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Forced Oscillators and their equation
Storyboard 
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
Definition
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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Phase shift
Image
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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Forced Oscillators and their equation
Storyboard
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.
Variables
Calculations
to 
,
then, select the variable: 
to 
Calculations
Variable
Given
Calculate
Target :
Equation
To be used
Equations
To simplify the solution of the differential equation
we use the solution
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
gives the equation
Examples
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:
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 $\omega$.
In the case of a damped oscillator without external forcing, the equation of motion is
In the case of external forcing, the force we define as
acts additionally on the system, leading to a modified equation of motion
In the case of an undriven damped oscillator, the equation of motion is
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
can be proposed.
If we use the equation of oscillation
and introduce it into
we obtain the equation of the forced oscillator in complex space
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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