# Dynamic Analysis and Visualization of Laminate Mindlin Plate

I am trying to make the jump from a FEM plot of a static load applied to Mindlin laminate plate to a surface visualization of various Mode Shapes of the Laminate Plate. For that, I understand that the two big steps are:

1. developing a Mass Matrix $[M]$; and

2. getting the eigenvectors and eigenvalues $[V]$ and $[D]$.

While all my textbooks tell me how to do this for a simple Mindlin plate, I don't know how to do it for a laminate Mindlin plate where the very top and bottom layers are of different thickness. All the resources I have found only deal with the most generic of situations. If anyone of you could explain to me what changes (if any) I need to make to the equations, that would be great.

After assembling my mass and stiffness matrices I solve the generalized eigenvalue problem

D, V = eig(stiffness(activeDof, activeDof), M_g(activeDof, activeDof)


But then I do not fully understand what I do with the results. Also, I don't get how to represent the modes in graph form. I know that I'm asking a lot, but if you could at least point me toward a good resource.

• Regarding your second question: it seems to be a duplicated question of this one. Apr 21, 2017 at 23:03

Regarding your first question. One option would be to try to use what is commonly done in laminates for Kirchoff-Love plates, I would not know how good is that approximation, though.

Another option, is to consider some effective properties and just replace your constitutive tensor in your equations. For example, for the orthotropic case you would have

$$\begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{12} & C_{22} & C_{23} \\ C_{13} & C_{23} & C_{33} \end{bmatrix} = \frac{1}{1-\nu_{12}\nu_{21}} \begin{bmatrix} E_1 & \nu_{12}E_2 & 0 \\ \nu_{21}E_1 & E_2 & 0 \\ 0 & 0 & 2G_{12}(1-\nu_{12}\nu_{21}) \end{bmatrix} \, ,$$

$$D_x \frac{\partial^4 w}{\partial x^4} + 2 D_{xy}\frac{\partial^4 w}{\partial x^2 \partial y^2} + D_y \frac{\partial^4 w}{\partial y^4} = q - \rho h \frac{\partial^2 w}{\partial t^2}$$ where

\begin{align} D_x & = \frac{2h^3 E_1}{3(1 - \nu_{12}\nu_{21})} \\ D_y & =\frac{2h^3 E_2}{3(1 - \nu_{12}\nu_{21})} \\ D_{xy} & = \frac{4h^3 G_{12}}{3} + \frac{2h^3 \nu_{21} E_1}{3(1 - \nu_{12}\nu_{21})} \, , \end{align}

for sure it would be more complex in the case of Mindlin-Reissner plates. You could probably check the references suggested below.

### References

1. Leissa, A. W., & Qatu, M. S. (2011). Vibrations of Continuous Systems.McGraw-Hill.

2. Reddy, J. N. (2004). Mechanics of laminated composite plates and shells: theory and analysis. CRC press.

• Thank you very much for you help! Your references certainly helped a lot! But I was wondering, is there a method to check if my mass matrix is correct? Because I figured out how to create a surface graph with the eigenvectors/values, but my mode shapes are inverted, compared to another reference mode shape. Normally, I would just add a minus sign to the z-values and call it a day, but if I increase the number of elements from 4x4 to 6x6, Mode 1 corrects itself, but Mode 2 to 4 stay inverted. What's going on? Apr 22, 2017 at 6:39
• @AdditionalPylons, the eigenvalues are determined up to a multiplicative constant. It might get less clear if you have degenerate modes (eigenvalues with multiplicity higher than 1), because any linear combination is an eigenvalue as well. Apr 22, 2017 at 14:08
• You are solving a problem of the form $(K - \omega^2 M) u = 0$. If you replace $u$ with $-u$, the result does not change. Therefore, inversion of the mode shape is not an indication of error. However, a translation of the mode shapes usually indicates a problem. Apr 22, 2017 at 21:47