How Can I Analyze Energy Dissipation in a Non-Linear Damping System?

[dmitryz]

Homework Statement

vinit = sqrt(2g*h); h = drop distance
vfinal = 0;

xinit = 0;
xfinal = 100mm;

a = g;

Issue: non-linear damping.
M*x'' - b*(x')^2 - k*x = 0;
b = 128*mu*(length fluid travels)*(D^4(piston)/[(D(hydraulic)^4)(orifice opening)]every book I've been reading on vibrations damping says there's no solution for v^2 damping. Currently reading "Influence of Damping in Vibration Isolation" and they give an equivalent linear damping coefficient as:
C(eq) = (D0)/(∏ω(z0)^2); D0 being energy dissipated per cycle, z0 being relative displacement.

Then they go into equivalent damping force being: γ*F; γ= (2/sqrt(∏))*gamma((n+2)/2)/gamma((n+3)/2)...very long story short. Is there anyway to do a stepwise energy dissipation of a mass/spring/damper problem? Can I use something like: initial energy in - energy to compress spring - energy dissipated by damper = 0.

the issue I think is with this is that I don't know how to figure out energy dissipated by a damper whose dependent on v^2...

can I define a function that says: this system was deflected by 0.1mm at this time and Z amount of energy was taken away from the initial impact. W energy was taken up by the spring, and X was taken by the damper. This is how much much energy was left over at the boundary of this iteration...

Im in analysis paralysis at the moment and I think I'm overthinking this...

 

[rude man]

Isn't x'' = a = g = constant? So
b(x')^2 + kx = ma?

 

[dmitryz]

Yes. The thing is figuring out what b value I need for any drop between 25mm and 1200mm. At a height of 1000mm what b do I need to be critically damped? At a height of 400mm what b do I need to be critically damped? and so on. Vibration teacher says that the impact is making this problem very difficult to solve. Because the damping is non-linear and because the area of the orifice is also changing (see fluids orifice measurements) this problem becomes non-uniform area, non-linear damping.

I have the non-uniform area calculations by using the hydraulic diameter: Dh = 4*(area of orifice)/(wetted perimeter). I have the damping coefficient calculations: 128*mu*L*(d(piston)^4)/(d(orifice)^4)...now I don't have the damping coefficient necessary for any given height so that I can get the orifice diameter required at that height.

 

[rude man]

Oops, my initial assumption is incorrect, on reflection (even though you agreed!).

I though the equation would be 1st order but it's not. And I have to cconcede that I wouldn't know how to approach it. What's the background material for this problem? I mean, numerical techniques or whatever? What's the complete statement of the problem?

 

[paranormal]

I understand your concerns about the non-linear damping and the difficulties in finding a solution for v^2 damping. It is true that many books and studies on vibration damping do not provide a solution for this type of damping.

However, there are alternative approaches that can be used to analyze and model the energy dissipation in a mass/spring/damper system. One approach is to use equivalent linear damping coefficients, as mentioned in the book you are currently reading. This approach can provide a simplified solution that can still accurately capture the effects of damping in the system.

Another approach is to use numerical methods, such as finite element analysis or computational fluid dynamics, to simulate the behavior of the system and calculate the energy dissipation at each step. This approach can provide more detailed and accurate results, but it may require more computational resources and expertise.

In either case, it is possible to define a function that calculates the energy dissipated by the damper at each step, taking into account the velocity and displacement of the system. This can be used to track the energy dissipation over time and analyze the overall behavior of the system.

In summary, while there may not be a direct solution for v^2 damping, there are alternative approaches that can be used to analyze and model the energy dissipation in a mass/spring/damper system. It is important to carefully consider the assumptions and limitations of each approach and choose the one that best fits your specific problem.