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Q1:Problem 3: Newton-Raphson method (MATLAB) Use the Newton-Raphson method to find the root of f(x) = c 0.5x (4-x) - 2. Employ initial guesses of ₁,1 = 2, ₁,2 = 6, and ₁,3 = 8. Explain your results. (3)See Answer
Q2:Problem 5: System of Nonlinear Equations (MATLAB) Given a system of nonlinear equations: u(x, y, z) = x(x + 2y + 5z - 4) = 0, v(x, y, z) = y(5x + y + 2z - 4) = 0, w(x, y, z) = 2(2x + 5y +z - 4) = 0, (5) (6) (7) 1. Derive all real solutions to this system of equations by hand. Clearly show your work. 2. Write a MATLAB script to obtain the root of this system of non-linear equations using the Newton-Raphson method. Use an initial guess of (x, y, z) = (4.4,-0.8,0.4). How many iterations are needed for the approximate error ca to go below € = 0.1%? 3. Use your solution to Problem 5.2 to generate all the analytical solutions from Problem 5.1 by modifying your initial guesses for (x, y, z).See Answer
Q3:2. Develop a function that implements the Secant method, called Secant_[your last name].m. The initial guesses should be input as a vector such that if the user inputs [xl, xu] it uses the regular secant method, if the user inputs a single value, [x0], the program uses the modified secant method with. Upload the .m file to canvas.See Answer
Q4:3. Develop a function that implements the Newton-Raphsom method, called NRM_[your last name].m, and upload it to canvas.See Answer
Q5:4. Develop a function that implements the Inverse Quadratic Interpolation method, called IQI_[your last name].m, and upload it to canvas.See Answer
Q6: Given the following block diagram, with G_{f}=10, G_{c}=20, G_{p 1}=\frac{2.0}{1.0 s^{2}+10.0 s+20.0}, G_{p 2}=\frac{5.0}{1.0 s^{2}+5.0 s+25.0}, H=1.0 s+6.0 а.(50%) Convert this system into state-space form, accounting for the multiple inputs and out-puts. There are several ways to transfer the block diagram into state-space, but let's try to remodelthe state-space from the beginning. \text { - The plant transfer functions give us the plant dynamics } \dot{x}=A x+B u \text { and } u=[u d]^{\wedge} T^{\prime \prime} The sensor H(s) relates the state to the output: y = Cx- - The feedback defines the control law: U(s) = G.(Gf * R(s) - Y(s)), which can be substituted intothe plant dynamics to find the closed-loop form. С.(50%) Simulate the response of the system using basic numerical integration. To a- Reference: r(t) = 5 - Random Noise, normally distributed, zero mean with o = 2: d(t) = 2 * rand() Note you already have the model of the system from the state-space, and this is a linear system.You can calculate the derivative and propagate the system directly. See Answer
Q7: On February 18th, 2021, NASA's Mars 2020 Preserverance Rover is planned for landing on Mars.A landing animation video can be seen here http://bit.ly/PreserveranceThe Sky Crane, which is responsible for gracefuly landing the rover on the designated site, can be modeled as a rigid body with thruster forces being controlled by a gimbal to produce thrusta t angles 0 as shown. a. (20%) Derive the equations of motion for the system (3 directions)The thruster angles and thrust forces are all independent input variables now.We wish to design a state-space controller to help the SkyCrane navigate to a desired position in space. b. (30%) Linearize the system and put it in state-space form, then design a controller via pole placement to achieve the following transient response characteristics - Ts = 2 s -\omega_{d}=20 \mathrm{rad} / \mathrm{s} c. (25%) Apply your controller on the nonlinear system (numerical integration) d. (25%) In reality, there are limits to the inputs. The thrust can not be negative, and there is a minimum thrust once the engine is ignited. The thruster gimbal can only operate within a specific angular range. Repeat part c with the following saturation limits: - Thrusts: 100N < F < 2000N Gimbal Angle Range: -45° < 0 < 45°-See Answer
Q8: Given a 2-DOF system with the following nonlinear equations of motion 3 x_{1}-2 x_{2}+12 \dot{x}_{1} \dot{x}_{2}-11 \dot{x}_{1}+20 \ddot{x}_{1}=5 f_{1}+3 f_{2} 2 x_{1}+0.9 x_{2}^{2}+12 \dot{x}_{2}+11 \ddot{x}_{2}=f_{1}+5 f_{2} a. (25%) Put the nonlinear equations in a vector form i = f(x, t, u). b. (25%) Simulate the natural response of the nonlinear system with the following initial condi- \text { tions } x_{0}=\left[\begin{array}{llll} 0.0 & 0.05 & 0.05 & 0.0 \end{array}\right]^{T} \text { for } t=0: 15 s c. (25%) Linearize the system (ignore the square and coupling terms) and put the system into the state space form x = Ax + Bu d. (25%) Design a full-state feedback controller using pole placement method, to stabilize the system. Test your controller on the linearized system first using Isim, then simulate the closed-loop system response using the linear and nonlinear model in a numerical integration setup.Assume you can control both inputs (What is the dimension of the gain matrix?) Your controller is a regulator, so observe the closed-loop response with r = 0 and initial conditions xo = | 15.0 0.7 0.510.0 Plot the response of all the 4 states, for both the linear and non-linear system in one subplot.Can you control the non-linear (actual) system with the same controller? Explain why or why not.See Answer
Q9: Rise:0~25 mm in 90° Dwell:at 25 mm in 45° Fall:25-0 mm in 90° Dwell:for the remainder Set up appropriate boundary conditions and determine the coefficients C, for the 7th-degree polynomial s=\sum_{i=0}^{7} C_{i}\left(\frac{\theta}{b}\right)^{n} \text { where } \theta \text { is the cam angle, } b \text { is the total angle of any segment, rise, fall, or dwell } (b) Develop a MATLAB program to compute and plot svaj diagrams, assume the cycle takes 2 sec.Parameters: The mass of the follower is 1.0 kg. The spring has a rate of 10 N/m, a damping ratio of 0.10,and a preload of 1.0 N. Find the follower force over one revolution. If there is a follower jump, re-specify the spring rate and preload to eliminate it. 1: Design a double-dwell cam to move a follower:See Answer
Q10: Problem 2: The figure shows a 4-bar linkage. Write a Matlab calculate and plot the angular displacements, angular velocities and angular accelerations of links 3 and 4, and x and y acceleration components of the coupler point P over the maximum range of motion possible. Given: w2 = 20rpm and az =0.program to See Answer
Q11: Problem 5:Write an M-file to evaluate the definite integral below using the composite trapezoidal rule with 1, 2 5,and 10 intervals. Compute the true relative percent error (s.) for each result. Provide a printout of your m-file and the command window showing the results. I=\int_{a}^{n}(8+4 \cos x) d xSee Answer
Q12: Problem 3: The offset crank-slider linkage has the dimensions given in the figure. Develop a MATLAB program to calculate and plot the accelerations AAand Ag in the global coordinate system for w 25 rad/sec CW and a2= 0rad/sec?. See Answer
Q13:Project Part-1: 1. Implement and test (show execution of) the continuous-time component representing the dynamic model of a car given in the Textbook. Use the following values in the model: m= 1450 kg, -63. Simulate the response for the case F-0, with initial conditions (0)-0, v(0)-15 m/sec; and the case F-550 N with initial conditions x(0)-0), and v(0)=0. Use Trapezoidal discrete approximation of derivative with simulation step At 0.10 sec. Plot the component responses generated from your simulation. 2. Now add the effect of graded road to the above car model and regenerate the car responses to road grade of 0-5deg, and 0-10deg and the case F 5SON with initial conditions x(0)-0, and v(0)-0 only. Plot the component responses generated by your simulation.See Answer
Q14:Q1 a) Use the colon operator to create the following row vector [ 1 2 3 4 5]. Save this as variable a but suppress the output in the command window. b) Use the transpose operator on a to create a column vector b, suppress the output in the command window. c) Calculate a x b using matrix multiplication and display the answer clearly in the command window. See Answer
Q15:Q2 a) Create a 3 x 3 matrix of random integers between 5 and 20. Save this as variable A and display it clearly in the command window. b) Create a copy of A called B (do not display it in the command window yet). Modify B so that the middle element is equal to 50, and display B clearly in the command window. c) Use elementwise operations to calculate 4²/B and display the answer clearly in the command window.See Answer
Q16:Figure explanation (see also comments in the script) The figure above shows three curves plotted in the same figure window. The blue curve y1 is the plot of the rational function The above formula estimates exp(-x) for small values of x. The red curve y2 is the plot of the decaying exponential function exp(-x). Finally, the black dash-dotted line y3 shows a horizontal line with y-coordinate 0.See Answer
Q17:Q2 a) Create a 3 x 3 matrix of random integers between 5 and 20. Save this as variable A and display it clearly in the command window. b) Create a copy of A called B (do not display it in the command window yet). Modify B so that the middle element is equal to 50, and display B clearly in the command window. c) Use elementwise operations to calculate 4²/B and display the answer clearly in the command window.See Answer
Q18:Q1 a) Use the colon operator to create the following row vector [ 1 2 3 4 5]. Save this as variable a but suppress the output in the command window. b) Use the transpose operator on a to create a column vector b, suppress the output in the command window. c) Calculate a x b using matrix multiplication and display the answer clearly in the command window. See Answer
Q19:Use MATLAB (using available scripts - links will be provided in lab instructions slides) to plot the streamlines: (i) around a circular object, and (ii) around a combination of uniform flow, a source and a sink (i.e., a Rankine body). You may also use the potential and stream functions from the textbook to determine the streamlines and plot using MATLAB.See Answer
Q20:1 Introduction Analyzing the Mariana Trench 1.1 The Mariana Trench The Mariana Trench is the deepest trench in the world, lying in the Pacific Ocean between Japan and Papua New Guinea. Because it is so deep, and the sides so steep, it is a unique environment on earth and the subject of much scientific inquiry. The United States National Oceanic and Atmospheric Administration (NOAA) has conducted a series of bathymetric investigations of the Mariana Trench, collecting high-resolution data of the region. This data is often difficult to work with, containing an extremely high number of data points. You've been asked to reduce the size of the data in a way that maintains the overall structure of the trench and perform some scientific analysis. 1.2 The Singular Value Decomposition To perform this science, we will need to compute the Singular Value Decomposition (SVD) of the dataset. The SVD is a special factorization of a given matrix A. Similarly to standard numbers, which can be factored several ways, for example, 123-4-3-2-2 6-4 11/11 a matrix can have many different factorizations, with many different purposes and benefits. The SVD factoriz A into three parts, 223 A = UEVT, where U and V are orthogonal matrices (that is, UTU = I, UUTI, VTV = I and VVT = I) and Σ is a diagonal matrix. A is mxn U is mxm Σ is mxn VT is nxn Each of the three factor matrices tells you information about the matrix. Σ contains the "singular values", which are the square roots of the eigenvalues of ATA, and the columns of V are the associated eigenvectors. U holds the same information about the eigenvectors of AAT. breaks a given matrix Physically speaking, this representation tells you how the matrix A acts on vectors. The columns of V represent a basis for R", and the columns of U are a basis for R". A is a matrix which maps the basis V to the basis U, scaled by the singular values on the diagonal of E. Written mathematically, that is AX = A (G₁V₁ +₂V₂ + ... + ₁₂V₁) = c V) = c2U + c2U,+...+c Στη 1.3 The Incomplete SVD This representation is helpful for more than just a deeper understanding of the matrix A, though. If we order the singular values and vectors so that Σ₁1 ≥ 22 ..., then we can make the approximation C₁E11Ū₁ + €₂22Ū₂ + ··· + CkEkkŰk + ... +€mΣmmŪm ≈ 1 Σ11Ū₁ + €₂Σ22Ū₂ + ... + CkEkkŰk for some k <m. That is, we are "cutting off" the action of the matrix A a little early, but we are confident that we have ordered the way the matrix A works so that we are losing as little information as possible. CONTINUED/nThis is called the Incomplete Singular Value Decomposition. The core idea behind using the Incomplete SVD is that the columns of U and V corresponding to large diagonal elements of Σ essentially matter more, and we can save a lot of energy by using tall, skinny approximations to U and V which throw away the columns we don't care about. A is mxn ≈ U is mxk A = importdata ('mariana_depth.csv'); lor = importdata ('mariana_longitude.csv lat importdata ('mariana_latitude.csv Σ is kxk 1.4 Acquiring the data NOAA hosts bathymetric datasets of the Mariana trench. A cleaned and lightly sampled version of that dataset has been posted to the Canvas page for this course. The units for the depths are meters with latitude and longitude in degrees. The original data set is 25 times the size given here, and is difficult to work with on personal computers. Download the three files mariana_depth.csv mariana_latitude.csv mariana_longitude.csv from the Project Module in Canvas and put them in your working directory. CSV files are easy to import into For example, in MATLAB one would use the commands VT is kxn coding languages. 2 Questions When writing your report, do not simply number and answer the questions below. sure you answer all the questions, but remember your write-up needs to have the look and feel of a scientific report. 2.1 Investigating the Trench 1. Import the data, and display it as an image in your report. Be sure to label your axes and have latitude increase to the north (in the positive y-direction) and longitude increase to the east (in the positive x-direction). 2. How deep is the deepest point sampled in the trench? What are its latitude and longitude? 3. The ocean floor can be nominally defined in this region at a depth of 6 kilometers. What is the average depth of the trench, that is, the mean depth among all points deeper than the ocean floor? 2.2 Computing Eigenvectors Before running your code on the trench depth data, it is good to develop the code using a smallish matrix for which you know the eigenvalues and eigenvectors. Do not include the testing data in the report. CONTINUED 1. Code your own algorithm to find the first eigenvector and eigenvalue of ATA, where A denotes the depth matrix. To do this, (a) First begin with a random guess vector of the correct size and magnitude one. (b) Apply ATA to that vector, and then divide the result by its magnitude to make sure it's still magnitude one. That is your updated guess. (c) Repeat the two steps above until the vector stops changing./nPut another way. (a) ₁ = a random unit vector (magnitude 1). (b) un+1 = A¹Aun/A¹Aun|| (c) Repeat until n+1- un is small. It should only take around ten steps. Note that the notation ||u| denotes the norm (magnitude) of the vector u that has N components, and is defined as Call the eigenvector you have found at the end V₁. Provide the eigenvalue and a plot of the eigenvector V₁ in your report. The z-axis of this plot should go from 1 to N, and the y-axis should be the corresponding value of the component of the eigenvector. Also provide a hypothesis for why this method of finding the eigenvector works. HINT: Note that the eigenvectors of ATA constitute a basis of R", and try expressing u, in terms of that basis. 2. We can also compute the i largest eigenvalues and associated eigenvectors of the system by using the fact that every eigenvector of a symmetric matrix (such as ATA) must be orthogonal to all the previous ones, with a process called Gram-Schmidt Orthogonal- ization. The basic idea is that we make sure that the current eigenvector we are building is orthogonal to every other eigenvector we have built so far. In other words, if we are currently working on constructing V, un, V₁ = ₁V₂ = = V₁-1=0. From a next-guess vector +1 = ATA, that we have not touched with this Gram-Schmidt method yet, it to the i previously computed eigenvectors V₁, V2, ... V, by computing un+1 = u n+1 i-1 -Σ(˜‚+ V‚) V₂. j=1 These components we are subtracting from +1 are called the "projection of +1 onto Using this idea, alter your code to compute the i= 50 largest eigenvalues and associated ization step should be done after you multiply your previous guess vector by ATA, but before you divide your new guess by its own magnitude. (a) Initialize V₁ to V50 as a matrix of zeros. (b) For i = 1 to 50, eigenvectors of ATA. This orthogonal- i. ₁ a random unit vector (magnitude 1) ii. +1A¹Au, iii. Un+1 = +1-EV iv. Un+1 = Un+1/|un+1|| v. repeat ii-iv until || un+1 - || is small (less than 10-² or 10-3 should be fine for us), and store the final un+1 as V₁. CUoulder Store the eigenvectors you compute as the columns of a 1440 x 50 matrix and call it V. Provide a semilog plot of the eigenvalues you find. 2.3 The Incomplete SVD Decomposition 1. Construct a 50 x 50 matrix with the square roots of the eigenvalues you found in the previous part on the diagonal. Call that matrix E. Likewise, define a matrix U so that each column of U is equal to A times the associated column of V, divided by the associated element of Σ. 2. The V, E and U matrices you have constructed constitute the incomplete SVD decomposition of the matrix A. The idea is that we've captured the relevant features of A without using so much storage. To test this, first calculate how many numbers we need to save in total between the U. Σ and V matrices we've built, and compare that to the total number of elements of A. 3. Now, provide a picture like you did of the matrix A in Question 2.1, but now of the matrix product UZVT. Describe the relationship between these two images. 4. Try using fewer columns of U and V and a similarly smaller and see how the image holds up. If your computer can handle it, try using more than 50. Describe how the number of singular values used affects the image. CONTINUED/n3 Submission Your group's report needs to be submitted to Gradescope as a single pdf file. Add all groups members to the Gradescope submission in addition to including them in the actual report. Any code that you include should be included within the single pdf as an appendix. Be sure to type up your final report. Further details on project write-ups can be found in the Project Information Modele in Canvas.See Answer

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