I think modal analysis technology can be said to be the most mature linear analysis technology in the field of dynamics at present, that is to say, the system it targets is an (approximate) linear system. Modal analysis makes use of the most basic property of linear system-homogeneity: if it is assumed that the response of the system to excitation A is x(a) and excitation B is x(b), then the response of the system to excitation c*a+d*b is c*x(a)+d*x(b). When we study vibration theory, we usually start with the single-degree-of-freedom system, and the properties of the single-degree-of-freedom system, such as undamped vibration frequency, damped vibration frequency, * * * vibration frequency, impulse excitation response, etc. It's easy to get.

So when we analyze a multi-degree-of-freedom system, we will wonder if we can turn this multi-degree-of-freedom system into an independent single-degree-of-freedom system in some way. Modal analysis is used to realize this process. Through modal theory, we can transform a system with n degrees of freedom in a generalized coordinate system (usually Cartesian coordinate system) into a modal coordinate system, and each degree of freedom in the modal coordinate system is independent of each other, that is, we can get n independent single-degree-of-freedom systems (real modal theory), which is called decoupling. What we get from the analysis, such as the frequency and mode of the system, is actually the property of each single-degree-of-freedom system after decoupling. Moreover, after obtaining the responses of each single-degree-of-freedom system, we can linearly combine them again and convert them back to the previous generalized coordinate system. The above is just my understanding. If there is anything wrong, please point out.