Transmissibility and Transfer Functions
The transmissibility Q of a single-dof system subject to base excitation can be expressed as
where r is the ratio of the excitation frequency to the resonant frequency, and ζ is the fraction of critical damping. At resonance, r = 1, and the relation reduces to
For a lightly-damped system, 1 + (ζ)² ≈ 1, so we get
SINGLE DEGREE OF FREEDOM
The response power spectral density (PSD) at the resonant frequency can be computed as
where Po is the response or output PSD, and Pi is the input PSD.
MULTIPLE DEGREES OF FREEDOM
In contrast, for a more general system (which may have multiple inputs and outputs, as well as an arbitrary number of degrees of freedom), the input and outputs can be related as
where Hja(ω) is the transfer function for input at dof a and output at dof j. It is tempting to try to apply Equation (2) to estimate the response of a MDOF system as though it behaved like a SDOF system, along the lines of Miles’ Relation. To investigate when that may or may not be appropriate, consider a 2-dof spring-mass system subject to base excitation, as shown in Figure 1. The transfer functions for this system are plotted in Figure 2. These transfer functions were generated by a frequency response solution in Nastran, with 4% critical modal damping applied. A SDOF system with 4% modal damping would have transmissibility Q = 12.5. The transfer function magnitude is listed in Table 1 for each degree of freedom and at each resonant frequency. In the spirit of Equation (1), an “equivalent” critical damping fraction is generated using
2-DOF SPRING-MASS SYSTEM
The first mode of the spring-mass system has the masses moving in phase with each other. In some sense, the masses are acting as one unit, and it could be argued that the system is behaving very much like a SDOF system. However, the masses have different equivalent damping. Mass 1 has somewhat more, and mass 2 has somewhat less equivalent damping than the modal fraction of critical damping.
Figure 1: A 2-dof spring-mass system subject to base excitation.
TRANSFER FUNCTIONS CHARACTERIZE 2-DOF SYSTEM…
The masses move out of phase with each other for the second mode of the spring-mass system. The difference between a SDOF and a MDOF system is, the relative motion between the degrees of freedom in the system. For this second mode, that relative motion is proportionally much greater than for the first mode, and it may be for that reason that the equivalent damping calculated for this mode is considerably higher than the modal fraction of critical damping.
Figure 2: Plots of transfer functions for the 2-dof system.
MDOF EQUIVALENT DAMPING REVEALS LIMITS OF SDOF APPROACH
When modal damping is applied to a MDOF system, the system decouples into a set of SDOF systems in modal coordinates, and it would probably be reasonable to apply Equations (1) and (2) to obtain the modal response if a modal input PSD were somehow applied. However, this example shows that the SDOF equations are limited, at best, in their applicability to general systems.
Table 1: Transfer function magnitude and “equivalent” damping for two modes and two measurement locations.