Nonlinear System Homework: Potential Energy & Stability of Fixed Points

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SUMMARY

The discussion focuses on a damped mass-on-a-spring system characterized by a restoring force defined by the equation F/m = -βx - αx³, where β represents the linear Hookean component and α denotes a nonlinear term. The participants explore the potential energy function and the stability of fixed points, emphasizing the need to integrate the restoring force to derive the potential energy. The fixed points are determined by setting the restoring force to zero, which is a valid approach for analyzing stability in this nonlinear system.

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  • Knowledge of damping effects in oscillatory systems
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Homework Statement


In this question we will consider a damped mass-on-a-spring system whose spring exhibits cubic deviations from Hooke’s law. We will consider a damped spring with a restoring force F such that

F/m=−βx−αx^3

where β is the “Hookian” part and α is a new nonlinear term. Unlike the usual spring constant k, β can have either sign. Use γ (not 2γ) as the damping parameter that multiplies x'.

1.) What does the potential energy function look like for this system? Consider cases where α and β are both positive and negative.
2.)Solve for the fixed points of this system and determine how their stability depends on α and β.

The Attempt at a Solution


[/B]
So the general equation of motion with damping and restoring force is: x'' + γx' + βx + αx^3 = 0
So I know the net force is F = -γx' - mβx - mαx^3. And in general F = - dU/dx (however this is true for conservative forces -hence I don't think it would apply here). So can I just integrate the for the Force? If not, I unsure how to approach this problem to get the potential energy function.
 
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The restoring force has potential that you can find by integrating the restoring force.
 
Last edited:
So then I was wondering, if I had to determine the fixed points of the system would setting the restoring force equal to zero and solving be sufficient ?
 

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