**SUMMARY**

The missing mass method in spectrum analysis within Ansys Mechanical compensates for the higher-frequency modes not included in the modal analysis, ensuring a more accurate assessment of structural response under dynamic loading. By accounting for the quasi-static effects of the unconsidered modes, this method improves the overall accuracy of dynamic analyses without the computational burden of calculating all high-frequency modes. The primary advantages include enhanced accuracy, reduced computational cost, making it a valuable tool for comprehensive structural analysis in dynamic design and evaluation.

In this blog the basic theory behind this method is presented as well as one simple example to compare results including this calculation to a model using only the first truncated modes.

**Theoretical Background**

Modal analysis is a fundamental tool in structural dynamics to determine the natural frequencies and corresponding mode shapes of a structure. Each mode represents a specific pattern of vibration that the structure can undergo. Response spectrum analysis evaluates the maximum response of a structure to a given set of ground motion inputs, typically provided in the form of a response spectrum. This analysis is particularly useful for seismic design and assessment.

Problem with Modal Truncation

High-frequency modes are typically less significant individually but can collectively impact the overall dynamic response of the structure. However, calculating all these modes can be computationally prohibitive. That’s why engineers often truncate the modal analysis to include only the most significant lower-frequency modes. This truncation introduces a potential error because the contribution of the high-frequency modes is neglected.

The Missing Mass Method

The missing mass method addresses this issue by accounting for the approximate contribution of the higher-frequency modes that are not included in the modal analysis. Here's how the method works in detail:

In a typical response under spectrum load, beyond the frequency f_{ZPA}, the rigid response dominates the dynamic response and will act in phase in all modes.

If we apply an acceleration field of S_{ZPA} on a static analysis can calculate what the total inertia force should be above f_{ZPA}.

And we know the contribution of each obtained mode.

Then we can subtract the sum of modal inertia forces from the total inertia force to obtain the “missing” inertia force.

Finally, the missing mass response will be:

And this value can be included in the total response of the Spectrum Analysis.

**Implementation in Ansys Mechanical**

Set up a modal analysis in Ansys Mechanical. Define the geometry, material properties, and boundary conditions, the number of modes to be calculated or the frequency range up to which modes should be included. Then execute the modal analysis to obtain the natural frequencies and mode shapes.

Set up the response spectrum analysis. This includes importing the response spectrum curve, which represents the seismic or dynamic input. Configure the analysis to include the relevant parameters, such as damping ratio and direction of loading.

Enable Missing Mass Option:

In the spectrum analysis settings, enable the option to include the missing mass correction. This ensures that Ansys Mechanical will automatically calculate the contribution of the higher-frequency modes not included in the modal analysis. Here, is necessary to define the S_{ZPA} value.

**EXAMPLE:**

A simple beam model has been constructed to show the difference between including the missing mass effect and using only the truncated modes. A tower with fixed supports in all base point is used for this example. In the modal analysis 50 modes were extracted from the solution. Although this is a high number of modes, the ratio of effective mass to total mass is low specially in Y direction (2.3%).

In the spectrum analysis an acceleration profile in Z direction to all supports is defined as follows:

Note in this case the ‘Missing Mass Effect’ property has been defined as ‘No’.

Other similar case has been setup with only one difference. Activating the missing mass effect and defining the S_{ZPA} acceleration value.

Results:

Once the results have been combined, is possible to evaluate displacement. In the left, the result without missing mass effect. Please note the difference with the right image where the missing mass effect has been considered. This will increase accuracy in calculation.

**CONCLUSION.**

Implementing the missing mass method in Ansys Mechanical offers several significant advantages that enhance the overall accuracy and reliability of structural response analysis under dynamic loading conditions. By accounting for the effects of higher-frequency modes that are typically excluded in modal analysis due to practical constraints, the missing mass method ensures that the full dynamic behavior of the structure is considered. This method reduces computational costs by avoiding the need to calculate an extensive number of high-frequency modes while still capturing their collective impact through a quasi-static correction. Overall, incorporating the missing mass method in Ansys Mechanical enables engineers to achieve a more precise and efficient evaluation of structural performance, ultimately leading to safer and more reliable designs.

###### Tags:

Modal Analysis, Multibody Dynamics, Structural Analysis, Computational effort, Modal Effective Mass, ANSYS Mechanical, Missing mass effectMay 28, 2024