In tools and software for vibroacoustic simulations many different quantities are used. The overview of all those different criteria shall help to avoid mistakes and confusion.
1.3 Two Degrees of Freedom Systems (2DOF)
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
Figure 1.11 Two degrees of freedom system. Source: Alexander Peiffer.
By introducing harmonic motion for u1=u1ejωt and u2=u2ejωt we get
neglecting the time dependence ejωt. It is practical to write this in matrix form:
In the following, the square brackets and the curly brackets denote a coefficient matrix and vector, respectively.
1.3.1 Natural Frequencies of the 2DOF System
We start with a simplified system without damping and external forces in order to get the natural frequencies of the system.
This equation can be rearranged so that it corresponds to the general eigenvalue problem
The non trivial solutions of this are given by:
This leads to the characteristic equation with λ=ω2
With ω12=ks1/m1, ω12=ks1/m1 and ωc2=ksc(m1+m2)m1m2 the solutions are:
The eigenvalues shall be entered into the equations to solve for {Ψi}.
