Tow Coordinate System


Product(s):SACS
Version(s):23
Area:Tow

Problem

How does the SACS Tow Co-ordinate System work within the program?

Solution

Let us walk through this scenario with a sample problem shared by one of our users.

We have prepared the following 4 sample files.

  1. Roll-Y: Here SACS default YXZ axes is considered. Roll angle & period along with heave coefficient is provided with 4 combinations of directions, with “V” as “Include Gravity Option” for all cases.
  2. Roll-Y_Pitch X: Here SACS default YXZ axes is considered. Roll angle & period along with heave coefficient is provided with 5 combinations of “Include Gravity Options”. Similarly, Pitch angle & period along with heave coefficient is provided with 5 combinations of “Include Gravity Options.
  3. Pitch-Y: Here SACS tow coordinate is used as XYZ axes. Pitch angle & period along with heave coefficient is provided with 4 combinations of directions, with “V” as “Include Gravity Option” for all cases.
  4. Roll-X_Pitch-Y: Here SACS tow coordinate is used as XYZ axes. Roll angle & period along with heave coefficient is provided with 5 combinations of “Include Gravity Options”. Similarly, Pitch angle & period along with heave coefficient is provided as below with 5 combinations of “Include Gravity Options”.

Now, for the first 2 cases, as the tow coordinate axes is as per SACS default coordinate system, we expect due to positive roll (about SACS Y-axis), negative sway will be generated along SACS global X-axis, based on right hand thumb rule. Similarly, due to positive pitch (about SACS X-axis) positive surge will be generated along SACS global Y-axis.

For the 3rd and 4th cases, as the tow coordinate axes is XYZ, based on SACS coordinate system, for positive roll (about SACS X-axis), negative sway will be generated along SACS global Y-axis. And for positive pitch (about SACS global Y-axis), positive surge will be generated along SACS global X-axis.

In SACS output results of Dynamic Loading Summation, you will see the above pattern.

If you refer SACS Tow Manual Section 2.2.3, then you will find that the default coordinate system is YXZ as shown in below snap.

And, when you are changing the Roll and Pitch directions, automatically you are moving to the XYZ system of tow coordinates, for which heave direction is downward. See below figure from SACS Tow Manual.

Hence, your output for heave is reversed. You need to follow SACS right-handed coordinate system and user defined axis cannot not be valid here. When you are providing +0.2g heave with XYZ tow coordinates, SACS automatically considering the heave along SACS global downward Z direction. The calculations are set according to this.

For a sample of calculation, let me take your Roll-Y & Pitch-X file, with “V” as include gravity option (RH-V Load condition)
Here, the inputs are as below:

ϴ = 20degree, t = 10s, a_heave = 0.2g.

Calculation of acceleration in horizontal direction:

Transitional acceleration = [a_sway-a_heave*sinϴ]*g = [0-0.2*sin(20*π/180)c]*g = -0.068*g

Acceleration due to gravity = -g*sinϴ = -0.342g

Total acceleration = [(-0.068) + (-0.342)]*g = -0.41g

Calculation of acceleration in vertical direction:

Transitional acceleration = a_heave*cosϴ = 0.2*[cos (20*π/180)c]*g = +0.188g

Acceleration due to gravity = (cosϴ-1)*g = [cos (20*π/180)c]*g = -0.0603g

(As option “V” includes gravity for surge and sway only, hence vertical component is subtracted as g* cosϴ-g)

Total acceleration = [(+0.188) + (-0.0603)]*g = 0.128g

 

Now, let us investigate the same load condition for Roll-X & Pitch-Y model.

ϴ = 20degree, t = 10s, a_heave = 0.2g.

Calculation of acceleration in horizontal direction:

Transitional acceleration = [a_sway-a_heave*sinϴ]*g = [0-(-0.2)*sin(20*π/180)c]*g = +0.068*g

(See the +0.2g heave input is considered negative as tow coordinate system is now XYZ)

Acceleration due to gravity = -g*sinϴ = -0.342g

Total acceleration = [(+0.068) + (-0.342)]*g = -0.274g

Calculation of acceleration in vertical direction:

Transitional acceleration = -a_heave*cosϴ = -(-0.2)*[cos (20*π/180)c]*g = 0.188g

Acceleration due to gravity = -(cosϴ-1)*g = -[cos (20*π/180)c]*g = +0.0603g

(In the above 2 equations, to maintain the right-hand thumb rule coordinate system sign convention, the heave components become negative to what we were seeing in the similar formulations of YXZ coordinate system)

Total acceleration = [(+0.188) + (+0.0603)]*g = 0.248g

So, overall, you see the acceleration components individually consists of translational, angular (here in our case 0) and due to gravity. For different coordinates they will have different sign conventions. When the individual components are added to get the total acceleration coefficient, they will automatically produce different results. Based on what coordinate system you are selecting and what input you are providing SACS will execute the calculations. SACS provides 6 set of combinations of tow coordinate systems. 3 of which are in line with SACS default tow coordinate system and the other 3 are different. The gravity component further gets altered based on what among the 5 options in “Include gravity option” you select. For “V” and “L” option, the gravity component will be subtracted.