# How to derive duffing equation for a particular system?

• great_sushi
In summary: The cubic term comes from the fact that the duffing equation is a 3rd order polynomial. So, the terms in the duffing equation are raised to various powers.
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 :(

Your method is basically correct. But, yes, you do need to be careful with the signs.

Generally, there would be a constant factor in front of the x3 term: βx3

I would recommend going ahead and putting in the driving force at the beginning.

The damping force is usually written -bx' rather than +bx' since the force opposes the velocity.

So, mx'' = -kx +βx3 -bx' + Fcos(ωt)

Then you can easily rearrange the terms to get Fcos(ωt) alone on one side.

You could just as well write the nonlinear part as -βx3 so that

mx'' = -kx -βx3 -bx' + Fcos(ωt)

Then you will get all positive terms when you solve for Fcos(ωt).

[Edit: Be careful when dividing through by the mass m. All terms get divided by m. You should get the Duffing equation .]

Last edited:
Oh thank you very much!
Apologies for asking the question twice, i didn't know if it was advanced or introductory..

So if I integrate the duffing equation with respect to x will I get the potential energy? Because the work = force* distance and so the force = work / distance.

great_sushi said:
So if I integrate the duffing equation with respect to x will I get the potential energy? Because the work = force* distance and so the force = work / distance.

Potential energy is only defined for forces that depend on position alone. (When you have motion in more than one dimension, there is an additional restriction.) Since you have forces that depend on velocity (damping force) and on time (driving force), there will not be a potential energy for the total force. However, there is a potential energy associated with the force -kx - βx3 which you can find by integrating with respect to x.

Ahhh I see that makes sense. So the integral of -kx + βx^3 = -1/2kx^2 + 1/4βx^4 = V(x) and in this case, that gives a double well potential.

The potential energy is the negative of the integral of the force. So, the signs in V(x) would be opposite. The shape will depend on whether β is positive or negative.

Oh great! I was wondering why my signs were the wrong way round. I didnt realize it was a negative integral.
Thank you very much, you have been extremely helpful :)

Can you tell me where the cubic term comes from? I know that it an odd power series but why?
Thanks

## 1. What is the Duffing equation and what does it represent?

The Duffing equation is a mathematical model used to describe the motion of a damped, driven oscillator. It is commonly used in physics and engineering to study systems with nonlinear behavior, such as a pendulum or a spring-mass system. The equation takes the form of a second-order differential equation and represents the dynamics of the system over time.

## 2. How is the Duffing equation derived for a particular system?

The Duffing equation can be derived using various methods, such as the Lagrangian approach or the Newtonian approach. The first step is to identify the forces acting on the system and write down the equations of motion. Then, the equations are simplified and transformed into the standard form of the Duffing equation. This process involves solving differential equations and applying mathematical techniques such as Taylor series expansion and harmonic balance method.

## 3. What are the assumptions made when deriving the Duffing equation?

The Duffing equation assumes that the system is conservative, meaning that energy is conserved. It also assumes that the damping force is linear and proportional to the velocity, and the driving force is sinusoidal. Additionally, the equation assumes that the system is described by a single degree of freedom, meaning it can be modeled using only one variable, such as displacement or angle.

## 4. Can the Duffing equation be applied to any system?

The Duffing equation is a general model that can be applied to a wide range of systems. However, it is most commonly used to describe systems with nonlinear behavior, such as those with large amplitude oscillations, hysteresis, and chaos. It is also important to note that the Duffing equation is an approximation and may not accurately represent the behavior of a real-world system.

## 5. How is the Duffing equation used in practical applications?

The Duffing equation has various applications in physics, engineering, and other fields. It is commonly used to study and understand the dynamics of nonlinear systems, such as electrical circuits, mechanical systems, and biological systems. It can also be used for predictive modeling and control of these systems, as well as for analyzing and designing structures to withstand vibrations and other dynamic forces.

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