Multiple Degree of Freedom Systems: Coupling of Equations of Motion
Figure 8.7: Simple two degree of freedom system
As mentioned, these equations are since both and appear in each of these equations. The implication with coupled equations is that they must be solved as a system. Neither of the equations can be solved individually.
Figure 8.8: Simpler two degree of freedom system
In contrast, consider the same system as above but with the central spring removed. This is still a TDOF system as two coordinates are still required to completely specify the positions of the masses. The two equations of motion for this system are
where* (*Note that in this case due to our choices for the system parameters. is a general result for uncoupled systems.)
and and are arbitrary constants. As can be seen, uncoupled systems are significantly easier to solve that coupled ones. Review the effort required to obtain the solution to equations (8.20) and compare that to the effort required to obtain the solution to those in (8.21). The difference is apparent, and will become more significant as systems with more degrees of freedom are considered. It is hopefully clear from this that it is preferable to deal with equations if possible.
Note that in matrix form, equations (8.20) and (8.21) are
respectively. As can be seen, uncoupled equations correspond to matrices (both mass and stiffness matrices must be diagonal). Any off-diagonal term leads to a coupling of the equations. In equation (8.23) the off-diagonal terms occur in the stiffness matrix, which is referred to as . The situation in which the mass matrix contains off-diagonal is termed . It is possible for both type of coupling to be present at the same time.
Coupling and Coordinates
For the two degree of freedom system in Figure (8.7), the equations of motion are coupled as we have seen
Consider what happens however if we look at the sum and difference of these equations:
If we now define new coordinates and as
(c) and (d) become simply
which are now . The solutions to these equations are therefore easy to write down as
and and are the arbitrary constants. Note that we get the same natural frequencies using this representation. Further, and can be found
from and as
These can also be written as
which is exactly the same solution determined previously for this system in (8.14). Here we found the solution using only uncoupled equations in terms of the “coordinates” and . The important point of this exercise is to recognize is that
Coupling is not a property of the system. Rather coupling is a consequence of the choice of coordinates used to describe the system.
Relationship to Mode Shapes
In the linear systems we have been discussing it is (almost) always possible to find a set of coordinates which uncouples the equations of motion. That is essentially what the mode shapes represent: linear combinations of the chosen coordinates that will uncouple the equations of motion:
In the current system, which is relatively simple, we were able to “guess” that the uncoupled coordinates would be and . In general guessing like this is not very feasible. It was done here only to illustrate the concepts involved. For a general vibration problem, the uncoupled coordinates (i.e. mode shapes) can be found by following the procedure outlined earlier. This will work for (almost) all multiple degree of freedom systems.