2c:null 2d:["$","title",null,{"children":"Section 4.3 - A Brief Explanation of Maneuvering Mathematical Model | Felipe Ferrari | Observable"}] 32:T5dc,md`This section gives an explanation about the maneuvering simulation model as an example of simulation adopted in the Vessel.js library and marks the end of this report as a guide for vessel.js training. As explained in the beginning of this chapter, the simulation differs from the analysis process, such as the one shown in the previews section, by inserting the time as a variant for the performance evaluation. The maneurvering model was constructed as a joint result from the Maneuvering Visualization project demanded by NTNU in the context of "Markom Project: Advanced data Analytics Framework for Ship Energy Efficiency (ADAF)" together with my Master Thesis propose. Here we are going to explain briefly the model and assumptions as part of this project tutorial. For a more detailed explanation regarding the model itself and its implications read please my Master Thesis. A pratical perception of the maneuvering model can be found in the following examples: * Manoeuvring Model; * ADAF Project; * Trondhein Scene Model. `33:Tc58,md` ## Loop Equation The maneuvering model is dependent of several methods which will result in the displacement of the ship. In summary the code will read the user command, calculate the resistance, take the thrust values according to the propeller series and store the consumption value for just then estimate the ship movement in that period. The loop is sketched by the Figure bellow:

${await FileAttachment("loop_render@2.png").image()}

Drawings offered by NTNU, propeller drawing from Kongsberg Website (https://www.kongsberg.com/maritime/products/, accessed in 2021-04-04).

The resistance ${tex`R_t`} is calculated accoding to the [[1]](#one), and the propeller thrust and torque are represeneted by the ${tex`K_t`} and ${tex`K_q`}, both previous explained in Section 4.1. The ship moving displacement is calculated by solving the following matricial equation: ${tex.block`\dot{\textbf{X}}_{6 \times 1} = \textbf{A}_{6 \times 6}\textbf{X}_{6 \times 1} + \textbf{B}_{6 \times 1}`} Where ${tex`\textbf{X}_{6 \times 1} = [\eta~V]^T`} with ${tex`\eta = [x~y~z]`} and ${tex`V = [u~v~\gamma]`}. The differencials are approximated acording to the Domand Prince method explained in [[2]](#two), using the numeric.js open library for calculation [[3]](#three). This method is similar to the application of Ode45 in matlab software, a method that is well extensivelly applied in industry. The coefficients ${tex`\textbf{A}_{6 \times 6}`} and ${tex`\textbf{B}_{6 \times 1}`} are derived from the second newtowns' law applied to the body. ${tex.block` \textbf{M}.\dot{V}+\textbf{N}(V).V = \textbf{F} `} The term of ${tex`\textbf{M}`} represents the inercial term of the rigid body plus the coriolis effect and ${tex`\textbf{N}`} is dependent of the hidrodynamic effects both in the rigid body and in the coriolis: ${tex.block` \begin{cases} \textbf{M} = M_{RB} + M_{A} \\ \textbf{N}(V) = C_{A}(V) + C_{RB}(V) + D(V) \end{cases} `} The model presented can be approximated under several functions, for a further explanation about the cited equations above please verify [[4]](#four). For vessel.js library the coefficients ${tex`\textbf{A}_{6 \times 6}`} and ${tex`\textbf{B}_{6 \times 1}`} can be provided from the user. In case they are not provided, the library will approximate the coefficients by the method explained in [[5]](#five). The matrix relate each other by the following expression: ${tex.block` \textbf{A}_{6 \times 6} = \begin{bmatrix} \textbf{null}_{3 \times 3} & R(\delta) \\ \textbf{null}_{3 \times 3} & -\textbf{M}^{-1}\textbf{N} \end{bmatrix} `} ${tex.block` \textbf{B}_{6 \times 1} = \begin{bmatrix} \textbf{null}_{3 \times 1} \\ \textbf{M}^{-1}\textbf{F} \end{bmatrix} `} Where the term ${tex`R(\delta)`} is the coordinate transformation matrix: ${tex.block`R(\delta) = \begin{bmatrix} cos(\varphi) & -\sin(\varphi) & 0 \\ \sin(\varphi) & cos(\varphi) & 0 \\ 0 & 0 & 1 \end{bmatrix} `} `34:Tbac,// @ferrari212 // This class requires the numeric.js class ManoeuvringMovement { constructor(manoeuvring) { if (typeof numeric !== "function") { console.error("Manoeuvring requires the numeric.js library."); } this.mvr = manoeuvring; this.manoeuvring = manoeuvring.manoeuvring; this.states = manoeuvring.states; this.getPropResult = manoeuvring.getPropResult; this.dt = 0; } // Insert this inside the ship maneuvring function setMatrixes(F = [0, 0, 0], yaw = 0) { // this.M_RB = numeric.add(this.M, this.I) let mvr = this.mvr; let h = mvr.hydroCoeff; let u = this.states.V.u; // In this case the M_A is constant and the value ws left as it is const M_A = [[0, 0, 0], [0, -h.Yvacc, -h.Yracc], [0, -h.Yvacc, -h.Nracc]]; const M = numeric.add(mvr.M_RB, M_A); // const M = [[ 1.1e5, 0, 0 ], // [ 0, 1.1e5, 8.4e4 ], // [ 0, 8.4e4, 5.8e6 ]] mvr.INVM = numeric.inv(M); // The value of N is in relation with the damping // mvr.INVMN = numeric.dot(numeric.neg(mvr.INVM), mvr.N) const { Cl, Cll, Clll } = mvr.dn; const N = this.manoeuvring.N || [ [0, 0, 0], [0, -Cl * h.Yvdn, this.manoeuvring.m * u - Cll * h.Yrdn], [0, -h.Yvacc * u - Cll * h.Nvdn, -h.Yracc * u - Clll * h.Nrdn] ]; mvr.INVMN = numeric.dot(numeric.neg(mvr.INVM), N); mvr.R = this.parseR(yaw); mvr.A = this.parseA(mvr.R, mvr.INVMN); const INVMF = numeric.dot(mvr.INVM, F); mvr.B = this.parseB(INVMF); } parseA(R, M) { var A = []; for (let i = 0; i < 6; i++) { A.push([0, 0, 0, 0, 0, 0]); } for (let i = 0; i < 6; i++) { for (let j = 0; j < 6; j++) { if (j < 3) { A[i][j] = 0; } else { A[i][j] = i < 3 ? R[i][j - 3] : M[i - 3][j - 3]; } } } return A; } parseB(INVMF) { return [0, 0, 0, INVMF[0], INVMF[1], INVMF[2]]; } parseR(yaw) { var trig = { cos: Math.cos(yaw), sin: Math.sin(yaw) }; return [[trig.cos, -trig.sin, 0], [trig.sin, trig.cos, 0], [0, 0, 1]]; } getDerivatives(V = { u: 0, v: 0, yaw_dot: 0 }) { let mvr = this.mvr; var X = [0, 0, 0, this.states.V.u, this.states.V.v, this.states.V.yaw_dot]; var X_dot = numeric.add(numeric.dot(mvr.A, X), mvr.B); return X_dot; } getDisplacements(dt) { let self = this; // Parse matrix V var X = [0, 0, 0, this.states.V.u, this.states.V.v, this.states.V.yaw_dot]; let sol = numeric .dopri( 0, dt, X, function(t, V) { return self.getDerivatives({ u: X[3], v: X[4], yaw_dot: X[5] }); }, 1e-8, 100 ) .at(dt); // Get global coordinates variation (dx, dy, dyaw) // Get local velocity (du, dv, dyaw_dot) this.states.DX = { x: sol[0], y: sol[1], yaw: sol[2] }; this.states.V = { u: sol[3], v: sol[4], yaw_dot: sol[5] }; this.states.yaw += this.states.DX.yaw; } }35:[{"id":"8221b45ddd1f15974ed6729f5cb3ac564d266a73c629242f455abed53bfb5299526c7215b47235f14764a472983c67809d2edceeb7d9c2229f039e2a871d91aa","href":"https://static.observableusercontent.com/files/8221b45ddd1f15974ed6729f5cb3ac564d266a73c629242f455abed53bfb5299526c7215b47235f14764a472983c67809d2edceeb7d9c2229f039e2a871d91aa","name":"loop_render@2.png","type":"image/png","lastModified":1620639718232.341,"size":157449},{"id":"ee03343d3cefa6d9679c0a2fcc917e436fb24c19b6c6d6ec2f5d01da2f12112c71771e5e4db882dd72cad016650a29ba992f8786bd5db576ca391fed86e1dbda","href":"https://static.observableusercontent.com/files/ee03343d3cefa6d9679c0a2fcc917e436fb24c19b6c6d6ec2f5d01da2f12112c71771e5e4db882dd72cad016650a29ba992f8786bd5db576ca391fed86e1dbda","name":"gunnerus.json","type":"application/json","lastModified":1643483884702.495,"size":51661},{"id":"6f869d80be2ed77d70c56b69717dc6f96e12b75ac51f6bb3c55a096d3ff169d79c49383e5c3e3c83d499aa59360df7c3e6db17bdd79b91fb182f5fb7f1949063","href":"https://static.observableusercontent.com/files/6f869d80be2ed77d70c56b69717dc6f96e12b75ac51f6bb3c55a096d3ff169d79c49383e5c3e3c83d499aa59360df7c3e6db17bdd79b91fb182f5fb7f1949063","name":"gunnerus_power_plant.json","type":"application/json","lastModified":1643484136853.579,"size":377},{"id":"77c9c41bb7da71d7e38f9074b26a715aeab9a0765bdfde681efb650552942d683ecb05e54fb5406b949a7b96686b87dd9010aa8eceb2181e12d00065d49ee260","href":"https://static.observableusercontent.com/files/77c9c41bb7da71d7e38f9074b26a715aeab9a0765bdfde681efb650552942d683ecb05e54fb5406b949a7b96686b87dd9010aa8eceb2181e12d00065d49ee260","name":"man_sample.json","type":"application/json","lastModified":1643484190842.299,"size":252}] 30:{"id":"c4bf3b15fe6cdfaf","slug":"simulation-model-a-brief-method","title":"Section 4.3 - 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A Brief Explanation of Maneuvering Mathematical Model","cells":[{"id":0,"value":"md`# Section 4.3 - A Brief Explanation of Maneuvering Mathematical Model`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":10,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":140,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":144,"value":"md`\n## Maneuvering Model\n\nTo feed the maneuvering model it is requested to pass by this order the ship, states, propeller, power system and then the maneuvering model. The ship and states were well discussed in Chapter 1 of this report. Here we are going just post the values of the parsed data for Gunnerus:\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":175,"value":"Gunnerus = FileAttachment(\"gunnerus.json\").json()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":177,"value":"ship = new Vessel.Ship(Gunnerus)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":181,"value":"shipState = new Vessel.ShipState(ship.designState.getSpecification())","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":184,"value":"md`\nOnce setted the ship object and the ship state, we introduce the propeller and wave such as decrivred in the Section 4.1 and Section 4.2.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":187,"value":"propeller = d3.json(\n \"https://raw.githubusercontent.com/shiplab/vesseljs/dev/examples/others/propeller_specifications/wag_4b_0.55a_1.2p.json\"\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":459,"value":"wave = new Vessel.WaveCreator()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":201,"value":"md`\nIt is writen bellow the power plant data structure used in gunnerus. The power plant follows a similar structure for different engines:\n~~~~json \n{\n\t\"main\": {\n\t\t\"noSys\": 1,\n\t\t\"etas\": 0.99,\n\t\t\"etag\": 0.95,\n\t\t\"engines\": [\n\t\t\t{\n\t\t\t\t\"MCR\": 500,\n\t\t\t\t\"rpmSpeed\": 239,\n\t\t\t\t\"weight\": 1900,\n\t\t\t\t\"a\": 8.64,\n\t\t\t\t\"b\": -23.76,\n\t\t\t\t\"c\": 228.96,\n\t\t\t\t\"polOrder\": 2\n\t\t\t},\n\t\t\t{\n\t\t\t\t\"MCR\": 500,\n\t\t\t\t\"rpmSpeed\": 239,\n\t\t\t\t\"weight\": 1900,\n\t\t\t\t\"a\": 8.64,\n\t\t\t\t\"b\": -23.76,\n\t\t\t\t\"c\": 228.96,\n\t\t\t\t\"polOrder\": 2\n\t\t\t}\n\t\t]\n\t}\n}\n~~~~\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":197,"value":"powerPlant = FileAttachment(\"gunnerus_power_plant.json\").json()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":221,"value":"md`\nThe coefficients ${tex`a`}, ${tex`b`} and ${tex`c`} represents the coefficients of the approximation equation for the motor Specific Fuel Consumption (SFOC) in ${tex`[g/kWh]`} according to the engine load percentage. The following graphic represent the present scenario consumption of the system for a equal percentage load in the engines:\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":246,"value":"chart = {\n const svg = d3.create(\"svg\").attr(\"viewBox\", [0, 0, (3 * width) / 2, height]);\n\n const labelsObj = [\n { title: \"Rt\", color: \"steelblue\" },\n { title: \"Rtadd\", color: \"green\" }\n ];\n\n svg.append(\"g\").call(xAxis);\n\n svg.append(\"g\").call(yAxis);\n\n // Draw the x and y axes labels.\n svg.append('g').call(xAxisLabel('Speed [knots]'));\n svg.append('g').call(yAxisLabel('Resistance [N]'));\n\n svg\n .append(\"path\")\n .datum(data)\n .attr(\"fill\", \"none\")\n .attr(\"stroke\", \"steelblue\")\n .attr(\"stroke-width\", 1.5)\n .attr(\"stroke-linejoin\", \"round\")\n .attr(\"stroke-linecap\", \"round\")\n .attr(\"d\", line1);\n\n return svg.node();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":335,"value":"md`\nFinally we must define the maneuvering json object man, which will represent the coefficients regarding the maneuvering model. The following object is an example for the maneuvering model:\n~~~~json\n{\n \"distHel\": 15,\n \"m\": null,\n \"I\": null,\n \"initial_yaw\": 0,\n \"initial_angle\": 0,\n \"M\": null,\n \"N\": [[ 0, 0, 0 ],[ 0, 5.5e4, 6.4e4 ],[ 0, 6.4e4, 1.2e7 ]],\n \"helRate\": 0.5,\n \"rudderRate\": 5,\n \"maxPropRot\": 210,\n \"maxTorque\": 400\n}\n~~~~\n\nThe values of the mass and inercia, represented by m and I are the constants from the rigid body, in case they are not provided they are approximated by the methods used inside the vessel.js, those two coefficients are used to construct the matrix M in case it is not especified. The N represent the hidrodynamic factor and also can be calculated throught pre made formulas in case it is not provided. The table bellow gives a breaf description for each parameter:\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":365,"value":"html`\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n

Parameter	Description	Unit
`distHel`	Longitudinal distance between the propellers and the center of gravity of the ship	${tex`m`}
`m`	Ship total mass	${tex`kg`}
`I`	Ship total inercia	${tex`kg \\times m^2`}
`M`	Matrix of masses in ${tex`3 \\times 3`}	${tex`-`}
`N`	Matrix of damping in ${tex`3 \\times 3`}	${tex`-`}
`helRate`	Helix rotation rate	${tex`RPM/s`}
`rudderRate`	Rudder angle change rate	${tex`[degree/s]`}

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":508,"value":"man = FileAttachment(\"man_sample.json\").json()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":417,"value":"md`\n## Mounting manoeuvring object\n\nWith the models estabilished before hand above, we can mount the maneuvering displacement model in chain by order stated above with the chain model provided bellow:\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":485,"value":"{\n mutable hullRes = new Vessel.HullResistance(ship, shipState, propeller, wave);\n mutable hullRes.writeOutput();\n\n mutable propellerInteraction = new Vessel.PropellerInteraction(\n ship,\n shipState,\n propeller\n );\n mutable propellerInteraction.writeOutput();\n\n mutable fuelCons = new Vessel.FuelConsumption(ship, shipState, powerPlant);\n mutable fuelCons.writeOutput();\n\n \n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":563,"value":"{\n mutable manoeuvring = new Vessel.Manoeuvring(\n ship,\n shipState,\n hullRes,\n propellerInteraction,\n fuelCons,\n man\n );\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":566,"value":"md`\n## Displacement calculation steps\n\nTo maintain the vessel.js without any external library dependecies the ManouverinModel class was created to handle the derivative calculations. The only argument required for the creation of the ManouverinModel is the manoeuvring object and the numeric:\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":584,"value":"manoeuvringMovement = new ManoeuvringMovement(manoeuvring)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":590,"value":"md`\nThe user can follow the following steps to calculate the displacement:\n\n- Set the advance speed by using the setSpeed method, passing the velocity in knots:\n~~~~js\nmanoeuvring.setSpeed(manoeuvringMovement.states.V.u * 1.96);\n~~~~\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":757,"value":"(manoeuvringMovement.states.V.u = 1)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":601,"value":"manoeuvring.setSpeed(manoeuvringMovement.states.V.u)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":610,"value":"md`\n- Set the cos and sin of the propeller angle in relation to the center line:\n~~~~js\n// This particular example uses a 45 degree propeller angle\nvar propellerAngle = Math.PI / 4;\nvar cos = Math.cos( propellerAngle );\nvar sin = Math.sin( propellerAngle );\n~~~~\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":639,"value":"angle = {\n let propellerAngle = Math.PI / 4;\n let cos = Math.cos(propellerAngle);\n let sin = Math.sin(propellerAngle);\n return { cos, sin };\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":668,"value":"md`\n- Read the propeller rotation state with the .getPropResult method:\n~~~~js\nvar rotationStates = manoeuvring.getPropResult(manoeuvringMovement.states.n);\n~~~~\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":680,"value":"rotationStates = {\n manoeuvringMovement.states.n = 10;\n var rotationStates = manoeuvring.getPropResult(manoeuvringMovement.states.n);\n return rotationStates;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":675,"value":"md`\n- Mont the forceVector by subtracting the longitudinal resistance and applying the angle transformations:\n\n${tex.block`F = [T \\cdot \\cos(\\theta) - Rt,~ T \\cdot \\sin (\\theta),~ T \\cdot \\sin(\\theta) \\cdot distHel]`}\n\n~~~~js\nvar Rt = manoeuvring.getRes(manoeuvringMovement.states.V.u);\n\nconst distHel = manoeuvringMovement.manoeuvring.distHel;\n\nvar forceVector = [\n rotationStates.Fp * cos - Rt,\n rotationStates.Fp * sin,\n rotationStates.Fp * sin * distHel\n ];\n~~~~\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":697,"value":"forceVector = {\n var Rt = manoeuvring.getRes(manoeuvringMovement.states.V.u);\n\n const distHel = manoeuvringMovement.manoeuvring.distHel;\n\n var forceVector = [\n rotationStates.Fp * angle.cos - Rt,\n rotationStates.Fp * angle.sin,\n rotationStates.Fp * angle.sin * distHel\n ];\n\n return forceVector;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":710,"value":"md`\n- Use the method setMatrixes to set internally the ${tex`\\textbf{A}_{6 \\times 6}`} and ${tex`\\textbf{B}_{6 \\times 1}`} matrices that requires the forceVector matrix defined above as first argument and the heeling angle, in this example we are using a heeling angle of ${tex`(3 \\cdot \\pi) / 2`}. Finnally we can get the displacements by using the method getDisplacements, which will pass the variable states. The results will be stored inside the manoeuvringMovement.states.DX:\n\n~~~~js\nmanoeuvringMovement.setMatrixes(forceVector, Math.PI / 4);\nmanoeuvringMovement.getDisplacements(manoeuvringMovement.dt);\n~~~~\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":768,"value":"{\n manoeuvringMovement.setMatrixes(forceVector, Math.PI / 4);\n manoeuvringMovement.getDisplacements(0.1);\n return manoeuvringMovement.states.DX;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":839,"value":"md`\n

[<< Previous](https://observablehq.com/@ferrari212/section-4-2-advance-resistance-analysis-pt-2-resistance-mod?collection=@ferrari212/from-hull-to-simulation) || Top ||\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":454,"value":"MutablesModel = md`### Mutables`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":452,"value":"mutable propellerInteraction = {\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":457,"value":"mutable hullRes = {\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":470,"value":"mutable fuelCons = {\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":494,"value":"mutable manoeuvring = {\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":233,"value":"GraphModels = md`### Graphics`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":237,"value":"data = {\n let result = [];\n\n function SFOC(l) {\n let totPower = 0;\n let engines = powerPlant.main.engines;\n\n engines.forEach(engine => {\n totPower += engine.a * l * l + engine.b * l + engine.c;\n });\n\n return { x: l, y: totPower };\n }\n\n let l = 0;\n\n do {\n result.push(SFOC(l));\n l = l + 0.01;\n } while (l <= 1);\n\n return result;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":249,"value":"line1 = d3\n .line()\n .x(d => x(d.x))\n .y(d => y(d.y))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":254,"value":"x = d3\n .scaleLinear()\n .domain(d3.extent(data, d => d.x))\n .range([margin.left, width - margin.right])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":256,"value":"y = d3\n .scaleLinear()\n .domain([d3.min(data, d => d3.min([d.y])), d3.max(data, d => d3.max([d.y]))])\n .nice()\n .range([height - margin.bottom, margin.top])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":259,"value":"xAxis = g =>\n g.attr(\"transform\", `translate(0,${height - margin.bottom})`).call(\n d3\n .axisBottom(x)\n .ticks(width / 50)\n .tickSizeOuter(0)\n )","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":262,"value":"xAxisLabel = label => g =>\n g\n .attr('transform', `translate(0,${height - margin.bottom})`)\n // Add label\n .append('text')\n .attr('class', 'axis-label')\n .text(label)\n .attr('x', margin.left + (width - margin.left - margin.right) / 2)\n .attr('y', (margin.bottom * 5) / 6) // Relative to the x axis.","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":264,"value":"yAxis = g =>\n g\n .attr(\"transform\", `translate(${margin.left},0)`)\n .call(d3.axisLeft(y))\n .call(g => g.select(\".domain\").remove())\n .call(g =>\n g\n .select(\".tick:last-of-type text\")\n .clone()\n .attr(\"x\", 3)\n .attr(\"text-anchor\", \"start\")\n .attr(\"font-weight\", \"bold\")\n .text(data.z)\n )\n .call(d3.axisLeft(y).tickSizeOuter(0))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":267,"value":"yAxisLabel = label => g =>\n g\n .attr('transform', `translate(${margin.left},0)`)\n // Add label\n .append('text')\n .attr('class', 'axis-label')\n .text(label)\n .attr('transform', 'rotate(-90)')\n .attr('x', -(margin.top + (height - margin.top - margin.bottom) / 2))\n .attr('y', (-margin.left * 3) / 4) // Relative to the y axis.","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":270,"value":"margin = ({ top: 20, right: 30, bottom: 50, left: 75 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":272,"value":"height = width / 2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":153,"value":"md`### Libraries`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":151,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":521,"value":"Vessel = require('@ferrari212/ntnu-vessel-dopri').catch(() => window[\"Vessel\"])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":572,"value":"numeric = require('numeric').catch(() => window[\"numeric\"])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":156,"value":"d3 = require(\"d3@5\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":56,"value":"md`### References`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":84,"value":"one = md`**[1] Holtrop, J. ** - 1984 - A statistical re-analysis of resistance and propulsion data, pg. 272–276. `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":59,"value":"two = md`\n**[2] On Dormand-Prince Method ** - Toshinori Kimura - September 24,2009 http://depa.fquim.unam.mx/amyd/archivero/DormandPrince_19856.pdf.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":62,"value":"three = md`\n**[3] Kt, Kq and Efficiency Curves for the Wageningen B-Series Propellers ** - Sébastien Loisel - Open source repository https://github.com/sloisel/numeric.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":130,"value":"four = md`\n**[4] Handbook Marine Craft hydrodynamics and motion control ** - Thor I. Fossen - John Wiley & Sons - United Kingdon.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":119,"value":"five = md`\n**[5] On an Emprirical Prediction of Hydrodynamic Coefficients for Modern Ship Hulls ** - Tae-II Lee and Kyoung-Soo Ahn and Hyoung-Suk Lee and Deuk-Joon Yum - Marsin 03.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1"}} 31:"$30:body"