User:Eml4507.s13.team2/Report6

Honor Pledge
On our honor, this problem was solved on our own with reference to Dr. Vu-Quoc's Fead.s13.sec53b and Fead.s13.sec54 lecture notes.

Given: A spring-mass-damper system
Given the spring-mass-damper system shown below:



and the following data for the masses and stiffness coefficients:

$$ m_{1}=3, m_{2}=2 $$ (1.1) $$ k_{1}=10, k_{2}=20, k_{3}=15 $$ (1.2)

Using this data, the mass and stiffness matrices (denoted by M and K, respectively) can be written as:

$$ M = \begin{bmatrix} m_{1} & 0\\ 0& m_{2} \end{bmatrix} $$ (1.3) $$ K = \begin{bmatrix} k_{1}+k_{2} & -k_{2}\\ -k_{2}& k_{2}+k_{3} \end{bmatrix} $$ (1.4) Using all of this, we are to use the method outlined in Dr. Vu Quoc's section 54 FEAD notes to transform the generalized eigenvalue problem (GEP), into a standard eigenvalue problem (SEP).

Solution
We start with the format for the GEP.

$$ Kx=\lambda Mx        $$ (1.5)

We want to transform this GEP into an SEP with a symmetric matrix. So first, we multiply both sides of Eq. 4.5 by M-1/2 to yield:

$$ M^{-1/2}Kx=\lambda M^{-1/2}Mx $$ (1.6)

Using matrix multiplication properties, the right side of Eq. 1.6 can be simplified to:

$$ \lambda M^{1/2}x $$ (1.7)

Now we will define a new vector x* as:

$$ x^{*}=M^{1/2}x\rightarrow x=M^{-1/2}x^{*} $$ (1.8)

Substituting Eq. 1.8 into Eqs. 1.6 and 1.7, we now get:

$$ (M^{-1/2}KM^{-1/2})x^{*}=K^{*}x^{*}=\lambda x^{*} $$ (1.9)

This is now a SEP for the new matrix K*, which is a symmetric matrix.

$$ K^{*}=M^{-1/2}KM^{-1/2} $$ (1.10)

So, using this method and the final result of it (Eq. 1.10), we can now transform our GEP with the data in the problem statement into a SEP. Using Eqs. 1.3 and 1.4, our mass and stiffness matrices are as follows:

$$ M = \begin{bmatrix} 3 & 0\\ 0& 2 \end{bmatrix} $$ (1.11) $$ K = \begin{bmatrix} 30 & -20\\ -20& 35 \end{bmatrix} $$ (1.12)

Using matrix properties, M-1/2 (rounding to 3 decimal places) is: $$ M = \begin{bmatrix} .577 & 0\\ 0& ..707 \end{bmatrix} $$ (1.13)

Plugging Eqs. 1.12 and 1.13 into Eq. 1.10, we can get a value for the matrix K* to be: $$ K^{*}=\begin{bmatrix} 10& -8.165\\ -8.165& 17.5 \end{bmatrix} $$ (1.14)

Using Eq. 1.9, we can now find the eigenvalues by using the equation of the eigenvalue: $$ det(K^{*}-\lambda I)=0 $$ (1.15) Where I for this particular problem is the 2x2 identity matrix: $$ I=\begin{bmatrix} 1 & 0\\ 0& 1 \end{bmatrix} $$ (1.16)

Honor Pledge
On our honor, this problem was solved using our groups previous solution R4.5 by modifying the code to be a standard eigen problem.

Problem Statement
Re-work problem 4.5/5.5 by transforming the general eigenvalue problem in a standard eigenvalue problem

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.

Because we want to transform this problem into a standard eigenvalue problem, we must transform K to K* using the formula: $$ K^{*}=\begin{bmatrix} 10& -8.165\\ -8.165& 17.5 \end{bmatrix} $$<p style="text-align:right"> (2.0)

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 from our own Problem 3.1 was used.

Given: 3-Bar Truss
A three-bar plane truss with identical members is presented, and is shown in the image below.

Element 2 is vertical, and all members make equal angles with each other. A force is applied at Node 1, while Nodes 2 and 3 can have non-zero or zero prescribed displacement degrees of freedom.



The following member properties are given:

$$ E = 206 \ GPa $$

$$ A = 1*10^{-4} \ m^2 $$

$$ L = 1 \ m $$

The following force and displacement DOFs are given:

$$ F = 20,000 \ N $$

$$ d_3= 2 \ cm $$

$$ d_4= -1 \ cm $$

$$ d_5= -3 \ cm $$

$$ d_6= 5 \ cm $$

$$ d_7= 0 \ cm $$

$$ d_8= 0 \ cm $$

Compute the reactions for non-zero displacement dofs
Compute the reactions for the case of non-zero prescribed displacement degrees of freedom.

Compute the reactions for zero displacement dofs
Compute the reactions for the case of zero prescribed displacement degrees of freedom.

Compare the reactions
Compare the reactions found for the two different cases.

Solving for Non-Zero Displacement DOFs
In order to solve for the desired values, the following MATLAB code from Problem 3.1 was used:

Output of MATLAB Code
The above code returns the following output:

k = 1.545e+007 -8.9201e+006 -1.545e+007  8.9201e+006 -8.9201e+006   5.15e+006  8.9201e+006   -5.15e+006 -1.545e+007 8.9201e+006   1.545e+007 -8.9201e+006 8.9201e+006  -5.15e+006 -8.9201e+006    5.15e+006 k = 7.7238e-026 1.2614e-009 -7.7238e-026 -1.2614e-009 1.2614e-009   2.06e+007 -1.2614e-009   -2.06e+007 -7.7238e-026 -1.2614e-009 7.7238e-026  1.2614e-009 -1.2614e-009  -2.06e+007  1.2614e-009    2.06e+007 k = 1.545e+007 8.9201e+006  -1.545e+007 -8.9201e+006 8.9201e+006   5.15e+006 -8.9201e+006   -5.15e+006 -1.545e+007 -8.9201e+006  1.545e+007  8.9201e+006 -8.9201e+006  -5.15e+006  8.9201e+006    5.15e+006 K = Columns 1 through 5 3.09e+007 1.8626e-009 -7.7238e-026 -1.2614e-009  -1.545e+007 1.8626e-009   3.09e+007 -1.2614e-009   -2.06e+007  8.9201e+006 -7.7238e-026 -1.2614e-009 7.7238e-026  1.2614e-009            0 -1.2614e-009  -2.06e+007  1.2614e-009    2.06e+007            0 -1.545e+007 8.9201e+006            0            0   1.545e+007 8.9201e+006  -5.15e+006            0            0 -8.9201e+006 -1.545e+007 -8.9201e+006           0            0            0 -8.9201e+006  -5.15e+006            0            0            0 Columns 6 through 8 8.9201e+006 -1.545e+007 -8.9201e+006 -5.15e+006 -8.9201e+006  -5.15e+006 0           0            0            0            0            0 -8.9201e+006            0            0 5.15e+006           0            0 0  1.545e+007  8.9201e+006 0 8.9201e+006    5.15e+006 R = 14142       14142            0            0            0            0            0            0 d = -0.028976    0.010785         0.02        -0.01        -0.03         0.05            0            0 rf = -2.6217e-011 -4.2816e+005 -3.6562e+005 2.1109e+005 3.5148e+005 2.0293e+005 forces = -4.2219e+005 -4.2816e+005 -4.0586e+005

The first, second, and third outputs k give the element stiffness matrices for element 1, 2, and 3, respectively.

K gives the global stiffness matrix.

R gives the RHS external force matrix.

d gives the global displacement DOFs.

rf gives the reaction force components.

forces gives the member forces.

Plot of Deformed and Undeformed Shape
To plot the figure, the following MATLAB code was used.

The resulting figure for non-zero displacement DOFs is shown below.



Output of MATLAB Code
By changing the prescribed displacement degrees of freedom in the MATLAB code ("ebcVals") to zero, the following results are obtained.

k = 1.545e+007 -8.9201e+006 -1.545e+007  8.9201e+006 -8.9201e+006   5.15e+006  8.9201e+006   -5.15e+006 -1.545e+007 8.9201e+006   1.545e+007 -8.9201e+006 8.9201e+006  -5.15e+006 -8.9201e+006    5.15e+006 k = 7.7238e-026 1.2614e-009 -7.7238e-026 -1.2614e-009 1.2614e-009   2.06e+007 -1.2614e-009   -2.06e+007 -7.7238e-026 -1.2614e-009 7.7238e-026  1.2614e-009 -1.2614e-009  -2.06e+007  1.2614e-009    2.06e+007 k = 1.545e+007 8.9201e+006  -1.545e+007 -8.9201e+006 8.9201e+006   5.15e+006 -8.9201e+006   -5.15e+006 -1.545e+007 -8.9201e+006  1.545e+007  8.9201e+006 -8.9201e+006  -5.15e+006  8.9201e+006    5.15e+006 K = Columns 1 through 4 3.09e+007 1.8626e-009 -7.7238e-026 -1.2614e-009 1.8626e-009   3.09e+007 -1.2614e-009   -2.06e+007 -7.7238e-026 -1.2614e-009 7.7238e-026  1.2614e-009 -1.2614e-009  -2.06e+007  1.2614e-009    2.06e+007 -1.545e+007 8.9201e+006            0            0 8.9201e+006  -5.15e+006            0            0 -1.545e+007 -8.9201e+006           0            0 -8.9201e+006  -5.15e+006            0            0 Columns 5 through 8 -1.545e+007 8.9201e+006  -1.545e+007 -8.9201e+006 8.9201e+006  -5.15e+006 -8.9201e+006   -5.15e+006 0           0            0            0            0            0            0            0   1.545e+007 -8.9201e+006            0            0 -8.9201e+006   5.15e+006            0            0 0           0   1.545e+007  8.9201e+006 0           0  8.9201e+006    5.15e+006 R = 14142       14142            0            0            0            0            0            0 d = 0.00045767  0.00045767            0            0            0            0            0            0 rf = -5.773e-013 -9428.1     -2988.6       1725.5       -11154      -6439.5 forces = -3450.9     -9428.1        12879

Plot of Deformed and Undeformed Shape
The resulting figure for non-zero displacement DOFs is shown below. Displacements were increased by a factor of 100 to clearly show the difference.



Comparing the reactions
For the non-zero case, the reactions are:

rf = -2.6217e-011 -4.2816e+005 -3.6562e+005 2.1109e+005 3.5148e+005 2.0293e+005

For the zero case, the reactions are:

rf = -5.773e-013 -9428.1    -2988.6      1725.5      -11154     -6439.5

As can be seen, the reactions for the zero displacement DOF case are several orders of magnitude smaller.

Honor Pledge
On our honor, this problem was solved on our own with the help of notes "kim.2008.chap2.pdf" and "Introduction to Finite Element Analysis and Design" by Kim and Sankar

Problem Statement
A space truss, consisting of a 25 truss member is subject to a constant force of 60,000lbs as shown in the figure. Each member has a Young's modulus E = 3 x 10^7 psi. From this a MATLAB code is to be developed to solve for the deformed shape. Nodes 7,8,9, and 10 are all constrained to the ground and do not move. The force is applied at nodes 1 and 2.



Solution
This gave k values of k(:,:,1) =

1.0e+007 *

3.1416        0         0   -3.1416         0         0         0         0         0         0         0         0         0         0         0         0         0         0   -3.1416         0         0    3.1416         0         0         0         0         0         0         0         0         0         0         0         0         0         0

k(:,:,2) =

1.0e+007 *

0.5964   0.2982   -0.7951   -0.5964   -0.2982    0.7951    0.2982    0.1491   -0.3976   -0.2982   -0.1491    0.3976   -0.7951   -0.3976    1.0602    0.7951    0.3976   -1.0602   -0.5964   -0.2982    0.7951    0.5964    0.2982   -0.7951   -0.2982   -0.1491    0.3976    0.2982    0.1491   -0.3976    0.7951    0.3976   -1.0602   -0.7951   -0.3976    1.0602

k(:,:,3) =

1.0e+007 *

0.5964  -0.2982    0.7951   -0.5964    0.2982   -0.7951   -0.2982    0.1491   -0.3976    0.2982   -0.1491    0.3976    0.7951   -0.3976    1.0602   -0.7951    0.3976   -1.0602   -0.5964    0.2982   -0.7951    0.5964   -0.2982    0.7951    0.2982   -0.1491    0.3976   -0.2982    0.1491   -0.3976   -0.7951    0.3976   -1.0602    0.7951   -0.3976    1.0602

k(:,:,4) =

1.0e+007 *

0.5964  -0.2982   -0.7951   -0.5964    0.2982    0.7951   -0.2982    0.1491    0.3976    0.2982   -0.1491   -0.3976   -0.7951    0.3976    1.0602    0.7951   -0.3976   -1.0602   -0.5964    0.2982    0.7951    0.5964   -0.2982   -0.7951    0.2982   -0.1491   -0.3976   -0.2982    0.1491    0.3976    0.7951   -0.3976   -1.0602   -0.7951    0.3976    1.0602

k(:,:,5) =

1.0e+007 *

0.5964   0.2982    0.7951   -0.5964   -0.2982   -0.7951    0.2982    0.1491    0.3976   -0.2982   -0.1491   -0.3976    0.7951    0.3976    1.0602   -0.7951   -0.3976   -1.0602   -0.5964   -0.2982   -0.7951    0.5964    0.2982    0.7951   -0.2982   -0.1491   -0.3976    0.2982    0.1491    0.3976   -0.7951   -0.3976   -1.0602    0.7951    0.3976    1.0602

k(:,:,6) =

1.0e+007 *

0        0         0         0         0         0         0    0.2724   -0.7263         0   -0.2724    0.7263         0   -0.7263    1.9369         0    0.7263   -1.9369         0         0         0         0         0         0         0   -0.2724    0.7263         0    0.2724   -0.7263         0    0.7263   -1.9369         0   -0.7263    1.9369

k(:,:,7) =

1.0e+007 *

0        0         0         0         0         0         0    0.2724    0.7263         0   -0.2724   -0.7263         0    0.7263    1.9369         0   -0.7263   -1.9369         0         0         0         0         0         0         0   -0.2724   -0.7263         0    0.2724    0.7263         0   -0.7263   -1.9369         0    0.7263    1.9369

k(:,:,8) =

1.0e+007 *

0        0         0         0         0         0         0    0.2724   -0.7263         0   -0.2724    0.7263         0   -0.7263    1.9369         0    0.7263   -1.9369         0         0         0         0         0         0         0   -0.2724    0.7263         0    0.2724   -0.7263         0    0.7263   -1.9369         0   -0.7263    1.9369

k(:,:,9) =

1.0e+007 *

0        0         0         0         0         0         0    0.2724    0.7263         0   -0.2724   -0.7263         0    0.7263    1.9369         0   -0.7263   -1.9369         0         0         0         0         0         0         0   -0.2724   -0.7263         0    0.2724    0.7263         0   -0.7263   -1.9369         0    0.7263    1.9369

k(:,:,10) =

1.0e+007 *

0        0         0         0         0         0         0    3.1416         0         0   -3.1416         0         0         0         0         0         0         0         0         0         0         0         0         0         0   -3.1416         0         0    3.1416         0         0         0         0         0         0         0

k(:,:,11) =

1.0e+007 *

0        0         0         0         0         0         0    3.1416         0         0   -3.1416         0         0         0         0         0         0         0         0         0         0         0         0         0         0   -3.1416         0         0    3.1416         0         0         0         0         0         0         0

k(:,:,12) =

1.0e+007 *

3.1416        0         0   -3.1416         0         0         0         0         0         0         0         0         0         0         0         0         0         0   -3.1416         0         0    3.1416         0         0         0         0         0         0         0         0         0         0         0         0         0         0

k(:,:,13) =

1.0e+007 *

3.1416        0         0   -3.1416         0         0         0         0         0         0         0         0         0         0         0         0         0         0   -3.1416         0         0    3.1416         0         0         0         0         0         0         0         0         0         0         0         0         0         0

k(:,:,14) =

1.0e+006 *

1.5457   3.4006    2.4732   -1.5457   -3.4006   -2.4732    3.4006    7.4814    5.4410   -3.4006   -7.4814   -5.4410    2.4732    5.4410    3.9571   -2.4732   -5.4410   -3.9571   -1.5457   -3.4006   -2.4732    1.5457    3.4006    2.4732   -3.4006   -7.4814   -5.4410    3.4006    7.4814    5.4410   -2.4732   -5.4410   -3.9571    2.4732    5.4410    3.9571

k(:,:,15) =

1.0e+006 *

1.5457  -3.4006    2.4732   -1.5457    3.4006   -2.4732   -3.4006    7.4814   -5.4410    3.4006   -7.4814    5.4410    2.4732   -5.4410    3.9571   -2.4732    5.4410   -3.9571   -1.5457    3.4006   -2.4732    1.5457   -3.4006    2.4732    3.4006   -7.4814    5.4410   -3.4006    7.4814   -5.4410   -2.4732    5.4410   -3.9571    2.4732   -5.4410    3.9571

k(:,:,16) =

1.0e+006 *

1.5457  -3.4006   -2.4732   -1.5457    3.4006    2.4732   -3.4006    7.4814    5.4410    3.4006   -7.4814   -5.4410   -2.4732    5.4410    3.9571    2.4732   -5.4410   -3.9571   -1.5457    3.4006    2.4732    1.5457   -3.4006   -2.4732    3.4006   -7.4814   -5.4410   -3.4006    7.4814    5.4410    2.4732   -5.4410   -3.9571   -2.4732    5.4410    3.9571

k(:,:,17) =

1.0e+006 *

1.5457   3.4006   -2.4732   -1.5457   -3.4006    2.4732    3.4006    7.4814   -5.4410   -3.4006   -7.4814    5.4410   -2.4732   -5.4410    3.9571    2.4732    5.4410   -3.9571   -1.5457   -3.4006    2.4732    1.5457    3.4006   -2.4732   -3.4006   -7.4814    5.4410    3.4006    7.4814   -5.4410    2.4732    5.4410   -3.9571   -2.4732   -5.4410    3.9571

k(:,:,18) =

1.0e+006 *

7.4814  -3.4006    5.4410   -7.4814    3.4006   -5.4410   -3.4006    1.5457   -2.4732    3.4006   -1.5457    2.4732    5.4410   -2.4732    3.9571   -5.4410    2.4732   -3.9571   -7.4814    3.4006   -5.4410    7.4814   -3.4006    5.4410    3.4006   -1.5457    2.4732   -3.4006    1.5457   -2.4732   -5.4410    2.4732   -3.9571    5.4410   -2.4732    3.9571

k(:,:,19) =

1.0e+006 *

7.4814   3.4006   -5.4410   -7.4814   -3.4006    5.4410    3.4006    1.5457   -2.4732   -3.4006   -1.5457    2.4732   -5.4410   -2.4732    3.9571    5.4410    2.4732   -3.9571   -7.4814   -3.4006    5.4410    7.4814    3.4006   -5.4410   -3.4006   -1.5457    2.4732    3.4006    1.5457   -2.4732    5.4410    2.4732   -3.9571   -5.4410   -2.4732    3.9571

k(:,:,20) =

1.0e+006 *

7.4814   3.4006    5.4410   -7.4814   -3.4006   -5.4410    3.4006    1.5457    2.4732   -3.4006   -1.5457   -2.4732    5.4410    2.4732    3.9571   -5.4410   -2.4732   -3.9571   -7.4814   -3.4006   -5.4410    7.4814    3.4006    5.4410   -3.4006   -1.5457   -2.4732    3.4006    1.5457    2.4732   -5.4410   -2.4732   -3.9571    5.4410    2.4732    3.9571

k(:,:,21) =

1.0e+006 *

7.4814  -3.4006   -5.4410   -7.4814    3.4006    5.4410   -3.4006    1.5457    2.4732    3.4006   -1.5457   -2.4732   -5.4410    2.4732    3.9571    5.4410   -2.4732   -3.9571   -7.4814    3.4006    5.4410    7.4814   -3.4006   -5.4410    3.4006   -1.5457   -2.4732   -3.4006    1.5457    2.4732    5.4410   -2.4732   -3.9571   -5.4410    2.4732    3.9571

k(:,:,22) =

1.0e+006 *

3.8684   3.8684    6.1894   -3.8684   -3.8684   -6.1894    3.8684    3.8684    6.1894   -3.8684   -3.8684   -6.1894    6.1894    6.1894    9.9030   -6.1894   -6.1894   -9.9030   -3.8684   -3.8684   -6.1894    3.8684    3.8684    6.1894   -3.8684   -3.8684   -6.1894    3.8684    3.8684    6.1894   -6.1894   -6.1894   -9.9030    6.1894    6.1894    9.9030

k(:,:,23) =

1.0e+006 *

3.8684  -3.8684    6.1894   -3.8684    3.8684   -6.1894   -3.8684    3.8684   -6.1894    3.8684   -3.8684    6.1894    6.1894   -6.1894    9.9030   -6.1894    6.1894   -9.9030   -3.8684    3.8684   -6.1894    3.8684   -3.8684    6.1894    3.8684   -3.8684    6.1894   -3.8684    3.8684   -6.1894   -6.1894    6.1894   -9.9030    6.1894   -6.1894    9.9030

k(:,:,24) =

1.0e+006 *

3.8684   3.8684   -6.1894   -3.8684   -3.8684    6.1894    3.8684    3.8684   -6.1894   -3.8684   -3.8684    6.1894   -6.1894   -6.1894    9.9030    6.1894    6.1894   -9.9030   -3.8684   -3.8684    6.1894    3.8684    3.8684   -6.1894   -3.8684   -3.8684    6.1894    3.8684    3.8684   -6.1894    6.1894    6.1894   -9.9030   -6.1894   -6.1894    9.9030

k(:,:,25) =

1.0e+006 *

3.8684  -3.8684   -6.1894   -3.8684    3.8684    6.1894   -3.8684    3.8684    6.1894    3.8684   -3.8684   -6.1894   -6.1894    6.1894    9.9030    6.1894   -6.1894   -9.9030   -3.8684    3.8684    6.1894    3.8684   -3.8684   -6.1894    3.8684   -3.8684   -6.1894   -3.8684    3.8684    6.1894    6.1894   -6.1894   -9.9030   -6.1894    6.1894    9.9030