Fundamentals of Modal Analysis
Modal analysis is the study of dynamic properties of linear structures such as resonance frequencies and structural modes of structure. It also determines the dynamic responses of the system in response to externally excited disturbances. These disturbances can be operational loads on the system from the working environment or else it can be undesirable excitations that cause damage to the system. Modal analysis performed on the system calculates the different modes of vibration the studied system can exhibit, and the respective natural frequencies associated with each mode. Each mode of vibration is associated with the respective natural frequency with which the entire system or component vibrates. The change in the shape exhibited by the system due to its natural frequencies is called the mode of vibration.
Figure 1: A simple prestressed modal analysis on drum cymbals.
How many modes of vibration does a system have?
The number of modes equals the number of degrees of freedom the system has. There are two modes of vibration while considering a simple mass damper and spring system. Whereas in case of a large system computed by Finite Element Analysis method, the components will be discretized into millions of nodes, so there will be millions of degrees of freedom, and in effect millions of modes of vibration for the entire system.
All individual components, systems and assemblies vibrate at a particular frequency without external loading and constraints. This specific frequency is termed as natural frequency and is solely a property of that component or system. This is true in the case of individual points in a body as well. Each single point can be treated as a body and has its own natural frequency as its property. The only factors affecting the natural frequency of the system are its mass and stiffness. The formula of natural frequency can be written as square root of stiffness divided by mass of system. Thus, increasing the stiffness or decreasing the mass of system leads to increase in natural frequency of system.
What is resonance?
As the system has its own natural frequency, if the external excitation frequency matches with the natural frequency of system, the system begins to vibrate at a large amplitude. This undesirable vibration leads to failure of the system. So, there is a need to keep the natural frequency of the system away from the range of frequency to which the system is subjected to. At times, the frequencies range the system is subjected to cannot be changed. So as system engineers, we should make the necessary alterations in the model to keep the natural frequencies of system away from that frequencies range. Normally, distribution of mass and components in the structure are redistributed, increased, or decreased to achieve necessary stiffness of system.
What are the types of modal analysis?
Free modal analysis and Prestressed modal analysis. In free modal analysis, there are no boundary or loading conditions insisted on the structure while carrying out. Thus, the structure exhibits the first six natural frequencies of the system as zero values. These six natural frequencies show the six rigid body motions of free unconstrained structure. These include the translations and rotations along the three axes. Prestressed modal analysis is carried out after carrying the base static structural analysis on the system. So, there will be additional stiffness on the system other than the structure stiffness, which further increases the system's natural frequency. In simple terms, this can be stated as tensile stress on the system due to loading conditions increases the stiffness and the natural frequency of system. Whereas the compressive stress developed during the base static analysis decreases the stiffness of system and its natural frequency.
([K] - w2 [M]) {Ş} = {0}: Free modal analysis
([K + S] - w2 [M]) {Ş} = {0}: Prestressed modal analysis. Here, [K] is stiffness matrix, w is natural frequency, [M] is mass matrix and {Ş} is mode shape. The above problem is solved as eigne value problem with w2 as eigen value and {Ş} is eigen vector.
How to determine the number of modes of vibrations to be extracted? Or, how to determine the critical modes of vibration which have a high impact on the system?
The answers to these questions are easily tackled by two important terms. 1. Participation factor and 2. Effective mass. The mode participation factor measures the amount of mass of complete system moving in each direction for each mode. So, there will be separate participation factors for each mode of vibrations. The significance of mode participation factor is that, higher the value of the mode participation factor, the more critical is that mode to get excited by the external excitations.
Mode participation factor = {Ş}T [M] {D}, where Ş is mode shape vector, M is mass matrix and D is unit displacement vector of excitation.
Effective mass is clearly the square of mode participation factor. The higher the value of effective mass, the more easily that mode gets excited.
How can the user know whether he/she has extracted all the significant modes of system? There is another parameter called “ratio of sum of effective masses to total mass” which is as important as previously mentioned ones. The closer this parameter to unity, the more significant modes are extracted. To obtain the ratio as unity, we need to extract all the modes of vibration, which is unnecessary and significantly increases the computational time. A value greater than 0.9 is far enough to get all the significant modes.
How to change the natural frequency once the significant mode of vibration coincides with the external excitation frequencies range?
The two ways of changing the natural frequency is by changing the mass or stiffness of the system. The term strain energy density comes into rescue to this problem. Once the modal analysis is carried out, the user can plot the strain energy density variation for the system. Areas of large strain energy density indicate the locations which should be stiffened to increase the frequency for that mode of vibration. Thus, the user can determine the area's large strain and increase the stiffness by modifying the structure at that zone. Similarly, areas of low strain energy density are the places where significant mass reduction can be done without affecting the natural frequency of system.
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