What is the meaning of "inverse problems in vibration"? I am asking you to tell me or give me a link which I can understand the meaning of inverse problems in vibration. As it is my first experience in facing the Inverse method please give me a hand as easy as possible.
 A: Inverse problem is the inverse of a forward problem which starts with the cause or source and then calculates the results. So the inverse problem would be to start with the results and then calculates the cause or source.
In the case of vibration the results would be oscillation or a wave.  So the inverse problem would be to start with the oscillation or a wave (the results) and then calculate (or determine) the cause or source. 
Seismic imaging used in oil exploration would be an example.  Pressure pulses are generated which reflect off of points in the ground (the cause or source) and then then echoes (the results) are collected.  An image of the reflecting sources is created by solving the inverse problem with the echoes as the input data.
Your eyes are another example:  They detect spherical waves (the results) propagating from the points on the object (source) you are seeing, then process the waves into an image of the object.  This is an inverse problem solved with the use of focusing lenses.
See for inverse problems in general
https://en.wikipedia.org/wiki/Inverse_problem
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field.
It is called an inverse problem because it starts with the results and then calculates the causes. This is the inverse of a forward problem, which starts with the causes and then calculates the results.
Inverse problems are some of the most important mathematical problems in science and mathematics because they tell us about parameters that we cannot directly observe. They have wide application in optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, and many other fields.
Also see
http://mathworld.wolfram.com/InverseProblem.html
