Hi,
It seems you are trying to study the electromechanical effect on a piezoelectric cantilever beam by plotting a graph such as frequency vs damped vibration and the effect of length, thickness, and width of beam vs frequency.
You can make use of the frequency response along with the physical properties of the beam. For the frequency response function, you can make use of the impedance of the system. For example, there can be two types of impedances: one mechanical impedance and another electrical impedance due to the piezoelectric effect. Refer to the example pseudo code for a better understanding:
% Define material and beam properties as per the requirements, for example:
rho = 7500; % Density (kg/m^3)
Y = 2e11; % Young's Modulus (Pa)
d31 = -190e-12; % Piezoelectric coefficient (C/N)
epsilon = 8.854e-12; % Permittivity of free space (F/m)
epsilon33 = 1000 * epsilon; % Relative permittivity
% Define beam dimensions
L = 0.1; % Length (m)
b = 0.01; % Width (m)
h = 0.001; % Thickness (m)
% Damping ratio
zeta = 0.01; % Damping ratio
% Define the frequency range for the analysis
f_min = 0;
f_max = 1000;
frequencies = linspace(f_min, f_max, 500);
% Assumed a simple single degree of freedom system for the cantilever beam
% and include piezoelectric coupling in the equivalent mass, stiffness, and
% damping matrix. In reality, these would be matrices for a discretized
% system, and the coupling terms would be spread throughout.
% Equivalent mass (m), stiffness (k), and electromechanical coupling (k_e)
m = rho * L * b * h;
k = 3 * Y * b * h^3 / (L^3);
k_e = d31 * b * h / L;
% Define the beams Frequency response function
function FRF = freq_response_func(frequencies, m, k, k_e, zeta, epsilon33, b, h)
omega = 2 * pi * frequencies;
FRF = k_e ./ (-m*omega.^2 + 1i*omega*zeta*k + k - k_e^2./(1i*omega*epsilon33*b*h));
end
% To study the effect of length, thickness, width on the frequency response,
% you would loop over a range of lengths, thicknesses, or widths and recalculate
% the frequency response for each case, then plot the results.
% The loop and plotting code would be similar to the length vs natural frequency
If you would like to study the beam individually, you can refer to the natural frequency and respective damped response. Where the first natural frequency of the beam can be given as:
(1.875^2/(2*pi*L^2))*sqrt((Y*h^2)/(12*rho));
For more information on the dynamics of the damped cantilever beam, refer to the below documentation:
