What is Missing mass method? The details are in the Theory Reference, but a short description of this approach is that we input the ZPA (Zero Period Acceleration), and the thought is that if the structure acts ‘rigidly’, we don’t see amplification at/above that frequency – if we input an acceleration “A” at the base at such high frequencies, then the response at other points should be “A” as well. The value we input into Workbench Mechanical is this acceleration value “A” – the Mechanical solver knows our total mass [M] and the effective mass based on the number of modes we extract [M’]. If we have this assumption that, at high frequencies, the input/output ratio is close to 1, then the ‘missing mass’ is based on the difference between [M] and [M’]. There is a force {F} associated with the acceleration A times the missing mass ([M] – [M’]), and we can get this response {R} as [K]^-1{F}. Think of this like an extra mode shape (although it’s not a mode – it’s a pseudo-static response component) that we then add in the mode combination phase. All of this is described in a more mathematically rigorous method in Theory Reference Equations 15-226 through 15-232 with the mode combination in Equation 15-233. Basically, if we included enough modes (e.g., effective modal mass very close to actual mass), we don’t need this method. However, if we don’t extract/include enough modes and we assume those missing modes are high-frequency response, then the Missing Mass effect tries to include the effect of these modes in an additional pseduo-static mode shape. When to use missing mass method? To address whether Missing mass response is needed the user needs to answer this question: Did I solve enough modes, or are the truncated modes not accurately capturing the mass of the system? A major cause of error in response spectrum analyses is not using sufficient modes to correctly capture the mass. Ideally, you want to compute enough modes to have a high percentage of effective mass, but in some cases, but this may be impractical (resource consuming), especially if the excitation spectrum is a much lower range. One can check the mass captured in the modal analysis, by looking at the cumulative mass fraction in the same direction of excitation as the response spectrum analysis. For example, let’s say that the user extracted 200 modes and the participation factor table in the modal analysis shows that these 200 modes captured 75% of the total mass. If the frequency range the user wants is more than adequately covered by the modes extracted, then the user may not wish to solve for 400 modes (these cases are often associated with a lot of mass at constrained supports (i.e., bulky constrained base) since to get the dynamic content, you need really high modes). Thus, using Missing Mass effect is ideal in this case since those very high frequency modes (outside of frequency range of interest) would be computationally expensive to extract, but without them, we’re missing 25% of the mass of the system in the modal content. If one is still unsure, use more modes and see if the response in the response spectrum analysis changes significantly.
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