Third Order Differential Equations (Mass-Spring-Damper-etc)

[Dnandrea]

So I'm not sure whether this should be posted in the Physics forum or not, but here goes:

I'm a Junior ME student at NC State taking a Vibrations course. We've gone over the general differential equation for mass-damper-spring systems where

M_eq⋅x'' + C_eq⋅x' + k_eq⋅x = F_extermal

What I was wondering is, if you had a third order equation, where x''' (jerks, I believe) was added in, what would be the "damping constant" equivalent that would be multiplied by x''' and how could this be interpreted in the real world?

If you follow units, I think it should be mass ⋅ time as the coefficient. For k_eq, if you assume N/m, can be canceled out into kg/s², C_eq into kg/s, and M_eq into kg. Following that trend, the coefficient would have to be kg⋅s.

Mass ⋅ time I've tried researching and came across something called "action" that was really interesting. I'm not sure whether I'm making this much harder than it needs to be. Any explanation?

Thanks!

 

[Hesch]

What I was wondering is, if you had a third order equation, where x''' (jerks, I believe) was added in, what would be the "damping constant" equivalent

As for a 2. order system, written in the Laplace domain, the characteristic equation could be expressed:

s² + 2ζω_ns + ω_n² = 0

where ω_n is the freqency [rad/s] and ζ is the damping ratio of a sinusoidal oscillation. If ζ<1, the roots of the equation will be complex, and as a complex root will have a conjugated root, the roots will have "used" both roots in the polynomium. There is no further to add. ( See fig. 2. )
As for a 3. order system, you could express the characteristic equation:

( s + a )( s² + 2ζω_ns + ω_n² ) = 0

so the same ζ is still there, in principle unchanged. Only a factor ( s + a ) has been added, which is just an overlayed exponential function, that has no damping ratio. The interpretation of the ζ is the same. ( See fig. 3. )

The charateristic equation for a 4. order system could be expressed:

( s² + 2ζ₁ω_n1s + ω_n1² )( s² + 2ζ₂ω_n2s + ω_n2² ) = 0

thereby having 2 different damping ratios and 2 frequencies. ( See fig. 4. )