To better understandmodal analysis it is important to distinguish between time domain, frequency domain and modal space.
Introduction to time domain, frequency domain and modal space
If we imagine that the ruler is excited at one end by a pulse and an accelerometer is recording its response at the other end, the response will contain all the modes of the ruler. The response will contain all the modes of the ruler. This time response can be converted into the frequency domain by performing a Fourier transform of the signal. Without going into the mathematical details of this transform, the frequency representation of the signal at this excitation is called the "frequency response function (FRF)". This graph shows peaks corresponding to the system's natural frequencies. At each of these eigenfrequencies there is a particular pattern called the "modal distortion". The total frequency response of the system comes from the "summation" of the different eigenmodes. Incidentally, this also applies in the time domain.
Analytical model of localised mass
The rule can be represented using a "localised mass analytical model". This model is usually defined using a set of equations with a coupling between the different points also called degrees of freedom. As the number of equations used to describe the system increases, solving the equations becomes increasingly complex. We often use matrices to help organise all the equations of motion describing the behaviour of the system. The size of the matrices depends on the number of equations. Mathematically, we perform an eigenvalue solution and use the modal transformation equation to convert these coupled equations into a set of decoupled degree-of-freedom systems described by diagonal matrices in a new coordinate system called "modal space".
Since we can divide the analytical model into a set of systems, we could determine the FRF for each of the degrees of freedom. In addition, we could also determine the time response for each of these systems.
To conclude
There is no difference between the time domain, the frequency domain and the modal space. Each domain presents the data from a different perspective. Depending on the need, it is easier to interpret the data in one domain than in the other. For example, to identify activated eigenmodes, it is easier to use a frequency representation than a time representation.)
