User:Eml4507.s13.team3.steiner/Team Negative Damping (3): Report 7

=Problem 7.1= On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
The free body diagram to the right was given for the problem.

Find
Solve 2-element frame system using same data for 2-bar truss system. Solve 2-element truss system.

Frame Problem
A basic form is shown below:

$$\begin{bmatrix} a_{1}&0&0&-a_{1}&0&0\\

0&12a_{2}&6La_{2}&0&-12a_{2}&6La_{2}\\

0&6La_{2}&4L^2a_{2}&0&-6La_{2}&2L^2a_{2}\\

-a_{1}&0&0&a_{1}&0&0&\\

0&-12a_{2}&-6La_{2}&0&12a_{2}&-6La_{2}\\

0&6La_{2}&4L^2a_{2}&0&-6La_{2}&4L^2a_{2}\\

\end{bmatrix} \begin{bmatrix}

u_{1}\\

v_{1}\\

t_{1}\\

u_{2}\\

v_{2}\\

t_{2}\\

\end{bmatrix}= \begin{bmatrix} f_{x1}\\

f_{y1}\\

c_{c1}\\

f_{x2}\\

f_{y2}\\

c_{c2}\\ \end{bmatrix} $$

Equations derived from stiffness equations:

$$a_{1}=\frac{E*A}{L}$$

$$a_{2}=\frac{E*I}{L^3}$$

Globle stiffness equation is shown below:

$$\begin{bmatrix} a^{(1)}_{1}&0&0&-a^{(1)}_{1}&0&0&0&0&0\\

0&12a^{(1)}_{2}&6La^{(1)}_{2}&0&-12a^{(1)}_{2}&6La^{(1)}_{2}&0&0&0\\

0&6La^{(1)}_{2}&4L^2a^{(1)}_{2}&0&-6La^{(1)}_{2}&2L^2a^{(1)}_{2}&0&0&0\\

-a^{(1)}_{1}&0&0&a^{(1)}_{1}+a^{(2)}_{1}&0&0&-a^{(2)}_{1}&0&0\\

0&-12^{(1)}a_{2}&-6La^{(1)}_{2}&0&12a^{(2)}_{2}-12a^{(1)}_{2}&6La^{(2)}_{2}-6La^{(1)}_{2}&0^{(2)}&-12a^{(2)}_{2}&6La^{(2)}_{2}\\

0&6La^{(1)}_{2}&2L^2a^{(1)}_{2}&0&6La^{(2)}_{2}-6La^{(1)}_{2}&4L^2a^{(1)}_{2}+4L^2a^{(2)}_{2}&0^{(2)}&-6La^{(2)}_{2}&2L^2a^{(2)}_{2}\\

0&0&0&-a^{(2)}_{1}&0&0&a^{(2)}_{1}&0&0\\

0&0&0&0&-12a^{(2)}_{2}&-6La^{(2)}_{2}&0&-12a^{(2)}_{2}&-6La^{(2)}_{2}\\

0&0&0&0&6La^{(2)}_{2}&2L^2a^{(2)}_{2}&0&-6La^{(2)}_{2}&4L^2a^{(2)}_{2}\\ \end{bmatrix}$$

Q matrix is shown below:

$$ Q= \begin{bmatrix}

u_{1}\\

v_{1}\\

t_{1}\\

u_{2}\\

v_{2}\\

t_{2}\\

u_{3}\\

v_{3}\\

t_{3}\\

\end{bmatrix}= \begin{bmatrix} 0\\

0\\

0\\

u_{2}\\

v_{2}\\

t_{2}\\ 0\\

0\\

t_{3}\\ \end{bmatrix}$$

Force matrix is show below:

$$ F= \begin{bmatrix}

f_{x1}\\

f_{y1}\\

c_{c1}\\

f_{x2}\\

f_{y2}\\

c_{c2}\\

f_{x3}\\

f_{y3}\\

c_{c3}\\

\end{bmatrix}= \begin{bmatrix} f_{x1}\\

f_{y1}\\

c_{c1}\\

0\\

P\\

0\\

f_{x3}\\

f_{y3}\\

0\\ \end{bmatrix}$$

Matrix for the system is shown below:

$$\begin{bmatrix} a^{(1)}_{1}&0&0&-a^{(1)}_{1}&0&0&0&0&0\\

0&12a^{(1)}_{2}&6La^{(1)}_{2}&0&-12a^{(1)}_{2}&6La^{(1)}_{2}&0&0&0\\

0&6La^{(1)}_{2}&4L^2a^{(1)}_{2}&0&-6La^{(1)}_{2}&2L^2a^{(1)}_{2}&0&0&0\\

-a^{(1)}_{1}&0&0&a^{(1)}_{1}+a^{(2)}_{1}&0&0&-a^{(2)}_{1}&0&0\\

0&-12^{(1)}a_{2}&-6La^{(1)}_{2}&0&12a^{(2)}_{2}-12a^{(1)}_{2}&6La^{(2)}_{2}-6La^{(1)}_{2}&0^{(2)}&-12a^{(2)}_{2}&6La^{(2)}_{2}\\

0&6La^{(1)}_{2}&2L^2a^{(1)}_{2}&0&6La^{(2)}_{2}-6La^{(1)}_{2}&4L^2a^{(1)}_{2}+4L^2a^{(2)}_{2}&0^{(2)}&-6La^{(2)}_{2}&2L^2a^{(2)}_{2}\\

0&0&0&-a^{(2)}_{1}&0&0&a^{(2)}_{1}&0&0\\

0&0&0&0&-12a^{(2)}_{2}&-6La^{(2)}_{2}&0&-12a^{(2)}_{2}&-6La^{(2)}_{2}\\

0&0&0&0&6La^{(2)}_{2}&2L^2a^{(2)}_{2}&0&-6La^{(2)}_{2}&4L^2a^{(2)}_{2}\\ \end{bmatrix} \begin{bmatrix} 0\\

0\\

0\\

u_{2}\\

v_{2}\\

t_{2}\\ 0\\

0\\

0\\ \end{bmatrix}= \begin{bmatrix} f_{x1}\\

f_{y1}\\

c_{c1}\\

0\\

P\\

0\\

f_{x3}\\

f_{y3}\\

0\\ \end{bmatrix}  $$

The reduced stiffness is $$ k= \begin{bmatrix} 5.75&0&0&0\\

0&0.622&0.619&0.625\\

0&0.619&-0.818&0.417\\

0&0.625&0.417&0.833\\

\end{bmatrix} $$

To solve for the displacements $$ \begin{bmatrix} 0\\

6\\

0\\

0\\

\end{bmatrix} = \begin{bmatrix} 5.75&0&0&0\\

0&0.622&0.619&0.625\\

0&0.619&-0.818&0.417\\

0&0.625&0.417&0.833\\

\end{bmatrix} \begin{bmatrix} u_{2}\\

v_{2}\\

t_{2}\\

t_{3}\\

\end{bmatrix} $$

After matrix math. $$ \begin{bmatrix} u_{2}\\

v_{2}\\

t_{2}\\

t_{3}\\

\end{bmatrix} = \begin{bmatrix} 0\\

24.5\\

7.32\\

-22.08\\

\end{bmatrix} $$

After solving for the forces the result is $$ \begin{bmatrix} fx_{1}\\

fy_{1}\\

C_{1}\\

fx_{3}\\

fy_{3}\\

\end{bmatrix} = \begin{bmatrix} 0\\

-594\\

-4118\\

0\\

-6.1\\

\end{bmatrix} $$

Below is a figure of the new deformed frame plotted with the undeformed frame.

Truss Problem


= Problem 7.2: 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.