Modeling Boat Velocity During Acceleration & Deceleration

[chartman]

We recently published a report modeling recreational planing boat velocity during runs from zero to top speed as a function of the scientific constant e.

The velocity curve was split into displacement and planing portions. Each portion was curve fit to the equation in the image below using three independent, meaningful variables.

1. A = asymptote of top speed in miles per hour
2. b = a time constant representing how fast the boat accelerates
3. t0 = a time zero representing when the boat took off with respect to the time axis

Coast down / coastdown data was similarly processed. This method was compared to a published 4th order polynomial fit in an SAE paper on boat coast down testing in relation to accident reconstruction.

Velocity, RPM, drag, acceleration, deceleration, propeller slip, take-off, transition zone, and distance were plotted for multiple boats and drive types including two speed drives.

The process allowed two boats to be virtually raced against one another or the same boat to be virtually raced against itself after changes / improvements were made.

The 142 page 5 Megabyte report is online at:
http://www.propellersafety.com/wordpress/wp-content/uploads/boat-acceleration-model-PDL.pdf

I welcome comments on how this method might be improved upon and if you find it useful or not.

gary

 

[tspeer]

There are fast dynamics and slow dynamics to a boat's motion. Most performance related dynamics are comparatively slow and can be considered to be quasi-steady. One method I've used to model the acceleration of sailboats is to apply a fictitious D'Alembert force in the velocity prediction program. This consists of an additional drag (or thrust) equal to the mass times the acceleration.

The VPP calculates the equilibrium performance with the additional drag as before. The result is a boatspeed polar diagram corresponding to a given level of acceleration. The conventional steady performance polar is thus a special case corresponding to zero longitudinal acceleration. By generating a family of polars corresponding to different levels of acceleration, one can build up a picture that shows how best to accelerate the boat to a given speed and how fast it will slow down when it turns from a faster point of sail to a slower one.

Here are two sets of made-up polars that illustrate how such polars can be used to compare two different boats. The boat in red has better steady-state performance to windward and leeward, but the boat in blue has better reaching performance. At low speeds, the red boat can out accelerate the blue boat, even on a reach.

By taking the points where the acceleration polars are tangent to the rings of constant boat speed, one can form a schedule of true wind angle vs boat speed for accelerating from low speed to maximum speed in minimum time.

In a way similar to Charman's diagrams, one can integrate the trajectory across the acceleration polars to compute the time it takes to achieve a given sailing condition. One interesting question is, which is the better choice - accelerate on a reach along the locus of maximum acceleration points until one gets to the best speed for upwind Vmg before rapidly turning up and transitioning along the constant speed circle, or follow the locus of best Vmg points for each acceleration polar to accelerate more slowly but be making progress to windward the whole time?

 

[DCockey]

as a function of the scientific constant e

What was the reason for using an exponential equation? The only rational I see in the paper is that exponential curves are used to model "charging a capacitor, heat transfer, or charging an air tank", and "the equations fit the data very well, allowing just 6 numbers to represent a complete top speed run".

Was any consideration given to how the propeller thrust and boat resistance/drag vary with boat speed?

 

[tspeer]

An exponential function has the property that the rate of change is proportional to the value of the function. They arise naturally as the solution to first order differential equations. A good example is population growth, where the increase in the population is proportional to the size of the population. The drag is proportional to velocity to some power, so the slower the velocity the slower the deceleration. That's why the exponential fit.

 

[DCockey]

Hydrodynamic drag may be approximated over some speed range as proportional to velocity to a power , particularly if the speed is well below the onset of planing or well above the onset of planing.

The thrust of a propeller turned by an IC engine will generally be more complicated than a simple approximation as proportional to velocity to a different power, though over a small enough speed range an approximation of thrust as a constant plus a factor proportional to velocity may be reasonable.

If the vessel speed is sufficient that aerodynamic drag is significant then the aerodynamic drag will be close to proportional to the velocity squared.

Acceleration = (Thrust - hydrodynamic drag - aerodynamic drag)/mass The solution of the this equation will generally not be a simple exponential.

An exponential fit over some speed range may be satisfactory depending on the speed range and the required accuracy.

"Added mass" effects which are a method of accounting for the deviation of hydrodnamic forces from the quasi-steady values when the vessel is accelerating are usually not significant for "performance" analysis which is consistent with Tom's comment in his post.

 

[chartman]

What was the reason for using an exponential equation? The only rational I see in the paper is that exponential curves are used to model "charging a capacitor, heat transfer, or charging an air tank", and "the equations fit the data very well, allowing just 6 numbers to represent a complete top speed run".
Was any consideration given to how the propeller thrust and boat resistance/drag vary with boat speed?

The planing curve going up to top speed immediately reminded me of an exponential curve I used a decade earlier to curve fit a hydraulic heat transfer situation on mobile construction equipment. To fit the boat data I used an ancient version of Sigma Plot and tried multiple kinds of equations including polynomials. Quite quickly it was obvious the curve needed split between the displacement and planing portion and the exponential fit was very good.

Also I had previously seen presentation by the U.S. Olympic bike racing technology folks on coast down studies to minimize drag based on position, helmets, etc. and they used an exponential fit. That was way back in the era the tear drop bike racing helmets were first being used.

No consideration was given to variability of thrust, resistance, drag at varying speed EXCEPT going up and coming down (running to top speed, then coasting down from there). The boat was also coasted down from multiple speeds to show it followed the same velocity curve as it did when it coasted down from top speed.

gary

 

[chartman]

Hydrodynamic drag may be approximated over some speed range as proportional to velocity to a power , particularly if the speed is well below the onset of planing or well above the onset of planing.

The thrust of a propeller turned by an IC engine will generally be more complicated than a simple approximation as proportional to velocity to a different power, though over a small enough speed range an approximation of thrust as a constant plus a factor proportional to velocity may be reasonable. ...

Acceleration = (Thrust - hydrodynamic drag - aerodynamic drag)/mass The solution of the this equation will generally not be a simple exponential.
An exponential fit over some speed range may be satisfactory depending on the speed range and the required accuracy.

Prior to the exercise of gathering and analyzing the data I would have agreed with you. But the data showed an exponential fit over the entire displacement mode of operation, then a different exponential fit over the entire planing mode, See the charts below. Those are actual velocity data points taken 10 times per second plotted against the exponential fitted to the data. The planing data gets a bit jittery at top speed as the operator was adjusting trim to find top speed. Looks very good over the entire displacement or planing ranges to me. Similar results including even better fits were found on several boats. The same goes for exponential curve fits to coast down data.

The report analyzes the residuals of the curve fit in comparison to work done by others on coast down data that were fit by 4th order polynomials.

This (the general belief that something as simple as an exponential curve fit would not work) is why the report was written. While it may not seem logical given all the variables involved and their interactions over a wide range of speeds, the exponential fit was found to work very well.

Others may find different results. I am just reporting many boats were curve fit with exponential curves with results like those below.

Similarly the report shows exceptional fit between the resulting exponential curve fits and the actual data during the transition zone between displacement and planing. The actual data is the orange balls in the two transition charts attached.

Back to the earlier discussion of why to fit with an exponential curve, the resulting coefficients really mean something. One is the asymptote of top velocity, one is take off time, the inverse of the other is the time constant Tau which can be used to predict velocity at certain times.



gary

 

[Kayakmarathon]

What was the reason for using an exponential equation?

In a simple mass-friction system, the time response of velocity to a step input is an exponential function. The time constant is related to mass and friction. Since the time constant is easy to measure, and mass is constant, then friction can be calculated. Drag in water is complex, so it can be simplified through a piece-wise linear approximation. Once the speed and friction are known, required power can be estimated.

 

[aziziyeng]

If thrust and resistance be linear function of velocity, velocity will be exponential function of time:
ma=T-R
mdv/dt=av-bv
dv/dt=[(a-b)/m]v
v=exp{[(a-b)/m]t+c}

 

[Alan Cattelliot]

I mostly agree with AZIZIYENG. There is nothing new under the sun, here. It gives no clue about the influence of the propeller diameter, step, rake, skew.... which is "hidden" behind the coefficients A and b (following the general expression given in page 2 of the report), together mixed with hydrodynamic drag coefficient of the hull's shape. Even with the same engine, ( same gears, same shaft, same lubrificant... ) with the same propeller (at the same place, with the same trim ... ), a given boat with the same loads but with difference in load placement will have different speed curves. It could eventually be that the planning mode of the boat could not be reached, due to a wrong load configuration.

More efforts should have been made to truly separate the thrust and the drag. There is a strong lack of theorical and technical background. For instance, when I read, on page 93 :"Tau is often associated with First Order Systems with one energy storage element (is used in many other models)", I really thanks my parents and my professors to have given to me such an educationnal degree that I can still find some meaning in that sentence, which is totally uncorrelated with any of the concepts developped in the whole publication. Page 26, I read : "A New Way to Estimate Thrust at Takeoff", which described elementary mechanics and data exploitations based on derivatives of speed acquisition .... Has any of the author of this publication done any experiment in his life ? Was it the first time they have been studying movements, trajectories and forces ? Even on this forum .... https://www.boatdesign.net/threads/do-these-measurements-of-drag-for-my-canoes-make-sense.67448/#post-936434 (original publication of the methodology used is given in the same thread). How is it that there is nothing about the boat windage ? It seems that there is some confusion of total drag with hydrodynamic drag. I wish I could figure out better what has been demonstrated with this publication, but I cannot find anywhere a exhaustive presentation of the experimental runs, nor any presentation of the experimental protocole. The description on page 40 is rather small, and gives no clues about how the experiment has been done. Thus It cannot be given any opinion on the results, nor compared them with other datas. Only on the methodology that is presented here can be discussed.

Finally, it seems to me very difficult to come to the same conclusions (1 to 19) as the authors of this publication, as given in page 97. Hard to swallow...