There are 3 degrees
of freedom in this problem since to fully characterize the
system we must know the positions of the three masses
(x1, x2, and
x3).
Three free
body diagrams are needed to form the equations of motion.
However, it is also possible to form
the coefficient matrices directly, since each parameter in a
mass-dashpot-spring system has a very distinguishable role.
Observing the above coefficient matrices, we found that all
diagonal terms are positive and contain terms that are directly
attached to the corresponding elements.
Furthemore, all non-diagonal terms are negative and
symmetric. They are symmetric since they are attached to
two elements and the effects are the same in these two elements (a
condition known as Maxwell's
Reciprocity Therorem). They are negative due to the
relative displacements/velocities of the two attached
elements.
In summary,
1.
Determine the number of degrees of freedom
for the problem; this determines the size of the mass,
damping, and stiffness matrices. Typically, one degree of
freedom can be associated with each mass.
2.
Enter the mass values (if associated with
a degree of freedom) into the diagonals of the mass matrix;
the exact ordering does not matter. All other values in the
mass matrix are zero.
3.
For each mass (associated with a degree of
freedom), sum the damping from all dashpots attached to that
mass; enter this value into the damping matrix at the diagonal
location corresponding to that mass in the mass matrix.
4.
Identify dashpots that are attached to two
masses; label the masses as m and n. Write down
the negative dashpot damping at
the (m, n) and (n, m) locations in
the damping matrix. Repeat for all dashpots. Any remaining
terms in the damping matrix are zero.
5.
For each mass (associated with a degree of
freedom), sum the stiffness from all springs attached to that
mass; enter this value into the stiffness matrix at the
diagonal location corresponding to that mass in the mass
matrix.
6.
Identify springs that are attached to two
masses; label the masses as m and n. Write down
the negative spring stiffness at
the (m, n) and (n, m) locations in
the stiffness matrix. Repeat for all springs. Any remaining
terms in the stiffness matrix are zero.
7.
Sum the external forces applied on each
mass (associated with a degree of freedom); enter this value
into the force vector at the row location corresponding to the
row location for that mass (in the mass matrix).
