Matlab code 6.3.2:

% plotting of linear damping (Eq. 6.3.10).
clc
clear all
a0=3;
ep=.5;
mu=.09;
t=0:0.1:100;
omega=1;
beta=-3.15;
a=a0*exp(-ep*mu*t);
u=a0*exp(-ep*mu*t).*cos(omega*t+beta);
plot(t,u,t,a,'--',t,-a,'--')
% title('SYSTEM WITH LINEAR DAMPING')
set(findobj(gca,'Type','line'),'Color','b','LineWidth',2);
set(gca,'FontSize',14)
xlabel('t','fontsize',14,'fontweight','b');
ylabel('u','fontsize',14,'fontweight','b');
grid on

Single degree of freedom system with quadratic damping.

Here, the equation of motion of the system can be written as
.........................................................................................(6.3.14)

Similar to viscous damping here also using KB method the solution can be written as

.........................................................................................................(6.3.15)

Here, a and β can be given by Eq. (6.3.5) and (6.3.6). Now using the expression for in Eq. (6.3.5) and (6.3.6), one may write

= ........................................................(6.3.16)

and (6.3.17)