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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.
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Forced oscillator
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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
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 }$ |
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Forced Oscillator Equation
Equation
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 }$ |
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Structure of the forced oscillator solution
Equation
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 )}$ |
can be proposed.
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Equation of the forced oscillator in complex space
Equation
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 $ |
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:
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Video: Forced Oscillators and their Equation