# Cantilever beam with disc at end (structural mechanics)

I am trying to figure out how I can assemble multiple finite elements to obtain the system mass matrix $\boldsymbol{M}$ and the stiffness matrix $\boldsymbol{K}$.

The situation is depicted in a picture (see below, assume the left side is not fixed). Now, I thought about splitting the body into a cantilever beam and a disc. The beam has four degrees of freedom in a planar analysis. I will refer to $q_1$ as the vertical displacement at the left side of the beam and $q_2$ is the deflection angle at the left side of the beam. At the right side of the beam, the vertical displace is denoted by $q_3$ and the deflection is denoted by $q_4$.

For the disc, I assume the degrees of freedom as depicted in the picture below. Hence, $q_{\text{S},1}$ and $q_{\text{S},3}$ for displacements and $q_{\text{S},2}$ and $q_{\text{S},4}$ for the angles of deflections (double headed arrow). It is obvious that $q_3=q_{\text{S},3}$ and $q_4 =q_{\text{S},4}$.

If I set up the mass matrix $\boldsymbol{M}_\text{B}$ for the beam I see that it has the format $4\times 4$. The same goes for the stiffness matrix $\boldsymbol{K}_\text{B}$. Assume that the mass matrix of the disc is given by $\boldsymbol{M}_\text{D}$ and the stiffness matrix is given by $\boldsymbol{K}_\text{D}$. How can I assemble the system mass matrix $\boldsymbol{M}$ and the stiffness matrix $\boldsymbol{K}$ from the previous matrices?

Feel free to write down the procedure on a sheet of paper, that would be good enough :). I also appreciate a resource that is explaining this procedure. • What are the damping parameters - dashpot symbol? – Solar Mike May 9 '18 at 11:30
• They are $d_\text{B}$ for the beam and $d_\text{D}$ for the disc. – MrYouMath May 9 '18 at 11:42
• Damping isn't part of the question, is it? Could you remove those from the diagram, as well as the constraint? Also maybe rename it to assembling system matrix from element matrices so it's helpful to others who don't have a beam and a disk. – user1318499 May 10 '18 at 0:03

The procedure for assembling the system mass and stiffness matrices are the same.

Repeat steps 1 and 2 below for each element matrix, $K_D$:

1) Expand the element matrix to have the same structure as the system matrix, putting zeros where there's no corresponding value in the element matrix. Do this by inserting each element matrix element into the expanded element matrix row with the same DOF that it came from and the column with the same DOF that it came from. For example:

If the DOFs in the system matrix $K$ are in this order:

$$qs2, q1, q2, q3, q4, qs1$$

and the DOFs in the element matrix $K_D$ are in this order:

$$qs1, qs2, q3, q4$$

and

$$K_D =\begin{bmatrix}a & b & c & d \\ e & f & g & h \\ i & j & k & l \\ m & n & o & p\end{bmatrix}$$

then the expanded element matrix would be

$$\begin{bmatrix}f & 0 & 0 & g & h & e \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ j & 0 & 0 & k & l & i \\ n & 0 & 0 & o & p & m \\ b & 0 & 0 & c & d & a \end{bmatrix}$$

2) Add (matrix addition) the expanded element matrix to the system matrix.

In real life, you wouldn't actually build the expanded matrix explicitly but just add the non-zero elements.