 to get
 * {| style="width:100%" border="0"

$$  \displaystyle P^{T}MP\ddot{X}+P^{T}CP\dot{X}+P^{T}KPX=P^{T}F $$     (6.12)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

Neglecting the damping (i.e., letting C=0) in order to simplify the problem, the solution to equation 6.10 becomes
 * {| style="width:100%" border="0"

$$  \displaystyle U=\Phi sin\omega (t-t_{o}) $$     (6.13) where ɸ is the eigenvector of order n, t is the time variable, to is the time constant, and ω is the frequency of vibration. Taking the second derivative yields
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \ddot{U}=-\omega^{2}\Phi sin\omega (t-t_{o}) $$     (6.14)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

And substituting into equation 6.12, we get
 * {| style="width:100%" border="0"

$$  \displaystyle K\Phi -\omega^{2}M\Phi=F $$     (6.15)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

Using the mass orthogonality of the eigenvectors, we can write
 * {| style="width:100%" border="0"

$$  \displaystyle \Phi^{T}K\Phi =\Omega^{2} $$     (6.16) and
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \Phi^{T}M\Phi =I $$     (6.17)
 * style="width:95%" |
 * style="width:95%" |
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Finally, using the relations of P = ɸ, plugging this into equation 6.11, and substituting into 6.12, we get the new equilibrium equation of :{| style="width:100%" border="0" $$  \displaystyle \ddot{X}+\Phi^{T}C\Phi\dot{X}+\Omega^{2}X=\Phi^{T}F $$     (6.18) Note: If C=0 (the simplified case), then the second term in equation 6.18 will go to zero.
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Honor Pledge
On our honor this problem was attempted alone using no outside sources other than the class notes located here.

Problem Statement
Consider the truss system shown above under free vibration with an initial displacement of F=5. Using the first 3 lowest modes, solve for the motion of the truss by modal superposition. Plot the time history for the vertical displacement of node 2 over 5 periods, i.e. 0<=t<=T=5T

Solution
By method of inspection, we know the following,
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$$  \displaystyle M = \left[\begin{array}{cc} m_{1} & 0 \\ 0 & m_{2} \\ \end{array} \right] $$     (7.0)
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$$  \displaystyle C = \left[\begin{array}{cc} (c_{1}+c_{2}) & -c_{2} \\ -c_{2} & (c_{2}+c_{3}) \\ \end{array} \right] $$     (7.1)
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$$  \displaystyle K = \left[\begin{array}{cc} (k_{1}+k_{2}) & -k_{2} \\ -k_{2} & (k_{2}+k_{3}) \\ \end{array} \right] $$     (7.2) From the problem statement we know that:
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$$  \displaystyle m_{1}=3, m_{2}=2 $$     (7.3)
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$$  \displaystyle c_{1}=\frac{1}{2}, c_{2}=\frac{1}{4}, c_{3}=\frac{1}{3} $$     (7.4)
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$$  \displaystyle k_{1}=10, k_{2}=20, k_{3}=15 $$     (7.5) Plugging 7.3-7.5 into equations 7.0-7.2 yields:
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$$  \displaystyle M = \left[\begin{array}{cc} 3 & 0 \\ 0 & 2 \\ \end{array} \right] $$     (7.6)
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$$  \displaystyle C = \left[\begin{array}{cc} 3/4 & -1/4 \\ -1/4 & 7/12 \\ \end{array} \right] $$     (7.7)
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$$  \displaystyle K = \left[\begin{array}{cc} 30 & -20 \\ -20 & 35 \\ \end{array} \right] $$     (7.8) The modal superposition method states:
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$$  \displaystyle \ddot{z}(t)+ \bar \Phi^T C \bar \Phi \dot{z}(t) + \Omega^2 z(t) = \bar \Phi^T R(t) $$     (7.9) By mass-orthogonality we know:
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$$  \displaystyle \overline{\phi} _{i}^{T}M\overline{\phi}_j=\delta _{ij} $$     (7.10) we also know that:
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$$  \displaystyle d(t)=\sum _{j=1}^{n}=z_j(t)\overline{\phi}_j $$     (7.11)
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