# Driven Harmonic Oscillator - Mathematical Manipulation of Equations

• phyzmatix
In summary, the problem involves expressing a forcing function in terms of e^{i\omega t} and separating the real and imaginary parts of the solution for x. The Euler Equation is used to do this, along with the identities for sin(A+B) and cos(A+B). By equating coefficients, the solution can be obtained.
phyzmatix
1. Homework Statement and the attempt at a solution

I'm not so sure if my problem lies with the physics or the mathematics. I have the distinct feeling that it's the latter and that I'm missing something elementary, but truly have no idea how to proceed.

Any advice will be appreciated.
Thanks!
phyz

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Express the forcing function in terms of

$$e^{i\omega\mbox{t}$$

then separate the real and imaginary parts of the solution for x. The real part is the desired solution.

I'm afraid I don't quite know what you mean...How do I write

$$F(t)=F_0 \sin{\omega t}$$

in terms of

$$e^{i\omega t}$$

?

Use the Euler Equation

$$e^{i\theta}=\cos{\theta}+i\sin{\theta}$$

and

$$e^{-i\theta}=\cos{\theta}-i\sin{\theta}$$

If you are not familiar with this, try using the identity for sin(A+B) and cos(A+B).

I really appreciate the help, but please bear with me as I try to wrap my head around this. I do know the Euler equation, but my understanding is that only the real part relates to SHM and since x as well as F(t) are given as functions of sine (not cosine), I don't know how to "bridge the gap" so to speak. I've tried using the identities for sine and cosine as you mention, but end up with massively intimidating equations involving $$\sin\phi$$ and $$\cos \phi$$ which doesn't really help as I don't know how to get rid of either...

After using the sin(A+B) and cos(A+B) identities, equate the coeffecients:

The sin(omega*t) coefficients on the right side of the equation are equal to F0 and the cos(omega*t) coefficients are equal to zero.

Last edited:
Finally the light! Thank you! I'm going to play with this and hopefully I won't get stuck again

## 1. What is a driven harmonic oscillator?

A driven harmonic oscillator is a type of physical system that experiences oscillatory motion in response to an external force or driving force. It is a simple model used to study the behavior of systems such as springs, pendulums, and electrical circuits.

## 2. How is a driven harmonic oscillator mathematically represented?

A driven harmonic oscillator can be mathematically represented by the differential equation: m(d^2x/dt^2) + kx = F(t), where m is the mass, k is the spring constant, x is the displacement from equilibrium, and F(t) is the external force as a function of time.

## 3. What is the role of damping in a driven harmonic oscillator?

Damping refers to the dissipation of energy in a driven harmonic oscillator, causing it to gradually come to rest. This can be represented mathematically by adding a damping term, b(dx/dt), to the differential equation.

## 4. How can we manipulate the equations of a driven harmonic oscillator to solve for variables?

To solve for variables in a driven harmonic oscillator, we can use techniques such as substitution, integration, and solving systems of equations. We can also use mathematical tools such as Laplace transforms to simplify the equations.

## 5. What are some real-life applications of a driven harmonic oscillator?

A driven harmonic oscillator has many practical applications, such as in musical instruments, shock absorbers, and seismographs. It is also used in fields such as engineering, physics, and mathematics to model and study various systems and phenomena.

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