I resolved the factor of two from Griffiths' discussion of dipoles in non-uniform electric fields. The force on a dipole in a non-uniform field is where is the difference in the field between the plus end and the minus end. Component wise, where d is a unit vector. This holds for y and z, the whole thing can also be written as . Since p=qd, we can write .
Jackson derives it differently by deriving the electrostatic energy of a dielectric from the energy of a collection of charges in free space. He then derives the change in energy of a dielectric placed in a fixed source electric field to derive that the energy density w is given by . This explicity explains the factor of two and is an interesting alternative explanation.
par.a = 75e-3/2; % radius [m]
par.h = 1.004e-3; % thickness [m]
par.E = 73.2e9; % Young's modulus [Pa]
par.nu = 0.155; % Poisson's ratio
par.rho = 2202; % density [kg/m^3]
%Calculate fundamental modes of the disk
[freqs, modes, shapes, x, y] = disk_frequencies(par, 10000, 1, 'shapes', 0.5e-3);
%Now we extract the force profile from the COMSOL model
function out = model
% Model exported on Jul 14 2017, 14:47 by COMSOL 188.8.131.522.
model = ModelUtil.create('Model');
fpro = zeros(6, 27);
no = 1;
for count = 1:.2:2
gap = strcat(num2str(count), ' [mm]')
model = fst2(gap);
fpro(no, :)= product(:);
no = no + 1;
I am posting a diagram of the geometric parameters that I swept. The only one not included is the vertical space between the ESD and sample that sweeps perpendicularly out of the image
From the data I have gathered from a variety of MATLAB sweeps, I think that the optimal geometry I can produce has the parameters in the attached image. Neither the original or optimized drawing is to scale. The gap between the arms of the electrodes should be 1.25 mm, the arm width 0.55 mm, the arm length 16 mm, and the offset of the arms 3.5 mm.
It is also optimal to place the ESD as close to the sample disk as can reasonably be achieved, at around 0.5 mm away. Since the force on the disk scales exponentially with the distance from the ESD, decreasing that gap is the most substantial way to impact the excitation. Decreasing the gap from 1 mm to .5 mm increases the excitation of the modes by approximately a factor of 8.
From my simulations, the shift in geometry alone still has a useful impact on the excitation. Modes 1 and 3 are the only two modes that are less excited by the new geometry, mode 1 is 10% weaker and mode 5 is 5% weaker. Modes 5 and 6 are nearly unaffected by the shift, mode 5 is 2% stronger and mode 6 is 5% stronger. Modes 7, 18 and 19 are outliers, 7 is excited by a factor of 7, 18 by a factor of 4 and 19 by a factor of 17. The rest of the modes are improved by between a factor of 1.5 and 3. For mode numbers, shapes, and frequencies a plot is included.