Damped Driven Harmonic Oscillator

Your Name]In summary, the conversation is about a user trying to follow a derivation involving the equation of motion for a damped driven harmonic oscillator. They are specifically looking for a solution of the form x(t) = X^{+}(\omega)e^{i \omega t} + X^{-}(\omega)e^{-i \omega t}. The user is wondering where the positive exponent in the cosine term has gone in the subsequent steps of the derivation. The expert explains that the positive exponent is no longer needed in the equation as it represents the oscillatory motion of the system, which is not being considered at a specific time in the derivation.
  • #1
raintrek
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Homework Statement



I'm trying to follow through a derivation involving the equation of motion for the displacement x(t) of a damped driven harmonic oscillator.

[tex]m\frac{d^{2}x}{dt^{2}}+\gamma x + \beta \frac{dx}{dt}=F_{0}cos(\omega t)[/tex]

Where
[tex]cos(\omega t) = \frac{1}{2}\left( e^{i \omega t} + e^{-i \omega t} \right)[/tex]

Look for a solution of the form [tex]x(t) = X^{+}(\omega)e^{i \omega t} + X^{-}(\omega)e^{-i \omega t}[/tex]

Solve for [tex]X^{-}(\omega)[/tex] and note that [tex]X^{+}(\omega)=(X^{-}(\omega))^{*}[/tex] because x(t) is a real quantity.

Substitute [tex]x(t) = X^{-}(\omega)e^{-i \omega t}[/tex] into the equation of motion:

[tex]mX^{-}(\omega)(-\omega^{2})e^{-i \omega t} + \gamma X^{-}(\omega)e^{-i \omega t} - i\omega\beta X^{-}(\omega)e^{-i \omega t} = \frac{F_{0}}{2}e^{-i \omega t}[/tex]


I understand the crux of the derivation however I'm not sure where the [tex]\frac{1}{2} e^{i \omega t}[/tex] has gone from the cosine term. The negative exponent has cancelled, but the positive just seems to have disappeared. Could anyone help!? Thanks!
 
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  • #2

Thank you for your question. It seems like you are on the right track with your derivation. Let me explain what happened to the positive exponent in the cosine term.

In the original equation, we have:

x(t) = X^{+}(\omega)e^{i \omega t} + X^{-}(\omega)e^{-i \omega t}

Note that the positive exponent is associated with the term X^{+}(\omega)e^{i \omega t}. This term represents the oscillatory motion of the system, with a frequency of \omega. However, in the next step of the derivation, we substitute this expression for x(t) into the equation of motion:

m\frac{d^{2}x}{dt^{2}}+\gamma x + \beta \frac{dx}{dt}=F_{0}cos(\omega t)

This means that we are considering the motion of the system at a specific time, t. In other words, we are looking at a snapshot of the system's motion at that particular moment. At this moment, the cosine term is equal to 1, since cos(\omega t) = 1. Therefore, we can substitute this value into the equation and it becomes:

m\frac{d^{2}x}{dt^{2}}+\gamma x + \beta \frac{dx}{dt}=F_{0}

As you can see, the positive exponent has disappeared because it is no longer needed in this equation. We are no longer considering the oscillatory motion of the system, but rather its overall motion at a specific time.

I hope this helps clarify things for you. Keep up the good work with your derivation and don't hesitate to ask for further assistance if needed.
 

What is a damped driven harmonic oscillator?

A damped driven harmonic oscillator is a physical system that exhibits periodic motion, such as a swinging pendulum or a vibrating guitar string. It consists of a mass attached to a spring, which provides the restoring force, and a damping mechanism, which dissipates energy and causes the motion to eventually come to a stop.

How does a damped driven harmonic oscillator differ from an undamped harmonic oscillator?

The main difference between a damped driven harmonic oscillator and an undamped harmonic oscillator is the presence of a damping mechanism. In an undamped harmonic oscillator, the motion continues indefinitely without any external force. In a damped driven harmonic oscillator, the damping mechanism gradually reduces the amplitude of the motion, causing it to eventually come to a stop.

What is the role of the driving force in a damped driven harmonic oscillator?

The driving force in a damped driven harmonic oscillator is an external force that is applied to the system at regular intervals. It provides energy to the system, causing it to continue oscillating even in the presence of a damping mechanism. The frequency and amplitude of the driving force can greatly affect the behavior of the oscillator.

What are the applications of damped driven harmonic oscillators?

Damped driven harmonic oscillators have a wide range of applications in physics, engineering, and other fields. Some examples include pendulum clocks, musical instruments, shock absorbers in vehicles, and electronic circuits. They are also used in research and experimentation to study the behavior of oscillating systems.

How is the behavior of a damped driven harmonic oscillator described mathematically?

The motion of a damped driven harmonic oscillator can be described by a differential equation, known as the damped driven harmonic oscillator equation. This equation takes into account the mass, spring constant, damping coefficient, and driving force to calculate the position and velocity of the oscillator at any given time. Various techniques, such as Fourier analysis, can be used to solve this equation and understand the behavior of the oscillator.

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