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Adjusted moments of inertia. #30

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Adjusted moments of inertia.
robertmcewen Feb 28, 2023
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140 changes: 68 additions & 72 deletions docs/docs/theory.md
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Expand Up @@ -21,22 +21,23 @@ the specific physical characteristics of the MBARI WEC are tabulated for referen
Each rigid body in the simulation has a "Link Frame" coordinate system in which all other characteristics of the
body are defined in for computational purposes. This link-frame coordinate system is often selected to be at the
location of a joint that connects the various bodies (which are also called links in the vernacular of Gazebo).
In the following sections, all quantities are defined in the body's link frame unless otherwise noted. Also, the
description below includes alternative notation for added mass, e.g. \(\\ \mu_{xx}\), in the style of Newman.
Also, the description below includes alternative notation for added mass, e.g. \(\\ \mu_{xx}\), in the style of Newman.


### Surface Buoy
| | Description | | Units |
|:----------------:|:----------------------------------------------------------|:----------------:|:---------:|
| \(m\) | Buoy Mass | 1400 | kg |
| \(V\) | Displacement (undisturbed buoy) | 1.0 | m\(^3\) |
| \(V\) | Displacement (undisturbed buoy) | 2.39 | m\(^3\) |
| \(\bf{x}_{COG}\) | Center of Gravity in Link Frame (x,y,z) | (0.0, 0.0, 2.03) | m |
| \(\bf{x}_{COB}\) | Center of Buoyancy in Link Frame (x,y,z) | (0.0, 0.0, 2.05) | m |
| \(\bf{x}_{COW}\) | Center of Waterplane in Link Frame, including PTO and cone| (0.0, 0.0, 2.27) | m |
| \(I_{xx}\) | Roll Moment of Inertia (MOI), origin at pivot | 7000 | kg m\(^2\)|
| \(I_{yy}\) | Pitch Moment of Inertia origin at pivot | 7040 | kg m\(^2\)|
| \(I_{zz}\) | Yaw Moment of Inertia, origin at pivot | 670 | kg m\(^2\)|
| \(A_{wp}\) | Waterplane Area (undisturbed buoy) | 5.0 | m\(^2\) |
| \(S_{x}\) | Second moment of area of water plane, about roll axis | 1.37 | m\(^4\) |
| \(S_{y}\) | Second moment of area of water plane, about pitch axis | 1.37 | m\(^4\) |
| \(I_{xx}\) | Roll Moment of Inertia (MOI) about center of mass | 1450 | kg m\(^2\)|
| \(I_{yy}\) | Pitch Moment of Inertia about center of mass | 1450 | kg m\(^2\)|
| \(I_{zz}\) | Yaw Moment of Inertia about center of mass | 670 | kg m\(^2\)|
| \(A_{wp}\) | Waterplane Area (undisturbed buoy) | 5.47 | m\(^2\) |
| \(X_{\dot{u}}\) | Surge Added Mass (\(\mu_{xx}\)) | 260 | kg |
| \(X_{\dot{q}}\) | Surge-Pitch Added Mass, origin at pivot (\(\mu_{xq}\)) | 370 | kg |
| \(Y_{\dot{v}}\) | Sway Added Mass ( \(\mu_{yy}\)) | 260 | kg |
Expand All @@ -54,88 +55,83 @@ description below includes alternative notation for added mass, e.g. \(\\ \mu_{

- Buoy Link Frame is located at base of the buoy bridle at the pivot.
- Unspecified stability derivative values (\(M_{\dot{r}}\), \(X_{uv}\), \(Y_{vu}\), etc) are zero.
- Inertias, Added mass inertias, and stability derivatives are all specified about the link frame origin, i.e. the pivot.
- Added mass moments of inertia and stability derivatives are all specified about the link frame origin, i.e. the pivot.
- Mass moments of inertia are specified about the center of mass.
- Added mass values are infinite frequency.
- Free-Surface Hydrodynamic Coefficients and Impulse Response Functions can be found <a href="https://www.google.com/") target="_blank">here</a>:

### Power Take-Off Device
| | Description | | Units |
|:-----------------:|:-------------------------------------------|:----------------:|:---------:|
| \(m\) | PTO Mass | 605 | kg |
| \(V\) | PTO Displacement | 1.0 | m\(^3\) |
| \(\bf{x}_{COG}\) | Center of Gravity in Link Frame (x,y,z) | (0.0, 0.0, -4.0) | m |
| \(\bf{x}_{COB}\) | Center of Buoyancy in Link Frame (x,y,z) | (0.0, 0.0, -3.0) | m |
| \(I_{xx}\) | Roll Moment of Inertia | 32600 | kg m\(^2\)|
| \(I_{yy}\) | Pitch Moment of Inertia | 32600 | kg m\(^2\)|
| \(I_{zz}\) | Yaw Moment of Inertia | 7 | kg m\(^2\)|
| \(I_{xy}\) | Roll-Pitch Product of Inertia | 0 | kg m\(^2\)|
| \(I_{xz}\) | Roll-Yaw Product of Inertia | -2 | kg m\(^2\)|
| \(I_{yz}\) | Pitch-Yaw Product of Inertia | -3 | kg m\(^2\)|
| \(X_{\dot{u}}\) | Surge Added Mass (\(\mu_{xx}\)) | 310 | kg |
| \(X_{\dot{q}}\) | Surge-Pitch Added Mass MOI (\(\mu_{xq}\)) | 1250 | kg m |
| \(Y_{\dot{v}}\) | Sway Added Mass (\(\mu_{yy}\)) | 310 | kg |
| \(Y_{\dot{p}}\) | Sway-Roll Added Mass MOI (\(\mu_{yp}\)) | -1250 | kg m |
| \(Z_{\dot{w}}\) | Heave Added Mass (\(\mu_{zz}\)) | 10 | kg |
| \(K_{\dot{p}}\) | Roll Added Mass MOI (\(\mu_{pp}\)) | 7040 | kg m\(^2\)|
| \(M_{\dot{q}}\) | Pitch Added Mass MOI (\(\mu_{qq}\)) | 7040 | kg m\(^2\)|
| \(X_{u\mid u \mid}\) | Surge Quadratic Drag | -1140 | kg/m |
| \(Y_{v\mid v \mid}\) | Sway Quadratic Drag | -1140 | kg/m |
| \(Z_{w\mid w \mid}\) | Heave Quadratic Drag | -50 | kg/m |
| \(K_{p\mid p \mid}\) | Roll Quadratic Drag | -195400 | kg m\(^2\)|
| \(M_{p\mid p \mid}\) | Pitch Quadratic Drag | -195400 | kg m\(^2\)|
| \(N_{r\mid r \mid}\) | Yaw Quadratic Drag | -50 | kg m\(^2\)|
| | Description | | Units |
|:-----------------:|:--------------------------------------------------------|:----------------:|:---------:|
| \(m\) | PTO Mass | 605 | kg |
| \(V\) | PTO Displacement | .205 | m\(^3\) |
| \(\bf{x}_{COG}\) | Center of Gravity in Link Frame (x,y,z) | (0.0, 0.0, -4.0) | m |
| \(\bf{x}_{COB}\) | Center of Buoyancy in Link Frame (x,y,z) | (0.0, 0.0, -3.0) | m |
| \(I_{xx}\) | Roll Moment of Inertia about center of mass | 3525 | kg m\(^2\)|
| \(I_{yy}\) | Pitch Moment of Inertia about center of mass | 3525 | kg m\(^2\)|
| \(I_{zz}\) | Yaw Moment of Inertia about center of mass | 10 | kg m\(^2\)|
| \(X_{\dot{u}}\) | Surge Added Mass (\(\mu_{xx}\)) | 310 | kg |
| \(X_{\dot{q}}\) | Surge-Pitch Added Mass MOI (\(\mu_{xq}\)) | 1250 | kg m |
| \(Y_{\dot{v}}\) | Sway Added Mass (\(\mu_{yy}\)) | 310 | kg |
| \(Y_{\dot{p}}\) | Sway-Roll Added Mass MOI (\(\mu_{yp}\)) | -1250 | kg m |
| \(Z_{\dot{w}}\) | Heave Added Mass (\(\mu_{zz}\)) | 10 | kg |
| \(K_{\dot{p}}\) | Roll Added Mass MOI (\(\mu_{pp}\)) | 7040 | kg m\(^2\)|
| \(M_{\dot{q}}\) | Pitch Added Mass MOI (\(\mu_{qq}\)) | 7040 | kg m\(^2\)|
| \(X_{u\mid u \mid}\) | Surge Quadratic Drag | -1140 | kg/m |
| \(Y_{v\mid v \mid}\) | Sway Quadratic Drag | -1140 | kg/m |
| \(Z_{w\mid w \mid}\) | Heave Quadratic Drag | -50 | kg/m |
| \(K_{p\mid p \mid}\) | Roll Quadratic Drag | -195400 | kg m\(^2\)|
| \(M_{p\mid p \mid}\) | Pitch Quadratic Drag | -195400 | kg m\(^2\)|
| \(N_{r\mid r \mid}\) | Yaw Quadratic Drag | -50 | kg m\(^2\)|

- PTO Link Frame is located at top attachment of the PTO (where connects to the buoy).
- Unspecified stability derivative values (\(M_{\dot{r}}\), \(X_{uv}\), \(Y_{vu}\), etc) are zero.
- Inertias, Added mass inertias, and stability derivatives are all specified about the link frame origin.
- Added mass values are infinite frequency.
- Added mass moments of inertia and stability derivatives are all specified about the link frame origin.
- Mass moments of inertia are specified about the center of mass.

### Piston
| | Description | | Units |
|:-----------------:|:-------------------------------------------|:-----------------:|:---------:|
| \(m\) | Piston Mass | 48.0 | kg |
| \(\bf{x}_{COG}\) | Center of Gravity in Link Frame (x,y,z) | (0.0, 0.0, -2.58) | m |
| \(I_{xx}\) | Roll Moment of Inertia | 100.0 | kg m\(^2\)|
| \(I_{yy}\) | Pitch Moment of Inertia | 100.0 | kg m\(^2\)|
| \(I_{zz}\) | Yaw Moment of Inertia | 5.0 | kg m\(^2\)|
| | Description | | Units |
|:-----------------:|:--------------------------------------------------------|:-----------------:|:---------:|
| \(m\) | Piston Mass | 48.0 | kg |
| \(\bf{x}_{COG}\) | Center of Gravity in Link Frame (x,y,z) | (0.0, 0.0, -2.58) | m |
| \(I_{xx}\) | Roll Moment of Inertia | 100.0 | kg m\(^2\)|
| \(I_{yy}\) | Pitch Moment of Inertia | 100.0 | kg m\(^2\)|
| \(I_{zz}\) | Yaw Moment of Inertia | 5.0 | kg m\(^2\)|

- The piston is contained with the PTO housing, so it has mass and moments of inertial, but contributes
no buoyancy, added mass, or quadratic fluid drag.

### Heave Cone
| | Description | | Units |
|:-----------------:|:-------------------------------------------|:----------------:|:---------:|
| \(m\) | Heave Cone Mass | 820 | kg |
| \(V\) | Heave Cone Displacement | 1.0 | m\(^3\) |
| \(\bf{x}_{COG}\) | Center of Gravity in Link Frame (x,y,z) | (0.0, 0.0, -1.25)| m |
| \(\bf{x}_{COB}\) | Center of Buoyancy in Link Frame (x,y,z) | (0.0, 0.0, -1.21)| m |
| \(I_{xx}\) | Roll Moment of Inertia | 4200 | kg m\(^2\)|
| \(I_{yy}\) | Pitch Moment of Inertia | 4200 | kg m\(^2\)|
| \(I_{zz}\) | Yaw Moment of Inertia | 610 | kg m\(^2\)|
| \(I_{xy}\) | Roll-Pitch Product of Inertia | 0 | kg m\(^2\)|
| \(I_{xz}\) | Roll-Yaw Product of Inertia | -1 | kg m\(^2\)|
| \(I_{yz}\) | Pitch-Yaw Product of Inertia | 0 | kg m\(^2\)|
| \(X_{\dot{u}}\) | Surge Added Mass (\(\mu_{xx}\)) | 720 | kg |
| \(X_{\dot{q}}\) | Surge-Pitch Added Mass MOI (\(\mu_{xq}\)) | 900 | kg m |
| \(Y_{\dot{v}}\) | Sway Added Mass (\(\mu_{yy}\)) | 720 | kg |
| \(Y_{\dot{p}}\) | Sway-Roll Added Mass MOI (\(\mu_{yp}\)) | -900 | kg m |
| \(Z_{\dot{w}}\) | Heave Added Mass: Doors Closed (\(\mu_{zz}\))| 9330 | kg |
| \(Z_{\dot{w}}\) | Heave Added Mass: Doors Open | 3000 | kg |
| \(K_{\dot{p}}\) | Roll Added Mass MOI (\(\mu_{pp}\)) | 3990 | kg m\(^2\)|
| \(M_{\dot{q}}\) | Pitch Added Mass MOI(\(\mu_{qq}\)) | 3990 | kg m\(^2\)|
| \(N_{\dot{r}}\) | Yaw Added Mass MOI (\(\mu_{rr}\)) | 10 | kg m\(^2\)|
| \(X_{u\mid u \mid}\) | Surge Quadratic Drag | -1580 | kg/m |
| \(Y_{v\mid v \mid}\) | Sway Quadratic Drag | -1580 | kg/m |
| \(Z_{w\mid w \mid}\) | Vertical Quadratic Drag: Doors Open | -3200 | kg/m |
| \(Z_{w\mid w \mid}\) | Vertical Quadratic Drag: Doors Closed| -3900 | kg/m |
| \(K_{p\mid p \mid}\) | Roll Quadratic Drag: | -4620 | kg m\(^2\)|
| \(M_{q\mid q \mid}\) | Pitch Quadratic Drag: | -4620 | kg m\(^2\)|
| \(N_{r\mid r \mid}\) | Yaw Quadratic Drag: | -50 | kg m\(^2\)|
| | Description | | Units |
|:-----------------:|:--------------------------------------------------------|:----------------:|:---------:|
| \(m\) | Heave Cone Mass | 820 | kg |
| \(V\) | Heave Cone Displacement | .12 | m\(^3\) |
| \(\bf{x}_{COG}\) | Center of Gravity in Link Frame (x,y,z) | (0.0, 0.0, -1.25)| m |
| \(\bf{x}_{COB}\) | Center of Buoyancy in Link Frame (x,y,z) | (0.0, 0.0, -1.21)| m |
| \(I_{xx}\) | Roll Moment of Inertia about the center of mass | 340 | kg m\(^2\)|
| \(I_{yy}\) | Pitch Moment of Inertia about the center of mass | 340 | kg m\(^2\)|
| \(I_{zz}\) | Yaw Moment of Inertia about the center of mass | 610 | kg m\(^2\)|
| \(X_{\dot{u}}\) | Surge Added Mass (\(\mu_{xx}\)) | 720 | kg |
| \(X_{\dot{q}}\) | Surge-Pitch Added Mass MOI (\(\mu_{xq}\)) | 900 | kg m |
| \(Y_{\dot{v}}\) | Sway Added Mass (\(\mu_{yy}\)) | 720 | kg |
| \(Y_{\dot{p}}\) | Sway-Roll Added Mass MOI (\(\mu_{yp}\)) | -900 | kg m |
| \(Z_{\dot{w}}\) | Heave Added Mass: Doors Closed (\(\mu_{zz}\)) | 9330 | kg |
| \(Z_{\dot{w}}\) | Heave Added Mass: Doors Open | 3000 | kg |
| \(K_{\dot{p}}\) | Roll Added Mass MOI (\(\mu_{pp}\)) | 3990 | kg m\(^2\)|
| \(M_{\dot{q}}\) | Pitch Added Mass MOI(\(\mu_{qq}\)) | 3990 | kg m\(^2\)|
| \(N_{\dot{r}}\) | Yaw Added Mass MOI (\(\mu_{rr}\)) | 10 | kg m\(^2\)|
| \(X_{u\mid u \mid}\) | Surge Quadratic Drag | -1580 | kg/m |
| \(Y_{v\mid v \mid}\) | Sway Quadratic Drag | -1580 | kg/m |
| \(Z_{w\mid w \mid}\) | Vertical Quadratic Drag: Doors Open | -3200 | kg/m |
| \(Z_{w\mid w \mid}\) | Vertical Quadratic Drag: Doors Closed | -3900 | kg/m |
| \(K_{p\mid p \mid}\) | Roll Quadratic Drag: | -4620 | kg m\(^2\)|
| \(M_{q\mid q \mid}\) | Pitch Quadratic Drag: | -4620 | kg m\(^2\)|
| \(N_{r\mid r \mid}\) | Yaw Quadratic Drag: | -50 | kg m\(^2\)|

- Heave-Cone Link Frame is located at top attachment of the Heave Cone (where it connects to the tether).
- Unspecified stability derivative values (\(M_{\dot{r}}\), \(X_{uv}\), \(Y_{vu}\), etc) are zero.
- Inertias, Added mass inertias, and stability derivatives are all specified about the link frame origin.
- Added mass values are infinite frequency.
- Added mass moments of inertia and stability derivatives are all specified about the link frame origin.
- Mass moments of inertia are specified about the center of mass.


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