% This m-file shows the frequency response functions of a structure
% (represented by its first mode) with and without a 
% tuned mass damper (TMD) tuned to its resonant frequency.
%
% The input parameters to the m-file are 
% the natural frequency of the structure, the mass
% of the structure, the mass ratio of the tMD to that of
% the structure, and the damping ratio of the TMD, as 
% shown below:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Define the parameters of the structure and TMD:
w=2*pi*40;		% mechanical resonace
M=10;			%Kg. mass of the first mode
zeta=0.005;						% damping ratio of the structure
% Note that for numerical purposes a very small structural 
% damping ratio of 0.5% is assumed. If your structure has a higher
% damping ratio, replace 0.005 in zeta to that of your
% structural damping ratio, e.g. 0.02 for 2% damping ratio.

b=0.1;				%mass ratio of TMD to structure (m/M)
zeta_tmd=0.3;		% damping ratio of the TMD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%The equations of motion, in Laplace domain, for a mass (M) 
%spring (K) system (resemblingthe first mode of a 
%structure) equipped with a tuned-mass-damper (m-c-k) is:
%
% |Ms^2+cs+(K+k)  -(cs+k)||x    | |F|    
% |                      ||     |=| |
% |-(cs+k)      ms^2+cs+k||x_tmd| |0|
%
%where s is the Laplace variable, F is the force disturbing 
%the structure, x is the displacement of the structure, and 
%x_tmd is the displacement of the mass of the tuned-mass_damper.
%
%Solving the above equation of motion for x results in the 
%transfer function
% 	   	x                   (ms^2+cs+k)
%   ----------- = -----------------------------------
%        F         (Ms^2+K)(ms^2+cs+k)+ms^2(cs+k)
% where x displacement of the structure subject to
% the disturbance force F.  Note that without
% the absorber m=0 the transfer function is that of the
% original structure, i.e.,
% 	   	x            1
%   ----------- = ----------
%        F         (Ms^2+K)

% Run the m-file with various damping ratios for the TMD and observe
% that for very small damping ratios, the sturctural mode, splits into
% two modes.  The higher the mass ratio m/M, the higher the
% spread between the two new splitted  modes.
%
% Very small damping ratios are not suitable when the
% objective of using TMD is adding damping to the structure.  However
% very small damping ratios are good when the goal is vibration 
% absorption, i.e., lowering the effect of forced vibration at a certain
% frequency such as the rpm of a motor installed on a structure.



K=M*w^2;		%stiffness of the structure
				% assuming modal mass is M Kg               
num_structure=[1];		% transfer function of the structure disp/(static disp.)
den_structure=[M zeta*2*sqrt(K*M)  K];


freq=10:.5:100;	% frequency range of interest
mag_structure=bode(num_structure,den_structure,2*pi*freq);


m=b*M;	%kg, mass of the TMD
w_tmd=w;	% natural frequency of the TMD is set equal to the structure
k=m*w_tmd^2;	% stiffness of the TMD is calculated so that natural
					% frequencies of TMD is the same as that of 
					%the structure
c=zeta_tmd*2*sqrt(k*m);		% damping coefficient of the TMD


num=[m  c k];	% transfer function of the structure with TMD
den=conv([M 0  K],[m c k])+ [0 m*c  m*k 0 0];

mag=bode(num,den,2*pi*freq);
plot(freq,20*log10(mag_structure),'-b',freq,20*log10(mag),'--m')
legend('structure', 'structure+TMD')
xlabel('Frequency, Hz');
ylabel('Magnitude of x/F');
title('FRF of displacment/Force with and without a TMD');