# What is the interpretation of 'ms' from the modalfit function?

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FHkit on 1 Dec 2023
Edited: Aiswarya on 22 Dec 2023
Hello,
I want to know what the 'ms' of the modalfit (Modal parameters from frequency-response functions - MATLAB modalfit - MathWorks Deutschland) Function is. In the documentation it says 'mode shape' and sometimes 'mode shape vector'.
I simulated a simple spring-mass-damper system (SDOF) and generated the FRF using modalfrf. When I use modalfit, ms is a complex number. - But when I calculate the mode shape vector by hand its not a complex number at all (and it shouldn't be).
So my question is: What actually is that 'ms' that I get when using modalfit?

Aiswarya on 22 Dec 2023
Edited: Aiswarya on 22 Dec 2023
The 'ms' calculated from 'modalfit' function is basically mode shape, which refers to the shape that a structure takes while vibrating at one of its natural frequencies. For a simple single-degree-of-freedom (SDOF) system, the mode shape is a measure of the relative displacement of the mass when the system vibrates at its natural frequency. The mode shapes returned by the modalfit are complex because they are derived by frequency response functions (frf) which are inherently complex. For interpreting complex mode shapes:
• Real part - The magnitude of the complex mode shape vector, i.e. the relative amplitude of motion at that point.
• Imaginary part - The phase angle of the complex mode shape vector, i.e. the phase relationship of the motion at that point relative to the input force.
When you calculate the mode shape by hand for a simple SDOF system, you consider only the amplitude of the displacement, which results in a real number. However, while experimentation with damped systems, the phase relationship is crucial for understanding how the structure deforms over time during vibration. Hence the imaginary part, which also represents the damped natural frequency.
An example for understanding the calculation for the same use case, spring-mass-damper system which has one mode (SDOF) is given in the following resource: https://www.mathworks.com/help/signal/ref/modalfrf.html?searchHighlight=modal%20analysis%20modalfit&s_tid=srchtitle_support_results_8_modal%20analysis%20modalfit#bvncuj_
Hope this helps!

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