An earlier post looked at the effect of damping on free vibrations. We looked at the equation
'' + γ u
' + k u = 0
where the coefficients m, γ, and k were all positive. But what if some of these terms are negative?
Let’s assume that m is positive. Otherwise multiply the equation above by -1. What happens if γ or k are negative?
The term γ, when positive, takes energy out of the system. A negative value of γ would be a term that adds energy, a sort of negative damping. The behavior of the solutions is determined by the eigenvalues of the system, that is the roots of the equation
m x2 + γ x + k = 0.
If γ is negative, the eigenvalues have positive real part and so the amplitude of the solutions increases exponentially. If γ2 < 4mk then the eigenvalues are complex and so the solutions have an oscillating component. If γ2 = 4mk then there is one repeated, positive eigenvalue. But if γ2 > 4mk the system has one positive and one negative eigenvalue. The solution corresponding to the negative eigenvalue decays exponentially. The other solution increases exponentially. The general solution is a linear combination of these two solutions. As time increases, only the exponentially increasing component of the solution matters because the effect of the other component goes to zero.
In theory, the solution could consist purely of the exponentially decaying component. But in practice, if there is even the tiniest component of the exponentially increasing solution, this component will eventually dominate. A numerical solution, for example, would eventually be dominated by the exponentially increasing solution.
Now what about negative springs? Instead of being a restoring force, a negative spring would be a sort of amplifier, reinforcing rather than resisting displacement. The discriminant γ2 – 4mk will be positive if k is negative. There will be no oscillation because the eigenvalues have no complex part. Also, there will be one positive and one negative eigenvalue, and so the solutions grow exponentially as described above.
Related: If you’re interested in differential equations, check out @diff_eq on Twitter.