System mass-arises

A system mass-arises is a mechanical system with a degree of freedom. It is consisted a mass fixed on a Ressort forced to move in only one direction. Its movement is due to three forces:

a force of F_R recall,
a force of F_A damping,
an external force F_E.

The system mass-arises is a simple subject of study within the framework of the oscillating harmonics.

Rectilinear oscillations of a mass subjected to the action of a spring

One can put in Oscillation a Masse subjected to the action of a Ressort. These oscillations can be, according to the cases, of the vertical oscillations or the horizontal oscillations (by using a device allowing to minimize the Frottement S on the support).

In both cases, the oscillations are harmonic: the function of the time of the position of the mass on both sides of the position of balance (static) is a function sine. Into the case of the vertical oscillator, the effect of the Pesanteur introduces only one translation of the static position of balance. The relation deduced from the application of the theorem of the center of inertia can be written:

$\ frac {d^2x} {dt^2} + \ omega_0^2 X = 0$ , with $\ omega_0 = \ sqrt {\ frac {K} {m}}$

$\ omega_0$ is called own pulsation Oscillateur harmonic. The solutions of the differential equation are form $x = x_0 \ sin (\ omega_0 T + \ varphi)$ , which is characteristic of a harmonic oscillator.

The period is independent of the Amplitude (Isochronisme of the oscillations): it depends only on the inertia of the system (mass m) and the characteristic of the force of recall (constant of stiffness K of the spring): $T = 2 \ pi \ cdot \ sqrt \ frac {m} {K}$

Note: this oscillator is subjected to the conservation of the mechanical energy: this one is form $\ frac {1} {2} m v^2 + \ frac {1} {2} K x^2 = E_0$
By deriving member with member the equation compared to time one finds the preceding differential equation.

Improvement

What precedes is valid if the mass of the spring is negligible compared to that of the mass which oscillates. The experiment shows that the period is closer to:

$T = 2 \ pi \ cdot \ sqrt \ frac {m + \ mu/3} {K}$ where ${\ driven \; /3} =$ one the third of the mass of the spring;
${m \;}$ = mass suspended on the spring;
$\ ~ {K \;}$ = the constant rubber band or stiffness of the spring.

Another improvement

This is again an approximation. A study supplements is in the external bonds. To seek: “Study of the period of oscillation of a spring”.
It is shown that the correct period of oscillation is: $T = \ frac {2 \ pi} {\ Omega} \ cdot \ sqrt \ frac {\ driven} {K}$

where $\ quad \ Omega \; \ quad$ is defined by the relation: $\ quad \ Omega \ cdot \ tan (\ Omega) = \ frac {\ driven} {m}$
${\ driven \;}$ = mass of the spring;
${m \;}$ = mass suspended on the spring;
$\ ~ {K \;}$ = the constant rubber band or stiffness of the spring.

A manner of calculating $\ quad \ Omega \ quad$ is to reiterate: $\ quad \ Omega = \ arctan (\ frac {\ driven} {m \ cdot \ Omega})$ while starting with: $\ quad \ Omega = \ sqrt \ frac {\ driven} {m + \ driven/3}$

See too

Internal bonds

oscillating Systems with a degree of freedom.
Examples of differential equations

External bonds

Juggling.ch, '' Physique in bulk '', to see: “Study of the period of oscillation of a spring”
univ-lemans.fr, '' helicoid Ressort ''

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