User:


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.

>Model

ID:(52, 0)


Forced oscillator
Image

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:

ID:(14098, 0)


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:

ID:(14102, 0)


Forced Oscillators and their equation
Description

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
Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
$A$
A
Amplitude of forced oscillation
m
$F_0$
F_0
Amplitude of the forcing force
N
$omega$
omega
Angular forcing frequency
rad/s
$z$
z
Complex number describing forced oscillation
m
$b$
b
Constant of the Viscose Force
kg/s
$x$
x
Elongation of the Spring
m
$F$
F
Forcing force
N
$\omega$
omega
Frecuencia angular del resorte
rad/s
$k$
k
Hooke Constant
N/m
$m_i$
m_i
Inertial Mass
kg
$phi$
phi
Swing phase
rad
$t$
t
Time
s

Calculations

First, select the equation:   to ,  then, select the variable:   to 
Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

 Variable   Given   Calculate   Target :   Equation   To be used


Equations

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 $

(ID 14103)

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:

(ID 14098)

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)

ID:(52, 0)