try fitting an accelerometer or a load cell to pick up the vibration or use a laser vibrometer (This is much easier.) to pick up the velocity of vibration.

Thanks Ranjan Bhattacharyya for the answer, but I want to calculate it not to measure it. Maybe you know how to theoretically calculate these vibrations?

Theoretically, you can calculate the displacement and frequency of vibration just by applying conservation of momentum and energy. If the disspitaion is also present, then you have to take care of energy loss. Still the calculation needs detailed nature of vibration (simple harmonic motion or damped).

theoretically the case that you use as an example could be considered an aproximation of a dirac delta, and as such it would exite all frecuencies with the same amplitude.

Given that, you could calculate this vibration by preforming a modal analisys and adding the response of the structure for N number of modes until you achived the desired precision.

Best regards

Theoretically first you should solve the problm of interaction of the two bodies. That wiil be one of the dynamical (displacements and forces) boundary conditions. Then you should solve the problem for free vibrations with initial conditions. For one-dementional case (two rods) it is a known problem/ The first part you better solve with the method of characteristics. The second part - by Fourier method. (You can see any book on Mathematical physics). In "two or three" - dementional case it is a quite a difficult problem to be solved numerically. These are cases when you use partial defferential equations.

The vibration of A will depend on the shape of A as well as the elastic properties of the material. Does your shape resemble one of the following?

(1) Solid sphere [eg. cricket ball or base ball]

(2) Spherical shell [eg. table-tennis ball, soccer ball]

(3) Solid cylinder [rod] or

(4) Cylindrical shell [pipe]

In the case of (3) or (4) the vibration will depend on whether it is struck at one end or on the side wall.