I got a lot of knowledge of you and my PhD starts getting well because of your precious help (especially the bright and friendly Professor Rouben Rostamian).
I am here seeking for help. I set up the interface matrix for a Winkler foundation in order to find the eigenfrequencies and plot the mode shapes of Figure 8 (a) and (b) from Article 1. Besides, I tried to expand Professor Rouben Rostamian codes to Figure 2 (a) and (b) - without axial force, Figure 3 from Article 2. Furthermore, I applied that codes in Figure 1(b) and (c), also Figure 3 (b) and (c) from Article 3. However, I could obtain the mode shape of Figure 8 (a) and (b) from Article 1 and I am somewhat mystified by my results.
Question: how I can obtain the matrix, characteristic equation, eigenfrequencies and the mode shapes of a beam when supported with Winkler foundation in the configurations I have mentioned above?
Book: Vibration of continuous systems-John Wiley & Sons Ltd: https://onlinelibrary.wiley.com/doi/book/10.1002/9781119424284
Article 1: Cazzani, A., 2013. On the dynamics of a beam partially supported by an elastic foundation: an exact solution-set. International Journal of structural stability and dynamics, 13(08), p.1350045.(https://doi.org/10.1142/S0219455413500454) In case you need the pdf file, let me know, please!
Article 2: Ghannadiasl, A. and Mofid, M., 2015. An analytical solution for free vibration of elastically restrained Timoshenko beam on an arbitrary variable Winkler foundation and under axial load. Latin American Journal of Solids and Structures, 12(13), pp.2417-2438.(https://www.scielo.br/pdf/lajss/v12n13/1679-7825-lajss-12-13-02417.pdf)
Article 3: Doyle, P.F. and Pavlovic, M.N., 1982. Vibration of beams on partial elastic foundations. Earthquake Engineering & Structural Dynamics, 10(5), pp.663-674.(https://doi.org/10.1002/eqe.4290100504) In case you need the pdf file, let me know, please!