# Difference between revisions of "Forest UCM Osc Driven"

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:<math> D x = f(t)</math> | :<math> D x = f(t)</math> | ||

− | ;Linear differential equations have | + | ;Linear differential equations have coefficients that can be constant or variable coefficients that can be transformed into constant coefficients. |

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:<math>D(ax) = aD(x)</math> and <math>D(x_1 + x_2) = D(x_1) + D(x_2)</math> | :<math>D(ax) = aD(x)</math> and <math>D(x_1 + x_2) = D(x_1) + D(x_2)</math> | ||

− | |||

==Solving the Inhomogeneous Diff. Eq. == | ==Solving the Inhomogeneous Diff. Eq. == | ||

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− | Since the solution of the 2nd order differential equation requires exactly two arbitrary | + | Since the solution of the 2nd order differential equation requires exactly two arbitrary constants, |

We can formulate the solution by tacking a particular solution onto the homogeneous solution that already has two arbitrary constants. | We can formulate the solution by tacking a particular solution onto the homogeneous solution that already has two arbitrary constants. | ||

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;Trick: If a differential equation has the cosine function as a solution then the sine function may also be a solution since the difference between the two is only a phase shift. | ;Trick: If a differential equation has the cosine function as a solution then the sine function may also be a solution since the difference between the two is only a phase shift. | ||

− | It must also be | + | It must also be true that |

:<math>D(y_p) = f_0 \sin ( \omega t)</math> | :<math>D(y_p) = f_0 \sin ( \omega t)</math> | ||

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:<math> {f_0} e^{i\delta} = A \left ( (\omega_0^2- \omega^2) + 2 i \beta \omega \right ) </math> | :<math> {f_0} e^{i\delta} = A \left ( (\omega_0^2- \omega^2) + 2 i \beta \omega \right ) </math> | ||

− | Since <math>f_0</math> and <math>A</math> are not imaginary | + | Since <math>f_0</math> and <math>A</math> are not imaginary variables |

− | The phase | + | The phase (<math>\delta</math>) must be due to the remaining complex term <math> \left ( (\omega_0^2- \omega^2) + 2 i \beta \omega \right )</math> |

− | To | + | To determine the angle <math>\delta</math> we construct a right triangle in the complex plane where the leg along the real axis is given by <math>(\omega_0^2- \omega^2)</math> and the other leg of the triangle is along the imaginary axis with a length of <math> 2 \beta \omega</math> |

:<math>\tan \delta = \frac{2 \beta \omega}{(\omega_0^2- \omega^2)}</math> | :<math>\tan \delta = \frac{2 \beta \omega}{(\omega_0^2- \omega^2)}</math> | ||

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with | with | ||

− | :<math>x_p = \mbox{Real} \left |z_p \right | = \frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 }} \cos(\omega t-\delta)</math> | + | :<math>x_p = \mbox{Real} \left |z_p \right | = \frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 }} \cos(\omega t-\delta)</math> |

==Complete Solution for the Driven Damped oscilaltor== | ==Complete Solution for the Driven Damped oscilaltor== |

## Latest revision as of 23:23, 14 October 2021

# Damped Oscillators driven by an external source

An external force must be supplied to do work on a damped oscillator in an amount that is equal to or greater than the work done by the dissipative force.

An external force (source) is added to the homogeneous differential equation making it inhomogenous

making it

where f(t) represents the external force (source) that depends on time divided by the objects mass.

## Differential equations in Operator form

In the previous sections we used the definition

to solve the second order linear differential equation.

Let's take this a step further with the following operator definition

then

becomes

- Linear differential equations have coefficients that can be constant or variable coefficients that can be transformed into constant coefficients.

- Note
- is a linear operator

meaning

the above is a property of differential calculus where

- and

## Solving the Inhomogeneous Diff. Eq.

To solve the Inhomogeneous problem we take advantage of the linear operator such that

Since the solution of the 2nd order differential equation requires exactly two arbitrary constants,

We can formulate the solution by tacking a particular solution onto the homogeneous solution that already has two arbitrary constants.

All we need to do is find the particular solution

that satisfiesand we will have a complete set of solutions.

Break the equation up into a Homogeneous solution and a Particular Solution

### Homogeneous Solution

Th Homogenous solution solved the equation

we know from the previous section that the homogenous solution has the form

where

### Case 1 : f(t) is sinusoidal

Consider the case where the driving force is a sinusoidal function

We seek a solution to the particular equation

- Remember
- is not necessarily at the natural (resonant) frequency

- Trick
- If a differential equation has the cosine function as a solution then the sine function may also be a solution since the difference between the two is only a phase shift.

It must also be true that

You can construct a complex solution now such that

the constant C is determine by substituting the solution into the equation

The amplitude of the solution is a complex number that may be cast in terms of a real amplitude times the complex exponential such that

where the real amplitude A is given by

#### What is delta?

Since

and are not imaginary variablesThe phase (

) must be due to the remaining complex termTo determine the angle

we construct a right triangle in the complex plane where the leg along the real axis is given by and the other leg of the triangle is along the imaginary axis with a length of- The final complex solution si

with

## Complete Solution for the Driven Damped oscilaltor

where