Pitch - Roll Order and Axes

[DCockey]

Is there a general convention in naval architecture for defining pitch and roll angles of a vessel when the pitch or roll angles are large? I've looked through several references and can't find anything. As long as the pitch angle is small the differences would be small. But if one or other of the angles becomes large then the differences will be significant.

Alternatives are:

1A) Roll, then Pitch with both axis fixed in space. Neither axis moves with the vessel.
1B) Pitch, then Roll about the longitudinal axis of the hull. The roll axis pitches with the vessel.
The final position is the same for 1A and 1B. The longitudinal axis remains in the same vertical plane as that of the undisturbed vessel, ie no yaw.

2A) Pitch, then Roll about an axis fixed in space. Neither axis moves with the vessel.
2B) Roll, then Pitch about the transverse axis of the hull, The pitch axis rolls with the vessel.
The final position is the same for 2A and 2B. The longitudinal axis is not in the same vertical plane as that of the undisturbed vessel, ie there is a yaw angle.

I'm assuming 1A/1B is more desirable due to no yaw.

[Ad Hoc]

hmmm..you're getting into the nitty gritty.

Basically naval architecture deals with
i) movement in flat clam conditions
ii) motion in a distrub seaway

then the other lesser know
iii) structural/vibration responses from other sources.

At times all 3 conditions can coexist. But can sensible anwsers be obtained from superpositioning?..this has been debate endless and is difficult to say with certainty.

But i) ii) and indeed iii) can be investiagted using linear theory.

So the coords are usually defined using "rigid body" coordinates, thus all linear and fixed.

[DCockey]

Thanks for the reply.

I'll try to clarify. My question is about describing vessel motion, ie rigid body motion. It's probably a little outside the scope of usual naval architecture which seems to generally consider only small pitch/trim angles. I'm looking at some very small boats and interested in hydrostatics with combinations of large roll angles large pitch/trim angles.

So the question is what is the position of a boat with say a pitch/trim angle of 10 deg and roll angle of 30 deg. Different postions result using the proceedures of 1A/B and 2A/B. In 1A and 2A the axis of rotation are fixed in space. In 1B and 2B the axis are fixed to the vessel.

I was wondering if there was a convention in case I want to use some software requiring input of roll and pitch/trim angles. Currently I'm moving the geometry "manually" and can work either way.

[Ad Hoc]

Originally Posted by DCockey

I'll try to clarify. My question is about describing vessel motion, ie rigid body motion. It's probably a little outside the scope of usual naval architecture which seems to generally consider only small pitch/trim angles..

This is dealt with in undergraduate and post graduate courses. But we study this simply to be aware of it and how to calculate it, since once done, very rarely done again, unless that is ones field of expertise. (I certianly have not)The mathematics is seriously complex.

Since we have the reference about the normal axis of the boat, X-Y-Z planes. But then you need to account for the translation of the vessel relative to that and is defined by another fixed orgin (earth) which is the "swing" to the azimuth, a "tilt" to the actual elevation and then the "heel" to the actual orientation.

When you go through the maths of these equations of motion, you can easily write 10-20 pages for just one solution. And..is is correct?

Thus when more than one orgin is used the equations of motion become mind boggling. Hence because of the complexity, we "study" this at Uni, but beyond that, everything becomes "linearised" and use a single datum.

The differences between the 2 approaches (apart from the complex maths involved) and their results is still questionable. There does not seem to be any reason, other than a PhD or software driven, to actually explore these equations in their fullest form. Since the difference in actual results are questionable to accuracy owing to all the attributes that must be taken into account, which are variable anyway.

Hence simple linear theory and fixed datum. It works and provides quick simple results with a high degree of accuracy.

[DCockey]

My mistake in using the word "motion". My current interest is hydrostatics. I know how to calculate the hydrostatics for the boat in an arbitrary position. My question is about which definition of the coordinate system to use to describe the position of the boat..

With an axis system fixed to the earth, does XX degs or roll and YY degs of pitch mean the position which is achieved by:
a) Rolling the boat by XX degs, then pitching/triming the rolled boat by YY degs?
- or -
b) Pitching/triming the boat by YY degs, then rolling the pitched boat by XX degs?

The resulting positions are different. Attached is a picture showing two boat like objects. The blue one was rolled 40 degrees, then pitched 10 degrees. The green one was pitched 10 degrees, then rolled 40 degrees.

The answer may be that there is no general convention, and I just need to pick one. But if there is a convention I'd like to use it to enable potential comparisons in the future.

[Ad Hoc]

Originally Posted by DCockey

My current interest is hydrostatics..

If your interest is in hydrostatics then there is only one convention. That is, equilibrium.

However you roll and/or pitch, the resulting potion must be in equilibrium.

There cannot exist 2 solutions..only one.

[DCockey]

Originally Posted by Ad Hoc

If your interest is in hydrostatics then there is only one convention. That is, equilibrium.

However you roll and/or pitch, the resulting potion must be in equilibrium.

There cannot exist 2 solutions..only one.

Agree, this is only a question about coordinate system definition used to describe the position of a boat in equilibrium.

[jehardiman]

What you are asking about is the cross-product order, and yes it makes a difference. You must use a right-hand axis system. the most common one for Naval Architecture is x=fwd, y=stbd, z=down. So you roll (x) then pitch (y) then yaw (z) to get the correct orientation.

[DCockey]

Thanks! That's what I was looking for.

[Ad Hoc]

Thanks JEH...i was getting confused what was the question being asked, as it wasn't clear to me.

DCockey see attached, pictorally, ref: any naval architecture book. (I assumed you had already looked at one, hence over thinking the simple co-ord question).

[DCockey]

Thanks. The coordinate system itself wasn't the question, but the order of rotations, ie the cross-product order, which JEH clarified.

[daiquiri]

It doesn't make sense to me. The order of rotations doesn't count because the final equilibrium position of the ship is always the same, independently of the procedural path which has led it to that position. It is a consequence of the conservative nature of the gravitational field. You can verify this by putting a small weight on a boat model (somewhere off from the cg, both laterally and longitudinally) and see how it settles in the water after a forced pitch and roll movements. It will always end up into the same heeled and pitched position.
If you get different equilibrium positions with different order of rotations, there's something you didn't keep constant in the process - the c.g. position, the displacement, for example. Or you didn't respect the colinearity of Weight and Buoyancy vectors.
Cheers

[Alik]

You need to define clearly what is pitch and what is heel. The sample of such definition is presented as I remember here: Hamamoto M., Akiyoshi T. Study on Ship Motions and Capsizing in Following Seas. (1st Report) // Journal of The Society of Naval Architects of Japan. No.147, 1980.

I believe You are taking heel and pitch as angles between reference 3D model and plane, but those should be Euler angles.

[DCockey]

JEHardiman provided the information I was looking for.

No question that the final equilibrium position is always the same irregardless of what definitions of roll and pitch (and yaw) are used. Perhaps a clearer version of the question I'm interested in follows:

Consider a vessel which has three orthangonal axes defined: longitudinal, transverse and vertical. Disturb the postion of the vessel so that it has rolled and pitched. Now measure and report the roll and pitch angles of the vessel in its new orientation. How should the roll and pitch angles be measured? There are several legitimate definitions which give different answers.

Next, given those definitions of pitch and roll, how is a model of the vessel moved so that it has the same orientation. The "order of rotations" is important and it depends on the definitions used.

For most of naval architecture this isn't a significant question. If yaw is ignored and the ptich angle is small then the differences between the various definitions are generally negligable. That may be why I didn't find it discussed in several naval architecture texts and references I consulted.

I'm looking at the stability of small boats when the passengers and/or cargo move so that the pitch angle is relatively large which is why I'm currently interested.

[Ad Hoc]

It appears you’re over thinking this, perhaps coming from an aero background??

As noted above a vessel when floating is hydrostatically in equilibrium. But it may exhibit 3 conditions, i)stable, ii) unstable and iii) neutral equilibrium.

Couple this with the fact that any vessel has 6 degrees of freedom. Taking the translation movements first.

A displacement, or translation in the x-axis (longitudinally - surge) leads to a situation which has no resultant force, this is neutral equilibrium. In other words, you push the vessel fwd, but there is no restoring force because it is still in equilibrium.
A displacement in the y-axis (transverse - sway), is the same result as x-axis.
A displacement in the z-axis (vertically - heave) is different, as there shall be a restoring force, buoyancy, once a displacement occurs. This returns the vessel back to its equilibrium state.

Looking at rotational.
Rotation about the x-axis, heel, results is a moment that acts on the vessel. This leads to a condition of either: stable, neutral or unstable equilibrium.
Rotation about the y-axis, trim, is similar to x-axis again.
Rotation about the z-axis, yaw, ends up with no resultant force/moment so the vessel is in a neutral equilibrium.

Thus it is clear that only rotations about the x and y-axis are to be considered because it is not in equilibrium as 3 possible solutions exist.

Thus for vessels all you need to address is the heel and trim.

You will note I am using heel, and trim and not roll or pitch. Since we are talking about a hydrostatic case, not dynamic. The 2 are very different hence has its own nomenclature for clarity.

The only reason why you’re getting confused is that you’re assuming something that is either not important/considered or that you unaware how to calculate the above at large angles of heel/trim. Since at small angles, it is a linear relationship. At large angles it is not. At large angles, it just requires more detailed computations, since the simple relations between heel and trim from hydrostatics tends to be lost owing to the change in waterplane area and submerged volume for buoyancy.

It is much easier to calculate the resulting conditions when a vessel experiences a large angle of heel than for that of a large trim. Also because longitudinal stability, GM(L) is significantly greater than GM(T), it is simple convention to calculate heel first, since the large heel may yield the result anyway!.