# External damping force in Mass Spring Damper system

Hi,

I am designing a wave energy converter, which has one degree of freedom - heaving motion. Meaning it will move only in vertical direction.
My system can be considered as a Mass Spring Damper system.

Thus my equation of motion is:

F=Fh+Fe

Where,

Fe= Excitation force (Wave force)
F=m*a (mass times acceleration)
Fh= -m(a)*a - b(hyd)*v - k*x (hydromechanical load)

b(hyd) - damping coefficient
v - velocity
k - spring coefficient
x - distance

However, I also want to add a power extracting device to my system, which can be considered a generator. I have read many papers and they usually add a external damping force to enable power extraction.

Fb(ext)=-b(ext)*v

Thus my new equation of motion would be:

F=Fh+Fb(ext)+Fe

And finally my question. I would like to understand the physical meaning behind the external damping force. Does it mean that it is just a resistance (load) applied to the system to enable power extraction?

SteamKing
Staff Emeritus
Homework Helper
If you want to generate electrical power with this system, you will need a way to turn the heaving motion (basically a translation) into rotary motion to spin a generator.

AlephZero
Homework Helper
The basic idea of modeling the system is that to extract energy from it, you need to create a non-conservative force, so that a graph of force against displacement around one cycle of motion encloses (an "indicator diagram") encloses some area. The area represents the amount of work done.

Assuming the force is proportional to velocity is one way to do that, and it has the advantage that the math works out easily. There are other ways - for example a "friction" force with constant amplitude that always acts in the opposite direction to the motion.

One of the basic issues here is that taking energy out of the system will reduce the amplitude of the motion, and there is an "optimum" amount of energy you can take out without reducing the amplitude so much that no energy can get in! This optimum point depends on the operating frequency of the machine (i.e. the frequency of the waves) compared with its resonant frequency.

Thank you for the answers. I also did a bit of research myself and my understanding of the external damping coefficient, b(ext), is that it it determines the damping force on the generator by increasing resistance and thus decreasing the velocity of the generator. Or the other way around, by decreasing the resistance you have an increase in velocity.
It turns out that the optimum condition is achieved when you set b(ext)=b(hyd) and when your system oscillates at a frequency which is equal to the natural frequency.