# Deriving the duffing equation?

• great_sushi
In summary, the conversation discussed a mass on a non-linear spring with damping. The restoring force of the spring was given by F=-kx+x^3, and this was set equal to Newton's second law F=mx'' = -kx+x^3. The damping, which is dependent on velocity, was then added to the equation. The system is also driven by a periodic force dependent on time. The final equation is x'' + bx' + w0*x + x^3 = Fcos(wt), and there was some confusion regarding the signs in the equation.
great_sushi
Say I have a mass m on a non linear spring k with some damping b.

I start with the restoring force of the spring F=-kx+x^3... the x^3 is the non linearity.

Set that equal to Newtons second law F=mx'' = -kx+x^3

Add in the damping which is dependent of velocity bx'

......mx'' = -kx + x^3 + bx'
......= x'' + k/m*x + x^3 + bx'
......= x'' + bx' + w0*x + x^3... Now because my system is driven I add in a periodic force dependent on t.

Fcos(wt) = x'' + bx' + w0*x + x^3

Is this the correct method? I may have gotten mixed up with my signs I tend to do that :(

We kindly ask that you not post the same question twice. Thank you.

## 1. What is the duffing equation?

The duffing equation is a nonlinear differential equation used to model a damped, driven harmonic oscillator. It is named after Georg Duffing, a German mathematician who studied this type of equation in the early 20th century.

## 2. How is the duffing equation derived?

The duffing equation can be derived from the equation of motion for a damped, driven harmonic oscillator by adding a nonlinear term that accounts for the restoring force becoming nonlinear at large amplitudes.

## 3. What are the applications of the duffing equation?

The duffing equation has applications in various fields, including physics, engineering, and mathematics. It can be used to model the behavior of many physical systems, such as pendulums, electrical circuits, and biological systems.

## 4. What are the challenges in solving the duffing equation?

One of the main challenges in solving the duffing equation is that it is a nonlinear equation, which means that there is no general solution that works for all cases. Instead, numerical methods or approximations must be used to solve the equation.

## 5. What are the implications of the duffing equation in chaos theory?

The duffing equation is often used as an example of a chaotic system, meaning that small changes in the initial conditions can lead to drastically different outcomes. This has implications for predicting the long-term behavior of certain systems and has been studied extensively in the field of chaos theory.

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