pfa

The governing differential equations or motion equations of the spring mass system can be obtained
by equating the forces on the individual masses
Consider Mass 1: As per the force equilibrium the motion equation can be written as,
̈
Consider Mass 2: As per the force equilibrium the motion equation can be written as,
̈
Consider Mass 3: As per the force equilibrium the motion equation can be written as,
̈
The above equations can be expressed in the Matrix form as follows:
[ ]{ ̈} [ ]
Where is the [ ]Mass Matrix, and[ ] is the stiffness matrix, ̈ is the acceleration vector and
[ ] is the displacement vector.
[ ]=[



] which is = [



]
[ ] [



] which is = [



] and { ̈} {


̈
̈
̈
} and {



}
Let [ ] be a Matrix such that [ ] [ ] [ ]
Program:
M=[1 0 0; 0 0.75 0; 0 0 1.25]; %% Mass matrix from motion eqn.
K=[6 -3 -3; -3 6 -3; -3 -3 6]; %%Stiffness matrix from motion
eqn.
A=inv(M)*K
[v,d]=eig(A); %Find Eigenvalues and vectors.
omega=sqrt(diag(-d)); %frequencies
x0=[1 1 1]' %Initial condition
gam=inv(v)*x0 % unknown coefficients
disp('A matrix'); disp(A)
disp('Eigenvalues');...