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The harmonic oscillator is also named as single degree of freedom (SDOF) system. Realistic systems consist of multiple degrees of freedom (MDOF). In order to keep things manageable we stay in a first step with two degrees of freedom. As for the SDOF case several phenomena can be treated exemplarily, especially the coupling effects. For presenting the idea of MDOF systems the start is done by two degrees of freedom (2DOF). The equations of motion for such a system as shown in Figure 1.11 are

(1.69)

(1.70)

Figure 1.11 Two degrees of freedom system. Source: Alexander Peiffer.

By introducing harmonic motion for u1=u1ejωt and u2=u2ejωt we get

(1.71)

(1.72)

neglecting the time dependence ejωt. It is practical to write this in matrix form:

(1.73)

In the following, the square brackets and the curly brackets denote a coefficient matrix and vector, respectively.