### Modal(Seismic) Analysis [Part 2]

In the previous posting, we have reviewed on equivalent static analysis, frequency response analysis, response spectrum analysis and vibration analysis. Why suddenly we want to find out about mode analysis?

Equivalent static analysis is calculated by changing seismic load, which is dynamic load, to static load. The static load is derived from the primary mode. In addition, the response spectral analysis is an analysis type considering various modes as well as the primary mode. In the end, vibration analysis can be performed with an essential comprehension of the mode.

In practice, modal analysis must always be performed in advance before performing vibration analysis since the vibration analysis uses the modal analysis results to obtain the required results. Therefore, the modal analysis is the basis of all dynamic analysis, and plays a very important role in understanding the noise vibration characteristics, especially in the mechanical field.

### So what is the modal analysis?

* Modal analysis, eigenvalue analysis, and free vibration analysis are the identical analysis.

Mode analysis is an analysis to understand the inherent characteristic such as natural frequency and mode shape of a structure in a free vibration state. It predicts the shape of the structure due to resonance or vibration. But what is free vibration? Before that, is it common to feel the vibration usually? In fact, all structures we live in are exposed to vibration loads. It is not so severe that we do not need to perform vibration analysis on all structures. Vibration… No matter how you think, only earthquake may come up with your mind. However earthquakes are the case of extreme vibration.

Let's take an example of vibration in real life. Could you feel the vibrations caused by the engine vibration even when you are stopping while moving in a car? I think that many people have already felt the vibration when using a computer. The fan is spinning in the main computer and generating vibration.

Figure 1. Vibration is the motion that continues to move when the initial force is applied based on the point of equilibrium.

Vibration is a status in which an object moves repeatedly based on an equilibrium state. These vibrations can be classified into free vibration and forced vibration. Let's look at an example of what each one means. First, free vibration means a phenomenon that vibrates with its inherent force without external force action, and forced vibration means that vibration is forcibly generated. For example, if you pull a stationary object in one direction and release it, the object will vibrate by itself for a while. These vibrations are called free vibrations. Forced vibrations can be considered as a phenomenon in which objects are shaken by hand continuously. Therefore, the free vibration vibrates with the inherent frequency and natural frequency of the object, however the forced vibration vibrates with the externally applied excitation frequency, that is, the holding frequency. Therefore, we perform the analysis in the free vibration state since we must obtain the natural frequency from the mode analysis.

Figure 2.  In mode analysis, there is no damping force and no external force. In static analysis, there is no inertial and damping force.

### Do you remember the equation of motion in linear static analysis?

Here, modal analysis is free vibration analysis without damping force and external force. Since there is no external force, it is not necessary to input the load condition separately. Therefore, we can express it as a differential equation to find the solution of the following equation.

To find the general solution of this differential equation, suppose the solution is {x} =  {Ø} sin(ωt),  in other words, {x} = {Ø}e^(iωt).

This equation is the eigenvalue problem, where ω2 is the eigenvalue, ω is the frequency, and {Ø} is the mode shape. Therefore, the purpose of modal analysis is to find the solution of this equation. In the end, the modal analysis is to find the frequency and mode shape where the inertia force and the elastic force are in the equilibrium state. Note that ω frequency is an angular frequency (rad / sec), which must be divided by 2π to obtain the natural frequency f [Hz]. (ω = 2∏f)

What is the relationship between the equilibrium of inertia force and restoring force and the vibration? In conclusion, if inertia force and restoring force are in equilibrium state, vibration occurs. To understand this, let’s look at what each force means.

Inertia force is the ability of an object to resist change and to maintain the current movement. The restoring force is the resilience that the structure returns to its original shape after deformation. Let's look at the relationship between vibration, inertial force and restoring force when we bend and then release the beam. If you bend the beam, it will try to return to its initial position due to restoring force. However, due to the inertia force to maintain the motion, the movement will continue without stopping at the initial position.

What if the restoring force is greater in this case? The first few times, it will go past the initial position and continue to move, but eventually it will return to the initial equilibrium status and stop moving.

So if the inertia force is larger, will it continue to exercise No. Since inertia force is the force to maintain the motion, it is difficult to change the direction of motion if the inertia force is greater than the restoring force. As a result, the vibration of the beam will gradually decrease. Therefore, when the inertia force and the restoring force are in equilibrium status, the vibration can be repeated infinitely at a constant speed without stopping the movement. This motion (vibration) speed is called the natural frequency of the structure and the motion (vibration) behavior is the mode shape.