External – and reaction forces can be distinguished as listed below, each with their component that will later on compose the equation of motion:
External: Hydrostatic force K_H∙ξ caused by the simple principle of buoyancy Excitation loads F_ex enforced by the action of the waves on a motionless body (Fe) Radiation force due to the induced oscillatory movement in the surrounding fluid, when no incident waves are present A∙ξ ̈+B∙ξ ̇
Reaction forces: Power take-off equipment (offering stiffness and damping) B_PTO∙ξ ̇+K_PTO∙ξ Foundation/mooring system (offering extra stiffness) K_A∙ξ End stop mechanism, which is beyond the scope of this paper
Equation of motion
All of these forces result in a harmonic motion ξ of the floating body, that
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This indicates the wave period for which the largest response of a certain device is found; as it incorporates the added mass A into the natural frequency.
In order for the buoys to acquire a displacement of volume of V, they need to have a mass equal to that of the displaced amount of water. From this relationship, the hydrostatic stiffness can be found as
K_H=N/∆z=gM/(V/S)=(gVρ_w)/(V/S)=gρ_w S=gρ_w πr^2
The resonance frequency will mainly depend upon the radius of the buoy as the hydrodynamic stiffness is proportional to the squared radius. When the latter increases, a lower resonance period is
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Starting the calculations while neglecting the added mass‚ should offer a good estimation of the actual value, which is then used as initial frequency. ω_n≈√(K_H/M) Power Absorption
The optimal PTO damping coefficient BPTO that will allow for maximized power absorption of a device at a given wave frequency; is given by:
B_PTO=√(B(ω)^2+1/ω^2 (-(m+A(ω)) ω^2+K_H)²)
The response amplitude operator can be calculated based upon the equation of motion; but this time written in the frequency domain with ξ(ω)=RAO(ω)∙exp(-iωt).
RAO(ω)=(F_ex (ω))/(-ω^2 (m+A(ω))-iω(B(ω)+B_PTO )+K_H )
Lastly, the mean power output of each geometry, for any PTO-damping a certain frequency, is obtained as the Power-Take-Off damping times the velocity of the buoy. p(ω)=1/2∙ω²∙RAO²∙B_PTO To find the absolute value of the mean power output; the normalized power is to be multiplied with the squared amplitude of the incoming waves; as the RAO is dimensionless and therefore accounts for an amplitude of unity.
P(ω)=A² ∙p(ω)=H²/4∙p(ω)
Hydrodynamic
Finally you will analyze the graphs you made by telling about the relationship between buoyancy and volume of the different boat hulls.
Preston, Raymond. "Definitions of Relative Wind, Lift, Weight, Drag, Thrust, and Side Force ." Definition of Aerodynamic Terms. N.p., n.d. Web. 18 Feb. 2014.
Bear in mind the amplitude cannot be larger than the extension caused. by the smallest mass applied to the spring as this would not allow the
Newton's Laws can be found in the textbook, Physics for Scientists and Engineers by Serway.
states that "a body immersed in a fluid is buoyed up by a force equal the weight
Determine the reaction order for Na2S2O3 using calculations described in the Background. Show your work. Note that your answer will probably not be an even whole number as it is in the examples.
waves. The waves at the front of the boat are compressed and are shorter than the longer, spread out waves at the back of the boat.
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Nature of wave: It is an electromagnetic wave as it does not necessarily require a medium for p...
the hull is called friction. This slows the boat down. If a boat hull is designed