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The considerations above show the concept of how to write the equation of motion in matrix form. The matrices of the equation of motion in the frequency domain follow a certain convention in order to separate mass, stiffness and damping effects. As in Equation (1.73) the equation of motion of every discrete linear mechanical system can be approximated by the following form:

(1.88)

or in the frequency domain as

(1.89)

The coefficients q_i of {q} are generic displacement degrees of freedom, for example the displacements u,v,w in x-, y- and z-directions at different positions. The first matrix [D] is called the dynamic stiffness matrix. The matrices in the parentheses are called mass matrix, damping matrix, stiffness matrix and proportional damping matrix. The solution of equation 1.89 with regard to {q} is called frequency response.

The stiffness matrix must not be confused with the dynamic stiffness matrix. The dynamic stiffness matrix is frequency dependent and includes all matrices whereas the stiffness matrix includes the real and frequency independent stiffness part of the equation of motion. For specific lumped elements like springs and dampers there exist simple sub matrices that can be used to set up the global matrix of a more complex set-up. For illustration, see the example network of masses and springs in Figure 1.17.

Figure 1.17 Multiple degrees of freedom network Source: Alexander Peiffer.

We speak about nodes for the different locations in space running over index i. Each node may have different degrees of freedom(DOF). In our case, there are two translational coordinates u and v. So the displacement and force vector is running over all DOF of all N nodes:

(1.90)

A recipe can be used to assemble the equations of motion in the above given matrix form. For this purpose we derive a set of rules that allows for creating those matrices. This is practical for the understanding of complex mechanical networks but also a good start for understanding the basics of finite element simulation which is also based on mathematical methods to create mass and stiffness matrices from a discretized model. The concept of nodes and DOF is also kept in the finite element simulation.