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Q1:Lab Sheet 6: Performance Analysis of a Three-Phase Induction Motor Objective: Analyze and calculate various performance parameters of a three-phase, 4-pole, 200V, 400Hz induction motor. Understand the effects of electrical properties on motor performance using MATLAB, Python, or Excel. Equipment and Software: MATLAB, Python, or Excel Calculator for manual verification Motor Specifications: Type: Three-phase, 4-pole induction motor Rated Voltage: 200V Frequency: 400Hz Stator Resistance (𝑅𝑠Rs ): 1.71 ohms Stator Reactance (𝑋𝑠Xs ): 0.12 ohms Rotor Resistance (𝑅𝑟Rr ): 0.5 ohms Rotor Reactance (𝑋𝑖′Xi′ ): 0.3 ohms Magnetizing Reactance (𝑋𝑚Xm ): 32 ohms Slip (𝑠s): 2.5% Friction Losses: 100W Negligible Core Losses Tasks: Calculate the Synchronous Speed: Use the formula: 𝑛𝑠=120×𝑓𝑃ns =P120×f Where 𝑓f is the frequency, and 𝑃P is the number of poles. Calculate the Motor Speed at Full Load: Considering the slip (𝑠s), use the formula: 𝑛=𝑛𝑠×(1−𝑠)n=ns ×(1−s) Determine the Total Impedance: Combine stator impedance, rotor impedance referred to the stator, and magnetizing impedance. Use parallel and series combinations as applicable. Calculate the Stator Current and Power Factor: Use Ohm’s law and impedance to find the current: 𝐼𝑠=𝑉𝑝ℎ𝑍𝑡𝑜𝑡𝑎𝑙Is =Ztotal Vph Determine the power factor from the impedance phase angle. Compute Stator and Rotor Copper Losses: 𝑃𝑠𝑐𝑢=3×𝐼𝑠2×𝑅𝑠Pscu =3×Is2 ×Rs 𝑃𝑟𝑐𝑢=3×𝐼𝑟2×𝑅𝑟Prcu =3×Ir2 ×Rr Calculate the Air Gap Power: 𝑃𝑔𝑎𝑝=𝑃𝑟𝑐𝑢/𝑠 Determine the Developed Power and Torque: Subtract rotor copper losses from the air gap power for developed power. Calculate the torque using the developed power and motor speed. Assess Input and Output Power: Input power calculation incorporating power factor. Output power considering friction losses. Evaluate Motor Efficiency: Ratio of output power to input power, expressed as a percentage. Instructions: Students may choose MATLAB, Python, or Excel to perform the calculations. Document all steps, formulas, and intermediate values. Discuss the impact of slip, stator, and rotor resistances on motor performance. Submission: Submit all calculations, code (if applicable) and a discussion on the implications of your findings on motor design and operation. Lab Sheet for Session 5 Objective: Analyse DC motor operations using MATLAB and perform a comprehensive load analysis for an aircraft's electrical system using Excel. Materials: MATLAB software Microsoft Excel Exercise 1: DC Motor Analysis Using MATLAB A DC shunt motor connected to a 460-V supply has an armature resistance of 0.15 Ω. Calculate: (a) The back e.m.f when the armature current is 120 A. (b) The armature current when the back e.m.f. is 447.4 V. A DC shunt motor connected to a 460-V supply takes an armature current of 120 A on full load. Calculate the back e.m.f. at this load given the armature resistance of 0.25 Ω. A 4-pole DC shunt motor takes an armature current of 150 A at 440 V, with an armature resistance of 0.15 Ω. Calculate the back e.m.f. at this load. A 28 V DC motor with an armature resistance of 0.3 ohm, rated at 800 rpm and 5 A, is used to drive a pump. Calculate the no-load speed. Exercise 2: Load Analysis Using Excel Using the provided aircraft electrical data and load profiles, compute the following using Excel: The electrical data for a military aircraft is as follows: Number of generators : 2 (split bus) Number of inverter (for emergency use only) : 1 Number of Transformer Rectifier Unit : 2 Number of battery : 1 Specs: Generator Voltage 115 Frequency 400 Power Factor 0.8 Configuration WYE Max Continuous Power Rating (kVA) 15 Interval Rating (5 s – KVA 24 Interval Rating (5 min – kVA) 18 Inverter Emergency Rating (kVA) 15 Voltage (AC) 115 Frequency 400 Power Factor 0.8 Configuration WYE TRU Rating (amps) 150 Output Voltage (DC) 28 Max Continuous Power Rating (kVA) 150 Interval Rating (5 s – KVA 600 Interval Rating (5 min – kVA) 225 Battery Rating (AH) 40 Output Voltage (DC) 28 There are seven flight profiles. G1 : Start-Up G2 : Taxi G3 : Take Off and Climb G4: Cruise G5 : Landing G6: Ground Alert G7: Double Gen Failure The various AC Loads are as follows: LRU Consumption (VA) Profiles Used Operating Time Fuel Booster Pump 1800 G1, G6 Continuous Fuel Transfer Pump 1628 G1, G6,G7 Continuous Amplifier 10 G1, G3,G7 Continuous Oxygen 9 G2, G3,G4,G7 Continuous Cabin Control 23 G2, G3,G5,G6 Continuous Cockpit Lighting 167 G2, G4,G5,G6 Continuous Heat 270 G1, G2, G5,G6 Continuous Radar 2500 G1, G4, G6 Continuous Avionics Computer 170 G1,G3, G4,G6,G7 Continuous Avionics Sys 1 2700 G3, G7 Continuous Avionics Sys 2 870 G5,G7 Continuous HUD 235 G3,G4,G5 Continuous Fuel Measurement 28 G4,G5 Continuous Weapon System 400 G2, G6 Continuous EW System 3410 G1,G2, G6,G7 Continuous Nav 1` 132 G1,G2, G4,G6,G7 Continuous Nav 2 85.7 G1,G5, G4,G6,G7 Continuous GPS 98.9 G5,G6,G7 Continuous 26 V transformer 82 G2,G5 Continuous Others 4300 G3 Continuous The various DC Loads are as follows: LRU Consumption (Amps) Profiles Used Operating Time VOR/ILS 2 G1, G7 Continuous EW1 5.5 G5, G7 Continuous EW2 1.52 G2, G5,G7 Continuous Controller 0.1 G2, G5, G6 Continuous J Box 0.5 G2, G4,G5,G6 Continuous Air Con 1.8 G3, G4,G5 Continuous Cockpit Lights 4.7 G2, G3,G5,G6,G7 Continuous Anti-Collision Lights 3.4 G1, G4, G3 Continuous Formation Light 0.2 G7, G4, G3 Continuous Landing Light 16.1 G6 Continuous EFIS 2.33 G6,G7 Continuous ADAHRS 0.6 G2 Continuous Transmitter 14.28 G4,G5,G6,G7 Continuous Receiver 3.57 G2, G3,G5,G6,G7 Continuous Standby 4 G3,G5,G6,G4 Continuous Comms Panel 0.3 G4, G3,G5,G6,G7 Continuous AWS 6.9 G2, G5,G7 Continuous SMD 5 G7 Continuous Master Comp 1 G7 Continuous Others 122 G1 Continuous AC Load Analysis: For each flight profile, compute the total consumption in VA. DC Load Analysis: For each flight profile, compute the total consumption in Amps. Evaluate whether the provided generators, inverter, and TRU can cater to these loads under different flight profiles. Questions: How does the back e.m.f. affect the performance of a DC motor? What is the impact of varying the load conditions on the electrical system of an aircraft? Can the aircraft's electrical system handle all operational scenarios without overloading any component? Lab Sheet for Session 4 Objective: Explore the Fourier series representation of square waves and perform detailed harmonic power analysis for inverters in electrical circuits. Materials: Microsoft Excel software Exercise 1: Fourier Series Analysis of a Square Wave Given a square wave of amplitude 3 V and frequency 50 Hz, use MATLAB to compute the Fourier series coefficients up to the 25th harmonic. Sum up to 12th harmonics and up to 2th 5 harmonics plot the two resulting waveform. Observe which waveform (12 vs 25 harmonics) looks more like the original square wave. Questions: Which waveform resembles the original square wave more closely, and why? How does increasing the number of harmonics affect the approximation of the square wave? Exercise 2: Power Analysis for Inverter Circuits Using Excel Using Microsoft Excel, perform a detailed power analysis for a half-bridge and a full-bridge inverter under the following conditions: Load = 20 ohms, DC voltage = 20V, frequency of switching = 200 Hz. Calculate up to the 100th harmonic. Columns: Harmonic Number, Frequency, Voltage at the Harmonic, Resistance, Inductive Reactance, Impedance, Current, Power Factor, and Power. Compute Total Power, Fundamental Power, Total Harmonic Power, and Total Harmonic Distortion (THD). Repeat for the two inverters for Load = 20 ohms, L = 159 mH, DC voltage = 20V, frequency of switching = 200 Hz. Questions: How does the addition of inductance affect the power distribution across the harmonics? What implications do your findings have for the design and operation of inverter systems?See Answer
Q2:Assignment 1 (10 marks) There are two questions in this assignment: Question 1 is an LP problem, Question 2 is an NLP problem. Hint: First establish the mathematical model based on the lecture slides. Then get start with MATLAB. Get familiar with the MATLAB program codes provided, L1_ex1_1_simple.m and L1_ex1_3_simple.m or L1_ex1_1_standard.m and L1_ex1_3_standard.m. The simple version of these two codes do not include how to plot the feasible areas and are therefore more suitable for students who are not very familiar with MATLAB. Try to understand the functions in the programs by using MATLAB online help and reading the necessary document. Then obtain the graphical solutions to the following two questions by making necessary changes to the codes. Please go to the tutorial classes to know more about MATLAB, the modelling, and the detailed requirements. You must complete the MATLAB quiz in Week 0 before submitting Assignment 1, otherwise your assignment will be marked as a 0. Question 1: The local bookstore must determine how many of each of the four new books on photonics it must order to satisfy the new interest generated in the discipline. Book 1 costs $50, will provide a profit of $10, and requires 3 inches of shelf space. Book 2 costs $70, will provide a profit of $12, and requires 8 inches of shelf space. Book 3 costs $80, will provide a profit of $8, and requires 2 inch of shelf space. Book 4 costs $65, will provide a profit of $15, and requires 2 inches of shelf space. Find the number of each type that must be ordered to maximize profit. Total shelf space is 620 inches. Total amount available for ordering is $9500. It has been decided to order at least 20 Book 3. (1) Establish the model of the optimisation problem. State clearly in words what the decision variables are, what the objective function and the equation/inequality constraints represent. (2) Suppose the number of Book 3 is half the number of Book 1, and the number of Book 4 is fixed as 13, establish the model of the new optimisation problem. (3) For the model established in step (2), obtain the graphical solution using MATLAB. (4) Change one of the parameters to cause the optimal solution (optimal design variables) changes and solve the problem again. 1 Question 2: Determine the objective function for building a minimum-cost cylindrical tank of volume 60m3 and the height should be at least 2m longer than the radius, it should be noted that the cylindrical tank does not have a lid. If the circular ends cost $12 per m2 , the cylindrical wall costs $5 per m2 , the reinforcement cost for the base is $20 per m2, and it costs $30 per m2 for the protecting material over the whole surface (double sides). Additionally, the base requires an extra 10% of its total cost for transportation and installation fees. (1) Establish the mathematical model of the optimisation problem. State clearly in words what the decision variables are, what the objective function and the equation/inequality constraints represent. (2) Draw the graphs and find the optimal solution (graphical solution). (3) Change one of the parameters to cause the optimal solution (optimal design variables) changes and solve the problem again. (4) If this tank needs a lid, rebuild the mathematical model and solve the problem again. 2 Assignment 1 Submission Guide · Attempt two questions: one linear programming problem (questions 1) and one non- linear programming problem (questions 2). · Write a short report in PDF document format. Submit the written code separately (MATLAB scripts) for each question you have attempted along with the report. Make sure to name the files with the question number and name the MATLAB script (.m extension file) with your student ID. Submit the files to Canvas TurnItIn using the multiple file upload option. For example, the three files could be: Assignment_1_DOM_13123456.PDF, Qestion_1_DOM_13123456.m and Qestion_2_DOM_13123456.m . File Upload Text Entry Upload a file, or choose a file you've already uploaded. File: Choose file No file chosen PDF report + Add Another File MATLAB scripts (.m file) Click here to find a file you've already uploaded Comments ... Cancel Submit Assignment . No marks will be given if you do not upload the MATLAB scripts along with the report. · The assignment is worth 10 marks, each question is worth 5 marks. · For question1: o For sub-question (1), write the model of the optimisation problem (0.5 marks), describe the decision variables, objective function, and equality/inequality constraints (1 mark) o Solve sub-question (2) (0.5 marks) o For sub-question (3), find the optimal solution by the code of MATLAB and give an image of the graphical solution (2 marks) 3 o For sub-question (4), find the optimal solution and write the values of the design variables, objective function, and equality/inequality constraints (0.5 marks). Given the corresponding code and the image of the graphical solution (0.5 marks) · For question 2: o For sub-question (1), write the model of the optimisation problem (0.5 marks), describe the decision variables, objective function, and equality/inequality constraints (1 mark) o Solve sub-question (2), find the optimal solution by the code of MATLAB and give an image of the graphical solution (1.5 marks) o For sub-question (3), find the optimal solution and write the values of the design variables, objective function, and equality/inequality constraints (0.5 marks). Given the corresponding code and the image of the graphical solution (0.5 marks) o If this tank needs a lid, rebuild the mathematical model (0.5 marks) and solve the problem again (0.5 marks). · Include the code with proper comments in line to explain each code line that you have written to solve each question and attach it with the report. . For clarity, we might ask you to explain what you have written and how the code works. No mark will be given if you cannot explain the results and the code. . Any suspicious plagiarism will be reported to the university for further investigation. And the mark will not be given until the investigation is finished. . As stated in the subject outline, "Work submitted late without an approved extension is subject to a late penalty of 10 per cent of the total available marks deducted per calendar day that the assessment is overdue (e.g. if an assignment is out of 40 marks, and is submitted (up to) 24 hours after the deadline without an extension, the student will have four marks deducted from their awarded mark). Work submitted after five calendar days is not accepted and a mark of zero is awarded". 4See Answer
Q3:12:11 LTE 29 < Assignment Details CMPSC 200 Programming for Engineers with MATLAB (SU24) Instructions: Complete Problem 1, Problem 2, and Problem 3 below. Remember that the work you submit must be your own. Create a single script (.m file) to complete this assignment. Unless directed otherwise, use meaningful variable names for each variable; do not use the default variable ans to store your results. For this assignment suppress your output with semi-colons ( ; ). Each problem (i.e. Problem 1, Problem 2, and Problem 3) should be in a separate cell, using the cell mode feature of MATLAB. Please remember to follow the Programming Style Convention on Canvas. When complete, please submit your code to the dropbox on Canvas; the grader will run it to view your output. Name your file like this: username_assignment10.m (example: bjs5332_assignment10.m). Your submission must be a single .m file. Problems: Please make sure to print your results for each of these problems using print statements. Submit Assignment « Previous Next 1 47 Dashboard Calendar To Do Notifications Inbox 12:11 LTE 29 < Assignment Details CMPSC 200 Programming for Engineers with MATLAB (SU24) Problems: Please make sure to print your results for each of these problems using print statements. 1. Consider a discrete function given by the values in Table 1. Table 1. A Discrete Function Point i 1 2 3 4 5 x-value 1 2 3 4 5 y-value -1 1 3 5 7 Recalling that for a linear fit the resulting equation should be of the form y = ax + b, find the linear curve fit using: a) The mathematical technique described in Lecture 23 b) The polyfit function 2. Consider the function b(x) = x - x2 + x3. Use the polyval function to evaluate b(x) at the point x = 2, then print the results to the Command Window. 3. Consider the function c(x) = x4 - 5x2 + 4. Find the zeros of the function, then print the results to the Command Window. Submit Assignment « Previous Next 1 47 Dashboard Calendar To Do Notifications Inbox 12:11 LTE 29 Assignment Details < CMPSC 200 Programming for Engineers with MATLAB (SU24) 1. Consider a discrete function given by the values in Table 1. Table 1. A Discrete Function Point i 1 2 3 4 5 x-value 1 2 3 4 5 y-value -1 1 3 5 7 Recalling that for a linear fit the resulting equation should be of the form y = ax + b, find the linear curve fit using: a) The mathematical technique described in Lecture 23 b) The polyfit function 2. Consider the function b(x) = x - x2 + x3 .Use the polyval function to evaluate b(x) at the point x = 2, then print the results to the Command Window. 3. Consider the function c(x) = x4 - 5x2 + 4. Find the zeros of the function, then print the results to the Command Window. Suggestion: You may find it helpful to plot the graph first to get a feel for where the zeros exist; there are exactly 4 zeros for this function. Submit Assignment « Previous Next 1 47 Dashboard Calendar To Do Notifications InboxSee Answer
Q4:Problem 4 - Curl The curl operator is defined as VXF = â [F - 2E) -a ( F - F)+ (x , y Ex) and may be thought of as "rotation" or the "amount a paddle-wheel will spin if put into the vector field". To demonstrate the curl operator, let us consider a situation where wind is blowing with a velocity given by the function v(Robs)={0, R2 (1 - Rz) + 0.5, 0) The curl of this velocity vector field is trivial to derive and is V XỬ = (R2, 0, -2Rx(R2 - 1) > 1. Produce a 3D vector field plot of the "wind" field in the following range: -1≤x≤1 -1 ≤y≤1 0≤z≤1 2. In the same plot, overlay a second vector field with represents the curl of the wind field.See Answer
Q5:/n/nCRN 54392 / 54397 Measurement Analysis and Assessment @ University of Salford Vibroacoustics Lab Core Aims . To measure the passive mobility of an assembly using an impact hammer and accelerometers. • To plot this data including validation checks (reciprocity, coherence) and standard error. . To measure and plot the operational velocities when the source part of the assembly is active. Extended Aims (level 7 students only) All the core aims plus: . To calculate the blocked forces the source assembly exerts at the interface. • To validate these blocked forces using the 'on board validation' approach. Background In this lab we introduce modal testing, also called system definition, structural testing, mobility measurement, and Frequency Response Function (FRF) measurement. Although we are using a mechanical device as our test object, there are strong parallels with the Room Acoustics Lab. The mechanical object responds to mechanical excitation by vibrating and radiating sound. In room acoustics we thought of the room as a filter, and similarly we can consider the mechanical structure as a filter whose properties we can extract by measurement. We are using an impulsive excitation here, but the system will filter any mechanical input in essentially the same way. A complication relative to typical Room Acoustics measurement is that we need to measure multiple forcing and multiple response points, thus the system become Multi-Input Multi-Output (MIMO). A diagram and photographs of the apparatus are shown in figures 1 and 2 respectively. Shaker (internal source excitation) Source Ys, fs Z Interface x Isolators Zı y Interface Target Vb Receiver Y Figure 1: Diagram showing the test apparatus in a disassembled state to distinguish: the source assembly (top), which includes a shaker that will be used in the operational case; the isolators (middle); and the receiver assembly (bottom), which includes the on-board validation point vb. Vibroacoustics Lab 1 CRN 54392 / 54397 Measurement Analysis and Assessment @ University of Salford Figure 2: Photo showing the apparatus. Left: top view, plus impact hammer. Right: side view, focussing on source assembly and isolators. Note that these photos do not show the complete set of accelerometers. Lab Brief The measurements will be performed using Dewesoft X software. The process is detailed in a video that can be found on Blackboard. Here is a non-exhaustive checklist to help you while you are in the lab: . Channel Setup > Analog In screen: ensure all sensors are detected (most will require power via IEPE), are functioning correctly and have the correct sensitivity data set (should be read from TEDS for the accelerometers but needs to be entered manually for the hammer). . Channel Setup > Modal Test screen: this is where the FRF measurement is set up. o Left column: Set test method and number of averages appropriately. Ensure the FFT length is set to be a power of 2 (required to match the FFT analysis used to analyse the operational data) and to give a suitable frequency resolution (df). In the Output channels box ensure 'FRF H1', 'Ordinary Coherence' and 'Intermediate Fourier Transforms' are all set to 'Used' (the latter is required so that FFT data of each hit is also saved separately, as is required for the specified analysis). o Excitation tab: There should be one excitation channel: the hammer. Set the trigger threshold, window type, and pre-trigger appropriately. o Response tab: Add all accelerometers and the hammer here (the latter is required so that Intermediate Fourier Transforms Measure > Modal Test screen: this is where the FRF measurement is performed (see video). • . Channel Setup > FFT Analysis screen: this is where the operational measurement is set up. o Enable this for all accelerometers. Set Output spectra to be complex and specify no averaging. . Measure > FFT Analysis screen: this is where the operational measurement is performed (see video). . Analyse > Data Files > Multifile Export: Use this to export your data to Matlab . mat files. Be sure to tick the Real and Imag complex export boxes so that data is exported with phase. The required analysis is specified below. The majority of this follows the tutorial script issued on Blackboard, which serves as a skeleton code for the lab and provides a summary of relevant theory. You are strongly advised to work through that script and complete the missing sections before the lab week. Core Analysis 1. Plot some FRF data checks - coherence and reciprocity for a few point combinations. 2. For one point combination, compute the FRFs for individual hits (using the data saved by the 'Intermediate Fourier Transforms' option) and plot this overlaid on the averaged FRFs output by Dewesoft. Compute Standard Error based on the coherence and use this to add ±95% confidence limits on the averaged FRF line (plotting as additional dashed lines is recommended). 3. Plot the spectrum of the operational velocity data. Please note that you should use the Vibroacoustics Lab Report Proforma if you are doing only the core analysis and submitting a Summary Lab Report. There are also marks for discussion & evaluation (see assessment brief). Vibroacoustics Lab 2 CRN 54392 / 54397 Measurement Analysis and Assessment @ University of Salford Extended Analysis (selected level 7 students only) All the above core analysis, plus: 4. Plot the magnitude of the blocked forces at each connection versus frequency. 5. Plot the result of the 'on-board validation', being the comparison of the directly measured operational vibration response Vb versus that computed via the blocked forces from 2. 6. Repeat step 5, but this time excluding rotations DoFs. For Full Lab Reports you are not required to use the issued Summary Lab Report Proforma. Instead, the report should be more along the lines of a standard academic report - see assessment brief for full details. You are advised to use the Salford report template (available on Blackboard) for this. Note that there are also marks for summarising the background theory, introducing the experiment, and discussion and evaluation. Vibroacoustics Lab 3See Answer
Q6:Write a live script AVG(X) which calculates the average of a column vector of any size. Create a vector which is the integers from 1 to 100. Also use the Excel “Average” function to get the average . Use your live script function to find the average. See below data for T (temp) vs time. Newton’s law of cooling says for a body cooling in an ambient T of Ta, dT dt = -k(T- T a ) Ta, T are in C, t is in minutes, and k is in min-1. Say T0 = 80 C and Ta = 20 C. The T vs time data is given on the spreadsheet. Using central difference calculate the derivate at 5, 10, 15 and 20 minutes Using forward difference calculate the derivate at 0 Using backward difference calculate the derivate at 25 minutes Evaluate k at the same time steps. (Just use Excel) make a plot of T vs time … and k vs We have a function f x = (x+ e -x - x 2 ) 4+ 3 x Create a live script function for this (MATLAB) Make a Table of f(x) vs x from 1 to 10 using your function Use the trapezoidal rule on Excel to calculate the integral from 1 to each entry Plot the integral vs x Curve fit the f(x) vs x plot with an exponential function and report the constants Integrate the equation in “e” by hand ( analytically) and enter the values from that analytical function in another column Use MATLAB to integrate the data (from part b) using the trapz function built in to MATLAB. (use help feature to investigate how on MATLAB (its very easy .. just create vectors x and y containing the data and say trapz(x,y). Use MATLAB to integrate the function itself . This is a little more difficult as you need to create the function, then integrate it. Again use the help feature to see how by searching “quadrature” in MATLABSee Answer
Q7:CHEE 205 In Class Challenge 1 CHEE 205 – In Class Challenge 8: Solving Systems of Equations in MATLAB Introduction Engineers frequently encounter systems of linear equations in their work. For example, in chemical and environmental engineering, these systems can represent mass balances around process units or interactions between pollutants in an environmental system. The ability to solve these systems is therefore crucial. In this challenge, we're going to solve a system of equations related to a process in a chemical plant. We have a system of reactors where the concentrations and the flow rates of the species involved need to be balanced at a steady state. Objective Our goal for this in-class challenge is to: Solve a system of linear equations that represent mass balances around a set of mixing tanks. The Challenge Consider a system where: Reactor 1 receives an inflow with a concentration of A at 4 M, and it sends out an outflow at a rate of x1 L/min. Reactor 2 receives the x1 L/min flow from Reactor 1 and also has an inflow at a rate of 3 L/min with a concentration of A at 2 M. Reactor 2 sends out an outflow at a rate of x2 L/min. Reactor 3 receives the x2 L/min flow from Reactor 2 and drains all of its inflow. Write and solve the mass balance for chemical A in each reactor. For example, for Reactor 1, the mass balance can be written as: 4 - x1*4 = 0 where x1 is the flow rate from Reactor 1 to Reactor 2. Similar equations can be written for Reactor 2 and 3. Write a script in MATLAB to solve this system of equations. Starter Code % Define the system matrix A and the right-hand side vector B % Solve the system % Display the results CHEE 205 – In Class Challenge 8: Solving Systems of Equations in MATLAB [Name] [Date] Submit Your Solution in the Below Mentioned Format Our goal for this in-class challenge is to: Solve a system of linear equations that represent mass balances around a set of mixing tanks. In Class Challenge Solution Format Code Output [Insert your code output here] MatLab Code [Copy and paste your code here] CHEE 205 In Class Challenge Introduction to MatLab and PythonSee Answer
Q8: Description In this assignment, you will work through several tasks within the study of the Electric Drives module to perform calculations and simulations in MATLAB/Simulink. You are expected to compare your results with the laboratory data provided in the below section and complete the tasks as appropriate. Introduction The torque of an asynchronous motor is a result of the interaction of flux and rotor flux. At a rated frequency, if the voltage is set to a certain value and only reduces the frequency, then therewill be a large magnetic flux and magnetic circuit saturation (severely, it will burn the motor). Therefore, the frequency and voltage must be changed proportionally. When changing the frequency, the output voltage of the AC drive should be controlled, to keep constant flux and avoid weak magnetic and magnetic saturation phenomena. The AC induction motor is somewhat more complex, to overcome its poor transient behaviour. L1 ULIL L2 L3 +U₁ Q1 Q3 Q5 R1 R2 R3 14 R4 K1 R5 14 Up R6 Q2 Q4 Q6 -Up Fig.1. Circuit diagram U V M W 3-Y Pair of poles, P = 2 Referring to the circuit diagram in Fig. 1, you are required to deliver the following tasks: 1. As a motor control engineer, how would you apply computational and analytical techniques to devise a strategy preventing high current in the lower speed range of a motor, specifically in relation to minimum voltage boosting? Discuss the limitations associated with the techniques employed. Support your answer from the technical literature. 2. Perform and describe the simulation model in MATLAB/Simulink software you would employ to obtain the speed/torque characteristics of a motor using the altered V/f characteristics at frequencies of 15, 30, 50, 65, and 80 Hz, based on the experimental setup in Fig. 1. Discuss the results. 3. In the context of the simulation in Fig. 1, how would you select and apply equipment and engineering technologies to obtain the speed/torque characteristics of the pump/ventilator at frequencies of 15, 30, and 50 Hz for both V/f and V/f2 characteristics? Discuss the results and the limitations of the chosen technologies. 4. From the data on the motor nameplate given in Fig. 2 and ignoring core, friction, windage, and stray losses, plot a torque-speed curve, and output power-speed curve for this motor, at rated terminal voltage and frequency. 5. Determine the torque, voltage, and current in the phase winding U for various frequencies in both standard and 87 Hz configurations under torque control. By using the appropriate software packages (MATLAB or Excel) to plot the graphs, analyse and explain, how measured torque and currents respond in operating points for the standard connection and 87 Hz configuration with a maximum corner frequency of 50 Hz. Evaluate the advantages and disadvantages of the 87 Hz method for actual industrial use, considering the complexities and uncertainties involved. VDE 0530 A/Y U: LN Lucas-Nuelle GmbH Germany-50170 Kerpen - Siemensstr. 2 SE2673-1H Mot. DS IP: 20 Is.KI. : F 230/400 VI: 4,15/2,4 A 1 kW cos: 1410 1/min f: 0,74 50 Hz 100 81,4% IE2 175 81,4% Made in Germany so 79,2% Fig. 2. Motor nameplate You are expected to perform independent research and further study as appropriate to aid completion. 3 Laboratory data results using Lucas-Nuelle: a) No-load Frequency (Hz) Max Torque (Nm) Speed (RPM) 15 4.0 440 30 4.2 890 50 4.4 1500 65 2.8 1950 80 1.9 2390 Table a b) For Pump An ActiveServo software was used to calculate the power factor and efficiency for the various frequency as shown in the table below. Frequency (Hz) 15 30 50 Load Torque Speed Power Factor Efficiency (Nm) (RPM) (cos) (%) 0.2 440 0.15 24 0.8 890 0.67 61 2.1 1410 0.87 75 Table b c) Operating Points with 87 Hz Configuration Frequency n(Syn) n(Bem) M Uuv lu (Hz) (per min) (per min) (Nm) (V) (A) 10 300 210 7.3 89 2.8 20 600 510 7.2 167 2.5 30 900 810 7.3 242 2.6 40 1200 1110 7.3 320 2.6 50 1500 1410 7.3 395 2.6 60 1800 1710 5.4 395 2.1 70 2100 2010 4.1 395 1.8 80 2400 2310 3.2 395 1.6 90 2700 2610 2.5 400 1.4 100 3000 2910 1.9 400 1.3 Table c n(Syn) is the synchronous speed of the motor; n(Bem) is the rated speed of the motor for the operating point according to its rated data; M is the motor torque; Uuv is the motor voltage between U and V-phase windings; lu is the motor current in the U-phase winding.See Answer
Q9: Assignment Outline: Active Cell Balancing Circuits in MATLAB Simulink Objective: To design and simulate various models of active cell balancing circuits using MATLAB Simulink, and to analyze their performance and efficiency. Introduction: 1. Overview of Cell Balancing: О Importance of cell balancing in battery packs. Differences between passive and active balancing. Benefits of active balancing: enhanced efficiency, extended battery life, and improved performance. 2. Types of Active Cell Balancing: О О о ○ Switched capacitor balancing. Inductor-based balancing. Transformer-based balancing. Resonant balancing. Assignment: Modeling Active Cell Balancing Circuits Using MATLAB Simulink Objective: To design, simulate, and analyze various models of active cell balancing circuits using MATLAB Simulink. The goal is to understand the impact of different balancing techniques on battery performance and efficiency. Introduction: Provide a brief overview of cell balancing, emphasizing its importance in battery management systems (BMS). Explain the differences between passive and active cell balancing, highlighting the advantages of active balancing methods. Tasks: 1. Literature Review: ○ Research different active cell balancing techniques. О Summarize the working principles, advantages, and disadvantages of each technique. 2. Modeling in MATLAB Simulink: о Task 1: Switched Capacitor Balancing о Design a switched capacitor cell balancing circuit. Simulate the circuit for a battery pack with 4 cells. Analyze the results and observe how the balancing affects the cell voltages. Task 2: Inductor-Based Balancing Design an inductor-based cell balancing circuit. Simulate the circuit for the same 4-cell battery pack. Compare the performance with the switched capacitor balancing. О О Task 3: Transformer-Based Balancing ◉ Design a transformer-based cell balancing circuit. Simulate and analyze its performance. Task 4: Resonant Balancing Design a resonant balancing circuit. Simulate and analyze its performance. 3. Analysis and Comparison: о Compare the efficiency, balancing speed, and complexity of the different circuits. ○ Discuss the trade-offs involved in each balancing method. о Provide graphical results and observations from the simulations. 4. Conclusion: ○ Summarize your findings. ○ Discuss which balancing method you find most effective and why. О Propose potential improvements for each balancing technique. Deliverables: 1. A detailed report including: ○ Introduction to cell balancing. О Literature review on active balancing techniques. О О O Explanation of the modeling process in MATLAB Simulink. Analysis and comparison of the simulation results. Conclusion and discussion. 2. MATLAB Simulink models for each balancing technique. 3. Simulation results (graphs, data tables, etc.). Submission Guidelines: • Submit the report in PDF format. Include the MATLAB Simulink model files in a zipped folder. • Ensure that the report is well-structured, with clear sections and sub-sections. Assessment Criteria: • Understanding of active cell balancing techniques. Accuracy and functionality of MATLAB Simulink models. Depth of analysis and comparison. Clarity and coherence of the report. Quality of graphical results and observations.See Answer
Q10:Y-Bus/Gauss-Seidel/Newton-Raphson Project Consider the following power system: 1 3 to *Line info: all lines X=0.05 p.u *Transformer info: X=0.1 p.u *Generator info: Bus 1: Slack bus, V=1.05 p.u Bus 2: V=1.02, Pg=50 MW *Load info: Bus 4: P=100 MW, Q=80 MVAR *Global Base Power: 100 MVA 1. Calculate the Y-bus Matrix. 2. Write Gauss-Seidel iteration equations to solve the power flow for this specific system. 3. Write a MATLAB script to solve the power flow using Gauss-Seidel for this specific system. 4. Write the Newton-Raphson iteration equation (with all elements such as Jacobian, etc..) for this system. 5. Write a MATLAB script to solve the power flow using Newton-Raphson for this specific system. 6. Solve the power flow using Power World. Compare the software solution with the previous two solutions. Provide your solution in a word document report. Also, provide the m-files for your MATLAB codes and Power World files.See Answer
Q11:Given an initial point (xo,yo), slope function dy dx = f(x, y), and a step size h: Euler's Method (Xn+1¹ Yn+1) = (xn + h, Yn + h * f (xn, Yn)) Improved Euler's Method slope₁ = f(xn, Yn) slope₂ = f(xn+h, yn + h * slope₁) (Xa+sı Ya+z) = (x₂ + h. Yn + (Xn+1, Xn +h (slope₁ + slope₂` 2/nSee Answer
Q12:Given an initial point (x,y), slope function=f(x,y), and a step size h: (Xn+1 Yn+1) Euler's Method (Xn+1 Yn+1) = (xn + h, yn + h* f(xn. Yn)) Improved Euler's Method slope₁ = f(xn. Yn) slope₂ = f(xn+h, yn + h * slope₁) > = (x₂ =xn+h, yn+h* (Xn+1, Yn+1) (slope,+ slopez) 2 Runge-Kutta Method slope₁ = f(xn, Yn) h h slope2 = f (x + 2Y₁+2+ sloper) = (x₂ + 1 = x₂ +h, yn + h * h h slope = f (x + 22. Yn +2 slopez) * slope4 = f(x₂ +h, yn + h * slopez) (slope₁ +2 slope₂ +2 slope3 + slope * 6See Answer
Q13:3. (30 points) The following equations show an example of building Jacobian matrix (Newton Raphson method) for power flow calculation. DAP DAP ANO ANO 20 Four submatrices of the Jacobian 0-APP-V,V,V, 0-0₂-0₂) 0-AQ, -Q.-VĖV,V, sin(8-88-9₂) J- BAP, AP, CAP 20, 20, CAP ZAP 20, 20 BAQ, BAQ, BAQ 20 20, V₁=1 6₁0 Diagonal Off-Diagonal ****** z=j0.25 (y=-j4) PAR Please write a MATLAB code to build Jacobian matrix for the lecture example. The expected results (Jacobian matrix) should match those in the lecture notes. Example) BAR av Z=10.2 (y=-j5) Q - Vi Σ VjYg sân (6 - 8;-)+sinu -V₁VjY₁jin (8-dj-dy) And z= 10.5 (y=-j2) 0.5+10.2 VjFij cos (-8-y)- Dag --Visin(8-0)-dyl --V, Σ VjY₁j (8₂ – 0₂ - ©y) + V²Y₂x -B₂-chul Ma Σ; Τι κάτι (0-1, - dog) + Nile και ένα [-13.2859 9.9010 0.9901 9.9010 -20.000 -1.9604 -0.9901 2.0000 -19.4040 Paz 1.0,V₂1.05 0.4+/0.2 -j9j5 j4 Yous 15-17 12 j2 -16] Boc=ImaglY2=₂3 Figure 2. DC PF model for problems 4-5.See Answer
Q14:2. (30 points) Build Y bus matrix using bus/line data in MATLAB. For example, import line and bus matrix and build Y bus matrix. Do not build y₁j (matrix component) manually (e.g., use "for" or "while" loop). Example) The expected results (Y bus) should match those in the lecture notes. (a) Phasor form 1 13.15052-84.7148° 3.3260293.8141" 9.9504295.7106° 3.3260293.8141° 13.15052-84.7148" 9.9504295.7106 9.9504/95.7106° 9.9504295.7106 19.80122-84.2606" (b) Rectangular form 1.2113-13.09461 -0.2212 +3.31861 -0.9901 +9.9010i -0.2212 +3.3186i 1.2113-13.0946i -0.9901 +9.9010i -0.9901 +9.9010i -0.9901 +9.9010i 1.9802 -19.7020iSee Answer
Q15:1. Recall the convergent series from Part #1: 1 1 1 1 1+5+52 +53 For this problem, write a script that performs the following: 1. Prompt the user for input in the form of the number of terms to compute. Check for valid input, and loop until the user enters a positive number greater than zero. If an invalid number is entered, display a message reminding the user to enter a positive integer. 2. Use fprintf to output to the command window each term. See below for an example of what I would like for the formatted output. Also, output to the command window the total as well as the number of terms, as shown below. + = 5|4 Example of the user entering 5 (five terms): >> Enter number of terms to compute: 5 1.000+ 0.200+ 0.040 +0.008 +0.002 = 1.250, number of terms = 5 Specify three decimal places for each term and the total, and no decimal places for the number of terms.See Answer
Q16:4. Let y magic(3). Use the function repmat() and the row/column augmentation technique to generate the matrix below. To get full points, you must use repmat in your solution. y y'See Answer
Q17: Mechatronics 5CCE2MCT Individual Coursework Project You are a mechatronics design engineer working for a firm that specialises in developing custom components for cinematic production. Your manager has sent you a parametric model of a two- axis gantry mechanism that was developed on a previous project. The gantry can move a camera both horizontally and vertically by rotating lead screw mechanisms attached to runners. Mechatronics 5CCE2MCT Individual Coursework Project Design & Analysis of a Two-axis Camera Gantry System Brief Your manager has asked you to design a new electromechanical actuation system composed of two geared DC motors that drive the leadscrew runners of the two-axis gantry system shown in Figure 1. She provided CAD components and Simulink starter models for the assembly, n.b. these files can be downloaded from the KEATS module page in the MATLAB project archive, camera_gantry.mlproj. She has also sent you a list of requirements that she has discussed with the customer appended to the end of this document. She encourages you to use the model as a starting point and welcomes further input on how to improve the mechanical design. You are responsible for: specifying DC motor, power supply, and gearbox characteristics ● design a digital motor controller demonstrate the effectiveness of your design. KING'S LONDON You are highly encouraged to brainstorm additional information about the context in which this mechanical system is to be used. ● Deliverables a 3-minute video recording in which you present the motor and mechanism design to an engineering design team. The video should contain an animation of the mechanism and an overview of the Simulink model and results. ● a 1-page written report presenting the results of your design analysis with a maximum of 2 page of supporting figures in appendix zip and upload Simulink model to KEATS Ⓒ2023 Department of Engineering, King's College London 1 Mechatronics 5CCE2MCT Individual Coursework Project ● camera ● y runners origin lead screw Z end support rails runner Figure 1: Camera Gantry System twoaxis_camera_gantry_start.slx Learning objectives Model the electromechanical system that actuates the gantry using a combination of mathematical, physical and data-driven methods and critique the choice of your Ⓒ2023 Department of Engineering, King's College London modelling approach Specify component parameters based on a design analysis of system requirements Implement and tune a feedback controller to control position and speed of the mechanism Test the controller design against multiple loading scenarios Conduct a design space study to optimise system-level performance Report and justify recommended design implementation Additional resources You are encouraged to complete laboratory exercises of weeks 22-26 to develop skills in modelling and control of mechatronics systems. KING'S LONDON 2 Mechatronics 5CCE2MCT Individual Coursework Project Marking Criteria Individual coursework submission accounts for 30% of module grade. Your submission will be scored with regards to its merits in six core areas: Area Science & Mathematics (20 marks) Engineering Analysis (20 marks) Engineering Design (20 marks) Engineering Context (10 marks) Engineering Practice (20 marks) Additional General skills (10 marks) Actions Justify modelling and control design approach Interpret design performance using mathematical and statistical techniques Critique choice for actuator and sensor characteristics and technology Apply engineering tools to solve the design task Conduct critical analysis to identify, classify and describe system performance compared to benchmark Adopt systems approach to improve on design Extract and evaluate pertinent data to solve unfamiliar problems Evaluate user needs and requirements Identify and work with design constraints and unknowns Communicate to a technical audience Deliver efficient, effective and robust design Identify and mitigate areas of risk Demonstrate design effectiveness in the context in which the system is applied Demonstrate effectiveness, clarity and originality of communication Ⓒ2023 Department of Engineering, King's College London KING'S LONDON 3 Mechatronics 5CCE2MCT Individual Coursework Project Requirements Your manager discussed these requirements with the customer. You are welcome to add you own requirements to this initial draft. Battery Power Supply Requirements Battery Voltage: 5-12V Battery Capacity: 2-5 Ah Battery Resistance: 0.1-10 Battery Lifespan: 15-20 minutes You can use 18650 Li-Ion batteries (example) Battery Voltage: 3.7V per battery* Battery Capacity: 2.6 Ah Battery Resistance: 250 m2 per battery *n.b. 1 battery is 3.7V (1S), 2 batteries is 7.4V (2S), 3 batteries is 11.1V (3S) and so on. leadScrewz.length 90 y positionZ. bias leadScrew.length leadScrew.lead position.bias Figure 2: Key geometric dimensions for the gantry gantryParams.m Ⓒ2023 Department of Engineering, King's College London N leadScrew. lead KING'S LONDON origin Mechatronics 5CCE2MCT Individual Coursework Project Lead Screw Linear travel per revolution of the screw 5-20 mm** **n.b. parameter saved as leadScrew.lead = 10; in the gantryParams.m script DC Motor Requirements Stall current: 0.2A-1A Stall torque: 500 g.cm No-load speed: 7000-21000 rpm Motor Sensor Requirements Hall Sensor Encoder with 24 counts per revolution Gantry Lift Requirements Lift a camera payload weight of 10-20 kg Camera Safety Requirements No collisions with frame Fail safe mechanisms for operation and calibration Maximum acceleration of 30 m/s² Camera Manipulation Requirements Cover an area of 0.75 by 0.4 m² Maximum jerk of 10 m/s³*** Suggestion: As the jerk is the derivative of the acceleration, this result is sensitive to the filtering of the acceleration signal. Use a first-order filter with time delay constant of 0.01 s when you take the derivative. System Response Requirements Rise time < 3-5 s per 100 mm travel Settling time < 5-7 s per 100 mm travel Overshoot < 2% Undershoot < 2% Steady-state error < 2% Suggestion: Perform a step-response between the two ends of the frame. Tracking Response Requirements Relative error < 3% Absolute error < 3% Suggestion: Perform a tracking-response using a sine wave or polynomial trajectory. College LONDON Ⓒ2023 Department of Engineering, King's College London 5See Answer
Q18: VIRTUAL KEYBOARD, PHONE, & FLYER CIS 4930/ CIS 6930 - HW #3 PARTI OVERVIEW & GOALS The goal of this assignment is to practice digitally creating and manipulating sounds, you will create a keyboard in MATLAB, using your knowledge from lecture. Each key will play a different note between C3 and C6. You will implement the keyboard to play a particular note when pressed, according to the frequencies specified on the chart below. You will include additional functionality that creates, plays, and saves a song (of your choice) as a .wav file. Lastly, you will practice using the MATLAB Spectrogram feature to analyze the spectral components of your song. MIDI number ನಟಿಸ935888899999988885888958 8 8 8೧೧&a8599…? 23 24 26 28 29 31 33 36 40 41 37 38 39 43 47 42 44 45 46 50 65 48 49 67 71 72 76 77 79 81 84 66 68 69 70 86 22 88 25 27 96 30 32 34 73 74 75 100 101 103 105 107 108 51 54 56 58 61 63 78 80 91 92 93 94 82 85 87 97 98 99 90 102 104 106 Note name AO ВО C1 D1 E1 F1 G1 Al B1 C2 D2 E2 F2 G2 A2 B2 F3 G3 A3 B3 C4 D4 E4 F4 G4 B4 C5 D5 E5 F5 G5 A5 B5 C6 D6 E6 F6 G6 A6 B6 C7 D7 E7 F7 G7 A7 B7 C8 Keyboard J. Wolfe, UNSW ASSIGNMENT Frequency 27.500 30.868 29.135 32.703 36.708 34.648 41.203 38.891 43.654 48.999 55.000 51.913 46.249 58.270 61.735 65.406 73.416 69.296 82.407 77.782 87.307 97.999 92.499 110.00 103.83 123.47 116.54 554.37 622.25 Period ms 698.46 783.99 739.99 830.61 880.00 987.77 932.33 1046.5 1174.7 1108.7 13 18.5 1244.5 1396.9 1568.0 1480.0 1760.0 1661.2 1975.5 1864.7 2093.0 2349.3 22 17.5 2637.0 2489.0 2793.0 3136.0 2960.0 3520.0 3322.4 3951.1 3729.3 4186.0 36.36 32.40 34.32 30.58 27.24 28.86 24.27 25.71 22.91 20.41 21.62 18.18 19.26 16.20 17.16 15.29 9.09 1 8.099 130.81 7.645 146.83 138.59 6.811 7.216 6.068 6.428 5.727 164.81 155.56 174.61 196.00 185.00 220.00 207.65 246.94 233.08 261.63 293.67 277.18 329.63 311.13 349.23 392.00 369.99 5.102 5.405 4.545 4.816 4.050 4.290 3.822 3.405 3.608 3.034 3.214 2.863 2.551 2.703 440.00 415.30 2.273 2.408 493.88 523.25 466.16 2.025 2.145 587.33 659.26 13.62 14.29 12.13 12.86 11.45 10.20 10.81 9.631 8.581 1.910 1.703 1.804 1.517 1.607 1.432 1.276 1.351 1.136 1.204 1.012 1.073 0.9556 0.85 13 0.9020 0.7584 0.8034 0.7159 0.6378 0.6757 0.5682 0.6020 0.5062 0.5363 0.4778 0.4257 0.45 10 0.3792 0.40 18 0.3580 0.3189 0.3378 0.2841 0.30 10 0.2531 0.2681 0.2389 :6 PART II Using MATLAB, implement a virtual keyboard 1. Please don't call it keyboard.m 2. Use notes from C3 to C6 3. Play as a sine wave 4. Each note's duration is your choice Pick a Song (30-45 seconds) 1. Some ideas: ● https://latouchemusicale.com/en/easiest-piano-songs-with- letters Make one button to play the song you chose and output the song to a wav file. 1. This will involve appending many sine waves together for different durations and saving the results to a .wav file Make a second button that graphs the spectrogram for your entire song Include the picture of your spectrogram (as an image file) OVERVIEW & GOALS You will use MATLAB to create an "old school" traditional analog telephone keypad. When the buttons are pressed, the correct tone is played for .5 seconds. Next, you will read five different mystery phone numbers (hint: “digit-by-digit”) and determine what phone number has been dialed. ASSIGNMENT Using MATLAB, implement a virtual telephone keypad 1. Pressing a single key of a traditional analog telephone keypad produces a telephony signaling event to the remote switching system. For touchtone service, the signal is a dual-tone multi-frequency signaling tone consisting of two simultaneous pure tone sinusoidal frequencies. The row in which the key appears determines the low-frequency component, and the column determines the high-frequency component. For example, pressing key 1 results in a signal composed of tones with frequencies 697 hertz (Hz) and 1209 Hz. 697 Hz. 770 Hz. 852 Hz. 941 Hz. guess-phone1.wav guess-phone2.wav 1209 Hz. guess-phone3.wav 1 guess-phone4.wav 4 7 1336 Hz. 2 (1 Decoding the phone numbers 1. Download the 5 phone numbers below. 5 8 1477 Hz. 2. As each button is pressed, it should play the corresponding tone 3. Each button's duration is .5 seconds 3 6 9 # guess-phone5.wav 2. Each digit is played for .5 seconds, sample rate is 44,100 3. For each mystery phone number, ● Read in each digit, determine its spectral content (hint: use a spectrogram) and note the dialed phone number Write down the full phone number (10 digits) SUBMISSION Upload to Canvas one (1) zip file that contains Part 1 (in its own separate folder, containing the relevant.m, .fig, .wav, and image files) Part 2 (in its own separate folder, containing the relevant .m, .fig, and text file containing the phone number guesses) Include a separate text file entitled “contributions.txt” describing who was responsible for each portion of the homework.See Answer
Q19:22. A truss is a structure made of mem- bers joined at their ends. For the truss shown in the figure, the forces in the 11 members are determined by solv- ing the following system of 11 equa- tions: 36 ft 1200 lb 800 lb 30 ft Ⓒ Ficos 50° +F₂=0, Fisin 50° - 400=0, -F₂ +F6=0 -Ficos 50° +F4 + F5cos 41° =0, F₁sin 50° +F3 + F5sin 41° - 800 = 0, -F5cos 41°-F6+F10 =0, Fssin 41° + F7-1200 = 0, 42 ft 400 lb 48 ft -F4+F8 + Focos 37° =0,-F₁-Fosin 37° = 0, -Focos 37°-F10-4933=0, Fosin 37° + F11 = 0 Write the equations in matrix form and use MATLAB to determine the forces in the members. A positive force means tensile force and a negative force means compressive force. Display the results in a table where the first column displays the member number and the second column displays the corresponding force.See Answer
Q20: Solve tasks 1.1.2, 1.1.3, 1.2.2, 1.2.3 and 2.2.2. Do not use any built-in functions in Matlab. Use only addition, subtraction, multiplication, division, square root, power, matrix inverse, matrix transpose, summation, mean, min, max, integral2, eig, load(), size(). For simplicity, we start with simple quadratical surfaces of the general formulation: z= f(x,y) = Ax2 + By2 + Cxy + Dx + Ey+ F. (1) The objective is to estimate the unknown parameters A, B, C, D, E and F using the observed points in the provided file 'quadratic surface.mat'. The fitting error is (2) which is minimized when a = 0, δε = 0, ΔΕ = 0, 0 = 0, δ£ = 0 and of = 0. Refer to the lecture notes', use linear regression to estimate the unknown parameters A, B, C, D, E and F. N Σ(Ax + By? + Cxy + Dx + Ey + F –z)2 t= Task 1.1.1 Derive six linear quati involving the unknown parameters A, B, C, D, E and F based on the error minimization condition. Then, reformulate these linear equations into the form of Eq.(3). Namely, derive the matrix XEROX6 and vector Y € Ro in Eq.(3). Completed task 1.1.1 Σε Σαν xy Σαν Σε Σχένι Σε xí ΓΧου Χοι ΧΟΖ Χος Χος Χος Χιο Χης Χης Χ13 Χ14 Χις 1X20 Χ21 Χ22 Χ23 Χ24 Χης 1X30 Χ31Χ32 X33 X34 Xas D ΙΧΑΟ ΧΑΙ ΧΑΖ Χ43 ΧΑΑ Χός E X50 X51 X52 X53 X54 X55] F Task 1.1.2 Estimate the unknown parameters A, B, C, D, E and F by solving Eq.(3). Task 1.1.3 Report the volume bounded by the estimated surface and the ground plane (z = 0) by evaluating the integral Josxeseisygy f(x,y)dxdy. Σχέν Σαν Exy yi C Σε λα λα λα λα λα Σαν Σαν xy x² Σχινι Σ Σ Σανι yi xiyi Yi y χένι Σκιν xi yi xi yi x²yi x? Xiyi Συ Σαν Στ y Σ» Σαν Σε Xiyi Xi XiVi y τω Yi y Xiyi Xi Yi N tooomi (3) II x? Zi y? Zi XiYiZi XiZi YiZi MI Zi The quadratic surface model assumes smooth and simple shapes. A bivariate cubic function can model more complex shapes of a bulk. In this context, the sensed surface is approximated by the following cubic function, z = f(x,y) = Ax3 + By + Cxy + Dxy2 + Ex? + Fy? + Gxy +Hx+ Iy + 1 The objective is to estimate the unknown parameters A, B, C, D, E,F,G,H, and J with the observed points in the provided file 'cubic surface.mat". The fitting error is (4) (5) which is minimized when of = 0, 0 = 0, 8€ = 0, 0 = 0,0€ = 0,8F = 0, 86 = 0, 95 = 0, f = 0 and af = 0. Task 1.2.1 Derive ten linear equations involving the unknown parameters A, B,C,D,E,F,G,H, I and J based on the error minimization condition. Then, reformulate these linear equations into the form of Eq.(6). Namely, derive the matrix XER 10x10 and vector Y€ Rio in Eq.(6). N (Ax3 + By + Cxy; + Dxy? + Ex? + Fy? + Gxy; + Hx; + lyi+1-z)2 ΓΧου Χοι Χοζ Χος Χου Χος Χος Χοτ Χοβ Χορζ ΓΑ Χ1 Χ12 Χ14 Χης Χίο Χιτ Χ18 Χιο B Χιο Χ21 Χ22 Χ24 Χης Χζο Χατ Χ28 Χ 20 Χ13 Χ23 X33 ΧΑ3 ΧΑΑ Χ31Χ32 X34 Χος ΧΑΙ ΧΑΖ ΧΑς Χ30 Χ3τ Χ38 Χ 30 ΧΑΟ Χιτ Χ18 Χιο Xss Xso Χιτ Χ18 Χιο Χος Χου Χοτ Χος Χο Xso Xs1 Xs2 X53 Xst | Χου Χοι Χοζ Χος Χολ ΙΧΤο Χτζ Χτζ ΧΤΑ Χτς Χτο Χτ ΧτΒ Χτο Χτι X80 X1 X2 X3 X4 X5 X86 X87 XB8 X89 LXgo Χοι Χοζ Χοζ Χολ Χος Χος Χοτ Χοβ Χορζ xy; 1X20 | X30 ΧΑΟ Σ Xiy Xiyi αν xfyi xy Σαν Σε Σαν Σαν xy _x?y? κέν Σαν Σαν Σ Σαν xy y xy x{y} xy xyi Xiyi x¡y? y² x Σχν xyi xy Σε κένι Σαν Σαν Σαν x Σκv? Σαν Σε Σχνι xy xy xiyi και Συ xy Σαν Σαν Σ Σαν Ση Xiyi y Σαν Σαν xy; x{y} x{y} yi Σαν Σ xy xy Σαν Σως Σαν Σαν Σαν Σα xty? x¡y{ x?y? xy x{y} Xiy Σε x xy xyi xy xfyi x?y? yi xy MI y D ULOI- E x} yi H | 11 Yg xyi xy xyi x Σ Σαν Σ [x²y? [x²y. Σxy? x} yi Σ xy Σ Xi Yi xyi xy Σε Συ Σαν Στέν Σαν Σχνι Σαν Σε x² yi Task 1.2.2 Estimate the unknown parameters A, B, C, D, E,F,G,H,I and J by solving Eq.(6). Task 1.2.3 Report the volume bounded by the estimated surface and the ground plane (z = 0) by evaluating the integral Josxetine fraisysyf(x,y)dxdy. XiVi Xi Xiyi y τω XiYi Yi xzi yi zi x²yizi x₁y} Zi xzi y} Zi N XiYiZi Xi Zi Σχετι τω Ζή The explicit representation z=f(x,y) introduced in Topic 1 has a significant limitation in efficiently representing arbitrary surfaces, particularly when a location (x,y) corresponds to multiple height z. In contrast, implicit represen- tation can easily represent arbitrary topological structures. In this method, the surface S is implicitly defined by the zero-level set of a function f(x.y.z), denoted as $ = {(x,y,z)|f(x,y,z) = 0}. For simplicity, here we constrain the form of f(x, y, z) to a quadratic function: f(x, y, z)= Ax² +By²+Cz²+Dxy+Eyz+Fxz+Gx+Hy+lz+J. The error on fitting an implicit representation is N c = = E(Ax} + By? + Cz? +Dxy + Ey,z, + Fxz, + Gx + Hy, + Iz + 1 – 02 i-1 which is minimized when f = 0, 0 = 0, 0 = 0,86 = 0, 8£ = 0,8£ = 0, 8c = 0, @f = 0, f = 0 and of = 0. Task 2.1 Derive ten linear equations involving the unknown parameters A,B,C,D,E,F,G,H, I and J based on the error minimization condition. Then, reformulate these linear equations into the form of Eq.(8). Namely, derive the matrix X € 110x10 in Eq.(8). Σχετι xzi Σχ xt x?y? xz x²³ yi ΓΧου Χοι Χοζ Χοζ Χος Χος Χος Χοτ Χος Χορτ [Α] 1Χ10 Χ1 Χ12 Χ13 Χ14 Χης Χιό Χιτ Χ18 Χιο B Χης Χης Χατ Χ28 Χο 1Χ20 Χ21 Χ22 Χ 23 C Χ24 X31 X32 X33 X34 X30 Χ35 Χ30 Χ3 X38 Χ30 D Χ40 ΧΑΙ ΧΑΖ ΧΑ3 ΧΑΑ ΧΑς ΧΑς Χιτ Χ18 Χιο | Xso Xει Χεζ Χεζ Xs4 Xss Xso Χιτ Χ18 Χιο ΙΧσο Χοι Χοζ Χοζ Χός Χος Χος Χοτ Χός Χερ ΙΧτο Χτι Χτζ Χτζ ΧΤΑ Χτς Χτο Χτ Χτ8 Χτο Xgo Χοι Χgz XB3 X84 Χος Χου Χοτ Xes Xeo LXgo Χοι Χοζ Χοζ Χολ Χος Χος Χοτ Χος Χαολ yz x Συνz? x? Yizi yizi αν Viz Σχετι Σχετινες Σχ x;Z Σχ Σαν Σε xiz x² yi Task 2.2 For the observed points in 'implicit_l.mat', estimate their corresponding parameters A, B, C, D, E, F,G,H,1 and J. Then, based on the value and signs of these parameters, classify the type of underlying surface. • Hint: directly solving Eq. (8) will result in a solution whose elements are all zero. To address this prob- lem, we can use the eigenvector of X corresponding to the smallest eigenvalue of X as an estimate of [A, B, C, D,E,F,G,H,1,JJT. The relationship between this eigenvector v and the corresponding ground truth parameters θ = [A,B,C,D,E,F,G,H,1,JJT is v = αθ + n where a is a scale factor and n denotes the model noise. Σ» Σαν Σνική xy y? Zi • For the observed points in file 'implicit_2.mati, estimate their corresponding parameters A, B, C, D, E, F,G,H,I and J. Then, based on the value and signs of these parameters, classify the type of underlying surface. • Note: Refer to Fig.2, recognize the surface type by analyzing the values and signs of surface parameters. You must not draw a conclusion via eyeballing. Στο x Σxy Ex?z Σχένι Σχέντι Σχει Στ yt yz x¡y? Σνακ Σαν Σ Στο Xiyiz? Viz Σ xiz xi yi Σχν? x?y? Σκινέζι Zi Zi Σνικ κινέζι χένιτι Σκινιζέ Σχινιτι x?yiZi yizi Σχετιν Σχένι Σχνι x² yi ουυοΙ· H yz || XiYiZi Στο Σχινι Συ XiVi - xri νέζι Σχινιτι yzi XiYiZi yız? xiz ....... xiz Σχένιαι Σκιντέ Σ Σχέτ xz XiZi Xiyiz? XiYiZi Σχέτι Σ. Σχινι Xi Yi Σ XiZi Xi WW x? Yi xiy y? Zi Σχινιτι XiYiZi Viz Σ z² xv? Σχινίζε Σχινι Xi Yi y? YiZi xzi Yi y? Zi Viz Σχει Στ XiZi Σ YiZi Στ Xiyi YiZi Zi XiZi 0 Xi Yi Zi Ellipsoid Eliptic parabolaid Non-degenerate real quadric surfaces Hyperbolic paraboloid Hyperboloid of one sheet or Hyperbolic hyperboloid Hyperboloid of two sheets or Eliptic hyperboloid 2² + + 12 # TSee Answer

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