PhD of Physic and Mathematics Dokukova N.A., Kaftaikina E.N., Konon N.P.

Belarus State University, Minsk, Belarus

N sync metronomes on a vibrating horizontal platform

The work is devoted to the description, calculation and physical generalizations of the oscillations of a large number of metronomes on a stationary at the initial moment of time, the horizontal platform. Such experiments are widely known in the scientific literature and presented on a video hosting YouTube.

Many metronomes that are installed on the free surface of the platform, backed by resilient springs General Coefficient C, is presented on the dynamic diagram of figure 1. They are taken from a state of rest, giving each of the bodies m_i, the initial deviation . Pendulums metronomes, moving some time with different amplitudes and phases, rocking the platform and, after a short time, begin to synchronize in frequency and phase fluctuations. If this is the same for all of the metronomes mode movements, consistent with displacement of a horizontal platform.

Picture 1 - Diagram of movements n metronomes oscillating on a horizontal platform

For the derivation and study of simple regularities of the influence of mechanical system parameters on the General oscillatory process, consider the linear dynamic chart in figure 1 and write down its equations of motions

,

, (1)

,

…

.

The system of equations (1) can be represented in the canonical form of the dynamic model [1 − 3]

, (2)

,

Where − moving masses ; − own and partial frequency platform and metronomes; − coefficients of elasticity of springs. Let additional external harmonic forces acting on the metronomes with the same g frequency, different amplitudes, and phases that characterize such factors as the precision of manufacture by the manufacturer of the device metronome,

The system equations of motion (2) consists of n+1 associated homogeneous differential equations are not separable variables, it is necessary to attach the initial conditions for its full resolution using the techniques developed in [1 - 5].

, , (3)

, , (4)

To simplify calculations we assume that all private and partial frequency n of identical metronomes ; . The laws of motion multi-element model (2) - (4) leads to the following vibrational modes

(5)

where , (7)

. (8)

Analytical formulas of undetermined coefficients corresponding solutions are summarized in table 1.

Table 1

№	Displacements of the beam coefficients, x(t)	metronome movement coefficients xj(t),
1	2	3
1	\
2	\
3
4
5

The formula of the laws of movement of the pendulums and the platform (5) to (8) and table 1 obtained personally by the author and presented here for the first time. Examples of implementation of the laws of motions (5)-(8) and table 1 can be horizontal oscillations of the London bridge in marching it result in a large number of people, the graph in figure 2. Due to the swinging bridge and in the steady state, people begin to move in unison and equally, the graph in figure 3.

Movement of identical metronomes in the number of n=250 are presented in figures 4-5. The initial conditions are the image amplitude. Result will be the same steady-state oscillatory displacement laws for all of the metronomes and horizontal displacement of the platform on which they are installed. Previously this physical effect has been described in experiments of Rayleigh [3].

Picture 2 - The law equally movements n=250 of people walking by London Bridge and the horizontal displacement of the bridge

Picture 3 - Established laws oscillatory movements n=250 of people walking by London Bridge and the horizontal displacement of the bridge

a) b)

Picture 4 - Laws n=250 metronomes movements with an alternating different initial amplitudes and identical natural frequencies (pic. a). And the in the arbitrary point in time (pic. b).

Picture 5 - Steady oscillating metronomes n=250 laws of motion and the horizontal movement of the platform

Literature:

1. Dokukova N.A., Konon P.N. General laws governing in mechanical vibratory systems// JEPT,– 2006.– V. 79.– N. 4.– P. 824-831.

2. Dokukova N.A., Martynenko M.D., Kaftaikina E.N. Nonlinear vibrations of hydraulic shock absorbers// Journal of Engineering Physics and Thermophysics.– 2008.– V. 81.– N. 6.– P. 1197-1200.

3. Dokukova N.A., Kaftaikina E.N. The synchronization of two linear oscillators// Materialy VII miedzynarodowej naukowi-praktycznej konferencji. Polska. Przemysl.– 2012.– V. 18. – P. 28 – 35.

4. Докукова Н.А., Кафтайкина Е.Н., Конон Н.П. О синхронных колебаниях многоэлементных линейных динамических систем// Materials of the X International scientific and practical conference, «Conduct of modern science».- V.21.- Sheffield. S Yorkshire. England.- 2014.- P. 25-32.

5. Докукова Н.А., Кафтайкина Е.Н., Конон Н.П. Синхронные колебания n - автономных осцилляторов на одной неупругой балке с жесткой заделкой// Материали за X международна научна практична конференция. Република България. София.- 2014.- Т. 18.- С. 19-25.