Vibroacoustic Simulation. Alexander Peiffer

u 1 Choose bold-italic u 2 EndBinomialOrMatrix equals StartBinomialOrMatrix bold-italic upper F Subscript x Baseline 1 Baseline Choose 0 EndBinomialOrMatrix"/> (1.84)

      with the following transfer function

       (1.85)

The result is non-dimensionalized by dividing through the static response u1(0)=Fx1/ksb.

      Assuming zero damping gives the characteristic equation for the combined resonances

       (1.86)

      With the resonance frequencies of each single system ω02=ksb/m, ωs2=ks/ms and the mass ratio μ=ms/m the resonance frequencies of the combined undamped system are given by

       (1.87)

Figure 1.15 shows the result for a master system with m = 0.1 kg and ksb=10 N/m and an additional DVA tuned to the same frequency as the master system with ms=0.02 kg and ks=2 N/m. Several curves for different critical damping ζ are given. From the undamped case we learn that the response can theoretically be reduced to zero but implicating two resonances at different frequencies. With additional damping the response can be diminished for a broad frequency range. The design of the best DVA is an optimisation task depending on several constraints as discussed in detail by Harris and Crede (1976). In the optimisation procedures issues such as total mass, DVA mass displacement, linearity range of the spring, and the dynamic must be considered.

      Figure 1.15 1DOF system with and without DVA. m = 0.1 kg, ms=0.02 kg, ksb=10 N/m, ks=2 N/m. Source: Alexander Peiffer.

      Figure 1.16 Frequency spread for DVA tuned to the same frequency depending on mass ratio. Source: Alexander Peiffer.

      1.4 Multiple Degrees of Freedom Systems MDOF

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.

      1.4.1 Assembling the Mass Matrix

      The mass matrix in multidimensional space follows from Newton’s law of point masses in free space.