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1 Introduction to Vibration in Systems with Two Degrees of Freedom
Many real world systems can be modeled with single degree of freedom systems. Either 1 system or multiple systems under the principle of superposition. In particular MIMO systems can often be decoupled into multiple SISO systems.
However, not all systems can be adequately modeled with a single degree of freedom so in this article we work through an example system with two degrees of freedom. The Beard text book works this example through the differential equations. A much simpler approach for the controls engineer is to skip the full blown solution to the differential equation. We skip the differential equation solution and all of that math by creating State Space equations directly from the equations of motion.
2 Equations of Motion for Vibration in Systems with Two Degrees of Freedom^{[1]}
A system model with two degrees of freedom is depicted in Figure 1. The two masses are connected by three springs to two walls and each other. There is no damping in the system.
If we consider the case where x_{1} > x_{2} then the free body diagrams become those seen in Figure 2. The equations of motion are
for body 1 
for body 2 
The same equations are obtained for x_{2} > x_{1}. In this case the direction of the central spring force is reversed.
As with all differential equations, we first assume a solution form. The above equations can be solved for the natural frequency and corresponding mode shapes by assuming a solution of the form
and 
Assumptions:
 x_{1} and x_{2} oscillate with the same frequency ω and are either in phase or π out of phase
This is a sufficient condition to make ω a natural frequency.
Substituting these solutions into the equations of motion we get
and
Since these solutions are true for all values on t,
and
A_{1} and A_{2} can be eliminated by writing
Eqn. 1 
This is the characteristic or frequency equation. Alternatively, we may write
Eqn. 2a 
Eqn. 2b 
Thus
and
Eqn. 3 
This result is the frequency equation and could also be obtained by multiplying out the above determinant (Eqn. (1)).
The solutions to Eqn. (3) give the natural frequencies of free vibration for the system in Figure 1. The corresponding mode shapes are found by substituting these frequencies, in turn, into either Eqn. 2a or 2b.
Consider the case when k_{1}=k_{2}=k, and m_{1}=m_{2}=m. The frequency equation is
expanding leads to
or 
Therefore, either
or 
Thus
rad/s and rad/s 
If
then 
then 
This gives the mode shapes corresponding to the frequencies ω_{1} and ω_{2}. Thus, the first mode of free vibration occurs at a frequency
Hz and 
this is, the bodies move in phase with each other and with the same amplitude as if connected by a rigid link. The second mode of free vibration occurs at a frequency
Hz and 
that is, the bodies move exactly out of phase with each other but with the same amplitude.
2.1 State Space Equations for Systems with Two Degrees of Freedom
If we revisit the original equations of motion for this system with two degrees of freedom
for body 1 
for body 2 
then we can form State Space equations directly from these equations.
For body 1 the equation of motion for this system with two degrees of freedom
For body 1 the equation of motion for this system with two degrees of freedom
The state space equations become
where:
 B, C, and D are unknown at this stage.
2.1.1 Most likely values for the rest of the State Space equations
If we assume a 1 dimensional force which is positive with positive x_{1} acting on m_{1} and an analogous force on x_{2} then
and
If we then also assume that the most likely desired outputs are x_{1} and x_{2} then
Therefore
3 Free Motion^{[2]}
The two modes of vibration can be written
and
where the ratio A_{1}/A_{2} is specified for each mode.
Since each solution satisfies the equation of motion, the general solution is
where A_{1}, A_{2}, ψ_{1}, and ψ_{2} are found from the initial conditions.
For example, for the system considered above, if one body is displaced a distance X and released,
and 
Remembering that in this system
 ,
 ,
 , and
we can write
and
Substituting the initial conditions x_{1}(0) = X and x_{2}(0) = 0 gives
and
that is
The remaining conditions give cos ψ_{1} = cos ψ_{2} = 0.
Hence
and
That is, both natural frequencies are excited and the motion of each body has two harmoinc components.
4 Notes
Beards, C. F. 1995 Engineering Vibration Analysis with Applications to Control Systems. ISBN 034063183X