User:Eml4507.s13.team3.steiner/Team Negative Damping (3): Report 5

=Problem 5.1= On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
$$ m_{1}=3 $$

$$ m_{2}=2 $$

$$ k_{1}=10 $$

$$ k_{2}=20 $$

$$ k_{3}=15 $$

Find
Solve by hand the gen. eigenvalue problem for the spring-mass-damper system on p.53-13, using the data for the masses in (2) p.53-13b, and the data for the stiffness coefficients in (4) p.53-13b.

Solution
First plug given values into the stiffness matrix:

$$ K= \begin{bmatrix} (k_{1}+k_{2}) & -k_{2}\\ -k_{2} & (k_{2}+k_{3})\\ \end{bmatrix}$$

$$ K= \begin{bmatrix} (10+20) & -20\\ -20 & (20+15)\\ \end{bmatrix}$$

$$ K= \begin{bmatrix} 30 & -20\\ -20 & 35\\ \end{bmatrix}$$

Next plug stiffness matrix in equation below:

$$[K-\gamma_I]x= 0$$

Take the determinant:

$$det \begin{Bmatrix} 30-\gamma & -20\\ -20 & 35-\gamma \end{Bmatrix} =[(30-\gamma)(35-\gamma)]-(-20)(-20)= 0$$

Simplifying:

$$1050-30\gamma-35\gamma+\gamma^2-400$$

$$\gamma^2-65\gamma+650=0$$

Using the Quadratic formula to solve:

$$\gamma_1=\frac{65+\sqrt{65^2-(4)(650)}}{2}$$

$$\gamma_2=\frac{65-\sqrt{65^2-(4)(650)}}{2}$$

Final gamma values:

$$\gamma_1=52.66$$

$$\gamma_2=12.34$$

To find the eigenvectors, first we:

$$[K-\gamma_I]x= [\begin{bmatrix} 30 & -20\\ -20 & 35\\ \end{bmatrix} - \gamma* \begin{bmatrix} 1 & 0\\ 0 & 1\\ \end{bmatrix}] \begin{bmatrix} x_{1}\\ x_{2}\\ \end{bmatrix} = 0 $$

To solve, lets set $$ x_{1} $$ equal to 1:

$$[K-\gamma_I]x= [\begin{bmatrix} 30 & -20\\ -20 & 35\\ \end{bmatrix} - 52.66* \begin{bmatrix} 1 & 0\\ 0 & 1\\ \end{bmatrix}] \begin{bmatrix} 1\\ x_{2}\\ \end{bmatrix} = 0 $$

$$[K-\gamma_I]x= [\begin{bmatrix} 30 & -20\\ -20 & 35\\ \end{bmatrix} + \begin{bmatrix} -52.66 & 0\\ 0 & -52.66\\ \end{bmatrix}] \begin{bmatrix} 1\\ x_{2}\\ \end{bmatrix} = 0 $$

$$[K-\gamma_I]x= [\begin{bmatrix} 30-52.66 & -20\\ -20 & 35-52.66\\ \end{bmatrix} \begin{bmatrix} 1\\ x_{2}\\ \end{bmatrix} = 0 $$

$$[K-\gamma_I]x= [\begin{bmatrix} -22.66 & -20\\ -20 & -17.66\\ \end{bmatrix} \begin{bmatrix} 1\\ x_{2}\\ \end{bmatrix} = 0 $$

Multiplying out the matrices, we obtain:

$$-20-17.66x_{2}=0$$

$$x_{2}= 1.1325$$

Using the same process we find $$ x_{1} $$:

$$[K-\gamma_I]x= [\begin{bmatrix} 30 & -20\\ -20 & 35\\ \end{bmatrix} - 52.66* \begin{bmatrix} 1 & 0\\ 0 & 1\\ \end{bmatrix}] \begin{bmatrix} x_{1}\\ 1\\ \end{bmatrix} = 0 $$

$$[K-\gamma_I]x= [\begin{bmatrix} -22.66 & -20\\ -20 & -17.66\\ \end{bmatrix} \begin{bmatrix} x_{1}\\ 1\\ \end{bmatrix} = 0 $$

Multiplying out the matrices, we obtain:

$$-22.66x_{1}-20=0 $$

$$x_{1}=0.8826 $$

= Problem 5.2 = On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Find: Three Lowest Eigen Pairs
Solve the generalized eigenvalue problem of the above truss. Display the results for the lowest 3 eigenpairs. plot and animate the lowest 3 mode shapes.

Solution
The mass matrix was computed by attributing to each node half of the mass of each element which is connect to that node. This is shown in the following code. Code presented in Problem 3.1 was used as a guideline.

%Mass Matrix Construction m = zeros(2*G,3*G); for i=1:G for j=1:E if(l1(j)==i || 10(j)==i) m(1*i-2,3*i-1)=(Ro(j)*Area(j)*L(j))/3 + m(i-2,i-2); m(1*i-1,3*i-1)=(Ro(j)*Area(j)*L(j))/3 + m(i-1,i-1); m(1*i,2*i)=(Ro(j)*Area(j)*L(j))/3 + m(2*i,3*i); m(1*i,2*i)=(Ro(j)*Area(j)*L(j))/3 + m(2*i,3*i); end end end end

k_red was determined using the code presented in Problem 3.1 from the inputs specified in Problem 3.6. It is given here.

%This declares the function and a few variables. function [k, Q, F, Q_bar, F_bar] = displacement(G, x, y, z, F_in, Q_in, E, l1, l2, Modulus, Area) %Element Computation L = zeros(1,E); l = zeros(1,E); m = zeros(1,E); n = zeros(1,E); k_elem = zeros(3,3,E); %This for loop computes the length, direction cosines, and the top left quarter of the for i=1:E L(i)=sqrt((x(l2(i))-x(l1(i)))^2+(y(l2(i))-y(l1(i)))^2+(z(l2(i))-z(l1(i)))^2); l(i)=(x(l2(i))-x(l1(i)))/L(i); m(i)=(y(l2(i))-y(l1(i)))/L(i); n(i)=(z(l2(i))-z(l1(i)))/L(i); k_elem(:,:,i) = (Modulus(i)*Area(i))/L(i)*[l(i)*l(i), l(i)*m(i), l(i)*n(i); m(i)*l(i), m(i)*m(i), m(i)*n(i); n(i)*l(i), n(i)*m(i), n(i)*n(i)]; end %This nested for loop constructs the stiffness matrix. %Stiffness Matrix Construction k = zeros(3*G,3*G); for i=1:E for j=1:3 for h=1:3 k(3*(l1(i)-1)+h,3*(l1(i)-1)+j) = k(3*(l1(i)-1)+h,3*(l1(i)-1)+j) + k_elem(h,j,i); k(3*(l1(i)-1)+h,3*(l2(i)-1)+j) = k(3*(l1(i)-1)+h,3*(l2(i)-1)+j) - k_elem(h,j,i); k(3*(l2(i)-1)+h,3*(l1(i)-1)+j) = k(3*(l2(i)-1)+h,3*(l1(i)-1)+j) - k_elem(h,j,i); k(3*(l2(i)-1)+h,3*(l2(i)-1)+j) = k(3*(l2(i)-1)+h,3*(l2(i)-1)+j) + k_elem(h,j,i); end end end %These for loops partition the stiffness matrix into sections corresponding %to known and unknown force and displacement components. The known force components %remain, separated into known and unknown displacement components. These are used %to solve for the unknown displacements. %remove rows and columns corresponding to 0 displacements. Partition %stiffness matrix for non-zero known displacements removed = 0; k_red = k;  k_dis = k;   for i=1:length(Q_in(:,1)) k_red(Q_in(i,1)-removed,:)=[]; k_dis(Q_in(i,1)-removed,:)=[]; k_red(:,Q_in(i,1)-removed)=[]; removed = removed + 1; end removed = 0; for i=1:length(F_in(:,1)) k_dis(:,F_in(i,1)-removed)=[]; removed = removed + 1; end %This computes the unknown displacements from the reduced k matrix. %Get unknown displacements Q_out=[F_in(:,1) inv(k_red)*(F_in(:,2)-k_dis*Q_in(:,2))]; %This concatenates the original known displacements with those newly computed and then %sorts them. %Construct (and order) full displacement matrix Q=[Q_out; Q_in]; sort_done=0; temp = zeros(1); while(sort_done==0) sort_done=1; for i=1:length(Q(:,1))-1 if(Q(i,1)>Q(i+1,1)) temp=Q(i,:); Q(i,:)=Q(i+1,:); Q(i+1,:)=temp; sort_done=0; end end end %This computes the unknown forces using the global stiffness matrix and the %now-known displacement matrix. %Get Force Matrix F = k*Q(:,2); %Get Member forces %Get local displacements Q_bar=zeros(2); for i=1:E T = [l(i) m(i) n(i) 0   0    0 0   0    0    l(i) m(i) n(i)]; Q_bar(:,i)=T*transpose([Q(3*l1(i)-2, 2) Q(3*l1(i)-1, 2) Q(3*l1(i), 2) Q(3*l2(i)-2, 2) Q(3*l2(i)-1, 2) Q(3*l2(i), 2)]); %This for loop translates the global displacements back to local coordinates %and then computes the internal forces for each element. %Get member forces k_bar= (Modulus(i)*Area(i))/L(i)*[1 -1 -1 1];       F_bar(:,i)=k_bar*Q_bar(:,i); end %This set of loops removes rows and columns associated with extra degrees of freedom. %remove rows of zeros if problem is 1D or 2D DOF = 3; %assume 3 degrees of freedom if(x==0) for i=1:G k(DOF*i-(1+i),:)=[]; k(:,DOF*i-(1+i))=[]; end DOF = DOF - 1; end if(y==0) for i=1:G k(DOF*i-i,:)=[]; k(:,DOF*i-i)=[]; end DOF = DOF - 1; end if(z==0) for i=1:G k(DOF*i-(i-1),:)=[]; k(:,DOF*i-(i-1))=[]; end DOF = DOF - 1; end %Plot truss before and after forces for i=1:E x_bef(1,i)=x(l1(i)); x_bef(2,i)=x(l2(i)); y_bef(1,i)=y(l1(i)); y_bef(2,i)=y(l2(i)); x_aft(1,i)=x(l1(i))+Q(1+3*(l1(i)-1),2); x_aft(2,i)=x(l2(i))+Q(1+3*(l2(i)-1),2); y_aft(1,i)=y(l1(i))+Q(2+3*(l1(i)-1),2); y_aft(2,i)=y(l2(i))+Q(2+3*(l2(i)-1),2); end plot(x_bef,y_bef,'b',x_aft,y_aft,'g') axis equal;

The eigenpairs were computed using the following code [v,d]=eig(k_red,m_red);

and the eigenvectors for columns one through three are:

v = Columns 1 through 10 -0.1326  -0.2423   -0.2690         0.4115   -0.2936    0.4200         -0.2413   -0.3732    0.2292         0.3965   -0.1693    0.3089        -0.2375   -0.3729   -0.3094        0.4167   -0.2315  -0.4766         -0.1599   -0.4869    0.2086         0.3897   -0.1968   -0.3290        -0.3295   -0.3900   -0.1101

The three lowest eigenpairs are as follows:

$$ \omega_{1}=7.12e5; \phi_{1}=\begin{bmatrix}-0.1326\\0.4115\\-0.2413\\\ 0.3965\\-0.2375\\0.4167\\-0.1599\\0.3897\\-0.3295\end{bmatrix};

\omega_{2}=2.22e6; \phi_{2}=\begin{bmatrix}-0.2423\\-0.2936\\-0.3732\\-0.1693\\-0.3729\\-0.2315\\-0.4869\\ -0.1968\\-0.3900\end{bmatrix}

\omega_{3}=4.47e6; \phi_{3}=\begin{bmatrix}-0.2690\\0.4200\\0.2292\\0.3089\\-0.3094\\-0.4766\\0.2086\\-0.3290\\-0.1101\end{bmatrix} $$



Animated Modes of the Plane Truss
Plotted below are the three eigen modes found with the matlab program. They are plotted with undeformed truss system shown in blue to compare the deformations from the different eigen modes.

CALFEM Verification
First, provide material data - modulus, cross sectional area, and the density.

>> E=100e9; A=.0001; rho=5000; >> ep=[E A rho*A];

Construct the Edof  matrix. Column 1 corresponds to the element number; columns 2 though 5 correspond to the degrees of freedom that correspond to the first and second local nodes respectively. Three degrees of freedom as given for each node - translation in the x and y directions and a moment. >> Edof =[ 1   1  2  3  4 2  1  2  5  6               3   3  4  5  6               4   5  6  9 10               5   5  6  7  8               6   3  4  9 10               7   3  4  7  8               8   7  8  9 10               9   9 10 11 12              10   7  8 11 12];

The construct the coordinate matrix. >> Coord = [ 0 0 1 0              1 1               2 0               2 1               3 0];

Construct the list of degrees of freedom. This describes the degrees of freedom assigned to each node (which is described by the row number). >> Dof =[ 1 2 3 4             5  6             7  8             9 10            11 12];

Construct the coordinate matrices. >> [Ex, Ey]=coordxtr(Edof,Coord,Dof,2);

Then generate and assemble the stiffness and mass matrices. >> K=zeros(12); >> for i=1:10 [k]=bar2e(Ex(i,:),Ey(i,:),ep); K=assem(Edof(i,:),K,k); end >> M=zeros(12); >> M = a_CALFEM_Mass_matrix(Edof, Coord, Ex, Ey, ep);

Denote the degrees of freedom that are known to be 0. >> b =[1 2  12];

Compute the eigenpairs for the stiffness and mass matrices. >> [La,Egv]=eigen(K,M,b);

The three lowest eigenpairs, which are identical to those given above, are as follows

$$ \omega_{1}=7.12e5; \phi_{1}=\begin{bmatrix}-0.1326\\0.4115\\-0.2413\\\ 0.3965\\-0.2375\\0.4167\\-0.1599\\0.3897\\-0.3295\end{bmatrix};

\omega_{2}=2.22e6; \phi_{2}=\begin{bmatrix}-0.2423\\-0.2936\\-0.3732\\-0.1693\\-0.3729\\-0.2315\\-0.4869\\ -0.1968\\-0.3900\end{bmatrix}

\omega_{3}=4.47e6; \phi_{3}=\begin{bmatrix}-0.2690\\0.4200\\0.2292\\0.3089\\-0.3094\\-0.4766\\0.2086\\-0.3290\\-0.1101\end{bmatrix} $$



=Problem 5.3= On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
Letting the coeffients of the eigenvectors look like the following matrices. $$x_1= \begin{Bmatrix} x_{11}\\ x_{12} \end{Bmatrix}$$ $$x_2= \begin{Bmatrix} x_{21}\\ x_{22} \end{Bmatrix}$$ $$ \begin{bmatrix} x_{1} & x_{2} \end{bmatrix} = \begin{bmatrix} x_{ij} \end{bmatrix} = \begin{bmatrix} x_{11}& x_{12}\\ x_{21}& x_{22} \end{bmatrix}$$ Now assume that $$x_{11}=x_{12}=1$$

Find
Find the eigenvectors for $$\gamma_1$$ and $$\gamma_2$$ when setting $$x_{11}=x_{12}=1$$ Plot the modes and create an animation of the moment of the modes

Solution
We find the eigenvectors from $$\gamma_1$$:

$$\gamma_1=4-\sqrt{5}$$ $$[K-\gamma_1I]x=$$ $$ \begin{bmatrix} -1+\sqrt{5} & -2\\ -2 & 1+\sqrt{5} \end{bmatrix}$$ $$ \begin{Bmatrix} x_1\\ x_2 \end{Bmatrix}$$ $$=$$ $$ \begin{Bmatrix} 0\\ 0 \end{Bmatrix}$$ Set $$x_{11}=x_{12}=1$$ $$(-1-\sqrt{5})x_1-(2)x_2=0$$ $$x_2=\frac{-1+\sqrt{5}}{2}$$ $$x_1= \begin{Bmatrix} 1\\ \frac{-1+\sqrt{5}}{2} \end{Bmatrix}$$

We find the eigenvectors from $$\gamma_2$$:

$$\gamma_2=4+\sqrt{5}$$ $$[K-\gamma_2I]x=$$ $$ \begin{bmatrix} -1-\sqrt{5} & -2\\ -2 & 1-\sqrt{5} \end{bmatrix}$$ $$ \begin{Bmatrix} x_1\\ x_2 \end{Bmatrix}$$ $$=$$ $$ \begin{Bmatrix} 0\\ 0 \end{Bmatrix}$$ Set $$x_{11}=x_{12}=1$$ $$(-1-\sqrt{5})x_1-(2)x_2=0$$ $$x_2=\frac{-1-\sqrt{5}}{2}$$ $$x_2= \begin{Bmatrix} 1\\ \frac{-1-\sqrt{5}}{2} \end{Bmatrix}$$ $$ \begin{bmatrix} x_1 & x_2 \end{bmatrix}= \begin{bmatrix} 1 & 1\\ \frac{-1+\sqrt{5}}{2} & \frac{-1-\sqrt{5}}{2} \end{bmatrix}$$



Animation of Mode 1

http://makeagif.com/i/Vi1kwS Animation of Mode 2 http://makeagif.com/i/ULkQrV

= Problem 5.4 = On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given: Plane Truss
Consider the plane truss in the figure above. The horizontal and vertical members have length L, while inclined members have length 1.414*L. Assume the Young's modulus E = 100 GPa, cross-sectional area A = 1.0 cm2, L = 0.3 m, and the density if 5000 kg/m3.

Find: Three Lowest Eigen Pairs
Assemble a lumped mass matrix and find the three lowest eigen pairs $$ \left ( \omega _{j},\phi _{j} \right ) $$ for $$ j = 1,2,3 $$. Then plot and animate the three lowest eigen pairs. Finally verify the results with CALFEM.

Solution
The mass matrix was computed by attributing to each node half of the mass of each element which is connect to that node. This is shown in the following code.

%Mass Matrix Construction m = zeros(3*G,3*G); for i=1:G for j=1:E if(l1(j)==i || l2(j)==i) m(3*i-2,3*i-2)=(Ro(j)*Area(j)*L(j))/2 + m(3*i-2,3*i-2); m(3*i-1,3*i-1)=(Ro(j)*Area(j)*L(j))/2 + m(3*i-1,3*i-1); m(3*i,3*i)=(Ro(j)*Area(j)*L(j))/2 + m(3*i,3*i); end end end

This is the resulting mass matrix

m = Columns 1 through 10 0.2561        0         0         0         0         0         0         0         0         0           0    0.2561         0         0         0         0         0         0         0         0           0         0    0.1500         0         0         0         0         0         0         0           0         0         0    0.1500         0         0         0         0         0         0           0         0         0         0    0.3311         0         0         0         0         0           0         0         0         0         0    0.3311         0         0         0         0           0         0         0         0         0         0    0.3311         0         0         0           0         0         0         0         0         0         0    0.3311         0         0           0         0         0         0         0         0         0         0    0.3311         0           0         0         0         0         0         0         0         0         0    0.3311           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0    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        0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0      0.3311         0         0         0         0         0         0         0         0         0           0    0.3311         0         0         0         0         0         0         0         0           0         0    0.3311         0         0         0         0         0         0         0           0         0         0    0.3311         0         0         0         0         0         0           0         0         0         0    0.3311         0         0         0         0         0           0         0         0         0         0    0.3311         0         0         0         0           0         0         0         0         0         0    0.3311         0         0         0           0         0         0         0         0         0         0    0.3311         0         0           0         0         0         0         0         0         0         0    0.3311         0           0         0         0         0         0         0         0         0         0    0.3311           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         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 21 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         0         0         0         0         0           0         0         0         0         0         0         0         0      0.3311         0         0         0         0         0         0         0           0    0.3311         0         0         0         0         0         0           0         0    0.3311         0         0         0         0         0           0         0         0    0.3311         0         0         0         0           0         0         0         0    0.1500         0         0         0           0         0         0         0         0    0.1500         0         0           0         0         0         0         0         0    0.2561         0           0         0         0         0         0         0         0    0.2561

and reduced mass matrix

m_red = Columns 1 through 10 0.2561        0         0         0         0         0         0         0         0         0           0    0.3311         0         0         0         0         0         0         0         0           0         0    0.3311         0         0         0         0         0         0         0           0         0         0    0.3311         0         0         0         0         0         0           0         0         0         0    0.3311         0         0         0         0         0           0         0         0         0         0    0.3311         0         0         0         0           0         0         0         0         0         0    0.3311         0         0         0           0         0         0         0         0         0         0    0.3311         0         0           0         0         0         0         0         0         0         0    0.3311         0           0         0         0         0         0         0         0         0         0    0.3311           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         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 11 through 20 0        0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0      0.3311         0         0         0         0         0         0         0         0         0           0    0.3311         0         0         0         0         0         0         0         0           0         0    0.3311         0         0         0         0         0         0         0           0         0         0    0.3311         0         0         0         0         0         0           0         0         0         0    0.3311         0         0         0         0         0           0         0         0         0         0    0.3311         0         0         0         0           0         0         0         0         0         0    0.3311         0         0         0           0         0         0         0         0         0         0    0.3311         0         0           0         0         0         0         0         0         0         0    0.3311         0           0         0         0         0         0         0         0         0         0    0.3311           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         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 21 through 25 0        0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0      0.3311         0         0         0         0           0    0.1500         0         0         0           0         0    0.1500         0         0           0         0         0    0.2561         0           0         0         0         0    0.2561

k_red was determined using the code presented in Problem 3.1 from the inputs specified in Problem 3.6. It is given here

k_red = 1.0e+07 * Columns 1 through 10 4.5118        0         0   -1.1785   -1.1785         0         0         0         0         0           0    7.8452    1.1785         0         0   -3.3333         0   -1.1785   -1.1785         0           0    1.1785    4.5118         0   -3.3333         0         0   -1.1785   -1.1785         0     -1.1785         0         0    7.8452    1.1785         0         0   -3.3333         0         0     -1.1785         0   -3.3333    1.1785    4.5118         0         0         0         0         0           0   -3.3333         0         0         0    7.8452    1.1785         0         0   -3.3333           0         0         0         0         0    1.1785    4.5118         0   -3.3333         0           0   -1.1785   -1.1785   -3.3333         0         0         0    7.8452    1.1785         0           0   -1.1785   -1.1785         0         0         0   -3.3333    1.1785    4.5118         0           0         0         0         0         0   -3.3333         0         0         0    7.8452           0         0         0         0         0         0         0         0         0    1.1785           0         0         0         0         0   -1.1785   -1.1785   -3.3333         0         0           0         0         0         0         0   -1.1785   -1.1785         0         0         0           0         0         0         0         0         0         0         0         0   -3.3333           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0   -1.1785           0         0         0         0         0         0         0         0         0   -1.1785           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         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 11 through 20 0        0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0   -1.1785   -1.1785         0         0         0         0         0         0         0           0   -1.1785   -1.1785         0         0         0         0         0         0         0           0   -3.3333         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0      1.1785         0         0   -3.3333         0   -1.1785   -1.1785         0         0         0      4.5118         0   -3.3333         0         0   -1.1785   -1.1785         0         0         0           0    7.8452    1.1785         0         0   -3.3333         0         0         0         0     -3.3333    1.1785    4.5118         0         0         0         0         0         0         0           0         0         0    7.8452    1.1785         0         0   -3.3333         0   -1.1785           0         0         0    1.1785    4.5118         0   -3.3333         0         0   -1.1785     -1.1785   -3.3333         0         0         0    7.8452    1.1785         0         0   -3.3333     -1.1785         0         0         0   -3.3333    1.1785    4.5118         0         0         0           0         0         0   -3.3333         0         0         0    7.8452    1.1785         0           0         0         0         0         0         0         0    1.1785    4.5118         0           0         0         0   -1.1785   -1.1785   -3.3333         0         0         0    7.8452           0         0         0   -1.1785   -1.1785         0         0         0   -3.3333    1.1785           0         0         0         0         0         0         0   -3.3333         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0   -1.1785   -1.1785   -3.3333           0         0         0         0         0         0         0   -1.1785   -1.1785         0    Columns 21 through 25 0        0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0     -1.1785         0         0         0         0     -1.1785         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0   -3.3333         0   -1.1785   -1.1785     -3.3333         0         0   -1.1785   -1.1785      1.1785         0         0   -3.3333         0      4.5118         0         0         0         0           0    3.3333         0         0         0           0         0    3.3333         0   -3.3333           0         0         0    4.5118    1.1785           0         0   -3.3333    1.1785    4.5118

The eigenpairs were computed using the following code [v,d]=eig(k_red,m_red);

The eigenvalues are

d = 1.0e+08 * 0.0017     0.0267      0.0702      0.1154      0.2452      0.3962      0.5060      0.5582      0.7074      0.9848      1.4803      1.6625      1.7264      2.0579      2.1076      2.3730      2.7265      2.9420      3.0592      3.0755      3.6054      3.7650      3.8342      4.3106      4.7174

and the eigenvectors are

v = Columns 1 through 10 -0.0082  -0.0745   -0.0352   -0.1585   -0.2044    0.1742    0.1920    0.0217   -0.0954   -0.2269     -0.0279    0.0227   -0.2178    0.0805    0.1314   -0.1703    0.0753    0.2817   -0.5222    0.2573     -0.0757   -0.4176   -0.0416   -0.7045   -0.6624    0.4822    0.2281   -0.1073   -0.0840   -0.0497      0.0364    0.0732   -0.1204    0.0655   -0.0171   -0.0784    0.3013    0.3254   -0.0828   -0.6379     -0.0677   -0.3541   -0.0089   -0.6326   -0.6564    0.5954    0.2226   -0.2685   -0.1359    0.2547     -0.0479    0.0973   -0.3907    0.1430    0.0754    0.0293    0.2220    0.1865   -0.7884    0.5187     -0.1982   -0.6512   -0.1023   -0.5029    0.2342   -0.6304   -0.3946    0.3616    0.0688   -0.2516      0.0645    0.0714   -0.2656   -0.0210   -0.1958   -0.0048    0.5684    0.4827   -0.1509   -0.7071     -0.1906   -0.6149   -0.0780   -0.5780    0.1042   -0.5400   -0.5738    0.3161   -0.1416    0.1913     -0.0605    0.1884   -0.5191    0.0564   -0.0719    0.0597   -0.1205    0.1428   -0.6628    0.4696     -0.3581   -0.6252   -0.1283    0.2317    0.5929    0.2045    0.6036   -0.3703    0.0850   -0.1956      0.0847    0.0153   -0.4221   -0.0962   -0.1715   -0.2322    0.4408    0.5933   -0.0022   -0.3405     -0.3514   -0.6228   -0.1195    0.1118    0.6874    0.0772    0.4545   -0.4834   -0.2591    0.0511     -0.0670    0.2603   -0.6115   -0.1301    0.0372    0.0198   -0.2481   -0.2984   -0.3558    0.0165     -0.5384   -0.3137   -0.0751    0.6068   -0.3313    0.4151   -0.3643    0.5202    0.2595   -0.0564      0.0974   -0.0603   -0.5663   -0.0277   -0.0325   -0.2105    0.4691    0.2200    0.3102    0.1624     -0.5332   -0.3368   -0.0809    0.5799   -0.1222    0.6072   -0.4223    0.4529   -0.0837   -0.0451     -0.0690    0.2941   -0.6722   -0.2590    0.2657    0.3276   -0.4922   -0.3530    0.0403   -0.4965     -0.7244    0.1825    0.0253    0.1509   -0.5316   -0.6657    0.2222   -0.4202    0.0626    0.0358      0.1038   -0.1201   -0.6710    0.1374   -0.1245   -0.0592    0.1074    0.0431    0.6890    0.4511     -0.7213    0.1553    0.0154    0.2186   -0.5329   -0.5115   -0.0379   -0.4467   -0.3304   -0.3669     -0.0690    0.2976   -0.6941   -0.2732    0.2987    0.3986   -0.6373   -0.4714    0.0591   -0.8916     -0.9052    0.6937    0.0971   -0.6251    0.4707    0.2516    0.2812    0.4534    0.5067    0.4254      0.1058   -0.1455   -0.7178    0.2440   -0.3146   -0.2400   -0.0672   -0.3789    0.7446    0.3424     -0.9045    0.6853    0.0940   -0.5927    0.4187    0.2067    0.2172    0.3395    0.3454    0.2369    Columns 11 through 20 1.1014  -0.8914   -0.9974   -0.1819    0.1967    0.3149   -0.5119    0.2138    0.3817    0.1753      0.4543    0.7334   -0.2702   -0.1383   -0.0740   -0.4877    0.3242    0.2377    0.6957   -0.3554     -0.1745    0.2136    0.6758   -0.0218    0.5027    0.3439   -0.4392    0.4407    0.4465   -0.2243      0.6242   -0.2441    0.4132   -0.0051    0.4234    0.0264    0.4876   -0.1787   -0.6039   -0.4932      0.0500    0.0514   -0.4903    0.1221   -0.5712   -0.4444    0.5852   -0.3695   -0.4719   -0.0071      0.2086    0.4463   -0.0845    0.1054   -0.0147    0.1962   -0.3945    0.2584   -0.2841   -0.0931     -0.0576   -0.3509    0.0194    0.4715   -0.6901   -0.1063   -0.0682    0.6136   -0.0116   -0.4823      0.1796    0.1619    0.4433    0.1059   -0.1613   -0.2685    0.2151   -0.1046    0.1118    0.2812      0.1909    0.2464   -0.0169   -0.5468    0.6510    0.0619    0.1365   -0.5678   -0.2093    0.4625     -0.1800   -0.5598    0.2431    0.0791   -0.0281    0.3354   -0.0249    0.1357   -0.6338   -0.0748     -0.2506    0.0247   -0.1562   -0.8120    0.1312   -0.2915    0.0560    0.6483   -0.4069    0.0052     -0.4969    0.1101   -0.2794   -0.0989   -0.3868    0.0472   -0.4748   -0.2024    0.0495    0.6645      0.1847   -0.0654    0.2228    0.8301   -0.0254    0.2754   -0.0445   -0.6555    0.3139    0.2547     -0.2371   -0.6909    0.2650   -0.1265    0.1354   -0.4360    0.4633    0.2069    0.4267    0.4374     -0.0607    0.2149   -0.3203    0.6818    0.4632    0.0528    0.3153    0.4967   -0.1925    0.4879     -0.6066   -0.1414   -0.5200   -0.0470    0.3008    0.3339    0.1010   -0.3204    0.0698   -0.4536     -0.0130   -0.2542    0.2926   -0.6362   -0.5317    0.0496   -0.2146   -0.5656    0.3473   -0.3666      0.0128    0.2045   -0.1038    0.0020   -0.0098   -0.0586    0.0209    0.0626    0.1455    0.2493      0.1774    0.1421    0.1188   -0.3085   -0.4443    0.8011    0.5128    0.1953    0.1094    0.2299      0.1087   -0.1101    0.0199    0.1185    0.2406   -0.0463    0.3521   -0.0926    0.3934   -0.4516     -0.2889   -0.0426   -0.2554    0.2232    0.4295   -0.6797   -0.4900   -0.3040   -0.0315   -0.4214      0.0384    0.8121   -0.4651    0.0275   -0.1907    0.8641   -0.0921   -0.1932   -0.3865   -0.6494      0.1965   -0.0466    0.1590    0.0948    0.0545   -0.4313   -0.5135   -0.1207   -0.3894   -0.1069      0.7058    0.2173    0.4619   -0.0338   -0.2980   -0.4386   -0.6744    0.0168   -0.4332    0.2940      0.0656   -0.0117    0.0355    0.0070    0.0028    0.0293    0.1165    0.0391    0.1467    0.0410    Columns 21 through 25 -0.1566   0.0900    0.0936    0.0623    0.0217     -0.2810   -0.5220    0.2496   -0.3958   -0.2705      0.0112   -0.1824   -0.0157   -0.1362   -0.0687      0.5242   -0.3961   -0.3624   -0.3426   -0.1427      0.1030    0.0044   -0.0593   -0.0024    0.0032      0.5762    0.4127   -0.5547    0.4536    0.4767      0.1044    0.0084   -0.0466    0.1170    0.0978     -0.5515    0.5187    0.4730    0.6375    0.3262     -0.0916    0.1777    0.0534    0.1013    0.0412     -0.4521    0.0048    0.6761   -0.1539   -0.6077     -0.0796    0.1712    0.0581   -0.0295   -0.1125      0.2157   -0.0109   -0.3895   -0.6625   -0.4835     -0.0380   -0.1358    0.0069   -0.1388   -0.0785     -0.1051   -0.0879   -0.4979   -0.3379    0.6621     -0.0776   -0.1527   -0.0157   -0.0724    0.1091      0.0328   -0.7005    0.3085    0.3889    0.5702      0.1244   -0.0659   -0.0549    0.0939    0.1042      0.5424   -0.3362    0.1250    0.7810   -0.6638     -0.0677   -0.1538   -0.1955    0.1912   -0.1148     -0.0239    0.8962   -0.3383    0.0108   -0.5548      0.0556    0.2329    0.1049   -0.0144   -0.1062     -0.8715    0.4842   -0.1723   -0.8311    0.5912     -1.1741   -0.6007   -1.0802    0.5228   -0.2720      0.0808   -0.3740    0.4022   -0.2698    0.4132      0.7308    0.4170    0.7836   -0.4913    0.3054

Therefore, the 3 lowest eigenpairs are as follows:

$$ \omega_{1}=1.7e5; \phi_{1}=\begin{bmatrix}-0.0082\\-0.0279\\-0.0757\\\ 0.0364\\-0.0677\\-0.0479\\-0.1982\\ 0.0645\\-0.1906\\-0.0605\\-0.3581\\ 0.0847\\-0.3514\\-0.0670\\-0.5384\\ 0.0974\\-0.5332\\-0.-0690\\-0.7244\\ 0.1038\\-0.7213\\-0.0690\\-0.9052\\ 0.1058\\-0.9045\end{bmatrix};

\omega_{2}=2.67e6; \phi_{2}=\begin{bmatrix}-0.0745\\ 0.0227\\-0.4176\\ 0.0732\\-0.3541\\ 0.0973\\-0.6512\\ 0.0714\\-0.6149\\ 0.1882\\-0.6252\\ 0.0153\\-0.6228\\ 0.2603\\-0.3137\\-0.0603\\-0.3368\\ 0.2941\\ 0.1825\\-0.1201\\ 0.1553\\ 0.2976\\ 0.6937\\-0.1455\\ 0.6853\end{bmatrix}

\omega_{3}=7.02e6; \phi_{3}=\begin{bmatrix}-0.0352\\-0.2178\\-0.0416\\-0.1204\\-0.0089\\-0.3907\\-0.1023\\-0.2656\\-0.0780\\-0.5191\\-0.1283\\-0.4221\\-0.1195\\-0.6115\\-0.0751\\-0.5663\\-0.0809\\-0.6722\\0.0253\\-0.6710\\0.0154\\-0.6941\\0.0971\\-0.7178\\0.0940\end{bmatrix} $$

Animated Modes of the Plane Truss
Plotted below are the three eigen modes found with the matlab program. They are plotted with undeformed truss system shown in blue to compare the deformations from the different eigen modes.

CALFEM Verification
First, provide material data - modulus, cross sectional area, and the density.

>> E=100e9; A=.0001; rho=5000; >> ep=[E A rho*A];

Construct the Edof  matrix. Column 1 corresponds to the element number; columns 2 though 5 correspond to the degrees of freedom that correspond to the first and second local nodes respectively. Three degrees of freedom as given for each node - translation in the x and y directions and a moment. >> Edof =[ 1   1  2  5  6 2  5  6  9 10               3   9 10 13 14               4  13 14 17 18               5  17 18 21 22               6  21 22 25 26               7   3  4  7  8               8   7  8 11 12               9  11 12 15 16              10  15 16 19 20              11  19 20 23 24              12  23 24 27 28              13   1  2  3  4              14   5  6  7  8              15   9 10 11 12              16  13 14 15 16              17  17 18 19 20              18  21 22 23 24              19  25 26 27 28              20   1  2  7  8              21   5  6 11 12              22   9 10 15 16              23  13 14 19 20              24  17 18 23 24              25  21 22 27 28];

The construct the coordinate matrix. >> Coord = [    0         0 0   0.3000              0.3000         0              0.3000    0.3000              0.6000         0              0.6000    0.3000              0.9000         0              0.9000    0.3000              1.2000         0              1.2000    0.3000              1.5000         0              1.5000    0.3000              1.8000         0              1.8000    0.3000];

Construct the list of degrees of freedom. This describes the degrees of freedom assigned to each node (which is described by the row number). >> Dof =[ 1   2 3   4            5    6            7    8            9    10           11    12           13    14           15    16           17    18           19    20           21    22           23    24           25    26           27    28];

Construct the coordinate matrices. >> [Ex, Ey]=coordxtr(Edof,Coord,Dof,2);

Then generate and assemble the stiffness and mass matrices. >> K=zeros(28); >> for i=1:25 [k]=bar2e(Ex(i,:),Ey(i,:),ep); K=assem(Edof(i,:),K,k); end >> M=zeros(28); >> M = a_CALFEM_Mass_matrix(Edof, Coord, Ex, Ey, ep);

Denote the degrees of freedom that are known to be 0. >> b =[1 3  4];

Compute the eigenpairs for the stiffness and mass matrices. >> [La,Egv]=eigen(K,M,b);

The three lowest eigenpairs, which are identical to those given above, are as follows

$$ \omega_{1}=1.7e5; \phi_{1}=\begin{bmatrix}-0.0082\\-0.0279\\-0.0757\\\ 0.0364\\-0.0677\\-0.0479\\-0.1982\\ 0.0645\\-0.1906\\-0.0605\\-0.3581\\ 0.0847\\-0.3514\\-0.0670\\-0.5384\\ 0.0974\\-0.5332\\-0.-0690\\-0.7244\\ 0.1038\\-0.7213\\-0.0690\\-0.9052\\ 0.1058\\-0.9045\end{bmatrix};

\omega_{2}=2.67e6; \phi_{2}=\begin{bmatrix}-0.0745\\ 0.0227\\-0.4176\\ 0.0732\\-0.3541\\ 0.0973\\-0.6512\\ 0.0714\\-0.6149\\ 0.1882\\-0.6252\\ 0.0153\\-0.6228\\ 0.2603\\-0.3137\\-0.0603\\-0.3368\\ 0.2941\\ 0.1825\\-0.1201\\ 0.1553\\ 0.2976\\ 0.6937\\-0.1455\\ 0.6853\end{bmatrix}

\omega_{3}=7.02e6; \phi_{3}=\begin{bmatrix}-0.0352\\-0.2178\\-0.0416\\-0.1204\\-0.0089\\-0.3907\\-0.1023\\-0.2656\\-0.0780\\-0.5191\\-0.1283\\-0.4221\\-0.1195\\-0.6115\\-0.0751\\-0.5663\\-0.0809\\-0.6722\\0.0253\\-0.6710\\0.0154\\-0.6941\\0.0971\\-0.7178\\0.0940\end{bmatrix} $$



=Problem 5.5= On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given: 2 Bar Truss



 * Data:
 * Young's Modulus Element Length  Cross Section Area  Force
 * $$ E^{(1)}=3 $$ $$ L^{(1)}=4 $$  $$ A^{(1)}=1 $$  $$ P=2 $$
 * $$ E^{(2)}=5 $$ $$ L^{(2)}=2 $$  $$ A^{(2)}=2 $$

Find: Eigenvectors & Eigenvalues
Plot the eigenvectors corresponding to the four zero eigenvalues then interpret the results.

Solution
Using the code displayed in Problem 5.4 the stiffness matrix was constructed and the eigenvalues and eigenvectors for the system were found.

Stiffness matrix K
The stiffness matrix could not be reduced for this problem because there weren't any known displacements.

Displacements and Forces: The following displacement matrix and force matrix results in "NaN" answers because the truss system is now a mechanism. Since a mechanism can potentially displace infinitely then there can not be a finite answer to the problem.

Eigenvectors and Eigenvalues: The eigenvectors and eigenvalues were calculated using $$Kv=\lambda v$$ which resulted in four zero eigenvalues. The matrix "v" corresponds to the eigenvectors and the matrix "d" corresponds to the eigenvalues.

The following matrices correspond to the four zero eigenvalues. $$ \mathbf{u}_{1}=\begin{bmatrix} -0.3337\\ -0.1582 \\ -0.0997\\ -0.5636\\ 0.6949\\ 0.2308\\ \end{bmatrix} $$ $$ \mathbf{u}_{2}=\begin{bmatrix} -0.5828\\ 0.7938\\ -0.1266\\ 0.0037\\ -0.1183\\ 0.0120\\ \end{bmatrix} $$

$$ \mathbf{u}_{3}=\begin{bmatrix} -0.3931\\ -0.4604\\ -0.6337\\ -0.0437\\ -0.4623\\ 0.1276\\ \end{bmatrix} $$

$$ \mathbf{u}_{4}=\begin{bmatrix} -0.1142\\ -0.0760\\ 0.1321\\ -0.5025\\ -0.1877\\ -0.8222\\ \end{bmatrix} $$

Animated Modes Corresponding to Zero Eigenvalues
The first mode is a pure mechanism because point at which the force is applied doesn't change along the x-axis and the two beams rotate around this point. The third and fourth modes are pure rigid body motions because the truss stays in the configuration that it was originally in and it's the node were the force is applied that rotates. The second mode is a linear combination of a pure mechanism and a pure rigid body.

Given: 3 Bar Truss



 * Data:
 * Young's Modulus Element Length  Cross Section Area  Force
 * $$ E^{(1)}=2 $$ $$ L^{(1)}=1 $$  $$ A^{(1)}=3 $$  $$ P=1 $$
 * $$ E^{(2)}=2 $$ $$ L^{(2)}=1 $$  $$ A^{(2)}=3 $$
 * $$ E^{(3)}=2 $$ $$ L^{(3)}=1 $$  $$ A^{(3)}=3 $$

Find: Zero Eigenvalues
Plot the corresponding zero eigenvalue.

Solution
Using the code displayed in Problem 5.4 the stiffness matrix was constructed and the eigenvalues and eigenvectors for the system were found.

Reduced Stiffness matrix K
The reduced stiffness matrix K is shown below.

Displacements and Forces: The following displacement matrix and force matrix results in "NaN" answers because the truss system is a mechanism. Since a mechanism can potentially displace infinitely then there can not be a finite answer to the problem.

Eigenvectors and Eigenvalues: The eigenvectors and eigenvalues were calculated using $$Kv=\lambda v$$ which resulted in one zero eigenvalues. The matrix "v" corresponds to the eigenvectors and the matrix "d" corresponds to the eigenvalues.

The following matrix correspond to the one zero eigenvalues. $$ \mathbf{u}_{1}=\begin{bmatrix} -0.7071\\ 0 \\ -0.7071\\ 0\\ \end{bmatrix} $$

Given: 4 Bar Truss



 * Data:
 * Young's Modulus Element Length  Cross Section Area  Force
 * $$ E^{(1)}=2 $$ $$ L^{(1)}=1 $$  $$ A^{(1)}=3 $$  $$ P=1 $$
 * $$ E^{(2)}=2 $$ $$ L^{(2)}=1 $$  $$ A^{(2)}=3 $$
 * $$ E^{(3)}=2 $$ $$ L^{(3)}=1 $$  $$ A^{(3)}=3 $$
 * $$ E^{(4)}=2 $$ $$ L^{(4)}=1.414 $$  $$ A^{(4)}=3 $$

Find: First Eigenvalue
Plot the first eigenvalue.

Solution
Using the code displayed in Problem 5.4 the stiffness matrix was constructed and the eigenvalues and eigenvectors for the system were found.

Reduced Stiffness matrix K
The reduced stiffness matrix K is shown below.

Displacements and Forces: The following displacement matrix and force matrix results in finite answers because the truss system is constrained.

Eigenvectors and Eigenvalues: The eigenvectors and eigenvalues were calculated using $$Kv=\lambda v$$ which resulted in four non-zero eigenvalues. The matrix "v" corresponds to the eigenvectors and the matrix "d" corresponds to the eigenvalues.

The following matrix correspond to the one of the four non-zero eigenvalues. $$ \mathbf{u}_{1}=\begin{bmatrix} 0.7370\\ 0 \\ 0.6498\\ -0.1860\\ \end{bmatrix} $$

Animated First Mode Eigenvector


= Problem 5.6 = On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
Consider the following numerical values for the MDOF system:

$$m_{1}=3, m_{2}=2$$

$$c_{1}=\frac{1}{2}, c_{2}=\frac{1}{4}, c_{3}=\frac{1}{3}$$

$$k_{1}=10, k_{2}=20, k_{3}=15$$

$$F_{1}(t)=0, F_{2}(t)=0$$

$$d_{1}(0)=-1, d_{2}(0)=2$$

$$d{}'_{1}(0)=0, d{}'_{2}(0)=0$$

Find
Integrate numerically the governing system of L2-ODEs-CC (equations of motion) in order to generate the time histories to determine the displacement matirx:

'''$$d(t)=\begin{Bmatrix}d_{1}(t) \\ d_{2}(t)\end{Bmatrix}$$'''

Solution
To use superposition of modal coordinates, a transformation matrix $$\mathbf \bar \Phi$$ must be found, such that the following are true:
 * $$\mathbf{\bar \Phi^T K \Phi}=\mathbf \Omega^2$$
 * $$\mathbf{\bar \Phi^T M \Phi}=\mathbf I$$

The MATLAB command "eig" is used to solve the generalized eigenvalue problem:

$$Kx=\lambda Mx$$

K = 30  -20 -20   35 M = 3     0 0    2 EDU>> [L,X]=eig(K,M) L = 4.7651 22.7349 X = -0.4860  -0.3116 -0.3817   0.5953

Where $$\lambda _{1}=\omega_1^2=4.7651, \lambda _{2}=\omega_2^2=22.7349$$

This gives the matrix of natural frequencies, $$\mathbf \Omega^2=\begin{bmatrix}\omega_1^2 & 0 \\ 0 & \omega_2^2\end{bmatrix}=\begin{bmatrix}4.7651 & 0 \\ 0 & 22.7349\end{bmatrix}$$.

The eigenvector matrix $$X$$ gives $$\mathbf \bar \Phi = \begin{bmatrix}-0.4860 & -0.3116 \\ -0.3817 & 0.5953 \end{bmatrix}$$

The modal superposition method is defined in the form:

$$\ddot{z}(t)+ \bar \Phi^T C \bar \Phi \dot{z}(t) + \Omega^2 z(t) = \bar \Phi^T R(t)$$

Where
 * $$C=\begin{bmatrix} 3/4 & -1/4 \\ -1/4 & 7/12 \end{bmatrix}$$
 * $$R(t)=\begin{Bmatrix}F_1(t)\\F_2(t)\end{Bmatrix}=\begin{Bmatrix}0\\0\end{Bmatrix}$$

Obtain damping matrix:
 * $$\bar \Phi^T C \bar \Phi=\begin{bmatrix}-0.4860 & -0.3817 \\ -0.3116 & 0.5953 \end{bmatrix} \begin{bmatrix} 3/4 & -1/4 \\ -1/4 & 7/12 \end{bmatrix} \begin{bmatrix}-0.4860 & -0.3116 \\ -0.3817 & 0.5953 \end{bmatrix}=\begin{bmatrix}0.1694 & 0.0236 \\ 0.0236 & 0.3723 \end{bmatrix}$$

Use M-orthogonality to find the initial conditions $$\dot z(0)$$ and $$z(0)$$ from $$\dot d(0)$$ and $$d(0)$$.
 * $$\dot z(0)=\bar \Phi^T M \dot d(0)=\begin{bmatrix}-0.4860 & -0.3817 \\ -0.3116 & 0.5953 \end{bmatrix} \begin{bmatrix}3 & 0 \\ 0 & 2 \end{bmatrix} \begin{Bmatrix}0 \\ 0 \end{Bmatrix}=\begin{Bmatrix}0 \\ 0 \end{Bmatrix}$$


 * $$z(0)=\bar \Phi^T M d(0)=\begin{bmatrix}-0.4860 & -0.3817 \\ -0.3116 & 0.5953 \end{bmatrix} \begin{bmatrix}3 & 0 \\ 0 & 2 \end{bmatrix} \begin{Bmatrix}-1 \\ 2 \end{Bmatrix}=\begin{Bmatrix}-0.0688 \\ 3.3160 \end{Bmatrix}$$

Solve $$z_i(t)$$ using the L2-ODE-CCs:
 * $$\ddot z_1(t) + 0.1694 \dot z_1(t) + 4.7651 z_1(t) = 0$$
 * $$\ddot z_2(t) + 0.3723 \dot z_2(t) + 22.7349 z_2(t) = 0$$

This results in the exact solutions:
 * $$z_1(t)=-0.0688 e^{-0.0847t} \cos 2.1813t + -0.002672 e^{-0.0847t} \sin 2.1813t $$
 * $$z_2(t)=3.3160 e^{-0.1862t} \cos 4.7645t + -0.129592 e^{-0.1862t} \sin 4.7645t $$

Multiply by the eigenvector to get the displacement matrix:
 * $$d(t)=\bar \Phi z(t)$$
 * $$d(t)=\begin{bmatrix}-0.4860 & -0.3116 \\ -0.3817 & 0.5953 \end{bmatrix} \begin{Bmatrix}-0.0688 e^{-0.0847t} \cos 2.1813t + -0.002672 e^{-0.0847t} \sin 2.1813t \\ 3.3160 e^{-0.1862t} \cos 4.7645t + -0.129592 e^{-0.1862t} \sin 4.7645t \end{Bmatrix}$$
 * $$d(t)=\begin{Bmatrix}0.033437 e^{-0.0847t} \cos 2.1813t + 0.001299 e^{-0.0847t} \sin 2.1813t + -1.03327 e^{-0.1862t} \cos 4.7645t + 0.040381 e^{-0.1862t} \sin 4.7645t \\ 0.026261 e^{-0.0847t} \cos 2.1813t + 0.00102 e^{-0.0847t} \sin 2.1813t + 1.97401 e^{-0.1862t} \cos 4.7645t + -0.077146 e^{-0.1862t} \sin 4.7645t \end{Bmatrix}$$

Find the fundamental frequency, $$\omega_1$$ and fundamental period, $$T_1$$:
 * $$\omega_1=\sqrt{4.7651}=2.1829 Hz$$
 * $$T_1=\frac{1}{\omega_1}=\frac{1}{2.1829}=0.4581 s$$

Let the time step size be $$dt$$
 * $$dt=\frac{T_1}{10}$$
 * $$dt=0.04581 s$$

Plot the time histories for $$d_1(t)$$, $$d_2(t)$$ for $$0 \le t \le T=5T_1$$:



= Problem 5.7: Plot the history for the vertical displacement at nodes of truss system = On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
Consider the truss under free vibration. The initial displacement is generated by applying a force at node 4:

$$ F = 5 $$

Find
Let the truss be released from static deflection with zero intial velocity. Use the first 3 lowest modes to solve for the motion of the truss by modal superposition.

Solution
A MATLAB function is created to establish all the parameters of the truss system:

function [G, x, y, z, F_in, Q_in, E, l1, l2, Modulus, Area, Ro, mode, mag] = Report5_truss1 G=6; x=[0 1 1 2 2 3]; y=[0 0 1 0 1 0]; z=[0 0 0 0 0 0]; E=10; l1=[1 1 2 3 3 2 2 4 5 4]; l2=[2 3 3 5 4 5 4 5 6 6]; F_in=[4 0; 5 0; 7 0; 8 0; 10 0; 11 -5; 13 0; 14 0; 16 0]; Q_in=[1 0; 2 0; 3 0; 6 0; 9 0; 12 0; 15 0; 17 0; 18 0;]; Modulus=[5 5 5 5 5 5 5 5 5 5]; Area=[.5 .5 .5 .5 .5 .5 .5 .5 .5 .5]; Ro=[2 2 2 2 2 2 2 2 2 2]; mode=1; mag=1;

The function truss is used to determine displacement and force: EDU>> [Q, F, Q_bar, F_bar, Q_in, k, k_red, Q_out, L, m, m_red, v, d, Q_red] = truss(G, x, y, z, F_in, Q_in, E, l1, l2, Modulus, Area,   Ro, mode, mag) Q =1.0000        0 2.0000        0    3.0000         0    4.0000    0.6667    5.0000   -4.4372    6.0000         0    7.0000    2.4254    8.0000   -4.3110    9.0000         0   10.0000    1.4596   11.0000   -6.8054   12.0000         0   13.0000    1.2183   14.0000   -5.3459   15.0000         0   16.0000    2.7929   17.0000         0   18.0000         0 F =-0.0000 1.6667        0   -0.0000    0.0000   -0.0000         0   -5.0000   -0.0000   -0.0000         0    3.3333 Q_bar = Columns 1 through 7 0        0   -4.4372    2.4254    4.7633   -2.6662    0.6667    0.6667   -1.3333   -4.3110    1.2183    5.8442   -2.9186    1.4596 Columns 8 through 10 -6.8054   4.6415    1.4596   -5.3459    1.9749    2.7929 F_bar = Columns 1 through 7 -1.6667   2.3570   -0.3156    3.0178   -1.9107    0.4463   -1.9822    1.6667   -2.3570    0.3156   -3.0178    1.9107   -0.4463    1.9822 Columns 8 through 10 -3.6489   4.7140   -3.3333    3.6489   -4.7140    3.3333 Q_in = 1    0     2     0     3     0     6     0     9     0    12     0    15     0    17     0    18     0 k = Columns 1 through 7 3.3839   0.8839   -2.5000         0   -0.8839   -0.8839         0    0.8839    0.8839         0         0   -0.8839   -0.8839         0   -2.5000         0    5.8839    0.8839         0         0   -2.5000         0         0    0.8839    3.3839         0   -2.5000         0   -0.8839   -0.8839         0         0    4.2678         0   -0.8839   -0.8839   -0.8839         0   -2.5000         0    4.2678    0.8839         0         0   -2.5000         0   -0.8839    0.8839    5.8839         0         0         0         0    0.8839   -0.8839   -0.8839         0         0   -0.8839   -0.8839   -2.5000         0         0         0         0   -0.8839   -0.8839         0         0         0         0         0         0         0         0         0   -2.5000         0         0         0         0         0         0         0  Columns 8 through 12 0        0         0         0         0         0         0         0         0         0         0   -0.8839   -0.8839         0         0         0   -0.8839   -0.8839         0         0    0.8839   -2.5000         0         0         0   -0.8839         0         0         0         0   -0.8839         0         0   -2.5000         0    3.3839         0   -2.5000         0         0         0    4.2678         0   -0.8839    0.8839   -2.5000         0    4.2678    0.8839   -0.8839         0   -0.8839    0.8839    3.3839   -0.8839         0    0.8839   -0.8839   -0.8839    0.8839 k_red = Columns 1 through 7 5.8839   0.8839         0         0   -2.5000         0   -0.8839    0.8839    3.3839         0   -2.5000         0         0   -0.8839         0         0    4.2678         0   -0.8839    0.8839   -2.5000         0   -2.5000         0    4.2678    0.8839   -0.8839         0   -2.5000         0   -0.8839    0.8839    5.8839   -0.8839         0         0         0    0.8839   -0.8839   -0.8839    3.3839         0   -0.8839   -0.8839   -2.5000         0         0         0    4.2678   -0.8839   -0.8839         0         0         0   -2.5000         0         0         0         0         0   -2.5000         0   -0.8839  Columns 8 through 9 -0.8839        0   -0.8839         0         0         0         0         0         0   -2.5000   -2.5000         0         0   -0.8839    4.2678    0.8839    0.8839    3.3839 Q_out = 4.0000   0.6667    5.0000   -4.4372    7.0000    2.4254    8.0000   -4.3110   10.0000    1.4596   11.0000   -6.8054   13.0000    1.2183   14.0000   -5.3459   16.0000    2.7929 L = Columns 1 through 7 1.0000   1.4142    1.0000    1.0000    1.4142    1.4142    1.0000  Columns 8 through 10 1.0000   1.4142    1.0000 m = Columns 1 through 7 1.2071        0         0         0         0         0         0         0    1.2071         0         0         0         0         0         0         0    2.2071         0         0         0         0         0         0         0    2.2071         0         0         0         0         0         0         0    2.4142         0         0         0         0         0         0         0    2.4142         0         0         0         0         0         0         0    2.2071         0         0         0         0         0         0         0         0         0         0         0         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 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         0         0         0         0         0         0         0    2.2071         0         0         0         0         0    2.4142         0         0         0         0         0    2.4142         0         0         0         0         0    1.2071         0         0         0         0         0    1.2071 m_red = Columns 1 through 7 2.2071        0         0         0         0         0         0         0    2.2071         0         0         0         0         0         0         0    2.4142         0         0         0         0         0         0         0    2.4142         0         0         0         0         0         0         0    2.2071         0         0         0         0         0         0         0    2.2071         0         0         0         0         0         0         0    2.4142         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 8 through 9 0        0         0         0         0         0         0         0         0         0         0         0         0         0    2.4142         0         0    1.2071 v = Columns 1 through 7 -0.0938  -0.1713   -0.1902    0.0382    0.2733    0.3282    0.2536    0.2910   -0.2076    0.2970    0.1640    0.2621   -0.2744   -0.0698   -0.1706   -0.2639    0.1621   -0.2909   -0.0575    0.1584   -0.3997    0.2804   -0.1197    0.2185    0.1947   -0.1395    0.4169    0.0356   -0.1679   -0.2637   -0.2188    0.2640    0.1216    0.0086   -0.1597    0.2946   -0.1637   -0.3370   -0.0109   -0.3929   -0.0685   -0.0534   -0.1131   -0.3443    0.1475   -0.1970   -0.1489   -0.1419    0.3993    0.2756   -0.1391   -0.2327   -0.2662    0.2738   -0.0349   -0.0422   -0.2330   -0.2757   -0.0779    0.4295   -0.1309   -0.2114   -0.0753  Columns 8 through 9 -0.2409   0.2693   -0.2344    0.0801   -0.0703    0.1071    0.1823   -0.0887   -0.0451   -0.4420   -0.2396    0.0803    0.0482   -0.1352    0.3404   -0.0103    0.4477    0.4895 d = Columns 1 through 7 0.0890        0         0         0         0         0         0         0    0.2779         0         0         0         0         0         0         0    0.5582         0         0         0         0         0         0         0    1.4125         0         0         0         0         0         0         0    2.3629         0         0         0         0         0         0         0    2.5171         0         0         0         0         0         0         0    2.7049         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 8 through 9 0        0         0         0         0         0         0         0         0         0         0         0         0         0    3.4896         0         0    4.8604 Q_red = 4.0000   0.6667    5.0000   -4.4372    7.0000    2.4254    8.0000   -4.3110   10.0000    1.4596   11.0000   -6.8054   13.0000    1.2183   14.0000   -5.3459   16.0000    2.7929

Next, $$ z(0)$$  is found: function [z_0] = modal_coord(v, m_red, Q_red) z_0=zeros(length(Q_red),1); for i=1:length(Q_red) z_0(i)=transpose(v(:,i))*m_red*Q_red(:,2); end

The resulting output is:

EDU>> [z_0] = modal_coord(v, m_red, Q_red) z_0 = -16.5465   2.9451    3.0186    0.0387    0.8314    0.1361    0.0987    0.3433   -0.0826

We solve for t, z and displacement using the modal_solve function: function [t,z, disp]=modal_solve(dof, j, t_0, t_f, z_0, zp_0, v) t_set=t_0:.5:t_f; z=zeros(length(t_set),dof); [t,z_res]=ode45('F1',[t_0,t_f],[z_0(1),zp_0(1)]); z(:,1)=interp1(t,z_res(:,2),t_set); for i=2:j [t,z_res]=ode45(strcat('F',num2str(i)),[t_0,t_f],[z_0(i),zp_0(i)]); z(:,i)=interp1(t,z_res(:,2),t_set); end disp=zeros(dof,length(t_set)); for i=1:dof disp(i,:)=z*v(:,i); end

To plot the vertical displacement at node 2, we plot the second column of the disp matrix versus t. The history for the vertical displacement at node 2 is shown in the graph.