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The goal of modal analysis in structural mechanics is to determine the natural mode shapes and frequencies of an object or structure during free vibration.It is common to use the finite element method (FEM) to perform this analysis because, like other calculations using the FEM, the object being analyzed can have arbitrary shape and the results of the calculations are acceptable.
In structural engineering, modal analysis uses the overall mass and stiffness of a structure to find the various periods at which it will naturally resonate.These periods of vibration are very important to note in earthquake engineering, as it is imperative that a building's natural frequency does not match the frequency of expected earthquakes in the region in which the building is to be ...
Vibration mode of a clamped square plate. The vibration of plates is a special case of the more general problem of mechanical vibrations.The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two.
Modal impact hammer with interchangeable tips and accompanying temporal and frequency responses An ideal impact to a structure is a perfect impulse, which has an infinitely small duration, causing a constant amplitude in the frequency domain; this would result in all modes of vibration being excited with equal energy.
Modal analysis is performed to identify the modes, and the response in that mode can be picked from the response spectrum. These peak responses are then combined to estimate a total response. A typical combination method is the square root of the sum of the squares (SRSS) if the modal frequencies are not close.
The formula for the variation around the mode (ModVR) is derived as follows: = = where f m is the modal frequency, K is the number of categories and f i is the frequency of the i th group. This can be simplified to = where N is the total size of the sample.
A good way to estimate the lowest modal vector (), that generally works well for most structures (even though is not guaranteed), is to assume () equal to the static displacement from an applied force that has the same relative distribution of the diagonal mass matrix terms. The latter can be elucidated by the following 3-DOF example.
Consider an example adopted from John Bitzan’s Transportation Economics Notes. Let X be a binary variable that is equal to 1 with probability γ, and equal to 0 with probability (1 − gamma). Then f(0) = (1 − γ) and f(1) = γ. Suppose that we have 5 observations of X, giving the sample {1,1,1,0,1}.