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).
The simplest vibratory system can be described by a single mass connected to a spring (and possibly a dashpot). The mass
is allowed to travel only along the spring elongation direction. Such systems are called Single Degree-of-Freedom
(SDOF) systems and are shown in the following figure,
The solution to the general SDOF equation of motion is shown in the damped SDOF discussion.
You can easily simulate the motion of spring/mass/damper systems using the powerful software package Interactive Physics from MSC Working Knowledge. It provides an easy-to-use, interactive, and discovery-oriented platform where thought experiments can be quickly reduced
to full simulation results.
