Below is a table of support positions for three-point mounts of rectangular plates to maximize the first natural frequency of vibration. If one has a plate with an aspect ratio not listed, then one can interpolate between the values. We will publish a more detailed post giving methods and results at a later date.
Silicon is one of the most commonly encountered materials in precision engineering (including silicon wafers and MEMS), but it poses a challenge in modeling as its material properties are anisotropic (orthotropic to be exact), meaning that the stiffness varies depending on the direction of loading relative to the crystal orientation.
A great reference for its elastic properties including when isotropic approximations are appropriate is the paper by Hopcroft et al (M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s Modulus of Silicon?,”
Journal of Microelectromechanical Systems, vol. 19, pp. 229-238, 2010. link to researchgate https://www.researchgate.net/publication/224124507_What_is_the_Young%27s_Modulus_of_Silicon). Additionally Hopcroft’s course notes are useful http://micromachine.stanford.edu/~hopcroft/Publications/Hopcroft_E_Si_v1p1.pdf