Forced Oscillators and their equation

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In the case of a forced oscillator an external force is applied on the oscillating mass. This can lead to the dough being slowed or accelerated.

If the force acts synchronously (at the same rate that the mass oscillates naturally) resonances arise that can increase the amplitude of the oscillation dramatically.

>Model

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Forced oscillator

Image

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A forced oscillator can be a system in which a mass attached to a spring is immersed in a viscous fluid, and the point where the spring is attached oscillates. This effect can be achieved by fixing the point to a rotating disk:

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Forcing force

Equation

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A simple way to model the external force is to assume that it has a magnitude of $F_0$ and undergoes oscillation with an arbitrary angular frequency $\omega$.

$ F = F_0 e^{ i \omega t }$

$F_0$
Amplitude of the forcing force
$N$
9993
$\omega$
Angular forcing frequency
$rad/s$
9989
$F$
Forcing force
$N$
9988
$t$
Time
$s$
5264

ID:(14099, 0)



Forced Oscillator Equation

Equation

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In the case of a damped oscillator without external forcing, the equation of motion is

$ m_i \displaystyle\frac{d^2 x }{d t ^2} + b \displaystyle\frac{d x }{d t } + k x = 0$



In the case of external forcing, the force we define as

$ F = F_0 e^{ i \omega t }$



acts additionally on the system, leading to a modified equation of motion

$ m_i \displaystyle\frac{d^2 x }{d t ^2} + b \displaystyle\frac{d x }{d t } + k x = F_0 e^{ i \omega t }$

$F_0$
Amplitude of the forcing force
$N$
9993
$\omega$
Angular forcing frequency
$rad/s$
9989
$b$
Constant of the Viscose Force
$kg/s$
5312
$x$
Elongation of the Spring
$m$
5313
$k$
Hooke Constant
$N/m$
5311
$m_i$
Inertial Mass
$kg$
6290
$t$
Time
$s$
5264

ID:(14100, 0)



Structure of the forced oscillator solution

Equation

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In the case of an undriven damped oscillator, the equation of motion is

$ z = x_0 e^{i \omega t }$



and it's important to note that the angular frequency corresponds to the system's own natural frequency. In our case, the angular frequency will match that of the system driving the oscillation. Apart from that, it's possible for the oscillation to exhibit a phase difference from the driving force. Hence, a solution in the form of

$ z = A e^{ i ( \omega t + \phi )}$

$A$
Amplitude of forced oscillation
$m$
9991
$\omega$
Angular forcing frequency
$rad/s$
9989
$z$
Complex number describing forced oscillation
$m$
9990
$\phi$
Swing phase
$rad$
9992
$t$
Time
$s$
5264

can be proposed.

ID:(14101, 0)



Equation of the forced oscillator in complex space

Equation

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If we use the equation of oscillation

$ z = A e^{ i ( \omega t + \phi )}$



and introduce it into

$ m_i \displaystyle\frac{d^2 x }{d t ^2} + b \displaystyle\frac{d x }{d t } + k x = F_0 e^{ i \omega t }$



we obtain the equation of the forced oscillator in complex space

$(- m_i \omega ^2 + i b \omega + m_i \omega_0 ^2 ) A e^{i \phi } = F_0 $

$A$
Amplitude of forced oscillation
$m$
9991
$F_0$
Amplitude of the forcing force
$N$
9993
$\omega$
Angular forcing frequency
$rad/s$
9989
$b$
Constant of the Viscose Force
$kg/s$
5312
$\omega_0$
Frecuencia angular del resorte
$rad/s$
9798
$m_i$
Inertial Mass
$kg$
6290
$\phi$
Swing phase
$rad$
9992

To simplify the solution of the differential equation

$ m_i \displaystyle\frac{d^2 x }{d t ^2} + b \displaystyle\frac{d x }{d t } + k x = F_0 e^{ i \omega t }$



we use the solution

$ z = A e^{ i ( \omega t + \phi )}$



and proceed to differentiate it with respect to time to obtain the velocity

$v = \displaystyle\frac{dz}{dt} = x_0 \displaystyle\frac{d}{dt}e^{i(\omega t + \phi)}=x_0 i \omega e^{i(\omega t + \phi)} = i\omega z$



and thus the second derivative, which is equal to the first derivative of velocity

$a = \displaystyle\frac{dv}{dt} = x_0 i \omega e^{i\omega t} \displaystyle\frac{d}{dt}e^{i(\omega t + \phi)} = - \omega^2 x_0 e^{i(\omega t + \phi)}= - \omega^2 z$



which, along with

$ \omega_0 ^2=\displaystyle\frac{ k }{ m_i }$



gives the equation

$(- m_i \omega ^2 + i b \omega + m_i \omega_0 ^2 ) A e^{i \phi } = F_0 $

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Phase shift

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Phase shift is a temporal displacement of an oscillation, meaning it starts either ahead of or behind its regular timing while maintaining the same shape:

ID:(14102, 0)



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